Volume 14 Issue 4 Article 3 12-31-2012 Three firms on a unit disk market: intermediate product differentiation Aljoša Feldin Follow this and additional works at: https://www.ebrjournal.net/home Recommended Citation Feldin, A. (2012). Three firms on a unit disk market: intermediate product differentiation. Economic and Business Review, 14(4). https://doi.org/10.15458/2335-4216.1236 This Original Article is brought to you for free and open access by Economic and Business Review. It has been accepted for inclusion in Economic and Business Review by an authorized editor of Economic and Business Review. 321 ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012 | 321–345 ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION alJoša fEldIn1 Received: 22 October 2012 Accepted: 18 December 2012 aBSTraCT: Irmen and Thisse (1998) demonstrate that two firms competing with multi- characteristic products differentiate them in one characteristic completely, while keeping them identical in all others. This paper shows that their min-…-min-max differentiation result is not robust with respect to the number of firms. A market setting that replicates their result in a duopoly, but fails to do so in a three firm oligopoly is identified. Sym- metric pure strategy equilibrium with three firms differentiating their products in two dimensions, but not completely in either of them, is a novel medium-medium differen- tiation result. Keywords: spatial competition, location-price game JEl classification: L11, L13, R39 ThrEE fIrmS on a unIT dISK marKET: InTErmEdIaTE produCT dIffErEnTIaTIon Models of discrete location choice commonly interpreted as modeling product differ- entiation, as well, have been explored in a variety of contexts.2 Irmen and Thisse (1998) provide the most comprehensive study of two firms competing with their products in an 1 University of Ljubljana, faculty of Economics, Ljubljana, Slovenia, e-mail: aljosa.feldin@ef.uni-lj.si 2 ReVelle and Eiselt (2005), and ReVelle, Eiselt, and Daskin (2008) provide a comprehensive survey of the field and collect an extensive bibliography covering various aspects and problems in this area, respectively. A part of the literature is interested in the extent of product differentiation that firms should employ, and the number of product dimensions they should use doing that. pioneering work with a linear duopoly model by hotelling (1929) offered a principle of minimum differentiation. hotelling’s contribution was revisited 50 years later, when d’Aspremont et al. (1979) showed that there is no price equilibrium in pure strategies when two firms are located too closely to each other. Using quadratic instead of linear transportation costs, they find unique market equilibrium with firms maximizing product differentiation. The direct demand effect that makes a firm move towards its opponent to capture its demand is followed by the opponent’s price cut. The latter overrides extra profit gained with new demand from moving towards the opponent. The negative strategic effect of igniting stronger competition induces firms to differentiate their products as much as they can. As it was shown later, a maximum differentiation result rests both on the form of consumers’ utility function (e.g. Economides, 1986), and the uniform distribution of their tastes within the product characteristics space (e.g. Neven, 1986, Tabuchi & Thisse, 1995). ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012322 n-dimensional product space.3 They investigate a unit hyper-cube market that is uni- formly populated by consumers and served by two firms. When consumers incur disu- tility that is quadratic in distance between a product variety that they would prefer the most and the variety bought, they show that it is always optimal for the two firms to differentiate their products in one dimension only, and doing so completely. This type of practice is referred to as min-…-min-max product differentiation. This paper departs from the Irmen in Thisse model in that it studies a market with three firms in a two-dimensional product space. We present a unit disk market uniformly populated by consumers that are served by firms operating one store each. firms choose respective store locations in the first stage and compete with prices in the second. There are some authors that explore how competition between more than two firms affects product differentiation. Salop (1979) and Economides (1989) look at a circular city model with location equilibria that place firms equidistantly. Economides (1993) provides price equilibrium characterizations for every location configuration in a linear city model, and Brenner (2005) presents location equilibria for different number of firms (from three to nine) in the same type of the market. Brenner shows that firms depart from a maximum differentiation result in that the store locations move towards the middle of the market. We address two questions. first, will more than two firms in a bounded, two dimen- sional product characteristics space differentiate their products less than completely, as we might suspect from Brenner (2005)? Brenner shows that a firm with two neighbors does not cut prices as drastically as in a duopoly case when it is approached by one of the neighbors. The reason is that it does not want to alter its optimal revenues from the other side, where it neighbors a firm that did not deviate from an equilibrium position. Consequently, in equilibrium even the two firms on the two outskirts of the city move to- wards the center. The direct demand effect outweighs the strategic price effect and extent of product differentiation is reduced. There are two reinforcing effects facilitating Bren- ner’s result. first, the area of confrontation between an intrusive firm and its victim is a single point, a marginal consumer between the two firms. Since the area of confrontation between the victim and its other side neighbor is of the same size, the incentive to cut its price to counter the intruder is offset in a large part by a lower price and suboptimal profits on the other side. Second, the other side neighbor anticipates lower prices; it re- duces its price as well. hence, a reduction of victim’s price does not translate in a sizeable increase of its demand on the other neighbor’s side and is not profitable. Consequently, a cut in a victim’s price is not large enough to keep the intrusive firm from moving in to capture a part of its demand. Consequently, firms locate closer to each other. The role of the two effects we have described is less obvious in our case of firms competing in two dimensions. Market configuration may be such that a firm that moves its store does that in a direction of two and not just one neighbor. The firms’ demands are now delineated by line segments of consumers that are indifferent between buying from any of the two 3 Irmen and Thisse are not the first to explore markets where consumers care about more than one product characteristic. Neven and Thisse (1990) and Tabuchi (1994) were the first to show that in a two dimensional product characteristics space two firms will never find it optimal to differentiate their products fully. There are equilibria in which products are completely differentiated along one dimension and identical in the other. A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 323 neighboring firms. A confrontation frontier for an intrusive firm may hence be longer than is a line segment between respective neighbors fighting the intruder. This means that a price cut by the neighbors does not necessarily be as pronounced as it was in a one-dimensional case but may still keep the intrusive firm away. As a result, it might be that firms stay on the outskirts of the market. On the other hand, we might see a free- rider effect, meaning that firms under attack would count on each other to counter the intruder with lower prices. This might lead to a price cut that is insufficient to keep the intruder on the edge of the market. Another question is whether firms find it optimal to differentiate their respective prod- ucts in more than one dimension in the first place. If the market was unbounded and consumers’ reservation prices finite, the answer is obviously positive. With a bounded market, the answer is not imminent and might depend on the shape of the market in general. We expect that firms will find it beneficial to leave the congested competition in one product characteristic at some point and will choose to differentiate their products in another one as well. Swann (1990) explores such a process with a simple model and simulations. Whenever the field of competition becomes too dense at least one firm en- dogenously finds it optimal to introduce a new product attribute. We first show that maximum differentiation in one dimension – and no differentiation in the other (Irmen and Thisse, 1998) – remains optimal in a duopoly. In our setting this means that the two firms position their stores on the perimeter of the disk, exactly opposite from each other. We then present two novel results. An oligopoly with three firms competing on the same market facilitates a pure strategy subgame perfect Nash equilibrium of our location-price game. The equilibrium has all three firms located at the same distance from the center of the disk, equidistant from each other. That means that we observe differentiation in two product characteristics, a result that extends the exist- ent literature. furthermore, firms do not choose full differentiation, but move inward, towards the center of the disk considerably. We find medium-medium type of product differentiation in a setting that yields a min-max differentiation result in a duopoly. This means that the conjecture based on Brenner (2005), given above, carries over to markets with more than two competitors and more than just one product characteristic. When firms are located close to the perimeter of the market, the positive demand effect of a radial deviation towards the center outweighs the negative strategic effect of rivals de- creasing their prices. Another interesting aspect of the model is that, while in a duopoly a social planner would have firms differentiating their products less extensively, the result reverses in a three- firm oligopoly. hence, some cooperative behavior or regulation on product specifica- tions would be beneficial both to firms and to society as a whole. The organization of the paper is as follows. We set up our model in Section 1, and present our results for a duopoly and a three-firm oligopoly in Sections 2 and 3, respectively. Section 4 considers welfare issues, and we make our conclusions in Section 5. proofs of Lemmas and propositions and all necessary derivatives are deferred to the Appendices. ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012324 1. ThE modEl We use the classical spatial model of an oligopoly. Consumers of a total mass π are uni- formly distributed on a unit disk.4 Each consumer has a unit demand for a homogeneous good produced by n firms on the market. We explore configurations in which all firms are at the same radial distance from the origin, R. Specifically, the firms are located at Li=(R,φi), i = 1,…, n. Firm 1 is always a counter-clockwise direction neighbor of Firm n, while Firm n−1 is a clockwise direction neighbor of Firm n. firms charge pi, i = 1, …, n, per unit of the good. If a consumer residing at point x buys the good from Firm i, she derives utility: consumer residi g at point x buys the good from Firm i, she derives util ty: u(p i ,L i ;x) = v − p i − d 2 x,L i ( ) . We assume that consumers incur traveling costs that are quadratic in the Euclidean distance, d x,L i ( ), traveled, and with cost per unit traveled normalized to one. The surplus v enjoyed from consuming the good is assumed to be large enough so that every consumer buys a good from one of the firms. Consumers maximize their utility by choosing the store they buy the good from optimally. Firms operate with symmetric constant marginal costs that we normalize to zero and are allowed to operate one store each. We consider a non-cooperative two-stage location-price game of the following form. Stage 1: Firms select locations on a unit disk. Stage 2: Firms observe chosen locations and compete in prices. Formally, firm i’s strategy space in the first period is ],[]1,0[ ππ−×= i L , i = 1,… , n, and each firm’s strategy in the second stage is += ℜ→× i n ii Lp 1 : . We seek subgame perfect Nash equilibria of this game: S * ={ }),(),...,,(),,( *** 2 * 2 * 1 * 1 nn pLpLpL . A natural candidate for equilibrium configuration has firms positioned at an equal distance from the origin, equidistantly along the circle they occupy. Specifically, we explore locations: L 1 * = (R, ππ n 2 +− ), L2 * = (R, ππ n 4 +− ), … , and L n * = (R,π ). 2 Two firms We first derive market equilibrium in a duopoly. Here we show that min-max result derived by Irmen and Thisse (1998) is also optimal strategy in our setting. In our setting that means that firms locate their stores on the perimeter of the disk, symmetrically across the origin. We search for an equilibrium that is symmetric in firms’ locations with Figure 1 showing a possible off- equilibrium configuration L 1 * = (R,0) and L 2 = (r,φ). Figure 1: Configuration of the market, two firms. L 1 * =(R, 0) L 2 =(r, φ) A=(1,α) B=(1,β) O z≡α −β We assume that consumers incur traveling costs that are quadratic in the Euclidean dis- tance, d(x, Li), traveled, and with cost per unit traveled normalized to one. The surplus v enjoyed from consuming the good is assumed to be large enough so that every consumer buys a good from one of the firms. Consumers maximize their utility by choosing the store they buy the good from optimally. firms operate with symmetric constant marginal costs that we normalize to zero and are allowed to operate one store each. We consider a non-cooperative two-stage location- price game of the following form. Stage 1: firms select locations on a unit disk. Stage 2: firms observe chosen locations and compete in prices. formally, firm i’s strategy space in the first period is consumer residing at point x buys the good from Firm i, she derives utility: u(p i ,L i ;x) = v − p i − d 2 x,L i ( ) . We assume that consumers incur traveling co ts that are quadratic in the Euclidean distance, d x,L i ( ), traveled, and with cost per unit traveled normalized to one. The surplus v enjoyed from consuming the good is assumed to be large e ough so that every consumer buys a good from one of the firms. Consumers maxi ize their utility by ch osing the store they buy the good from optimally. Firms opera e with symmetric constant marginal cos s th we normalize to zero and are allowed to operate one store each. We consider a non-cooperative two-stage location-price game of the following form. Stage 1: Firms select locations on a unit disk. Stage 2: Firms observe chosen locations and compete in prices. Formally, firm i’s strategy space in the first p r ],[]1,0[ ππ−×= i L , i = 1,… , n, and each firm’s strategy in the second stage is += ℜ→× i n ii Lp 1 : . We seek subgame perfect Nash equilibria of this game: S * ={ }),(),...,,(),,( *** 2 * 2 * 1 * 1 nn pLpLpL . A natural candidate for equilibrium configuration has firms positioned at an equal distance from the origin, equidistantly along the circle they occupy. Specifically, we explore locations: L 1 * = (R, ππ n 2 +− ), L2 * = (R, ππ n 4 +− ), … , and L n * = (R,π ). 2 Two firms We first derive market equilibrium in a duopoly. Here we show that min-max result derived by Irmen and Thisse (1998) is also optimal strategy in our setting. In our setting that means that firms locate their stores on the perimeter of the disk, symmetrically across the origin. We search for an equilibrium that is symmetric in firms’ locations with Figure 1 showing a possible off- equilibrium configuration L 1 * = (R,0) and L 2 = (r,φ). Figure 1: Configuration of the market, two firms. L 1 * =(R, 0) L 2 =(r, φ) A=(1,α) B=(1,β) O z≡α −β , i , , each firm’s strategy in the second stage is consumer residing at point x buys the good from Firm i, she derives utility: u(p i ,L i ;x) = v − p i − d 2 x,L i ( ) . We assume that consumers incur traveling costs that are quadratic in the Euclidean distance, d x,L i ( ), traveled, nd with cost per unit traveled normalized to one. The surplus v enjoyed from consuming the good is assumed to be large enough so that every consumer buys a good from one of the firms. Consumers maximize their utility by choosing the store they buy the good from optimally. Firms operate with symmetric constant marginal costs that we normalize to zero and are allowed to operate one store each. We consider a non-cooperative two-stage location-price game of the following form. Stage 1: Firms select locations on a unit disk. Stage 2: Firms obs rve chosen locations and compete in price . Formally, firm i’s strategy space in the first period is ],[]1,0[ ππ−×= i L , i = 1,… , n, and each firm’s strategy in th second stage is += ℜ→× i n ii Lp 1 : . We seek subgame perfect Nash equilibria of this game: S * ={ }),(),...,,(),,( *** 2 * 2 * 1 * 1 nn pLpLpL . A natural candidate for equilibrium configuration has firms positioned at an equal distance from the origin, equidistantly along the circle they occupy. Specifically, we explore locations: L 1 * = (R, ππ n 2 +− ), L2 * = (R, ππ n 4 +− ), … , and L n * = (R,π ). 2 Two firms We first derive market equilibrium in a duopoly. Here we sho that min-max result derived by Irmen and Thisse (1998) is also optimal strategy in our setting. In our setting that means that firms locate their stores on the perimeter of the disk, symmetrically across the origin. We search for an equilibrium that is symmetric in firms’ locations with Figure 1 showing a possible off- equilibrium configuration L 1 * = (R,0) and L 2 = (r,φ). Figure 1: Configuration of the market, two firms. L 1 * =(R, 0) L 2 =(r, φ) A=(1,α) B=(1,β) O z≡α −β . e e perfect Nash equilibria of this game: consumer residing at point x buys th good from Firm i, she derives utility: u(p i ,L i ;x) = v − p i − d 2 x,L i ( ) . We assu e that consumers incur traveling costs that are quadratic in the E clidean distance, d x,L i ( ), traveled, and with cost per unit traveled normalized to one. The surplus v enjoyed from consuming the good is assumed to be large enough so that every consumer buys a good from one of the firms. Consumers maximize their utility by choosing the store they buy the good from opti ally. Firms operate with symmetric constant marginal costs that we normalize to zero and are allowed to operate one store each. We consider a non-cooperative two-stage location-price game of the following form. Stage 1: Firms select locations on a unit disk. Stage 2: Firms observe chose locations and compete in prices. Formal y, firm i’s tr tegy space in the first period is ],[]1,0[ ππ−×= i L , i = 1,… , n, and each firm’s strategy in the second stage is += ℜ→× i n ii Lp 1 : . We seek subgame perfect Nash equ libria of this a S * ={ }),(),...,,(),,( *** 2 * 2 * 1 * 1 nn pLpLpL . A natural candidate for equilibrium configuration has firms positioned at an equal distance from the origin, equidistantly along the circle they occupy. Specifically, we explore locations: L 1 * = (R, ππ n 2 +− ), L2 * = (R, ππ n 4 +− ), … , and L n * = (R,π ). 2 Two firms We first derive market equilibriu in a duopoly. Here we show that min-max result derived by Irmen and Thisse (1998) is also optimal strategy in our setting. In our setting that means that firms locate their stores on the perimeter of the disk, symmetrically across the origin. We search for an equilibrium that is symmetric in firms’ locations with Figure 1 showing a possible off- equilibrium configuration L 1 * = (R,0) and L 2 = (r,φ). Figure 1: Configuration of the market, two firms. L 1 * =(R, 0) L 2 =(r, φ) A=(1,α) B=(1,β) O z≡α −β atural candidate for equilibrium configuration has firms positioned at an equal distance from the origin, equidistantly along the circle they occupy. S ecifically, we explore locations: L1 *= (R, –π + –2n π), L2* = (R, –π + – 4 n π)), … , and Ln* = (R, π). 2. Two fIrmS We first derive market equilibrium in a duopoly. here we show that min-max result derived by Irmen and Thisse (1998) is also opt mal strategy in our setting. In our setting that means that firms locate their stores o the pe imeter of the disk, symmetrically across the origin. We search for an equilibrium that is symmetric in firms’ locations with figure 1 showing a possible off-equilibrium configuration L1 *= (R,0) and L2 = (r,φ). 4 polar symmetry is used to avoid non-differentiable demand functions that arise in a rectangular market when a line of consumers indifferent between buying from two neighboring stores touches the corner of a market. Moreover, there is no symmetry to exploit in the market with three firms on a square. A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 325 figure 1: Configuration of the market, two firms. Line AB in figure 1 represents consumers that are indifferent between buying a product from either of the two firms given their locations and product prices. AB is perpendicu- lar to the line connecting firms’ locations L1 * and L2. It is closer to the firm that sets the higher of the two respective prices. AB crosses the perimeter of the disk at angles a and b. Firm 1’s demand is the area between a and b, (a−b)/2, reduced by the area of a tri- angle ABO, which is Line AB in Figure 1 represents consumers that are indifferent between buying a product from either of t tw firms given their locations and product prices. AB is perpendicular to the line connecting firms’ loc tions L 1 * and L 2 . It is closer to the firm that sets the higher of the two respective prices. AB crosses the perimeter of th l s α and β. Firm 1’s demand is the area between α and β, (α−β)/2, reduced by the area of a triangle ABO, which is 22 cossin βαβα −− ⋅ = 2/)sin( βα − . We set βα −≡z and state the firms’ demands: ( )zzD sin 2 1 1 −= , and ( )zzDD sin2 2 1 12 +−=−= ππ , (1) Firms’ second-stage profits given locations chosen in the first stage are: Π 1 );,( Rr φ = p1 );,( Rr φ ·D1 );,( Rr φ and Π2 );,( Rr φ = p2 );,( Rr φ ·D2 );,( Rr φ . (2) Note that 1 p , 2 p , α , β , and z are all functions of firms’ respective locations. Second- stage profit maximization with respect to prices yields the following result. Lemma 1: When firms choose locations L 1 * = (R,0) and L 2 * = (R,π) in the first stage of the game, they both charge p * = π ·R in the second. The price firms charge in the second-stage of the game increases in distance from the origin, that is in the distance between their stores or their products. This shows a common incentive dictating the need for product differentiation. Next, we want to see whether the proposed structure of firms’ first stage locations can be supported in equilibrium. We show there is R ]1,0(∈ , such that given Firm 1’s location, L1 * , Firm 2’s location, L 2 * , is optimal. It turns out that this R equals one; firms locate their stores on the perimeter of the disk. Also, they will be positioned symmetrically across the origin. To see this, we look at respective effects deviations in radial and polar directions have on Firm 2’s profits. These effects are: dΠ 2 d r = ∂Π 2 ∂ p 2 ⋅ d p 2 d r + p 2 ⋅ ∂D 2 ∂r + ∂D 2 ∂ p 1 ⋅ d p 1 d r       , and (3)         ⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π φφφφ d pd p DD p d pd pd d 1 1 22 2 2 2 22 . (4) The first term on the right hand side (RHS henceforth) of both (3) and (4) is zero due to profit maximization with respect to p 2 in the second period, so we are left with the direct (demand) effect and indirect (strategic) effect of each move, which are the first and second terms in parentheses, respectively. We first derive the demand and strategic effects of a small radial move on Firm 2’s profit. = sin(a – b)/ . z ≡ a – b and st t fi s’ demands: (1) firms’ second-stage profits given locations chosen in the first stage are: (2) Note that p1, p2, a, b, and z are all functions of firms’ respective locations. Second-stage profit maximization with respect to prices yields the following result. Lemma 1: When firms choose locations L1 * = (R,0) and L2 * = (R,π) in the first stage of the game, they both charge p* = π ·R in the second. The price firms charge in the second-stage of the game increases in distance from the origin, that is in the distance between their stores or their products. This shows a com- mon incentive dictating the need for pr duct differentiation. Next, we want to see whether the proposed structure of firms’ first stage locations can be supported in equilibrium. We show there is R∈ (0,1], such that given Firm 1’s location, L1 *, Firm 2’s location, L2 *, is optimal. It turns out that this R equals one; firms locate their stores on the perimeter of the disk. Also, they will be positioned symmetrically across Line AB in Figure 1 represents con umers that are ind fferent between buyi g product from either of he two firms giv n their locations and product prices. AB is perpendicular to the line connecting firms’ locati ns L 1 * and L 2 . It i closer to the firm that s ts the higher of the two respective prices. AB crosses the perim ter of the disk at angles α and β. Firm 1’s demand is the area between α and β, (α−β)/2, reduced by the area of a triangle ABO, which is 22 cossin βαβα −− ⋅ = 2/)sin( βα − . We set βα −≡z and state the firms’ demands: ( )zzD sin 2 1 1 −= , and ( )zzDD sin2 2 1 12 +−=−= ππ , (1) Firms’ second-stag profit giv n l cations chosen n the fir t stage are: Π 1 );,( Rr φ = p1 );,( Rr φ ·D1 );,( Rr φ and Π2 );,( Rr φ = p2 );,( Rr φ ·D2 );,( Rr φ . (2) Note that 1 p , 2 p , α , β , and z are all functions of firms’ respective locations. Second- stage profit maximization with respect to prices yields the following result. Lemma 1: When firms choose locations L 1 * = (R,0) and L 2 * = (R,π) in the first stage of the game, they both charge p * = π ·R in the second. The price firms charge in the second-stage of the game increases in distance from the origin, that is in the dista between their stores or th ir ts. This shows c mmon incentive dictating the need for product differentiation. Next, we want to see whether the proposed structure of firms’ first stage locations can be supported in equilibrium. We show there is R ]1,0(∈ , such that given Firm 1’s location, L1 * , Firm 2’s location, L 2 * , is optimal. It turns out hat this R equals one; firms ocate the r stores on the perimeter of the disk. Also, they ill be position d symm trically cross the origin. To see this, we look at respective effects deviations in radi l and polar directions have on Firm 2’s profits. These effects are: dΠ 2 d r = ∂Π 2 ∂ p 2 ⋅ d p 2 d r + p 2 ⋅ ∂D 2 ∂r + ∂D 2 ∂ p 1 ⋅ d p 1 d r       , and (3)         ⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π φφφφ d pd p DD p d pd pd d 1 1 22 2 2 2 22 . (4) The first term on the right hand side (RHS hencefort ) of bo h (3) a d (4) is zero due to profit maximization with respect to p 2 i th s ond period, so we are left with the direct (demand) effect and indirect (strategic) effect of each move, which are the first and second terms in parentheses, respectively. We first derive the demand and strategic effects of a small radial move on Firm 2’s profit. Line AB in Figure 1 represents consumers that are indifferent between buying a product from either of the two firms given their locations and product prices. AB is perpendicular to the line connecting firms’ locations L 1 * and L 2 . It is closer to the firm that sets the higher of the two respective prices. AB crosses the perimeter of the disk at angles α and β. Firm 1’s demand is the area between α and β, (α−β)/2, reduced by the area of a triangle ABO, which is 22 cossin βαβα −− ⋅ = 2/)sin( βα − . We set βα −≡z and state the firms’ demands: ( )zzD sin 2 1 1 −= , and ( )zzDD sin2 2 1 12 +−=−= ππ , (1) Firms’ second-stage profits given locations chosen in the first stage are: Π 1 );,( Rr φ = p1 );,( Rr φ ·D1 );,( Rr φ and Π2 );,( Rr φ = p2 );,( Rr φ ·D2 );,( Rr φ . (2) Note that 1 p , 2 p , α , β , and z are all functions of firms’ respective locations. Second- stage profit maximization with respect to prices yields the following result. Lemma 1: When firms choose locations L 1 * = (R,0) and L 2 * = (R,π) in the first stage of the game, they both charge p * = π ·R in the second. The price firms charge in the second-stage of the game increases in distance from the origin, that is in the distance between their stores or their products. This shows a common incentive dictating the need for product differentiation. Next, w wa t to see whether the propos d structure of firms’ first stag locations can be supported in equilibrium. We show there is R ]1,0(∈ , such that given Firm 1’s location, L1 * , Firm 2’s location, L 2 * , is opt al. It turns out that this R equals one; firms locate their sto es on the perimeter of the disk. Also, they will be positioned symmetrically across the origin. To see this, we look at respective effects deviations in radial and polar directions have on Firm 2’s profits. These effects are: dΠ 2 d r = ∂Π 2 ∂ p 2 ⋅ d p 2 d r + p 2 ⋅ ∂D 2 ∂r + ∂D 2 ∂ p 1 ⋅ d p 1 d r       , and (3)         ⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π φφφφ d pd p DD p d pd pd d 1 1 22 2 2 2 22 . (4) The first term on the right hand side (RHS henceforth) of both (3) and (4) is zero due to profit maximization with respect to p 2 in the second period, so we are left with the direct (demand) effect and indirect (strategic) effect of each move, which are the first and second terms in parentheses, respectively. We first derive the demand and strategic effects of a small radial move on Firm 2’s profit. ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012326 the origin. To see this, we look at respective effects deviations in radial and polar direc- tions have on Firm 2’s profits. These effects are: (3) (4) The first term on the right hand side (RhS henceforth) of both (3) and (4) is zero due to profit maximization with respect to p2 in the second period, so we are left with the direct (demand) effect and indirect (strategic) effect of each move, which are the first and sec- ond terms in parentheses, respectively. We first derive the demand and strategic effects of a small radial move on Firm 2’s profit. Lemma 2: When L1 * = (R,0) and L2 * = (R,π), R∈ (0,1]: (a) Lemma 2: When L 1 * = (R,0) and L 2 * = ( , ), ],0(∈R : (a) 1 2 −= ∂ ∂ r D , (b) Rp D 2 1 1 2 = ∂ ∂ , and (c) R dr dp 3 2 2 1 += π . Part (a) quantifies Firm 2’s direct gain in demand it captures from its rival by moving towards its location, while (b) and (c) show that such a move affects Firm 2’s demand adversely through a rival’s price cut. These effects are such that both firms would like to locate their stores on the perimeter of the disk when they are separated by angle π. Lemma 3: For any pair of locations L 1 * = (R,0) and L 2 * = (R,π), ]1,0(∈R , firms find it profitable to move farther away from the origin. Finally, we argue that the angle separating the two firms is exactly π. Lemma 4: For any R, φ = π is optimal for Firm 2. Lemmas 3 and 4 prove Proposition 1. Proposition 1: When in the first stage of the game two firms position their stores on the perimeter of the disk symmetrically across its origin; the equilibrium price they charge in the second-stage is π. Furthermore, every firm’s location is a local best response to their rival’s location. Due to the complexity of the problem we are not able to show that each firm’s location is also a global best response to an opponent’s location analytically. We hence do it numerically. We fix Firm 1’s location at R=1 and φ 1 = 0, and vary Firm 2’s location across the disk. We solve for the second-stage price equilibrium for every configuration numerically and calculate Firm 2’s profit. Figure 2 shows that the proposed location r=1 and φ 2 = π is actually Firm 2’s global best response to Firm 1’s location choice. The profit attained there is π 2/2. , , , , . , . . , , , , , . , . , . . . , . . . , . . . . (c) Lemma 2: When L 1 * = (R,0) and L 2 * = (R,π), ]1,0(∈R : (a) 1 2 −= ∂ ∂ r D , (b) Rp D 2 1 1 2 = ∂ ∂ , and (c) R dr dp 3 2 2 1 += π . Part (a) quantifies Firm 2’s direct gain in demand it captures from its rival by moving towards its location, while (b) and (c) show that such a move affects Firm 2’s demand adversely through a rival’s price cut. These effects are such that both firms would like to locate their stores on the perimeter of the disk when they are separated by angle π. Lemma 3: For any pair of locations L 1 * = (R,0) and L 2 * = (R,π), ]1,0(∈R , firms find it profitable to move farther away from the origin. Finally, we argue that the angle separating the two firms is exactly π. Lemma 4: For any R, φ = π is optimal for Firm 2. Lemmas 3 and 4 prove Proposition 1. Proposition 1: When in the first stage of the game two firms position their stores on the perimeter of the disk symmetrically across its origin; the equilibrium price they charge in the second-stage is π. Furthermore, every firm’s location is a local best response to their rival’s location. Due to the complexity of the problem we are not able to show that each firm’s location is also a global best response to an opponent’s location analytically. We hence do it numerically. We fix Firm 1’s location at R=1 and φ 1 = 0, and vary Firm 2’s location across the disk. We solve for the second-stage price equilibrium for every configuration numerically and calculate Firm 2’s profit. Figure 2 shows that the proposed location r=1 and φ 2 = π is actually Firm 2’s global best response to Firm 1’s location choice. The profit attained there is π 2/2. . part (a) quantifies Firm 2’s direct gain in demand it captures from its rival by moving towards its location, while (b) and (c) show that such a move affects Firm 2’s demand adversely through a rival’s price cut. These effects are such that both firms would like to locate their stores on the p rimeter of the disk when they ar separated by angle π. Lemma 3: for any pair of locations L1 *= (R,0) and L2 *= (R,π), R∈ (0,1], firms find it profit- able to move farther away from the origin. finally, we argue that the angle separating the two firms is exactly π. Lemma 4: for any R, φ = π is optimal for firm 2. Lemmas 3 and 4 prove proposition 1. proposition 1: When in the first stage of the game two firms position their stores on the perimeter of the disk symmetrically across its origin; the equilibrium price they charge in the second-stage is π. furthermore, every firm’s location is a local best response to their rival’s location. Due to the complexity of the problem we are not able to show that each firm’s location is also a global best response to an opponent’s location analytically. We hence do it numeri- cally. We fix Firm 1’s location at R=1 and φ1 = 0, and vary Firm 2’s location across the disk. We s lve for the seco d-s age price equilibrium for every configuration numerically and calculate Firm 2’s profit. figure 2 sh ws hat the proposed location r=1 and φ2 = π is ac- tually Firm 2’s global best response to Firm 1’s location choice. The profit attained there is π 2/2. Line AB in Figure 1 represents consumers that are indifferent between buying a product from either of the two firms given their locations and product prices. AB is perpendicular to the line connecting firms’ locations L 1 * and L 2 . It is closer to the firm that sets the higher of the two respective prices. AB crosses the perimeter of the disk at angles α and β. Firm 1’s demand is the area between α and β, (α−β)/2, reduced by the area of a triangle ABO, which is 22 cossin βαβα −− ⋅ = 2/)sin( βα − . We set βα −≡z and state the firms’ demands: ( )zzD sin 2 1 1 −= , and ( )zzDD sin2 2 1 12 +−=−= ππ , (1) Firms’ second-stage profits given locations chosen in the first stage are: Π 1 );,( Rr φ = p1 );,( Rr φ ·D1 );,( Rr φ and Π2 );,( Rr φ = p2 );,( Rr φ ·D2 );,( Rr φ . (2) Note that 1 p , 2 p , α , β , and z are all functions of firms’ respective locations. Second- stage profit maximization with respect to prices yields the following result. Lemma 1: When firms choose locations L 1 * = (R,0) and L 2 * = (R,π) in the first stage of the game, they both charge p * = π ·R in the second. The price firms charge in the second-stage of the game increases in distance from the origin, that is in the distance between their stores or their products. This shows a common incentive dictating the need for product differentiation. Next, we want to see whether the proposed structure of firms’ first stage locations can be supported in equilibrium. We show there is R ]1,0(∈ , such that given Firm 1’s location, L1 * , Firm 2’s location, L 2 * , is optimal. It turns out that this R equals one; firms locate their stores on the perimeter of the disk. Also, they will be positioned symmetrically across the origin. To see this, we look at respective effects deviations in radial and polar directions have on Firm 2’s profits. These effects are: dΠ 2 d r = ∂Π 2 ∂ p 2 ⋅ d p 2 d r + p 2 ⋅ ∂D 2 ∂r + ∂D 2 ∂ p 1 ⋅ d p 1 d r       , and (3)         ⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π φφφφ d pd p DD p d pd pd d 1 1 22 2 2 2 22 . (4) The first term on the right hand side (RHS henceforth) of both (3) and (4) is zero due to profit maximization with respect to p 2 in the second period, so we are left with the direct (demand) effect and indirect (strategic) effect of each move, which are the first and second terms in parentheses, respectively. We first derive the demand and strategic effects of a small radial move on Firm 2’s profit. A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 327 figure 2: Firm 2’s profits given the location of its store. Source: Own calculations. We now state our first result, which links this work to the existent literature of two firms competing in a multi-characteristic product space. This gives us a valid reference point with which to compare our subsequent results. Result 1: Two firms positioning their stores on the perimeter of the disk, symmetrically across its origin in the first stage of the game, and setting p=π in the second is a subgame perfect Nash equilibrium of the game. This finding is in the spirit of Neven and Thisse (1990), Tabuchi (1994), and Irmen and Thisse (1998). Two firms offer products that are fully differentiated in one characteristic, while identical in the other. 3. ThrEE fIrmS We add another firm to the model and search for symmetric equilibrium. We find that firms separate their stores in both dimensions, and interestingly, do not choose to locate them on the perimeter anymore, but relocate them towards the origin noticeably. given locations L1 * = (R,−π/3), L2* = (R,π/3), and L3 = (r,φ), firms compete in prices, p1, p2, and p3. A particular market configuration is shown in figure 3. ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012328 figure 3: Configuration of the market, three firms. Store locations and prices define the boundaries of market areas covered by firms. There are three line segments, DA, DB, and DC, representing buyers indifferent between buy- ing from Firms 1 and 2, Firms 2 and 3, and Firms 3 and 1, respectively. These segments are needed in determining the demand functions firms face. They all join in one point, D, which is due to the fact that the delineating lines must be straight. point D = (x, y) represents a consumer indifferent between buying from any of the three firms. points A = (1, a), B = (1, b), and C = (1, γ) stand for consumers on the perimeter indifferent between buying from respective firms. With the knowledge of x, y, a, b, and γ, which, as well as p1, p2, and p3, are all functions of r, φ, and R, we can write the demand functions firms face.5 Firm 1’s demand equals the area of the disk between a and γ reduced for the area covered by triangles OCD and ODA. The other two firms’ demands are obtained similarly: 5 for the derivation of x, y, a, b, and γ see the proof of Lemma 5. Figure 3: Configuration of the market, three firms. Store locations and prices define the boundaries of market areas covered by firms. There are three line segments, DA, DB, and DC, representing buyers indifferent between buying from Firms 1 and 2, Firms 2 and 3, and Firms 3 and 1, respectively. These segments are needed in determining the demand functions firms face. They all join in one point, D, which is due to the fact that the delineating lines must be straight. Point D = (x, y) represents a consumer indifferent between buying from any of the three firms. Points A = (1, α), B = (1, β), and C = (1, γ) stand for consumers on the perimeter indifferent between buying from respective firms. With the knowledge of x, y, α, β, and γ, which, as well as ,and,, 321 ppp are all functions of Rr and ,,φ , we can write the demand functions firms face. 4 Firm 1’s demand equals the area of the disk between α and γ reduced for the area covered by triangles OCD and ODA. The other two firms’ demands are obtained similarly: ( ))sin()sin( 2 1 1 δαγδγα −−−−−= ddD , ( ))sin()sin( 2 1 2 δβδααβ −−−+−= ddD , and ( ))sin()sin(2 2 1 3 γδδβγβπ −+−++−= ddD . The sine of the difference rule and definitions of x and y yield: ( ))sin(sin)cos(cos 2 1 1 γαγαγα −−−+−= yxD , (5) A=(1,α) L 2 * = (R,π/3) L1 * =(R,−π/3) O B = (1,β) C = (1,γ) L 3 = (r,φ) D = (x,y) = (d·sinδ, d·cosδ) A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 329 The sine of the difference rule and definitions of x and y yield: (5) (6) (7) The firms’ profit functions are: (8) Second-stage optimal prices are derived from the system of necessary conditions ob- tained from these profits. The system is nonlinear and its general analytical solution for any possible Firm 3’s location, L3, is therefore out of reach. however, we are looking for symmetric equilibrium, so the derivation of optimal prices is straightforward. Lemma 5: When three firms in the first stage of the game position their stores R away from the origin, equidistantly from one another, the optimal Nash equilibrium price they charge in the second stage is ( ))sin(sin)cos(cos 2 1 2 αβαβαβ −−−+−= yxD , and (6) ( ))sin(sin)cos(cos2 2 1 3 γβγβγβπ −+−−+−= yxD . (7) The firms’ profit functions are: );,();,();,( 111 RrDRrpRr φφφ ⋅=Π , );,();,();,( 222 RrDRrpRr φφφ ⋅=Π , and (8) );,();,();,( 333 RrDRrpRr φφφ ⋅=Π . Second-stage optimal prices are derived from the system of necessary conditions obtained from these profits. The system is nonlinear and its general analytical solution for any possible Firm 3’s location, L 3 , is therefore out of reach. However, we are looking for symmetric equilibrium, so the derivation of optimal prices is straightforward. Le ma 5: When three firms in the first stage of t game position their stores R away from the origin, equidistantly from one another, the optimal Nash equilibrium price they charge in the s p * = π R 3 3 . The system of first-order conditions derived in the proof of Lemma 5 (A.8-A.10) completely characterizes price competition in the second stage given that Firm 3 chooses its location reasonably close to the proposed L 3 * . 5 It remains to be seen whether such a configuration is optimal in the first stage of the game. We rewrite Firm 3’s profit function with rivals’ store positions being L 1 * and L 2 * , and firms charging equilibrium prices in the second stage: ));,(),;,(),;,(,;,();,();,( 321333 RrpRrpRrpRrDRrpRr φφφφφφ ⋅=Π . We are interested in two derivatives:         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π rd pd p D rd pd p D r D p rd pd prd d 2 2 31 1 33 3 3 3 33 , and (9)         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π φφφφφ d pd p D d pd p DD p d pd pd d 2 2 31 1 33 3 3 3 33 . (10) Again, the first term in RHS of both (9) and (10) equals zero. The first term in parentheses in both equations represents the direct or demand effect of the deviation in a respective variable, while the last two represent the indirect or strategic effects of such a deviation through competitors’ prices. We show that there exists a distance from the origin, R, such that if firms locate there equidistantly from one another, the necessary conditions for symmetric equilibrium The system of first-order conditions derived in the proof of Lemma 5 (A.8-A.10) com- pletely characterizes price competition in the second stage given that Firm 3 chooses its location reasonably close to the propose L3 *.6 It remains to be seen whether such a configuration is optimal in the first stage of the game. We rewrite Firm 3’s rofit function with rivals’ store positions being L1 * and L2 *, and firms charging equilibrium prices in the second stage: We are interested in two derivatives: (9) 6 If Firm 3 located its store at the top of the disk, the configuration of the demands would have changed and quantities defined in figure 3 would not be valid anymore. Figure 3: Configuration of the market, three firms. Store locations and prices define the boundaries of market areas covered by firms. There are three line segments, DA, DB, and DC, representing buyers indifferent between buying from Firms 1 and 2, Firms 2 and 3, and Firms 3 and 1, respectively. These segments are needed in determining the demand functions firms face. They all join in one point, D, which is due to the fact that the delineating lines must be straight. Point D = (x, y) represents a consumer indifferent between buying from any of the three firms. Points A = (1, α), B = (1, β), and C = (1, γ) stand for consumers on the perimeter indifferent between buying from respective firms. With the knowledge of x, y, α, β, and γ, which, as well as ,and,, 321 ppp are all functions of Rr and ,,φ , we can write the demand functions firms face. 4 Firm 1’s demand equals the area of the disk between α and γ reduced for the area covered by triangles OCD and ODA. The other two firms’ demands are obtained similarly: ( ))sin()sin( 2 1 1 δαγδγα −−−−−= ddD , ( ))sin()sin( 2 1 2 δβδααβ −−−+−= ddD , and ( ))sin()sin(2 2 1 3 γδδβγβπ −+−++−= ddD . The sine of the difference rule and definitions of x and y yield: ( ))sin(sin)cos(cos 2 1 1 γαγαγα −−−+−= yxD , (5) A=(1,α) L 2 * = (R,π/3) L1 * =(R,−π/3) O B = (1,β) C = (1,γ) L 3 = (r,φ) D = (x,y) = (d·sinδ, d·cosδ) ( ))sin(sin)cos(cos 2 1 2 αβαβαβ −−−+−= yxD , and (6) ( ))sin(sin)cos(cos2 2 1 3 γβγβγβπ −+−−+−= yxD . (7) The firms’ profit functions are: );,();,();,( 111 RrDRrpRr φφφ ⋅=Π , );,();,();,( 222 RrDRrpRr φφφ ⋅=Π , and (8) );,();,();,( 333 RrDRrpRr φφφ ⋅=Π . Second-stage optimal prices are derived from the system of necessary conditions obtained from these profits. The system is nonlinear and its general analytical solution for any possible Firm 3’s location, L 3 , is therefore out of reach. However, we are looking for symmetric equilibrium, so the derivation of optimal prices is straightforward. Lemma 5: When three firms in the first stage of the game position their stores R away from the origin, equidistantly from one another, the optimal Nash equilibrium price they charge in the second stage is p * = π R 3 3 . The system of first-order conditions derived in the proof of Lemma 5 (A.8-A.10) completely characterizes pric competition in the second stage give that Firm 3 chooses its location reasonably close to the proposed L 3 * . 5 It remains o be seen whether such a configuration is optimal in the first stage of the game. We rewrite Firm 3’s profit function with rivals’ store positions being L 1 * and L 2 * , and firms charging equilibrium pric s in the econd stage: ));,(),;,(),;,(,;,();,();,( 321333 RrpRrpRrpRrDRrpRr φφφφφφ ⋅=Π . We are interested in two derivatives:         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π rd pd p D rd pd p D r D p rd pd prd d 2 2 31 1 33 3 3 3 33 , and (9)         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π φφφφφ d pd p D d pd p DD p d pd pd d 2 2 31 1 33 3 3 3 33 . (10) Again, the first term in RHS of both (9) and (10) equals zero. The first term in parentheses in both equations represents the direct or demand effect of the deviation in a respective variable, while the last two represent the indirect or strategic effects of such a deviation through competitors’ prices. We show that there exists a distance from the origin, R, such that if firms locate there equidistantly from one another, the necessary conditions for symmetric equilibrium ( ))sin(sin)cos(cos 2 1 2 αβαβαβ −−−+−= yxD , and (6) ( ))sin(sin)cos(cos2 2 1 3 γβγβγβπ −+−−+−= yxD . (7) The firms’ profit functions are: );,();,();,( 111 RrDRrpRr φφφ ⋅=Π , );,();,();,( 222 RrDRrpRr φφφ ⋅=Π , and 8) );,();,();,( 333 RrDRrpRr φφφ ⋅=Π . Second-stage optimal prices are derived from the system of necessary conditions obtained from these profits. The system is nonlinear and its general analytical solution for any possible Firm 3’s location, L 3 , is therefore out of reach. However, we are looking for symmetric equilibrium, so the derivation of optimal prices is straightforward. Lemma 5: When three firms in the first stage of the game position their stores R away from the origin, equidistantly from one another, the optimal Nash equilibrium price they charge in the second stage is p * = π R 3 3 . The system of first-order conditions derived in the proof of Lemma 5 (A.8-A.10) completely characterizes price competition in the second stage given that Firm 3 chooses its location reasonably close to the proposed L 3 * . 5 It remains to be seen whether such a configuration is optimal in the first stage of the game. We rewrite Firm 3’s profit function with rivals’ store positions being L 1 * and L 2 * , and firms charging equilibrium prices in the second stage: ));,(),;,(),;,(,;,();,();,( 321333 RrpRrpRrpRrDRrpRr φφφφφφ ⋅=Π . We are interested in two derivatives:         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π rd pd p D rd pd p D r D p rd pd prd d 2 2 31 1 33 3 3 3 33 , and (9)         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π φφφφφ d pd p D d pd p DD p d pd pd d 2 2 31 1 33 3 3 3 33 . (10) Again, the first term in RHS of both (9) and (10) equals zero. The first term in parentheses in both equations represents the direct or demand effect of the deviation in a respective variable, while the last two represent the indirect or strategic effects of such a deviation through competitors’ prices. We show that there exists a distance from the origin, R, such that if firms locate there equidistantly from one another, the necessary conditions for symmetric equilibrium ( ))sin(sin)cos(cos 2 1 2 αβαβαβ −−−+−= yxD , and (6) ( ))sin(sin)cos(cos2 2 1 3 γβγβγβπ −+−−+−= yxD . (7) The firms’ profit functions are: );,();,();,( 111 RrDRrpRr φφφ ⋅=Π , );,();,();,( 222 RrDRrpRr φφφ ⋅=Π , and (8) );,();,();,( 333 RrDRrpRr φφφ ⋅=Π . Second-stage optimal prices are derived from the system of necessary conditions obtained from these profits. The system is nonlinear and i s g ne al analytical solut o for any possible Firm 3’s location, L 3 , is therefore out of reach. However, we are looking for symmetri equilibrium, so the derivation of optimal prices s straightforward. Lemma 5: When thr e firms in t e first stage of the game position their stores R away fr m the origin, equidistantly from one another, ptimal Nash equilibriu price they charge in the second stage is p * = π R 3 3 . The system of first-order conditions derived in the proof of Lemma 5 (A.8-A.10) completely characterizes price competition in the second st g given that Firm 3 ch oses its lo ation reasonably close to the proposed L 3 * . 5 It rem ins to be seen whether such a configuration i optimal in the first stage of the game. We rewrite Firm 3’s profit function with rivals’ store positions being L 1 * and L 2 * , and firms charging equilibrium prices in the second stage: ));,(),;,(),;,(,;,();,();,( 321333 RrpRrpRrpRrDRrpRr φφφφφφ ⋅=Π . We are interested in two derivativ s:         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π rd pd p D rd pd p D r D p rd pd prd d 2 2 31 1 33 3 3 3 33 , and (9)         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π φφφφφ d pd p D d pd p DD p d pd pd d 2 2 31 1 33 3 3 3 33 . (10) Again, the first term in RHS of both (9) and (10) equals zero. The first term in parentheses in both equations represents the direct or demand effect f the deviation in a spective variable, while the last two represent the indirect or strat gic effec s f such a deviation through competitors’ prices. We show that there exists a distance from the origin, R, such that if firms locate there equidistantly from one another, the nec ssary conditions f r symmetric quilibrium ( ))sin(sin)cos(cos 2 1 2 αβαβαβ −−−+−= yxD , and (6) ( ))sin(sin)cos(cos2 2 1 3 γβγβγβπ −+−−+−= yxD . (7) Th fi s’ rofit functions are: );,();,();,( 111 RrDRrpRr φφφ ⋅=Π , );,();,();,( 222 RrDRrpRr φφφ ⋅=Π , and (8) );,();,();,( 333 RrDRrpRr φφφ ⋅=Π . Second- tag optimal prices are derived from the system of necessar c nditions obtained from these profits. Th system is nonlinear and its general a alytical solu ion for any possible Firm 3’s location, L 3 , is therefore out of reac . Ho ever, we are looking for symmetric equilibrium, so the derivation of optimal prices is straightforward. Lemma 5: When three firms in the first st ge of the game osition their stores R away from the origin, equidistantly from one another, the optimal Nash equilibrium price they charge in the second stage is p * = π R 3 3 . Th system of first-order c itions derived in the proof f Lemma 5 (A.8-A.10) completely charac eriz s price competition in the second stage given that Firm 3 chooses it location reasonably close o the proposed L 3 * . 5 It remains to be seen whether such a configuration is optimal in the first stage of the game. We rew te F rm 3’ profit function with rivals’ store positions being L 1 * and L 2 * , and firms charging equilibrium prices in the second stage: ));,(),;,(),;,(,;,();,();,( 321333 RrpRrpRrpRrDRrpRr φφφφφφ ⋅=Π . We are interested in two derivatives:         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π rd pd p D rd pd p D r D p rd pd prd d 2 2 31 1 33 3 3 3 33 , and (9)         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π φφφφφ d pd p D d pd p DD p pd pd d 2 2 31 1 33 3 3 3 33 . (10) Again, th first term in RHS of both (9) and (10) equals zero. The first term in parentheses in both equatio s represents the direct or demand effe t of the deviation in a respective variable, while the last tw represent th indirect or strategic eff cts of such a devi ion through compet tors’ prices. We show tha ther exists a distance from the origin, R, such that if firms locate there equidistantly from one another, the necessary conditions for symmetric equilibrium ( ))sin(sin)cos(cos 2 1 2 αβαβαβ −−−+−= yxD , and (6) ( ))sin(sin)cos(cos2 2 1 3 γβγβγβπ −+−−+−= yxD . (7) The firms’ profit functions are: );,();,();,( 111 RrDRrpRr φφφ ⋅=Π , );,();,();,( 222 RrDRrpRr φφφ ⋅=Π , and (8) );,();,();,( 333 RrDRrpRr φφφ ⋅=Π . ec -stage optimal pric are derived from th system of necessary conditions obtain d fro t se profits. The system is nonlinear and its g n al an lytical solution for any possible Firm 3’s location, L 3 , is theref re out of rea h. However, we are looking for symmetric equilibrium, so the derivation of optimal prices is straightforward. Lemma 5: Whe three firms in the fi s stage of the gam position their stores R away from the origin, equidistantly from one another, the optimal Nash equilibrium price they charge in the second stage is p * = π R 3 3 . Th system of first-order co ditions d riv d in the proof of Lemma 5 (A.8-A.10) completely c aracterizes price com tition in the second stage given that Firm 3 chooses its loc ti reasonably close to the proposed L 3 * . 5 It remains to be seen whether such a configuration is optimal in the first stage of the game. We r wr te Fir ’s profit functi wi h rivals’ store positions being L 1 * and L 2 * , and firms charging equilibrium prices in the second stage: ));,(),;,(),;,(,;,();,();,( 321333 RrpRrpRrpRrDRrpRr φφφφφφ ⋅=Π . We are interested in two derivatives:         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π rd pd p D rd pd p D r D p rd pd prd d 2 2 31 1 33 3 3 3 33 , and (9)         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π φφφφφ d pd p D d pd p DD p pd pd d 2 2 31 1 33 3 3 3 33 . (10) Again, th first term in RHS of both (9) and (10) equals zero. The first term in parentheses in both equations represents the direct or dem nd effect f the devi tion n a respective variable, while the last two represent the indir ct or trategic effects f such a deviation t rough compe itors’ price . We show that there exists a distance fr m the origin, R, such that if f rms locate there equidistantly from one another, the necessary conditions for symmetric equilibrium ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012330 (10) Again, the first term in RhS of both (9) and (10) equals zero. The first term in parentheses in both equations represents the direct or demand effect of the deviation in a respective variable, while the last two represent the indirect or strategic effects of such a deviation through competitors’ prices. We show that there exists a distance from the origin, R, such that if firms locate there equidistantly from one another, the necessary conditions for symmetric equilibrium are satisfied. The derivatives we need to determine the de- mand and strategic effects in (9) are collected in Lemma 6. Lemma 6: When L1 * = (R,−π/3), L2* = (R,π/3) and L3* = (R,π): (a) are satisfied. The derivatives we need to determine the demand and strategic effects in (9) are collected in Lemma 6. Lem 1 * / 2 * 3 * 32 14 3 R R r D − −= ∂ ∂ , (b) 32 1 2 3 1 3 Rp D p D = ∂ ∂ = ∂ ∂ , and (c) dr dp dr dp 21 = = ( ) 3390 3236399 2 π πππ − −+−+− R . Part (a) considers the direct demand effect. If R > 0.25 Firm 3 would like to position its store closer to the origin as far as this effect is concerned. This way it captures the opponents’ demand around the center of the disk. For R < 0.25 the effect reverses, Firm 3 loses demand to competitors and would like to move away from the origin. 