Rational Expectations Equilibrium in Economies with Uncertain Delivery

Transcription

1 Rational Expectations Equilibrium in Economies with Uncertain Delivery João Correia-da-Silva Faculdade de Economia. Universidade do Porto. PORTUGAL. Carlos Hervés-Beloso RGEA. Facultad de Económicas. Universidad de Vigo. SPAIN. April 28th, 2007 Abstract. This paper studies general equilibrium with private and incomplete state verification. All trade is agreed ex ante, that is, before private information is received. Agents buy lists of bundles that give them the right to receive one of the bundles. With rational expectations, that is, knowledge of the selection mechanism, agents can predict which bundle will be delivered in each state of nature, and thus equate the utility of the list with the utility of this bundle. We show by counter-example that rational expectations equilibrium may fail to exist. We also show that a perfectly informed trader with arbitrarily small endowments is enough to guarantee existence of a rational expectations equilibrium. Keywords: General equilibrium, Differential information, Rational expectations, Uncertain delivery, Lists of bundles. JEL Classification Numbers: C62, D51, D82. João Correia-da-Silva (joao@fep.up.pt) acknowledges support from CEMPRE, Fundação para a Ciência e Tecnologia and FEDER. Carlos Hervés-Beloso (cherves@uvigo.es) acknowledges support from Ministerio de Ciencia y Tecnología and FEDER. 1

2 1 Introduction In chapter 7 of his Theory of Value, Debreu (1959) showed how the general equilibrium model could be extended to the case of uncertainty under public state verification. All that is needed is to consider a generalized notion of a commodity that also includes in its description the state of nature in which it is delivered (Arrow, 1953). The model becomes equivalent to the model without uncertainty: prices for the contingent commodities are announced, and agents choose the vector of contingent claims (specifying a bundle to be consumed in each of the possible states of nature) that they prefer, among those that satisfy their budget restriction. Then, the state of nature is publicly announced and agents receive the bundle that they bought for contingent delivery in the announced state. We are interested in studying the implications of differential information, in the form of private state verification. While keeping the basic structure of the model, we assume that each agent will only be able to verify (for example: in a court of law, for contracts to be enforced) that the state of nature belongs to a certain set. Our basic assumption is that if an agent bought different bundles for delivery in two states and is not able to verify whether the true state is one or the other, then the agent has to accept delivery of any of these two bundles. For example, consider an agent that cannot verify in a court of law whether the true state is 1 or 2, but nevertheless has bought A1 (delivery of good A in state 1) and B2 (delivery of good B in state 2). Then: if state 1 has occurred, the agent can receive A or B; and if state 2 occurred, the agent can also receive the same bundles A or B. Observe that this assumption implies that the set of alternatives that may be delivered is always the same in states that an agent cannot distinguish (a condition that is usually known as measurability ). This context may be dealt with by considering that objects of choice are lists such that the agents have the right to receive one of the bundles in the list. Notice that this concept extends Arrow s (1953) notion of contingent goods. Without loss of generality, we can restrict our attention to lists that are measurable with respect 2

3 to each agent s private information, since, as above, any non-measurable choice can be converted into a measurable one. It is important to understand that this measurability restriction (on lists ) is not an actual restriction on trade, but a consequence of incomplete state verification. We have introduced this general market structure in two previous papers (2006, 2007). Prices of the contingent lists are announced, and agents choose the vector of contingent lists that they prefer in their budget set. Then, the agents are able to verify in which set of their information partition lies the true state of nature, and receive one of the alternatives in the list that they bought for delivery in these states. Notice that agents cannot choose which alternative is delivered. On the contrary, they have to accept any of the alternatives. This market structure can be interpreted as consisting of many competitive brokers, who offer lists in exchange for the agent s endowments, and then trade among themselves in conditions of perfect information (a state of nature is internally announced to the brokers). In this internal market, an Arrow-Debreu equilibrium is generated, consisting of an allocation between brokers, together with equilibrium prices for the contingent goods. Brokers are profit maximizers, therefore, each broker buys (in the internal market) the cheapest of the alternatives in the list selected by the agent that the broker is dealing with. A no arbitrage condition implies that the price of a list should be equal to the price of the delivered bundle (brokers have no profits). The (internal) equilibrium defines, then, (1) the equilibrium prices of the lists; and (2) which of the alternative bundles is delivered to each agent. In previous papers, we have studied prudent expectations equilibrium (2006) and subjective expectations equilibrium (2007). The meaning of prudent expectations is that agents expected to receive the worst possible bundle in a list. The beliefs of the agents with subjective expectations about the probabilities of delivery of the different alternatives depend on the prices that they observe (perfectly or imperfectly), and on the alternatives specified in the list. In this paper, we study the existence of a rational expectations equilibrium. 3

