Uncertainty in Equilibrium

Uncertainty in Equilibrium Larry Blume May 1, 2007 1 Introduction The state-preference approach to uncertainty of Kenneth J. Arrow (1953) and Gérard Debreu (1959) lends itself rather easily to Walrasian general equilibrium theory (although see Maurice Allais (1953) for another early attempt at reconciling uncertainty and general equilibrium). Uncertainty is represented by states of the world. Commodities are indexed not just by their physical characteristics (pizza or beer) but also by the state of the world in which they are available (pizza and the Orioles win, pizza and the Orioles lose). Utility-maximizing traders trade state contingent claims, that is, a claim to a pizza slice if the Orioles win, a claim to a case of beer if they lose. The market prices these claims by to clear the market, just as it prices guns and butter. In particular, existence and the welfare theorems are inherited from the general framework, but must be interpreted anew here. In addition, there are interesting questions concerning the implications for equilibrium prices and allocations of those assumptions about preferences and endowments which are natural in the uncertainty framework. 2 Preferences The simplest model to study is one in which there is only one physical commodity, say wealth, concerning which claims in several states are traded. Let S denote a finite set of states. The consumption set for consumer i will be X i = R S, wherein the vector x = (x 1, x 2..., x S ) represents a claim to wealth x s in state s. These consumption bundles are called contingent claims. These contingent 1

2 claims are contracts that promise delivery in the specified state. When these contingent claims are claims on units of account (as opposed to pizza and beer) they are often known as Arrow securities. Each individual i has a utility function u i : X i R. We will make the usual assumptions: 1. u i is continuous, 2. u i is strictly increasing, 3. u i is strictly quasi-concave. This model incorporates expected utility maximizers and other probabilistically sophisticated decision rules, but it also incorporates individuals whose preferences are not consistent with any particular measure of belief. Important preference relations include: EU maximization: u i (s) = s π s v(x s ), π a probability function on S and v : R R a payoff function. Maximin: u i (x) = min s α s x s with α 0. Maxmin EU: u i (x) = min p P s π s v(x s ). Choquet EU: A capacity is a set function ν from the subsets of S to R such that ν( ) = 0, ν(s) = 1 and A B implies ν(a) ν(b). The Choquet integral is a means of integrating with nonadditive set functions. Let u i (x) = v(x s )dν. Minimax Regret: Given a set B of consumption bundles and a payoff function v : R R, the regret R(x s ) = max y B v(y s ) v(x s ). Then u i (x) = max s R(x s ). Notice that in a GE context, this would be price-dependent. It is tempting to interpret convexity as risk aversion, but notice that so far it is hard to do because we have introduced no notion of probability. More generally, if we have L physical commodities and S states, then we actually trade L S contingent claims: Pizza when the Orioles win, pizza when they lose, beer when the Orioles win, and beer when they lose.

3 2.1 Expected Utility Suppose that everyone is an expected utility maximizer with respect to a common probability distribution π. That is, for all i, u i (x) = s π s v i (x s ), where v is strictly concave. For bundle x, denote by x the bundle which pays off identically E p x = s π s x s. The strict concavity of u implies that u i ( x) > u i (x) for all x X i. This is a consequence of Jensen s inequality. The 45 degree line in x 2 B C A x 1 Figure 1: EU Preferences figure 1 is the set of all claims such that x 1 = x 2, that is, the set of sure things. It is easy to compute that the MRS on the diagonal is π 1 /π 2, the odds ratio of state 2 to state 1. Suppose that point A is a given consumption bundle. The straight line through A is an iso-expected value line; all points on it have the same expected value. The coordinates of point B give its expected value. Point C is the sure thing on the indifference curve on which A sits. Point C measures the certainty equivalent of A, the the difference B C represents the risk premium associated with holding bundle A. The curvature of the indifference curve measures attitudes towards risk. The more curvature, the greater the risk premium. 3 The Arrow-Debreu Economy Arrow-Debreu anticipates that all trading happens at time 0, and there is a complte set of forward markets.

