General equilibrium in a heterogeneous-agent incomplete-market economy with many consumption goods and a risk-free bond
aa r X i v : . [ ec on . T H ] J un A Bewley-Huggett model with many consumption goods
Bar Light ∗ June 18, 2019
Abstract :We study a pure-exchange incomplete markets model with heterogeneous agents.In each period, the agents choose how much to save and which bundle of goodsto consume while their endowments are fluctuating. We focus on a competitivestationary equilibrium (CSE) in which the wealth distribution is invariant, the agentsmaximize their expected discounted utility, and both the prices of goods and theinterest rate are market-clearing. Our main contribution is to extend some generalequilibrium results to an incomplete markets setting. Under mild conditions on theagents’ preferences, we show that the aggregate demand for goods depends onlyon their relative prices and we prove the existence of a CSE. When the agents’preferences can be represented by a CES (constant elasticity of substitution) utilityfunction with an elasticity of substitution that is higher than or equal to one, we provethat the CSE is unique. Under the same preferences, we show that a higher inequalityof endowments does not change the equilibrium prices of goods, and decreases theequilibrium interest rate.Keywords: Arrow-Debreu model; general equilibrium; heterogeneous agents; Bewley models;dynamic economies. ∗ Graduate School of Business, Stanford University, Stanford, CA 94305, USA. e-mail: [email protected]
Introduction
We extend the classic Arrow-Debreu model to a dynamic incomplete markets general equilibriummodel with a continuum of agents, in which each agent has an individual state that correspondsto his wealth level. There is an infinite number of periods and in each period agents participatein a pure-exchange Arrow-Debreu model (as in the seminal paper by Arrow and Debreu (1954)).Each agent has a different wealth level and different preferences over consumption bundles, andthus the agents are heterogeneous in the static pure-exchange Arrow-Debreu model. In eachperiod, given the agents’ wealth level and their preferences over consumption bundles, the agentsdecide how much to spend on a bundle of goods to be consumed in that period, which bundleof goods to consume, and how much to save for future consumption. In the tradition of theBewley models (see Ljungqvist and Sargent (2012) for a textbook treatment), the markets areincomplete. The agents face uninsurable idiosyncratic risk and can transfer assets from oneperiod to another only by saving in a risk-free bond. In each period, the agents receive arandom endowment vector. We assume that all random shocks are idiosyncratic, ruling outaggregate random shocks that are common to all agents. As in Arrow and Debreu (1954), theagents are price takers, that is, the agents take the prices of goods and the risk-free bond’s rateof return as given. In this paper we focus on a pure-exchange economy without production, andhence, the model studied in this paper is closely related to Huggett’s model (Huggett, 1993). In Huggett’s model, and in most similar Bewley models that are used in applied work, there isonly one consumption good. In contrast, in the model presented in this paper, there are manyconsumption goods and each good has a price. Thus, in addition to the standard inter-temporalsavings decision, the agents also make a static decision of how to allocate their spending betweenthe different consumption goods.This key feature of the model allows us to examine questions relating to wealth distribution,the prices of goods, and the risk-free bond’s rate of return in the framework of a Bewley-Huggettmodel. Such questions cannot be examined in a Bewley model with one consumption good. Bewley models feature rich heterogeneity and are used to study many economic phenomena. For similar models with one consumption good see Lucas (1980), Geanakoplos et al. (2014), andHu and Shmaya (2019). In Bewley (1986) there are multiple consumption goods but the interest rate is fixed and is assumed to be0 (see also Karatzas et al. (1994)). In this paper the interest rate is determined in equilibrium as in Huggett(1993). There is a vast literature on asset pricing and wealth inequality in different models than the model pre-sented in this paper (for example, Judd et al. (2003), Blume and Easley (2006), Krueger and Lustig (2010),and Kubler and Schmedders (2015), just to name a few). In particular, Bewley models are often used to study wealth inequality (see Heathcote et al. (2009),De Nardi (2015), and Benhabib and Bisin (2016), for surveys). We prove the uniqueness of a CSE for the special case that the agents’ preferences overbundles can be represented by a CES (constant elasticity of substitution) utility function withan elasticity of substitution that is equal to or higher than one (see Theorem 2). This assumptionon the agents’ preferences implies that the consumption goods are gross substitutes, i.e., thedemand for each consumption good increases with the prices of the other consumption goods. Wenote that the standard argument for proving the uniqueness of an equilibrium in the static Arrow-Debreu model cannot be applied in our setting, since the aggregate demand for consumptiongoods is not necessarily increasing with the interest rate and the aggregate demand for savings isnot necessarily increasing with prices of consumption goods. Thus, the excess demand functiondoes not necessarily satisfy the gross substitutes property.Our main result regarding the wealth distribution’s influence on the prices of goods andon the interest rate is Theorem 3. We prove that if the agents’ preferences over bundles canbe represented by a CES utility function with an elasticity of substitution that is higher thanone, then an increase in the risk of the random future endowments (in the sense of the convexstochastic order) changes the CSE in the following way: the interest rate decreases, and theprices of goods do not change. In the classic Arrow-Debreu model the result that the prices ofgoods do not change when the wealth inequality is higher is intuitive because the demand foreach good is linear in wealth. Thus, the demand for each good does not change when the wealthinequality is higher. In our setting, under a CES utility function, the marginal propensity toconsume is decreasing, so the demand for each good is concave in wealth. An increase in the riskof the random future endowments increases the aggregate savings because of the precautionarysavings effect. Thus, the aggregate demand for each good decreases. At the same time, adecrease in the interest rate decreases the aggregate savings because of the substitution effect.It turns out that in general equilibrium, the precautionary savings effect and the substitutioneffect offset each other exactly and the prices of goods do not change.The rest of the paper is organized as follows. Section 2 presents the model. In Section 2.1we define the CSE. In Section 3 we present the main results of this paper. In Section 3.1 weestablish the existence of a CSE. In Section 3.2 we provide conditions that ensure the uniquenessof a CSE. In Section 3.3 we discuss how the wealth distribution influences the prices of goodsand the interest rate. In Section 3.4 we compare the current paper to recent work on mean fieldgames. In Section 3.5 we extend the model to ex-ante heterogeneous agents. In Section 4 weprovide final remarks, followed by an Appendix containing proofs. See Davila et al. (2012), Shanker (2017), Nuno and Moll (2018), and Park (2018) for a study of welfaremaximization in Bewley models. The model
There is a continuum of agents of measure 1. Every agent has an individual state. We assumefor now that the agents are ex-ante identical. In Section 3.5 we extend the model to ex-anteheterogeneous agents. There are n goods. Let Y i be a random variable that describes theevolution of good i ’s endowment. We assume that Y i has a finite support Y i and a probabilitymass function q i ( y ) := Pr( Y i = y ) for all 1 ≤ i ≤ n and y ∈ Y i . Let Y = Y × ... × Y n and let q ( y ) = q ( y , . . . , y n ) := Pr( Y = y , . . . , Y n = y n ) be the joint probability mass function wherewe denote elements in R n by bold letters. In each period t = 1 , , . . . the agents receive anendowment vector y ∈ Y with probability q ( y ). We assume that y ≫ y ∈ Y , where y ≫ y i > i = 1 , . . . , n . We refer to q as the endowments process.Denote the agents’ wealth at time t = 1 by a (1). In each period t = 1 , , . . . , the agentsreceive an an endowment vector y ( t ). After receiving their endowment vector, the agents choosea bundle of goods to consume in that period ( x ( t ) = ( x ( t ) , . . . , x n ( t )) ∈ R n + ) and choose howmuch to save in a risk-free bond for future consumption. The price of good x i ( t ) is given by p i ( t ) >
0, so the price of a bundle x ( t ) is p ( t ) · x ( t ) where p ( t ) · x ( t ) := P p i ( t ) x i ( t ) denotes thescalar product of two elements in R n . The agents’ savings rate of return is 1 + r ( t ) where r ( t ) isthe interest rate in period t . The agents are price takers, i.e., they take the sequence of prices( p ( t ) , r ( t )) ∞ t =1 as given. If an agent’s wealth at time t is a ( t ), the agent’s wealth at time t + 1when y ( t + 1) is the realized endowment vector is a ( t + 1) = (1 + r ( t ))( a ( t ) − p ( t ) · x ( t )) + p ( t + 1) · y ( t + 1) . We assume that the agents can borrow, and the borrowing limit is given by b ( t ). Thus, a ( t ) − p ( t ) · x ( t ) ≥ b ( t ) for each period t . We assume that the borrowing limit in period t is givenby a fraction of the the natural borrowing limit, that is, b ( t ) = (1 − ψ ) − min y ∈Y p ( t ) · y r ( t ) where0 < ψ < ψ is higher then the borrowing constraint is tighter.For a discussion of the natural borrowing constraint see Aiyagari (1994). For now we assumethat p ( t ) ≫ r ( t ) > t . We note that under the borrowing constraint b ( t ), theequilibrium interest rates satisfy r ( t ) > p ( t ) ≫ C ( a, p ( t )) = [ b ( t ) min { a, P ni =1 p i ( t ) b } ] the interval from which an agent maychoose his level of savings when his wealth is a and the prices of goods are p ( t ). P ni =1 p i ( t ) b All the results in this paper can be extended to the case that Y i has a compact support. As usual, the positive cone of R n is denoted by R n + , i.e., R n + = { x = ( x , . . . , x n ) : x j ≥ j = 1 , . . . , n } .
