Image Anal Stereol 2011;30:143-151 Original Research Paper FRACTALS AND SELF-SIMILARITY IN ECONOMICS: THE CASE OF A STOCHASTIC TWO-SECTOR GROWTH MODEL Davide La Torre1, Simone Marsiglio2 and Fabio Privileggi2 1 Department of Economics, Business and Statistics, Universita degli studi di Milano, via Conservatorio 7, I-20122 Milano, Italy; 2Department of Public Policy and Public Choice Polis, Universita del Piemonte Orientale Amedeo Avogadro, viaCavour 84, I-15121 Alessandria, Italy e-mail: davide.latorre@unimi.it; simone.marsiglio@sp.unipmn.it; fabio.privileggi@sp.unipmn.it (Accepted August 30, 2011) ABSTRACT We study a stochastic, discrete-time, two-sector optimal growth model in which the production of the homogeneous consumption good uses a Cobb-Douglas technology, combining physical capital and an endogenously determined share of human capital. Education is intensive in human capital as in Lucas (1988), but the marginal returns of the share of human capital employed in education are decreasing, as suggested by Rebelo (1991). Assuming that the exogenous shocks are i.i.d. and affect both physical and human capital, we build specific configurations for the primitives of the model so that the optimal dynamics for the state variables can be converted, through an appropriate log-transformation, into an Iterated Function System converging to an invariant distribution supported on a generalized Sierpinski gasket. Keywords: fractals, iterated function system, self-similarity, Sierpinski gasket, stochastic growth. INTRODUCTION Mandelbrot (1982) in his seminal work presented the first description of self-similar sets, namely sets that may be expressed as unions of rescaled copies of themselves. He called these sets fractals, because their (fractional) Hausdorff-Besicovitch dimensions exceeded their (integer-valued) topological dimensions. The Cantor set, the von Koch snowflake curve and the Sierpinski gasket are some of the most famous examples of such sets. Hutchinson (1981) and, shortly thereafter, Barnsley and Demko (1985) and Barnsley (1989) showed how systems of contractive maps with associated probabilities, referred to as Iterated Function Systems (IFS), can be used to construct fractal, self-similar sets and measures supported on such sets. These sets and measures are attractive fixed points of fractal transform operators. After these pioneering papers, applications of IFS theory in several fields have been widely developed, eventually landing, at the end of the last century, also into Economics. As a matter of fact, economists are intrinsically reluctant to accept the idea that economic dynamics may generate fractals. A first breakthrough has been introduced by Boldrin and Montrucchio (1986), who showed that complicated (chaotic) optimal dynamics can occur in deterministic concave intertemporal optimization models when the discount factor is small enough. This result opened a new chapter in mainstream Economics, starting a huge literature aimed at studying complexity and chaos in almost all economic fields. Prominent, but by no means exhaustive,1 references are Montrucchio (1994), Nishimura and Yano (1995), Brock and Hommes (1997) and, more recently, Gardini et al. (2009), who exploited the IFS framework to construct a deterministic OLG-model converging to a fractal attractor. A decade later complex behavior started to be investigated in stochastic concave intertemporal optimization models as well. Montrucchio and Privileggi (1999) borrowed from the literature on fractal images generation (specifically, from the 'Collage Theorem' by Hutchinson, 1981; Barnsley, 1989; Vrscay, 1991) to show that standard stochastic concave optimal growth models may exhibit optimal trajectories which are random processes converging to singular invariant distributions supported on fractal sets regardless of the discount factor. Such economies have optimal dynamics defined by IFS with linear maps. Mitra et al. (2004) investigated a simple one-sector growth model with two random shocks