4 3.1 Budget Constraint The market prices each contingent claim. The idea is that, by trading claims, individuals can move wealth between different states. Time trivially exists in this model. Agents trade at time 0, and at time 1 the uncertainty is resolved, the state is realized, and the claims are paid off. We suppose that consumers have pre-existing claims in the states, their endowments e i. The price of a unit claim in state s ( a dollar in state s ) is p s. Then the budget constraint is B(p, e) = {x : X i : s p s x s s p s e s }. Since agents trade at date 0, contingent claims and money really do change hands, even though physical commodities only change hands after the state is realized. 3.2 No Aggregate Uncertainty Suppose that in every state s, i e is = e which is independent of s. In this case there is no aggregate uncertainty Changing the state just redistributes a fixed amount of wealth among the consumers. In any optimal allocation, each consumer must bear no risk. That is, consumers fully insure. Theorem 1. The allocation x is Pareto optimal if and only if for each i and each pair of states s and t, x is = x it. Proof. Notice that giving each individual the expected value of his endowment is feasible and a Pareto improvement over the original allocation. Theorem 2. The equilibrium price ratio p s /p t equals the odds ratio π s /π t. Proof. This follows from the previous theorem, the First Welfare Theorem and a computation of the MRS. The picture for two consumers is that with no aggregate uncertainty, the Edgeworth box is square, and the contract curve is the diagonal because on the diagonal every consumer has an MRS equal to the odds ratio. Without common beliefs, the indifference curves will cross on the 45 degree line, and every consumer will end up holding more in those states she thinks is most likely. Although full insurance

5 is possible, no one takes it because in equilibrium insurance is super-fair from each individual s point of view. The proof of theorem 1 makes it clear that the conclusion should hold true for preferences more general than Eu preferences. For instance, if preferences are maxmin, the conclusion still holds. If people have identical beliefs and always prefer a lottery s sure thing expected value, to the lottery, then the theorem holds. 3.3 Aggregate Uncertainty Suppose the aggregate endowment is higher in state 1 than in state 2. The Edgeworth box is no longer a square. The optimal allocation must lie in between the two 45 degree lines. Then p 1 /p 2 < π 1 /π 2. MWG interpret this with equiprobable states to mean that the higher priced asset is that which pays off when the good is in least supply; that asset whose payoff is negatively correlated with the aggregate return. 4 Sequential Trade Equilibria of Plans, Prices and Price Expectations The Arrow-Debreu model requires the existence of LS forward markets, so that at date 0 a consumer can buy consumption of any good in any state delivered at date 1. Suppose we were to open spot markets at date 1 for the commodities after the state had been realized. No further trading would take place because all necessary trades have already been achieved through the forward markets. If, on the other hand, some futures markets had been closed initially, retrading would subsequently be required on the spot markets, and even then equilibria may not be ex ante efficient. Arrow (1953, linked on the web page) that in some cases, optima could be achieved through a set of forward and futures markets. Here is one case: At least one commodity can be traded contingently at date 0 and spot prices in each state are correctly anticipated. At t = 0 consumers have date 1 price expectations: p s in state s. Suppose to that claims to commodity 1 in each state are traded forward. The date 0 price of a unit of commodity 1 in state s at