5s an upper bound on savings that ensure compactness of the state space where b >
0. Weassume that the maximal level of savings that an agent can have is bounded to avoid technicaldifficulties that arise in dynamic programming with unbounded rewards. In a standard incomefluctuation problem, one can find sufficient conditions on the utility function that ensure thatthe upper bound on savings never binds (see Li and Stachurski (2014), A¸cıkg¨oz (2018) andreferences therein).We assume that the agents’ preferences over bundles are represented by a utility function U : R n + → R . For x , x ′ ∈ R n we write x ≥ x ′ if x i ≥ x ′ i for all i = 1 , ..., n . We say that U isincreasing if x ≥ x ′ implies U ( x ) ≥ U ( x ′ ). We say that U is strictly increasing if x > x ′ implies U ( x ) > U ( x ′ ).Throughout the paper, we assume the following conditions on the utility function. Assumption 1
The utility function U is strictly increasing, continuously differentiable, strictlyconcave, and ∂U (0) ∂x j = ∞ for some ≤ j ≤ n . Let A be the set of possible wealth levels that an agent can have, and let A t := A × . . . × A | {z } t times . Astrategy π for the agents is a function that assigns to every finite history a t = ( a (1) , ..., a ( t )) ∈ A t a feasible bundle x ( t ). A strategy π induces a probability measure over the space of all infinitehistories. We denote the expectation with respect to that probability measure by E π .When the agents follow a strategy π and the sequence of prices is given by ( p ( t ) , r ( t )), theirexpected present discounted value is V π ( a ) = E π (cid:16) ∞ X t =1 β t − U ( π ( a (1) , . . . , a ( t )) (cid:17) , where a (1) = a is the initial wealth and 0 < β < V ( a ) = sup π V π ( a ) . That is, V ( a ) is the maximal expected utility that an agent can have when his initial wealth is a . We call V the value function. The probability measure on the space of all infinite histories A N is uniquely defined (see for exampleBertsekas and Shreve (1978)). .1 Competitive stationary equilibrium In this section we define a competitive stationary equilibrium (CSE). We first introduce somenotations that are necessary in order to define a CSE. In a CSE the prices of goods and theinterest rate are constant over time. For the rest of the section we assume that ( p ( t ) , r ( t )) =( p , r ) for all t ∈ N .We denote by b the agents’ savings in the next period. When the agents’ wealth is a , theirnext period’s savings are b ∈ C ( a, p ), and the prices of goods are p , then the set of consumptionbundles available to the agents is given by X ( a − b, p ) = { x ∈ R n + : p · x = a − b } . We sometimeschange variables and define c = a − b to the total consumption of the agents.The minimal level of wealth that an agent can have when p ≫ r > a ( p , r ) =(1 + r ) b + min y ∈Y p · y and the maximal level of wealth that an agent can have is a ( p , r ) :=(1 + r ) P ni =1 p i b + max y ∈Y p · y , so the set of possible wealth levels that an agent can have A ( p , r ) = [ a ( p , r ) , a ( p , r )] is compact for all p ≫ r >
0. For the rest of the section weassume that p ≫ r > p ≫ r > A ( p , r ),we can use standard dynamic programming arguments to solve the agents’ problem. Let B ( A )be the space of all bounded real-valued functions defined on a set A . For any p ≫ r > T : B ( A ( p , r )) → B ( A ( p , r )) by T f ( a, p , r ) = max b ∈ C ( a, p ) max x ∈ X ( a − b, p ) U ( x ) + β X y ∈Y q ( y ) f ((1 + r ) b + p · y , p , r ) . The value function V is the unique fixed point of T , i.e., there is a unique function V ∈ B ( A ( p , r )) such that T V = V . We denote by x ∗ ( a − b, p ) the demand function of an agent, i.e., x ∗ ( a − b, p ) = argmax x ∈ X ( a − b, p ) U ( x ) . Note that given the choice of the next’s period savings b , the decision of how to distribute thespending a − b between the different consumption goods is a static decision. Also note that x ∗ is single-valued since U is strictly concave and continuous. We denote by g ( a, p , r ) the savings The Banach-fixed point theorem (see Theorem 3.48 in Aliprantis and Border (2006)) shows that T hasa unique fixed point. Standard dynamic programming arguments (e.g., Blackwell (1965)) show that the valuefunction V is the unique fixed point of T . g ( a, p , r ) = argmax b ∈ C ( a, p ) U ( x ∗ ( a − b, p )) + β X y ∈Y q ( y ) V ((1 + r ) b + p · y , p , r ) . (1)For a set K ⊆ R n we denote by P ( K ) the set of all probability measures on K and by B ( K )the Borel sigma-algebra on K . Define M λ ( D ; p , r ) = Z A ( p ,r ) X y ∈Y q ( y )1 D ((1 + r ) g ( a, p , r ) + p · y ) λ ( da ; p , r ) , (2)for any D ∈ B ( A ( p , r )) where 1 D is the indicator function of the set D ∈ B ( A ( p , r )). M λ ∈P ( A ( p , r )) describes the next period’s wealth distribution, given that the current wealth dis-tribution is λ ∈ P ( A ( p , r )) and the prices are ( p , r ). A wealth distribution µ ∈ P ( A ( p , r )) iscalled an invariant wealth distribution if µ = M µ .We now define a CSE.
Definition 1
A competitive stationary equilibrium consists of prices ( p , r ) , a savings policyfunction g , a demand function x ∗ , and a wealth distribution µ ∈ P ( A ( p , r )) such that(i) Given the prices ( p , r ) , the savings policy function g and the demand function x ∗ areoptimal for the agents. That is, g satisfies equation (1) and x ∗ ( a − g ( a, p , r ) , p ) = argmax x ∈ X ( a − g ( a, p ,r ) , p ) U ( x ) . (ii) Given the prices ( p , r ) , µ is an invariant wealth distribution. That is, µ ∈ P ( A ( p , r )) satisfies µ = M µ .(iii) For each good ≤ i ≤ n , the aggregate supply of good i equals the aggregate demand forgood i : Z A ( p ,r ) x ∗ i ( a − g ( a, p , r ) , p ) µ ( da ; p , r ) = X y i ∈Y i q i ( y i ) y i (iv) The aggregate supply of savings equals the aggregate demand for savings: Z A ( p ,r ) g ( a, p , r ) µ ( da ; p , r ) = 0 . The first equilibrium condition says that agents choose a demand function and a savingspolicy function to maximize their expected discounted utility. The second equilibrium conditionsays that the wealth distribution induced by the agents’ savings policy function is invariant.8he third equilibrium condition says that the aggregate demand for good i equals the aggregatesupply of good i . The fourth equilibrium condition says that the aggregate savings in theeconomy are 0, i.e., the supply of savings equals the demand for savings. The third and fourthequilibrium conditions require that the prices ( p , r ) are market-clearing prices. The naturalinterpretation of the stationary equilibrium prices are that the prices represent average prices(see Huggett (1993)). An alternative to CSE is a competitive recursive equilibrium (see Miao(2006)). A competitive recursive equilibrium is a sequence of prices ( p ( t ) , r ( t )), and a sequence ofmeasures ( λ ( t )) such that the savings and consumption decisions are optimal for the agents; theprices ( p ( t ) , r ( t )) are market-clearing prices for every period t ; and the wealth distribution followsthe law of motion defined by equation (2). Clearly, if the initial agents’ wealth distribution isinvariant, then the CSE is also a competitive recursive equilibrium. In this paper we focus ona CSE. The existence result presented in the next section can be applied to the competitiverecursive equilibrium case as well. The analysis and computation of a competitive recursiveequilibrium is generally much harder than the analysis and computation of a CSE.We note that the model presented in this paper is closely related to Huggett’s model (Huggett,1993) and Bewley’s model (Bewley, 1986). In Huggett’s model there is only one consumptiongood and only the interest rate is determined in equilibrium. In Bewley’s model there are manyconsumption goods and their prices are determined in equilibrium but the interest rate is fixedand is not determined in equilibrium. In the model presented in this paper, however, thereare many consumption goods, and both their prices and the interest rate are determined inequilibrium. Our model also generalizes the static pure-exchange Arrow-Debreu model to anincomplete markets setting where the agents can transfer assets from one period to another onlyby investing in a risk-free bond.
In this section we present our main results. In Section 3.1 we state our existence result andprovide the main idea behind the proof. In Section 3.2 we provide conditions that ensure theuniqueness of a competitive stationary equilibrium (CSE). In Section 3.3 we discuss how anincrease in the risk of the endowments process influences the equilibrium prices of goods andthe equilibrium interest rate when the agents’ preferences can be represented by a CES utilityfunction. In Section 3.4 we compare our model to mean field equilibrium models. In Section 3.5 For general existence results of a competitive recursive equilibrium with aggregate shocks seeBrumm et al. (2017).
9e extend the model to include ex-ante heterogeneous agents.
The main theorem of this section is the following:
Theorem 1
Suppose that Assumption 1 holds. Then, there exists a competitive stationaryequilibrium.
To prove the theorem, we use a well-known excess demand approach (e.g., Debreu (1982)).Under Assumption 1, the savings policy function g ( a, p , r ) is single-valued and continuous.Furthermore, there exists a unique invariant wealth distribution µ ( da ; p , r ) for all ( p , r ) ∈ P where P is the non-empty and convex set defined in equation (3) below. We define an excessdemand function ζ ( p , r ) from P ⊆ R n +1+ into R n +1 , where ζ i ( p , r ) = Z A ( p ,r ) x ∗ i ( a − g ( a, p , r ) , p ) µ ( da ; p , r ) − X y i ∈Y i q i ( y i ) y i is the excess demand for good i , i = 1 , ..., n , and ζ n +1 ( p , r ) = − Z A ( p ,r ) g ( a, p , r ) µ ( da ; p , r )is the excess demand for savings. The excess demand function ζ : P → R n +1 is defined by ζ ( p , r ) = ( ζ ( p , r ) , . . . , ζ n ( p , r ) , ζ n +1 ( p , r )) . Note that if ζ ( p , r ) = then ( p , r ) are equilibrium prices, µ ( · ; p , r ) is the equilibrium invariantwealth distribution, x ∗ ( a − g ( a, p , r ) , p ) is the equilibrium demand function, and g ( a, p , r ) is theequilibrium savings policy function.We extend a well-known result from the static Arrow-Debreu model to the incomplete mar-kets model studied in this paper. We show that the aggregate demand for goods and the aggre-gate supply of goods depend only on their relative prices. In particular, the next Propositionshows that if ( p , r ) are equilibrium prices then ( θ p , r ) are also equilibrium prices for all θ > Proposition 1
Fix p ≫ , r > with (1 + r ) β < , and θ > . Then i) θg ( a, p , r ) = g ( θa, θ p , r ) and x ∗ ( a − g ( a, p , r ) , p ) = x ∗ ( θa − g ( θa, θ p , r ) , θ p ) for all a .(ii) ζ i ( θ p , r ) = ζ i ( p , r ) for ≤ i ≤ n and ζ n +1 ( θ p , r ) = θζ n +1 ( p , r ) .Thus, if ( p , r ) are equilibrium prices then ( θ p , r ) are also equilibrium prices. We note that the excess demand for savings is homogeneous of degree one in the prices ofgoods, while the excess demand for good k is homogeneous of degree zero for all 1 ≤ k ≤ n .These results rely on the fact that θC ( a, p ) = C ( θa, θ p ) for all θ > C ( a, p ) = [(1 − ψ ) − min y ∈Y p · y r , min { a, n X i =1 p i b } ]is the interval from which an agent may choose his level of savings and θA = { θx : x ∈ A } forany set A . That is, if the agent can save an amount b given the wealth level a and the prices p , then the agent can save an amount θb given the wealth level θa and the prices θ p . This isreasonable in our setting since all the prices in our model are real prices.Also note that if the bonds are not in zero net supply than it is not true that if ( p , r ) areequilibrium prices then ( θ p , r ) are also equilibrium prices for all θ >
0. This follows since theaggregate demand for savings is homogeneous of degree one in the prices of goods.From Proposition 1, if ( p , r ) are equilibrium prices then ( θ p , r ) are also equilibrium pricesfor all θ >
0. Thus, we can normalize the prices of the goods. More precisely, the search forequilibrium prices can be confined to sets that contain at least one element from the half-ray { θ p : θ > } .We define the sets Λ = { ( p , r ) ∈ R n + × R + : P ni =1 p i + r = β − } and P = { ( p , r ) ∈ Λ : p ≫ , r > } . (3)Note that if ( p , r ) ∈ P then (1 + r ) β < p , r ) ∈ P such that ζ ( p , r ) = , we show that the excess demand function is continuous, satisfiesWalras’ law and satisfies suitable boundary conditions, and we apply a well-known Propositionthat guarantees the existence of at least one vector ( p , r ) ∈ P that satisfies ζ ( p , r ) = (seeProposition 2 in the Appendix). It is well know that if (1 + r ) β ≥ emark 1 We prove the existence of an equilibrium with a strictly positive interest rate. Anequilibrium with a strictly positive interest rate exists since we assume that the borrowing con-straint tends to minus infinity as the interest rate tends to zero (a similar observation is madeon page 673 in Aiyagari (1994)).