whose optimal path is defined by a linear IFS which, for some values of parameters, converges to a singular distribution supported on a Cantor set. They also characterized singularity versus absolute continuity of the invariant probability in terms of (almost) all 1For a recent and quite comprehensive survey on complex dynamics arising in non-competitive economies see Bischi et al. (2010) and the references listed there. parameters' values. Mitra and Privileggi (2004, 2006) further generalized that model and eventually (2009) provided an estimate of the Lipschitz constant for the (nonlinear) maps of the IFS defining the optimal policy in a class of stochastic one-sector optimal growth models in the Brock and Mirman (1972) tradition. This result yields sufficient conditions for the model to converge to a singular distribution supported on a generalized Cantor set directly in terms of the parameters' values. In this paper we consider a neoclassic stochastic, discrete-time, two-sector growth model in which production of a unique homogeneous good depends on both physical and human capital through a Cobb-Douglas technology, while education requires only human capital, as suggested by Lucas (1988). However, we modify the Lucas (1988) framework by postulating that the marginal returns of the human capital employed in education are decreasing, thus embedding Rebelo (1991) assumption. Production in both sectors is multiplicatively affected by random i.i.d. shocks taking on a finite number of values. Our main contribution is to provide sufficient conditions on the parameters of the model - namely, on the exponents of the Cobb-Douglas production function and of the human capital production function, and on the values of random shocks - such that the IFS corresponding to the optimal policy function converges to a unique invariant distribution supported on a (generalized) Sierpinski gasket. Hence, this result can be seen as a further extension of the approach pursued by Mitra and Privileggi (2004,2006,2009) for the one-sector growth model to a multi-sector growth model under uncertainty. In Section "Iterated function systems" the main results from the IFS theory are briefly recalled. In Section "The model", the model is stated and the optimal dynamics are explicitly computed. Section "Conjugate linear IFSP" contains the central contribution of this paper: a linear IFS conjugate to the true optimal dynamics is constructed and sufficient conditions for its attractor to be a Sierpinski gasket supporting the unique invariant distribution of the economy are provided directly in terms of parameters of the model. Finally, in Section "Examples of Sierpinski gasket-like attractors" a few examples of economies converging to differently shaped Sierpinski gaskets are described, while Section "Conclusions" reports some concluding remarks. All proofs are gathered in the Appendix. ITERATED FUNCTION SYSTEMS Iterated Function Systems allow to formalize the notion of self-similarity or scale invariance of some mathematical object. Hutchinson (1981) and Barnsley and Demko (1985) showed how systems of contractive maps with associated probabilities can be used to construct self-similar sets and measures. In the IFS literature, these are called IFS with probabilities (IFSP) and are based on the action of a contractive Markov operator on the complete metric space of all Borel probability measures endowed with the Monge-Kantorovich metric. Applications of these methods can be found in image compression, approximation theory, signal analysis, denoising, and density estimation (see, e.g., Freiberg et a/., 2011; Kunze et al., 2007; lacus and La Torre, 2005a,b; La Torre et al, 2006; La Torre and Mendivil, 2008, 2009; La Torre and Vrscay, 2009; La Torre et al, 2009; Mendivil and Vrscay, 2002a,b). In what follows, let (X, d) be a complete metric space and w = {wi,..., wa?