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Radner Equilibrium: Definition and Equivalence with Arrow-Debreu

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Radner Equilibrium: Definition and Equivalence with Arrow-Debreu Radner Equilibrium: Definition and Equivalence with Arrow-Debreu Equilibrium Econ3030 Fall2020 Lecture 11, November 12 Outline 1 Sequential Trade and Arrow Securities 2 Radner Equilibrium 3 Equivalence between Arrow-Debreu and Radner Equilibria 4 Assets and Asset Markets 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. Afterwards, as time and uncertainty unfold, these promises are carried through exactly as planned, without changes. What if tomorrow there were also markets for the L goods? Is there an incentive to trade in these spot markets? 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 period 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. 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 The main role of state contingent commodities is to allow welfare transfers across states... If one can have achieve that same objective with just a few state-contingent markets. Then only some state-contingent markets are needed, and most trade takes place in spot markets. Radner Equilibrium Ken Arrow proposed the following model. Individuals make decisions now and in the future only one commodity is traded both now and in the future (and therefore used to transfer wealth across states), while all other commodities are traded only in markets that open after uncertainty is resolved. Today’sdecisions depend on forecasts about the future: trade is sequential, and 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. Afterwards, 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: having fewer markets does not matter. Sequential Trading and Arrow Securities Date 0 Only one physical is used as 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. Date 1 After some state occurs, there are markets where commodities trade at spot prices (these prices typically differ across states). Individuals decide how much to trade depending on (i) the spot prices, and (ii) how much they have traded in the state-contingent commodity that corresponds to the realized state (that determines part of their income). 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. Sequential Trading and Arrow Securities LS Consider an exchange economy with Xi = R+ . Notation S zi = (z1i , .., zSi ) R denotes i’strades in the state-contingent commodity; 2 this Arrow security specifies amounts to be delivered, or received, of commodity 1 in each state. note: these trades can be negative or positive. S q = (q1, .., qS ) R denotes the prices of the Arrow securities. 2 LS xi = (x1i , .., xSi) R denotes i’s consumption plans vector; 2 L xsi = (x1si , .., xLsi ) R denotes i’sexpected consumption in state s; 2 S note: x1i = (x11i , .., x1Si ) R represents expected trade in commodity 1, the only commodity for which2 there are also date 0 markets. LS p = (p1, .., pS ) R is the vector of expected prices; L 2 ps R is the expected price vector for the L goods in state s. 2 Expected prices and expected consumption plans are elements of RLS . Note: that xi and p are expected: they represent planned choices and prices. Consumers make choices and plans given current and expected prices. There is no date 0 consumption (for simplicity). Optimization Given current prices (q) and expected prices (p), each individual maximizes utility. A plan includes current trades in the state-contingent commodity (zi ) and expected spot market purchases (xi ). Consumers’Choices at time 0 At date 0, consumer i solves the following maximization problem max E [u (x)] S LS zi R ,x R 2 2 + subject to q zi 0 and ps xsi ps !si + p1s zsi for each s · ≤ · ≤ · budget constraint at time 0 expected budget constraints at time 1 | {z } | {z } There will be S budget constraints at date 1. From the point of view of date 0 these constraints are “in expectation”: xsi is expected consumption; ps are expected prices. Note: date 1 trading of good 1 has three components: buy the desired amount for immediate consumption (x1si ), sell the endowment (!1si ), and sell the realized state-contingent “outcome”of date 0 trades (zsi ). Time 0 Budget Constraint Date 0 The budget constraint at date 0 is S qs zsi 0 ≤ s=1 X The individual has no income at time 0, hence her trades cannot have positive cost. Since the price vector is non-negative, if she promises to buy good 1 in state s (zsi > 0), she must also promise to sell it in some other state t (zti < 0). She cannot promise to spend more than she makes one could add a positive date 1 endowment (and date zero consumption), without affecting this logic. The time 0 budget constraint is homogeneous of degree zero in prices. Therefore, we can normalize time 0 prices. Time 1 Budget Constraints Date 1 The budget constraints at time 1 are L L pls xlsi pls !lsi + p1s zsi for each s = 1, ..., S ≤ l=1 l=1 X X At time 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 ps1 = 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 zsi < !1si for some s: i sells money ‘short’(she promises to deliver more than she will have); if so, her other income must compensate. 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 s0 dollar. If state s occurs, she uses the extra dollar to buy commodities; if state s 0 occurs, she has one less dollar to buy commodities. Radner Equilibrium In equilibirum, expected prices must equal realized prices, and expected trades must equal actual trades: everything happens exactly as planned. Definition S A Radner equilibrium is composed by state-contingent commodities prices q R , S L LS 2 trades zi R , spot prices ps R for each s, and consumption xi R for each i such2 that: 2 2 1 for each individual, zi and xi solve S (i): s=1 qszsi 0 S ≤ max si usi (xsi ) subject to S L s=1 zi R ,xsi R P 2 2 + P (ii): p xsi p !si + p zsi for all s s · ≤ s · 1s 2 all markets clear: I I I z 0 and x !si for all s si ≤ si ≤ i=1 i=1 i=1 X X X 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. 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 LSI LS Suppose the allocation x R and the prices p R++ constitute an 2 2 S Arrow-Debreu equilibrium. Then, there are prices q R++ and trades SI 2 z = (z1, .., z) R for the state-contingent commodity such that: z, q, I 2 x , and spot prices ps for each s form a Radner equilibrium. 2 LSI SI S Suppose consumption plans x R and z R and prices q R++ and L 2 2 2 ps R++ for all s constitute a Radner equilibrium. Then, there are S strictly 2 positive numbers 1, .., S such that the allocation x and the LS state-contingent commodities price vector (1p1, .., S pS) R++ form an Arrow-Debreu equilibrium. 2 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. From Arrow-Debreu to Radner First, choose p1s = qs for each s (we can do this because...).
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