In this section we prove the uniqueness of a CSE for the special case where the utility functionis given by: U ( x ) = P ni =1 α i x γi for some 0 < γ < α i > P ni =1 α i = 1, i.e., the agents’preferences over bundles can be represented by a CES utility function with an elasticity ofsubstitution that is higher than one.There is a vast literature that provides sufficient conditions to ensure the uniqueness of anequilibrium in the standard static pure-exchange Arrow-Debreu model. The property thatthe demand for each good increases with the prices of the other goods (”gross substitutesproperty”) usually plays a crucial role in proving the uniqueness of an equilibrium in the staticArrow-Debreu model. Given the gross substitutes property, an easy argument shows that theequilibrium must be unique. This fact led most of the previous literature on the uniqueness ofan equilibrium to find conditions on agents’ preferences that ensure that the gross substitutesproperty holds. While the gross substitutes property remains a crucial property in proving theuniqueness of an equilibrium in the dynamic incomplete markets Arrow-Debreu model consideredin this paper also, the standard argument that proves the uniqueness of an equilibrium does notapply. The reason is that the aggregate demand for goods does not necessarily increase with theinterest rate, and the aggregate demand for savings does not necessarily increase with the pricesof goods. Thus, the excess demand function does not necessarily have the gross substitutesproperty. We show that when the agents’ preferences are represented by a CES utility functionwith an elasticity of substitution that is higher than one the CSE is unique even when the excessdemand does not have the gross substitutes property.It is well known and easy to check that when the agents’ preferences are represented by aCES utility function, the indirect utility function v ( a, b, p ) = max x ∈ X ( a − b, p ) U ( x )is given by a constant relative risk aversion (CRRA) utility function. The CRRA utility func- For a survey of the work done on the uniqueness of equilibrium, see Arrow and Hahn (1971), Mas-Colell(1991), and Kehoe (1998). For recent results, see Toda and Walsh (2017) and Geanakoplos and Walsh (2018),and references therein.
Theorem 2
Assume that U ( x ) = P ni =1 α i x γi for some < γ < , α i > , P ni =1 α i = 1 . Thenthere exists a unique competitive stationary equilibrium. If the agents’ preferences can be represented by a Cobb-Douglas utility function, i.e., U ( x ) = P ni =1 α i ln( x i ) for α i > P ni =1 α i = 1, then the same proof as the proof of Theorem 2 showsthat there exists a unique CSE in this case, as well. Note that in this case, the indirect utilityfunction corresponds to a log utility function, which is often used in the quantitative literature(for example, see Aiyagari (1994) and Krusell et al. (2010)).The conditions on the agents’ preferences that ensure uniqueness are restrictive. However,uniqueness results in Bewley models are rare, and a multiplicity of equilibria can arise even underthe standard specifications of the model (for examples of the multiplicity of equilibria see Toda(2017) and A¸cıkg¨oz (2018)). Even in a static Arrow-Debreu model, a multiplicity of equilibriacan easily arise. Kubler and Schmedders (2010a) and Kubler and Schmedders (2010b) provideexamples of multiplicity in the case that the agents’ preferences can be represented by a CESutility function, and also provide a general method of finding all the equilibria in semi-algebraicArrow-Debreu models.
In this section we show that if the agents’ preferences are represented by a CES utility functionwith an elasticity of substitution that is higher than one, then an increase in the risk of theendowments process (in the sense of the convex stochastic order) does not change the equilibriumprices of goods, and decreases the equilibrium interest rate.In response to an increase in the risk of the endowments process, we show that the partialequilibrium wealth inequality is higher in the sense of the convex stochastic order. That is, for afixed interest rate and fixed prices of goods, the wealth inequality is higher when the endowments A natural question that arises is: under what conditions is there a finite number of equilibria? (seeDebreu (1970) for an answer to this question in the static Arrow-Debreu model). A related question can beasked about the stability of the CSE (see Arrow and Hurwicz (1958) and Arrow et al. (1959)). We did notexplore these directions in the current paper. In addition, for a fixed interest rate r and prices of goods p , theprecautionary savings effect increases the aggregate savings, and thus the aggregate expenditureon goods decreases. Since the goods are normal, the decrease in the aggregate expenditure ongoods implies that the aggregate demand for each good decreases.We note that this is different from the static Arrow-Debreu model where riskier endowmentsdo not change the demand for each good. In the static Arrow-Debreu model the demand for eachgood is linear in wealth while in our setting the demand for each good is concave in wealth sincethe marginal propensity to consume is decreasing. Thus, in the dynamic incomplete marketsArrow-Debreu model the equilibrium prices might change in response to an increase in the riskof the endowments process. The prices of goods, however, do not change at all in the new CSE.While the interest rate decreases, the negative effect of this decrease on the aggregate savingsis exactly offset by the positive effect on the aggregate savings of an increase in the risk of theendowments process. In other words, the negative substitution effect on the aggregate savingsand the positive precautionary effect on the aggregate savings are equal.We now introduce notations that are needed to state the main theorem of this section.For two probability measures λ , λ we define the partial order (cid:23) I − CX by λ (cid:23) I − CX λ if andonly if R f ( a ) λ ( da ) ≥ R f ( a ) λ ( da ) for every convex and increasing function f . Similarly,we write λ (cid:23) CX λ if and only if R f ( a ) λ ( da ) ≥ R f ( a ) λ ( da ) for every convex function f .We say that the endowments process q is riskier than the endowments process q ′ if q (cid:23) CX q ′ . With slight abuse of notation, we add the argument q to the functions defined above,when q ( y ) is the probability of receiving the endowment vector y ∈ Y . For example, we write µ ( · ; p , r, q ) for the invariant wealth distribution, g ( a, p , r, q ) for the savings policy function, and x ∗ ( a − g ( a, p , r, q ) , p ) for the demand function. Theorem 3
Assume that U ( x ) = P ni =1 α i x γi for some < γ < , α i > , P ni =1 α i = 1 . Assumethat the endowments process q is riskier than the endowments process q ′ . Then(i) The partial equilibrium wealth inequality is higher under q than under q ′ , i.e., µ ( · ; p , r, q ) (cid:23) I − CX µ ( · ; p , r, q ′ ) for all ( p , r ) ∈ P . In addition, if ( p ( q ) , r ( q )) are equilibrium prices under the en-dowments process q then µ ( · ; p ( q ) , r ( q ) , q ) (cid:23) CX µ ( · ; p ( q ) , r ( q ) , q ′ ) .(ii) The equilibrium prices of goods do not change, i.e., p ( q ) = p ( q ′ ) . The equilibrium interestrate is lower under q than under q ′ , i.e., r ( q ′ ) ≥ r ( q ) . The convexity of the savings policy function follows from the CES assumption and is not easy to establishfor general utility functions.
14e note that when the endowments process q is riskier than the endowments process q ′ thenthe total supply of each consumption good does not change and the relative total supply of eachconsumption good does not change either (see more details in the proof of Theorem 3). Thisfact plays a major rule in the proof of Theorem 3, in particular, in proving that the prices ofconsumption goods do not change.When the the agents’ preferences are not represented by a CES utility function, Theorem 3does not necessarily hold. Since the excess demand function does not satisfy a gross substituteproperty we are not able to prove comparative statics results for a utility function that is nota CES utility function. It would be interesting to explore the connection between the prices ofgoods and the risk of the endowment process for different utility functions. Mean field equilibrium models have been popularized in the recent literature in operationsresearch, economics and optimal control. In a mean field model, the agents’ utility functionsand the evolution of the agents’ states depend on the distribution of the other agents’ states. Ina mean field equilibrium, each agent maximizes his expected discounted payoff, assuming thatthe distribution of the other agents’ states is fixed. Given the agents’ strategy, the distributionof the agents’ states is an invariant distribution of the Markov process that governs the dynamicsof the agents’ states. While the notion of a mean field equilibrium is conceptually similar to thenotion of a CSE, we cannot write the dynamic incomplete markets Arrow-Debreu model studiedin this paper as a discrete-time mean field model. This is because the market-clearing conditions(see conditions (iii) and (iv) in Definition 1) are not consistent with the definition of a mean fieldequilibrium. Thus, we cannot apply the recent existence, uniqueness and comparative staticsresults developed for discrete-time mean field equilibrium models (e.g., Adlakha and Johari(2013), Acemoglu and Jensen (2015), and Light and Weintraub (2018)).
In this section we extend the model described in Section 2 to the case of ex-ante heterogeneousagents. We assume that the agents are heterogeneous in their preferences over consumptionbundles as well as in their endowments. Assume that before the process starts, each agenthas a type θ ∈ Θ. For simplicity we assume that Θ is a finite set. Each agent’s type is fixedthroughout the horizon. An agent with type θ ∈ Θ has preferences that are represented by For example, see Lasry and Lions (2007), Weintraub et al. (2008), and Iyer et al. (2014).