4 We assume that agents observe prices for all the contingent commodities, without any informational restriction, and that they know the selection mechanism. As a result, they can predict which bundle is going to be selected for delivery (the cheapest) in each state of nature. In this setting, the market appears as a friendly and truthful mechanism. Prudent expectations were associated to the opposite paradigm, as agents expected to receive the worst alternative bundle to the agent. Since, by observing prices, agents can calculate what will be the delivered bundle (the cheapest), they know which consumption bundle results from each of the lists. Therefore, instead of choosing among lists, they can choose directly among these resulting consumption bundles. These bundles are those that satisfy a sort of endogenous incentive compatibility restrictions. Consider an agent who does not distinguish between states s and t. For a state-contingent consumption plan, (x s, x t ), to be delivered, it must be such that p s x s p s x t and p t x t p t x s. If these incentive compatibility conditions are not satisfied, then the agent will not receive x s in state s and x t in state t. An agent with rational expectations chooses among bundles which are incentive compatible in this sense. The choice set depends, therefore, on the equilibrium prices and on each agent s private information. If the correspondence between prices and incentive compatible bundles were continuous, then equilibrium existence would be guaranteed. However, lower hemi-continuity fails when prices in some state are null or when prices in different states are collinear (it is easy to see that the correspondence is upper hemi-continuous). Collinearity does not interfere with equilibrium existence because, in this case, it can be shown that (having convex preferences) agents choose the same bundle for delivery in both states, and in this case the incentive compatibility restrictions are obviously satisfied in equality. We give a simple example of non-existence of equilibrium caused by null prices. In the presence of differential information, prices may be null, even if state-contingent preferences are strictly monotonic. There may be some state in which resources are abundant, but such that no agent can verify that it has occurred. As a result, no agent is willing to buy commodities contingent on the occurrence of this state. 4

5 Introducing a perfectly informed agent removes this problem, because this agent can verify the occurrence of any state. We do this in a way that imposes a lower bound in prices, simply by assuming this agent s preferences to be linear. The main result in this paper establishes existence of equilibrium in an economy with this additional trader. The rest of the paper is organized as follows: in section 2 the model is presented; in section 3 an example of non-existence is given; in section 4 a small but perfectly informed trader is defined; and in section 5 existence of equilibrium is established (with this additional trader in the economy). 5

6 2 The Model We consider a finite number of agents (i = 1,..., n), commodities, (l = 1,..., L), and states of nature (Ω = {1,..., S}, indexed by s and also by t when necessary). The economy extends over two time periods, τ {0, 1}, with uncertainty about which state of nature will occur in the second period. Trade agreements are made at τ = 0 (all trade is ex ante). At τ = 1, agents receive their private information, trade agreements are enforced, and consumption takes place. The private information of agent i is represented by a partition of the set of states of nature, P i, such that agent i can distinguish states that belong to different sets of the partition. The set of states that agent i does not distinguish from state s is denoted P i (s). A function that is constant across elements of P i is said to be P i -measurable. Consumption of agent i in state s is x s i IR L +, and the contingent consumption plan of agent i is x i = (x s i ) s Ω IR SL +. A list is a finite set of bundles, indexed by k = 1,..., K. If an agent wants to guarantee delivery of a precise bundle, all the selected alternatives must be equal. The list selected by agent i for delivery in state s is denoted x s i IR KL +, with the k th alternative being x sk i agent i is x i IR SKL +. IR SL +. The plan of state-contingent lists selected by The economy extends over two time periods. In the first, taking prices as given, agents trade their state-contingent endowments for P i -measurable vectors of statecontingent lists, x i = ( x 1 i, x 2 i,..., x S i ), specifying the bundles that the market may deliver in each state of nature. In the second period, agents receive their information, and consume one of the bundles in the list that corresponds to the state of nature that occurs. bundles, x sk i, in the list x s i. If state s occurs, then agent i receives one of the Agents agree to receive more than they are entitled to. With the delivery of a bundle x sj i IR L + in state s, the market keeps the contract for delivery of any list 6