6 Part (b) quantifies the positive effect competitors’ prices have on the demand captured by Firm 3. The most demanding task is to evaluate the effect Firm 3’s radial deviation has on opponents’ prices (part (c)). In order to simplify a very complex exercise we exploit the symmetry of the problem extensively. It is clear that replies in prices of both competitors must be identical when Firm 3 moves along the vertical axis. We therefore use only necessary conditions for profit maximization with respect to own prices for Firms 2 and 3 (A.9-A.10) in the proof of part (c). It can be readily verified that drdp 1 , drdp 2 , and drdp 3 are positive for any R∈[0,1], that is, when Firm 3 moves towards the origin, prices decrease as competition toughens, and vice versa. These results, when compared to those for a duopoly (Lemma 2), offer some idea for what follows. Suppose all three firms were located on the perimeter of the disk, and Firm 3 contemplated a small radial move towards the origin. The line of marginal consumers affected by this move is of length two (BO and CO; see Figure 3). The same is true for the duopoly case. Hence, direct demand effects should be very similar in both cases. They are −1 in the duopoly and −0.87 with three firms. 7 Furthermore, we derive the elasticity of the firm’s demand with respect to its radial distance from the origin when all the firms are on the perimeter of the disk. The results are −0.64 and −0.83 for the duopoly and three-firm oligopoly, respectively. At current demands, a firm in a three-firm case gains relatively more than in the duopoly when it moves its store towards the origin (∆R < 0). We also derive the elasticity of opponents’ prices with respect to the firm’s radial distance from the origin. 8 It is 0.71 in the duopoly and 0.41 in the three-firm oligopoly. Rivals’ price cut response to a firm moving towards the origin in the first stage of the game will be weaker in the three-firm case than in the duopoly. This is because a price cut by one of the two firms that have not moved would not only affect the aggressive rival, but the other neighbor as well. This would provoke a response from a peaceful rival and would lead to lower profits made on consumers not affected by the aggressive firm. We have illustrated the incentive a firm in a three-firm oligopoly has when R = 1, when compared to the duopoly. It will gain relatively more demand directly and will be punished by relatively less severe a price cut by rivals in the second stage. If a firm residing at R = 1 in the duopoly had an incentive to , (b) are satisfied. The derivatives we need to determine the demand and strategic effects in (9) are collected in Lemma 6. Lemma 6: When L 1 * = (R,−π/3), L 2 * = (R,π/3) and L 3 * = (R,π): (a) 32 14 3 R R r D − −= ∂ ∂ , 32 1 2 3 1 3 Rp D p D = ∂ ∂ = ∂ ∂ , and (c) dr dp dr dp 21 = = ( ) 3390 3236399 2 π πππ − −+−+− R . Part (a) considers the direct demand effect. If R > 0.25 Firm 3 would like to position its store closer to the origin as far as this effect is concerned. This way it captures the opponents’ demand around the center of the disk. For R < 0.25 the effect reverses, Firm 3 loses demand to competitors and would like to move away from the origin. 6 Part (b) quantifies the positive effect competitors’ prices have on the demand captured by Firm 3. The most demanding task is to evaluate the effect Firm 3’s radial deviation has on opponents’ prices (part (c)). In order to simplify a very complex exercise we exploit the symmetry of the probl m extensively. It is clear that replie in prices of both competitors must be identic l when Fi 3 moves along the vertical axis. W therefore use only necessary conditions for profit maximization with respect to own prices for Firms 2 and 3 (A.9-A.10) in the proof of part (c). It can be readily verified th t drdp 1 , drdp 2 , and drdp 3 are positive for any R∈[0,1], that is, when Firm 3 moves towards the origin, prices decrease as competition toughens, and vice versa. These results, when compared to those for a duopoly (Lemma 2), offer some idea for what follows. Suppose all three firms were located on the perimeter of the disk, and Firm 3 contemplated a small radial move towards the origin. The line of marginal consumers affected by this move is of length two (BO and CO; see Figur 3). The same is tru for the duopoly case. Hence, direct demand effects should be very similar in both cases. They are −1 in the duopoly and −0.87 with three firms. 7 Furthermore, we derive the elasticity of the firm’s demand with respect to its radial distance from the origin when all the firms are on the perimeter of the disk. The results are −0.64 and −0.83 f r the duopoly and thre -firm oligopoly, respectively. At current d mands, a firm in a three-firm case gains rel tively more than in the duopoly when it moves its store towards the origin (∆R < 0). We also derive the elasticity of opponents’ prices with respect to the firm’s radial distance from the origin. 8 It is 0.71 in the duopoly and 0.41 in the three-firm oligopoly. Rivals’ price cut response to a firm movi g towards the origin in the first stage of the game will be weaker in the three-firm case than in the duopoly. This is because a price cut by one of the two firms that have not moved would not only affect the aggressive rival, but the other neighbor as well. This would provoke a response from a peaceful rival and would lead to lower profits made on consumers not affected by the aggressive firm. We have illustrated the incentive a firm in a three-firm oligopoly has when R = 1, when compared to the duopoly. It will gain relatively more demand directly and will be punished by relatively less severe a price cut by rivals in the second stage. If a firm residing at R = 1 in the duopoly had an incentive to , and (c) are satisfied. The derivativ s we ne d to determine the demand and strategic effects in (9) are collected in Lemma 6. Lemma 6: When L 1 * = (R,−π/3) L 2 * = (R,π/3) and L 3 * = (R,π): (a) 32 14 3 R R r D − −= ∂ ∂ , (b) 2 1 2 3 1 3 Rp D p D = ∂ ∂ = ∂ ∂ ( ) dr dp dr dp 21 = = ( ) 3390 236399 2 π πππ − −+−+− R . Part (a) considers he direct demand effect If R > 0.25 Firm 3 would like to po ition its store close to the origin a far as thi effect is concerned. This way it captu es the opponents’ demand arou d the center of the disk. For R < 0.25 h effect reverses, Firm 3 loses demand to competitors and would lik to move away from the origin. 6 Part (b) quantifies th positive eff ct com etitors’ prices ave on the deman captured by Firm 3. The most demanding task is o evaluate he effect Firm 3’s radial deviation has on op onents’ prices (part (c)). In order to simplify a very complex exercise we exploit the symm try of the problem extensive y. I is cl ar that repli in prices f both competitors must be identical when Firm 3 moves along the vertical axis. We therefore use only necessary conditions for prof t maximization wi h respect to own prices for Firms 2 and 3 (A.9-A.10) in the proof of part (c). It can be readily v rified that r 1 , rdp 2 , nd drdp 3 ar positive for any R∈[0,1], that is, when Firm 3 moves towards the origin, p ices decrease as competition toughens, and vice versa. These results, when compar t those for a duopoly (Lemma 2), offer some idea for what follow . Suppos all three firms w re located on the perim ter of the disk, and Firm 3 contemplated small radial move towards the origin. The line of marginal consum rs affected by this move is of length two (B and CO; see Figure 3). The same is true f r the duopoly case. H nce, direct demand effects should be ve y similar in both cases. They are −1 in the duopoly and −0.87 with three firms. 7 Furthermo e, we derive the elasticity of the firm’s demand with respect to its r dial distanc from the origin when all the firms are on the perim ter of the disk. The results are −0.64 and −0.83 f r the duopoly and three-firm oligo oly, respec ively. At current demands, firm in a three-firm case gains relatively more th n in the d opoly wh n it moves its store towards the origin (∆R < 0). We also derive the elasticity of op onents’ prices wi h respect to the f rm’s r dial distanc from the origin. 8 It is 0.71 in the duopoly a d 0.41 in the three-firm oligopoly. Rivals’ price cut response to a firm moving towards the origin in the first stage of the game will be weaker in the three-firm case than in the duopoly. This is because a price cut by one f the two firms that have not m ved w u d not only affect the aggressive rival, but th other neighbor as well. This would provoke a response from a peaceful rival and would lead to lower profits made on co sumers not affected by the agg essive firm. We have illustrated he incentive firm in a three-firm oligopoly has when R = 1, when compared to the duopoly. It wi l gain relatively more emand directly and will be punished by relativ ly l ss severe a price cut by rivals in the second stage. If a f rm r iding at R = 1 in the duopoly had an incentive to . part (a) considers the direct demand effect. If R > 0.25 Firm 3 would like to position its store closer to the origin as far as this effect is concerned. This way it captures the oppo- nents’ demand around the center of the disk. for R < 0.25 the effect reverses, Firm 3 loses demand to competitors and would like to move away from the origin.7 part (b) quantifies the positive effect competitors’ prices have on the demand captured by Firm 3. The anding task is o evalua e the ffect Firm 3’s r dial deviation has on p- ponents’ prices (part (c)). In order to simplify a very complex exercise we exploit t e symmetry of the problem extensively. It is clear that replies in prices of both competitors must be identical when Firm 3 moves along the vertical axis. We therefore use only nec- essary conditions for profit maximization with respect to own prices for Firms 2 and 3 (A.9-A.10) in the proof of part (c). It can be readily verified that dp1/dr, dp2/dr, and dp3 are positive for any R∈[0,1], that is, when Firm 3 moves towards the origin, prices decrease as competition toughens, and vice versa. These results, when compared to those f r a duopoly (Lemma 2), ffer some idea for what follows. Suppose all three firms were located on the perimeter of the disk, and Firm 3 contemplated a small radial move towards the origin. The line of marginal consumers affected by this move is of length two (BO and CO; see figure 3). The sa e is true for the duopoly case. hence, direct demand effects should be very similar in both cases. They are −1 in the duopoly and −0.87 with three firms.8 furthermore, we derive the elasticity 7 point D move along ve tical axes towards the top of the disk, so Firm 3 gains some ew customers from the oponents (see figure 3). At the same time line segments DC and DB rotate toward each other, which means that Firm 3 loses some customers on the outskirts of the market. The total effect is negative for R < 0.25. 8 from part (a) in Lemmas 2 and 6. ( ))sin(sin)cos(cos 2 1 2 αβαβαβ −−−+−= yxD , and (6) ( ))sin(sin)cos(cos2 2 1 3 γβγβγβπ −+−−+−= yxD . (7) The firms’ profit functions are: );,();,();,( 111 RrDRrpRr φφφ ⋅=Π , );,();,();,( 222 RrDRrpRr φφφ ⋅=Π , and (8) );,();,();,( 333 RrDRrpRr φφφ ⋅=Π . Second-stage optimal prices are derived from the system of necessary conditions obtained from these profits. The system is nonlinear and its general analytical solution for any possible Firm 3’s location, L 3 , is therefore out of reach. However, we are looking for symmetric equilibrium, so the derivation of optimal prices is straightforward. Lemma 5: When three firms in the first stage of the game position their stores R away from the origin, equidistantly from one another, the optimal Nash equilibrium price they charge in the second stage is p * = π R 3 3 . The system of first-order conditions derived in the proof of Lemma 5 (A.8-A.10) completely characterizes price competition in the second stage given that Firm 3 chooses its location reasonably close to the proposed L 3 * . 5 It remains to be seen whether such a configuration is optimal in the first stage of the game. We rewrite Firm 3’s profit function with rivals’ store positions being L 1 * and L 2 * , and firms charging equilibrium prices in the second stage: ));,(),;,(),;,(,;,();,();,( 321333 RrpRrpRrpRrDRrpRr φφφφφφ ⋅=Π . We are interested in two derivatives:         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π rd pd p D rd pd p D r D p rd pd prd d 2 2 31 1 33 3 3 3 33 , and (9)         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+⋅ ∂ Π∂ = Π φφφφφ d pd p D d pd p DD p d pd pd d 2 2 31 1 33 3 3 3 33 . (10) Again, the first term in RHS of both (9) and (10) equals zero. The first term in parentheses in both equations represents the direct or demand effect of the deviation in a respective variable, while the last two represent the indirect or strategic effects of such a deviation through competitors’ prices. We show th t th re exists a istanc fro the origin, R, such that if firms locate there equidistantly from one another, the necessary conditions for symmetric equilibrium A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 331 of the firm’s demand with respect to its radial distance from the origin when all the firms are on the perimeter of the disk. The results are −0.64 and −0.83 for the duopoly and three-firm oligopoly, respectively. At current demands, a firm in a three-firm case gains relatively more than in the duopoly when it moves its store towards the origin (∆R < 0). We also derive the elasticity of opponents’ prices with respect to the firm’s radial distance from the origin.9 It is 0.71 in the duopoly and 0.41 in the three-firm oligopoly. Rivals’ price cut response to a firm moving towards the origin in the first stage of the game will be weaker in the three-firm case than in the duopoly. This is because a price cut by one of the two firms that have not moved would not only affect the aggressive rival, but the other neighbor as well. This would provoke a response from a peaceful rival and would lead to lower profits made on consumers not affected by the aggressive firm. We have il- lustrated the incentive a firm in a three-firm oligopoly has when R = 1, when compared to the duopoly. It will gain relatively more demand directly and will be punished by rela- tively less severe a price cut by rivals in the second stage. If a firm residing at R = 1 in the duopoly had an incentive to move even farther away from the origin we expect this not to be the case with three firms anymore. Lemma 7 presents an interior radial distance from the origin for the three firms positioned equidistantly from each other, which does not make them want to relocate in radial direction. Lemma 7: When move even farther away from the origin we expect this not to be the case with three firms anymore. Lemma 7 presents an interior radial distance from the origin for the three firms positioned equidistantly from each other, which does not make them want to relocate in radial direction. Lemma 7: h R * = )5476.0( 38288 231572 2 ≈ − −+ π ππ and firms are located equidistantly, no firm finds it profitable to locally deviate from it. Proof of Lemma 7 yields another result. Symmetric configuration with maximum distance between three firms, i.e. maximum differentiation, is never optimal. Corollary 1: The three firms located on the perimeter of the disk, equidistantly from each other, is not a subgame perfect Nash equilibrium of the game. Lemma 5 shows that locating at the perimeter of the disk will still yield the highest possible prices and profits in a symmetric configuration, but these are not sustainable since capturing the opponents’ demand around the origin is too tempting; a classic prisoner dilemma on an oligopoly market. Next, we show that, when firms are positioned equidistantly, none of them has an incentive to locally move along a polar direction. Lemma 8: For any R, φ = π is locally optimal for Firm 3. Results from Lemmas 5, 7, and 8 are summarized in Proposition 2. Proposition 2: When in the first stage of the game the three firms locate their stores R * = )5476.0( 38288 231572 2 ≈ − −+ π ππ away from origin, equidistantly from each other, the equilibrium price they charge in the second stage is πR * 3 3 . Furthermore, every firm’s location is a local best response to rivals’ locations. We lack analytical proof that every firm’s location is in fact a global best reply to rivals’ locations given the price competition in the second stage of the game. We therefore solve the fi s are located equidistantly, no firm finds it profitable to locally deviate from it. proof of Lemma 7 yields another result. Symmetric configuration with maximum dis- tance between three firms, i.e. maximum differentiation, is never optimal. Corollary 1: The three firms located on the perimeter of the disk, equidistantly from each other, is not a subgame perfect Nash equilibrium of the game. Lemma 5 shows that locating at the perimeter of the disk will still yield the highest pos- sible prices and profits in a symmetric configuration, but these are not sustainable since capturing the opponents’ demand around the origin is too tempting; a classic prisoner dilemma on an oligopoly market. Next, we show that, when firms are positioned equidistantly, none of them has an incen- tive to locally move along a polar direction. Lemma 8: for any R, φ = π is locally optimal for firm 3. Results from Lemmas 5, 7, and 8 are summarized in proposition 2. 9 from part (c) in Lemmas 2 and 6. ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012332 proposition 2: When in the first stage of the game the three firms locate their stores R* move even farther away from the origin we expect this not to be the case with three firms anymore. Lemma 7 presents an interior radial distance from the origin for the three firms positioned equidistantly from each other, which does not make them want to relocate in radial direction. Lemma 7: When R * = )5476.0( 38288 231572 2 ≈ − −+ π ππ and firms are located equidistantly, no firm finds it profitable to locally deviate from it. Proof of Lemma 7 yields another result. Symmetric configuration with maximum distance between three firms, i.e. maximum differentiation, is never optimal. Corollary 1: The three firms located on the perimeter of the disk, equidistantly from each other, is not a subgame perfect Nash equilibrium of the game. Lemma 5 shows that locating at the perimeter of the disk will still yield the highest possible prices and profits in a symmetric configuration, but these are not sustainable since capturing the opponents’ demand around the origin is too tempting; a classic prisoner dilemma on an oligopoly market. Next, we show that, when firms are positioned equidistantly, none of them has an incentive to locally move along a polar direction. Lemma 8: For any R, φ = π is locally optimal for Firm 3. Results from Lemmas 5, 7, and 8 are summarized in Proposition 2. Proposition 2: When in the first stage of the game the three fir s locate their stores R * = )5476.0( 38288 231572 2 ≈ − −+ π ππ away from origin, equidistantly from each other, the equilibrium price they charge in the second stage is πR * 3 3 . Furthermore, every firm’s location is a local best response to rivals’ locations. We lack analytical proof that every firm’s location is in fact a global best reply to rivals’ locations given the price competition in the second stage of the game. We therefore solve the igin, equidistantly from each other, the equilibrium price they charge in the second stage is move even farther away from the origin we expect this not to be the case with three firms anymore. Lem a 7 prese ts an int rior radial distance from the origin or the three firms positioned quidistantly f om each other, which does not make them want to relocate in radial direction. Lemma 7: When R * = )5476.0( 38288 231572 2 ≈ − −+ π ππ and firms are located equidistantly, no firm finds it profitable to locally deviate from it. Proof of Lemma 7 yields another result. Symmetric configuration with maximum distance between three firms, i. . maximum diffe entiation, is never optimal. Corollary 1: The three firms located on the perimeter of the disk, equidistantly from each other, is not a subgame p rfect Nash equilibrium of the game. Lemma 5 shows that locating at the perimeter of the disk will still yield the highest possible prices and profits i a symmet c configuration, but these ar not sustainable since capturing the pponent ’ demand around the rigin is too tempting; a classic prisoner dilemma on an oligopoly market. Next, we show that, when firms are positioned equidistantly, none of them has an incentive to l cally move along a polar d rection. Lemma 8: For any R, φ = π is locally optimal for Firm 3. Results from Lemmas 5, 7, and 8 are summarized in Proposition 2. Proposition 2: When in the first stage of the game the three firms locate their stores R * = )5476.0( 38288 231572 2 ≈ − −+ π ππ away from origin, equidistantly from each other, the equilibrium price they charge in the secon πR * 3 3 . Furthermore, every firm’s location is a local best response to rivals’ locations. We lack analytical proof that every firm’s location is in fact a global best reply to rivals’ loc tions given the pric competiti n in the second stage of th game. We therefore solve the . f er ore, every firm’s locat on is a local best response to rival ’ locations. We lack analytical proof that every firm’s location is in fact a global best reply to rivals’ locations given the price competition in the second stage of the game. We therefore solve the system of first-order conditions for second-stage profit maximization with respect to prices (A.8-A.10) for an array of Firm 3’s locations numerically to derive optimal profits. We find (see figure 4) that locations we propose are globally optimal, and state our next result. Result 2: Three firms positioning their stores at R* = system of first-order conditions for second-stage profit maximization with respect to prices (A.8- A.10) for an array of Firm 3s locations numerically to derive optimal profits. We find (see Figure 4) that locations we propose are globally optimal, and state our next result. Result 2: Thr e firms positioni t res at R * )5476.0( 38288 231572 2 ≈ − −+ π ππ , equidistantly from each other in the first stage of the game, and setting p * = πR * 3 3 in the second, is a subgame perfect Nash equilibrium of the game. There are two novel perspectives on product differentiation to this result. First, in a setting that leads to differentiation in one characteristic only in duopoly, which is consistent with the existing literature, the three firms find it optimal to differentiate their products in two characteristics. Second, firms do not differentiate their products as much as they could have. In a setting that yields familiar a min-max differentiation result in the duopoly, a medium-medium type of product differentiation is observed. When all three firms are located on the perimeter of the disk, a radial deviation towards the center of the disk by a firm is followed by rivals’ price cut that is less severe than the one observed in duopoly. Therefore, positive direct demand effect of such a move outweighs the negative strategic effect of rivals’ price cuts, and firms move closer together. We can speculate on whether the min-min-…-min part of the product differentiation result by Irmen and Thisse (1998) could be observed in a three-firm market with additional product characteristics. Firms may not want to differentiate their products in any additional characteristics, since they find max-max differentiation to be excessive in two dimensions, already. This suggests that firms may have no need to differentiate their products in another, third, dimension, since even the possibilities in two were not exhausted. We therefore predict that the min-min-…-min part of a product differentiation result is going to hold in our setting as well, which is what Feldin (2001) formally shows. This hypothesis rests on the market being uniformly populated by consumers. If there were some areas with higher population density there would be obviously more differentiation, since firms would try to tailor their products to meet the tastes of these different groups of customers. , equi- distantly from each other in the first stage of the game, and setting system of first-orde conditions for second-stage profit maximization with respect to prices (A.8- A.10) for an ar ay of F rm 3s locations numerically to derive optimal profi s. We find (see Figure 4) that locations we propose are globally optimal, and state our next result. Result 2: Three firms positioning their stores at R * = )5476.0( 38288 231572 2 ≈ − −+ π ππ , equidistantly from eac other in the first stage of the game, and se ting p * = πR * 3 3 in the second, is a subgame perfect Nash equilibrium of the game. There are two novel perspectives on product differentiation to this result. First, in a setting that l ds o differ ntiation in one cha acteristic only n duop ly, which is consistent with the existing l terature, he three firms find it optimal to ifferentiate their products in two characteristics. Second, fir s do not differentiate their products as much as they could have. In a se ting that yields familiar a min-max differ ntiation resul in the duopoly, a medium-medium type of pro uct diffe entiation is observed. When all hree firms are located on the peri eter of the disk, a radial deviation towards the center of e disk by firm is followed by rivals’ price cut that is less severe than the on observed in duopol . Therefore, p sitive direct demand effect of such a move outweighs th negative strategic effect of rivals’ price cuts, an firms move closer t g ther. We can speculate on whether the min-min-…-min part of the product differentiation result by Irmen and Thiss (1998) could be observed in a three-firm ma ket with additional prod c ch racter tics. Firms may not want to differentiate their products in any additional charac eristics, since they find max-max differentiation to be excessive in two dimensions, lr ady. This sugg sts that firms may hav no need to differ nt at their product in another, third, dimension, since even the possibilities in two w not x austed. We theref re predict that the mi -mi -…-mi part f a product differentiation result is going to hold in our se ting as w ll, which is what Feldin (2001) formally shows. Thi hypothesis rests on the market being uniformly populated by consumers. If there were some area with higher population density there would be obviously more differen iation, since firms would try to tailor their products to me t the tastes f these different groups of customers. e second, is a subgame perfect Nash equilibrium of the game. There are two novel perspectives on product differentiation to this result. first, in a setting that leads to differentiation in one characteristic only in duopoly, which is con- sistent with the existing literature, the three firms fi it op imal to differenti te heir products in two characteristics. Second, firms do not different ate their produc s as much as they could have. In a set ing that yi lds familiar a min-max differentiation result in the duopoly, a medium-medium type f product differentiation is observed. When all three firms are located on the perimeter of the disk, a radial deviation to- wards the center of the disk by a firm is followed by rivals’ price cut that is less severe than the one observed in duopoly. Therefore, positive direct demand effect of such a move outweighs the negative strategic effect of rivals’ price cuts, and firms move closer together. We can speculate on whether the min-min-…-min part of the product differentiation result by Irmen and Thisse (1998) could be observed in a three-firm market with ad- ditional product characteristics. firms may not want to differentiate their products in any additional characteristics, since they find max-max differentiation to be excessive in two dimensions, already. This suggests that firms may have no need to differentiate their products in a other, third, dimension, since even the possibilities i two were not exhausted. We th refore predict that t min-m n-…-min pa t of a product differe tia- ti n result is going to hold n our s tti g as well, which s what feldin (2001) formally shows. This hypothesis rests n the market being niformly populated by consu ers. If there were some areas with higher population density there would be obviously more differentiation, since firms would try to tailor their products to meet the tastes of these different groups of customers. A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 333 figure 4: Firm 3’s profits with respect to location of its store and given equilibrium loca- tions of opponents. Source: Own calculations. 