15 utility function U ( x , θ ) and receives an endowment y ( θ ) with a probability q ( y ( θ )) in eachperiod. Let φ be the probability mass function over the type space; φ ( θ ) is the mass of agentswhose type is θ ∈ Θ. Adding the argument θ ∈ Θ to the functions defined in Section 2, we canmodify the definitions of Section 2 to include the ex-ante heterogeneity of agents. For example, g ( a, p , r, θ ) is the savings policy function of type θ agents and x ∗ ( a − g ( a, p , r, θ ) , p , θ ) is thedemand function of type θ agents.Let A h = R × Θ be an extended state space for the model with ex-ante heterogeneous agents.If an agent’s extended state is a h = ( a, θ ) ∈ A h then the agent’s wealth is a and his type is θ .Let λ h be a probability measure over the extended state space, i.e., λ h ∈ P ( A h ).Define the Markov kernel Q h (( a, θ ) , D × E ) = X y ∈Y q ( y ( θ ))1 D ((1 + r ) g ( a, p , r, θ ) + p · y ( θ ))1 E ( θ )for any D × E ∈ B ( R ) × Θ . The Markov kernel Q h describes the evolution of the extendedstate. That is, when the agent’s wealth is a and his type is θ , the probability that the nextperiod’s pair of wealth-type will lie in D × E ∈ B ( R ) × Θ is given by Q h (( a, θ ) , D × E ).Define M λ h ( D × E ; p , r ) = Z X y ∈Y q ( y ( θ ))1 D ((1 + r ) g ( a, p , r, θ ) + p · y ( θ ))1 E ( θ ) λ h ( d ( a, θ ); p , r ) , for any D × E ∈ B ( R ) × Θ . M λ h ∈ P ( A h ) describes the next period’s wealth-types distribution,given that the current wealth-types distribution is λ h ∈ P ( A h ) and the prices are ( p , r ). Awealth-types distribution µ h ∈ P ( A h ) is called an invariant wealth-types distribution if µ h = M µ h .These definitions map the model with ex-ante heterogeneous agents to the model with ex-antehomogeneous agents that we considered in Section 2. We can define a competitive stationaryequilibrium as in Definition 1. A competitive stationary equilibrium consists of prices ( p , r ),savings policy functions g , demand functions x ∗ , and a wealth-types distribution µ h ∈ P ( A h )such that the savings policy function g and the demand function x ∗ are optimal for each type θ , the wealth-types distribution µ h is invariant, and the prices of goods and the interest rate aremarket-clearing (see conditions (iii) and (iv) in Definition 1).We note that if U ( x , θ ) satisfies Assumption 1 for each θ , then Theorem 1 holds and thereexists a CSE. The proof is similar to the proof of Theorem 1 so we omit the details.16 Final remarks
In this paper we study a dynamic incomplete markets Arrow-Debreu model which combines aHuggett-Bewley model with the classic static pure-exchange Arrow-Debreu model. We studya competitive stationary equilibrium where the prices of consumption goods and the interestrate are market clearing. Under mild conditions on the agents’ preferences, we prove thatthe aggregate demand for consumption goods is homogeneous of degree 0, while the aggregatedemand for savings is homogeneous of degree 1 (see Proposition 1). We prove the existence of acompetitive stationary equilibrium (CSE) (see Theorem 1). We provide conditions that ensurethe uniqueness of a CSE. Under a CES utility function, we discuss how a riskier endowmentsprocess affects wealth inequality, the prices of goods and the interest rate. We prove that if theagents’ preferences can be represented by a CES utility function with an elasticity of substitutionthat is equal to or higher than one, then there exists a unique CSE (see Theorem 2), and thata riskier endowments process increases the partial equilibrium wealth inequality, decreases theequilibrium interest rate, and does not change the equilibrium prices of goods (see Theorem 3).It remains an open question whether Theorem 2 and Theorem 3 can be extended to differentutility functions. Many other open questions remain concerning the CSE. For example, studyingthe stability of a CSE awaits future research.
In this section we prove Proposition 1.Recall that for a set K we denote by P ( K ) the set of all probability measures defined on K . We endow P ( R ) with the topology of weak convergence. We say that λ n ∈ P ( R ) convergesweakly to λ ∈ P ( R ) if for all bounded and continuous functions f : R → R we havelim n →∞ Z R f ( a ) λ n ( da ) = Z R f ( a ) λ ( da ). Proposition 1.
Fix p ≫ , r > with (1 + r ) β < , and θ > . Then(i) θg ( a, p , r ) = g ( θa, θ p , r ) and x ∗ ( a − g ( a, p , r ) , p ) = x ∗ ( θa − g ( θa, θ p , r ) , θ p ) for all a .(ii) ζ i ( θ p , r ) = ζ i ( p , r ) for ≤ i ≤ n and ζ n +1 ( θ p , r ) = θζ n +1 ( p , r ) .Thus, if ( p , r ) are equilibrium prices then ( θ p , r ) are also equilibrium prices. Proof. (i) Recall that a function f ( a, p , r ) is homogeneous of degree l ≥ a, p ) if f ( θa, θ p , r ) =17 l f ( a, p , r ) for all θ >
0. We now show that g is homogeneous of degree 1 in ( a, p ).Fix p ≫ θ > a ∈ R , and r ∈ (0 , β − f ( a, p , r ) is homogeneous of degree 0 in ( a, p ). We have T f ( a, p , r ) = max b ∈ C ( a, p ) max x ∈ X ( a − b, p ) U ( x ) + β X y ∈Y q ( y ) f ((1 + r ) b + p · y , p , r )= max θb ∈ C ( θa,θ p ) max x ∈ X ( θa − θb,θ p ) U ( x ) + β X y ∈Y q ( y ) f ((1 + r ) θb + θ p · y , θ p , r )= max z ∈ C ( θa,θ p ) max x ∈ X ( θa − z,θ p ) U ( x ) + β X y ∈Y q ( y ) f ((1 + r ) z + θ p · y , θ p , r )= T f ( θa, θ p , r ) . Thus,
T f ( a, p , r ) is homogeneous of degree 0 in ( a, p ). The first and fourth equalities follow fromthe definition of T f . The second equality follows from the facts that X ( a − b, p ) = X ( θa − θb, θ p ), b ∈ C ( a, p ) if and only if θb ∈ C ( θa, θ p ), and f ((1 + r ) θb + θ p · y , θ p , r ) = f ((1 + r ) b + p · y , p , r )for θ > n = 1 , , . . . , T n f is homogeneous of degree 0. From standarddynamic programming arguments, T n f converges to V uniformly. Since the set of functionsthat are homogeneous of degree 0 is closed under uniform convergence, V is homogeneous ofdegree zero.Let h ( a, b, p , r, f ) := v ( a, b, p ) + β X y ∈Y q ( y ) f ((1 + r ) b + p · y , p , r ) , where v ( a, b, p ) = max x ∈ X ( a − b, p ) U ( x ).Note that v is homogeneous of degree 0 in ( a, b, p ) since X ( a − b, p ) = X ( θa − θb, θ p ). Thus, h ( a, b, p , r, f ) is homogeneous of degree 0 in ( a, b, p ) whenever f is homogeneous of degree 0in ( a, p ). Since V is homogeneous of degree zero in ( a, p ) we conclude that h ( a, b, p , r, V ) is Recall that C ( a, p ) = [(1 − ψ ) − min y ∈Y p · y r , min { a, P ni =1 p i b } ]. a, b, p ). We have h ( θa, θg ( a, p , r ) , θ p , r, V ) = h ( a, g ( a, p , r ) , p , r, V )= max b ∈ C ( a, p ) h ( a, b, p , r, V )= V ( a, p , r )= V ( θa, θ p , r )= h ( θa, g ( θa, θ p , r ) , θ p , r, V ) . The single-valuedness of the savings policy function g (see Lemma 1) implies that g ( θa, θ p , r ) = θg ( a, p , r ). We conclude that g is homogeneous of degree 1 in ( a, p ).The following chain of equalities show that the demand function x ∗ is homogeneous of degree0 in ( a, p ): x ∗ ( a − g ( a, p , r ) , p ) = argmax x ∈ X ( a − g ( a, p ,r ) , p ) U ( x )= argmax x ∈ X ( θa − θg ( a, p ,r ) ,θ p ) U ( x )= argmax x ∈ X ( θa − g ( θa,θ p ,r ) ,θ p ) U ( x )= x ∗ ( θa − g ( θa, θ p , r ) , θ p ) . (ii) We say that a probability measure λ ( · ; p , r ) is homogeneous of degree l ≥ p if forevery continuous and bounded function f ( a, p ) that is homogeneous of degree l in ( a, p ) and all θ > Z f ( a, θ p ) λ ( da ; θ p , r ) = θ l Z f ( a, p ) λ ( da ; p , r ) . (4)We now show that the invariant wealth distribution µ is homogeneous of degree l ≥ p .Assume that the probability measure λ ( · ; p , r ) is homogeneous of degree l in p . Let f ( a, p )be a continuous and bounded function that is homogeneous of degree l in ( a, p ) and let θ > ≫
0, and r ∈ (0 , β − θ l Z f ( a, p ) M λ ( da ; p , r ) = θ l Z X y ∈Y q ( y ) f ((1 + r ) g ( a, p , r ) + p · y , p ) λ ( da ; p , r )= Z X y ∈Y q ( y ) f ((1 + r ) g ( a, θ p , r ) + θ p · y , θ p ) λ ( da ; θ p , r )= Z f ( a, θ p ) M λ ( da ; θ p , r ) . The first and last equalities follow from Equation (5) (see Lemma 4). The second equalityfollows from the facts that e f ( a, p ) := P y ∈Y q ( y ) f ((1 + r ) g ( a, p , r ) + p · y , p ) is homogeneous ofdegree l in ( a, p ) and λ is homogeneous of degree l in p . We conclude that for all k , M k λ ishomogeneous of degree l in p .From Lemma 2, M k λ converges weakly to µ for all ( p , r ) such that p ≫
0, and r ∈ (0 , β − f ( a, p ) that is homogeneous of degree l in ( a, p ),we have Z f ( a, θ p ) µ ( da ; θ p , r ) = lim k →∞ Z f ( a, θ p ) M k λ ( da ; θ p , r )= lim k →∞ θ l Z f ( a, p ) M k λ ( da ; p , r )= θ l Z f ( a, p ) µ ( da ; p , r ) . Thus, µ is homogeneous of degree l in p .From the fact that g ( a, p , r ) is a continuous function on R × R n ++ × (0 , β −
1) (see Lemma1) that is homogeneous of degree 1 in ( a, p ) and the fact that A ( p , r ) is compact for all p ≫ r > ζ n +1 ( θ p , r ) = Z g ( a, θ p , r ) µ ( da ; θ p , r ) = θ Z g ( a, p , r ) µ ( da ; p , r ) = θζ n +1 ( p , r ) . Similarly, since for all 1 ≤ i ≤ n the function x ∗ i ( a − g ( a, p , r ) , p ) is a continuous function that20s homogeneous of degree 0 in ( a, p ), we have ζ i ( θ p , r ) = Z x ∗ i ( a − g ( a, θ p , r ) , θ p ) µ ( da ; θ p , r ) − X y i ∈Y i q i ( y i ) y i = Z x ∗ i ( a − g ( a, p , r ) , p ) µ ( da ; p , r ) − X y i ∈Y i q i ( y i ) y i = ζ i ( p , r ) . Thus, if ( p , r ) are equilibrium prices, i.e., ζ ( p , r ) = 0, then ζ ( θ p , r ) = 0; and so ( θ p , r ) are alsoequilibrium prices. In this section we prove the existence of a competitive stationary equilibrium.