6 date 1 is q s. The maximization problem is: max x R LS +, z R S u i(x) s.t. i) q s z s 0, (1) s ii) p s x s p s e is + p 1s z s for all s. Notice that one can buy (positive) or sell (negative) good 1 at date 0. If z s < e i1s, then the consumer is selling more than he owns of good 1 in state s. This is a short sale. Constraint i) is the date 0 budget constraint, and date 1 is the date 1 budget constraint. Definition 1. The tuple ( q, p, (z i, x i ) I i=1) is an equilibrium of plans, prices and price expectations if 1. for all i, (z i, x i ) solves (1); 2. for all s and i, i z is 0 and i x is i e is. Prices can be normalized separately for each of the S+1 budget constraints. We take p 1s = 1 and s q s = 1. This equilibrium concept is distinct from the Arrow-Debreu equilibrium. Nonetheless, the set of equilibrium allocations for both concepts is the same. Theorem 3. (i) If (p, x ) is an Arrow-Debreu equilibrium, then there are prices q and forward trades z such that (q, p, z, x ) is an equilibrium of plans, prices and price expectations. (ii) If (q, p, z, x ) is an equilibrium of plans, prices and price expectations, then there are scalars µ s > 0 such that ( (µ s p s) s S, x ) is an Arrow Debreu equilibrium. Proof. Let q s = p 1s. The difference between the two equilibria are the two budget sets: B AD i = {(x 1,..., x S ) R LS + : s p s(x s e is ) 0} B R i = {(x 1,..., x S ) R LS + : there are z 1,..., z s ) such that s q s z s 0 and for all s p s (x s e is p 1s z s }. We will show in each case that the two budget sets are identical. If so, an optimal choice in one is an optimal choice in the other.

7 (i) Choose x i B AD and let q s = p 1s. Let z is = (1/p 1s )p s (x is e is ). Then s q s z is = s p s(x is e is ) 0 and, for all s, p 1s z is = p s (x is e is ), so x B R. On the other hand, choose x i B R, and let z i denote the associated forward claims that make x i affordable in each state. Then s p s (x is e is ) s p 1s z is = s q s z is 0. So x i B AD. The clearing of goods markets in Arrow-Debreu equilibrium implies that the goods markets clear in Radner equilibrium. Also, z is = 1 p i 1s p s (x is e is ), i and the sum is 0 again because of Arrow-Debreu market clearing. (ii) Choose µ s such that µ s p 1s = q s. Then B R becomes {x i : there is a z i s.t. s q s z is 0 and µ s p s (x is e is ) q s z is for all s} This implies s µ s p s (x is e is ) 0, and so x i is in B AD. Goods market clear because x is a Radner equilibrium allocation. 5 Real Assets In the sequence economy, very simple instruments were used for transferring wealth across periods. We can consider more general kinds of assets for moving wealth. All assets are denominated in units of good 1. An asset is described by its asset return vector r = (r s ) s S. 5.1 Describing Assets Examples: Sure Thing: r = (1, 1,..., 1). A unit of the asset guarantees 1 unit of good 1 in each state. Arrow Security: r = (0,..., 0, 1, 0,..., 0). The asset pays off 1 unit only in state s, and 0 otherwise.

8 European Call: A vanilla derivative asset. A call on the asset with return vector r and strike price c is an opportunity to by the asset at a fixed price c at time 1 when the state is known, but before the asset pays out. It will only be exercised when it is worthwhile to pay c for the return, so r(c) = ( max{0, r 1 c},..., max{0, r S c} ). Cash-or-Nothing Call: A European cash-or-nothing binary call is an exotic derivative which pays a fixed amount of money if it expires in the money and nothing otherwise. For such an asset on the underlier with payout vector r with the strike price c and payout d, the return vector is r(c, d) = ( ) d1 rs c s S. The structure of asset returns is conveniently described by the S K asset returns matrix R, where r ks describes the amount of good 1 delivered by asset k in state s. A portfolio of the set of K assets is a vector z = (z 1,..., z K ). The dividend of portfolio z is how much it will pay off in each state: d s = K k=1 z kr k s is the dividend in state s. Asset prices are a vector q = (q 1,..., q K ). 5.2 Radner Equilibrium We need to modify the definition of equilibrium from the previous section. At t = 0 consumers have date 1 price expectations: p s in state s. Suppose to that claims to commodity 1 in each state are traded forward. The date 0 price of a unit of commodity 1 in state s at date 1 is q s. The maximization problem is: max u i (x) x R LS +, z R S s.t. i) q k z k 0, (2) k ii) p s x s p s e is + p 1s z k rs k for all s. k Definition 2. The tuple ( q, p, (z i, x i ) I i=1) is a Radner asset market equilibrium if 1. for all i, (z i, x i ) solves (2); 2. for all s, k and i, i z ik 0 and i x is i e is. Every security bought by some trader must be sold issued by someone else. If someone buys an asset, he is long in the asset. If someone sells an asset, she is short in the asset. The market clearing condition is that assets are in zero net supply.