4. wElfarE analySIS In this section we compare the extent of product differentiation in our competitive mar- ket to a social optimum for a duopoly and a three-firm oligopoly. A social planner who cares about well-being of all agents in the market would simply minimize the traveling costs borne by buyers. Each of n firms on the market will satisfy π/n of the market de- mand in a symmetric configuration. Every such piece of the pie can be split into two halves symmetrically over the line segment connecting the firm’s location and the center of the disk. One such part of the disk is presented in figure 5. To find the social opti- mum, we look at where a particular firm should have been in order to minimize the total traveling costs that consumers, from such a half of the disk, bear. figure 5: Derivation of socially optimal product differentiation. ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012334 The cost that a consumer residing at (ρ,φ) is faced with when buying from the firm lo- cated at R is: The cost that a consumer residing at (ρ,φ) is faced with when buying from the firm located at R is: 22222 cos2)cos(sin);,( ρφρφρφρφρ +−=−+= RRRRc . The total cost borne by buyers from this portion of the disk is then: ∫ ∫ ⋅= n ddRcRC π φρρφρ 0 1 0 ),,()( . This integrates to: C(R) = n R n R ππ sin 3 2 4 12 2 −⋅ + . Minimize this expression with respect to R to get the socially optimal distance from the center of the disk for the firms, R S : π π 3 sin2 n S n R = . We compare socially optimal locations with competitive equilibrium ones derived in Sections 3 and 4 in Table 1. Table 1: Social optimum vs. competitive equilibrium. n R S R * 2 0.4244 1 3 0.5513 0.5476 Source: Own calculations This is analogous to what Brenner (2005) finds. Intense price competition in a duopoly drives firms to the edge of the market, and that is too far apart from a social standpoint. In a three-firm oligopoly the situation reverses. Product differentiation becomes too weak when compared to the social optimum. Price competition becomes less intense and firms move in on each other’s demands. This gives rise to an interesting situation. It is in the social planner’s interest to induce cooperation among firms. She might pass a regulation demanding the firms locate R S away from the origin. This would help the firms to partly resolve their prisoner’s dilemma. They would now be able to charge higher prices; therefore, they would be willing to accept such a regulation. 5 Concluding remarks We have presented some novel findings on the extent of product differentiation that firms in an oligopoly employ. Contrary to the literature on duopoly markets, we show that three firms . The total cost borne by buyers from this portion of the disk is then: The cost that a consumer residing at (ρ,φ) is faced with when buying from the firm located at R is: 22222 cos2)cos(sin);,( ρφρφρφρφρ +−=−+= RRRRc . Th t tal cost borne by buyers from this portion f the disk i then: ∫ ∫ ⋅= n ddRcRC π φρρφρ 0 1 0 ),,()( . This integrates to: C(R) = n R n R ππ sin 3 2 4 12 2 −⋅ + . Minimize this expression with respect to R to get the socially optimal distance from the center of the disk for the firms, R S : π π 3 sin2 n S n R = . We compare socially optimal locations with competitive equilibrium ones derived in Sections 3 and 4 in Table 1. Table 1: Social optimum vs. competitive equilibrium. n R S R * 2 0.4244 1 3 0.5513 0.5476 Source: Own calculations This is analogous to what Brenner (2005) finds. Intense price competition in a duopoly drives firms to the edge of the market, and that is too far apart from a social standpoint. In a three-firm oligopoly the situation reverses. Product differentiation becomes too weak when compared to the social optimum. Price competition becomes less intense and firms move in on each other’s demands. This gives rise to an interesting situation. It is in the social planner’s interest to induce cooperation among firms. She might pass a regulation demanding the firms locate R S away from the origin. This would help the firms to partly resolve their prisoner’s dilemma. They would now be able to charge higher prices; therefore, they would be willing to accept such a regulation. 5 Concluding remarks We have presented some novel findings on the extent of product differentiation that firms in an oligopoly employ. Contrary to the literature on duopoly markets, we show that three firms . This integrates to: C(R) = The cost that a consumer residing at (ρ,φ) is faced with when buying from the firm located at R is: 22222 cos2)cos(sin);,( ρφρφρφρφρ +−=−+= RRRRc . The total cost borne by buyers from this portion of the disk is then: ∫ ∫ ⋅= n ddRcRC π φρρφρ 0 1 0 ),,()( . Thi i s t : ( ) n R n R ππ sin 3 2 4 12 2 −⋅ + . Minimize this expression with respect to R to get the socially optimal distance from the center of the disk for the firms, R S : π π 3 sin2 n S n R = . We compare socially optimal locations with competitive equilibrium ones derived in Sections 3 and 4 in Table 1. Table 1: Social optimum vs. competitive equilibrium. n R S R * 2 0.4244 1 3 0.5513 0.5476 Source: Own calculations This is analogous to what Brenner (2005) finds. Intense price competition in a duopoly drives firms to the edge of the market, and that is too far apart from a social standpoint. In a three-firm oligopoly the situation reverses. Product differentiation becomes too weak when compared to the social optimum. Price competition becomes less intense and firms move in on each other’s demands. This gives rise to an interesting situation. It is in the social planner’s interest to induce cooperation among firms. She might pass a regulation demanding the firms locate R S away from the origin. This would help the firms to partly resolve their prisoner’s dilemma. They would now be able to charge higher prices; therefore, they would be willing to accept such a regulation. 5 Concluding remarks We have presented some novel findings on the extent of product differentiation that firms in an oligopoly employ. Contrary to the literature on duopoly markets, we show that three firms i ize this expre sion with respect to R to get the socially optimal distance from the center of the disk for the firms, RS: . co pare socially optimal locations with competitive equilibrium ones derived in Sections 3 and 4 in Table 1. Table 1: Social optimum vs. competitive equilibrium. n RS R* 2 0.4244 1 3 0.5513 0.5476 Source: Own calculations This is analogous to what Brenner (2005) finds. Intense price competition in a duopoly drives firms to the edge of the market, and that is too far apart from a social stand- point. In a three-firm oligopoly the situation reverses. product differentiation becomes too weak when compared to the social optimum. price competition becomes less intense and firms move in n each other’s emands. This gives rise to an interesting situ tion. It s in the s cial planner’s intere t to in e coop ration among firms. She might pass a regulation demanding the firms locate RS away from the origin. This would help the firms to partly resolve their prisoner’s dilemma. They would now b able to charge higher prices; therefore, they would be willing to accept such a regulation. 5. ConCludIng rEmarKS We have presented some novel findings on the extent of product differentiation that firms in an oligopoly employ. Contrary to the literature on duopoly markets, we show that three firms will use two product characteristics to separate their produc s from rivals’. In a setting that usually yields min-max differentiation we find three firms to utilize a medium-medium type of differentiation. It remains to be seen how even a larger number of firms affects product differentiation in multi-characteristic space. A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 335 The demand side of the model is the sole driver of our results. If firms had to cover some R&D costs to introduce new varieties of products or improve some of the characteristics, which certainly is the case in reality, differentiation would be even weaker. We therefore note that firms deciding to introduce new characteristics to their products might want to be careful as not to engage in excessive differentiation that would not yield the maximum possible profits. Also, the fact that differentiation possibilities are not exhausted leads us to think that three firms will not want to use any new product characteristics to differ- entiate themselves from competitors. The first step towards such a min-…-min-medium- medium differentiation result is presented in feldin (2001). references Brenner, S. (2005). hotelling games with Three, four, and More players. Journal of Regional Science, 45 (4), 851–864. doi: 10.1111/j.0022-4146.2005.00395.x D’Aspremont, C.J., gabszewicz J., & Thisse J-f. (1979). On hotelling’s Stability in Competition. Econometrica, 47, 1145–1150. Retrieved from http://www.jstor.org/stable/1911955 Economides, N. (1986). Minimal and Maximal product Differentiation in hotelling’s Duopoly. Economics Letters, 21, 67–71. doi: 10.1016/0165-1765(86)90124-2 Economides, N. (1989). Symmetric Equilibrium Existence and Optimality in Differentiated product Markets. Journal of Economic Theory, 47, 178–194. doi: 10.1016/0022-0531(89)90108-7 Economides, N. (1993). hotelling’s ‘Main Street’ with More than Two Competitors. Journal of Regional Sci- ence, 33 (3), 303–319. doi: 10.1111/j.1467-9787.1993.tb00228.x feldin, A. (2001). Product Differentiation: How Many Dimensions? faculty of Economics Ljubljana Working paper Series, 121, 1–18. hotelling, h. (1929). Stability in Competition. The Economic Journal, 3, 41–57. Retrieved from http://www. jstor.org/stable/2224214 Irmen, A. & Thisse, J-f. (1998). Competition in Multi-Characteristic Spaces: hotelling was Almost Right. Journal of Economic Theory, 78, 76–102. doi: 10.1006/jeth.1997.2348 Neven, D.J. (1986). On hotelling’s Competition with Non-Uniform Customer Distributions. Economics Let- ters, 21 (2), 121–126. doi: 10.1016/0165-1765(86)90049-2 Neven, D.J. & Thisse, J.-f. (1990). On Quality and Variety Competition. In: gabszewicz, J.J., Richard, J.-f. & Wolsey, L. (Eds), Economic Decision Making: Games, Econometrics, and Optimization. Contributions in the honour of Jacques H. Dreze (pp. 175–199). Amsterdam: North-holland. ReVelle, C.S., & Eiselt, h.A. (2005). Location Analysis: A Synthesis and Survey. European Journal of Opera- tional Research, 165, 1–19. doi: 10.1016/j.ejor.2003.11.032 ReVelle, C.S., Eiselt, h.A. & Daskin, M.S. (2008). A Bibliography for Some fundamental problem Catego- ries in Discrete Location Science. European Journal of Operational Research, 184, 817–848. doi: 10.1016/j. ejor.2006.12.044 Salop, S. (1979). Monopolistic Competition with Outside goods. Bell Journal of Economics, 10, 141–156. Re- trieved from http://www.jstor.org/stable/3003323 ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012336 Swann, p. (1990). product Competition and the Dimensions of the product Space. International Journal of Industrial Organization, 8, 281–295. doi: 10.1016/0167-7187(90)90021-R Tabuchi, T. (1994). Two-stage Two-dimensional Spatial Competition between Two firms. Regional Science and Urban Economics, 24, 207–227. doi: 10.1016/0166-0462(93)02031-W Tabuchi, T. & Thisse, J-f. (1995). Asymmetric Equilibria in Spatial Competition. International Journal of Industrial Organization, 13, 213–227. doi: 10.1016/0167-7187(94)00449-C A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 337 appEndIx a: proofS of lEmmaS and propoSITIonS proof of Lemma 1: We first determine a and b that we need for the demand functions. They are given by the indifference conditions for the two consumers residing in points A and B: Appendix A: Proofs of Lemmas and Propositions Proof of Lemma 1: We first determine α nd β that we need for the emand functions. They are given by the indifference con e t o consumers residing in points A and B: 22 2 22 1 )coscos()sinsin()cos(sin xrxrpxRxp −+−+=−++ φφ , where x = α, β. This equation simplifies to (A.1) and by using the fact that in symmetric equilibrium (if exists) r =R and φ = π, to (A.2): 22 12 cos)cos(2sinsin2 RrppxRrxr −+−=⋅−+⋅ φφ . (A.1) 12 cos4 ppxR −=⋅− (A.2) We state the necessary conditions for profit maximization with respect to own prices for Firm 1 and Firm 2 (from (2)): ( ) ( ) 0)cos 2 sin 2 1 11 1 =⋅−⋅+− pp zzz p zz and ( ) ( ) .0cos 2 sin2 2 1 22 2 =⋅+−⋅++− pp zzz p zzπ (A.3) Firms’ behavior in prices will be symmetric, hence, it is enough to use only first equation in (A.3), the necessary condition for Firm 1. Since in equilibrium z =π, we get: .02 1 1 =+ p zpπ All derivatives we need in this section are collected in Appendix B. Using (B.1) completes the proof. Q.E.D. Proof of Lemma 2: Since firms will charge symmetric prices: α = π/2, β = −π/2, and z = π. (a) In equilibrium line AB crosses the origin, therefore, any firm’s infinitesimal radial move towards origin, −dr, yields new customers in the area 2×dr/2, hence, raising demand by dr. (b) Using (1) and derivative (B.1) the partial derivative in question is: R zzzz p D ppp 2 1 )cos( 2 1 111 1 2 =−=⋅+−= ∂ ∂ . (c) Denote the total derivative of any variable x with respect to r by x′ for the length of this appendix. The derivative in question is obtained from the system of first-order conditions (A.3) we used to derive equilibrium prices. Since this system must hold for any pair of locations, the total differentiation with respect to r yields 1 p′ . The total derivatives are (Note: sin z = 0 and cos z = −1 in equilibrium): 0 11 1 =′⋅+⋅′+′ pp zpzpz and 0 22 2 =′⋅+⋅′+′ pp zpzpz . The derivatives needed in above system are collected in (B.1)-(B.3). It becomes: 0424 21 =++′+′− πRpp , and 0442 21 =+−′−′ πRpp , , r a b This equation simplifies to (A.1) and by using the fact that in symmetric equilibrium (if exists) r =R and φ = π, to (A.2): (A.1) (A.2) We state the necessary conditions for profit maximization with respect to own prices for Firm 1 and Firm 2 (from (2)): and (A.3) firms’ behavior in prices will be symmetric, hence, it is enough to use only first equation in (A.3), the necessary condition for Firm 1. Since in equilibrium z =π, we get: All derivatives we ne d in this section are co ected in Appendix B. U ing (B.1) completes the proof. Q.E.D. proof of Lemma 2: Since firms will charge symmetric prices: a = π/2, b = −π/2, and z = π. (a) In equilibrium line AB crosses the origin, therefore, any firm’s infinitesimal radial move towards origin, −dr, yields new customers in the area 2×dr/2, hence, raising demand by dr. (b) Using (1) and derivative (B.1) the partial derivative in question is: Appendix A: Proofs of Lemmas and Propositions Proof of Lemma 1: We first determine α and β that we need for the demand functions. They are given by the indifference conditions for the two consumers residing in points A and B: 22 2 22 1 )coscos()sinsin()cos(sin xrxrpxRxp −+−+=−++ φφ , where x = α, β. This equation simplifies to (A.1) and by using the fact that in symmetric equilibrium (if exists) r =R and φ = π, to (A.2): 22 12 cos)cos(2sinsin2 RrppxRrxr −+−=⋅−+⋅ φφ . (A.1) 12 cos4 ppxR −=− (A.2) We state the necessary conditions for profit maximization with respect to own prices for Firm 1 and Firm 2 (from (2)): ( ) ( ) 0)cos 2 sin 2 1 11 1 =⋅−⋅+− pp zzz p zz and ( ) ( ) .0cos 2 sin2 2 1 22 2 =⋅+−⋅++− pp zzz p zzπ (A.3) Firms’ behavior in prices will be symmetric, hence, it is enough to use only first equation in (A.3), the necessary condition for Firm 1. Since in equilibrium z =π, we get: .02 1 1 =+ p zpπ All derivatives we need in this section are collected in Appendix B. Using (B.1) completes the proof. Q.E.D. Proof of Lemma 2: Since firms will charge sym etric prices: α = π/2, β = −π/2, and z = π. (a) In equilibrium line AB crosses the origin, therefore, any firm’s infinitesimal radial move towards origin, −dr, yields new customers in the area 2×dr/2, hence, raising demand by dr. (b) Using (1) and derivative (B.1) the partial d rivative in question is: R zzzz p D ppp 2 1 )cos( 2 1 111 1 2 =−=⋅+−= ∂ ∂ . (c) Denote the total derivative of any variable x with respect to r by x′ for the length of this appendix. The derivative in question is obtained from the system of first-order conditions (A.3) we used to derive equilibrium prices. Since this system must hold for any pair of locations, the total differentiation with respect to r yields 1 p′ . The total derivatives are (Note: sin z = 0 and cos z = −1 in equilibrium): 0 11 1 =′⋅+⋅′+′ pp zpzpz and 0 22 2 =′⋅+⋅′+′ pp zpzpz . The derivatives needed in above system are collected in (B.1)-(B.3). It becomes: 0424 21 =++′+′− πRpp , and 0442 21 =+−′−′ πRpp , (c) Denote the total derivative of any variable x with res ect to r by ′ for the length of this app ndix. Th derivative in question is obtained from the syst m f first-order conditions (A.3) we used to derive equilibrium prices. Since this system must h ld for any pair of locations, the t tal differentiation with respect to yields 1p′ . The total derivatives are (Note: sin z = 0 and cos z = −1 in equilibrium): Appendix A: Proofs of Lemmas and Propositions Proof of Lemma 1: We first determine α and β that we need for the demand functions. They are given by the indifference conditions for th two consumers residing in points A and B: 22 2 22 1 )coscos()sinsin()cos(sin xrxrpxRxp −+−+=−++ φφ , where x = α, β. This equation simplifies to (A.1) and by using the fact that in symmetric equilibrium (if exists) r =R and φ = π, to (A.2): 22 12 c s)cos(2sinsin2 RrppxRrxr −+−=⋅−+⋅ φφ . (A.1) 12 cos4 ppxR −=⋅− (A.2) We state the n cessary conditions for profit maximization with respe t to own prices for Firm 1 and Firm 2 (from (2)): ( ) ( ) 0)cos 2 sin 2 1 11 1 =⋅−⋅+− pp zzz p zz and ( ) ( ) .0cos 2 sin2 2 1 22 2 =⋅+−⋅++− pp zzz p zzπ (A.3) Firms’ behavior in prices will be symmetri , hence, it is enough to use only first equation in (A.3), the necessary condition for Firm 1. Since in equilibrium z =π, we get: .02 1 1 =+ p zpπ All derivatives we need in this section are collected in Appendix B. Using (B.1) completes the proof. Q.E.D. Proof of Lemma 2: Since firms will charge symmetric prices: α = π/2, β = −π/2, and z = π. (a) In equilibrium line AB crosses the origin, therefore, any firm’s infinitesimal radial move towards origin, −dr, yields new customers in the area 2×dr/2, nce, raising dem by dr. (b) Using (1) and derivative (B.1) the partial derivative in question is: R zzzz p D ppp 2 1 )cos( 2 1 111 1 2 =−=⋅+−= ∂ ∂ . (c) Denote the total derivative of any variable x with respect to r by x′ for the length of this appendix. The derivative in question is obtained from the system of first-order conditions (A.3) we used to derive equilibrium prices. Since this syste must hold for any pair of locations, the total differentiat on with respect to r yields 1 p′ . The total de tives are (Note: sin z = 0 and cos z = −1 in equilibrium): 0 11 1 =′⋅+⋅′+′ pp zpzpz and 0 22 2 =′⋅+⋅′+′ pp zpzpz . The derivatives needed in above system are collected in (B.1)-(B.3). It becomes: 0424 21 =++′+′− πRpp , and 0442 21 =+−′−′ πRpp , and Ap endix A: Pro fs of Lemmas and Propositions Pro f of Lemma 1: We first determine α and β that we ne d for the demand functions. They are given by the indif erence conditions for the two consumers residing in points A and B: 22 2 22 1 )coscos()sinsin()cos(sin xrxrpxRxp −+−+=−++ φφ , where x = α, β. This equation simplifies to (A.1) and by using the fact that in symmetric equilibrium (if exists) r =R and φ = π, to (A.2): 22 12 cos)cos(2sinsin2 RrppxRrxr −+−=⋅−+⋅ φφ . (A.1) 12 cos4 ppxR −=⋅− (A.2) We state the neces ary conditions for profit maximization with respect to own prices for Firm 1 and Firm 2 (from (2) : ( ) ( 0)cos 2 sin 2 1 11 1 =⋅−⋅+− pp zzz p zz and ( ) ( ) .0cos 2 sin2 2 1 22 2 =⋅+−⋅++− pp zzz p zzπ (A.3) Firms’ behavior in prices will be symmetric, hence, it is enough to use only first equation in (A.3), the neces ary condition for Firm 1. Since in equilibrium z =π, we get: .02 1 1 =+ p zpπ All derivatives we ne d in this section are collected in Ap endix B. Using (B.1) completes the pro f. Q.E.D. Pro f of Lemma 2: Since firms will charge symmetric prices: α = π/2, β = −π/2, and z = π. (a) In equilibrium line AB cros es the origin, therefore, any firm’s infinitesi al radial move towards origin, −dr, yields new customers in t e area 2×d /2, hence, raising d mand by dr. (b) Using (1) and derivative (B.1) the partial derivative in question is: R zzzz p D ppp 2 1 )cos( 2 1 111 1 2 =−=⋅+−= ∂ ∂ . (c) Denote the total derivative of any variable x with respect to r by x′ for the length of this ap endix. The derivative in question is obtained from the system of first-order conditions (A.3) we used to derive equilibrium prices. Since this system ust hold for any pair of locations, the total dif erenti tion w h respect to r yields 1 p′ . The t tal derivati are (Note: sin z = 0 and cos z = −1 in equilibrium): 0 11 1 =′⋅+⋅′+′ pp zpzpz an 0 22 2 =′⋅+⋅′+′ pp zpzpz . The derivatives ne ded in above system are collected in (B.1)-(B.3). It becomes: 0424 21 =++′+′− πRpp , and 0442 21 =+−′−′ πRpp , . Appendix A: Proofs of Lem as and Propositions Proof of Lemma 1: We first determine α and β that we need for t demand functions. They are given by the indifference conditions for the two consumers residing in points A and B: 22 2 22 1 )coscos()sinsin()cos(sin xrxrpxRxp −+−+=−++ φφ , where x = α, β. This equation simplifies to (A.1) and by using the fact that in symmetric equilibrium (if exists) r =R a d φ = π, to (A.2): 22 12 cos)cos(2sinsin2 RrppxRrxr −+−=⋅−+⋅ φφ . (A.1) 12 cos4 ppxR −=⋅− (A.2) We state the necessary conditions for profit maximization with respect to own prices for Firm 1 and Firm 2 (from (2)): ( ) ( ) 0)cos 2 sin 2 1 11 1 =⋅−⋅+− pp zzz p zz and ( ) ( ) .0cossin2 2 1 22 2 =⋅+−⋅++− pp zzz p zzπ ( .3) Firms’ behavior in prices will be symmetric, hence, it is enough to use only first equation in (A.3), the necessary ondition for Firm 1. Since in equilibri m z =π, we get: .02 1 1 =+ p zpπ All derivatives we need in this section are collected in Appendix B. Using (B.1) completes the proof. Q.E.D. Proof of Lemma 2: Since firms will charge symmetric prices: α = π/2, β = −π/2, and z = π. (a) In equilibrium line AB crosses the origin, therefore, any firm’s infinitesimal radial move towards origin, −dr, yield new ustom rs in the a ea 2×dr/2, hence, raising demand by r. (b) Using (1) and derivative (B.1) the partial derivative in questio is: R zzzz p D ppp 2 1 )cos( 2 1 111 1 2 =−=⋅+−= ∂ ∂ . (c) Denote th tot l d rivative of ny v riable x with respect to r by x′ for the length of this appendix. The derivative in question is obtained from the system of first-order conditions (A.3) we used to derive equilibrium prices. Since this system must hold for any pair of locations, the total differentiation with respect to r yields 1 p′ . The total derivatives are (Note: sin z = 0 and cos z = −1 i equilibrium): 0 11 1 =′⋅+⋅′+′ pp zpzpz and 0 22 2 =′⋅+⋅′+′ pp zpzpz . Th derivativ s needed in above system ar collected in (B.1)-(B.3). It becom s: 0424 21 =++′+′− πRpp , and 0442 21 =+−′−′ πRpp , Appendix A: Proofs of Lemmas and Propositions Proof of Lemma 1: We first determine α and β that we need for the demand functions. They are given by the indifference conditions for the two consu ers residing in points A and B: 22 2 22 1 )coscos()sinsin()cos(sin xrxrpxRxp −+−+=−++ φφ , whe e x = α, β. This equation simplifies to (A.1) and by using the fact that in symmetric equilibrium (if exists) r =R and φ = π, to (A.2): 22 12 cos)cos(2sinsin2 RrppxRrxr −+−=⋅−+⋅ φφ . (A.1) 12 s4 ppxR −=⋅− (A.2) We state the necessary conditions for profit maximization with respect to own prices for Firm 1 and Firm 2 (from (2)): ( ) ( ) 0)cos 2 sin 2 1 11 1 =⋅−⋅+− pp zzz p zz and ( ) ( ) .0cos 2 sin 2 1 22 2 =⋅+−⋅++− pp zzz p zzπ (A.3) Firm ’ behavior in prices will be symme r c, hence, it is en ugh to use only first equation in (A.3), the necessary condition for Firm 1. Si ce in equilibrium z =π, we get: .02 1 1 =+ p zpπ All derivatives we need in this section are collected in Appendix B. Using (B.1) completes the pr . Q.E.D. Proof of Lemma 2: Since firms will charge symmetric prices: α = π/2, β = −π/2, and z = π. (a) I equilibrium line AB cro ses th origin, th refore, any firm’s infinitesimal radial move towards origin, −dr, yields new custom s in the rea 2×dr/2, he ce, ra sing demand by dr. (b) Using (1) an derivative (B.1) the partial derivative in question is: R zzzz p D ppp 2 1 )cos( 2 1 111 1 2 =−=⋅+−= ∂ ∂ . (c) Denote the total derivative of any variable x with respect to r by x′ for the length of this appendix. The derivative in question is obtained from the system of first-order conditions (A.3) we used to derive quilibrium prices. Since this system must hold for any pair of locations, the total differentiation with respect to r yields 1 p′ . The total derivatives are (Note: sin z = 0 and cos z = −1 in equilibrium): 0 11 1 =′⋅+⋅′+′ pp zpzpz nd 0 22 2 =′⋅+⋅′+′ pp zpzpz . The derivatives needed in above system are collected in (B.1)-(B.3). It becomes: 0424 21 =++′+′− πRpp , and 0442 21 =+−′−′ πRpp , Appe dix A: Proofs of Lemmas and Propositions Proof of Lemma 1: We first etermine α and β tha e need for th demand functions. They are given by the indifference conditions for the two consumers residing in points A and B: 2222 1 )coscos()sinsin()cos(sin xrxrpxRxp −+−+=−++ φφ , where x = α, β. This equation simplifies to (A.1) and by using the fact that in symmetric equilibrium (if exists) r =R and φ = π, to (A.2): 22 12 cos)cos(2sinsin2 RrppxRrxr −+−=⋅−+⋅ φφ . (A.1) 12 cos4 ppxR −=⋅− (A.2) We state the neces ary conditions for profit maximization with respect to own prices for Firm 1 and Firm 2 (from (2)): ( ) ( ) 0)cos 2 sin 2 1 11 1 =⋅−⋅+− pp zzz p zz and ( ) ( ) .0cos 2 sin2 2 1 22 2 =⋅+−⋅++− pp zzz p zzπ (A.3) Firms’ behavior in prices will be symmetric, hence, it is enough to use only first equation in (A.3), the necessary cond tion for Firm 1. Since in equilibrium z =π, we get: .02 1 1 =+ p zpπ All derivatives we need in this section are collected in Appendix B. Using (B.1) completes the proof. .E.D. Proof of Lemma 2: Since firms will charge symmetric prices: α = π/2, β = −π/2, and z = π. (a) In equilibrium line AB crosses the origin, therefore, any firm’s infinitesimal radial move towards origin, −dr, yields new customers in the area 2×dr/2, hence, raising demand by dr. (b) Using (1) and derivative (B.1) the partial derivative in question is: R zzzz p D ppp 2 1 )cos( 2 1 111 1 2 =−=⋅+−= ∂ ∂ . (c) Den te the total de ivative of any variable x with respect to r by x′ fo the length of this appendix. Th der vative in question is obtained from he system of first-order conditions (A.3) we used to derive equilibrium prices. Since this system must hold for any pair of locations, the total differentiation with respect to r yields 1 p′ . The total derivatives are (Note: sin z = 0 and cos z = −1 in equilibrium): 0 11 1 =′⋅+⋅′+′ pp zpzpz and 0 22 2 =′⋅+⋅′+′ pp zpzpz . The derivatives needed in above system are collected in (B.1)-(B.3). It becomes: 0424 21 =++′+′− πRpp , and 0442 21 =+−′−′ πRpp , ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012338 The derivatives needed in above system are collected in (B.1)-(B.3). It becomes: Appendix A: Proofs of Lemmas and Propositions Proof of Lemma 1: We first determine α and β that we need for the demand functions. They are given by the indifference conditions for the two consumers residing in points A and B: 22 2 22 1 )coscos()sinsin()cos(sin xrxrpxRxp −+−+=−++ φφ , where x = α, β. This equation simplifies to (A.1) and by using the fact that in symmetric equilibrium (if exists) r =R and φ = π, to (A.2): 22 12 cos)cos(2sinsin2 RrppxRrxr −+−=⋅−+⋅ φφ . (A.1) 12 cos4 ppxR −=⋅− (A.2) We state the necessary conditions for profit maximization with respect to own prices for Firm 1 and Firm 2 (from (2)): ( ) ( ) 0)cos 2 sin 2 1 11 1 =⋅−⋅+− pp zzz p zz and ( ) ( ) .0cos 2 sin2 2 1 22 2 =⋅+−⋅++− pp zzz p zzπ (A.3) Firms’ behavior in prices will be symmetric, hence, it is enough to use only first equation in (A.3), the necessary condition for Firm 1. Since in equilibrium z =π, we get: .02 1 1 =+ p zpπ All derivatives we need in this section are collected in Appendix B. Using (B.1) completes the proof. Q.E.D. Proof of Lemma 2: Since firms will charge symmetric prices: α = π/2, β = −π/2, and z = π. (a) In equilibrium line AB crosses the origin, therefore, any firm’s infinitesimal radial move towards origin, −dr, yields new customers in the area 2×dr/2, hence, raising demand