7 in X s (x sj i ), defined as: X s (x sj i ) = { x s i = ( x s1 i,..., x sk i ) IR KL + : k s.t. x sk i x sj i }. A more explicit definition of the same correspondence is: K X s (x s i ) = {(IR L +) k 1 [0, x s i ] (IR L +) K k }. 1 k=1 Each agent chooses a P i -measurable vector of contingent lists, x i = ( x 1 i,..., x S i ), thus it makes sense to extend this correspondence between deliveries and original lists to the whole set of states of nature. Delivery of the consumption plan x i = (x 1 i,..., x S i ) IR SL + keeps the contract for delivery of any plan of lists in X(x i ), defined as: 2 X(x i ) = X s (x 1 i ) X s (x 2 i )... X s (x S i ). As usual, the prices of the contingent commodities are normalized to the simplex: { } S L p SL + = p IR SL + : p sl = 1. s=1 l=1 The price of a plan of state-contingent lists, x i, is the price of the cheapest plan of state-contingent bundles, x i, that keeps the contract for the delivery of x. It is enough to determine the prices of the contingent goods (primitives). The prices of lists (derivatives) follow as a consequence. The price of a list, p( x i ), is: p s ( x s i ) = min{p s x sk i }; k S S p( x i ) = p s ( x s i ) = min{p s x sk i }. k s=1 s=1 Therefore, the budget restriction faced by agent i is: 1 In this definition, [0, x s i ] denotes the set of bundles ys i such that 0 ys i xs i. For example, in state s, with two alternatives (K = 2) and a single commodity: x s i = 1 implies X s (x s i ) = {[0, 1] IR + } {IR + [0, 1]}. This formulation makes it clear that X is a continuous correspondence, because it is defined by finite unions and finite products of continuous correspondences. 2 When we talk about delivery of a plan, the reader should be aware that only the coordinate that corresponds to the state of nature that occurs will be delivered. 7

8 B i (e i, p) = { x i IR SKL + : } S min{p s x sk i } p e i. k s=1 In an economy with uncertain delivery, E (e i, u i, P i, µ i ) n i=1: - A partition of Ω, P i, represents the private information of agent i. The set of states that agent i does not distinguish from state s is denoted P i (s). - Agents assign subjective probabilities to the different states of nature. To each state s, corresponds a prior probability µ s i, with S s=1 µ s i = 1. - For each state s, a rational expectations function, R s i ( x s i, p) : [0, T ] KL SL [0, T ] L, returns the bundle that agent i expects to receive, which is simply the cheapest bundle among the alternatives in the list. In case of a tie, we assume that the market delivers any of the bundles that the agent prefers (the utility maximizer among the cheapest bundles). 3 - Preferences are the same in undistinguished states, represented by a vector of Von Neumann-Morgenstern (1944) utility functions u s i : IR L + IR +, which are assumed to be continuous, weakly monotone and concave. The objective function combines beliefs with preferences for consumption: Ũ i ( x i, p) = Ss=1 µ s i u s i (R s i ( x s i, p)). - The initial endowments are constant across undistinguished states, and strictly positive: e s i 0 for all s Ω. The problem of agent i is to maximize the expected utility function, restricted to the budget set. S max Ũ i ( x i, p) = max µ s x i B i (e i,p) x i B i u s i (Ri s ( x s i, p)). i (e i,p) s=1 When several alternatives have the same price, the market is assumed to deliver the alternative that the agent prefers. Since the agent can calculate which alternative is delivered in each state of nature, the problem of the consumer can be 3 It will be clear that, as a result, it is enough for the consumption plan to satisfy the endogenous incentive compatibility in equality. 8