9 5.3 Arbitrage One requirement of equilibrium is that equilibrium asset prices must satisfy a no arbitrage condition. Let R be the S K asset returns matrix, and let z denote any portfolio: Rz = 0 implies q z = 0 Rz 0 implies q z > 0 where here, y 0 means at least as big as but not 0 in all components. Suppose the first condition fails. Then there is a portfolio which has no effect on period 1 wealth and which generates cash in period 0. This cash can be used to buy good 1 in each period. An arbitrarily large position in this portfolio will generate arbitrarily large wealth in each state s. If the second condition fails, there is a portfolio which costs 0 (or less) and which generates arbitrarily large returns in some states and negative returns in no states. We formalize this as follows: Definition 3. A vector q of asset prices is arbitrage-free if there is no portfolio z such that the inequality system q z 0, R z R + /{0}. The first condition implies that q is a linear combination of the columns of R. Together with the first inequality, the second inequality implies that q is a non-negative linear combination of the columns of R. A formal proof derives this fact as a consequence of the separating hyperplane theorem. If preferences are non-satiated, then asset prices must be arbitrage free. Otherwise the is a costless asset portfolio that generates non-negative returns in every state and positive returns in some state s means that demand in state s will be unbounded. One might ask, does Radner equilibrium have any other implications for prices. The answer is, no. 5.4 Complete Markets We say that markets are complete if agents can insure each state separately, that is, if they can trade assets in such a way as to affect the payoff in one specific state without affecting the payoff in other states. When markets are complete the individual s decision problem in an asset economy is the same as in a contingent claim economy. Complete markets are important because this is precisely the condition which guarantees the equivalence of Radner and Arrow-Debreu equilibria. Definition 4. An asset returns structure is complete if rank R = S.

10 This condition, which means that there is a set of S assets whose returns vectors are linearly independent, means that every distribution w = (w 1,..., w S ) of wealth to the S states can be achieved by some portfolio: Rz = w has a solution for all w. Theorem 4. Suppose that the asset return structure is complete. Then 1. If (p, x ) is an Arrow-Debreu equilibrium, then there are asset prices q and a portfolio allocation z such that (q, p, z, x ) is an equilibrium of plans, prices and price expectations. 2. If (q, p, z, x ) is an equilibrium of plans, prices and price expectations, then there are scalars µ s > 0 such that ( (µ s p s) s S, x ) is an Arrow-Debreu equilibrium. The intuition is that so long as the asset structure is complete, we can find for any state s a portfolio that delivers 1 unit of good s in that state, and 0 in every other state. Thus anything that can be achieved by trading Arrow securities can be achieved by trading these portfolios. Furthermore, since trade in Arrow securities generates the maximal possible set of good-1 distributions achievable from any asset structure, what can be achieved by the complete asset structure can be achieved by Arrow securities. Finally, if we can price Arrow securities we can price every other security by arbitrage: The price of security k is s r sk ˆq s, where ˆq s is the price of the Arrow security which pays off in state s. In otherwords, trading in the asset structure R is just like trading Arrow securities, except that we have changed the names. See MWG for the details. We can do this even more generally. In this discussion we have constrained assets to pay off only in good 1. Since we are free to normalize prices state by state, we can take p 1s = 1 for all s without loss of generality, and so the matrix of quantity returns gives the matrix of wealth returns. Suppose that assets paid off in bundles of commodities rather than in good 1 alone; that is, r sk R L. We care about assets only through the wealth vectors they can achieve. The matrix of wealth returns is now R such that R sk = p s r sk. Whether or not the asset structure is complete, that the rank of R is S, now depends upon the equilibrium prices. This leads to all kinds of interesting phenomena. 5.5 Incomplete Markets The important fact about an asset return matrix is the set of possible wealth vectors it can achieve. If it can achieve any wealth vector, if range R = R S, then markets are complete. The existence of

