Arrow-Debreu Equilibrium

Transcription

1 Arrow-Debreu Equilibrium Econ 2100 Fall 2017 Lecture 23, November 21 Outline 1 Arrow-Debreu Equilibrium Recap 2 Arrow-Debreu Equilibrium With Only One Good 1 Pareto Effi ciency and Equilibrium 2 Properties of Interior Equilibria

2 Trade and Uncertainty Consumers know that, depending on the state, their wealth may fluctuate. This is because consumer i s initial endowment changes with the state s. maybe i s endowment is large in one state and small in another; or maybe i s endowment of desirable goods is large in some states and small in others. Without trade, consumers cannot insure themselves against these income and consumpion fluctuations. Trade allows for insurance. They they can sell their initial endowment in some states, and use the proceeds to buy consumption in other states; or they can sell some their initial endowment of particular goods in states in which they will have a lot of those, and use the proceeds to buy more of other goods i those states; or a combination of all these things. Markets for state-contingent goods make these trades possible.

3 Contingent Claims The set of dated events equals Γ = {(t, A) : 0 t T, A P t, for all t}}, where in a pair (t, A) the letter t is the date of occurrence of the event A. The partition P t is the set of events that occur up to time t. As information increases over time, then P t+1 refines P t, for all t. A contingent claim is an agreement to deliver or receive an amount of a specified commodity in a specified dated event. The set of all contingent claims is the set of all vectors of quantities of commodities in all dated events: R Γ L = {x : x is a function from Γ to R L} A dated event contingent commodity vector x R Γ L is a title to receive the commodity vector x (t,a) R L if and only if dated event (t, A) occurs. This specifies a quantity of each commodity l for each dated event: x (t,a),l, where (t, A) is a dated event in Γ, and l is one of the L commodities. A contingent claim is a title to receive or deliver some amount of the good if and only if a particular dated event occurs. Consumers and firms trade contingent claims (elements of R Γ L ) at given market prices.

4 Arrow-Debreu Economy With Complete Markets For all j, the firm production set is Y j R Γ N. For all i, preferences are over R Γ N + and the initial endowment is ω i R+ Γ N. There are Γ S markets open at date 1: p l(t,a) 0 is the price of one unit of good l to be delivered in dated event (t, A). Agents can trade any amount of contingent claims at prices p (t,a) R L +. Today agents exchange promises (contingent claims) for money. Definition A (x, y ) R Γ N + R Γ N and p R Γ N + are an Arrow-Debreu equilibrium if 1 For every j, y j 2 For every i, x i x i i x i for all x i 3 All markets clear: I I ω i + i=1 x i i=1 R SL maximizes profits: p yj p y j for all y j Y j R Γ N R SL is maximal in the budget set: x i X i R Γ N + : p x i p ω i + J j=1 y j J j=1 θ ji ( p y j ) with p l(t,a) = 0 when the inequality is strict

5 An Example: Exchange Economy Two individuals, one good, and two states In this exchange economy: I = 2, L = 1, S = 2, and T = 1 (there is no time). Each consumer s utility function is state independent and differentiable Initial endowments are π 1i u i (x 1i ) + π 2i u i (x 2i ) with i = A, B ω A = (1, 0) and ω B = (0, 1) There is no aggregate uncertainty: the aggregate endowment is the same in both states: ω = ω A + ω B = (1, 1) ; The only uncertainty is about who is the rich consumer. In an Edgeworth box, the certainty lines (what is that?) overlap. What is special about allocations on the certainty line? They provide Full Insurance.