9 written as a decision among bundles instead of lists, under a set of incentive compatibility conditions that will be defined below. The set of bundles that satisfy these deliverability conditions, under prices p, is denoted Φ(p): S max U i(x i ) = max µ s i u s i (x s i ). 4 x i Φ i (p) B i (e i,p) x i Φ i (p) B i (e i,p) s=1 A rational expectations equilibrium of the economy with uncertain delivery is a pair, (x, p ), composed by a price system p and an allocation x = (x 1,..., x n). These are such that, for every agent i: (1) The bundle x i maximizes expected utility, U i (x i ), in the choice set, Φ i (p ) B i (e i, p ). (2) The bundles selected for delivery, x i, do not violate the endogenous incentive compatibility restrictions. That is, for all s and t P i (s), p s x s i p s x t i. (3) The allocation, x, is feasible. That is, i x i i e i = e T. Taking prices as given, each agent trades its initial endowments, e i, for a vector of bundles that maximizes expected utility, U i (x i ), belonging to the endogenous incentive compatible set, Φ i (e i, p ), and to the budget set. 5 4 Remember that the price of the list and the price of the delivered bundle coincide. 5 Remember that if state s occurs, agent i can only claim the right to receive one of the bundles x t i with t P i(s). Possible deliveries from a vector of state-contingent lists, x i, are given by a P i -measurable vector of lists, with the set of alternatives in each state s being x s i = t Pi(s){ x t i}. 9

10 3 Non-existence of equilibrium - an example Consider an economy in which two agents trade a single good under uncertainty. There are three states of nature, and their future endowments depend on the state of nature that occurs: e A = (100, 100, 1) and e B = (1, 100, 100). Agents only observe their endowments. P A = {{1, 2}; {3}} and P B = {{1}; {2, 3}}. Consumption must be positive, and a significant level of risk aversion induces agents to trade ex ante. U i : IR SL + IR; S U i (x i ) = µ s i x s i. s=1 For simplicity, assume that the different states occur with objective and publicly known probabilities: µ = (µ 1, µ 2, µ 3 ) = (0.45, 0.1, 0.45). Prices in states 1 and 3 must be strictly positive, or else the demands of agent B and A would be infinite for the corresponding contingent goods. With strictly positive prices for all the contingent goods, if agents selected different consumption levels in states that they did not distinguish, then, they would end up receiving the cheapest of the alternatives, which would be the lowest consumption level. In this case, we must have: x A = (x 12 A, x 12 A, x 3 A) and x B = (x 1 B, x 23 B, x 23 B ). Since agents are at the frontier of their budget sets: (p 1 + p 2 )x 12 A + p 3 x 3 A = 100(p 1 + p 2 ) + p 3 ; p 1 x 1 B + (p 2 + p 3 )x 23 B = p (p 2 + p 3 ). Adding the two: 10

11 p 1 (x 12 A + x 1 B) + p 2 (x 12 A + x 23 B ) + p 3 (x 3 A + x 23 B ) = 101p p p 3. For this to be an equilibrium, the allocation must be feasible: x 12 A + x 1 B 101; x 12 A + x 23 B 200; x 3 A + x 23 B 101. With strictly positive prices, the conditions are verified in equality. This implies that the allocation is of the form: x A = (x 12 A, x 12 A, x 3 A) = (x 3 A + 99, x 3 A + 99, x 3 A); x B = (x 1 B, x 23 B, x 23 B ) = (x 1 B, x 1 B + 99, x 1 B + 99). The only individually rational allocation of this form corresponds to the initial endowments. There is no trade. But are agents maximizing their utility levels? x A = (100, 100, 1); U(x A ) = = 5.95; x B = (1, 100, 100). U(x B ) = = Suppose that p 1 = p 3. Agent 1 can trade consumption in s 1 for consumption in s 3. But consuming less in s 1 implies that delivery in s 2 will also be of this lower quantity. In any case, the agent can select: x 1 = (x 12 1, x 12 1, x 3 1 ) = (81, 81, 20). The corresponding utility level is: U(x 1) = = In the case with asymmetric prices (p 1 p 3 ), the same trade is even more favorable for one of the agents. We reached a contradiction, implying that there is no equilibrium with strictly positive prices. With p 2 = 0, an alternative bundle can be big enough to violate feasibility and still be incentive compatible. The incentive compatibility restriction is not relevant because it is of the form 0 x 2 0 x s. Agents can choose a consumption level for state 2 that is big enough to violate feasibility and still desire to increase it. There cannot be a rational expectations equilibrium with p 2 = 0. 11

