Dynamics, Games and Science I

We present a brief survey of focal decomposition stressing how this subject relates naturally to some a priori unrelated mathematical and physical subjects.

We present an infinite dimensional space of

C

1 + 

smooth conjugacy classes of circle diffeomorphisms that are

C

1 + 

fixed points of renormalization. We exhibit a one-to-one correspondence between these

C

1 + 

fixed points of renormalization and

C

1 + 

conjugacy classes of Anosov diffeomorphisms.

We analyze the main dynamical properties of the evolutionarily stable strategy (

ℰ????

) for asymmetric two-population games of finite size and its corresponding replicator dynamics. We introduce a definition of

ℰ????

for two-population asymmetric games and a method of symmetrizing such an asymmetric game. We show that every strategy profile of the asymmetric game corresponds to a strategy in the symmetric game, and that every Nash equilibrium (

??ℰ

) of the asymmetric game corresponds to a (symmetric)

??ℰ

of the symmetric version game. We study the (standard) replicator dynamics for the asymmetric game and we define the corresponding (non-standard) dynamics of the symmetric game. We claim that the relationship between

??ℰ

,

ℰ????

and the stationary states (

????

) of the dynamical system for the asymmetric game can be studied by analyzing the dynamics of the symmetric game.

We study an economy with heterogenous workers and firms as a two population game, in normal form, and its evolutionary dynamics implied by strategic complementarities. The population of firms is distributed in two groups, innovative and non innovative, while workers need to choose between two strategies, acquiring skills or remaining unskilled. Without having knowledge of the firms’ distribution, a worker reviews her strategy by asking herself whether it is worth it to change behavior or not. Rational choice on her part is taken, hereafter, to imply that she will choose the strategy which she expect to yield the greatest payoff, on the basis of her beliefs and the current state of the economy. By imitating successful agents, if the initial shares of innovative firms and skilled agents are “too small”, an economy eventually lead into a poverty trap. Hence, when an economy is close to a poverty trap, rationality may act as an actual obstacle to a take-off.

The Theory of Planned Behavior studies the decision-making mechanisms of individuals. We propose the Bayesian-Nash Equilibria as one, of many, possible mechanisms of transforming human intentions in behavior. This process corresponds to the best strategic individual decision taking in account the collective response. We built a game theoretical model to understand the role of leaders in decision-making of individuals or groups. We study the characteristics of the leaders that can have a positive or negative influence over others behavioral decisions.

We describe some recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors

Λ

introduced in [26]: (1) there exists an invariant foliation whose leaves are forward contracted by the flow; (2) there exists a positive Lyapunov exponent at every orbit; (3) attractors in this class are expansive and so sensitive with respect to initial data; (4) they have zero volume if the flow is

C

2

, or else the flow is globally hyperbolic; (5) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (6) the hitting time associated to a geometric Lorenz attractor satisfies a logarithm law; (7) the rate of large deviations for the physical measure on the ergodic basin of a geometric Lorenz attractor is exponential.

Quantum Information is a new area of research which has been growing rapidly since last decade. This topic is very close to potential applications to the so called Quantum Computer. In our point of view it makes sense to develop a more “dynamical point of view” of this theory. We want to consider the concepts of entropy and pressure for “stationary systems” acting on density matrices which generalize the usual ones in Ergodic Theory (in the sense of the ThermodynamicFormalism of R. Bowen, Y. Sinai and D. Ruelle). We consider the operator

ℒ

acting on density matrices ρ ∈ 

ℳ

N

over a finite

N

-dimensional complex Hilbert space

ℒ

(ρ) : =  ∑

i

 = 1

k

tr

(

W

i

ρ

W

i

 ∗ 

)

V

i

ρ

V

i

 ∗ 

, where

W

i

and

V

i

,

i

 = 1, 2, 

…k

are operators in this Hilbert space.

ℒ

is not a linear operator. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the

V

i

(. )

V

i

 ∗ 

 = : 

F

i

(. ),

i

 = 1, 2, 

…

, 

k

, play the role of the inverse branches (acting on the configuration space of density matrices ρ) and the

W

i

play the role of the weights one can consider on the IFS. We suppose that for all ρ we13pc]First author considered as corresponding author. Please check. have that ∑

i

 = 1

k

{ tr}

(

W

i

ρ

W

i

 ∗ 

) = 1. A family

W

: = {

W

i

}

i

 = 1, 

…

, 

k

determines a Quantum Iterated Function System (QIFS)

ℱ

W

,

ℱ

W

 = {

ℳ

N

, 

F

i

, 

W

i

}

i

 = 1, 

…

, 

k

. 