Theorem 1.
Suppose that Assumption 1 holds. Then, there exists a competitive stationaryequilibrium.
Recall that the sets Λ and P are given by Λ = { ( p , r ) ∈ R n + × R + : P ni =1 p i + r = β − } and P = { ( p , r ) ∈ Λ : p ≫ , r > } . The excess demand function ζ : P → R n +1 is given by ζ ( p , r ) = ( ζ ( p , r ) , . . . , ζ n ( p , r ) , ζ n +1 ( p , r ))where for i = 1 , . . . , n , ζ i ( p , r ) = Z A ( p ,r ) x ∗ i ( a − g ( a, p , r ) , p ) µ ( da ; p , r ) − X y i ∈Y i q i ( y i ) y i is the excess demand for good i , and ζ n +1 ( p , r ) = − Z A ( p ,r ) g ( a, p , r ) µ ( da ; p , r )is the excess demand for savings. Note that if ζ ( p , r ) = then ( p , r ) are equilibrium prices, µ ( · ; p , r ) is the equilibrium wealth distribution, x ∗ ( a − g ( a, p , r ) , p ) is the equilibrium demandfunction, and g ( a, p , r ) is the equilibrium savings policy function.For a proof of the following well-known proposition, see, for example, Theorem 1.4.8 in21liprantis et al. (1990)). Proposition 2
For a function ζ ( · ) = ( ζ ( · ) , ..., ζ n +1 ( · )) from P into R n +1 assume that:(i) ζ is continuous.(ii) ζ ( p , r ) satisfies Walras’ law, i.e., ( p , r ) · ζ ( p , r ) = 0 for all ( p , r ) ∈ P .(iii) ( p q , r q ) → ( p , r ) ∈ Λ \ P with { p q , r q } ⊆ P imply lim q →∞ k ζ ( p q , r q ) k = ∞ .(iv) { p q , r q } ⊆ P , ( p q , r q ) → ( p , r ) = ( p , . . . , p n , r ) and p k > imply that the sequence { ζ k ( p q , r q ) } of the k th components of { ζ ( p q , r q ) } is bounded. Similarly, r > implies that thesequence { ζ n +1 ( p q , r q ) } is bounded.(v) The excess demand function ζ ( p , r ) is bounded from below, i.e., there exists ξ > suchthat ζ i ( p , r ) ≥ − ξ for all ≤ i ≤ n + 1 and all ( p , r ) ∈ P .Then there exists at least one vector ( p , r ) ∈ P that satisfies ζ ( p , r ) = . The idea of the proof of Theorem 1 is to show that the conditions of Proposition 2 hold.Lemmas 1 and 2 show that the excess demand function ζ ( p , r ) is a well-defined function. InLemmas 4 and 5 we prove that the excess demand function is continuous. In Lemma 6 we provethat the excess demand function satisfies Walras’ law. In Lemma 7 and Lemma 8 we prove thatthe excess demand function satisfies the boundness and boundary conditions (conditions (iii),(iv) and (v) of Proposition 2). Thus, all the conditions of Proposition 2 hold and there exists aCSE. Lemma 1
The savings policy function g ( a, p , r ) is single-valued and continuous in ( a, p , r ) .The value function V ( a, p , r ) is continuous in ( a, p , r ) , increasing in a , and strictly concave in a . Proof.
Note that v ( a, b, p ) = max x ∈ X ( a − b, p ) U ( x ) is strictly concave in ( a, b ). To see this, let( a , b ) ∈ R , ( a , b ) ∈ R , γ ∈ [0 , a γ = γa + (1 − γ ) a , and b γ = γb + (1 − γ ) b .We have v ( a γ , b γ , p ) = max x ∈ X ( a γ − b γ , p ) U ( x ) ≥ U ( γ x ∗ ( a − b , p ) + (1 − γ ) x ∗ ( a − b , p )) > γU ( x ∗ ( a − b , p )) + (1 − γ ) U ( x ∗ ( a − b , p ))= γv ( a , b , p ) + (1 − γ ) v ( a , b , p ) . For a more general version of this proposition, see Debreu (1982) and Hildenbrand and Kirman (2014). For x ∈ R n we write k x k = n P j =1 | x i | . γ x ∗ ( a − b , p ) + (1 − γ ) x ∗ ( a − b , p ) ∈ X ( a γ − b γ , p ). The second inequality follows from the fact that U is strictly concave. We concludethat v is strictly concave in ( a, b ). Furthermore, since U is continuous and X ( a − b, p ) isa continuous correspondence, i.e., X is upper hemicontinuous and lower hemicontinuous, themaximum theorem (see Theorem 17.31 in Aliprantis and Border (2006)) implies that v ( a, b, p )is continuous. Since U is increasing, v is increasing in a . Now standard dynamic programmingarguments show that g ( a, p , r ) is single-valued and continuous in ( a, p , r ) and that V ( a, p , r ) iscontinuous, as well as strictly concave and increasing in a (see Chapter 9 in Stokey and Lucas(1989)). Lemma 2
For every ( p , r ) ∈ P there exists a unique invariant wealth distribution µ ( · ; p , r ) ∈P ( A ( p , r )) . Furthermore, for all λ ( · ; p , r ) ∈ P ( A ( p , r )) , the sequence of measures { M n λ } con-verges weakly to µ ( · ; p , r ) ∈ P ( A ( p , r )) . Proof.
Fix ( p , r ) ∈ P . Define the Markov chain Q ( a, D ) = X y ∈Y q ( y )1 D ((1 + r ) g ( a, p , r ) + p · y ) . for any D ∈ B ( A ( p , r )) where 1 D is the indicator function of the set D ∈ B ( A ( p , r )).We prove a more general result than the result stated in Lemma 2: we show that the Markovchain Q is uniformly ergodic. The proof follows a similar line to the proofs in Rabault (2002)and in Benhabib et al. (2015), so we only provide a sketch of the proof. The Markov chain Q is said to satisfy the Doeblin condition if there exists a positive integer n , ǫ > υ on A ( p , r ) such that Q n ( a, D ) ≥ ǫυ ( D ) for all a ∈ A ( p , r ) and all D ∈ B ( A ( p , r )). Under Assumption 1, the arguments in Proposition 3.1 inRabault (2002) yield that the borrowing constraint binds with positive probability after a finitenumber of periods for any initial wealth level a ∈ A ( p , r ). In other words, for any initial wealthlevel a ∈ A ( p , r ), we have g ( a, p , r ) = b with a positive probability after a finite number ofperiods. Thus, if we define the probability measure υ ( D ) = P y ∈Y q ( y )1 D ((1 + r ) b + p · y ),we can find a positive integer n and ǫ > Q n ( a, D ) ≥ ǫυ ( D ) for all a ∈ A ( p , r )and all D ∈ B ( A ( p , r )). We conclude that Q satisfies the Doeblin condition. From the factsthat M : P ( A ( p , r )) → P ( A ( p , r )) is continuous (see a more general result in Lemma 4) and Recall that the Markov chain Q is called uniformly ergodic if it has an invariant distribution µ andsup D ∈B ( A ( p ,r )) | Q n ( a, D ) − µ ( D ) | ≤ M ρ n for some ρ < M < ∞ and for all n ∈ N , a ∈ A ( p , r ). Clearly,if Q is uniformly ergodic then Lemma 2 holds. See also Schechtman and Escudero (1977), Ma et al. (2018), and Foss et al. (2018). ( A ( p , r )) is compact in the weak topology (since A ( p , r ) is compact), Schauder’s fixed-pointtheorem (see Corollary 17.56 in Aliprantis and Border (2006)) implies that M has at least onefixed point. That is, Q has at least one invariant distribution. A Markov chain that has aninvariant distribution and satisfies the Doeblin condition is uniformly ergodic (see Theorem 8in Roberts et al. (2004)). This completes the proof the Lemma.We say that w n : R → R converges continuously to w if w n ( a n ) → w ( a ) whenever a n → a .Lemma 3 provides a bounded convergence theorem with varying measures. For a proof, seeTheorem 3.3 in Serfozo (1982). We will use this Lemma to prove the continuity of the excessdemand function.
Lemma 3
Assume that w n : R → R is a uniformly bounded sequence of functions. If w n : R → R converges continuously to w and λ n ∈ P ( R ) converges weakly to λ ∈ P ( R ) then lim n →∞ Z w n ( a ) λ n ( da ) = Z w ( a ) λ ( da ) . Lemma 4
The unique invariant wealth distribution µ is continuous in ( p , r ) on P , i.e., if { p n , r n } converges to ( p , r ) , then µ ( p n , r n ) converges weakly to µ ( p , r ) . Proof.
