Radner Equilibrium: Definition and Equivalence with Arrow-Debreu Equilibrium

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1 Radner Equilibrium: Definition and Equivalence with Arrow-Debreu Equilibrium Econ 2100 Fall 2017 Lecture 24, November 28 Outline 1 Sequential Trade and Arrow Securities 2 Radner Equilibrium 3 Equivalence between Arrow-Debreu and Radner Equilibria 4 Assets and Asset Markets

2 Timing of Trades in Arrow-Debreu Remark In an Arrow-Debreu economy, all decisions are made at date 0. Individuals exchange promises to deliver and receive quantities of the goods according to the state that is realized. Tomorrow, one of the states occurs, and these promises are carried through exactly as planned, without changes. What if tomorrow, after the state is known, there were also markets for the L goods? Is there an incentive to trade in these spot markets?

3 Spot Markets Do Not Matter in Arrow-Debreu Fact Given an Arrow-Debreu equilibrium, there are no incentives to trade in spot markets. Proof. Suppose not: consumers can find a mutually beneficial trade in some state t. Thus, there exist a feasible allocation that all consumers pefer weakly to the equilibrium bundle, with at least one consumer preference being strict. consumers trade only if they do better. This new allocation is feasible, and Pareto dominates the Arrow-Debreu equilibrium. But this is impossible, because the First Welfare Theorem holds. This conclusion is disappointing as most real world markets are spot markets, not the forward markets imagined by an Arrow-Debreu economy.

4 Spot and Forward Markets Remark Most real world markets are spot, not forward. Objective Reconcile this observation with a general equilibrium model with uncertainty similar to Arrow-Debreu. Idea Since the main role of state contingent commodities is to allow welfare transfers across states one can have similar transfers with just a few state-contingent markets. This is because not all state-contingent commodities markets are needed provided there is another way to transfer wealth across states.

5 Radner Equilibrium Ken Arrow proposed the following model. Individuals make decisions today and tomorrow only one good is traded both today and tomorrow (and therefore used to transfer wealth across states), while all other commodities are traded only tomorrow (these markets open after uncertainty is resolved). Today s decisions depend on forecasts about tomorrow: trade is sequential, so expectations are crucial. This equilibrium concept is called Radner equilibrium. Main Idea Replace forward markets with expectations about future spot markets. Given these expectations, consumers make decisions about welfare transfers across states at date 0. When date 1 arrives, consumers trade in markets for physical goods. Main Result: Radner and Arrow-Debreu Are Equivalent If expectations are correct, and individuals have some effective way to transfer wealth across states, a Radner equilibrium is equivalent to an Arrow-Debreu equilibrium.

6 Date 0 Sequential Trading and Arrow Securities Only one physical good becomes a state-contingent commodity: money. Amounts of money are traded for delivery if and only if a particular state occurs. These are called Arrow securities (they are financial assets). Date 1 After some state occurs, there are spot markets where physical goods trade at spot prices (these prices can differ across states). Individuals decide how much to trade of each good depending on (i) the spot prices, and (ii) how much they have traded in the state-contingent commodity that corresponds to the realized state. Expectations Are Crucial Date 0 trades reflect what consumers think will happen in the spot markets. To decide how much to trade of the state-contingent commodity individuals make consumption plans for each possible state; these plans depend on their forecasts of the future spot prices.

7 Sequential Trading and Arrow Securities Consider an exchange economy with X i = R LS +. Notation z i = (z 1i,.., z Si ) R S denotes i s trades in the state-contingent commodity; these contracts specify amounts to be delivered, or received, of commodity 1 in each of the s states. note: these date zero trades can be negative or positive. q = (q 1,.., q S ) R S denotes the prices of the Arrow securities. x i = (x i1,.., x is ) R LS denotes i s consumption plans vector; x si = (x 1si,.., x Lsi ) R L denotes i s expected consumption in state s; x 1i = (x 11i,.., x 1Si ) R S represents expected trade in commodity 1, the only commodity for which there are also date 0 markets. p = (p 1,.., p S ) R LS is the vector of expected prices; p s R L is the expected price vector for the L goods in state s. Expected prices and expected consumption plans are elements of R LS. Note: that many items are expected, they represent planned choices. The consumer makes and plans choices given current and expected prices. There is no date 0 consumption (for simplicity).