We present for a general audience the state of the art on the generic properties of

C

2

Hamiltonian dynamical systems.

In recent years various methods from the field of artificial intelligence (AI) have been applied to economic problems. The subarea of multiagent systems (MAS) is particularly useful as it enables to simulate individuals or organizations and various interactions among them. In this paper we investigate a scenario with a set of agents, each belonging to a certain sector of activity (e.g. agriculture, clothing, health sector etc.). The agents produce, consume goods or services in their area of activity. Besides, our model includes also the resource of

free time

. The goods and resources are exchanged on a market governed by auction, which determines the prices of all goods. We discuss the problem of developing an

adaptive

producer that exploits

reward-based learning

. This facet enables the agent to exploit previous information gathered and adapt its production to the current conditions. We describe a set of experiments that show how such information can be gathered and explored in decision making. Besides, we describe a scheme that we plan to adopt in a full-fledged experiments in near future.

We present a tourism model where the choice of a tourism resort by a tourist depends not only on the product offered in the resort, but depends also on the characteristics of the other tourists present in the resort. In order to explore the effect of the types of the tourists in the allocation of tourists across resorts, we introduce a game theoretical model and we describe some relevant Nash equilibria.

In this paper we explore results that establish a link between dynamical systems and computability theory (not numerical analysis). In the last few decades, computers have increasingly been used as simulation tools for gaining insight into dynamical behavior. However, due to the presence of errors inherent in such numerical simulations, with few exceptions, computers have not been used for the nobler task of proving mathematical results. Nevertheless, there have been some recent developments in the latter direction. Here we introduce some of the ideas and techniques used so far, and suggest some lines of research for further work on this fascinating topic.

Our main interest is to study the relevant biological thresholds that appear in epidemic and immunological dynamical models. We compute the thresholds, of the SIRI epidemic models, that determine the appearance of an epidemic disease. We compute the thresholds, of a Tregs immunological model, that determine the appearance of an immune response.

In this chapter we present some global convergence theorems for difference equations and systems. We also present specific examples of difference equations and systems to illustrate how most of these theorems apply.

The synchronization of a network depends on a number of factors, including the strength of the coupling, the connection topology and the dynamical behaviour of the individual units. In the first part of this work, we fix the network topology and obtain the synchronization interval in terms of the Lyapounov exponents for piecewise linear expanding maps in the nodes. If these piecewise linear maps have the same slope ± 

s

everywhere, we get a relation between synchronizability and the topological entropy. In the second part of this paper we fix the dynamics in the individual nodes and address our work to the study of the effect of clustering and conductance in the amplitude of the synchronization interval.

We consider a recently proposed generalisation of the Kimura equation, a Fokker–Planck type equation describing the evolution of

p

(

x

, 

t

), the probability of finding a fraction

x

of mutants at time

t

in a population evolving according to standard models in evolutionary biology. We present a detailed description of the solution, and we show that it naturally divides in two different time scales: the first determined by the drift (the natural selection), the second by the diffusion (the genetic drift).

We examine models of yield curves through chaotic dynamical systems whose dynamics can be unfolded using non-linear embeddings in higher dimensions. We refine recent techniques used in the state space reconstruction of spatially extended time series in order to forecast the dynamics of yield curves. We use daily LIBOR GBP data (January 2007–June 2008) in order to perform forecasts over a one-month horizon. Our method outperforms random walk and other benchmark models on the basis of mean square forecast error criteria.

This is a brief survey of recent results on the KAM stability of quasiperiodic dynamics using renormalization of vector fields.