by dr. (b) Using (1) and derivative (B.1) the partial derivative in question is: R zzzz p D ppp 2 1 )cos( 2 1 111 1 2 =−=⋅+−= ∂ ∂ . (c) Denote the total derivative of any variable x with respect to r by x′ for the length of this appendix. The derivative in question is obtained from the system of first-order conditions (A.3) we used to derive equilibrium prices. Since this system must hold for any pair of locations, the total differentiation with respect to r yields 1 p′ . The total derivatives are (Note: sin z = 0 and cos z = −1 in equilibrium): 0 11 1 =′⋅+⋅′+′ pp zpzpz and 0 22 2 =′⋅+⋅′+′ pp zpzpz . The derivatives needed in above system are collected in (B.1)-(B.3). It becomes: 0424 21 =++′+′− πRpp , and 0442 21 =+−′−′ πRpp , , and Appendix A: Proofs of Lemmas nd Prop sitions Proof Lemma 1: We first determine α and β that we n ed for the d mand functions. They are given by the indifference onditions for the two c nsumers residing in points A and B: 22 2 22 1 )coscos()sinsin()cos(sin xrxrpxRxp −+−+=−++ φφ , where x = α, β. This equation simplifies to (A.1) and by using the fact that in symmetric equilibr um (if exists) r =R and φ = π, to (A.2): 22 12 cos)cos(2sinsin2 RrppxRrxr −+−=⋅−+⋅ φφ . (A.1) 12 cos4 ppxR −=⋅− (A.2) We state the n cessary conditions for p ofit maxim zation with respect to own prices for Firm 1 and Firm 2 (from (2)): ( ) ( ) 0)cos 2 sin 2 1 11 1 =⋅−⋅+− pp zz p zz and ( ) ( ) .0cos 2 sin2 2 1 22 2 =⋅+−⋅++− pp zz p zzπ (A.3) Firms’ behavior in prices will be symmetric, hence, it is enough to use only first equation i (A.3), the n cessary condition for Firm 1. Since in equilibr um z =π, we g t: .02 1 1 =+ p zpπ All derivatives we n ed in this section are collect d in Appendix B. Using (B.1) completes the proof. Q.E.D. Proof Lemma 2: Since firms will charge symmetric prices: α = π/2, β = −π/2, and z = π. (a) In equilibr um line AB crosses the origin, therefore, any firm’s inf itesimal radi l move towards origin, −dr, yields new customers in the are 2×dr/2, hence, raising demand by dr. (b) Using (1) and derivative (B.1) the partial derivative in question is: R zzz p D ppp 2 1 )cos( 2 1 111 1 2 =−=⋅+−= ∂ ∂ . (c) Denote the total derivative of any vari ble x with respect to r by x′ for the l ngth of this appendix. The d rivative in question is obtained from the system of first-order conditions (A.3) we used to derive equilibr um prices. Since this system ust hold for any pair of locations, the total differentia on with respect to r yields 1 p′ . The total derivatives are (Note: sin z = 0 and cos z = −1 in equilibr um): 0 11 1 =′⋅+⋅′+′ pp zpzpz and 0 22 2 =′⋅+⋅′+′ pp zpzpz . The d rivatives need in above system are collect d in (B.1)-(B.3) It becomes: 0424 21 =++′+′− πRpp , and 0442 21 =+−′−′ πRpp , , with the unique solution t i e s l tion Rp 3 2 2 1 +=′ π and Rp 3 2 2 2 −=′ π . Q.E.D. Proof of Lemma 3: Use the results from Lemma 2 and put them in parentheses in RHS of (3) to get the sign of the effect a radial move has on Firm 2’s profits: R43 2 π +− >0, ]1.0(∈∀R . Q.E.D. Proof of Lemma 4: The demand effect of any move in a polar direction is zero, since the new line of indifferent consumers (line AB) must pass through the origin (firms will charge identical prices). The demands firms face are unaffected by such a move, but, any such move that brings both firms closer together makes them more competitive and lowers the price they charge. The result follows. Q.E.D. Proof of Lemma 5: First we characterize x, y, α, β, and γ, which define the firms’ demands. Coordinates of point D, x and y solve the system: 2 3 2 32 2 3 2 31 )cos()sin())cos(())sin(( yRxRpyRxRp −+−+=−+−+ −− ππππ , 2 3 2 32 22 3 )cos()sin()cos()sin( yRxRpyrxrp −+−+=−+−+ ππ φφ , We note that sinα = x and we proceed with β. It solves: 2 3 2 32 22 3 )coscos()sinsin()coscos()sinsin( βββφβφ ππ −+−+=−+−+ RRprrp . And, γ solves: 22 3 2 3 2 31 )coscos()sinsin()coscos()sinsin( γφγφγγ ππ −+−+=−+−+ −− rrpRRp . Last four equations simplify and lead to: 32 sin 12 R pp x − == α , (A.4) )cos2(32 )(sin2)222(3 12 22 213 RrR pprRrpppR y − −−−+−− = φ φ , (A.5) 22 23 cos)cos2(sin)3sin2( RrppRrRr −+−=−+− βφβφ , (A.6) 22 13 cos)cos2(sin)3sin2( RrppRrRr −+−=−++ γφγφ . (A.7) The system of first-order conditions for profit maximization derived from (8), which we mainly need for future reference, is: and ith the unique solution Rp 3 2 2 1 +=′ π Rp 3 2 2 2 −=′ π . Q.E.D. Proof of Lemma 3: Use the results from Lemma 2 and put them in parentheses in RHS of (3) to get the sign of the ef ect a radial move has on Firm 2’s profits: R43 2 π +− >0, ]1.0(∈∀R . Q.E.D. Proof of Lemma 4: The demand ef ect of any move in a polar direction is zero, since the new line of indif erent consumers (line AB) must pas through the origin (firms wil charge identical prices). The demands firms face are unaf ected by such a move, but, any such move that brings both firms closer together makes them more competitive and lowers the price they charge. The result fol ows. Q.E.D. Proof of Lemma 5: First we characterize x, y, α, β, and γ, which define the firms’ demands. Coordinates of point D, x and y solve the system: 2 3 2 32 2 3 2 31 )cos()sin())cos(())sin(( yRxRpyRxRp −+−+=−+−+ −− ππππ , 2 3 2 32 22 3 )cos()sin()cos()sin( yRxRpyrxrp −+−+=−+−+ ππ φφ , e note that sinα = x and we proce d with β. It solves: 2 3 2 32 22 3 )coscos()sinsin()coscos()sinsin( βββφβφ ππ −+−+=−+−+ RRprrp . And, γ solves: 22 3 2 3 2 31 )coscos()sinsin()coscos()sinsin( γφγφγγ ππ −+−+=−+−+ −− rrpRRp . Last four equations simplify and lead to: 32 sin 12 R pp x − == α , (A.4) )cos2(32 )(sin2)222(3 12 22 213 RrR pprRrpppR y − −−−+−− = φ φ , (A.5) 22 23 cos)cos2(sin)3sin2( RrppRrRr −+−=−+− βφβφ , (A.6) 22 13 cos)cos2(sin)3sin2( RrppRrRr −+−=−++ γφγφ . (A.7) The system of first-order conditions for profit maximization derived from (8), which we mainly ne d for future reference, is: . Q.E.D. proof of Lemma 3: Use the results from Lemma 2 and put them in parentheses in RhS of (3) to get the sign of the effect a radial move has on Firm 2’s profits: with the unique solution Rp 3 2 2 1 +=′ π and Rp 3 2 2 2 −=′ π . Q.E.D. Pro of Lemma 3: Us the re ults from Lemma 2 and put them in par ntheses in RHS of (3) to get the sign of the ffect a radial move has on Firm 2’s profits: R43 2 π +− >0, ]1.0(∈∀R . Q.E.D. Proof of Lemma 4: The demand effect of any move in a polar direction is zero, since the new line of indifferent consumers (line AB) must pass through the origin (firms will charge identical prices). The demands firms face are unaffected by such a move, but, any such move that brings both firms closer together makes them more competitive and lowers the price they charge. The result follows. Q.E.D. Proof of Lemma 5: First we characterize x, y, α, β, and γ, which define the firms’ demands. Coordinates of point D, x and y solve the system: 2 3 2 32 2 3 2 31 )cos()sin())cos(())sin(( yRxRpyRxRp −+−+=−+−+ −− ππππ , 2 3 2 32 22 3 )cos()sin()cos()sin( yRxRpyrxrp −+−+=−+−+ ππ φφ , We note that sinα = x and we proceed with β. It solves: 2 3 2 32 22 3 )coscos()sinsin()coscos()sinsin( βββφβφ ππ −+−+=−+−+ RRprrp . And, γ solves: 22 3 2 3 2 31 )coscos()sinsin()coscos()sinsin( γφγφγγ ππ −+−+=−+−+ −− rrpRRp . Last four equations simplify and lead to: 32 sin 12 R pp x − == α , (A.4) )cos2(32 )(sin2)222(3 12 22 213 RrR pprRrpppR y − −−−+−− = φ φ , (A.5) 22 23 cos)cos2(sin)3sin2( RrppRrRr −+−=−+− βφβφ , (A.6) 22 13 cos)cos2(sin)3sin2( RrppRrRr −+−=−++ γφγφ . (A.7) The system of first-order conditions for profit maximization derived from (8), which we mainly need for future reference, is: Q.E.D. proof of Lemma 4: The demand effect of any move in a polar direction is zero, since the new line of indifferent consumers (line AB) must pass through the origin (firms will charge identical prices). The demands firms face are unaffected by such a move, but, any such move that brings both firms closer together makes them more competitive and low- ers the price they charge. The result follows. Q.E.D. proof of Lemma 5: first we characte ize x, y, a, b, and γ, which define the firms’ d - ma ds. Coordinates of p int D, x nd y solve the system: with the unique solution Rp 3 2 2 1 +=′ π and Rp 3 2 2 2 −=′ π . Q.E.D. Proof of Lemma 3: Use the results from Lemma 2 and put them in parentheses in RHS of (3) to get the sign of the effect a radial move has on Firm 2’s profits: R43 2 π +− >0, ]1.0(∈∀R . Q.E.D. Proof of Lemma 4: The demand effect of any move in a polar direction is zero, since the new line of indifferent consumers (line AB) must pass through the origin (firms will charge identical prices). The demands firms face are unaffected by such a move, but, any such move that brings both firms closer together makes them more competitive and lowers the price they charge. The result follows. Q.E.D. Proof of Lemma 5: First we characteriz x, y, α, β, and γ, which define the firms’ demands. Coo dinates of point D, x and y solve the system: 2 3 2 32 2 3 2 31 )cos()sin())cos(())sin(( yRxRpyRxRp −+−+=−+−+ −− ππππ , 2 3 2 32 22 3 )cos()sin()cos()sin( yRxRpyrxrp −+−+=−+−+ ππ φφ , We note that sinα = x and we proceed with β. It solves: 2 32 22 3 )cossinsin()coscos(sinsin( βββφβφ π −+=−+−+ Rprrp . And, γ solves: 22 3 2 3 2 31 )coscos()sinsin()coscos()sinsin( γφγφγγ ππ −+−+=−+−+ −− rrpRRp . Last four equations simplify and lead to: 32 sin 12 R pp x − == α , (A.4) )cos2(32 )(sin2)222(3 12 22 213 RrR pprRrpppR y − −−−+−− = φ φ , (A.5) 22 23 cos)cos2(sin)3sin2( RrppRrRr −+−=−+− βφβφ , (A.6) 22 13 cos)cos2(sin)3sin2( RrppRrRr −+−=−++ γφγφ . (A.7) The system of first-order conditions for profit maximization derived from (8), which we mainly need for future reference, is: , with the unique solution Rp 3 2 2 1 +=′ π and Rp 3 2 2 2 −=′ π . Q.E.D. Proof of Lemma 3: Use the results from Lemma 2 and put them in parentheses in RHS of (3) to get the sign of the effect a radial move has on Firm 2’s profits: R43 2 π +− >0, ]1.0(∈∀R . Q.E.D. Proof of Le ma 4: The demand effect of any move in a polar direction is zero, since the new line of indifferent consumers (line AB) must pass through the origin (firms will charge identical prices). The demands firms face ar una fected by such a move, but, a y uch move that brings both firms closer together mak s the more competitive and lowers the price they charge. The result follows. Q.E.D. Proof of Lemma 5: First we characterize x, y, α, β, and γ, which define the firms’ demands. Coordinates of point D, x and y solve the system: 2 3 2 32 2 3 2 31 )cos()sin())cos(())sin(( yRxRpyRxRp −+−+=−+−+ −− ππππ , 2 3 2 32 22 3 )cos()sin()cos()sin( yRxRpyrxrp −+−+=−+−+ ππ φφ , We note that si α = we proceed with β. It solves: 2 3 2 32 22 3 )coscos()sinsin()coscos()sinsin( βββφβφ ππ −+−+=−+−+ RRprrp . And, γ olves: 22 3 2 3 2 31 )coscos()sinsin()c scos()sinsin( γφγφγγ ππ −+−+=−+−+ −− rrpRRp . Last four equations simplify and lead to: 32 sin 12 R pp x − == α , (A.4) )cos2(32 )(sin2)222(3 12 22 213 RrR pprRrpppR y − −−−+−− = φ φ , (A.5) 22 23 cos)cos2(sin)3sin2( RrppRrRr −+−=−+− βφβφ , (A.6) 22 13 cos)cos2(sin)3sin2( RrppRrRr −+−=−++ γφγφ . (A.7) The system of first-order conditi ns for profit maximization derived from (8), which we mainly need for future reference, is: , We note that sin a = x and we proceed with b. It solves: with the unique solution Rp 3 2 2 1 +=′ π and Rp 3 2 2 2 −=′ π . Q.E.D. Proof of Lemma 3: Use the results from Le ma 2 and put them in parentheses in RHS of (3) to get the sign of the effect a radial move has on Firm 2’s profits: R43 2 π +− >0, ]1.0(∈∀R . Q.E.D. Proof o L mma 4: The de and effect of ny m ve in a p lar direction is zero, since the new line indifferent consu ers (line AB) must pass through the origin (firms will charge identical prices). The demands fir s face are naffected by such a m ve, but, any suc move that bring both firm closer together makes them more competitive and lowers the price they charge. The result follows. Q.E.D. Pro of Lemma 5: First we characterize x, y, α, β, and γ, which define the firms’ demands. Coordinates of point D, x and y solve the system: 2 3 2 32 2 3 2 31 )cos()sin())cos(())sin(( yRxRpyRxRp −+−+=−+−+ −− ππππ , 2 3 2 32 22 3 )cos()sin()cos()sin( yRxRpyrxrp −+−+=−+−+ ππ φφ , We note that sinα = x and we proceed with β. It solves: 2 3 2 32 22 3 )coscos()sinsin()coscos()sinsin( βββφβφ ππ −+−+=−+−+ RRprrp . And, γ solves: 22 3 2 3 2 31 )coscos()sinsin()coscos()insin( γφγφγγ ππ −+−+=−+−+ −− rrpRRp . Last four equations sim lify and lead to: 32 sin 12 R pp x − == α , (A.4) )cos2(32 )(sin2)222(3 12 22 213 RrR pprRrpppR y − −−−+−− = φ φ , (A.5) 22 23 cos)cos2(sin)3sin2( RrppRrRr −+−=−+− βφβφ , (A.6) 22 13 c s)cos2(sin)3sin2( RrpprRr −+−=++ γγφ . (A.7) The system of first-order conditions for profit maximization derived from (8), which we mainly need for future refe ence, is: . And, γ solves: with the unique solutio Rp 3 2 2 1 +=′ π and Rp 3 2 2 2 −=′ π . Q.E.D. Proof of Lemma 3: Use the results fro Le a 2 and put the in parentheses in RHS of (3) to get the sign of the effect a radial ove as ir ’s r fits: R43 2 π +− >0, ]1.0(∈∀R . Q.E.D. P of of Lemma 4: The demand effe t lar direc ion is zero, since the new line of indifferent consum s (line ) t t t e origin (firms will c arge identical prices). The demands fir s face r ff t s ch a ove, but, any such move that brings both firms clo er together akes the ore co petitive and lowers the price they charge The result follows. Q.E.D. Proof of Lemma 5: First we characterize x, y, α, β, and γ, which define the firms’ demands. Coordinates of point D, x and y solve the system: 2 3 2 32 2 3 2 31 )cos()sin())cos(())sin(( yRxRpyRxRp −+−+=−+−+ −− ππππ , 2 3 2 32 22 3 )cos()sin()cos()sin( yRxRpyrxrp −+−+=−+−+ ππ φφ , We note that sinα = x and we proceed with β. It solves: 2 3 2 32 22 3 )coscos()sinsin()coscos()sinsin( βββφβφ ππ −+−+=−+−+ RRprrp . And, γ solves: 22 3 2 3 2 31 )coscos()sinsin()coscos()sinsin( γφγφγγ ππ −+−+=−+−+ −− rrpRRp . Last four equations simplify and lead to: 32 sin 12 R pp x − == α , (A.4) )cos2(3 )(sin2)222(3 12 22 213 RrR pprRrpppR y − −−−+−− = φ φ , (A.5) 22 23 cos)cos(sin)3sin2( RrpRrRr −+−=−+− βφβφ , (A.6) 22 13 cos)cos2(sin)3sin2( RrppRrRr −+−=−++ γφγφ . (A.7) The system of fi st-order onditions for profit maximization derived from (8 , which we mainly need for future reference, is: . Last four equations simplify and lead to: with the unique solution Rp 3 2 2 1 +=′ and Rp 3 2 2 2 −=′ π . Q.E.D. Proof of Lemma 3: Use the results from Lemma 2 and put the in parentheses in RHS of (3) to get the sign of the effect a radial move has on Firm 2’s profits: R43 2 π +− >0, ]1.0(∈∀R . Q.E.D. Proof of Lemma 4: The demand effect of any move in a p lar direction is zero, since the new line of indifferent consumers (line AB) must pass through the origin (fir s will charge identical prices). The demands firms face are unaffected by such a move, but, any such move that brings both firms closer toge her mak s them more competitive and lowers the price they charge. The result foll w . Q.E.D. Proof of Lemma 5: First we characterize x, y, α, β, and γ, which define the firms’ demands. Coordinates of oint D, x and y solve the system: 2 3 2 32 2 3 2 31 )cos()sin())cos(())sin(( yRxRpyRxRp −+−+=−+−+ −− ππππ , 2 3 2 32 22 3 )cos()sin()cos()sin( yRxRpyrxrp −+−+=−+−+ ππ φφ , We note that sinα = x and we proceed with β. It solves: 2 3 2 32 22 3 )cosco()sinsin()coscos()sinsin( βββφβφ ππ −+−+=−+−+ RRprrp . And, γ solves: 22 3 2 3 2 31 )coscos()sinsin()coscos()sinsin( γφγφγγ ππ −+−+=−+−+ −− rrpRRp . Last four equa ions simplify and lead to: 32 sin 12 R pp x − =α , (A.4) )cos2(32 )(sin2)222(3 12 22 213 RrR pprRrpppR y − −−−+−− = φ φ , (A.5) 22 23 cos)cos2(sin)3sin2( RrppRrRr −+−=−+− βφβφ , (A.6) 22 13 co)cos2(sin)3sin2( RrppRRr −+−=−++ γφγφ . (A.7) The system of first-order conditions for profit maximization derived from (8), which we mainly need for future reference, is: , (A.4) with the unique solution Rp 3 2 2 1 +=′ π and Rp 32 2 −=′ π . Q.E.D. Proof of Lemma 3: Use the results from Lemma 2 and put them in parentheses in HS of (3) to get the sign of the effect a radial move has on Fir ’s profits: R43 2 π +− >0, ]1.0(∈∀R . Q.E.D. Proof of Lem a 4: The demand effect of any ve in a polar directi n is zero, since the new line of indifferent consu ers (line AB) must pass through the origin (firms will charge identical prices). The demands firms face are unaffected by such a move, but, any such move that brings both firms closer together akes the more competitive and lowers the price they charge. The result follows. Q.E.D. Proof of Lemma 5: First we characterize x, y, α, β, and γ, which define the firms’ demands. Coordinates of point D, x and y solve the system: 2 3 2 32 2 3 2 31 )cos()sin())cos(())sin(( xRpyRxRp +−+=−+−+ −− ππππ , 2 3 2 32 22 3 )cos()sin()cos()sin( yRxRpyrxrp −+−+=−+−+ ππ φφ , We note that sinα = x and we proceed with β. It solves: 2 3 2 32 22 3 )coscos()sinsin()coscos()sinsin( βββφβφ ππ −+−+=−+−+ RRprrp . And, γ solves: 22 3 2 3 2 31 )coscos()sinsin()coscos()sinsin( γφγφγγ ππ −+−+=++ −− rrpRRp . Last four equations simplify and lead to: 32 sin 12 R pp x − == α , (A.4) )cos2(32 )(sin2222(3 12 22 213 RrR pprRrpppR y − −−−+−− = φ φ , ( .5) 22 23 cos)cos2(sin)3sin2( RrppRrRr −+−=−+− βφβφ , (A.6) 22 13 cos)co2(sin)3sin2( RrppRrRr −+−=−++ γφγφ . (A.7) The system of first-order conditions for profit maximization derived from (8), which we mainly need for future reference, is: , (A.5) with the unique solution Rp 32 1 +=′ π and Rp 32 2 −=′ π . Q.E.D. Proof of Lemma 3: Us the esults fr m Lemma 2 and put them in parentheses in RHS of (3) to get the sign of the effect a radial move has on Firm 2’s profits: R43 2 π +− >0, ]1.0(∈∀R . Q.E.D. Proof of L m a 4: The demand effect of any move in a polar direction is zero, since the new line of indifferent onsumers (line AB) must pass through the origin (firms will charge identical prices). The demands firms face are unaffected by such a move, but, any such move that brings both firms closer together makes them more competitive and lowers the price they charge. The result follows. Q.E.D. Proof of Lemma 5: First we charact rize x, y, α, β, and γ, which define the firms’ demands. Coordinates of point D, x and y solve the system: 2 3 2 32 2 3 2 31 )cos()sin())cos(())sin(( yRxRpyRxRp −+−+=−+−+ −− ππππ , 2 3 2 32 22 3 )cos()sin()cos()sin( yRxRpyrxrp −+−+=−+−+ ππ φφ , We note that sinα = x and we proceed with β. It solves: 3 2 32 22 3 coscos()sinsin()coscos()sini βββφβφ ππ −+−+=−+− RRprr And, γ solves: 22 3 2 3 2 31 )coscos()sinsin()coscos()sinsin( γφγφγγ ππ −+−+=−+−+ −− rrpRRp . Last four equations simplify and lead to: 32 sin 12 R pp x − == α , (A.4) )cos2(32 )(sin2)222(3 12 22 213 RrR pprRrpppR y − −−−+−− = φ φ , (A.5) 22 23 cos)cos2(sin)3sin2( RrppRrRr −+−=−+− βφβφ , (A.6) 22 13 cos)cos2(sin)3sin2( RrppRrRr −+−=−++ γφγφ . (A.7) The system of first-order conditions for profit maximization derived from (8), which we mainly need for future reference, is: , (A.6) with the unique solution Rp 3 2 2 1 +=′ π and Rp 3 2 2 2 −=′ π . Q.E.D. Proof of Lemma 3: Use the results from Lemma 2 and put them in parentheses in HS of (3) to get the sign of the effect a radial move has on Firm 2’s profits: R43 2 π +− >0, ]1.0(∈∀R . Q.E.D. Proof of Lemma 4: The demand effect of any move in a polar direction is zero, since the new line of indifferent consumers (line AB) m st pass through t e origin (firms will c arge identical prices). The demands firms face are unaffected by such a move, but, any such move that brings both firms clo er together m k s them more competitive and low rs the price they c rge The result follow . Q.E.D Pr of of Lemma 5: First we characterize x, y, α, β, and γ, which define the firms’ demands. Coordinates of point D, x and solve the syst m: 2 3 2 3 2 3 2 31 )cos()sin())cos(())sin(( xRpyRxRp +−+=−+−+ −− ππππ , 2 3 2 32 22 3 )cos()sin()cos()sin( yRxRpyrxrp −+−+=−+−+ ππ φφ , We note that sinα = x and we proceed with β. It solves: 2 3 2 32 22 3 )coscos()sinsin()coscos()sinsin( βββφβφ ππ −+−+=−+−+ RRprrp . And, γ solves: 22 3 2 3 2 31 )coscos()sinsin()coscos()sinsin( γφγφγγ ππ −+−+=−+−+ −− rrpRRp . Last four equations simplify and lead to: 32 sin 12 R pp x − == α , (A.4) )cos2(32 )(sin2)222(3 12 22 213 RrR pprRrpppR y − −−−+−− = φ φ , (A.5) 2 23 cos)cos2(sin)3sin2( RrppRrRr −+−=−+− βφβφ , (A.6) 22 13 cos)cos2(sin) RrppRr −+−=−++ γφγ . (A.7) The system of fi st-order onditions for profit maximization derived from (8), which we mainly need for future reference, is: . (A.7) A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 339 The system of first-order conditions for profit maximization derived from (8), which we mainly need for future reference, is: ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 11111 111 1 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx γγααγαγγαα γαγαγαγαγα , (A.8) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 22222 222 2 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx ααββαβααββ αβαβαβαβαβ , (A.9) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos2 33333 333 3 =⋅−⋅+−+⋅−⋅+ −−+−+−+−−+− ppppp ppp yyx xpyx γγββγβγγββ γβγβγβγβγβπ . (A.10) We use the assumed symmetry of equilibrium configuration to find the optimal prices. We first note that in equilibrium, x = 0, y = 0, α = 0, β = 2π/3, and γ = −2π/3, which immediately simplifies (A.10): ( ) 03 3 2 333 3 =++−+ ppp yp γβ π . Next, symmetry yields 33 pp βγ −= . Remaining two partial derivatives are in (B.8) and (B.14). We get: 03 3 1 32 2 3 2 3 =   −     −+ RR p π , which gives the result. Q.E.D. Proof of Lemma 6: (a) Partially differentiate (7) with respect to r to get: 9 ( ))sin(sin)cos(cos 2 1 γβγβγβ −+−−+−= rrrrr yxD . Again, symmetry, and partials (B.4), (B.9), and (B.15) give: 32 14 32 4 32 24 2 1 3 3 2 32 12 2 2 1 R R R R R R R R D r − −=         − − −=         − − −= . (b) We proceed similarly as in part (a), to get: ( ))sin(sin)cos(cos 2 1 22222 γβγβγβ −+−−+−= ppppp yxD . From (A.7) we note that 0 2 = p γ (the indifference line CD is unaffected by Firm 2’s price change). With partials (B.4), (B.7) and (B.14) we obtain: 32 1 3 6 1 32 1 2 1 2 RRR D p =         += . , (A.8) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 11111 111 1 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx γγααγαγγαα γαγαγαγαγα , (A.8) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 22222 222 2 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx ααββαβααββ αβαβαβαβαβ , (A.9) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos2 33333 333 3 =⋅−⋅+−+⋅−⋅+ −−+−+−+−−+− ppppp ppp yyx xpyx γγββγβγγββ γβγβγβγβγβπ . (A.10) We use the assumed symmetry of equilibrium configuration to find the optimal prices. We first note that in equilibrium, x = 0, y = 0, α = 0, β = 2π/3, and γ = −2π/3, which immediately simplifies (A.10): ( ) 03 3 2 333 3 =++−+ ppp yp γβ π . Next, symmetry yields 33 pp βγ −= . Remaining two partial derivatives are in (B.8) and (B.14). We get: 03 3 1 32 2 3 2 3 =   −     −+ RR p π , which gives the result. Q.E.D. Proof of Lemma 6: (a) Partially differentiate (7) with respect to r to get: 9 ( ))sin(sin)cos(cos 2 1 γβγβγβ −+−−+−= rrrrr yxD . Again, symmetry, and partials (B.4), (B.9), and (B.15) give: 32 14 32 4 32 24 2 1 3 3 2 3 12 2 2 1 R R RRR D r − −=         − − −=         − − −= . (b) We proceed similarly as in part (a), to get: ( ))sin(sin)cos(cos 2 1 22222 γβγβγβ −+−−+−= ppppp yxD . From (A.7) we note that 0 2 = p γ (the indifference line CD is unaffected by Firm 2’s price change). With partials (B.4), (B.7) and (B.14) we obtain: 32 1 3 6 1 32 1 2 1 2 RRR D p =         += . , (A.9) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 11111 111 1 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx γγααγαγγαα γαγαγαγαγα , (A.8) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 22222 222 2 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx ααββαβααββ αβαβαβαβαβ , (A.9) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos2 33333 333 3 =⋅−⋅+−+⋅−⋅+ −−+−+−+−−+− ppppp ppp yyx xpyx γγββγβγγββ γβγβγβγβγβπ . (A.10) We use the assumed symmetry of equilibrium configuration to find the optimal prices. We first note that in equilibrium, x = 0, y = 0, α = 0, β = 2π/3, and γ = −2π/3, which immediately simplifies (A.10): ( ) 03 3 2 333 3 =++−+ ppp yp γβ π . Next, symmetry yields 33 pp βγ −= . Remaining two partial derivatives are in (B.8) and (B.14). We get: 03 3 1 32 2 3 2 3 =   −     −+ RR p π , which gives the result. Q.E.D. Proof of Lemma 6: (a) Partially differentiate (7) with respect to r to get: 9 ( ))sin(sin)cos(cos 2 1 γβγβγβ −+−−+−= rrrrr yxD . Again, symmetry, and partials (B.4), (B.9), and (B.15) give: 32 14 32 4 32 24 2 1 3 3 2 32 12 2 2 1 R R R R R R R R D r − −=         − − −=         − − −= . (b) We proceed similarly as in part (a), to get: ( ))sin(sin)cos(cos 2 1 22222 γβγβγβ −+−−+−= ppppp yxD . From (A.7) we note that 0 2 = p γ (the indifference line CD is unaffected by Firm 2’s price change). With partials (B.4), (B.7) and (B.14) we obtain: 32 1 3 6 1 32 1 2 1 2 RRR D p =         += . . (A.10) We use the assumed symmetry of equilibrium configuration to find the optimal prices. We first note that in equilibrium, x = 0, y = 0, a = 0, b = 2π/3, and γ = −2π/3, which im- mediat ly simplifies (A.10): ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)c s(cos 11111 111 1 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx γγααγαγγαα γαγαγαγαγα , (A.8) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 22222 222 2 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx ααββαβααββ αβαβαβαβαβ , (A.9) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos2 33333 333 3 =⋅−⋅+−+⋅−⋅+ −−+−+−+−−+− ppppp ppp yyx xpyx γγββγβγγββ γβγβγβγβγβπ . (A.10) We us the assumed symmetry of equilibrium configuration to find the optimal prices. We first note that in equilibrium, x = 0, y = 0, α = 0, β = 2π/3, and γ = −2π/3, which i ediately simpl fies (A.10): ( ) 03 3 2 333 3 =++−+ ppp yp γβ π . Next, symmetry yields 33 pp βγ −= . Remaining two partial derivatives are in (B.8) and (B.14). We get: 03 3 1 32 2 3 2 3 =   −     −+ RR p π , which gives the result. Q.E.D. Proof of Lemma 6: (a) Partially differentiate (7) with respect to r to get: 9 ( ))sin(sin)cos(cos 2 1 γβγβγβ −+−−+−= rrrrr yxD . Again, symmetry, and parti ls (B.4), (B.9), and (B.15) give: 32 14 32 4 32 24 2 12 32 12 2 2 1 R R R R R R R R D r − −=         − − −=         − − −= . (b) We proceed similarly as in part (a), to get: ( ))sin(sin)cos(cos 2 1 22222 γβγβγβ −+−−+−= ppppp yxD . From (A.7) we note that 0 2 = p γ (the indifference line CD is unaffected by Firm 2’s price change). With partials (B.4), (B.7) and (B.14) we obtain: 32 1 3 6 1 32 1 2 1 2 RRR D p =         += . . ext, sy metry yields ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 11111 111 1 =⋅−⋅−−⋅−⋅− −+−+−−−+− ppppp ppp yx xpyx γγααγαγγαα γαγαγαγαγα , (A.8) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 22222 222 2 =⋅−⋅−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx ααββαβααββ αβαβαβαβαβ , (A.9) ( 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos2 33333 333 3 =⋅−⋅+−+⋅−⋅+ −−+−+−+−−+− ppppp ppp yyx xpyx γγββγβγβ γβγβγβγβγβπ . We use the assumed sym etry of equilibrium configuration to find the opti al ri first note tha in equilibr um, x = 0, y = 0, α = 0, β = 2π/3, and γ = −2π/3, which i i simplifies (A.10): ( ) 03 3 2 333 3 =++−+ ppp yp γβ π . Next, sym t 33 pp βγ −= . Remaining two partial derivatives are in (B.8) and (B.14). We get: 03 3 1 32 2 3 2 3 =   −     −+ RR p π , which gives the result. Q.E.D. Pr of of Lemma 6: (a) Partially differ ntiate (7) with respect to r to get: 9 ( ))sin(sin)cos(cos 2 1 γβγβγβ −+−−+−= rrrrr yxD . Again, symmetry, and partials (B.4), (B.9), and (B.15) give: 32 14 32 4 32 24 2 1 3 3 2 32 12 2 2 1 R R R R R R R R D r − −=         − − −=         − −= . (b) We proc ed similarly as in part (a), to get: ( ))sin(sin)cos(cos 2 1 22222 γβγβγβ −+−−+−= ppppp yxD . From (A.7) we note that 0 2 = p γ (the indifference line CD is unaffected by Firm 2’s price change). With partials (B.4), (B.7) and (B.14) we obtain: 32 1 3 6 1 3 1 2 1 2 RRR D p =         += . . t partial derivatives are in (B.8) and ( . ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 11111 111 1 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx γγααγαγγαα γαγαγαγαγα , (A.8) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 22222 222 2 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx ααββαβααββ αβαβαβαβαβ , (A.9) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos2 33333 333 3 =⋅−⋅+−+⋅−⋅+ −−+−+−+−−+− ppppp ppp yyx xpyx γγββγβγγββ γβγβγβγβγβπ . (A.10) We use the assumed symmetry of equilibrium configuration to find the optimal prices. We first note that in equilibrium, x = 0, y = 0, α = 0, β = 2π/3, and γ = −2π/3, which immediately simplifies (A.10): ( ) 03 3 2 333 3 =++−+ ppp yp γβ π . t, s etry ields 33 pp βγ −= . Remaini g two parti l derivat es are in (B.8) and . . et: 03 3 1 32 2 3 =   −     RR , which gives the result. Q.E.D. r f f e a 6: (a) Partially differentiate (7) with respect to r to get: 9 ( ))sin(sin)cos(cos 2 1 γβγβγβ −+−−+−= rrrrr yx . Again, symmetry, and partials (B.4), (B.9), and (B.15) give: 32 14 32 4 32 24 2 1 3 3 2 32 12 2 2 1 R R R R R R R R D r − −=         − − −=         − − −= . (b) We proceed similarly as in part (a), to get: ( ))sin(sin)cos(cos 2 1 22222 γβγβγβ −+−−+−= ppppp yxD . From (A.7) we note that 0 2 = p γ (the indifference line CD is unaffected by Firm 2’s price change). With partials (B.4), (B.7) and (B.14) we obtain: 32 1 3 6 1 32 1 2 1 2 RRR D p =         += . , i i t lt. .E. . proof of Lemma 6: (a) partially differentiate (7) with respect to r to get:10 ( ) 0)cos(cos)sin(sin)sin )cos(cos)sin(sin)coss 11111 111 1 =⋅−⋅−−−⋅−⋅ −+−+−−− ppppp ppp yy xpy γγαγαγγ γαγαγαγ , (A.8) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos( s 22222 222 2 =⋅−⋅−−−⋅−⋅− −+−+−−− ppppp ppp yyx xpy ααββαβααββ αβαβαβα , (A.9) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos2 33333 333 3 =⋅−⋅+−+⋅−⋅+ −−++−+−−+− ppppp ppp yyx xpyx γγββγβγγββ γβγβγβγβγβπ . (A.10) e use the as umed sym etry of equilibr um configuration to find the optimal prices. We first note tha in equilibrium, x = 0, y = 0, α = 0, β = 2π/3, and γ = −2π/3, which immediately simplifies (A.10): ( ) 03 3 2 333 3 =++−+ ppp yp γβ π . Next, symmetry yields 33 pp βγ −= . Remaining two partial derivatives are in (B.8) and (B.14). We get: 03 3 1 32 2 3 2 3 =   −     −+ RR p π , which gives the result. Q.E.D. Proof of Lemma 6: (a) Partially differentiate (7) with respect to r to get: 9 ( ))sin(sin)cos(cos 2 1 γβγβγβ −+−−+−= rrrrr yxD . Again, symmetry, and partials (B.4), (B.9), and (B.15) give: 32 14 32 4 32 24 2 1 3 3 2 32 12 2 2 1 R R R R R R R R D r − −=         − − −=         − − −= . (b) We proceed similarly as in part (a), to get: ( ))sin(sin)cos(cos 2 1 22222 γβγβγβ −+−−+−= ppppp yxD . From (A.7) we note that 0 2 = p γ (the indifference line CD is unaffected by Firm 2’s price change). With partials (B.4), (B.7) and B.14) we obtain: 32 1 3 6 1 32 1 2 1 2 RRR D p =         += . . gain, symmetry, and partials (B.4), (B.9), and (B.15) give: y e g −p P oo o L m D ( ) i ilarly as i part (a), to get: ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 11111 111 1 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx γγααγαγγαα γαγαγαγαγα , (A.8) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 22222 222 2 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx ααββαβααββ αβαβαβαβαβ , (A.9) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos2 33333 333 3 =⋅−⋅+−+⋅−⋅+ −−+−+−+−−+− ppppp ppp yyx xpyx γγββγβγγββ γβγβγβγβγβπ . (A.10) use the assumed symmetry of equilibrium configuration to find the optimal prices. We first note that in equilibrium, x = 0, y = 0, α = 0, β = 2π/3, and γ = −2π/3, which immediately simplifies (A.10): ( ) 03 3 2 333 3 =++−+ ppp yp γβ π . Next, symmetry yields 33 pp βγ −= . Remaining two partial derivatives are in (B.8) and (B.14). We get: 03 1 32 2 3 2 3 =   −     −+ RR p π , which gives the result. Q.E.D. Proof of Lemma 6: (a) Partially differentiate (7) with respect to r to get: 9 ( ))sin(sin)cos(cos 2 1 γβγβγβ −+−−+−= rrrrr yxD . Again, sy etry, and partials (B.4), (B.9), and (B.15) give: 32 14 32 4 32 24 2 1 3 2 32 11 RR R R r − −=         − − −=        − − . r eed si ilarly as in part (a), to get: ( ))sin(sin)cos(cos 2222 γβγβγ −+−−+ pppp yx . r ( . ) e note that 0 2 = p γ (the indiff rence line CD is unaffected by Firm 2’s price change). ith partials (B.4), (B.7) and (B.14) we obtain: 32 1 3 6 1 32 1 2 1 2 RRR D p =        += . f ( .7) e te t at ( ))s( s)si(si)si(si )s( s)si(si)s( s 11111 111 1 ⋅⋅⋅⋅ ppppp ppp γγγγγ γγγγγ , ( . ) ( ))s( s)si(si)si( i )s( s)si(si)s( s 22222 222 2 ⋅⋅⋅⋅ ppppp ppp , ( . ) ( ))s( s)si(si)si(si )s( s)si(si)s( s 33333 333 3 ⋅⋅⋅⋅ ppppp ppp γγγγγ γγγγγ . ( . ) se t ss s tr f ili ri fi r ti t fi t ti l ri s. first t t t i ili ri , , , , / , γ / , i im i t l si lifi s ( . ): ( ) 333 3 ppp γ . t, s tr i l s 33 pp γ . i i t rti l ri ti s r i ( . ) ( . ) t: 3        , i i s t r s lt. . . . r f f : ( ) rti ll iff r ti t ( ) it r s t t r t t: 9 ( ))si(si)s( s γγγ rrrrr . i , s tr , rtials ( . ), ( . ), ( . ) i : 3 3 r                 . r s l l rt ( ), t : ( ))si(si)s( s 22222 γγγ ppppp . 2 p γ (t i iff r li is ff t ir ’s ri ). it rti ls ( . ), ( . ) ( . ) t i : 2 p         . t e ff e ce e aff c e y ir 2’s price c nge). ith partials (B.4), (B.7) and (B.14) we obtain: ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 11111 111 1 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp pp yyx xpyx γγααγαγγαα γαγαγαγαγα , (A.8) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos 22222 222 2 =⋅−⋅−−−⋅−⋅− −+−+−−−+− ppppp ppp yyx xpyx ααββαβααββ αβαβαβαβαβ , (A.9) ( ) 0)cos(cos)sin(sin)sin(sin )cos(cos)sin(sin)cos(cos2 33333 333 3 =⋅−⋅+−+⋅−⋅+ −−+−+−+−−+− ppppp ppp yyx xpyx γγββγβγγββ γβγβγβγβγβπ . (A.10) e use the assu ed sy etry of equilibriu configuration to find the opti al prices. e f rst not that in equilibriu , x = 0, y = 0, α = 0, β = 2π/3, and γ = −2π/3, which i ediately si plifies (A.10): ( ) 03 3 2 333 3 =++−+ ppp yp γβ π . Next, sym etry yields 33 pp βγ −= . Re aining two partial derivatives are in (B.8) and (B.14). e get: 03 3 1 32 2 3 2 3 =   −   −+ RR p π , which gives the result. Q.E.D. f e a 6: (a) Partially differentia e (7) with resp ct to r to get: 9 ( ))sin(sin)cos(cos γβγβγβ −+−−+ rrrr yx . i , s etry, and partials (B.4), (B.9), and (B.15) give: 32 14 32 4 32 24 2 1 3 3 2 32 12 2 2 R R R R R R R R r − −=     − − −=     − − . (b) e proceed si ilarly as in part (a), to get: ( ))sin(sin)cos(cos 2 1 22222 γβγβγβ −+−−+−= ppppp yxD . Fro (A.7) we note that 0 2 = p γ (the indifference line CD is unaffected by Firm 2’s price change). ith partials (B.4), 7) and (B.14) we obtain: 32 1 3 6 1 32 1 2 1 2 RRR D p =     += . . 10 We denote partial derivatives with the variable with respect to which we differentiate in subscript and omit the index of the firm, since we always work with Firm 3. ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012340 (c) The reaction in first two firms’ prices to a radial move by Firm 3 will be identical; hence, we can disregard Firm 1’s behavior. Totally differentiate (A.9) and (A.10) with respect to r to get: (c) The reaction in first two fir s’ prices to a radial move by Firm 3 will be identical; hence, we can disregard Firm 1’s behavior. Totally differentiate (A.9) and (A.10) with respect to r to get: ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)cos(cos 2222 2222 222222 22 =⋅−⋅′−′⋅−′⋅−−′− ⋅−⋅′−′⋅−′⋅−−′+ ′−′+−−−+−′+ −′−−′+′−′ pppp pppp pppppp yyy xxx pyxp yx ααββααββαβ ααββααββαβ αβαβαβαβ αβαβαβ , ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)cos(cos 3333 3333 333333 33 =⋅−⋅′+′⋅−′⋅+−′+ ⋅−⋅′+′⋅−′⋅+−′− ′+′−+−+−−+−′+ −′+−′−′+′− pppp pppp pppppp yyy xxx pyxp yx γγββγγββγβ γγββγγββγβ γβγβγβγβ γβγβγβ . Next, we assume the first equality from part (c) of present Lemma and observe from (B.5) and (A.4) that 0=′x and 0=′α . We also use the second part of (B.5), which says that the total derivative with respect to r of partial derivative of x, and, hence by (A.5) also of α, with respect to any of the three prices is 0. Note again that α = 0 in equilibrium, use (B.4) and symmetry of the problem to simplify the above system: ( ) ( )( ) 02332 332232 222222 2222 2 =−−′−′+′−′−′+ −−−′+′−′ pppppp pppp yyyxp yxpy αββββ αββ ( ) ( ) 0323232 333333 3 =′−′−′+′−++−′+′+′− pppppp yyypypy βββββ . We next note that by (A.5) 222 cos ppp xx == αα in equilibrium. The remaining derivatives are collected in (B.7), (B.8), (B.11), (B.12), (B.13), (B.14), (B.17), and (B.18). Applying all these we get the final form of the system: −54 ′p 2 +18 ′p 3 + 36R − 9 +π 3 ′p 2 − ′p 3 − 2R + 3( ) = 0 , 18 ′p 2 − 36 ′p 3 − 36R + 9 +π 3 − ′p 2 + ′p 3 + 2R + 3( ) = 0, with a solution: ′p 2 = −9 + 9π 3 −π 2 + 36 − 2π 3( )R 90 − 3π 3 , and ′p 3 = 18+12π 3 −π 2 − 72 − 4π 3( )R 90 − 3π 3 . Q.E.D. Proof of Lemma 7: Collect results from Lemma 6 to calculate the expression in parentheses of (9) and equate it with zero: , (c) The reaction in first two firms’ prices to a radial move by Firm 3 will be identical; hence, we can disregard Firm 1’s behavior. Totally differentiate (A.9) and (A.10) with respect to r to get: ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)cos(cos 2222 2222 222222 22 =⋅−⋅′−′⋅−′⋅−−′− ⋅−⋅′−′⋅−′⋅−−′+ ′−′+−−−+−′+ −′−−′+′−′ pppp pppp pppppp yyy xxx pyxp yx ααββααββαβ ααββααββαβ αβαβαβαβ αβαβαβ , ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)cos(cos 3333 3333 333333 33 =⋅−⋅′+′⋅−′⋅+−′+ ⋅−⋅′+′⋅−′⋅+−′− ′+′−+−+−−+−′+ −′+−′−′+′− pppp pppp pppppp yyy xxx pyxp yx γγββγγββγβ γγββγγββγβ γβγβγβγβ γβγβγβ . Next, we assume the first equality from part (c) of present Lemma and observe from (B.5) and (A.4) that 0=′x and 0=′α . We also use the second part of (B.5), which says that the total derivative with respect to r of partial derivative of x, and, hence by (A.5) also of α, with respect to any of the three prices is 0. Note again that α = 0 in equilibrium, use (B.4) and symmetry of the problem to simplify the above system: ( ) ( )( ) 02332 332232 222222 2222 2 =−−′−′+′−′−′+ −−−′+′−′ pppppp pppp yyyxp yxpy αββββ αββ ( ) ( ) 0323232 333333 3 =′−′−′+′−++−′+′+′− pppppp yyypypy βββββ . We next note that by (A.5) 222 cos ppp xx == αα in equilibrium. The remaining derivatives are collected in (B.7), (B.8), (B.11), (B.12), (B.13), (B.14), (B.17), and (B.18). Applying all these we get the final form of the system: −54 ′p 2 +18 ′p 3 + 36R − 9 +π 3 ′p 2 − ′p 3 − 2R + 3( ) = 0 , 18 ′p 2 − 36 ′p 3 − 36R + 9 +π 3 − ′p 2 + ′p 3 + 2R + 3( ) = 0, with a solution: ′p 2 = −9 + 9π 3 −π 2 + 36 − 2π 3( )R 90 − 3π 3 , and ′p 3 = 18+12π 3 −π 2 − 72 − 4π 3( )R 90 − 3π 3 . Q.E.D. Proof of Lemma 7: Collect results from Lemma 6 to calculate the expression in parentheses of (9) and equate it with zero: Next, we assume the first equality from part (c) of present Lemma and observe from (B.5) and (A.4) that ( ) The reaction in first two firms’ prices to a radial move by Firm 3 will be identical; hence, we can disregard Firm 1’s behavior. Totally differentiate (A.9) and (A.10) with respect to r to get: ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)cos(cos 2222 2222 222222 22 =⋅−⋅′−′⋅−′⋅−−′− ⋅−⋅′−′⋅−′⋅−−′+ ′−′+−−−+−′+ −′−−′+′−′ pppp pppp pppppp yyy xxx pyxp yx ααββααββαβ ααββααββαβ αβαβααβ αβαβα , ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)cos(cos 3333 3333 333333 33 =⋅−⋅′+′⋅−′⋅+−′+ ⋅−⋅′+′⋅−′⋅+−′− ′+′−+−+−−+−′+ −′+−′−′+′ pppp pppp pppppp yyy xxx pyxp yx γγββγγββγβ γγββγγββγβ γβγβγβγβ γβγβγ . Next, we assume the first equality from part (c) of present Lemma and observe from (B.5) a 0=′x and 0=′α . We also use the second part of (B.5), which says that the total derivative with respect to r of partial derivative of x, and, hence by (A.5) also of α, with respect to any of the three prices is 0. Note again that α = 0 in equilibrium, use (B.4) and symmetry of the problem to simplify the above system: ( ) ( )( ) 02332 332232 222222 2222 2 =−−′−′+′−′−′+ −−−′+′−′ pppppp pppp yyyxp yxpy αββββ αββ ( ) ( ) 0323232 333333 3 =′−′−′+′−++−′+′+′− pppppp yyypypy βββββ . We ext note that by (A.5) 222 cos ppp xx == αα in equilibrium. The remaining derivatives are collected in (B.7), (B.8), (B.11), (B.12), (B.13), (B.14), (B.17), and (B.18). Applying all these we get the final form of the system: −54 ′p 2 +18 ′p 3 + 36R − 9 +π 3 ′p 2 − ′p 3 − 2R + 3( ) = 0 , 18 ′p 2 − 36 ′p 3 − 36R + 9 +π 3 − ′p 2 + ′p 3 + 2R + 3( ) = 0, with a solution: ′p 2 = −9 + 9π 3 −π 2 + 36 − 2π 3( )R 90 − 3π 3 , and ′p 3 = 18+12π 3 −π 2 − 72 − 4π 3( )R 90 − 3π 3 . Q.E.D. Proof of Lemma 7: Collect results from Lemma 6 to calculate the expression in parentheses of (9) and equate it with zero: a ( ) r ti i fir t t fir ’ ri t r i l i ill i ti l; , i r r i ’ i r. t ll iff r ti t ( . ) ( . ) it r t t t t: ( ) ( ))()()i( i )i( i)i( i)( )i( i)( )i( i)( 2222 2222 222222 22 ⋅⋅′′⋅′⋅′ ⋅⋅′′⋅′⋅′ ′′′ ′′′′ pppp pppp pppppp β , ( ) ( ))()()i( i )i( i)i( i)( )i( i)( )i( i)( 3333 3333 333333 33 ⋅⋅′′⋅′⋅′ ⋅⋅′′⋅′⋅′ ′′′ ′′′′− pppp pppp pppppp β . t, t fir t lit fr rt ( ) f r t r fr ( . ) at ′ ′ . l t rt f ( . ), i t t t t t l ri ti it r t t f rti l ri ti f , , ( . ) l f , it r t t f t t r ri i . t i t t i ili ri , ( . ) tr f t r l t i lif t t : ( ) ( )( ) 222222 2222 2 ′′′′′ ′′′ pppppp pppp ( ) ( ) 333333 3 ′′′′′′′ pppppp . n t t t t ( . ) 222 ppp i ili ri r i i ri ti r ll t i ( . ), ( . ), ( . ), ( . ), ( . ), ( . ), ( . ), ( . ). l i ll t t t fi l f r f t t : ′ 2 ′ 3 ′ 2 ′ 3 ( ) , ′ 2 ′ 3 ′ 2 ′ 3 ( ) , it l ti : ′ 2 2 ( ) , ′ 3 2 ( ) . . . . r f f : ll t r lt fr t l l t t r i i r t f ( ) t it it r : . e also use t of ( . ), ic says tha the total derivat ve with respect to of pa ti l d rivative of x, and, hence by (A.5) also of a, with respect o any of the three prices s 0. Note again that a = 0 in equilibrium, use (B.4) and symmetry of the problem to simplify the above system: (c) The reaction in first two firms’ prices to a rad l move by Firm 3 will be identical; hence, w can disregard Firm 1’s behavior. Totally differentiate (A.9) and (A.10) with respect to r to get: ( ) ( ) 0)cos(cos)co(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)cos(cos 2222 2222 222222 22 =⋅−⋅′−′⋅−′⋅−′ ⋅−⋅′−′⋅−′⋅−−′+ ′−′+−−−+−′+ −′−−′+′−′ pppp pppp pppppp yyy xxx pyxp yx ααββααββαβ ααββααββαβ αβαβαβαβ αβαβαβ , ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)cos(cos 3333 3333 333333 33 =⋅−⋅′+′⋅−′⋅+−′+ ⋅−⋅′+′⋅−′⋅+−′− ′+′−+−+−−+−′+ −′+−′−′+′− pppp pppp pppppp yyy xxx yxp yx γγββγγββγβ γγββγγββγβ γβγβγβγβ γβγβγβ . Next, we assume the first equality from part (c) of presen Lemma and observe from (B.5) and (A.4) tha 0=′x and 0=′α . We also use the second part of (B.5), which says that th total derivative wit r spect to r of par ial derivative of x, and, hence by (A.5) also of α, with respect o any of the three r ces is 0. Note again that α = 0 in equilibrium, use (B.4) and symmetry of the problem to simplify the above system: ( ) ( )( ) 02332 332232 222222 2222 2 =−−′−′+′−′−′+ −−−′+′−′ pppppp ppp yyyxp yxpy αββββ αβ ( ) ( ) 02232 333333 3 =′−′−′+′−++−′+′+′− pppppp yyypypy βββββ . We next note that by (A.5) 222 cos ppp xx == αα in equilibrium. The remaining derivatives ar coll cted in (B.7), (B.8), (B.11), (B.12), (B.13), (B.14), (B.17), and (B.18). Applying all these we get the final form of the system: −54 ′p 2 +18 ′p 3 + 36R − 9 +π 3 ′p 2 − ′p 3 − 2R + 3( ) = 0 , 18 ′p 2 − 36 ′p 3 − 36R + 9 +π 3 − ′p 2 + ′p 3 + 2R + 3( ) = 0, with a solution: ′p 2 = −9 + 9π 3 −π 2 + 36 − 2π 3( )R 90 − 3π 3 , and ′p 3 = 18+12π 3 −π 2 − 72 − 4π 3( )R 90 − 3π 3 . Q.E.D. Proof of Lemma 7: Collect results from Lemma 6 to calculate the expression in parentheses of (9) and equate it with zero: . e xt ote that by (A.5) (c) The reaction in first two firms’ prices to a radial move by Firm 3 will be identical; hence, we can disregard Firm 1’s behavior. Totally differentiate (A.9) and (A.10) with respect to r to get: ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)(cos 2222 2222 222222 22 =⋅−⋅′−′⋅−′⋅−−′− ⋅−⋅′−′⋅−′⋅−−′+ ′−′+−−−+−′+ −′−−′+′−′ pppp pppp pppppp yyy xxx pyxp yx ααββααββαβ ααββααββαβ αβαβαβαβ αβαβαβ , ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)(cos 3333 3333 333333 33 =⋅−⋅′+′⋅−′⋅+−′+ ⋅−⋅′+′⋅−′⋅+−′− ′+′−+−+−−+−′+ −′+−′−′+′− pppp pppp pppppp yyy xxx pyxp yx γγββγγββγβ γγββγγββγβ γβγβγβγβ γβγβγβ . Next, we assume the first equality from part (c) of present Le ma and observe from (B.5) and (A.4) that 0=′x and 0=′α . We also use the second part of (B.5), which says that the total derivative with respect to r of partial derivative of x, and, hence by (A.5) also of α, with respect to any of the three prices is 0. Note again that α = 0 in equilibrium, use (B.4) and sy metry of the problem to simplify the above system: ( ) ( )( ) 02332 332232 222222 2222 2 =−−′−′+′−′′+ −−−′+′−′ pppppp pppp yyyxp yxpy αββββ αββ ( ) ( ) 0323232 333333 3 =′−′−′+′−++−′+′+′− pppppp yyypypy βββββ . We next note t 222 cos ppp xx == αα in equilibrium. The remaining derivatives are collected in (B.7), (B.8), (B.11), (B.12), (B.13), (B.14), (B.17), and (B.18). Applying all these we get the final form of the system: −54 ′p 2 +18 ′p 3 + 36R − 9 +π 3 ′p 2 − ′p 3 − 2R + 3( ) = 0 , 18 ′p 2 − 36 ′p 3 − 36R + 9 +π 3 − ′p 2 + ′p 3 + 2R + 3( ) = 0, with a solution: ′p 2 = −9 + 9π 3 −π 2 + 36 − 2π 3( )R 90 − 3π 3 , and ′p 3 = 18+12π 3 −π 2 − 72 − 4π 3( )R 90 − 3π 3 . Q.E.D. Proof of Le ma 7: Collect results from Le ma 6 to calculate the expression in parentheses of (9) and equate it with zero: i ili rium. The remai i deriva- tiv s are collected in (B.7), (B.8), (B.11), (B.12), (B.13), (B.14), (B.17), and (B.18). Applying all th se we get the final form of the system: (c) The reaction in f st two firms’ prices to a radial mov by Firm 3 will be identical; hence, we can disregard Firm 1’s behavior. Totally differentiate (A.9) and (A.10) with respect to r to get: ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)cos(cos 2222 2222 222222 22 =⋅−⋅′−′⋅−′⋅−−′− ⋅−⋅′′⋅′⋅−′+ ′−′+−+−′ −′−−′+′−′ pppp pppp pppppp yyy xxx pyxp yx ααββααββαβ ααββ βαβαβαβ αβαβαβ , ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sinsincos(cos )sin(sin)cos(cos )sin(sin)cos(cos 3333 3333 333333 33 =⋅−⋅′+′⋅−′⋅+−′+ ⋅−⋅′+′⋅′⋅′− ′+′−+−+−−+−′+ −′+−′−′+′− pppp pppp pppppp yyy xxx yxp yx γγββγγββγβ γγββγγ γβγβγβγβ γβγβγβ . Next, e assum the first equality from part (c) of pres t Lemma and observe from (B.5) and (A.4) tha 0=′x and 0=′α . We lso use the second part of (B.5), which say that the t tal derivative with respect to r of partial derivative of x, and, hence by (A.5) also of α, with respect to any of the three prices is 0. Note again that α = 0 in equilibriu , use (B.4) and symmetry of the problem to simplify the above system: ( ) ( )( ) 02332 33232 222222 2222 2 =−−′−′+′−′−′+ −−−′+′−′ pppppp pppp yyyxp yxpy αββββ αβ ( ) ( ) 0323232 333333 3 =′−′−′+′−++−′+′+′− pppppp yyypypy βββββ . We next note that by (A.5) 222 cos ppp xx == αα in equilibrium. The remaining derivatives are coll cted in (B.7), (B.8), (B.11), (B.12), (B.13), (B.14), (B.17), and (B.18). Applying all these we get the final form of the system: −54 ′p 2 +18 ′p 3 + 36R − 9 +π 3 ′p 2 − ′p 3 − 2R + 3( ) = 0 , 18 ′p 2 − 36 ′p 3 − 36R + 9 +π 3 − ′p 2 + ′p 3 + 2R + 3( ) = 0, with a solution: ′p 2 = −9 + 9π 3 −π 2 + 36 − 2π 3( )R 90 − 3π 3 , and ′p 3 = 18+12π 3 −π 2 − 72 − 4π 3( )R 90 − 3π 3 . Q.E.D. Proof of Lemma 7: Collect esults from Lemma 6 to calculate the expression in parentheses of (9) and equate it with zero: , (c) The reaction in first two firms’ prices to a radial move by Firm 3 will be identical; hence, we can disregard Firm 1’s behavior. Totally differentiate (A.9) and (A.10) with respect to r to get: ( ) ( ) 0)cos(cos)co(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)cos(cos 2222 2222 222222 22 =⋅−⋅′−′⋅−′⋅−′ ⋅−⋅′−′⋅−′⋅−−′+ ′−′+−−−+−′+ −′−−′+′−′ pppp pppp pppppp yyy xxx pyxp yx ααββααββαβ ααββααββαβ αβαβαβαβ αβαβαβ , ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin(sin)cos(cos 3333 3333 333333 33 =⋅−⋅′+′⋅−′⋅+−′+ ⋅−⋅′+′⋅−′⋅+−′− ′+′−+−+−−+−′+ −′+−′−′+′− pppp pppp pppppp yyy xxx yxp yx γγββγγββγβ γγββγγββγβ γβγβγβγβ γβγβγβ . Next, we assume the first equality from part (c) of present Lemma and observe from (B.5) and (A.4) that 0=′x and 0=′α . e also use the s cond part of (B.5), whi h says that the t tal derivative with r spect to r of partial derivative of x, and, hence by (A.5) also of α, with resp ct to any of the thr e prices is 0. Note again that α = 0 in equilibriu , use (B.4) and symmetry of the problem to simplify the above system: ( ) ( )( ) 02332 332232 222222 2222 2 =−−′−′+′−′−′+ −−−′+′−′ pppppp pppp yyyxp yxpy αββββ αββ ( ) ( ) 0323232 333333 3 =′−′−′+′−++−′+′+′− pppppp yyypypy βββββ . e next note that by (A.5) 222 cos ppp xx == αα in equilibrium. The remaining derivatives are collected in (B.7), (B.8), (B.11), (B.12), (B.13), (B.14), (B.17), and (B.18). Applying all these we get the final form of the system: −54 ′p 2 +18 ′p 3 + 36R − 9 +π 3 ′ 2 − ′ 3 −( ) , 18 ′p 2 − 36 ′p 3 − 36R + 9 +π 3 − ′p 2 + ′p 3 + 2R + 3( ) = 0, with a solution: ′p 2 = −9 + 9π 3 −π 2 + 36 − 2π 3( )R 90 − 3π 3 , and ′p 3 = 18+12π 3 −π 2 − 72 − 4π 3( )R 90 − 3π 3 . Q.E.D. Proof of Lemma 7: Collect results from Lemma 6 to calculate the expression in parentheses of (9) and equate it with zero: with a solution: ( ) The action n first two firms’ prices to a radial move by Firm 3 will be id ntical; hence, we can disregard Firm 1’s behavior. Totally differentiate (A.9) and (A.10) with respect to r to get: ( ) ( ) 0)cos(cos)cos(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin( in)cos(cos 2222 2222 222222 22 =⋅−⋅′−′⋅−′⋅−−′− ⋅−⋅′−′⋅−′⋅−−′+ ′−′+−+−+ −′−−′+′−′ pppp pppp pppppp yyy xxx pyxp yx ααββααββαβ ααββααββαβ αβαβαβαβ αβαβαβ , ( ) ( ) 0)cos(c s)cos(co)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin( in)cos(cos 3333 3333 333333 33 =⋅−⋅′+′⋅−′⋅+−′+ ⋅−⋅′+′⋅−′⋅+−′− ′+′+−+−−+− −′+−′−′+′− pppp pppp pppppp yyy xxx pyxp yx γγββγγββγβ γγββγγββγβ γβγβγβγβ γβγβγβ . Next, e ssume the first equality from part (c) of present Lemma and observe from (B.5) and (A.4) that 0=′x and 0=′α . We lso use the second part of (B.5), which say that e t tal derivative with respect to r of partial deriva ive of x, and, hence by (A.5) also of α, with r spect to any of the three prices is 0. N te again that α = 0 in equilibrium, use (B.4) and symmetry of the problem to simplify the above system: ( ) ( )( ) 02332 332232 222222 2222 2 =−−′−′+′−′−′+ −−−′+′−′ pppppp pppp yyyxp yxpy αββββ αββ ( ) ( ) 0323232 333333 3 =′−′−′+′−++−′+′+′− pppppp yyypypy βββββ . We next te that by (A.5) 222 cos ppp xx == αα in equilibrium. The remaining derivatives are collected in (B.7), (B.8), (B.11), (B.12), (B.13), (B.14), (B.17), and (B.18). Applying all these we get the final form of the system: −54 ′p 2 +18 ′p 3 + 36R − 9 +π 3 ′p 2 − ′p 3 − 2R + 3( ) = 0 , 18 ′p 2 − 36 ′p 3 − 36R + 9 +π 3 − ′p 2 + ′p 3 + 2R + 3( ) = 0, ith a solution: ′p 2 = −9 + 9π 3 −π 2 + 36 − 2π 3( )R 90 − π 3 , and ′p 3 = 18+12π 3 −π 2 − 72 − 4π 3( )R 90 − 3π 3 . Q.E.D. Proof of Lemma 7: Collect results from Lemma 6 to calculate the expression in parentheses of (9) and equate it with zero: , (c) The reacti n in first two firms’ prices to a radi l move by Firm 3 will be identical; hence, we can disregard Firm 1’s behavior. Totally differentiate (A.9) and (A.10) with respect to r to get: ( ) ( ) 0)cos(cos)co(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin( in)cos(cos 2222 2222 222222 22 =⋅−⋅′−′⋅−′⋅−′ ⋅−⋅′−′⋅−′⋅−−′+ ′−′+−+−′+ −′−−′+′−′ pppp pppp pppppp yyy xxx pyxp yx ααββααββαβ αββααββαβ αβαβαβαβ αβαβαβ , ( ) ( ) 0)cos(cos)co(cos)sin(sin )sin(sin)sin(sin)cos(cos )sin(sin)cos(cos )sin( in)cos( 3333 3333 333333 33 =⋅−⋅′+′⋅−′⋅+−′+ ⋅−⋅′+′⋅−′⋅+−′− ′+′+−+−−+−′ −′+−′−′+′ pppp pppp pppppp yyy xxx yxp yx γγββγγββγβ γββγγββγβ γβγβγβγβ βγβγβ . N xt, we assume the first equality fr m part (c) o pres nt Lemma and observe rom (B.5) and (A.4) that 0=′x and 0=′α . We also use e second part of (B.5), which says that the total derivative wit res ect o r of partial der vative of x, nd, hence by (A.5) also of α, with respect o any of the three rices is 0. Note again that α = 0 in equilibrium, use (B.4) and symmetry of the problem to simplify the above system: ( ) ( )( ) 02332 332232 222222 2222 2 =−−′′+′−′−′+ −−′+′−′ pppppp pppp yyyxp xpy αββββ αββ ( ) ( ) 033232 333333 3 =′−′−′+′−++−′+′+′− pppppp yyypypy βββββ . We next ote that by (A.5) 22 cos ppp xx == αα in equilibrium. The remaining derivatives ar coll ct d in (B.7), (B.8), (B.11), (B.12), (B.13), (B.14), (B.17), and (B.18). Applying all these we get the final form of the system: −54 ′p 2 +18 ′p 3 + 6R − 9 +π 3 ′p 2 − ′p 3 − 2 + 3( ) = 0 , 18 ′p 2 36 ′p 3 − 36R + 9 +π 3 − ′p 2 + ′p 3 + 2R + 3( ) = 0, with a solution: ′p 2 = −9 9π 3 −π 2 + 36 − 2π 3( )R 90 − 3π 3 , and ′p 3 = 18+12π 3 −π 2 − 72 − 4π 3( )R 90 − 3π 3 . Q.E.D. Proof of Lemma 7: Collect results from Lemma 6 to calculate the expression in parentheses of (9) and equate it with zero: Q.E.D. A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 341 proof of Lemma 7: Collect results from Lemma 6 to calculate the expression in parenthe- ses of (9) and equate it with zero: − 4R * −1 2R * 3 + 2 2R * 3 ⋅ −9 + 9π 3 −π 2 + 36 − 2π 3( )R * 90 − 3π 3 = 0. Simple algebra yields the result. However, to establish that we have really found a firm’s local best response to the rival locations we must check the second order condition also. The second order derivative of Firm 3’s profit with respect to r is derived from (9):           ⋅ ∂ ∂ +⋅ ∂ ∂ +⋅ ′         ∂ ∂ +⋅ ′         ∂ ∂ + ′         ∂ ∂ ⋅+         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+         ⋅ ∂ Π∂ = Π 2 2 2 2 3 2 1 2 1 32 2 31 1 33 3 2 2 31 1 3333 3 3 2 3 2 rd pd p D rd pd p D rd pd p D rd pd p D r D p rd pd p D rd pd p D r D dr dp rd pd pdr d rd d (A.11) At r = R * both terms in the first row of (A.11) are zero. First one due to optimizing behavior in the second stage of the game, and the second one due to the expression in parentheses being zero at r = R * (F.O.C.). It remains to evaluate the sign of the big expression in parentheses in the second row of (A.11). We prove a series of claims. Claim 1: 0 3 < ′       ∂ ∂ r D . We go back to the proof of Lemma 6(a) to obtain the relevant part of the partial derivative of interest. Using the symmetry of the problem and the fact that 0≡ r x we get: ββ sin rrr yD +−= . Hence, ( ) ββ ′⋅−′+′−= ′ rrrr yyD 2 1 2 3 . The total derivatives are in (B.9), (B.10), (B.16) and (B.18). Putting them all together yields: ( ) 32 12 3 2 2 1 9 2 2 3 34 12 23 R Rpp RR R D r −+′−′ +− + −= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 3167.1−≈ ′ r D . Claim 2: 0 2 3 < ′         ∂ ∂ p D and 0 1 3 < ′         ∂ ∂ p D . We take the partial derivative from Lemma 6(b) and proceed similarly as in previous claim: ββ sin 222 ppp yD +−= . So, ( ) ββ ′⋅−′+′−= ′ 2222 2 1 2 3 pppp yyD . Simple algebra yields the result. however, to establish that we have really found a firm’s local best response to the rival locations we must check the second order condition also. The second order derivative of Firm 3’s profit with respect to r is derived from (9): (A.11) At r = R* both terms in the first row of (A.11) are zero. first one due to optimizing behav- ior in the second stage of the game, and the second one due to the ex ression in paren- theses being zero at r = R* (f.O.C.). It remains to evaluate the sign of the big expression in parentheses in the second row of (A.11). We prove a series of claims. Claim 1: − 4 * −1 2R * 3 + 2 2R * 3 ⋅ −9 + 9π 3 −π 2 + 36 − 2π 3( )R * 90 − 3π 3 = 0. Simple algebra yields the result. However, to establish that we have really found a firm’s local best response to the rival locations e must check the second order condition also. The second order derivative of Firm 3’s profit with respect to r is derived from (9):           ⋅ ∂ +⋅ ∂ +⋅ ′         ∂ +⋅ ′         ∂ + ′         ∂ ∂ ⋅+         ⋅ ∂ +⋅ ∂ + ∂ ∂ ⋅+         ⋅ ∂ Π = Π 2 2 2 2 3 2 1 2 1 32 2 31 1 33 3 2 2 31 1 3333 3 3 2 3 2 rd p p D rd p p D rd p p D rd p p D r D p rd p p D rd p p D r D dr p rd p pdr d rd d (A.11) At r = R * both terms in the first row of (A.11) are zero. First one due to optimizing behavior in the second stage of the game, and the second one due to the expression in parentheses being zero at r = R * (F.O.C.). It remains to evaluate the sign of the big expression in parentheses in the second row of (A.11). We prove a series of claims. 