12 4 A catalyzer To avoid the existence of zero prices, suppose that there is an agent that has a very small endowment, but that is perfectly informed and has a linear utility function. This agent will induce a lower bound in prices. Below a certain price level, the agent would select a consumption level that violated feasibility for the whole economy. Let agent n + 1 be perfectly informed about the state of nature: P n+1 = {{1}, {2},..., {S}}. Let the utility of this small agent be a linear function of consumption, u s n+1(x s n+1) = v s n+1 x s n+1. Then, the objective function is: S S L U n+1 (x n+1 ) = µ s n+1vn+1 s x s n+1 = µ s n+1 vn+1x sl sl n+1. s=1 s=1 l=1 The endowments of this agent can be arbitrarily small. He will use all its endowments to buy a single contingent good, the one with the highest ratio between utility and price: µ s n+1v sl n+1/p sl. The quantity that the agent will buy is: x sl n+1 = p e n+1 p sl. It is obvious that a sufficiently small p sl will induce x sl n+1 > e sl T, violating feasibility. The demand of this agent will exceed the endowments of all the agents taken together. In practice, the effect of introducing this agent is to impose a strictly positive lower bound on equilibrium prices. 12

13 5 Existence of equilibrium 5.1 A sequence of economies In order to establish existence of equilibrium, we construct a sequence of economies. In these economies, the choice set is not constrained to satisfy the endogenous incentive compatibility restrictions. But violation of these restrictions implies an utility penalty. The penalty is a linear function of the greatest of the differences between the cheapest bundles and the bundles that are delivered. In the economy E j, the utility penalty imposed on agent i is equal to: Z j i (x i, p) = j max t P i (s) {ps x s i p s x t i}. Since s P i (s) it follows that the maximum is at least zero, thus penalties are never negative. Penalties increase along the sequence of economies, and this is actually the only difference between the economies in the sequence. Observe that, for sufficiently high j, the penalty is big enough for a bundle to have less utility than the initial endowments. In the economy E j, the utility functions of the agents are: S U j i (x i, p) = U i (x i ) j µ s i max t P s=1 i (s) {ps x s i p s x t i} = { } S = µ s i u s i (x s i ) j max t P s=1 i (s) {ps x s i p s x t i}. It is clear that, for any j IN, the utility functions are continuous in prices and bundles, (x i, p). The maximum of linear functions is a convex function, and multiplying a convex function by a negative constant, j, yields a concave function. Thus, the objective function U j i (x i, p) is also concave. The fact that the utility functions are also continuous functions of prices does not interfere with existence of equilibrium. 6 6 With price dependent preferences, it is known that equilibrium exists (Arrow and Hahn, 1971). In the context of economies with uncertain delivery, see our previous paper (2007). 13

14 The utility functions are the only thing that varies along the sequence of economies. The sequence of economies has a sequence of equilibria, (x j, p j ) j IN, in the compact set that contains the total endowments of the economy, [0, e T ] n SL, with e T = i e i. 7 There exists a subsequence that converges. For the limit, (x, p ), to be a Rational Expectations Equilibrium, the following conditions must be satisfied: (1) Feasibility: i x i i e i = e T ; (2) Budget restriction: i : p x i p e i ; (3) Incentive compatibility: i : x i Φ i (p ); (4) Optimality: i : x i B(p, e i ) Φ i (p ) U i (x i ) U i (x i ). 5.2 The first three conditions Conditions (1) and (2) are easy consequences of the fact that (x, p ) is the limit of a sequence of equilibria. (1) The set of feasible allocations is closed, and the limit allocation, x i, is the limit of a sequence of feasible allocations, therefore it is feasible. (2) The limit allocation, x i, is the limit of a sequence of allocations in the sequence of budget sets. It is straightforward to see that it also belongs to the limit budget set. Suppose that x i does not satisfy agent i s budget restriction. Let α = 3 e T + 1, and select ɛ > 0 such that p x i p e i = αɛ. Choosing a sufficiently high j, we can guarantee that d(x, x j ) < ɛ and d(p, p j ) < ɛ. With p j = p + dp, x j = x i + dx i, and manipulating: (p +dp) (x i +dx i ) (p +dp) e i = p x i p e i +p dx i +dp x i +dp dx i dp e i = = αɛ + (p + dp) dx i + dp (x i e i ) > αɛ ɛ ɛ 3 e T = 0. 7 The equilibrium allocation is always in [0, e T ] n because it is feasible. 14