We provide an extension to Merton’s famous continuous time model of optimal consumption and investment, in the spirit of previous works by Pliska and Ye, to allow for a wage earner to have a random lifetime and to use a portion of the income to purchase life insurance in order to provide for his estate, while investing his savings in a financial market consisting of one risk-free security and an arbitrary number of risky securities whose diffusive terms are driven by a multi-dimensional Brownian motion. The wage earner’s problem is to find the optimal consumption, investment, and insurance purchase decisions in order to maximize expected utility of consumption, of the size of the estate in the event of premature death, and of the size of the estate at the time of retirement. Dynamic programming methods are used to obtain explicit solutions for the case of constant relative risk aversion utility functions, and60pt]First author considered as corresponding author. Please check. some new results are presented together with the corresponding economic interpretations.

We present a survey of some of the most updated results on the dynamics of periodic and almost periodic difference equations.

In this paper we present a brief survey of the role played by Richard Thompson’s group

F

in the study of the dynamical classification of certain conformal repellers, as first described in de Faria et al. (Contemp. Math., 355:166–1855, 2004). We exhibit a faithful and discrete action of

F

in the asymptotic Teichmüller spaces of such conformal repellers. An important ingredient to monitor such actions is the complex scaling function of the repeller, defined on its dual Riemann surface. We ask for generalizations to more general repellers, and formulate some open questions.

We present a model of an Edgeworthian exchange economy where two goods are traded in a market place. For a specific class of random matching Edgeworthian economies, the expectation of the limiting equilibrium price coincides with that of related Walrasian economies. The novelty of our model is that we associate a bargaining skill factor to each participant which brings up a game alike the prisoner’s dilemma into the usual Edgeworth exchange economy. We analyze the effect of the bargaining skill factors in the variations of the individual amount of goods and in the increase of the value of their utilities. Finally, we let the bargaining skills of the participants evolve along the trades and we study their variation.

This article presents a dynamical analysis of several traffic phenomena, applying a new modelling formalism based on the embedding of statistics and Laplace transform. The new dynamic description integrates the concepts of fractional calculus leading to a more natural treatment of the continuum of the Transfer Function parameters intrinsic in this system. The results using system theory tools point out that it is possible to study traffic systems, taking advantage of the knowledge gathered with automatic control algorithms.

We present a brief summary of the conclusions of our work on the set of orbits of the planar circular restricted three body problem which undergo consecutive close encounters with the small primary, or orbits of second species. In this study, the value of the Jacobi constant is fixed, and we consider consecutive close encounters which occur within a maximal time interval. With these restrictions, the full set of orbits of second species is found numerically from the intersections of the stable and unstable manifolds of the collision singularity on the surface of section that corresponds to passage through the pericentre. A “skeleton” of this set of curves can be computed from the solutions of the two-body problem. The set of intersection points found in this limit corresponds to the S-arcs and T-arcs of Hénon’s classification which verify the energy and time constraints, and can be used to construct an alphabet to describe the orbits of second species. We find periodic orbits that combine S-type and T-type quasi-homoclinic arcs and we determine the symbolic dynamics of the full set of orbits of second12pc]First author considered as corresponding author. Please check. species.

We study the asymptotic distribution of the partial maximum of observable random variables evaluated along the orbits of some particular dynamical systems. Moreover, we show the link between Extreme Value Theory and Hitting Time Statistics for discrete time non-uniformly hyperbolic dynamical systems. This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa.

This paper analyzes the dynamic properties of a standard New Keynesian monetary policy model in which private agents expectations are formed under a learning mechanism while the central bank believes they follow the hypothesis of rational expectations. By assuming a gain sequence that is asymptotically constant, explicit local and global stability conditions are derived. The main results are that stability is guaranteed even in cases in which full convergence to the rational expectations equilibrium is not attainable; furthermore, endogenous business cycles are likely to arise.

In this paper we report on some recent results for mean field models in discrete time with a finite number of states. These models arise in situations that involve a very large number of agents moving from state to state according to certain optimality criteria. The mean field approach for optimal control and differential games (continuous state and time) was introduced by Lasry and Lions (C. R. Math. Acad. Sci. Paris, 343(9):619–625, 2006; 343(10):679–684, 2006; Jpn. J. Math., 2(1):229–260, 2007). The discrete time, finite state space setting is motivated both by its independent interest as well as by numerical analysis questions which appear in the discretization of the problems introduced by Lasry and Lions. We address existence, uniqueness and exponential convergence18pc]First author considered as corresponding author. Please check to equilibrium results.