1: 0 3 < ′       ∂ ∂ r D . We go back to the proof f L mma 6(a) to obtain the relevant part of the partial derivative of interest. Using the symmetry of the problem and the fact that 0≡ r x we get: ββ sin rrr yD +−= . Hence, ( ) ββ ′⋅−′+′−= ′ rrrr yyD 2 1 2 3 . The total derivatives are in (B.9), (B.10), (B.16) and (B.18). Putting them all together yields: ( ) 32 12 3 2 2 1 9 2 2 3 34 12 23 R Rpp RR D r −+′−′ +− + −= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 3167.1−≈ ′ r D . Claim 2: 0 2 3 < ′         ∂p D and 0 1 3 < ′         ∂p D . We take the partial derivative from Lemma 6(b) and proceed similarly as in previous claim: ββ sin 222 ppp yD +−= . So, ( ) ββ ′⋅−′+′−= ′ 2222 2 1 2 3 pppp yyD . We go back to the proof of Lemma 6(a) to obtain the relevant part of the partial de- rivative of interest. Using the symmetry of the problem and the fact that − 4R * −1 2R * 3 + 2 2R * 3 ⋅ −9 + 9π 3 −π 2 + 36 − 2π 3( )R * 90 − 3π 3 = 0. Simple algebra yields the result. However, to establish that we have really found a firm’s local best response to the rival locations we must check the second order condition also. Th second order derivative of Firm 3’s profit wi h respect to r is derived from (9):           ⋅ ∂ ∂ +⋅ ∂ ∂ +⋅ ′         ∂ ∂ +⋅ ′         ∂ ∂ + ′         ∂ ∂ ⋅+         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+         ⋅ ∂ Π∂ = Π 2 2 2 2 3 2 1 2 1 32 2 31 1 33 3 2 2 31 1 3333 3 3 2 3 2 rd pd p D rd pd p D rd pd p D rd pd p D r D p rd pd p D rd pd p D r D dr dp rd pd pdr d rd d (A.11) At r = R * both terms in the first row of (A.11) are zero. First one due to optimizing behavior in the second stage of the game, and the second one due to the expression in parentheses being zero at r = R * (F.O.C.). It remains to evaluate the sign of the big expression in parentheses in the second row of (A.11). We prove a series of claims. Claim 1: 0 3 < ′       ∂ ∂ r D . We go back to the proof of Lemma 6(a) to obtain the relevant part of the partial derivative of interest. Using the symmetry of the problem and the fact t t 0≡ r x we get: ββ sin rrr yD +−= . Hence, ( ) ββ ′⋅−′+′−= ′ rrrr yyD 2 1 2 3 . The total derivatives are in (B.9), (B.10), (B.16) and (B.18). Putting them all together yields: ( ) 32 12 3 2 2 1 9 2 2 3 34 12 23 R Rpp RR R D r −+′−′ +− + −= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 3167.1−≈ ′ r D . Claim 2: 0 2 3 < ′         ∂ ∂ p D and 0 1 3 < ′         ∂ ∂ p D . We take the partial derivative from Lemma 6(b) and proceed similarly as in previous claim: ββ sin 222 ppp yD +−= . So, ( ) ββ ′⋅−′+′−= ′ 2222 2 1 2 3 pppp yyD . e t: − 4R * −1 2R * 3 + 2 2R * 3 ⋅ −9 + 9π 3 − 2 + 36 − 2π 3( )R * 90 − 3π 3 = 0. Simple algebra yields the result. However, to establis that we have really found firm’s local best res se t the rival l ations we must heck the second order condition also. The second order ivative f Firm 3’s profit with respect to r is de ive f om (9):           ⋅ ∂ ∂ +⋅ ∂ ∂ ⋅ ′         ∂ +⋅ ′         ∂ ∂ + ′         ∂ ∂ ⋅+         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+         ⋅ ∂ Π∂ = Π 2 2 2 2 3 2 1 2 1 32 2 31 1 33 3 2 2 31 1 3333 3 3 2 3 2 rd pd p D rd pd p D prd pd p D r D p rd pd p D rd pd p D r D dr dp rd pd pdr d rd d (A.11) At r = R * both terms in the first row of (A.11) are zer . First one du t optimizing behavior in the second stage of the game, and the second one due to the expr ssion in parenth ses being zero at r = R * (F.O.C.). It remains to evaluate the ign of the big xpression in parenth es in the second row of (A.11). We pr ve a series of claims. Claim 1: 0 3 < ′       ∂ ∂ r D . We go back to the pro f of Lemma 6(a) t t i the rel vant part of the partial derivative of interest. Using the symmetry of t e problem and t fact that 0≡ r x we get: ββ sin rrr yD +−= . Hence, ( ) ββ ′⋅−′+′−= ′ rrr yyD 2 1 2 3 . The total derivatives are in (B.9), (B.10), (B.16) and (B.18). Putting them all together yields: ( ) 32 12 3 2 2 1 9 2 2 3 34 12 23 R Rpp RR R D r −+′−′ +− + −= ′ . Evaluating this at R = R * and u ing result from Lemma 6(c) yields ( ) 3167.1−≈ ′ r D . Claim 2: 0 2 3 < ′         ∂ ∂ p D and 0 1 3 < ′         ∂ ∂ p . We take the partial derivative from L mma 6(b) and proceed similarly as in previous cl im: ββ sin 222 ppp yD +−= . So, ( ) ββ ′⋅−′+′−= ′ 2222 2 1 2 3 pppp yyD . . hence, − 4R * −1 2R * 3 + 2 2R * 3 ⋅ −9 + 9π 3 −π 2 + 36 − 2π 3( )R * 90 − 3π = 0. Simple alg bra yields the r sult. Ho ever, to es ablish that w hav real y found a firm’s local best r pon e to the rival locati ns we must check h econd rder condition also. Th econd rder erivativ f Firm 3’s profit with espect o i derived from (9):         ⋅ ∂ ∂ +⋅ ∂ ∂ +⋅ ′         ∂ ∂ +⋅ ′         ∂ ∂ + ′         ∂ ∂ ⋅+     ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ⋅+         ⋅ ∂ Π∂ = Π 2 2 2 2 3 2 1 2 1 32 2 31 1 33 3 2 2 31 1 3333 3 3 2 3 2 rd pd p D rd pd p D rd pd p D rd pd p D r D p rd pd p D rd pd p D r D dr dp rd pd pdr d rd d (A.11) At r = R * both terms in the first row of (A.11) are zero. First one du to optimiz ng b havior in the second stage of the game, and th second e du to the xpr ssion in parenth ses be ng zero at r = R * (F.O.C.). It remains to evaluate th sig of the big expr ssion in parenth ses in the second row of (A.11). We prove a s rie of claims. Claim 1: 0 3 < ′       ∂ ∂ r D . e go back to the proof of Lemma 6(a) to btain he rel vant par of the partial deriva ive of interest. U ing the symmetry of t e proble and the fact tha 0≡ r x we get: ββ sin rrr yD +−= . Hence, ( ) ββ ′⋅−′+′−= ′ rrrr yyD 2 1 2 3 . The total derivatives ar in (B.9), (B.10), (B.16) and (B.18). Putting hem all toge her yi lds: ( ) 32 12 3 2 2 1 9 2 2 3 34 12 23 R Rpp RR R D r −+′−′ +− + −= ′ . Evaluating his at R = * and using result from Lemma 6(c) yields ( ) 3167.1−≈ ′ r D . Claim 2: 0 2 3 < ′         ∂ ∂D and 0 1 3 < ′         ∂ ∂ p D . We tak the partial derivativ from Lemma 6(b) and proce d similarly s in pr vious cl im: ββ sin 222 ppp yD +−= . So, ( ) ββ ′⋅−′+′−= ′ 2222 2 1 2 3 pppp yyD . . The total derivatives are in (B.9), (B.10), (B.16) and (B.18). putting them all together yields: − 4R * −1 2R * 3 + 2 2R * 3 ⋅ −9 + 9π 3 −π 2 + 36 − 2π 3( )R * 90 − 3π 3 = 0. Simple algebra yields the result. However, to establish that e have really found a firm’s local best response to the rival locations we must check the second order condition also. The second order derivative of Firm 3’s profit with respect to r is erived from (9):           ⋅ ∂ ∂ +⋅ ∂ ∂ +⋅ ′         ∂ ∂ +⋅ ′         ∂ ∂ + ′         ∂ ∂ ⋅+         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+         ⋅ ∂ Π∂ = Π 2 2 2 2 3 2 1 2 1 32 2 31 1 33 3 2 2 31 1 3333 3 3 2 3 2 rd pd p D rd pd p D rd pd p D rd pd p D r D p rd pd p D rd pd p D r D dr dp rd pd pdr d rd d (A.11) At r = * both terms in the first row of (A.11) are zero. First one due to optimizing behavior in the second stage of the game, and the second one due to the expression in parentheses being zero at r = R * (F.O.C.). It remains to evaluate the sign of the big expression in parentheses in the second row of (A.11). We prove a series of claims. Claim 1: 0 3 < ′       ∂ ∂ r D . We go back to the proof of Lemma 6(a) to obtain the relevant part of the partial derivative of interest. Using the symmetry of the problem and the fact that 0≡ r x we get: ββ sin rrr yD +−= . Hence, ( ) ββ ′⋅−′+′−= ′ rrrr yyD 2 1 2 3 . The total derivatives are in (B.9), (B.10), (B.16) and (B.18). Putting them all together yields: ( ) 32 12 3 2 2 1 9 2 2 3 34 12 23 R Rpp RR R D r −+′−′ +− + −= ′ . Ev luating this at R = R * and using result from Lemma 6(c) yields ( ) 3167.1−≈ ′ r D . Claim : 0 2 3 < ′         ∂ ∂ p D and 0 1 3 < ′         ∂ ∂ p D . We take the artial derivative from Lemma 6(b) and proceed similarly as in previous claim: ββ sin 222 ppp yD +−= . So, ( ) ββ ′⋅−′+′−= ′ 2222 2 1 2 3 pppp yyD . . l ting this at R = R* and using result from Le ma 6(c) yields − 4R * −1 2R * 3 + 2 2R * 3 ⋅ −9 + 9π 3 −π 2 + 36 − 2π 3( )R * 90 − 3π 3 = 0. Simple algebra yields th result. Ho ver, to establis that we hav really found a firm’s local best response o the rival locations we must check th second order con ition also. Th second order derivative of Firm 3’s profit with respect to r is derived from (9):           ⋅ ∂ ∂ +⋅ ∂ ∂ +⋅ ′         ∂ ∂ +⋅ ′         ∂ ∂ + ′         ∂ ∂ ⋅+         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+         ⋅ ∂ Π∂ = Π 2 2 2 2 3 2 1 2 1 32 2 31 1 33 3 2 2 31 1 3333 3 3 2 3 2 rd pd p D rd pd p D rd pd p D rd pd p D r D p rd pd p D rd pd p D r D dr dp rd pd pdr d rd d (A.11) At r = R * bo h terms in the first row of (A.11) ar zero. First one due to opt mizing behavior in th second stage of the game, and th second one due o the expressio in parentheses being zero at r = R * (F O.C.). It remains to evalua e the sign of the big expressio in parentheses in the second row of (A.11). We prove a s ries of claims. Claim 1: 0 3 < ′       ∂ ∂ r D . We go back o th pro of Le ma 6(a) to obtain th relevant part of the partial derivative of interest. Using the sy metry of the problem and the fact that 0≡ r x we get: ββ sin rrr yD +−= . Hence, ( ) ββ ′⋅−′+′−= ′ rrrr yyD 2 1 2 3 . The total derivatives are in (B.9), (B.10), (B.16) and (B.18). Putting them all together yields: ( ) 32 12 3 2 2 1 9 2 2 3 34 12 23 R Rpp RR R D r −+′−′ +− + −= ′ . Evaluating this at R = R * and using result fr ( ) 3167.1−≈ ′ r D . Claim 2: 0 2 3 < ′         ∂ ∂ p D and 0 1 3 < ′         ∂ ∂ p D . We take the partial derivative from Le ma 6(b) and proceed s mi arly as in previous claim: ββ sin 222 ppp yD +−= . So, ( ) ββ ′⋅−′+′−= ′ 2222 2 1 2 3 pppp yyD . . l i 2: − 4 * −1 2 * 3 + 2 2 * 3 ⋅ −9 + 9π 3 −π 2 + 36 − 2π 3( ) * 90 − 3π 3 = 0. Si ple algebra yields the result. o ever, to establish that we have rea ly found a fir ’s local best response to the rival locations e ust check the second order condition also. he second order derivative of ir 3’s profit ith respect to r is derived fro (9):           ⋅ ∂ ∂ +⋅ ∂ ∂ +⋅ ′         ∂ ∂ +⋅ ′         ∂ ∂ + ′         ∂ ∂ ⋅+         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+         ⋅ ∂ ∂ = 2 2 2 2 3 2 1 2 1 32 2 31 1 33 3 2 2 31 1 3333 3 3 2 3 2 rd pd prd pd prd pd prd pd pr p rd pd prd pd prdr dp rd pd pdr d rd d ( .11) t r R * both ter s in the first ro of ( .11) are zero. First ne due to pti izing behavior in the second stage of the ga e, and the second one due to the xpression in parenthes being zero at r * (F. . .). It re ains to evaluate the sign of the big expression in parenthes in the second ro of ( .11). e prove a series of clai s. lai 1: 0 3 < ′       ∂ ∂ r . e go back to the proof of e a 6(a) to btain the rel vant part of the partial derivative of inter st. sing the sy etry of the proble and the fact that 0≡ r x e get: ββ sin rrr y+−= . ence, ( ) ββ ′⋅−′+′−= ′ rrrr yy 2 1 2 3 . he to al derivatives are in ( .9), ( .10), ( .16) and ( .18). Pu ting the a l together yields: ( ) 32 12 3 2 2 1 9 2 2 3 34 12 23 pp r −+′−′ +− + −= ′ . va ua this at * and using result fro e a 6(c) yields ( ) 3167.1−≈ ′ r . a 0 2 3 < ′         ∂ ∂ p and 0 1 3 < ′         ∂ ∂ p . e take the partial derivative fro e a 6(b) and proceed si ilarly as in previous clai : sin 222 ppp y . So, ( ) ββ ′⋅−′+′−= ′ 2222 2 1 2 3 pppp yy . − 4R * −1 2R * 3 + 2 2R * 3 ⋅ −9 + 9π 3 −π 2 + 36 − 2π 3( )R * 90 − 3π 3 = 0. Simple algebra yields the result. However, to establish that we have really found a firm’s local best response to the ival locations w mus check the s cond order condition also. The second order derivative of Firm 3’s profit with respect to r is derived from (9):         ⋅ ∂ ∂ +⋅ ∂ ∂ +⋅ ′       ∂ ∂ +⋅ ′       ∂ ∂ + ′       ∂ ∂ ⋅+       ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+       ⋅ ∂ Π∂ = Π 2 2 2 2 3 2 1 2 1 32 2 31 1 33 3 2 2 31 1 3333 3 3 2 3 2 rd pd p D rd pd p D rd pd p D rd pd p D r D p rd pd p D rd pd p D r D dr dp rd pd pdr d rd d (A.11) At r = R * b th t rms in the first row of (A.11) are zero. Fi st one due to optimizing behavior in the second stage of the game, and the second one du to the expressio in parentheses being zero at r = R * (F.O.C.). It remains to evaluate the sign of th big expre sion in parentheses in the second row of (A.11). We prove a series of claims. Claim 1: 0 3 < ′       ∂ ∂ r D . We go back to the pr of of Lemma 6(a) to obtain the relevant part of the partial derivative of interest. Using the symmetry of the problem and the fact that 0≡ r x we get: ββ sin rrr yD +−= . Hence, ( ) ββ ′⋅−′+′−= ′ rrrr yyD 2 1 2 3 . The total derivatives are in (B.9), (B.10), (B.16) and (B.18). Putting them all together yields: ( ) 32 12 3 2 2 1 9 2 2 3 34 12 23 R Rpp RR R D r −+′−′ +− + −= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 3167.1−≈ ′ r D . Claim 2: 0 2 3 < ′       ∂ ∂ p D and 0 1 3 < ′       ∂ ∂ p D . We take the partial derivative from Lemma 6(b) and proceed similarly as in previous claim: ββ sin 222 ppp yD +−= . So, ( ) ββ ′⋅−′+′−= ′ 2222 2 1 2 3 pppp yyD . ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012342 We take the partial derivative from Lemma 6(b) and proceed similarly as in previous claim: − 4R * −1 2R * 3 + 2 2R * 3 ⋅ −9 + 9π 3 −π 2 + 36 − 2π 3( )R * 90 − 3π 3 = 0. Simple algebra yields the result. However, to establish that we have really found a firm’s local best response to the rival locations we must check the second order condition also. The second order derivative of Firm 3’s profit with respect to r is derived from (9):           ⋅ ∂ ∂ +⋅ ∂ ∂ +⋅ ′         ∂ ∂ +⋅ ′         ∂ ∂ + ′         ∂ ∂ ⋅+         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+         ⋅ ∂ Π∂ = Π 2 2 2 2 3 2 1 2 1 32 2 31 1 33 3 2 2 31 1 3333 3 3 2 3 2 rd pd p D rd pd p D rd pd p D rd pd p D r D p rd pd p D rd pd p D r D dr dp rd pd pdr d rd d (A.11) At r = R * both terms in the first row of (A.11) are zero. First one due to optimizing behavior in the second stage of the game, and the second one due to the expression in parentheses being zero at r = R * (F.O.C.). It remains to evaluate the sign of the big expression in parentheses in the second row of (A.11). We prove a series of claims. Claim 1: 0 3 < ′       ∂ ∂ r D . We go back to the proof of Lemma 6(a) to obtain the relevant part of the partial derivative of interest. Using the symmetry of the problem and the fact that 0≡ r x we get: ββ sin rrr yD +−= . Hence, ( ) ββ ′⋅−′+′−= ′ rrrr yyD 2 1 2 3 . The total derivatives are in (B.9), (B.10), (B.16) and (B.18). Putting them all together yields: ( ) 32 12 3 2 2 1 9 2 2 3 34 12 23 R Rpp RR R D r −+′−′ +− + −= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 3167.1−≈ ′ r D . Claim 2: 0 2 3 < ′         ∂ ∂ p D and 0 1 3 < ′         ∂ ∂ p D . We take the partial derivative fro Lemma 6(b) and proceed similarly as in previous claim: ββ sin 222 ppp yD +−= . So, ( ) ββ ′⋅−′+′−= ′ 2222 2 1 2 3 pppp yyD . . o, − 4R * −1 2R * 3 + 2 2R * 3 ⋅ −9 + 9π 3 π 2 + 36 − 2 3( )R * 90 − 3π 3 = 0. Simple algebra yi lds the result. However, to establish that we have really found a firm’s loc l best response to the rival locations we must check the second order ition als . The second order derivativ of Firm 3’s profit with respect to r is derived from (9):           ⋅ ∂ ∂ +⋅ ∂ ∂ +⋅ ′         ∂ ∂ +⋅ ′        ∂ ∂ + ′         ∂ ∂ ⋅+         ⋅ ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅+         ⋅ ∂ ∂ = Π 2 2 2 2 3 2 1 2 1 32 2 31 1 33 3 2 2 31 1 3333 3 3 2 3 2 rd pd p D rd pd p D rd pd p D rd pd p D r D p rd pd p D rd pd p D r D dr dp d pd pdr d rd d (A.11) At r = R * both terms in t e first row of (A.11) are zero. First one due to optimizing behavior in the second stage of the g me, and the seco on due to the expression in arentheses being zero at r = R * (F.O.C.). It remains to evaluate the sign of the bi expression in arentheses in the second row of (A.11). We prove a series of claims. Claim 1: 0 3 < ′       ∂ ∂ r D . We go back to the proof f L mma 6(a) to obtain the releva t part of the partial derivative of interest. Usi g the symmetry of the problem and the fact that 0≡ r x we get: ββ sin rrr yD +−= . Hence, ( ) ββ ′⋅−′+′−= ′ rrrr yyD 2 1 2 3 . The total derivatives are in (B.9), (B.10), (B.16) and ( . 8). Putting them all together yields: ( ) 32 12 3 2 2 1 9 2 2 3 34 12 23 R Rpp RR R D r −+′−′ +− + −= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 3167.1−≈ ′ r D . Claim 2: 0 2 3 < ′         ∂ ∂ p D and 0 1 3 < ′         ∂ ∂ p D . take the partial derivative from Lemma 6(b) and proceed similarly as in previou claim: ββ sin 222 ppp yD +−= . So, ( ) ββ ′⋅−′+′−= ′ 2222 2 1 2 3 pppp yyD . . Derivatives needed are in (B.7), (B.11), (B.17), and (B.18). We get: Derivatives needed are in (B.7), (B.11), (B.17), and (B.18). We get: ( ) 32 12 6 1 2 1 9 1 2 3 34 1 23 222 R Rpp RRR D p −+′−′ ⋅−−−= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 8121.0 2 −≈ ′ p D . Due to the symmetry it must also be ( ) 8121.0 1 −≈ ′ p D Claim 3: 0 2 2 2 < dr pd and 0 2 1 2 < dr pd . To prove this claim analytically we would have to solve the system (A.8)-(A.10) without asserting r = R to get 1 p′ , 2 p′ , and 3 p ′ as functions of r. The dependence of relevant derivatives on r from Appendix B shows that such analytical solution is out of reach. We proceed as follows. We solve the system of the first-order conditions (A.8)-(A.10) for Firms 1 and 2 located R * away from the origin, while varying Firm 3’s location, r, around R * , numerically. The results are: 991583.0)005.0()005.0( * 2 * 1 =−==−= RrpRrp , 993237.0)()( * 2 * 1 ==== RrpRrp , and 994882.0)005.0()005.0( * 2 * 1 =+==+= RrpRrp . We are interested in two differences representing the first-order derivatives of p 2 with respect to r at R * −0.0025 and R * +0.0025: 3* 2 * 2 10654.1)005.0()( − ×=−=−= RrpRrp 3* 2 * 2 10645.1)()005.0( − ×==−+= RrpRrp . We see that in vicinity of R * the first-order derivative of rivals’ prices with respect to r is decreasing in r, hence, the second order derivative must be negative. This completes the proof of Claim 3. One can verify that 1 p′ and 2 p′ presented in Lemma 6 (c) are positive when r=R * , while Lemma 6(b) says that 13 pD ∂∂ and 23 pD ∂∂ are also positive. Using the three claims above, we see that every single summand in the big parentheses in the second row of (A.11) is negative. Hence, 0 2 3 2 < Π dr d at r = R * . We have found a local maximum of Firm 3’s profits with respect to r. Q.E.D. Proof of Lemma 8: We are going to show that the expression in the parentheses in (10) is zero. First, it is obvious that 0 3 =∂∂ φD , since the same number of customers won by a move along polar direction from one neighbor is lost to the other one. Second, we claim φφ ddpddp 21 −= . Clearly, the neighbor who becomes closer by such a move becomes more aggressive and reduces its price and vice versa, the neighbor that is now farther away becomes less aggressive and raises its price. The magnitude of the two derivatives i g this at R = R* and using result from Le ma 6(c) yields Derivatives ne ed are in (B.7), (B.11), (B.17), and (B.18). W get: ( ) 32 12 6 1 29 1 2 3 34 1 23 222 R Rpp RRR D p −+′−′ ⋅−−−= ′ . Evaluating this at R = R * and using result fr m ( ) 8 21.0 2 −≈ ′ p D . Due to the symmetry it must also be ( ) 8 21.0 1 −≈ ′ p D Claim 3: 0 2 2 2 < dr pd and 0 2 1 2 < dr pd . To prove this claim nalytically we would have t solve the system (A.8)-(A.10) without asserting r = R to get 1 p′ , 2 p′ , and 3 p ′ as functions of r. Th d pendence of relevant derivatives on from Appendix B shows that such nalytical solution is out of reach. We proceed as f llows. We solve the system of the first-order conditions (A.8)-(A.10) for Firms 1 and 2 located R * way from the origin, while varying Firm 3’s location, r, around R * , numerically. Th results are: 991583.0)005.0()005.0( * 2 * 1 =−==−= RrpRrp , 99 237.0)()( * 2 * 1 ==== RrpRrp , and 994882.0)005.0()005.0( * 2 * 1 =+==+= RrpRrp . We are inter sted in two differences represe ting the first-order derivatives of p 2 with respect to r at R * −0.0025 and R * +0.0025: 3* 2 * 2 10654.1)005.0()( − ×=−=−= RrpRrp 3* 2 * 2 10645.1)()005.0( − ×==−+= RrpRrp . W see that in vicinity of R * the first-order derivative of rivals’ prices with respect to r is decreasing in r, hence, th second order derivative must b negative. This completes the pro of Claim 3. One can verify that 1 p′ and 2 p′ presented in Lemma 6 (c) are positive when r=R * , while Lemma 6(b) says that 13 pD ∂∂ and 23 pD ∂∂ are also positive. Using the three claims above, we see that very single summand in the big parentheses in th second row of (A.11) is negative. Hence, 0 2 3 2 < Π dr d at r = R * . We have found a local maxi um of Firm 3’s profits with respect to r. Q.E D. Pro of Lemma 8: We are going to show that the expressio in the parentheses in (10) is zero. Firs , it is obvious that 0 3 =∂∂ φD , since the same number of customers won by a move along polar direction from o neighbor is lost o the other one. Second, we claim φφ ddpddp 21 −= . Clearly, th neighbor who becomes closer by such a mov becomes more aggressive and reduces its price and vic versa, th neighbor that is now farther way becomes less aggressive and raises its price. The magnitude of the two derivatives to the symmetry it must also be Derivatives ne ded are in (B.7), ( .11), (B.17), and (B.18). We get: ( ) 32 12 6 1 2 1 9 1 2 3 34 1 23 222 R Rpp RRR D p −+′−′ ⋅−−−= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 8121.0 2 −≈ ′ p D . Due to the symmetr it ( ) 8121.0 1 −≈ ′ p D Claim 3: 0 2 2 2 < dr pd and 0 2 1 2 < dr pd . To prove this claim analytically we would have to solve the system (A.8)-(A.10) without asserting r = R to get 1 p′ , 2 p′ , and 3 p ′ as functions of r. The dependence of relevant derivatives on r from Appendix B shows that such analytical solution is out of reach. We proceed as follows. We solve the system of the first-order conditions (A.8)-(A.10) for Firms 1 and 2 located R * away from the origin, while varying Firm 3’s location, r, around R * , numerically. The results are: 991583.0)005.0()005.0( * 2 * 1 =−==−= RrpRrp , 993237.0)()( * 2 * 1 ==== RrpRrp , and 994882.0)005.0()005.0( * 2 * 1 =+==+= RrpRrp . We are interested in two differences representing the first-order derivatives of p 2 with respect to r at R * −0.0025 and R * +0.0025: 3* 2 * 2 10654.1)005.0()( − ×=−=−= RrpRrp 3* 2 * 2 10645.1)()005.0( − ×==−+= RrpRrp . We see that in vicinity of R * the first-order derivative of rivals’ prices with respect to r is decreasing in r, hence, the second order derivative must be negative. This completes the proof of Claim 3. One can verify that 1 p′ and 2 p′ presented in Lemma 6 (c) are positive when r=R * , while Lemma 6(b) says that 13 pD ∂∂ and 23 pD ∂∂ are also positive. Using the three claims above, we see that every single summand in the big parentheses in the second row of (A.11) is negative. Hence, 0 2 3 2 < Π dr d at r = R * . We have found a local maximum of Firm 3’s profits with respect to r. Q.E.D. Proof of Lemma 8: We are going to show that the expression in the parentheses in (10) is zero. First, it is obvious that 0 3 =∂∂ φD , since the same number of customers won by a move along polar direction from one neighbor is lost to the other one. Second, we claim φφ ddpddp 21 −= . Clearly, the neighbor who becomes closer by such a move becomes more aggressive and reduces its price and vice versa, the neighbor that is now farther away becomes less aggressive and raises its price. The magnitude of the two derivatives Claim 3: i tives needed ar in (B.7), (B.1 ), (B. , d (B.1 . : ( ) 32 12 6 1 2 1 9 1 2 3 34 1 23 222 R Rpp RR D p −+′−′ ⋅−−−= ′ . Evaluating this at R = R * and using res lt from Lemma 6(c) yields ( ) 8121.0 2 −≈ ′ p D . Due to the symmetr it ust also be ( ) 8121.0 1 −≈ ′ p D : 0 2 2 2 < dr pd and 0 2 1 2 < dr pd . To prove this claim analytically we would have to sol the system (A.8)-(A.10) without asserting r = R to get 1 p′ , 2 p′ , and 3 ′ as functions of r. The depend nce of r levant d ri tives on r from Appendix B shows that such analytical solution is out of reach. We proceed as follows. We solve the system of the first-order conditions (A.8)-(A.10) for Firms 1 and 2 located R * away from the origin, while varying Firm 3’s location, r, rou d R * , numerically. The results are: 991583.)005.0()005.0( * 2 * 1 =−==−= RrpRrp , 993237.0)()( * 2 * 1 ==== RrpRrp , and 994882.)005.0()005.0( * 2 * 1 =+==+= RrpRrp . We are interest d in two ifferences r pres nting th first-order derivatives of p 2 with respect to r at R * −0.0025 and R * +0.0025: 3* 2 * 2 10654.1)005.0()( − ×=−=−= RrpRrp 3* 2 * 2 10645.1)()005.0( − ×==−+= RrpRrp . We see that in vicinity of R * the first-order derivative of rivals’ prices with r pect to r is decreasing in r, hence, the second order derivative must b negativ . This completes the proof of Claim 3. One can verify that 1 p′ and 2 p′ presented in Lemma 6 (c) are positiv when r=R * , while Lemma 6(b) says that 13 pD ∂∂ and 23 pD ∂ are also positive. U ing the three claims above, we see that ev ry singl summand in the big par ntheses i the second row f (A.11) is negative. Hence, 0 2 3 2 < Π dr d at r = R * . We have found a local maximu of Fir 3’s profits with re pect to r. Q.E.D. Proof of Lemma 8: We re going to show that the expression in the parentheses i (10) is zero. First, it is obviou that 0 3 =∂∂ φD , since the same number of customers w n by a move along polar direction from e neighbor is lost to the other one. Second, w claim φφ ddpddp 21 −= . Clearly, the neighbor who ecomes closer by such a move becom s more aggressive and reduces its price and vi versa, the neighbor that is now farther away becomes l ss aggressive and raises its price. The magnitude of the tw derivatives . r e t is claim analytically we would ha e to s l t s st ( .8)-( .1 ) ithout asserting r = R to get p1ʹ, p2ʹ, and p3ʹ as f nctions of r. The depe dence of relevant deriva- tives on r from Appendix B hows tha su h analyt cal solution is out of r ach. We pro- ceed as follows. We solve the system of he first order conditions (A.8)-(A.10) for Firms 1 and 2 located R* awa fro the rigin, while varying Fir 3’s location, r, around R*, numerically. The results are: Derivatives needed are in (B.7), (B.11), (B.17), and (B.18). We get: ( ) 32 12 6 1 2 1 9 1 2 3 34 1 23 222 R Rpp RRR D p −+′−′ ⋅−−−= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 8121.0 2 −≈ ′ p D . Due to the symmetry it must also be ( ) 8 21.0 1 −≈ ′ p D Claim 3: 0 2 2 2 < dr pd a d 0 2 1 2 < dr pd . To prove this claim analytically we would have to solve the system (A.8)-(A.10) without asserting r = R to get 1 p′ , 2 p′ , and 3 p ′ as functions of r. The dependence of relevant derivatives on r from Appendix B shows that such analytical solution is out of rea h. W proceed as follows. We solve the system of the first-order conditions (A.8)-(A.10) for Firms 1 and 2 located R * away from the origin, while varying Fi m 3’s location, r, around R * , numerically. The results are: 991583.0)005.0()005.0( * 2 * 1 =−==−= RrpRrp , 993237.0)()( * 2 * 1 ==== RrpRrp , and 994882.0)005.0()005.0( * 2 * 1 =+==+= RrpRrp . We are interested in two differences representing the first-order derivatives of p 2 with respect to r at R * 0.0025 and R * + 25: 3* 2 * 2 10654.1)005.0()( − ×=−=−= RrpRrp 3* 2 * 2 10645.1)()005.0( − ×==−+= RrpRrp . We see that in vicinity of R * the first-order derivative of rivals’ prices with respect to r is decreasing in r, hence, the second order derivative must be negative. This completes the proof of Claim 3. One can verify that 1 p′ and 2 p′ presented in Lemma 6 (c) are positive when r=R * , while Lemma 6(b) says that 13 pD ∂∂ and 23 pD ∂∂ are also positive. Using the three claims above, we see that every single summand in the big parentheses in the second row of (A.11) is negative. Hence, 0 2 3 2 < Π dr d at r = R * . We have found a local maximum of Firm 3’s profits with respect to r. Q.E.D. Proof of Lemma 8: We are going to show that the expression in the parentheses in (10) is zero. First, it is obvious that 0 3 =∂∂ φD , since the same number of customers won by a move along polar directi n fro one neighbor is lost to the other one. Second, we cl im φφ ddpddp 21 −= . Clearly, the neighbor w o becomes cl ser by such a move becomes more aggressive and reduces its price and vice versa, the neighbor that is now farther away becomes less aggressive and raises its price. The magnitude of the two derivatives We are interested in two differences representing the first-order derivatives of p2 with respect to r at R*−0.0025 and R*+0.0025: Derivatives needed are in (B.7), (B.11), (B.17), and (B.18). We get: ( ) 32 12 6 1 2 1 9 1 2 3 34 1 23 222 R Rpp RRR D p −+′−′ ⋅−−−= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 8121.0 2 −≈ ′ p D . Due to the symmetry it must also be ( ) 8121.0 1 −≈ ′ p D Claim 3: 0 2 2 2 < dr pd and 0 2 1 2 < dr pd . To prove this claim analytically we would have to solve the system (A.8)-(A.10) w thout asserting r = R to get 1 p′ , 2 p′ , and p ′ as functions of r. The dependence of relevant derivatives on r from Appendix B shows that such a alytical solution is out of reach. We proceed as follows. We solve the system of the first-order conditions (A 8)-(A.10) for Fi ms 1 and 2 located R * away from the origin, while varying Firm 3’s location, r, around R * , numerically. The results are: 991583.0)005.0()005.0( * 2 * 1 =−==−= RrpRrp , 993237.0)()( * 2 * 1 ==== RrpRrp , and 994882.0)005.0()0 5.0( * 2 * 1 =+==+= RrpRrp . We are interested in two differences representing the first-order derivatives of p 2 with respect to r at R * −0.0025 and R * +0.0025: 3* 2 * 2 10654.1)005.0()( − ×=−=−= RrpRrp 3* 2 * 2 10645.1)()005.0( − ×==−+= RrpRrp . We see that in vicinity of R * the first-o der d rivative of rivals’ prices with r spect to is decreasing i r, h nce, t e second order d rivative must be neg tive. This compl tes the proof of Claim 3. One can verify that 1 p′ and 2 p′ pr sented in Lemma 6 (c) are positive when r=R * , while Lemma 6(b) says that 13 pD ∂∂ and 23 pD ∂∂ are also p sitive. Us ng the three claims above, we see that every single summand in the big parentheses in the second row of (A.11) is negative. Hence, 0 2 3 2 < Π dr d at r = R * . We have found a local maximum of Firm 3’s profits with respect to r. Q.E.D. Proof of Lemma 8: We are going to show that the expression in the par ntheses in (10) is zero. First, it is obvious that 0 3 =∂∂ φD , since the same number of customers won by a move along polar direction from one n ighbor i lost to the other one. Second, we claim φφ ddpddp 21 −= . Clearly, the neighbor who becomes closer by such a move becomes more aggressive and reduces its price and vic versa, the neig bor that is now farther away becomes less aggressive and raises its price. The magnitude of the two derivatives e see that in vicin t of R* the first-ord r derivativ of rivals’ prices with respect to r is decreasing in r, hence, the second order derivative must be negative. This completes the proof of Claim 3. One can verify that p1ʹ and p2ʹ presented in Lemm 6 (c) are p si iv when r=R *, whil Lemma 6(b) says that Derivatives needed are in (B.7), (B.11), (B.17), and (B.18). We get: ( ) 32 12 6 1 2 1 9 1 2 3 34 1 23 222 R Rpp RRR D p −+′−′ ⋅−−−= ′ . Eva uating this at R = R * and using result from Lemma 6(c) yields ( ) 8121.0 2 −≈ ′ p D . Due to the symmetry it must also be ( ) 8121.0 1 −≈ ′ p D Claim 3: 0 2 2 2 < dr pd and 0 2 1 2 < dr pd . To prov this claim analytically we would have to solve the system (A.8)-(A.10) without asserting r = R to get 1 p′ , 2 p′ , and 3 p ′ as functions of r. The dependence of relevant derivatives on r from Appendix B hows that such a alytical solution is out of reach. We proceed as follows. We solve the system of the first-order conditio s (A.8)-(A.10) for Firms 1 and 2 located R * away from the origin, while varying Firm 3’s location, r, around R * , numerically. The results are: 991583.0)005.0()005.0( * 2 * 1 =−==−= RrpRrp , 993237.0)()( * 2 * 1 ==== RrpRrp , and 994882.0)0 5.0()005.0( * 2 * 1 =+=+= RrpRrp . We are intereste in two differences representing the first-order derivatives of p 2 with respect to r at R * −0.0025 and R * +0.0025: 3* 2 * 2 10654.1)005.0()( − ×=−=−= RrpRrp 3* 2 * 2 10645.1)()005.0( − ×==−+= RrpRrp . We see that in vicinity f R * the first-order derivative of rival ’ prices with respect t r is decrea ing in r, hence, the second order derivative must b nega ive. This completes the proof of Claim 3. On can verify that 1 p′ and 2 p′ r sente in Lem 6 (c) are positive hen r=R * , while Le ma 6(b) says t 13 pD ∂∂ and 23 pD ∂∂ are also positive. Using the three claims above, we see that every single summand in the big parentheses in the second row of (A.11) is negative. Hence, 0 2 3 2 < Π dr d at r = R * . We have found a local maximum of Firm 3’s profits with respect to r. Q.E.D. Proof of Lemma 8: We are going to s ow that the expression in the parentheses in (10) is zero. First, it is obvious that 0 3 =∂∂ φD , since the same number of customers won by a move al g polar d rection from one neighbor is lost to th oth one. Second, we claim φφ ddpdp 21 −= . Clearly, the neighbor who ecomes closer by such a move more a gre sive and reduce its price and vice versa, t neighbor that i now farther away becomes less aggressive and raises its price. The magnitude of the two derivatives a 2 l 2 1 , n 0= s e t p t er d b s s ls positive. Using the three claims above, we see that every single summand in the big parentheses in the second row of (A.11) is negative. hence, Derivatives needed are in (B.7), (B.11), (B.17), and (B.18). We get: ( ) 32 12 6 1 2 1 9 1 2 3 34 23 222 R Rpp RRR D p −+′−′ ⋅−−−= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 8121.0 2 −≈ ′ p . Due to the symmetry it must also be ( ) 8121.0 1 −≈ ′ p D Claim 3: 0 2 2 2 < dr pd nd 0 2 1 2 < dr pd . To prove this claim analytically we would have to solve the syste (A.8)-(A.10) without asserting r = R to get 1 p′ , 2 p′ , and 3 p ′ as functio s of r. Th dependence of relevant derivatives on r from Appendix B shows that s ch analytical solution is out of re ch. We proceed as follows. We solv the system f the first-order conditions (A.8)-(A.10) for Firms 1 and 2 located R * away from the ori in, while varying Firm 3’s location, r, around R * , numerically. The results are: 991583.0)005.0()005.0( * 2 * 1 =−==−= Rrpr , 993237.0)()( * 2 * 1 ==== RrpRrp , and 994882.0)005.0()05.0( * 2 * 1 =+==+= RrpRrp . We are interested in two differenc s representing the first- rder derivatives of p 2 with respect to r at R * − .0025 and R * +0.0025: 3* 2 * 2 10654.1)005.