} be a family of injective contraction maps w; : X ^ X, to be referred to as an A/^-map IFS. Let Ci G (0,1) denote the contraction factor of Wi and define c = max/gji^ ^f} Q- Note that c G (0,1). Associated with the IFS mappings wi,...,wn there is a set-valued mapping w : Jf {X) ^ Jf (X) defined over the space Jf (X) of all non-empty compact sets in X as n w (1) i=i where w; (5) = {w; (x) : x G 5} is the image of S under Wi, for i = l,...,N. A set C X is said to be an invariant set of w if it is compact and it is invariant under Fq. 1, that is, it satisfies w{Sw) = S^. If in addition, the contractive mappings w; are assumed to be similitudes, Le., if we assume that there exist numbers q G (0,1) such that d{wi{x),wi{y))=cid{x,y), x,yeX, i=l,...,N, the invariant set S^, is said to be self-similar. In Jf^(X) it is possible to define the so-called Hausdorff distance dn between compact sets which reads as dH{A,B) = max{supinf(i(x,)'),supinf(i(x,)')} and it can be proved that {,Jf{X),dH) is a complete metric space (see Hutchinson, 1981). Theorem 1 (Hutchinson, 1981) w is a contraction mapping on the metric space (Jf (X), dn), that is dniw (A) ,w{B))< cdH {A,B) , VA,ß G JT (X) . We have the following corollary from the Banach fixed point theorem. Corollary 1 There exists a unique compact set A G Jf (X), such that w{A) = A, which is called the attractor of the IPS w. Moreover, for any S e Jf (X), dH{w''{S),A) ^Oasn^^. The latter property provides a construction method of approximating a fractal. The equation w (A) = A obviously implies that A is self-tiling, i.e., A is the union of (distorted) copies of itself. Let ^ {X) be the space of probability measures defined on the cr-algebra (X) of Borel measurable subsets of X and define for some a G X the set (X) = M G : / d{a,x)d^{x) < oo Note that for any ;U-integrable function u : X it holds that Notice that the definition of Jl\ (X) does not depend on the choice of a (if the integral is finite for a certain a G X then it is finite for all a G X). For v G Jl\ (X), we define the Monge-Kantorovich distance as follows V) = sup / - V) : / G ^ip^ (X) where J^ip^ is the set of all Lipschitz functions with Lipschitz constant equal to 1. It can be proved that {.M\{X),dM) is a complete metric space under the Monge-Kantorovich metric provided X be a separable complete metric space. Furthermore, if X is compact, then Jl (X) = Jl\ (X) and both are compact metric spaces under the Monge-Kantorovich distance (see Barnsley etal.,im%). Let p = {pi,p2,-- -jPn), 0 < Pi <1,1 0, ho > 0, zo G {qi,q2,1}, Vo G {r, 1} are given, Eo denotes expectation at time r = 0, 0 < j3 < 1 is the discount factor, kt and ht denote physical and human capital at time t,Q < a < \ and 0 < ^ < 1. The Bellman equation associated to the problem defined in Eq. 2 reads as: V {kt,ht,zt,rit) =max Ct,u, \nct ßEtV ikt+i,ht+i,zt+i,rit+i] (4) Proposition 1 1. The solution V (k, /z, z, 17) of the Bellman equation in Eq. 4 is given by: V{k,h,z,ri) = e + dk\nk+dh\nh + e^lnz+e^lnT7 , (5) where the constants Oh, Oz drj are defined as follows: dk = 07 = a l-ajS' 1 dh = en = \-a l-aß){\-ßy il-a)ß l-ocß' " {\-aß){\-ß) (l-aß) , (i-a)ß' ElnT7 (6) 2. The optimal policy rules for consumption and share of human capital allocated to physical production are respectively given by: ut = l-ß<^, :tkfh] l-a (7) (8) while physical and human capital follow the (optimal) dynamics defined by: ■,k»hl l-a (9) The proof is reported in the Appendix. An argument parallel to that described on pp. 273277 in Stokey and Lucas (1989) establishes that the function V{k,h,z,ri) defined in Eq. 5 is actually the value function of problem in Eq. 2. ^Consider, for example, the global financial crisis triggered in 2009: both the productive and education sector have been strongly damaged by the faUing prices in the stock market. ^ A similar approach has been pursued by Bethmann (2007) in a Lucas-Uzawa model of endogenous growth. Thanks to the log-Cobb-Douglas specification of the model, both the value function V (•,•,•,•) and the optimal policy of the problem defined in Eq. 2 can be explicitly computed by applying the "guess and verify" method^ to the Bellman