In this work I introduce a classical example of an Interacting Particle System: the Simple Exclusion Process. I present the notion of hydrodynamic limit, which is a Law of Large Numbers for the empirical measure and an heuristic argument to derive from the microscopic dynamics between particles a partial differential equation describing the evolution of the density profile. For the Simple Exclusion Process, in the Symmetric case (

p

 = 1 ∕ 2) we will get to the heat equation while in the Asymmetric case (

p

≠1 ∕ 2) to the inviscid Burgers equation. Finally, I introduce the Central Limit Theorem for the empirical measure and the limiting process turns out to be a solution of a stochastic partial differential equation.

We consider the α re-scaled PSI20 daily index positive returns

r

(

t

)

α

and negative returns ( − 

r

(

t

))

α

called, after normalization, the α positive and negative fluctuations, respectively. We use the Kolmogorov–Smirnov statistical test as a method to find the values of α that optimize the data collapse of the histogram of the α fluctuations with the truncated Bramwell–Holdsworth–Pinton (BHP) probability density function (pdf)

f

{ BHP}

and the truncated generalized log-normal pdf

f

LN

that best approximates the truncated BHP pdf. The optimal parameters we found are α

{ BHP}

 + 

 = 0. 48, α

{ BHP}

 − 

 = 0. 46, α

LN

 + 

 = 0. 50 and α

LN

 − 

 = 0. 49. Using100pt]First author considered as corresponding author. Please check. the optimal α

′s

we compute analytic approximations of the probability distributions of the normalized positive and negative PSI20 index daily returns

r

(

t

). Since the BHP probability density function appears in several other dissimilar phenomena, our result reveals a universal feature of the stock exchange markets.

The goal of this article to give exposition of results demonstrating deep interrelation between topological classification of Dynamical Systems with nontrivially recurrent invariant manifolds and topological classification of standard objects existing on ambient manifold. One can see how the purely topological constructions, very pathological at first glance, appear naturally in Dynamical Systems.

This work aims to provide a short description of some old and new results on the theory of stability and related concepts for discrete dynamical systems. The emphasis is posed on noninvertible maps.

In this survey, I summarize some work towards understanding of differential rigidity and smooth conjugacy in one-dimensional dynamics. In particular, I focus on those dynamical systems that have critical points and on those dynamical systems that have only

$${C}^{1+\alpha },0 < \alpha < 1$$

, smoothness.

In this paper we propose a contingent claim pricing scheme between two counterparties in an incomplete one period market. According to our approach the two counterparties of a non-marketed contingent claim select a pair of pricing kernels, in order to agree on a common price, by minimizing their joint regret function, which quantifies the departure from their initial beliefs. The joint regret function is a convex combination of entropy-like or norm-dependent functionals. The relevant optimization problem is posed in terms of a partially finite convex programming problem in the space of pricing kernels.

We introduce a class of infinite dimensional replicator dynamics in the form of nonlinear and non local integrodifferential equations. We study the properties of the steady state of the equation and their connections with Nash equilibria of the game as well as the global stability of the steady state using techniques from the theory of variational inequalities and infinite dimensional dynamical systems.

A new model of the BE for binary reactive mixtures is here proposed with the aim of describing symmetric reversible reactions without a barrier, assuming appropriate reactive cross sections without activation energy and introducing suitable improvements in the elastic and reactive collision terms. The resulting model assures the correct balance equations and law of mass action, as well as good consistency properties for what concerns equilibrium and entropy inequality. Moreover the non-equilibrium effects induced by the chemical reaction on the distribution function are explicitly determined in a flow regime of slow chemical reaction.

We consider dynamical gene-environment networks under ellipsoidal uncertainty and discuss the corresponding set-theoretic regression models. Clustering techniques are applied for an identification of functionally related groups of genes and environmental factors. Clusters can partially overlap as single genes possibly regulate multiple groups of data items. The uncertain states of cluster elements are represented in terms of ellipsoids referring to stochastic dependencies between the multivariate data variables. The time-dependent behaviour of the system variables and clusters is determined by a regulatory system with (affine-) linear coupling rules. Explicit representations of the uncertain multivariate future states of the system are calculated by ellipsoidal calculus. Various set-theoretic regression models are introduced in order to estimate the unknown system parameters. Hereby, we extend our

Ellipsoidal Operations Research

previously introduced for gene-environment networks of strictly disjoint clusters to possibly overlapping clusters. We analyze the corresponding optimization problems, in particular in view of their solvability by interior point methods and semidefinite programming and we conclude with a discussion of structural frontiers and future research challenges.