First note that for every bounded and measurable function f : R → R and for all ( p , r )such that p ≫ r > Z f ( a ) M λ ( da ; p , r ) = Z X y ∈Y q ( y ) f ((1 + r ) g ( a, p , r ) + p · y ) λ ( da ; p , r ) . (5)To see this, note that if f = 1 D then Equality (5) holds from the definition of M . A standardargument shows that Equality (5) holds for any bounded and measurable f .Assume that { p n , r n } ⊆ P converges to ( p , r ) ∈ P . Let { µ ( p k , r k ) } be a subsequence of { µ ( p n , r n ) } that converges weakly to µ ( p , r ). Let f : R → R be a bounded and continuousfunction. From the continuity of the savings policy function g , we havelim k →∞ f ((1 + r k ) g ( a k , p k , r k ) + p k · y ) = f ((1 + r ) g ( a, p , r ) + p · y )whenever lim k →∞ ( a k , p k , r k ) = ( a, p , r ).Let us define w k ( a ) = P y ∈Y q ( y ) f ((1 + r k ) g ( a, p k , r k ) + p k · y ) and w ( a ) = P y ∈Y q ( y ) f ((1 + r ) g ( a, p , r ) + p · y ). Then, w k ( a ) is a uniformly bounded sequence of functions that convergescontinuously to w ( a ). See Feinberg et al. (2019) for a more general result of this type. k →∞ Z f ( a ) µ ( da ; p k , r k ) = lim k →∞ Z X y ∈Y q ( y ) f ((1 + r k ) g ( a, p k , r k ) + p k · y ) µ ( da ; p k , r k )= lim k →∞ Z w k ( a ) µ ( da ; p k , r k )= Z w ( a ) µ ( da ; p , r )= Z f ( a ) M µ ( da ; p , r ) . Since µ ( p k , r k ) converges weakly to µ ( p , r ), we also havelim k →∞ Z f ( a ) µ ( da ; p k , r k ) = Z f ( a ) µ ( da ; p , r ) . Thus, Z f ( a ) M µ ( da ; p , r ) = Z f ( a ) µ ( da ; p , r )which implies that µ = M µ , since µ and M µ are Borel probability measures that agree on allopen sets. From Lemma 2, µ is the unique fixed point of M , and thus, µ = µ .We conclude that each subsequence of { µ ( p n , r n ) } that converges weakly at all convergesweakly to µ ( p , r ). Furthermore, since A ( p , r ) is compact, for all ( p , r ) ∈ P we can assume thatthe supports of µ ( p n , r n ) and µ ( p , r ) are contained in a compact set so the sequence { µ ( p n , r n ) } is a tight sequence of probability measures. Thus, µ ( p n , r n ) converges weakly to µ ( p , r ) (see theCorollary after Theorem 25.10 in Billingsley (2008)). Lemma 5 ζ ( p , r ) is continuous on P . Proof.
Assume that the sequence { p n , r n } ⊆ P converges to ( p , r ) ∈ P . Fix i such that1 ≤ i ≤ n . Define w n ( a ) = x ∗ i ( a − g ( a, p n , r n ) , p n ) and w ( a ) = x ∗ i ( a − g ( a, p , r ) , p ). Thecontinuity of x ∗ i and of g imply that w n converges continuously to w , i.e., w n ( a n ) → w ( a )whenever a n → a . The sequence of functions { w n ( a ) } is bounded (see Lemma 8). Using Lemma25 and the fact that µ ( p n , r n ) converges weakly to µ ( p , r ) (see Lemma 4) yieldlim n →∞ ζ i ( p n , r n ) = lim n →∞ Z w n ( a ) µ ( da ; p n , r n ) − X y i ∈Y i q i ( y i ) y i = Z w ( a ) µ ( da ; p , r ) − X y i ∈Y i q i ( y i ) y i = ζ i ( p , r ) . Thus, ζ i ( p , r ) is continuous for 1 ≤ i ≤ n . A similar argument shows that ζ n +1 ( p , r ) is continu-ous. We conclude that ζ ( p , r ) is continuous. Lemma 6 ζ ( p , r ) satisfies Walras’ law, i.e., ( p , r ) · ζ ( p , r ) = 0 for all ( p , r ) ∈ P . Proof.
Fix ( p , r ) ∈ P . Equation (5) implies that Z aµ ( da ; p , r ) = Z X y ∈Y q ( y )((1 + r ) g ( a, p , r ) + p · y ) µ ( da ; p , r )= (1 + r ) Z g ( a, p , r ) µ ( da ; p , r ) + X y ∈Y q ( y ) p · y . Note that P y ∈Y q ( y ) p · y = n P i =1 p i P y i ∈Y i q i ( y i ) y i . To see this, let Y = { y , . . . , y l } and reason asfollows: X y ∈Y q ( y ) p · y = q ( y ) p · y + . . . + q ( y l ) p · y l = n X i =1 p i l X j =1 q ( y j ) y ji = n X i =1 p i X y i ∈Y i q i ( y i ) y i . From the agents’ budget constraints, we have p · x ∗ ( a − g ( a, p , r ) , p ) = a − g ( a, p , r ).The last equation implies n X i =1 p i Z x ∗ i ( a − g ( a, p , r ) , p ) µ ( da ; p , r ) = Z ( a − g ( a, p , r )) µ ( da ; p , r ) . p , r ) · ζ ( p , r ) = n X i =1 p i Z x ∗ i ( a − g ( a, p , r ) , p ) µ ( da ; p , r ) − n X i =1 p i X y i ∈Y i q i ( y i ) y i − r Z g ( a, p , r ) µ ( da ; p , r )= Z ( a − g ( a, p , r )) µ ( da ; p , r ) − n X i =1 p i X y i ∈Y i q i ( y i ) y i − r Z g ( a, p , r ) µ ( da ; p , r )= Z aµ ( da ; p , r ) − (1 + r ) Z g ( a, p , r ) µ ( da ; p , r ) − X y ∈Y q ( y ) p · y = 0 , which proves that ζ ( p , r ) satisfies Walras’ law. Lemma 7
The excess demand function ζ ( p , r ) is bounded from below, i.e., there exists ξ > such that ζ i ( p , r ) ≥ − ξ for all ≤ i ≤ n + 1 and all ( p , r ) ∈ P . Proof.
We have ζ i ( p , r ) ≥ − P y i ∈Y i q i ( y i ) y i for all 1 ≤ i ≤ n and all ( p , r ) ∈ P . Thus, ζ i isbounded from below for all 1 ≤ i ≤ n .Since g ( a, p , r ) is bounded from above by n P i =1 p i b we have R g ( a, p , r ) µ ( da ; p , r ) ≤ n P i =1 p i b , so ζ n +1 ( p , r ) = − Z g ( a, p , r ) µ ( da ; p , r ) ≥ − n X i =1 p i b ≥ − b (1 /β − p , r ) ∈ P . We conclude that the excess demand function is bounded from below. Lemma 8 (i) ( p q , r q ) → ( p , r ) ∈ Λ \ P with { p q , r q } ⊆ P imply lim q →∞ k ζ ( p q , r q ) k = ∞ .(ii) { p q , r q } ⊆ P , ( p q , r q ) → ( p , r ) = ( p , . . . , p n , r ) and p k > imply that the sequence { ζ k ( p q , r q ) } of the k th components of { ζ ( p q , r q ) } is bounded. Similarly, r > implies that thesequence { ζ n +1 ( p q , r q ) } is bounded. Proof. (i) Suppose that ( p q , r q ) → ( p , r ) = ( p , . . . , p n , r ) where ( p , r ) ∈ Λ \ P . We considertwo different cases.Case I: We have r q → r = 0. In this case the borrowing constraint tends to minus infinityand it follows from the same arguments as in page 673 in Aiyagari (1994) thatlim q →∞ Z g ( a, p q , r q ) µ ( da ; p q , r q ) = −∞ . Thus, we have lim q →∞ ζ n +1 ( p q , r q ) = ∞ which implies that lim q →∞ k ζ ( p q , r q ) k = ∞ .27ase II: We have r >
0. In this case ( p , r ) ∈ Λ \ P implies that p k = 0 for some 1 ≤ k ≤ n .Since the utility function U is strictly increasing, a standard argument shows that the demandfor some good tends to infinity, and thus, lim q →∞ P ni =1 x i ( a − g ( a, p q , r q ) , p q ) = ∞ (see forexample Theorem 1.4.6 in Aliprantis et al. (1990)). We conclude that lim q →∞ k ζ ( p q , r q ) k = ∞ .(ii) Assume that { p q , r q } is a sequence of strictly positive prices satisfying the conditionsof the Lemma where p q = ( p q , . . . , p qn ). Since p k > ≤ k ≤ n and ( p q , r q ) → ( p , r ), we infer that there exists some ǫ > p qk > ǫ for all q . We can assume that P y ∈Y q ( y ) p q · y ≤ M for all q .We have p qk Z A ( p q ,r q ) x ∗ k ( a − g ( a, p q , r q ) , p q ) µ ( da ; p q , r q ) ≤ Z A ( p q ,r q ) ( a − g ( a, p q , r q )) µ ( da ; p q , r q ).The last inequality implies that Z A ( p q ,r q ) x ∗ k ( a − g ( a, p q , r q ) , p q ) µ ( da ; p q , r q ) ≤ R A ( p q ,r q ) ( a − g ( a, p q , r q )) µ ( da ; p q , r q ) p qk = R A ( p q ,r q ) r q g ( a, p q , r q ) µ ( da ; p q , r q ) + P y ∈Y q ( y ) p q · y p qk ≤ r q P ni =1 p qi b + P y ∈Y q ( y ) p q · y p qk ≤ (1 /β − b + Mǫ .
The equality follows from Equation (5). The second inequality follows since g ( a, p q , r q ) ≤ P p qi b for all a ∈ A ( p q , r q ). Therefore, the sequence { ζ k ( p q , r q ) } is bounded for 1 ≤ k ≤ n .Now assume that r >
0. In this case, we can assume that there exists δ > r q > δ for all q . We can also assume that (1 − ψ ) min y ∈Y p q · y ≤ M for all q .Using the borrowing constraint, we have − Z A ( p q ,r q ) g ( a, p q , r q ) µ ( da ; p q , r q ) ≤ (1 − ψ ) min y ∈Y p q y r q ≤ Mδ .
Therefore the sequence { ζ n +1 ( p q , r q ) } is bounded from above. From Lemma 7, { ζ n +1 ( p q , r q ) } isbounded from below. The proof of the Lemma is completed.We proved that the excess demand function ζ satisfies the properties of Proposition 2. Thus,a CSE exists. 28 .3 The uniqueness of a competitive stationary equilibrium In this section we prove Theorem 2.
Theorem 2.
Assume that U ( x ) = P ni =1 α i x γi for some < γ < , α i > , P ni =1 α i = 1 . Thenthere exists a unique competitive stationary equilibrium. Proof.