8 Optimization Given current prices (q) and expected prices (p), each individual maximizes utility. A plan includes current trades in the state-contingent commodity (z i ) and expected spot market purchases (x i ). Consumers Choices at time 0 At date 0, consumer i solves the following maximization problem subject to q z i 0 }{{} budget constraint at time 0 max U (x i ) z i R S,x R LS + and p s x si p s ω si + p 1s z si for each s }{{} expected budget constraints at time 1 There will be S budget constraints at date 1. These constraints are in expectation (from the point of view of date 0): x si is expected consumption; p s are expected prices. Note that date 1 trading of good 1 has three components: buy the desired amount for consumption, sell the endowment, and sell the realized state-contingent outcome of date 0 trades.

9 Date 0 Budget Constraint Date 0 The budget constraint at date 0 is S q s z si 0 s=1 The individual must engage in zero net trades of the state-contingent commodity. Since the price vector is non-negative, if she promises to buy good 1 in state s (z si > 0), she must also promise to sell it in some other state t (z ti < 0). This is because she has no income at date 0 and she cannot promise to spend more than she makes one could add a positive date 0 endowment (and consumption), without affecting this logic. The date 0 budget constraint is homogeneous of degree zero in prices. Therefore, we can normalize date zero prices (sum up to one).

10 Date 1 Date 1 Budget Constraints The budget constraints at time 1 are L L p ls x lsi p ls ω lsi + p 1s z si l=1 l=1 for each s = 1,..., S At date 1, consumer i trades in the spot markets corresponding to the realized state; her wealth reflects the outcome of previous trades. The budget constraint is homogeneous of degree zero in spot prices. Here, we normalize by assuming that p s1 = 1 for each s. Thus, an Arrow security promises to deliver one unit of money (good 1). Since there are no restrictions on z, we may have z t < ω 1ti for some t: i sells money short (she promises to deliver more than she will have); if so, she must buy some money on the spot market. the ability to sell short is limited: consumption cannot be negative. Arrow securities allow wealth transfers across states: at date 0, a consumer can buy a state s dollar and pay for it with a state t dollar. If state s occurs, she uses the extra dollar to buy other goods; if state t occurs, she has one less dollar to buy other goods.

11 Radner Equilibrium In equilibirum, expected prices must equal realized prices, and expected trades must equal actual trades. Everything happens exactly as planned. Definition A Radner equilibrium is composed by state-contingent commodities prices q R S, trades zi R S, spot prices ps R L for each s, and consumption xi R LS for each i such that: 1 for each individual, z i max U (x i ) z i R S,x i R LS + 2 all markets clear: I zsi 0 i=1 and x i subject to and solve (i) : S s=1 q s z si 0 (ii) : p s x si p s ω si + p 1s z si I xsi i=1 I i=1 ω si for all s for each s Spot and forward markets must clear at consumption plans that maximize individuals utility, given current and expected prices; the expected prices equal the realized prices that clear the spot markets.

12 Radner and Arrow-Debreu Are Equivalent Under the rational expectations hypothesis implicit in Radner equilibirum, planned behavior equals actual behavior and the timing of decisions is unimportant. Proposition (Equivalence of Arrow-Debreu and Radner equilibria) 1 Suppose the allocation x R LSI and the prices p R LS ++ constitute an Arrow-Debreu equilibrium. Then, there are prices q R S ++ and trades z = (z1,.., z I ) RSI for the state-contingent commodity such that: z, q, x, and spot prices ps for each s form a Radner equilibrium. 2 Suppose consumption plans x R LSI and z R SI and prices q R S ++ and p R LS ++ constitute a Radner equilibrium. Then, there are S strictly positive numbers µ 1,.., µ S such that the allocation x and the state-contingent commodities price vector (µ 1 p1,.., µ S ps ) RLS ++ form an Arrow-Debreu equilibrium. 1 An Arrow-Debreu equilibrium becomes a Radner equilibrium if one appropriately chooses trades and prices of the state contingent commodity. 2 A Radner equilibrium becomes an Arrow Debreu equilibrium if one appropriately modifies spot prices to make them state-contingent prices. The proof only needs to show that the two budget sets are the same.

13 From Arrow-Debreu to Radner First, choose p 1s = q s for each s (we can do this because...). Write the Arrow-Debreu budget set as { } S Bi AD = x i R LS + : p s (x si ω si ) 0 Write the Radner budget set as s=1 S s=1 q sz si 0 Bi R = x i R LS + : there is z i R S s.t. and p s (x s ω si ) p 1s z si for all s We need to show these two sets are the same.

14 From Arrow-Debreu to Radner { Suppose x Bi AD = x i R LS + : } S s=1 p s (x si ω si ) 0. For each s let Then S q s z si = s=1 z si = 1 p 1s p s (x s ω si ). S p 1s z si = s=1 S p s (x s ω si ) 0 s=1 and Thus x B R i = p s (x s ω si ) = p 1s z si x i R LS + : z i R S s.t. for all s S s=1 q sz si 0 and p s (x s ω si ) p 1s z si for all s.