6 An Example: Exchange Economy Pareto Optimality with two individuals, one good, and two states Marginal rates of substitution must be equal: If consumers have the same probabilities, π 1A = π 1B and π 1A π 2A The MRS condition becomes u A (x 1A) u A (x 2A) = u B (x 1B ) u B (x 2B ). Pareto optimal allocations do not depend on probabilities. π 1A u A (x 1A) π 2A u A (x 2A) = π 1B u B (x 1B ) π 2B u B (x 2B ) = π 1B π 2B. Equilibirum with two individuals, one good, and two states Utility maximization requires: π 1A u A (x 1A) π 2A u A (x 2A) = p1 p 2 These are similar to the conditions for Pareto optimality. If π 1A = π 1B the probabilities, again, drop out. and π 1B u B (x 1B ) π 2B u B (x 2B ) = p1 p 2 π 1A u A (x 1A) π 2A u A (x 2A) = p 1 p 2 = π 1B u B (x 1B ) π 2B u B (x 2B ) and thus π 1Au A (x 1A) π 2A u A (x 2A) = π 1B u B (x 1B ) π 2B u B (x 2B ). Equilibrium allocations do not depend on probabilities either, but the equilibrium prices do.

7 Another Exchange Economy Example Another economy with two individuals, one good, and two states Same economy as before, except for the initial endowments Initial endowments are ω A = (2, 0) and ω B = (0, 1) There is aggregate uncertainty: the aggregate endowment differ in the two states; the economy is richer in state 1. In an Edgeworth box, the certainty lines no longer overlap. Can we have full insurance of both consumers? The conditions for an allocation to be Pareto optimality (and an equilibrium) are the same as before. Should Pareto optimal allocations be different from the previous example? How about the competitive equilibirum allocations and the prices?

8 The One Good Timeless Economy From now on, we will focus on a special case of the general Arrow-Debreu model: there is only one future time period (T = 1 and no consumption at t = 0), there is only one commodity (L = 1), and there are no firms (except for the usual free-disposal technology). The first restriction is for notational simplicity, while the second one implies we can think about the one good being money. This is similar to what happened when we talked about uncertainty, where we thought about the utility fuction being an equivalent to the indirect utility function of standard consumer theory. Agents preferences follow the subjective expected utility model: x i, y i X i R S + x i i y i if and only if π si u si (x si ) π si u si (y si ) where u si ( ) represents the utility i derives from the money in state s, and π si is the probability consumer i assigns to state s. If the utility function is state-independent x i, y i X i R SL + x i i y i if and only if π si u i (x si ) π si u i (y si )

9 General Equilibirum and Financial Markets An exchange economy with L = 1 is a good approximation for an economy in which only financial securities are traded. In financial markets, agents trade particular contracts that are denominated in money. In this economy, a consumption bundle x i = (x 1i,.., x Si ) represents the promise to deliver some quantities of the good in each state. This good can be thought of as money; consumption bundles are financial instruments: Prices measure how much agents have to pay for the delivery of one unit of the only good (money) in state s. These are sometimes called state prices since they measure the value of money in each state. Assume that individuals have (weakly) concave vnm utility functions which are differentiable and strictly increasing. Use the first and second welfare theorem for the differentiable economy to characterize Pareto optimal allocations and equilibria.

10 Pareto Optimality Pareto Optimality A Pareto optimal allocation solves the planner s problem I max λ i U i (x i ) where U i (x i ) = i=1 π si u si (x si ) Using the results from a few classes back, we have the following. Pareto optimality in the Arrow-Debreu economy In an Arrow-Debreu economy, an allocation is Pareto optimal if and only if it solves: [ I ] λ i π si u si (x si ) subject to max x si 0,...,S I x si = i=1 i=1 I i=1 ω si for s = 1,..., S

11 Properties of Pareto Optimal Consumption Bundles Necessary conditions for Pareto optimality At an interior solution ˆx si, consumption is strictly positive in each state, and we have the following first order condition: λ i π si u si (ˆx si ) = µ s for s = 1,..., S for i = 1,..., I. where µ s is the Lagrange multiplier associated the feasibility condition of state s. Summing over states, one gets π si u si (ˆx si ) = λ i µ s for i = 1,..., I. Dividing the first order condition above by this expression we obtain: λ i π si u si (ˆx si ) λ i π si u si (ˆx si ) = π si u si (ˆx si ) π si u si (ˆx si ) = µ s µ s for s = 1,..., S and for i = 1,..., I. The right hand side does not depend on i... comparison with the equilibirum condition shortly.