15 This means that x j does not satisfy the budget restriction of E j. Contradiction. (3) The limit allocation, x, satisfies the incentive compatibility restrictions in the limit economy. To see this, suppose that x violated one of the restrictions by more than δ > 0, then, for a sufficiently high j, x j would also violate the same restriction by more than δ. That is, for t P s i, j 0 IN: p s x s > p s x t + δ p sj x sj > p sj x tj + δ, for all j > j 0. Utility among feasible allocations is bounded by U i (e T ), so we can consider a j that is sufficiently high for jδ > U i (e T ) U i (e i ). It would follow that U j i (x j ) < U i (x j ) jδ < U i (x j ) U i (e T ) + U i (e i ) < U i (e i ) = U j i (e i ), which is a contradiction. 5.3 The fourth condition The difficult part of the proof is to verify that the limit, (x, p ), maximizes the utility (4) of the agents in the incentive compatible budget set, B(p, e i ) Φ i (p ). The fact that Φ i is not lower hemicontinuous could prevent (x, p ) from being optimal. There could be an incentive compatible bundle y i B(p, e i ) Φ i (p ) that is not even nearly incentive compatible in any of the economies in the sequence. In spite of having a low utility level for high j, this bundle may be optimal in the original economy, and, in this case, (x, p ) would not be an equilibrium. The strategy to prove that (4) holds is to pick a point, y i in B(p, e i ) Φ i (p ), that is a possible optimum (being preferred to x i ) and find that a neighbor of y i also belongs to B(p j, e i ) Φ i (p j ), for large j. This would contradict that (x j, p j ) is an equilibrium, because the neighbor of y i would also be preferred to x j i in the economy E j. A preliminary remark Suppose that prices for delivery in s and in t P i (s) are parallel: p s = ap t. The two incentive compatibility conditions yield an equality. 15

16 p s y s i p s y t i p t y t i p t y s i ap t y s i ap t y t i p t y t i p t y s i p s y s i = p s y t i p t y t i = p t y s i The two consumption vectors, y s i and y t i, cost the same in both states. The utility functions u s i and u t i are also equal, because s and t belong to the same element of the agent s partition of information. If u s i (y s i ) > u s i (y t i), then the agent would be better off selecting y s i for consumption in both states. Thus, we must have u s i (y s i ) = u s i (y t i). Since the utility functions are concave, the agent is not worse off consuming the average bundle in both states. Notice that if the original vector satisfies the incentive compatibility conditions, then this average vector also does. Define the consumption vector x i by modifying y i, considering this average bundle whenever there are two parallel prices. Therefore, we will have x s i = x t i whenever p s = ap t. An auxiliary lemma We suppose, then, that there exists a x i B(p, e i ) Φ i (p ) such that U i (x i ) > U i (x i ), with x s i = x t i whenever p s = ap t. The neighbor x i = (1 δ)x i is still preferred to x i (for small δ > 0), also belongs to Φ i (p ), and belongs to the interior of the budget sets in E and E j (for high j). U i (x i) > U i (x i ); p x i < p e i ; p j x i < p j e i. Since the utility functions are continuous, there exists a radius ɛ > 0 such that the neighbors of x i are still preferred to x and, therefore, to x j, for high j (according to U i, which does not include the possible utility penalty): x i B(x i, ɛ) U i (x i ) > U i (x j i ), for sufficiently high j. We are assuming that the bundle x i satisfies the incentive compatibility conditions for the equilibrium prices p. Consider, without loss of generality, the following element of the agent s information partition: P i (s) = {1,..., s}. The incentive compatibility conditions for delivery in these states are: 16