11 Arrow-Debreu equilibrium implies the existence of Radner equilibrium, and the set of Radner and Arrow-Debreu equilibrium allocations are identical. A market in which rank R < S is said to be incomplete. This can come about in two ways: There could be fewer than S assets, and assets could pay off in consumption bundles and the value of consumption bundles at equilibrium prices is such that the asset return matrix does not have full rank. In this case the space of wealth vectors that are achievable through portfolio purchase is not R S, but a vector space of lower dimension. A simple example of an economy with incomplete markets is given by assuming two states and a single asset. Suppose the asset transfers 1 unit of good 1 in each state that is, it is a sure thing. Suppose the economy has only one good, and that in state i consumer i has 2 units of the good, while in state j = i he has only 1. Suppose that consumers have identical preferences, and indentical beliefs in which each state is equally likely. Take p s 1 and q = 2. Then consuming one s initial endowment is the best point in the budget set. This is a Radner equilibrium, but not an Arrow-Debreu equilibrium. We can see this because in Radner equilibrium the consumers are not fully insured, although there is no aggregate risk. The following issues arise in Radner equilibria with incomplete markets: 1. Non-existence of equilibria. 2. Non-optimality of equilibria. 3. Existence of Pareto ranked equilibria. 4. Adding markets may make everyone worse off. Non-Existence: Suppose that asset 1 pays off 1 unit of good a in each of two states, and asset two pays off 1 unit of the other good in each state. The two states are equally likely. The price of good 1 in each state is taken to be 1. The price of good 2 in state s is p s. Then the asset return matrix is [ ] 1 1 R =. p 1 p 2 The rank of R is is 2 if p 1 = p 2, and 1 if p 1 = p 2. Suppose there are two consumers. Consumer i has Cobb-Douglas preferences α i log x 1 + (1 α i ) log x 2 in state 1 and β i log x 1 + (1 β i ) log x 2 in state 2. Consumer 1 has endowment (1, 0) in each state, while consumer 2 has endowment (0, 1) in each state. If the rank of R is 1, then

12 there will be no trade in the assets. Consequently, each market will be independent. The equations determining equilibrium are α 1 + α 2 p 1 = 1 p 1 = 1 α 1 α 2 β 1 + β 2 p 2 = 1 p 2 = 1 β 1 β 2 If these ratios are different, then p 1 = p 2, and so asset return matrix has full rank. Suppose next that the asset return equation has full rank. To find the Radner equilibrium prices, find the Arrow-Debreu prices. Take q sk = 1 for s = k = 1. π 1 α 1 (1 + q 21 ) + π 1 α 2 (q 12 + q 22 ) = 1 π 1 (1 α 1 )(1 + q 21 ) + π 1 (1 α 2 )(q 12 + q 22 ) = 1 π 2 β 1 (1 + q 21 ) + π 2 β 2 (q 12 + q 22 ) = q 21 Suppose it is the case that q 21 /1 = q 22 /q 12. Then the asset return matrix will be singular. This will happen if the equations are consistent. An equivalent system is π 1 α 1 (1 + q 21 ) + π 1 α 2 (1 + q 21 )q 12 = 1 π 1 (1 α 1 )(1 + q 21 ) + π 1 (1 α 2 )(1 + q 21 )q 12 = 1 π 2 β 1 (1 + q 21 ) + π 2 β 2 (1 + q 21 )q 12 = q 21 π 1 α 1 r + π 1 α 2 rq 12 = 1 π 1 (1 α 1 )r + π 1 (1 α 2 )rq 12 = 1 (π 1 α 1 + π 2 β 1 )r + (π 1 α 2 + π 2 β 2 )rq 12 = r where r = 1 + q 21. A sufficient condition for consistency is the existence of an r > 1 such that π 1 α 1 + π 2 β 1 π 1 α 1 = π 1α 2 + π 2 β 2 π 1 α 2 and a sufficient condition for this is that β 1 /α 1 = β 2 /α 2. = r