Since the savings policy function g , the demand function x ∗ , and the invariant wealthdistribution µ are unique given fixed prices ( p , r ), it is enough to show that the prices ( p , r )that clear the market are unique in order to prove the uniqueness of a CSE. The proof involvesa number of steps. Step 1.
If ( p , r ) and ( p ′ , r ′ ) are equilibrium prices, then p = p ′ . Suppose, in contradiction,that there are equilibrium prices ( p , r ) and ( p ′ , r ′ ) such that p ′ = p , and p and p ′ are notlinearly independent. From Proposition 1, we can normalize the prices such that p ≥ p ′ and p ′ k = p k = 1 for some 1 ≤ k ≤ n . We have Z aµ ( da ; p , r ) = X y ∈Y q ( y )((1 + r ) Z g ( a, p , r ) µ ( da ; p , r ) + p · y ) = X y ∈Y q ( y ) p · y . The first equality follows from Equation (5). The second equality follows from the fact that( p , r ) are equilibrium prices.Similarly, R aµ ( da ; p ′ , r ′ ) = P y ∈Y q ( y ) p ′ · y . Since y ≫ R aµ ( da ; p ′ , r ′ ) < R aµ ( da ; p , r ).Using the fact that R g ( a, p ′ , r ′ ) µ ( da ; p ′ , r ′ ) = R g ( a, p , r ) µ ( da ; p , r ) = 0 we have Z ( a − g ( a, p ′ , r ′ )) µ ( da ; p ′ , r ′ ) < Z ( a − g ( a, p , r )) µ ( da ; p , r ) . Since the utility function is in the constant elasticity of substitution class, it is well known andeasy to check that x ∗ ( a − g ( a, p , r ) , p ) = ( z ( p )( a − g ( a, p , r )) , ..., z n ( p )( a − g ( a, p , r )) where z i ( p ) is a positive function for each i = 1 , . . . , n . Thus, x ∗ i is linear in the total expenditure a − g ( a, p , r ) for all i . From the assumption that the elasticity of substitution is higher thanone, the demand for each good increases with the prices of the other goods. Since p ≥ p ′ and p ′ k = p k = 1 we have z k ( p ) ≥ z k ( p ′ ). 29e have Z x ∗ k ( a − g ( a, p , r ) , p ) µ ( da ; p , r ) = Z z k ( p )( a − g ( a, p , r )) µ ( da ; p , r ) > Z z k ( p ′ )( a − g ( a, p ′ , r ′ )) µ ( da ; p ′ , r ′ )= Z x ∗ k ( a − g ( a, p ′ , r ′ ) , p ′ ) µ ( da ; p ′ , r ′ ) , which leads to the contradiction 0 = ζ k ( p , r ) > ζ k ( p ′ , r ′ ) = 0. Step 2. g ( a, p , r ) is increasing and convex in a for all ( p , r ) ∈ P . It is easy to check thatthe indirect utility function v ( a, b, p ) = max x ∈ X ( a − b, p ) U ( x ) is given by v ( a, b, p ) = ( a − b ) γ z ( p )where z ( p ) is a positive function. The indirect utility function is a constant relative risk aversionutility function and thus the savings policy function is convex in a (for example, we can applyTheorem 4 in Jensen (2017) or the results in Huggett (2004)).To show that g is increasing in a , note that v ( a, b, p ) has increasing differences in ( a, b ) (recallthat a function v is said to have increasing differences in ( a, b ) if for all a ≥ a and b ≥ b wehave v ( a , b , p ) − v ( a , b , p ) ≥ v ( a , b , p ) − v ( a , b , p )). Thus, the function v ( a, b, p ) + β X y ∈Y q ( y ) V ((1 + r ) b + p · y , p , r )has increasing differences in ( a, b ) as the sum of functions with increasing differences. NowTheorem 6.1 in Topkis (1978) implies that g ( a, p , r ) is increasing in a . Step 3. g ( a, p , r ) is increasing in r on I = (0 , /β −
1) for all ( a, p ). The proof of this resultfollows from similar arguments to the arguments in the proof of Theorem 1 in Light (2017).Since the current setting is different from the setting in Light (2017) we provide the proof here.Assume that f ( a, p , r ) is a bounded function that is increasing, concave and continuouslydifferentiable in a with the following properties: (i) f has increasing differences in ( a, r ); (ii) af a ( a, p , r ) is increasing in a on R + (for a function f we denote by f a the derivative of f withrespect to a ). Let r > r ′ . We have(1 + r ) f a ((1 + r ) b + p · y , p , r ) ≥ (1 + r ′ ) f a ((1 + r ′ ) b + p · y , p , r ) ≥ (1 + r ′ ) f a ((1 + r ′ ) b + p · y , p , r ′ ) . The first inequality follows from property (ii) if b >
0, and from the concavity of f if b ≤ To see this, let a = (1 + z ) b + p · y . Then af a ( a, p , r ) = b (1 + z ) f a ((1 + z ) b + p · y , p , r ) + p · y f a ((1 + z ) b + p · f ((1 + r ) b + p · y , p , r ) with respect to b is increasing in r . We conclude that f ((1 + r ) b + p · y , p , r )has increasing differences in ( b, r ). Thus, the function v ( a, b, p ) + β X y ∈Y q ( y ) f ((1 + r ) b + p · y , p , r )has increasing differences in ( b, r ) as the sum of functions with increasing differences. Recallthat C ( a, p , r ) = [(1 − ψ ) − min y ∈Y p · y r , min { a, P ni =1 p i b } ] is the interval from which an agent maychoose his level of savings. Note that C is ascending in r (i.e., r ≥ r , b ∈ C ( a, p , r ), and b ′ ∈ C ( a, p , r ) imply max { b, b ′ } ∈ C ( a, p , r ) and min { b, b ′ } ∈ C ( a, p , r )). Theorem 6.1 inTopkis (1978) implies that g f ( a, p , r ) := argmax b ∈ C ( a, p ,r ) v ( a, b, p ) + β X y ∈Y q ( y ) f ((1 + r ) b + p · y , p , r )is increasing in r . The envelope theorem (see Benveniste and Scheinkman (1979)) implies that T f is differentiable and (
T f ) a ( a, p , r ) = v a ( a, g f ( a, p , r ) , p ) when a − g ( a, p , r ) > v has increasing differences in ( a, b ) and that g f ( a, p , r ) ≥ g f ( a, p , r ′ )yield ( T f ) a ( a, p , r ) = v a ( a, g f ( a, p , r ) , p ) ≥ v a ( a, g f ( a, p , r ′ ) , p ) = ( T f ) a ( a, p , r ′ ) . Thus,
T f has increasing differences in ( a, r ). Let a ≥
0. We have a ( T f ) a ( a, p , r ) = av a ( a, g f ( a, p , r ) , p )= aγ ( a − g f ( a, p , r )) γ − z ( p )= aa − g f ( a, p , r ) γ ( a − g f ( a, p , r ) γ z ( p ) . Since
T f is concave in a (see Lemma 1) for a ≥ a ′ we have γ ( a − g f ( a, p , r )) γ − z ( p ) = ( T f ) a ( a, p , r ) ≤ ( T f ) a ( a ′ , p , r ) = γ ( a ′ − g f ( a ′ , p , r )) γ − z ( p )which implies that the function a − g f ( a, p , r ) is increasing in a . We conclude that the function y , p , r ). The facts that af a ( a, p , r ) is increasing in a on R + and f a is decreasing in a imply that (1 + z ) f a ((1 + z ) b + p · y , p , r ) is increasing in z on I . Note that if f a is strictly decreasing, then (1 + r ) f a ((1 + r ) b + p · y , p , r )is strictly increasing in r . a − g f ( a, p , r )) γ is increasing in a . Furthermore, the function aa − g f ( a, p ,r ) is increasing in a on R + . Thus, a ( T f ) a ( a, p , r ) is increasing on R + as the product of two positive increasing functions.Define f n = T n f := T ( T n − f ) for n = 1 , , ... where T f := f . We conclude that f n ( a, p , r )is bounded, concave, increasing, and differentiable in a with increasing differences in ( a, r ), andthat af na ( a, p , r ) is increasing in a on R + for all n . The argument above shows that g f n ( a, p , r )is increasing in r for all n . Theorem 3.8 and Theorem 9.9 in Stokey and Lucas (1989) show that g f n converges pointwise to g . Thus, the savings policy function g is increasing in r . Furthermore,lim n →∞ f na ( a, p , r ) = lim n →∞ γ ( a − g f n ( a, p , r )) γ − z ( p ) = γ ( a − g ( a, p , r )) γ − z ( p ) = V a ( a, p , r ) . Thus, aV a ( a, p , r ) is increasing in a on R + and has increasing differences in ( a, r ). The sameargument as the argument above shows that the savings policy function g is increasing in r . Step 4.