17 p 1 x 1 i p 1 x 2 i ;... p 1 x 1 i p 1 x s i ; p 2 x 2 i p 2 x 1 i ;... p 2 x 2 i p 2 x s i ; p s x s i p s x 1 i ;... p s x s i p s x s 1 i. p 1 x 2 i p 1 x 1 i = k 12 0;... p 1 x s i p 1 x 1 i = k 1s 0; p 2 x 1 i p 2 x 2 i = k 21 0;... p 2 x s i p 2 x 2 i = k 2s 0; p s x 1 i p s x s i = k s1 0;... p s x s 1 i p s x s i = k s,s 1 0. It should be clear that this reasoning extends to all the elements of P i. Keep in mind that we seek a neighbor of x i that belongs to Φ i (p j ) and that therefore contradicts the fact that the allocation x j is an equilibrium of E j. This would prove (4) by contradiction. Let d(x i, x i) < ɛ. Consider a sufficiently high j for U i (x i ) > U i (x j i ) and also for d(p j, p ) < ɛ. Let dx i = x i x i and dp j = p j p. Manipulating, for example, the first condition: p 1 x 2 i p 1 x 1 i = (p j1 dp j1 ) (x 2 i dx 2 i ) (p j1 dp j1 ) (x 1 i dx 1 i ) = k 12 p j1 x 2 i p j1 x 1 i = k 12 +p j1 dx 2 i +dp 1 (x 2 i dx 2 i ) p j1 dx 1 i dp 1 (x 1 i dx 1 i ) p j1 x 2 i p j1 x 1 i > k 12 ɛ ɛ( e T + ɛ) ɛ ɛ( e T + ɛ) p j1 x 2 i p j1 x 1 i > k 12 2ɛ 2ɛ( e T + ɛ) = k 12 2ɛ( e T ɛ). Let ɛ 2 = 2ɛ( e T ɛ) > 0. We have: p j1 x 2 i p j1 x 1 i > k 12 ɛ 2. Let k min be the minimum over the set of strictly positive k ab. Choosing ɛ small enough to make ɛ 2 < k min guarantees that the strict inequalities for x i and p 17

18 remain strict for any x i B(x i, ɛ) and p j (with j large enough). If all inequalities still held, then U j i (x i ) = U i (x i ) > U i (x j i ) U j i (x j i ) and we would have a contradiction (the consumption bundle in the equilibrium sequence, x j i, would not be a maximizer of U j i, because x i would be preferred to x j i ). The difficulty lies in checking that the inequalities which are not strict at (x i, p ) are still satisfied at (x i, p j ) (with j large enough). ( Let γ ab = 1 p a p b ) p a, and let γ be the minimum of the strictly positive γ ab. Since we have a lower bound on equilibrium prices, γ ab is not zero, unless p a p b prices p a and p b are parallel. Consider a small deviation in prices, that is, a j that is large enough for: d(p j, p ) < ɛ 3 = ɛγ 8 e T. Keep x i sufficiently close to x i in order to preserve the strict inequalities, and select displacements parallel to prices: dx a i = ɛ 2 p a. p a Consider an inequality that is not strict, and suppose that the corresponding prices are not parallel. For example: p a x b i = p a x a i, implying that k ab = 0. Let s verify that this generic incentive compatibility condition is still verified in E j. p ja x b i p ja x a i = (p a + dp ja ) (x b i + dx b i) (p a + dp ja ) (x a i + dx a i ) = = p a (x b i + dx b i) + dp ja (x b i + dx b i) p a (x a i + dx a i ) dp ja (x a i + dx a i ) = = p a dx b i + dp ja (x b i + dx b i) p a dx a i dp ja (x a i + dx a i ) > > p a dx b i ɛ 3 ( e T + ɛ) p a dx a i ɛ 3 ( e T + ɛ) = = p a dx b i p a dx a i 2ɛ 3 ( e T + ɛ) > p b p a > p a ɛ + 2 p b p a ɛ 4ɛ 2 p a 3 e T = p a p b p a p a = ɛ 2 p a p b p a + ɛ 2 p a p a p a 4ɛ 3 e T = = ɛ 2 p a p a p a p b p a p a p a ɛ 2 p a p b p a ɛ γ = 2 = ɛ p a p b (1 2 p a p b ) p a ɛ γ 0 2 In sum, this displacement dx i implies that: 18

19 p ja x b i p ja x a i > 0. The condition is still verified, then. We need to check that this is also true if prices p a and p b are parallel. In this case: x a i = x b i and also dx a i = dx b i. Thus, x a i = x b i and the conditions remain satisfied in equality. All incentive compatibility conditions are satisfied, therefore: U j i (x i ) = U i (x i ) > U i (x j i ) U j i (x j i ). The consumption bundle in the equilibrium sequence, x j i, does not maximize U j i, because x i is preferred. This contradiction proves (4). Existence of a rational expectations equilibrium is, therefore, guaranteed if there is a lower bound on the prices of the contingent goods. For this bound to appear, we included an arbitrarily small, but perfectly informed trader in the economy. 19