Increasing globalization has created a realization by most countries that macroeconomic policy coordination is useful. However, the economic models advanced so far for this purpose do not adequately address the major questions that are of interest for policy coordination. The need for policy coordination arises from differences in resource endowments and differences in savings and investment that have implications for long run growth. The models that are currently in use are only of a comparative static nature based on open IS-LM framework with dynamics brought in as disequilibrium dynamics. It is suggested here that dynamic multi-sectoral multi-regional input output model with current input and capital input matrices may be used along with compatible spatial price equilibrium relationships and error correcting disequilibrium dynamics as the model. Some suggestions are offered to weaken the sufficiency conditions for the existence of a solution to Linear Quadratic Differential Game and some economic examples are discussed.

Period doubling cascades are observed at transition to chaos in many models used in the sciences and in physical experiments. These period doubling cascades are very well understood in one-dimensional dynamics. In particular, the microscopic geometrical properties of the attractors do not depend on the actual system, they are universal. Moreover, the attractors of two different maps are smoothly conjugate, they are rigid. Strongly dissipative Hénon maps describe parts of the dynamics of systems close to a homoclinic tangency and are often observed in various models. For these maps the transition to positive entropy also occurs along period doubling cascades. These strongly dissipative Hénon maps can be considered as perturbations of one-dimensional systems. Indeed, some of the universal geometrical properties of the one-dimensional systems are present in the Hénon maps. However, they appear in a much more delicate form: in a probabilistic sense the geometry of the Hénon attractors is the same as their one-dimensional counter part. This phenomenon is revered to as probabilistic universality and rigidity.

Fractional Calculus (FC) goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades. It has been recognized the advantageous use of this mathematical tool in the modelling and control of many dynamical systems. Having these ideas in mind, this paper discusses a FC perspective in the study of the dynamics and control of several systems. The paper investigates the use of FC in the fields of controller tuning, legged robots, electrical systems and digital circuit synthesis.

Existence of an invariant circle for any orientation-preserving 2D map whose orbit under renormalisation remains forever in a certain bounded subset is proved. The construction dates back to 1984. It was stimulated by a preprint by David Rand doing the same for the dissipative case. To include the general case, notably area-preserving, required a variation on his idea.

In this survey we describe the dynamics generated by expectation revisions in intertemporal economic models. Local uniqueness of equilibrium (like stationary states, cycles or sunspot equilibrium) is shown to be a necessary and sufficient condition to obtain stability for the dynamics of expectation revisions. For that reason, such equilibria are called Expectational Stable (E-Stable). Finally, we show how a stationary state which is E-Stable may exhibit a two period cycle bifurcation which is also E-Stable.

In this paper we will review several results on the dynamics of circle maps. This includes the theory of circle diffeomorphism where the combinatorial aspects goes back to Poincaré followed by the topological description of the dynamics by Denjoy and the geometric aspects by Herman and Yoccoz. In this case the dynamics is either periodic or quasi-periodic. The dynamics of non-invertible circle maps is much more complicated. The special case of covering maps of the circle is well understood. We will also consider maps with critical points and exhibit parametrized families of such maps that contains essentially all possible dynamical behavior. In the boundary between these two types of dynamical systems we have the critical circle maps also discussed here.

A spherical ball of radius δ rests on an oriented surface

S

embedded in

ℝ

3

and has a positive orthonormal frame attached to it. The

states

of the ball are the elements of the 5-dimensional manifold

$$M = S \times \mathrm{ SO}(3)$$

and a

move

is a smooth path on

M

corresponding to a rolling of the ball on

S

without slipping. The moves without slipping or twisting along geodesics are called

pure moves

. Rolling ball problems on

S

are mainly related to the search of

N

(

S

), the minimum number of moves (or moves without twisting, or pure moves) sufficient to reach continuously any final state starting at a given initial state. We mention some results and conjectures relative to the case of a unitary ball (δ = 1) rolling on surfaces of revolution; important cases are: plane, sphere, cylinder and surfaces parallel to Delaunay. The dynamics giving the moves without slipping of the rolling ball problems are non-holonomic, preserve a volume and lead, in certain cases, to the existence of minimal surfaces immersed in

M

.