()( − ×=−=−= RrpRrp 3* 2 * 2 10645.1)()005.0( − ×==−+= RrpRrp . We see that i v ci ity of R * the first-order d rivativ of rivals’ prices with respect to r is de reasing i r, hence, th second ord r deriv t must be nega ive. This completes the proof of Claim 3. O e can verify that 1 p′ and 2 p′ presented in Lem a 6 (c) a e positive when r=R * , while Lemma 6(b) says that 13 pD ∂∂ and 23 pD ∂∂ are also positive. Using the three claims above, we se that every single summa d in the big parentheses in the second row of (A.11) is negative. Hence, 0 2 3 2 < Π dr d at r = R * . We have found a local maximum of Firm 3’s profits with respect to r. Q.E.D. Proof of Lemma 8: We re going to show that expression in the parentheses in (10) is zer . First, it is obviou that 0 3 =∂∂ φD , since t same number of customers won by a move along polar dir ction from one neighbor is lost to the other one. Secon , we cl im φφ ddpddp 21 −= . Clearly, th neighbor who becomes closer by such a move b comes more aggressive and re uces its price and vice vers , the neighbor that is now farther away becomes less aggre sive and raises its price. T magnitude of the two derivatives at r = R*. We have found a local maximum of Firm 3’s profits with respect to r. Q.E.D. A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 343 proof of Lemma 8: We are going to show that the expression in the parentheses in (10) is zero. first, it is obvious that Derivatives needed are in (B.7), (B.11), (B.17), and (B.18). We get: ( ) 32 12 6 1 2 1 9 1 2 3 34 1 23 222 R Rpp RRR D p −+′−′ ⋅−−−= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 8121.0 2 −≈ ′ p D . Due to the symmetry it must also be ( ) 8121.0 1 −≈ ′ p D Claim 3: 0 2 2 2 < dr pd and 0 2 1 2 < dr pd . To prove this claim analytically we would have to solve the system (A.8)-(A.10) without asserting r = R to get 1 p′ , 2 p′ , and 3 p ′ as functions of r. The dependence of relevant derivatives on r from Appendix B shows that such analytical solution is out of reach. We proceed as follows. We solve the system of the first-order conditions (A.8)-(A.10) for Firms 1 and 2 located R * away from the origin, while varying Firm 3’s location, r, around R * , numerically. The results are: 991583.0)005.0()005.0( * 2 * 1 =−==−= RrpRrp , 993237.0)()( * 2 * 1 ==== RrpRrp , and 994882.0)005.0()005.0( * 2 * 1 =+==+= RrpRrp . We are interested in two differences representing the first-order derivatives of p 2 with respect to r at R * −0.0025 and R * +0.0025: 3* 2 * 2 10654.1)005.0()( − ×=−=−= RrpRrp 3* 2 * 2 10645.1)()005.0( − ×==−+= RrpRrp . We see that in vicinity of R * the first-order derivative of rivals’ prices with respect to r is decreasing in r, hence, the second order derivative must be negative. This completes the proof of Claim 3. One can verify that 1 p′ and 2 p′ presented in Lemma 6 (c) are positive when r=R * , while Lemma 6(b) says that 13 pD ∂∂ and 23 pD ∂∂ are also positive. Using the three claims above, we see that every single summand in the big parentheses in the second row of (A.11) is negative. Hence, 0 2 3 2 < Π dr d at r = R * . We have found a local maximum of Firm 3’s profits with respect to r. Q.E.D. Proof of Lemma 8: We are going to show at the express on in the parentheses in (10) is zero. First, it is obvio 0 3 =∂∂ φD , since the same number of customers won by a move along polar direction from one neighbor is lost to the other one. Second, we claim φφ ddpddp 21 −= . Clearly, the neighbor who becomes closer by such a move becomes more aggressive and reduces its price and vice versa, the neighbor that is now farther away becomes less aggressive and raises its price. The magnitude of the two derivatives , i sa e number of customers won by a move along polar direction from one neighbor is lost to the other one. Second, we claim Derivatives needed are in (B.7), (B.11), (B.17), and (B.18). We get: ( ) 32 12 6 1 2 1 9 1 2 3 34 1 23 222 R Rpp RRR D p −+′−′ ⋅−−−= ′ . Evaluating this at R = R * and using result from Lemma 6(c) yields ( ) 8121.0 2 −≈ ′ p D . Due to the symmetry it must also be ( ) 8121.0 1 −≈ ′ p D Claim 3: 0 2 2 2 < dr pd and 0 2 1 2 < dr pd . To prove this claim analytically we would have t solve the syst m (A.8)-(A.10) without asserting r = R to get 1 p′ , 2 p′ , and 3 p ′ as functions of r. The dependence of relevant derivatives on r from App ndix B shows that such analytical solution is out of reach. We proceed as follows. We solve the system of t e first-order conditions (A.8)-(A.10) for Firms 1 and 2 located R * away from the origin, while varying Firm 3’s location, r, around R * , numerically. The results are: 991583.0)005.0()005.0( * 2 * 1 =−==−= RrpRrp , 993237.0)()( * 2 * 1 ==== RrpRrp , and 994882.0)005.0()005.0( * 2 * 1 =+==+= RrpRrp . We ar interested in two differences representing the first-order derivatives of p 2 with respect to r at R * −0.0025 and R * +0.0025: 3* 2 * 2 10654.1)005.0()( − ×=−=−= RrpRrp 3* 2 * 2 10645.1)()005.0( − ×==−+= RrpRrp . We se that in vicinity of R * the first-order derivative of rivals’ prices with respect to r is decreasing in r, hence, the second order derivative must be negative. This completes the proof of Claim 3. One can verify that 1 p′ and 2 p′ presented in Lemma 6 (c) are positive when r=R * , while Lemma 6(b) says that 13 pD ∂∂ and 23 pD ∂∂ are also positive. Using the three claims above, we see that every single summand in the big parentheses in the second row of (A.11) is negative. Hence, 0 2 3 2 < Π dr d at r = R * . We have found a local maximum of Firm 3’s profits with respect to r. Q.E.D. Pro f of Lemma 8: We are going to show that th expressio in the parenthes s in (10) is zero. First, it is obvious that 0 3 =∂∂ φD , since t e same number of customers won by a move along polar direction from one neighbor is lost to the other on . , φφ ddpddp 21 −= . Clearly, the neighbor who becomes closer by such a move becomes more aggres ive and reduces its price and vice versa, the neighbor that is now farther away becomes less aggressive and raises its price. The magnitude of the two derivatives . learl , t es closer by such a move b comes m aggressive and reduces its price and vice versa, the neighbor that is no farther away becom less ggres ive and raises its price. The magnitu of the two derivatives is the same due to symmetry of the proposed configuration. The result fol- lows. Q.E.D. ECONOMIC AND BUSINESS REVIEW | VOL. 14 | No. 4 | 2012344 appEndIx B: dErIvaTIvES We collect all the derivatives needed in preceding Appendix in subsections B.1 and B.2 for duopoly and three-firm oligopoly markets, respectively. B.1 Two firms In equilibrium r = R, φ =π, and a = π/2. from (A.2): is the same due to symmetry of the proposed configuration. The result follows. Q.E.D. Appendix B: Derivatives We collect all th derivatives needed in preceding Appendix in subsections B.1 and B.2 for duopoly and three-firm oligopoly markets, respectively. B.1 Two firms In equilibrium r = R, φ =π, and α = π/2. F . : RRr pp 4 1 sin)(2 1 21 −= ⋅+ −=−= α αα and R zz pp 2 1 21 −=−= (B.1) From (B.1): 2222 8 1 )(2 1 sin)(4 sin2¸ 21 RRrRr pp = + = ⋅+ =′−=′ α α αα and 2 4 1 21 R zz pp =′−=′ (B.2) From (A.1): R Rpp Rr rpp 4 2 sin)(2 2 1212 +′−′ = ⋅+ +′−′ =′ α α and R Rpp z 2 2 12 +′−′ =′ (B.3) B.2 Three firms From (A.4): ,0,0, 32 1 321 ==−=−= rppp xx R xx (B.4) From (A.4) and (B.4): 0, 32 321 12 =′=′=′ ′−′ =′ ppp xxx R pp x , (B.5) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 1 = + = − +− = φ φ , (B.6) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 2 = + = − −− = φ φ , (B.7) From (A.5): RRrRrR R y p 3 1 2 1 )cos2(32 32 3 −= + −= − = φ , (B.8) From (A.5): 3 2 2 2 )cos2(32 43 −= + −= − ⋅ = Rr r RrR rR y r φ , (B.9) From (B.9): RRr rRr y r 9 2 )2( 4)2(2 2 −= + −+ −=′ , (B.10) is the same due to symmetry of the proposed configuration. The result fol ows. Q.E.D. Appendix B: Derivatives e c l ect al the derivatives needed in preceding App ndix in subsections B.1 and B.2 for duopoly and three-firm oligopoly markets, respectively. B.1 Two firms In equilibrium r = R, φ =π, and α = π/2. From (A.2): RRr pp 4 1 sin)(2 1 21 −= ⋅+ −=−= α αα and R zz pp 2 1 21 −=−= (B.1) From (B.1): 2222 8 1 )(2 1 sin)(4 sin2¸ 21 RRrRr pp = + = ⋅+ =′−=′ α α αα and 2 4 1 21 R zz pp =′−=′ (B.2) From (A.1): R Rpp Rr rpp 4 2 sin)(2 2 1212 +′−′ = ⋅+ +′−′ =′ α α and R Rpp z 2 2 12 +′−′ =′ (B.3) B.2 Three firms From (A.4): ,0,0, 32 1 321 ==−=−= rppp xx R xx (B.4) From (A.4) and (B.4): 0, 32 321 12 =′=′=′ ′−′ =′ ppp xxx R pp x , (B.5) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 1 = + = − +− = φ φ , (B.6) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 2 = + = − −− = φ φ , (B.7) From (A.5): RRrRrR R y p 3 1 2 1 )cos2(32 32 3 −= + −= − = φ , (B.8) From (A.5): 3 2 2 2 )cos2(32 43 −= + −= − ⋅ = Rr r RrR rR y r φ , (B.9) From (B.9): RRr rRr y r 9 2 )2( 4)2(2 2 −= + −+ −=′ , (B.10) (B.1) from (B.1): is the same due to symmetry of the proposed configuration. The result follows. Q.E.D. Appendix B: Derivatives We collect all the derivatives needed in preceding Appendix in subsections B.1 and B.2 for duopoly and three-firm oligopoly markets, respectively. B.1 Two firms In equilibrium r = R, φ =π, and α = π/2. From (A.2): RRr pp 4 1 sin)(2 1 21 −= ⋅+ −=−= α αα and R zz pp 2 1 21 −=−= (B.1) F (B.1 : 2222 8 1 )(2 1 sin)(4 sin2¸ 21 RRrRr pp = + = ⋅+ =′−=′ α α αα and 2 4 1 21 R zz pp =′−=′ (B.2) From (A.1): R Rpp Rr rpp 4 2 sin)(2 2 1212 +′−′ = ⋅+ +′−′ =′ α α and R Rpp z 2 2 12 +′−′ =′ (B.3) B.2 Three firms From (A.4): ,0,0, 32 1 321 ==−=−= rppp xx R xx (B.4) From (A.4) and (B.4): 0, 32 321 12 =′=′=′ ′−′ =′ ppp xxx R pp x , (B.5) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 1 = + = − +− = φ φ , (B.6) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 2 = + = − −− = φ φ , (B.7) From (A.5): RRrRrR R y p 3 1 2 1 )cos2(32 32 3 −= + −= − = φ , (B.8) From (A.5): 3 2 2 2 )cos2(32 43 −= + −= − ⋅ = Rr r RrR rR y r φ , (B.9) From (B.9): RRr rRr y r 9 2 )2( 4)2(2 2 −= + −+ −=′ , (B.10) and is the same due to sym etry of the proposed configuration. The result follows. Q.E.D. Ap endix B: Derivatives We collect all the derivatives needed in preceding Ap endix in subsections B.1 and B.2 for duopoly and three-firm oligopoly markets, respectively. B.1 Two firms In equilibrium r = R, φ =π, and α = π/2. From (A.2): RRr pp 4 1 sin)(2 1 21 −= ⋅+ −=−= α αα and R zz pp 2 1 21 −=−= (B.1) From (B.1): 2222 8 1 )(2 1 sin)(4 sin2¸ 21 RRrRr pp = + = ⋅+ =′−=′ α α αα and 2 4 1 21 R zz pp =′−=′ (B.2) From (A.1): R Rpp Rr rpp 4 2 sin)(2 2 1212 +′−′ = ⋅+ +′−′ =′ α α and R Rpp z 2 2 12 +′−′ =′ (B.3) B.2 Three firms From (A.4): ,0,0, 32 1 321 ==−=−= rppp xx R xx (B.4) From (A.4) and (B.4): 0, 32 321 12 =′=′=′ ′−′ =′ ppp xxx R pp x , (B.5) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 1 = + = − +− = φ φ , (B.6) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 2 = + = − −− = φ φ , (B.7) From (A.5): RRrRrR R y p 3 1 2 1 )cos2(32 32 3 −= + −= − = φ , (B.8) From (A.5): 3 2 2 2 )cos2(32 43 −= + −= − ⋅ = Rr r RrR rR y r φ , (B.9) From (B.9): RRr rRr y r 9 2 )2( 4)2(2 2 −= + −+ −=′ , (B.10) (B.2) from (A.1): is the same due to symmetry of the proposed configuration. The result follows. Q.E.D. Appendix B: Derivatives We collect all the derivatives need d in preceding Appendix in subsections B.1 and B.2 for duopoly and three-firm oligopoly markets, respectively. fi s In equilibrium r = R, φ =π, and α = π/2. From (A.2): RRr pp 4 1 sin)(2 1 21 −= ⋅+ −=−= α αα and R zz pp 2 1 21 −=−= (B.1) From (B.1): 2222 8 1 )(2 1 sin)(4 sin2¸ 21 RRrRr pp = + = ⋅+ =′−=′ α α αα and 2 4 1 21 R zz pp =′−=′ (B.2) F ( .1): R Rpp Rr rpp 4 2 sin)(2 2 1212 +′−′ = ⋅+ +′−′ =′ α α and R Rpp z 2 2 12 +′−′ =′ (B.3) B.2 Three firms From (A.4): ,0,0, 32 1 321 ==−=−= rppp xx R xx (B.4) From (A.4) and (B.4): 0, 3 321 12 =′=′=′ ′−′ =′ ppp xxx R pp x , (B.5) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )co2(32 sin23 1 = + = − +− = φ φ , (B.6) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )co2(32 sin23 2 = + = − −− = φ φ , (B.7) From (A.5): RRrRrR R y p 3 1 2 1 )cos(32 32 3 −= + −= − = φ , (B.8) From (A.5): 3 2 2 2 )cos2(32 43 −= + −= − ⋅ = Rr r RrR rR y r φ , (B.9) From (B.9): RRr rRr y r 9 2 )2( 4)2(2 2 −= + −+ −=′ , (B.10) a is the sa e due to sy etry of the proposed configuration. The result fol o s. .E. . ppendix B: erivatives e col ect al the derivatives needed in preceding ppendix in subsections B.1 and B.2 for duopoly and three-fir oligopoly arkets, respectively. B.1 Two firms In equilibriu r = R, φ =π, and α = π/2. Fro ( .2): RRr pp 4 1 sin)(2 1 21 −= ⋅+ −=−= α αα and R zz pp 2 1 21 −=−= (B.1) Fro (B.1): 2222 8 1 )(2 1 sin)(4 sin2¸ 21 RRrRr pp = + = ⋅+ =′−=′ α α αα and 2 4 1 21 R zz pp =′−=′ (B.2) From (A.1): R Rpp Rr rpp 4 2 sin)(2 2 1212 +′−′ = ⋅+ +′−′ =′ α α nd R Rpp z 2 2 12 +′−′ =′ (B.3) B.2 Three fir s Fro ( .4): ,0,0, 32 1 321 ==−=−= rppp xx R xx (B.4) Fro ( .4) and (B.4): 0, 32 321 12 =′=′=′ ′−′ =′ ppp xxx R pp x , (B.5) Fro ( .5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 1 = + = − +− = φ φ , (B.6) Fro ( .5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 2 = + = − −− = φ φ , (B.7) Fro ( .5): RRrRrR R y p 3 1 2 1 )cos2(32 32 3 −= + −= − = φ , (B.8) Fro ( .5): 3 2 2 2 )cos2(32 43 −= + −= − ⋅ = Rr r RrR rR y r φ , (B.9) Fro (B.9): Rr rRr y r 9 2 )2( 4)2(2 2 −= + −+ −=′ , (B.10) (B.3) Th fi s f ( .4): is the sa e due to sy etry of the proposed configuration. he result follo s. . . . ppendix : erivatives e collect all the derivatives needed in preceding ppendix in subsections .1 and .2 for duopoly and three-fir oligopoly ark ts, respectively. . fir s In equilibriu r , , and /2. ro ( .2): r pp si)( 21 ⋅ and zz pp 2 1 21 ( .1) ro ( .1): 2222 8 1 )(2 1 sin)(4 sin2¸ 21 rr pp ⋅ ′′ and 2 4 1 21 zz pp ′′ ( .2) ro ( .1): pp r rpp 4 2 sin)(2 2 1212 ′′ ⋅ ′′ ′ and z 12 ′′ ′ ( .3) .2 hree fir s ro ( .4): ,,, 321 rppp xxxx ( .4) ro ( .4) and ( .4): 0, 3 321 12 ′′′ ′′ ′ ppp xxx pp x , ( .5) ro ( .5): rr r y p 6 1 )2(2 1 )co2(32 sin23 1 , ( .6) ro ( .5): rr r y p 6 1 )2(2 1 )co2(32 sin23 2 , ( .7) ro ( .5): rr y p 3 1 2 1 )cos2(32 32 3 , ( .8) ro ( .5): 3 2 2 2 )cos2(32 43 ⋅ r r r r y r , ( .9) ro ( .9): r rr y r 9 2 )2( 4)2(2 2 ′ , ( .10) (B.4) from (A.4) and (B.4): is the same due to symmetry of the proposed configuration. The result follows. Q.E.D. Appendix B: Derivatives We collect all the derivatives needed in preceding Appendix in subsections B.1 and B.2 for duopoly and three-firm ol gopoly markets, resp ctively. B.1 Two firms In equilibrium r = R, φ =π, and α = π/2. From (A.2): RRr pp 4 1 sin)(2 1 21 −= ⋅+ −=−= α αα and R zz pp 2 1 21 −=−= (B.1) From (B.1): 2222 8 1 )(2 1 sin)(4 sin2¸ 21 RRrRr pp = + = ⋅+ =′−=′ α α αα and 2 4 1 21 R zz pp =′−=′ (B.2) From (A.1): R Rpp Rr rpp 4 2 sin)( 2 1212 +′−′ = ⋅+ +′−′ =′ α α and R Rpp z 2 2 12 +′−′ =′ (B.3) B.2 Three firms From (A.4): ,0,0, 32 1 321 ==−=−= rppp xx R xx (B.4) F ( . d (B.4): 0, 32 321 12 =′=′=′ ′−′ =′ ppp xxx R pp x , (B.5) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 1 = + = − +− = φ φ , (B.6) From (A.5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 2 = + = − −− = φ φ , (B.7) From (A.5): RRrRrR R y p 3 1 2 1 )cos2(32 32 3 −= + −= − = φ , (B.8) From (A.5): 3 22 )cos2(32 43 −= + −= − ⋅ = Rr r RrR rR y r φ , (B.9) From (B.9): RRr rRr y r 9 2 )2( 4)2(2 2 −= + −+ −=′ , (B.10) , (B.5) fro (A.5): is the sa e due to sy etry of the proposed configuration. The result follo s. .E. . ppendix : erivatives e collect all the derivatives needed in preceding ppendix in subsections .1 and .2 for duopoly and three-fir oligopoly arkets, resp ctively. .1 T o fir s In equilibriu r = R, φ π, and α = π/2. Fro ( .2): RRr pp 4 1 sin)(2 1 21 −= ⋅+ −=−= α αα and R zz pp 2 1 21 −=−= ( .1) Fro ( .1): 2222 8 1 )(2 1 sin)(4 sin2¸ 21 RRrRr pp = + = ⋅+ =′−=′ α α αα and 2 4 1 21 R zz pp =′−=′ ( .2) Fro ( .1): R Rpp Rr rpp 4 2 sin)(2 2 1212 +′−′ = ⋅+ +′−′ =′ α α and R Rpp z 2 2 12 +′−′ =′ ( .3) .2 Three fir s Fro ( .4): ,0,0, 32 1 321 ==−=−= rppp xx R xx ( .4) Fr ( . ) a d ( .4): 0, 32 321 12 =′=′=′ ′−′ =′ ppp xxx R pp x , ( .5) F ( . : RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 1 = + = − +− = φ φ , ( .6) Fro ( .5): RRrRrR rR y p 6 1 )2(2 1 )cos2(32 sin23 2 = + = − −− = φ φ , ( .7) Fro ( .5): RRrRrR R y p 3 1 2 1 )cos2(32 32 3 −= + −= − = φ , ( .8) Fro ( .5): 3 2 2 2 )cos2(32 43 −= + −= − ⋅ = Rr r RrR rR y r φ , ( .9) Fro ( .9): RRr rRr y r 9 2 )2( 4)2(2 2 −= + −+ −=′ , ( .10) , (B.6) fro ( .5): is t e sa e e t s etr f t e r se c fi rati . e res lt f ll s. . . . e i : eri ati es e c llect all t eri ati es ee e i rece i e i i s secti s . a . f r l a t ree-fir li l ar ets, res ecti el . . fir s I e ili ri r , , a / . r ( . ): r pp si)( 21 ⋅ a zz pp 21 ( . ) r ( . ): 2222 )(si)( si¸ 21 rr pp ⋅ ′′ a 221 zz pp ′′ ( . ) r ( . ): r r si)( 1212 ′′ ⋅ ′′ ′ a z 12 ′′ ′ ( . ) . ree fir s r ( . ): ,,, 321 rppp xxxx ( . ) ro ( .4) an ( . ): , 321 12 ′′′ ′′ ′ ppp xxxx , ( . ) ( .5): rr r y p )()c s( si 1 ( . ) ( . ): rr r y p )()c s( si 2 , ( . ) r ( . ): rr y p )c s( 3 , ( . ) r ( . ): )c s( ⋅ r r r r y r , ( . ) r ( . ): r rr y r )( )( 2 ′ , ( . ) (B.7) f . : h du o y y o h p opo d on gu on h u o o pp nd x v v o h d v v n d d n p d ng pp nd x n ub on 1 nd 2 o duopo y nd h o gopo y k p v y 1 o n qu b u φ nd 2 o 2 4 1 n2 1 nd 2 1 1 o 1 8 1 2 1 n4 n2 nd 4 1 2 o 1 pppp 4 2 n2 2 nd pp 2 2 3 2 h o 4 00 32 1 4 d 4 0 32 pp 5 6 1 22 1 o232 n23 φ φ , 6 6 1 22 1 o232 n23 φ φ 7 o 5 3 1 2 1 o232 32 φ 8 o 5 3 2 2 2 o232 43 φ 9 o 9 9 2 2 422 10 , (B.8) f . : ( (B.9) fr (B.9): i t t tr f t r fi r ti . r lt f ll . . . . i : ri ti ll t ll t r ti i r i i i ti . . f r l t r -fir li l r t , r ti l . . fi I ili ri , , / . r ( . ): pp i)( 21 ⋅ pp 21 ( . ) ro ( . ): 2222 )(i)( i¸ 21 pp ⋅ ′′ 221 pp ′′ ( . ) r ( . ): i)( 1212 ′′ ⋅ ′′ ′ 12 ′′ ′ ( . ) . fi r ( . ): ,,, 321 rppp ( . ) r ( . ) ( . ): , 321 12 ′′′ ′′ ′ ppp , ( . ) r ( . ): p )()( i 1 , ( . ) r ( . ): p )()( i 2 , ( . ) r ( . ): p )( 3 , ( . ) r ( . ): )( ⋅ r , ( . ) r ( . ): r )( )( 2 ′ , ( . ) (B.10) from (B.6) and (B.7): F ( . : 222 9 1 )2( 1 )cos2(32 cos23 21 RRrRrR R yy pp −= + − = − ⋅ =′=′ φ φ , (B.11) From (B.8): 222 9 2 )2( 2 )cos2( cos2 3 RRrRr y p = + = − −=′ φ φ , (B.12) From (A.5): R Rppp Rr rppp RrR rpppR y 6 42 )cos2(2 42 )cos2(32 )42(3 213213213 +′−′−′ −= − +′−′−′ = − +′−′−′ =′ φφ , (B.13) From (A.6): 32 1 )(3 1 )2( 1 sin)cos2(cos)3sin2( 1 2 3 2 3 32 RRrRrR RrRr pp −= + −= ++ −= −−− −=−= βφβφ ββ , (B.14) From (A.6): 32 12 )2( 12 sin)cos2(cos)3sin2( coscos22 2 3 2 3 R R RrR r RrRr r r − = ++ − = −−− − = βφβφ βφ β , (B.15) From (B.15): 34 12 12 3)12(232 )(3 12 2 R R R RR Rr r r + = −−⋅ = ′         + − =′β , (B.16) From (B.14): ( ) 34 1 )(3 3 sin)cos2(cos)3sin2( cos)cos2(sincos2sin3cossin2 22 232 RRr RrRr RrR pp = + = −−− ′⋅−−−′⋅+ =′=′ βφβφ ββφβφβββφ ββ , (B.17) From (A.6): 32 12 )(3 12 sin)cos2(cos)3sin2( coscos22 232323 R Rpp Rr rpp RrRr rpp −+′−′ = + −+′−′ = −−− −+′−′ =′ βφβφ βφ β . (B.18) , (B.11) from (B.8): From (B.6) and (B.7): 222 9 1 )2( 1 )cos2(32 cos23 21 RRrRrR R yy pp −= + − = − ⋅ =′=′ φ φ , (B.11) F ( . : 222 9 2 )2( 2 )cos2( cos2 3 RRrRr y p = + = − −=′ φ φ , (B.12) From (A.5): R Rppp Rr rppp RrR rpppR y 6 42 )cos2(2 42 )cos2(32 )42(3 213213213 +′−′−′ −= − +′−′−′ = − +′−′−′ =′ φφ , (B.13) From (A.6): 32 1 )(3 1 )2( 1 sin)cos2(cos)3sin2( 1 2 3 2 3 32 RRrRrR RrRr pp −= + −= ++ −= −−− −=−= βφβφ ββ , (B.14) From (A.6): 32 12 )2( 12 sin)cos2(cos)3sin2( coscos22 2 3 2 3 R R RrR r RrRr r r − = ++ − = −−− − = βφβφ βφ β , (B.15) From (B.15): 34 12 12 3)12(232 )(3 12 2 R R R RR Rr r r + = −−⋅ = ′         + − =′β , (B.16) From (B.14): ( ) 34 1 )(3 3 sin)cos2(cos)3sin2( cos)cos2(sincos2sin3cossin2 22 232 RRr RrRr RrR pp = + = −−− ′⋅−−−′⋅+ =′=′ βφβφ ββφβφβββφ ββ , (B.17) From (A.6): 32 12 )(3 12 sin)cos2(cos)3sin2( coscos22 232323 R Rpp Rr rpp RrRr rpp −+′−′ = + −+′−′ = −−− −+′−′ =′ βφβφ βφ β . (B.18) , (B.12) A. fELDIN | ThREE fIRMS ON A UNIT DISk MARkET: INTERMEDIATE pRODUCT DIffERENTIATION 345 from (A.5): From (B.6) and (B.7): 222 9 1 )2( 1 )cos2(32 cos23 21 RRrRrR R yy pp −= + − = − ⋅ =′=′ φ φ , (B.11) From (B.8): 222 9 2 )2( 2 )cos2( cos2 3 RRrRr y p = + = − −=′ φ φ , (B.12) Fr ( .5): R Rppp Rr rppp RrR rpppR y 6 42 )cos2(2 42 )cos2(32 )42(3 213213213 +′−′−′ −= − +′−′−′ = − +′−′−′ =′ φφ , (B.13) From (A.6): 32 1 )(3 1 )2( 1 sin)cos2(cos)3sin2( 1 2 3 2 3 32 RRrRrR RrRr pp −= + −= ++ −= −−− −=−= βφβφ ββ , (B.14) From (A.6): 32 12 )2( 12 sin)cos2(cos)3sin2( coscos22 2 3 2 3 R R RrR r RrRr r r − = ++ − = −−− − = βφβφ βφ β , (B.15) From (B.15): 34 12 12 3)12(232 )(3 12 2 R R R RR Rr r r + = −−⋅ = ′         + − =′β , (B.16) From (B.14): ( ) 34 1 )(3 3 sin)cos2(cos)3sin2( cos)cos2(sincos2sin3cossin2 22 232 RRr RrRr RrR pp = + = −−− ′⋅−−−′⋅+ =′=′ βφβφ ββφβφβββφ ββ , (B.17) From (A.6): 32 12 )(3 12 sin)cos2(cos)3sin2( coscos22 232323 R Rpp Rr rpp RrRr rpp −+′−′ = + −+′−′ = −−− −+′−′ =′ βφβφ βφ β . (B.18) , (B.13) f ( .6): . ) a d (B.7): 222 9 1 )2( 1 )cos2(32 cos23 21 RRrRrR R yy pp −= + − = − ⋅ =′=′ φ φ , (B.11) . ): 222 9 2 )2( 2 )cos2( cos2 3 RRrRr y p = + = − −=′ φ φ , (B.12) . : R Rppp Rr rppp RrR rpppR y 6 42 )cos2( 42 )cos2(32 )42(3 213213213 +′−′−′ −= − +′−′−′ = − +′−′−′ =′ φφ , (B.13) . ): 32 1 )(3 1 )2( 1 sin)cos2(cos)3sin2( 1 2 3 2 3 32 RRrRrR RrRr pp −= + −= ++ −= −−− −=−= βφβφ β , (B.14) . ): 32 12 )2( 12 sin)cos2(cos)3sin2( coscos22 2 3 2 3 R R RrR r RrRr r r − = ++ − = −−− − = βφβφ βφ , (B.15) . ): 34 12 12 3)12(232 )(3 12 2 R R R RR Rr r r + = −−⋅ = ′         + − =′β , (B.16) . ): ( ) 34 1 )(3 3 sin)cos2(cos)3sin2( cos)cos2(sincos2sin3cossin2 22 232 RRr RrRr RrR pp = + = −−− ′⋅−−−′⋅+ =′=′ βφβφ ββφβφβββφ ββ , (B.17) . ): 32 12 )(3 12 sin)cos2(cos)3i coscos22 232323 R Rpp Rr rpp RrR rpp −+′−′ = + −+′−′ = −−− −+′−′ βφβ βφ . (B.18) (B.14) fro (A.6): Fro (B.6) and ( .7): 222 9 1 )2( 1 )cos2(32 cos23 21 rr yy pp −= + − = − ⋅ =′=′ φ φ , ( .11) Fro (B.8): 222 9 2 )2( 2 )cos2( cos2 3 rr y p = + = − −=′ φ φ , ( .12) F . ppp r rppp r rppp y 6 42 )cos2(2 42 )cos2(32 )42(3 213213213 +′−′−′ −= − +′−′−′ = − +′−′−′ =′ φφ , ( .13) Fro (A.6): 32 1 )(3 1 )2( 1 sin)cos2(cos)3sin2( 1 2 3 2 3 32 rr rr pp −= + −= ++ −= −−− −=−= βφβφ ββ , ( .14) F ( .6): 32 12 )2( 12 sin)cos2(cos)3sin2( coscos22 2 3 2 3 r r rr r r φφ φ β , ( .15) Fro (B.15): 34 12 12 3)12(232 )(3 12 2 r r r ⋅ ′         ′ , ( .16) Fro (B.14): ( ) 34 1 )(3 3 sin)cos2(cos)3sin2( cos)cos2(sincos2sin3cossin2 22 232 r rr r pp = + = −−− ′⋅−−−′⋅+ =′=′ βφβφ ββφβφβββφ ββ , ( .17) Fro (A.6): 32 12 )(3 12 sin)cos2(cos)3sin2( coscos22 232323 pp r rpp rr rpp −+′−′ = + −+′−′ = −−− −+′−′ =′ βφβφ βφ β . ( .18) , (B.15) f m ( . : r m ( . ) ( . ): 222 )()( 21 pp ⋅ ′′ , ( . ) r m ( . ): 222 )()( 3 p ′ , ( . ) ro ( .5): )()( )( 213213213 ′′′′′′′′′ ′ , ( . ) r m ( . ): )()( i)()i( 2 3 2 3 32 pp , ( . ) rom (A. ): )(i)()i( 2 3 2 3 r − = ++ − = −−− − = ββ β ( . ) r ( . ): )( )( 2 r + = −−⋅ = ′ + − =′β , ( . ) r m ( . ): ( ) )( i)()i )(iii 22 232 pp ′⋅′⋅ ′′ , ( . ) r m ( . ): )(i)()i( 232323 ′′′′′′ ′ . ( . ) , (B.16) from (B.14): From (B.6) and (B.7): 222 9 1 )2( 1 )cos2(32 23 21 RRrRrR R yy pp −= + − = − ⋅ =′=′ φ , (B.11) From (B.8): 222 9 2 )2( 2 )cos2( cos2 3 RRrRr y p = + = − −=′ φ φ , (B.12) From (A.5): R Rppp Rr rppp RrR rpppR y 6 42 )cos2(2 42 )cos2(32 )42(3 21321321 +′−′−′ −= − +′−′−′ = − +′−′−′ =′ φφ , (B.13) From (A.6): 32 1 )(3 1 )2( 1 sin)cos2(cos)3sin2( 1 2 3 2 3 32 RRrRrR RrRr pp −= + −= ++ −= −−− −=−= βφβφ ββ , (B.14) From (A.6): 32 1 )2( 12 sin)cos2(cos)3sin2( coscos22 2 3 2 3 RRrR r RrRr r r − = ++ − = −−− − = βφβφ βφ β , (B.15) From (B.15): 34 12 12 3)12(232 )(3 12 2 RR RR Rr r r + = −−⋅ = ′         + − =′β , (B.16) From (B.14): ( ) 34 1 )(3 3 sin)cos2(cos)3sin2( cos)cos2(sincos2sin3cossin2 22 232 RRr RrRr RrR pp = + = −−− ′⋅−−−′⋅+ =′=′ βφβφ ββφβφβββφ ββ , (B.17) From (A.6): 32 12 )(3 12 sin)cos2(cos)3sin2( coscos22 232323 R Rpp Rr rpp RrRr rpp −+′−′ = + −′−′ = −−− −+′−′ =′ βφβφ βφ β . (B.18) , (B.17) from (A.6): From (B.6) and (B.7): 222 9 1 )2( 1 )cos2(32 cos23 21 RRrRrR R yy pp −= + − = − ⋅ =′=′ φ φ , (B.11) From (B.8): 222 9 2 )2( 2 )cos2( cos2 3 RRrRr y p = + = − −=′ φ φ , (B.12) From (A.5): R Rppp Rr rppp RrR rpppR y 6 42 )cos2(2 42 )cos2(32 )42(3 213213213 +′−′−′ −= − +′−′−′ = − +′−′−′ =′ φφ , (B.13) From (A.6): 32 1 )(3 1 )2( 1 sin)cos2(cos)3sin2( 1 2 3 2 3 32 RRrRrR RrRr pp −= + −= ++ −= −−− −=−= βφβφ ββ , (B.14) From (A.6): 32 12 )2( 12 sin)cos2(cos)3sin2( coscos22 2 3 2 3 R R RrR r RrRr r r − = ++ − = −−− − = βφβφ βφ β , (B.15) From (B.15): 34 12 12 3)12(232 )(3 12 2 R R R RR Rr r r + = −−⋅ = ′         + − =′β , (B.16) From (B.14): ( ) 34 1 )(3 3 sin)cos2(cos)3sin2( cos)cos2(sincos2sin3cossin2 22 232 RRr RrRr RrR pp = + = −−− ′⋅−−−′⋅+ =′=′ βφβφ ββφβφβββφ ββ , (B.17) Fr ( .6): 32 12 )(3 12 sin)cos2(cos)3sin2( coscos22 232323 R Rpp Rr rpp RrRr rpp −+′−′ = + −+′−′ = −−− −+′−′ =′ βφβφ βφ β . (B.18) . (B.18)