equation (Eq. 4). CONJUGATE LINEAR IFSP The optimal dynamics for the physical and human capital in Eq. 9 have the form of products of powers, suggesting that a logarithmic transformation of both variables kt and ht may yield an equivalent conjugate system which is linear in the transformed variables. Specifically, a suitable transformation of Eq. 9 may lead to a contractive IFSP converging to a unique invariant distribution supported on some fractal attractor in accordance with Corollaries 1 and 2 of Section "Iterated function systems". The following proposition shows that, for specific sets of values for parameters a, (j),qi,q2 an r, ahnear system conjugate to Eq. 9 exists defining a IFSP that converges to an invariant distribution supported on a (generalized) Sierpinski gasket with vertices (0,0), (1/2,1) and (1,0). The proof is reported in the Appendix. Rewriting the IFSP in (16) as Xt+i =axt + Yt yt+i + Proposition 2 Assume that a ^ (p and let ~a-(l) r = exp l-a (21n^2 -In^i) it is immediately seen that the three values (0,0), ((l-a)/2,(l-(^)) and ((l-a),0) taken on by the (conjugate) random vector correspond respectively to the three scenarios {q\j), {qiA) and (1,1) for the original random values (z, t]) discussed in Section "The Model". The mild restriction required in Proposition 2 precludes the possibility of generating the standard Sierpinski gasket with vertices (0,0), (1/2,1) and (1,0) through Eq. 16, as its construction postulates that a = ^ = 1/2 must hold. In this sense, we say that the atQ-actor of Eq. 16 is a generalized Sierpinski gasket. As it is clear from the proof, condition in Eq. 10 turns out to be the key restriction needed to construct the dynamics in Eq. 16 equivalent to those in Eq. 9. (10) Then the one-to-one logarithmic transformation {kt,ht) (xtjt) defined by: Xt = Pa\nkt+Pb\r).ht+Pc yt = Pd\nht+Pe, (11) with Pa = Pc=l l-a 1 Ph = (1-ar In Pd = 2 In 2( 1/2 and (p >\/2 with at least one sQ-ict inequality, all Q-iangles in each prefractal overlap, as shown in Fig. lb for a = 0.5 and yt) ixt+i,yt+i}=W2ix,,y,) = {ax, + {l-a)/2,(l>y, + {l-(l>)) {xt+i, yt+1 )=w2{xt,yt) = {axt + {l-a),dh ^_ while the envelope conditions read as: dk aßdkzk"-\uhy-" h zk»{uhY-"-c ' {\-a)ßekZk«{uh)-'' zk» (uh)^'"-C From Eq. 19 we get: 1 u^ßm c = l+ßQk zk"{uh) l-a (20) (21) (22) (23) which, when plugged into Eq. 21, after some algebra leads to: Using Eqs. 23 and 24 into Eq. 22, again after some algebra yields: ' (l-aß){\-ßPd + (ppe + 4 ■ (25) Let us focus on the first equation in Eq. 25. Substituting kt+i and ht+i as in the first equation of Eq. 9, rearranging terms and after dropping the common terms apalnkt such equation becomes: Pain \^aß{l-ß(l)Y-" + p,ln [{ß-a)pb]lnht = Yt -Palnzt -Pblnrit. (26) In order to let the constant Pc be independent of ht in the equation above, we need that {I —a)Pa + {(p — a) Pb = 0, so that the last term in the LHS cancels out and, under the assumption that a / we have: l-a Using Eq. 27, Eq. 26 boils down to: (27) In aß{l-ß^) l-a l-a = 7t- a-(j) (l-a)pc l-a \pa Inz, a — (j) lnT7i Pa. (28) As the LHS in Eq. 28 is constant, we can use the three values Yt = 0, = (l-a)/2 and Yt = (l-a), corresponding respectively to {zt,r\t) = Iqi^r), {zt,rit) = (^2,1) and {zt,rit) = (1,1) for the original shocks, and write: 1-«, In^i H---Inr a-(l) 1-« 1 Pa = --Paln^2 = 1 - a. From the second equation, using Eq. 27 we easily get Pa and Pb as in Eq. 12. Note, however, that the first equation on the left must hold as well, which, consistently with Pa = — (1 — a)/ (2In<^2), is equivalent to condition in Eq. 10. As a matter of fact, condition in Eq. 10 is the key assumption to let Eq. 28 - or, equivalentiy, Eq. 26 - be independent of ht. Substituting = 1 - a [corresponding to (zt,rit) = (1,1)] and Pa as in Eq. 12 into Eq. 28 easily yields Pc as in Eq. 13. As far as the second equation in Eq. 25 is concerned, substituting /z^+i as in the second equation of Eq. 9, rearranging terms and after dropping the common terms (pPdlnht such equation becomes: Pdln[{ß