12 Properties of Interior Pareto Optimal Allocations Necessary conditions for Pareto optimality An interior Pareto optimal allocation ˆx must satisfy π si u si (ˆx si ) π si u si (ˆx si ) = π sj u sj (ˆx sj ) π sj u sj (ˆx for s = 1,..., S sj ) for any i and j For any two states s and t this yields π si u si (ˆx si ) π si u si (ˆx si ) = π sj u sj (ˆx sj ) π sj u sj (ˆx sj ) for any i and j and π ti u ti (x ti ) S π si u si (ˆx si ) = π tj u tj (x tj ) S π sj u sj (ˆx sj ) for any i and j Therefore: π si u si (ˆx si ) π ti u ti (ˆx ti ) = π sj u sj (ˆx sj ) π tj u tj (ˆx tj ) for any i and j and s and t

13 Properties of Interior Pareto Optimal Allocations Necessary conditions for Pareto optimality An interior Pareto optimal allocation ˆx must satisfy π si u si (ˆx si ) π ti u ti (ˆx ti ) = π sj u sj (ˆx sj ) π tj u tj (ˆx for any i and j and s and t tj ) If π si = π sj for all s, then probabilities drop out and we are left with the marginal utility ratio for states s and t Use this fact to make conclusions about Pareto optima: u si (ˆx si ) u ti (ˆx ti ) u sj (ˆx sj ) u tj (ˆx tj ) which by monotonicity implies ˆx si ˆx ti 0 ˆx sj ˆx tj 0 A Pareto optima consumption bundle is comonotonic across agents: (x si x ti )(x sj x tj ) 0

14 Properties of Interior Competitive Equilibria Equilibrium in the Arrow-Debreu economy At an equilibrium x, p each consumer i solves max π si u si (x si ) x i subject to p x i p ω i and x i 0 The first order conditions evaluated at the optimum yield π si u si (x si ) = ηi p s for s = 1,..., S for i = 1,..., I. where η i is the Lagrange multiplier connected to consumer i budget constraint.

15 Equilibrium and Pareto Optimality Pareto optimality necessary conditions λ i π si u si (ˆx si ) = µ s for s = 1,..., S and i = 1,..., I. Arrow-Debreu equilibirum necessary conditions π si u si (x si ) = ηi p s for s = 1,..., S and i = 1,..., I. Welfare Theorems Since the welfare theorems hold: ηi = 1 and µ s = ps for s = 1,..., S and i = 1,..., I. λi Therefore π si u si (x si ) S π ti u ti (x ti ) = µ s µ t = p s p t for s = 1,..., S and i = 1,..., I. The right hand side is the same for everyone: prices are established in the market.

16 Equilibrium and Pareto Optimality The welfare theorems yield the following result ( ) π 1i u 1i (x 1i ) π 1j u 1j x 1j ( ) π ti u ti (x ti ) π tj u tj x tj π si u si (x si ) ( ) π sj u sj x sj π ti u ti (x ti ) = ( ) π tj u tj x = tj π Si u Si (x Si ) ( ) π Sj u Sj xsj π ti u ti (x ti ) ( ) π tj u tj xtj The better-than sets must have a common support. This support is given by the (normalized) price vector. p 1 p s p t p t ps p t

17 Equilibrium, Pareto Optimality, and Probabilities The support conditions define a probability distribution over the state space. Why? Each element of the vector ( ) π 1i u 1i (x 1i ) S π ti u ti (x ti ),, π si u si (x si ) S π ti u ti (x ti ),, π Si u Si (x Si ) S π ti u ti (x ti ) is a number between 0 and 1; and π si u si (x si ) S π ti u ti (x ti ) = 1 These are sometimes called risk-neutral probabilities. Why? If the vnm utility index is linear and state independent, then u si (xsi ) = constant π si u si (xsi ) πti u ti (x ti ) = π si constant π = πti constant si πti Therefore, the equilibrium state prices are equal to a common risk-neutral probability distribution. think of it as the probability distribution over the state space assigned by an hypothetical risk-neutral agent (the market).

18 Next Week Timing of Trades. Securities and Asset Prices.