15 From Arrow-Debreu to Radner Let x B R i = x i R LS + : z i R S s.t. S s=1 q sz si 0 and p s (x s ω si ) p 1s z si for all s Then, for some z i R S we have S q s z si 0 and p s (x s ω si ) p 1s z si for all s s=1 Summing over s yields S p s (x s ω si ) s=1 Therefore x B AD i = S p 1s z si = s=1 S q s z si 0 s=1 { x i R LS + : S s=1 p s (x si ω si ) 0 }..

16 From Arrow-Debreu to Radner We have shown that the budget set for Radner and Arrow-Debreu are the same. An Arrow-Debreu equilibrium yields a Radner equilibrium If x, p is an Arrow-Debreu equilibrium, then x, z, q = (p 11,..., p 1S ), and z si = 1 is a Radner equilibrium. Why? p 1s p s (x si ω si ) If the budget sets are the same, maximizing utility in Bi AD implies maximizing utility in Bi R. The spot markets clear because the Arrow-Debreu markets clear. The state contingent markets clear since I zsi = i=1 I 1 p i=1 1s ps (xsi ω si ) = 1 p p1s s ( I ) (xsi ω si ) 0 i=1

17 From Radner to Arrow-Debreu Choose µ s such that µ s p 1s = q s for each s. Next, show the budget sets are the same. Prove the rest as homework assignment.

18 Asset Markets in General Equilibrium Next, we model financial assets (rather than state contingent commodities), in a setup similar to Radner s. A unit of an asset gives the holder the right to receive some payment in the future. By convention, good 1 is the unit of accounts for assets. Definition An asset is a title to receive r s units of good 1 at date 1 if and only if state s occurs. An asset is completely characterized by its return vector r = (r 1,.., r S ) R S r s is the dividend paid to the holder of a unit of r if and only if state s occurs. Definition The return matrix R is an S K matrix whose kth column is the return vector of asset k. That is: r 11.. r k1.. r K R = r 1s.. r ks.. r Ks r 1S.. r ks.. r KS

19 Assets: Examples Example An asset that delivers one unit of good 1 in all states: r = (1,.., 1) If there is only one good, L = 1, this the risk-free (or safe) asset. Example Why is this not safe with many goods? Because the price of good 1 can change from state to state. An asset matters insofar as it can be transformed into consumption goods; the rate at which one can do this depends on relative prices. An asset that delivers one unit of good 1 in one state, and zero otherwise: This is called an Arrow security. r = (0,.., 1,.., 0)

20 Derivative Assets: Example Assets whose returns are defined in terms of other assets are called derivatives. These are very common in financial markets. Example A European Call Option on asset r at strike price c gives its holder the right to buy, after the state is revealed but before the returns on the asset are paid, one unit of asset r at price c. What is the return vector of this European call option? The option will be exercised only if r s > c since in the opposite case one loses money (equality does not matter); hence If r = (1, 2, 3, 4), then r (c) = (max {0, r 1 c},.., max {0, r S c}) r (1.5) = (0, 0.5, 1.5, 2.5) r (2) = (0, 0, 1, 2) r (3) = (0, 0, 0, 1)

21 Budget Constraints with Asset Markets Given an asset matrx R, one can define prices and holdings of each asset. q = (q 1,.., q K ) R K are the asset prices, where q k is the price of asset k. z i = (z 1i,.., z Ki ) R K are consumer i s holdings of each asset. This is called a portfolio: it shows how many units of each asset i owns. Assets are traded at time 0, while returns are realized at time 1. At that time, agents decide how much to consume and they trade on spot markets. As in Radner, i s budget constraints are K K q k z ki 0 and p s x si p s ω si + p 1s z ki r sk k=1 } {{ } time 0 k=1 for each s } {{ } time 1 Income at time 1 is given by the value of endowment plus the income one obtains by selling the returns of the assets one owns. As usual, one can normalize the spot price of good one to be 1.

22 Assets vs State-contingent Commodity in Radner Question What is the difference between these budget constraints K K q k z ki 0 and p s x si p s ω si + p 1s z ki r sk k=1 and the ones in a Radner equilibrium? k=1 for each s Here, the dividens of the assets are given; one focuses only on the portfolio choice. In Radner, the dividends are constructed by the consumers choice of trades in the state-contingent commodity. Formally, in Radner one implicitly assumes S different assets, each with returns r s = 1 in state s and zero otherwise. If S = K, we can write Radner using the z and r above: S zs Radner = z s r s. s=1