If ( p , r ) and ( p , r ′ ) are equilibrium prices with r > r ′ then R g ( a, p , r ) µ ( da ; p , r ) > R g ( a, p , r ′ ) µ ( da ; p , r ′ ).Let r > r ′ . We first show that g ( a, p , r ) > g ( a, p , r ′ ) for all a ∈ e A , and all p ≫ e A = { a : g ( a, p , r ′ ) ∈ int C ( a, p , r ′ ) } is the set of wealth levels such that the optimal savingsdecision is interior. Suppose, in contradiction, that g ( a, p , r ′ ) = g ( a, p , r ) for some a ∈ e A . Since V is differentiable and strictly concave in a (see Lemma 1), the arguments in Step 3 imply thatthe function (1 + r ) V a ((1 + r ) b + p · y , p , r ) is strictly increasing in r . The first order conditionimplies that0 = − z ( p ) γ ( a − g ( a, p , r ′ )) γ − + β (1 + r ′ ) X y ∈Y q ( y ) V a ((1 + r ′ ) g ( a, p , r ′ ) + p · y , p , r ′ ) < − z ( p ) γ ( a − g ( a, p , r )) γ − + β (1 + r ) X y ∈Y q ( y ) V a ((1 + r ) g ( a, p , r ) + p · y , p , r ) ≤ e λ , e λ ∈ P ( R ) we define the partial order (cid:23) I by e λ (cid:23) I e λ if and only if R f ( a ) λ ( da ) ≥ To see this, note that a − g f ( a, p , r ) := k ( a ) is concave since g f is convex in a . Thus, for a ′ > a ≥ k ( a ′ ) − k ( a ) a ′ − a ≤ k ( a ′ ) − k (0) a ′ − . Rearranging and using the fact that k (0) > a ′ k ( a ′ ) ≥ ak ( a ) . f ( a ) λ ( da ) for every increasing function f .Assume that λ ( · , p , r ) (cid:23) I λ ( · , p , r ′ ). Then, for every increasing function f we have Z f ( a ) M λ ( da ; p , r ) = Z X y ∈Y q ( y ) f ((1 + r ) g ( a, p , r ) + p · y ) λ ( da ; p , r ) ≥ Z X y ∈Y q ( y ) f ((1 + r ′ ) g ( a, p , r ′ ) + p · y ) λ ( da ; p , r ) ≥ Z X y ∈Y q ( y ) f ((1 + r ′ ) g ( a, p , r ′ ) + p · y ) λ ( da ; p , r ′ )= Z f ( a ) M λ ( da ; p , r ′ ) . The equalities follow from Equation (5) (see Lemma 4). The first inequality follows from thefact that g is increasing in r . The second inequality follows from the facts that g is increasing in a and λ ( · ; p , r ) (cid:23) I λ ( · ; p , r ′ ). We conclude that M k λ ( · ; p , r ) (cid:23) I M k λ ( · ; p , r ′ ) for all k = 1 , , ... .From Lemma 2, the sequence { M k λ } converges weakly to µ for all ( p , r ). Since (cid:23) I is closedunder weak convergence, we conclude that µ ( · ; p , r ) (cid:23) I µ ( · ; p , r ′ ).Suppose that ( p , r ) and ( p , r ′ ) are equilibrium prices with r > r ′ . We have Z g ( a, p , r ) µ ( da ; p , r ) > Z g ( a, p , r ′ ) µ ( da ; p , r ) ≥ Z g ( a, p , r ′ ) µ ( da ; p , r ′ ) . The first inequality follows from the fact that g is strictly increasing in r on e A (and we have µ ( e A ; p , r ) > p , r ) are equilibrium prices). The second inequality follows from the factsthat g is increasing in a and µ ( · ; p , r ) (cid:23) I µ ( · ; p , r ′ ). Step 5.
Suppose that ( p , r ) and ( p ′ , r ′ ) are equilibrium prices. From Step 1, we know that p ′ = p . From Step 4, if r > r ′ then 0 = R g ( a, p , r ) µ ( da ; p , r ) > R g ( a, p , r ′ ) µ ( da ; p , r ′ ) = 0which is a contradiction. We conclude that ( p , r )= ( p ′ , r ′ ). Thus, there is at most one CSE. Iteasy to see that Assumptions 1 is satisfied so Theorem 1 implies that there exists at least oneCSE. We conclude that there is a unique CSE. In this section we prove Theorem 3.
Theorem 3
Assume that U ( x ) = P ni =1 α i x γi for some < γ < , α i > , P ni =1 α i = 1 . Assumethat the endowments process q is riskier than the endowments process q ′ . Then(i) The partial equilibrium wealth inequality is higher under q than under q ′ , i.e., µ ( · ; p , r, q ) (cid:23) I − CX ( · ; p , r, q ′ ) for all ( p , r ) ∈ P . In addition, if ( p ( q ) , r ( q )) are equilibrium prices under the en-dowments process q then µ ( · ; p ( q ) , r ( q ) , q ) (cid:23) CX µ ( · ; p ( q ) , r ( q ) , q ′ ) .(ii) The equilibrium prices of goods do not change, i.e., p ( q ) = p ( q ′ ) . The equilibrium interestrate is lower under q than under q ′ , i.e., r ( q ′ ) ≥ r ( q ) . Proof. (i) Fix ( p , r ) ∈ P . Assume that the endowments process q is riskier than the endowmentsprocess q ′ . From Theorem 2 in Light (2018), we can show that g ( a, p , r, q ) ≥ g ( a, p , r, q ′ ) for all( a, p , r ) ∈ A × P .Suppose that λ ( · ; p , r, q ) (cid:23) I − CX λ ( · ; p , r, q ′ ). Then for every convex and increasing function f we have Z f ( a ) M λ ( da ; p , r, q ) = Z X y ∈Y q ( y ) f ((1 + r ) g ( a, p , r, q ) + p · y ) λ ( da ; p , r, q ) ≥ Z X y ∈Y q ′ ( y ) f ((1 + r ) g ( a, p , r, q ) + p · y ) λ ( da ; p , r, q ) ≥ Z X y ∈Y q ′ ( y ) f ((1 + r ) g ( a, p , r, q ′ ) + p · y ) λ ( da ; p , r, q ) ≥ Z X y ∈Y q ′ ( y ) f ((1 + r ) g ( a, p , r, q ′ ) + p · y ) λ ( da ; p , r, q ′ )= Z f ( a ) M λ ( da ; p, r, q ′ ) . The equalities follow from Equation (5) (see Lemma 4). The first inequality follows from thefact that f ((1 + r ) g ( a, p , r, q ) + p · y ) is convex in y as the composition of a convex and increasingfunction with a convex function. The second inequality follows from the facts that g ( a, p , r, q ) ≥ g ( a, p , r, q ′ ) and f is increasing. The third inequality follows from the fact that g is convex andincreasing in a (see Step 2 in the proof of Theorem 2), which implies that f ((1+ r ) g ( a, p , r )+ p · y )is convex and increasing in a , and from the fact that λ ( · ; p , r, q ) (cid:23) I − CX λ ( · ; p , r, q ′ ).We conclude that M k λ ( · ; p , r, q ) (cid:23) I − CX M k λ ( · ; p , r, q ′ ) for all k = 1 , , ... . From Lemma2, the sequence { M k λ } converges weakly to µ for all ( p , r ). Since under our assumptions(see Theorem 1.5.9 in M¨uller and Stoyan (2002)) (cid:23) I − CX is closed under weak convergence, weconclude that µ ( · ; p , r, q ) (cid:23) I − CX µ ( · ; p , r, q ′ ).Now assume that ( p ( q ) , r ( q )) are equilibrium prices under the endowment process q , so Z g ( a, p ( q ) , r ( q )) µ ( da ; p ( q ) , r ( q ) , q ) = 0 . (cid:23) CX q ′ and the linearity of the function p · y imply that P q ( y ) p · y = P q ′ ( y ) p · y . Wehave Z aµ ( da ; p ( q ) , r ( q ) , q ) = X y ∈Y q ( y )((1 + r ( q )) Z g ( a, p ( q ) , r ( q )) µ ( da ; p ( q ) , r ( q ) , q ) + p ( q ) · y )= X y ∈Y q ( y ) p ( q ) · y = X y ∈Y q ′ ( y ) p ( q ) · y = Z aµ ( da ; p ( q ) , r ( q ) , q ′ ) . We proved that µ ( · ; p ( q ) , r ( q ) , q ) (cid:23) I − CX µ ( · ; p ( q ) , r ( q ) , q ′ ) and R aµ ( da ; p ( q ) , r ( q ) , q ) = R aµ ( da ; p ( q ) , r ( q ) , q ′ ).This implies that µ ( · ; p ( q ) , r ( q ) , q ) (cid:23) CX µ ( · ; p ( q ) , r ( q ) , q ′ ) (see Theorem 1.5.3 in M¨uller and Stoyan(2002)).(ii) Assume that ( p ( q ) , r ( q )) and ( p ( q ′ ) , r ( q ′ )) are equilibrium prices. Suppose, in contradic-tion, that p ( q ′ ) = p ( q ). From Proposition 1 we can normalize the prices and set p ( q ) ≥ p ( q ′ )and p ′ k = p k = 1 for some 1 ≤ k ≤ n .We have x ∗ ( a − g ( a, p , r, q ) , p ) = ( z ( p )( a − g ( a, p , r, q )) , ..., z n ( p )( a − g ( a, p , r, q )) where z i ( p ) is a positive function and z ( p ) ≥ z ( p ′ ) (see Step 1 of the Proof of Theorem 2).Since R aµ ( da ; p ( q ) , r ( q ) , q ) = R aµ ( da ; p ( q ) , r ( q ) , q ′ ) we have Z x ∗ k ( a − g ( a, p ( q ) , r ( q ) , q )) , p ) µ ( da ; p ( q ) , r ( q ) , q ) = z k ( p ( q )) Z ( a − g ( a, p ( q ) , r ( q ) , q )) µ ( da ; p ( q ) , r ( q ) , q )= z k ( p ( q )) Z aµ ( da ; p ( q ) , r ( q ) , q )= z k ( p ( q )) Z aµ ( da ; p ( q ) , r ( q ) , q ′ ) > z k ( p ( q ′ )) Z aµ ( da ; p ( q ′ ) , r ( q ′ ) , q ′ )= Z x ∗ k ( a − g ( a, p ( q ′ ) , r ( q ′ ) , q ′ )) µ ( da ; p ( q ′ ) , r ( q ′ ) , q ′ ) . The inequality follows from the same argument as in Step 1 of the proof of Theorem 2. Since q (cid:23) CX q ′ , we have q i (cid:23) CX q ′ i for all 1 ≤ i ≤ n (see Theorem 3.4.4. In M¨uller and Stoyan(2002)). Recall that q i (cid:23) CX q ′ i implies that P q ′ i ( y i ) y i = P q i ( y i ) y i . Thus, 0 = ζ k ( p ( q ) , r ( q ) , q ) >ζ k ( p ( q ′ ) , r ( q ′ ) , q ′ ) = 0 which is a contradiction. We conclude that p ( q ) = p ( q ′ ).35ow assume, in contradiction, that r ( q ) > r ( q ′ ). We have0 = Z g ( a, p ( q ) , r ( q ) , q ) µ ( da ; p ( q ) , r ( q ) , q ) > Z g ( a, p ( q ) , r ( q ′ ) , q ) µ ( da ; p ( q ) , r ( q ′ ) , q ) ≥ Z g ( a, p ( q ) , r ( q ′ ) , q ) µ ( da ; p ( q ) , r ( q ′ ) , q ′ ) ≥ Z g ( a, p ( q ′ ) , r ( q ′ ) , q ′ ) µ ( da ; p ( q ′ ) , r ( q ′ ) , q ′ ) = 0which is a contradiction. The first (strict) inequality follows from Step 4 of the proof of Theorem2. The second inequality follows from the facts that g is convex in a and µ ( · ; p , r, q ) (cid:23) CX µ ( · ; p , r, q ′ ). The third inequality follows from the facts that g ( a, p , r, q ) ≥ g ( a, p , r, q ′ ) and p ( q ) = p ( q ′ ). We conclude that r ( q ′ ) ≥ r ( q ). References
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