This article concerns some global stability aspects of a class of models introduced by Nowak and Bangham that describe in a fairly successful way the initial phases of the HIV dynamics in the human body as well as some generalizations that take into account mutations. We survey recent results implying that the biologically meaningful positive solutions to such models are all bounded and do not display periodic orbits. For the mutationless cases the dynamics is characterized in terms of certain dimensionless quantities, the so-called basic reproductive rate and the basic defense rate. As a consequence, we infer that the finite dimensional models under consideration cannot account, without further modifications, for the third phase of the HIV infection. We conclude by suggesting a modification that according to our numerical simulations may describe the collapse of the infected patient.

We introduce the yes–no decision model, where individuals can make the decision yes or no. We characterize the coherent and uncoherent strategies that are Nash equilibria. Each decision tiling indicates the way coherent and uncoherent Nash equilibria co-exist and change with the relative decision preferences of the individuals for the yes, or no, decision. There are 289 combinatorial classes of decision tilings, described by the decision bussola, what shows the high complexity of making decision.130pt]The first author “Alberto A. Pinto” has been considered as corresponding author. Please check.

This paper develops a theoretical framework to study spatial price competition in a Hotelling-type network game. Each firm

i

is represented by a node of degree

k

i

, where

k

i

is the number of firm

i

’s direct competitors (neighbors). We investigate price competition á la Hotelling with complete and incomplete information about the network structure. The goal is to investigate the effects of the network structure and of the uncertainty on firms’ prices and profits. We first analyze the benchmark case where each firm knows its own degree as well as the rivals’ degree. Then, in order to understand the role of information in the price competition network, we also analyze the incomplete information case where each firm knows its type (i.e. number of connections) but not the competitors’ type.

This paper presents a discussion of the closing lemma, its origins and development.

Roughly speaking, Peixoto’s foundationalworks in the global theory of ordinary dif- ferential equations corresponds to the papers [17–19] which are nowadays referred to as Peixoto’s Theorem.

In this survey we will discuss some recent results on a certain class of dynamical systems, called

fictitious play

which are associated to game theory. Here we simply aim to show that the dynamics one encounters in these systems is unusually rich and interesting. This paper does not require a background in game theory.

We show that typical ODE’s sharing the obvious algebraic property of Euler’s equation exhibit finite time blowup. In a subsequent paper (Sullivan, Topology, Algebra and Algebraic Topology of Incompressible, Frictionless 3D Fluid Motion in the Georgia International Topology Conference Proceedings (to appear in 2010)) we study other finite dimensional analogues of Euler’s equation which have no finite time blowup.

Given a space-time and a continuous medium with elastic properties described by a 3-dimensional material space, one can ask whether they are compatible in the context of relativistic elasticity. Here a non-static, spherically symmetric spacetime metric is considered and we investigate the conditions for that metric to correspond to different 3-dimensional material metrics.

An investment game with an incumbent and an entrant is examined. The profit flows involve two uncertain factors: (1) the basic level of demand of the market observed only by the incumbent and (2) the fluctuation of the demand described by a geometric Brownian motion which is common to both firms. In our model, the incumbent enters into the market earlier than the entrant. The high demand type of the incumbent can invest earlier than the low demand type. This earlier investment, however, reveals the information, so that the entrant would accelerates the timing of the investment by observing the incumbent’s timing of the entry and it reduces the monopolistic profit of the incumbent. Thus, the incumbent who knows the high demand may delay the timing of the investment to hide the information strategically. I characterize this signaling effect and investigate the real option values of both firms.

This paper presents a framework to assess demand of durable products recognizing their indivisibility. The paper builds on the household production framework recognizing that consumers demand for product characteristics and this demand can be satisfied by purchasing durables that combined with variable inputs to generate these characteristics or renting services that provide these characteristics. One example is buying a washer and dryer or going to a laundromat. Our analysis recognizes heterogeneity among consumers and suggests that some segments of the population will buy the durables while others will rent. We derived demand for the durables and associated variable inputs by aggregating over population. We show the demands are affected by prices and income distribution parameters.