20 Appendix: The endogenous incentive compatible set correspondence The set of bundles satisfying the incentive compatibility restrictions depends on the prevailing prices. Consider the correspondence from prices to the set of incentive compatible bundles: Φ i : Ωl + IR Ωl { Φ i (p) = + ; x IR Ωl + : ω s, p s x s = min t P i (s) {ps x t } If this correspondence were continuous, we could apply the theorem of existence of social equilibrium, yielding the result we seek: existence of rational expectations equilibrium in economies with uncertain delivery. In finite dimensional Euclidean spaces, upper hemicontinuity of Φ i at p 0 means that, given an arbitrary open set, V, containing Φ i (p 0 ), there exists δ > 0 such that for all p B(p 0, δ), we have Φ i (p) V. The correspondence is closed-valued since all the restrictions are inequalities which are not strict. With a compact range space, that is, in a bounded economy (for example, by the total initial endowments in the economy) a correspondence is upper hemicontinuous if and only if it has closed values. Therefore, Φ i is upper hemicontinuous. In finite dimensional Euclidean spaces, lower hemicontinuity of Φ i at p 0 means that given an arbitrary open set, V, intersecting Φ i (p 0 ), there exists δ > 0 such that for all p B(p 0, δ), the image Φ i (p) also intersects V. The correspondence under study, Φ i, is not lower hemicontinuous. Below is an example where this type of continuity fails. Consider an economy with two goods, A and B, and two states of nature, s and t. Let p 0 = (p s 0, pt 0 ) = (pas 0, pbs 0 ; pat 0, pbt 0 ) = ( 1 4, 1 4 ; 1 4, 1 4 ). The bundle x 0 = (1, 0; 0, 1) belongs to the incentive compatible set, since: p s 0 xs 0 ps 0 xt , and p t 0 xt 0 pt 0 xs }. Delivering (1, 0) in state s and (0, 1) in state t does not violate incentive compatibility because both bundles have the same price in both states. 20

21 A small perturbation in prices can make (0, 1) cheaper in state s and (1, 0) cheaper in state t. Consider an open ball around x 0 with radius 0 < ɛ < After a perturbation in prices to p = ( δ, 1 4 δ, 1 4 δ, δ), this ball does not intersect the incentive compatible set. Suppose that there existed a vector dx = (ɛ As, ɛ Bs, ɛ At, ɛ Bt ) such that x = (1 + ɛ As, ɛ Bs ; ɛ At, 1 + ɛ Bt ) is inside that open ball and belongs to the incentive compatible set: (1) ( δ, 1 4 δ) (1 + ɛas, ɛ Bs ) ( δ, 1 4 δ) (ɛat, 1 + ɛ Bt ) ( δ)(1 + ɛas ) + ( 1 4 δ)ɛbs ( δ)ɛat + ( 1 4 δ, )(1 + ɛbt ) ɛas + δ + δɛ As ɛbs δɛ Bs 1 4 ɛat + δɛ At ɛbt δ δɛ Bt 1 4 (ɛas + ɛ Bs ɛ At ɛ Bt ) + δ(ɛ As ɛ Bs ɛ At + ɛ Bt ) 2δ; (2) ( 1 4 δ, δ) (ɛat, 1 + ɛ Bt ) ( 1 4 δ, δ) (1 + ɛas, ɛ Bs ) ( 1 4 δ)ɛat + ( δ)(1 + ɛbt ) ( 1 4 δ)(1 + ɛas + ( δ)ɛbs ) 1 4 (ɛat ɛ Bt 1 ɛ As ɛ Bs ) + δ( ɛ At ɛ Bt + ɛ As ɛ Bs (ɛat + ɛ Bt ɛ As ɛ Bs ) + δ( ɛ At + ɛ Bt + ɛ As ɛ Bs ) 2δ. Adding the two inequalities, we obtain: (1 + 2) δ(ɛ As ɛ Bs ɛ At + ɛ Bt ) 2δ ɛ As ɛ Bs ɛ At + ɛ Bt 2. Which is impossible, because ɛ As ɛ Bs ɛ At + ɛ Bt 4ɛ > This is an example of the failure of lower hemicontinuity of the incentive compatible set correspondence. When prices are equal in two states, a small perturbation may induce a significant change in the incentive compatible set. 21

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