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; ∈ 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. ∈ LS xi = (x1i , .., xSi) R denotes i’s consumption plans vector; ∈ L xsi = (x1si , .., xLsi ) R denotes i’sexpected consumption in state s; ∈ S note: x1i = (x11i , .., x1Si ) R represents expected trade in commodity 1, the only commodity for which∈ there are also date 0 markets. LS p = (p1, .., pS ) R is the vector of expected prices; L ∈ ps R is the expected price vector for the L goods in state s. ∈ 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 ∈ ∈ + subject to

q zi 0 and ps xsi ps ωsi + 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 (ω1si ), 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 ωlsi + 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 < ω1si 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 ∈ trades zi∗ R , spot prices ps∗ R for each s, and consumption xi∗ R for each i such∈ that: ∈ ∈

1 for each individual, zi∗ and xi∗ solve S (i): s=1 qs∗zsi 0 S ≤ max πsi usi (xsi ) subject to S L s=1 zi R ,xsi R P ∈ ∈ + P (ii): p∗ xsi p∗ ωsi + p∗ zsi for all s s · ≤ s · 1s 2 all markets clear: I I I z∗ 0 and x ∗ ωsi 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 ∈ ∈ S Arrow-Debreu equilibrium. Then, there are prices q∗ R++ and trades SI ∈ z∗ = (z1∗, .., z∗) R for the state-contingent commodity such that: z∗, q∗, I ∈ 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 ∈ ∈ ∈ ps∗ R++ for all s constitute a Radner equilibrium. Then, there are S strictly ∈ positive numbers µ1, .., µS such that the allocation x ∗ and the LS state-contingent commodities price vector (µ1p1∗, .., µS pS∗) R++ 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. From Arrow-Debreu to Radner

First, choose p1s = qs for each s (we can do this because...).

Write the Arrow-Debreu budget set as S AD LS Bi = xi R+ : ps (xsi ωsi ) 0 ∈ · − ≤ ( s=1 ) X Write the Radner budget set as S qs zsi 0 s=1 ≤ R  LS S P  Bi = xi R+ : there is zi R s.t. and   ∈ ∈    ps (xsi ωsi ) p1s zsi for all s  · − ≤      We need to show these two sets are the same. From Arrow-Debreu to Radner

AD LS S Suppose x Bi = xi R+ : ps (xsi ωsi ) 0 . ∈ ∈ s=1 · − ≤ n o For each s let P 1 zsi = ps (xs ωsi ) . p1s · − Then S S S

qs zsi = p1s zsi = ps (xs ωsi ) 0 · − ≤ s=1 s=1 s=1 X X X and ps (xs ωsi ) = p1s zsi for all s · − Thus S s=1 qs zsi 0 R LS S ≤ x Bi = xi R+ : zi R s.t. and . ∈  ∈ ∃ ∈ P  ps (xs ωsi ) p1s zsi for all s  · − ≤    From Arrow-Debreu to Radner

S s=1 qs zsi 0 R LS S ≤ Let x Bi = xi R+ : zi R s.t. and . ∈  ∈ ∃ ∈ P  ps (xs ωsi ) p1s zsi for all s  · − ≤  S Then, for somezi R we have  S ∈

qs zsi 0 and ps (xs ωsi ) p1s zsi for all s ≤ · − ≤ s=1 X Summing over s yields S S S

ps (xs ωsi ) p1s zsi = qs zsi 0 · − ≤ ≤ s=1 s=1 s=1 X X X AD LS S Therefore x Bi = xi R+ : ps (xsi ωsi ) 0 . ∈ ∈ s=1 · − ≤ n P o 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 1 x ∗, z∗, q∗ = (p11∗ , ..., p1∗S ) , and zsi∗ = ps∗ (xsi∗ ωsi ) p1∗s · − is a Radner equilibrium.

Why? AD If the budget sets are the same, maximizing utility in Bi implies maximizing R utility in Bi . The spot markets clear because the Arrow-Debreu markets clear. The state contingent markets clear since I I I 1 1 zsi∗ = ps∗ (xsi∗ ωsi ) = ps∗ (xsi∗ ωsi ) 0 p∗ · − p∗ · − ≤ i=1 i=1 1s 1s i=1 ! X X X From Radner to Arrow-Debreu

Choose µs such that µs p1s = qs for each s. Next, show the budget sets are the same.

Prove the rest as homework assignment.

Are the µs unique? Why do we care about that? The Importance of Arrow Securities

The propositon says that Arrow-Debreu equilibrium and Radner equilibrium are equivalent, provided there are enough Arrow securities.

Arrow-Debreu and Radner are equivalent An Arrow-Debreu equilibrium can be made into a Radner equilbrium. A Radner equilibrium can be made into an Arrow-Debreu equilibrium.

Then the timing of trades is not that important. Equilibrium outcomes are the same if we imagine all trade taking place at time 0, or if we imagine that Arrow securities are traded at that time and the other trades occur in spot markets. Having all the state-contingent commodities trades is not that important. Equilibrium outcomes are the same if we imagine a full set of state-contingent commodities, or if we only imagine one Arrow security per state. Asset Markets in General Equilibrium Next, we model ﬁnancial assets (rather than state contingent commodities). A unit of an asset gives the holder the right to receive some future payment. By convention, good 1 is the unit of accounts for assets.

Deﬁnition An asset is a title to receive pre-speciﬁed amounts of good 1 at date 1.

An asset is completely characterized by its return vector S rk = (rk1, .., rkS ) R ∈ rks is the dividend paid to the holder of a unit of rk 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: × r11 .. rk1 .. rK 1 ......   R = r1s .. rks .. rKs  ......     r1S .. rkS .. rKS    A row indicates the returns of all assets in one particular state. 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.

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.

Example An asset that delivers one unit of good 1 in one state, and zero otherwise: r = (0, .., 1, .., 0) This is an Arrow security. 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 rs > c since in the opposite case one loses money (equality does not matter); hence r (c) = (max 0, r1 c , .., max 0, rS c ) { − } { − } If r = (1, 2, 3, 4), then r (1.5) = (0, 0.5, 1.5, 2.5)

r (2) = (0, 0, 1, 2)

r (3) = (0, 0, 0, 1) Budget Constraints with Asset Markets Given an asset matrx R, one can deﬁne prices and holdings of each asset.

K q = (q1, .., qK ) R are the asset prices, where qk is the price of asset k. ∈ K zi = (z1i , .., zKi ) R are consumer i’sholdings 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 in spot markets.

As in Radner, i’sbudget constraints are K K

qk zki 0 and ps xsi ps ωsi + p1s zki rsk for each s ≤ · ≤ · k=1 k=1 X X time 0 time 1 | {z } | {z } 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. Assets vs State-contingent Commodity in Radner

Question What is the diﬀerence between these budget constraints K K

qk zki 0 and ps xsi ps ωsi + p1s zki rsk for each s ≤ · ≤ · k=1 k=1 X X and the ones in a Radner equilibrium? Here, assets’dividends 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 rs = 1 in state s and zero otherwise. If S = K, we can write Radner using the z and r above: S Radner zs = zs rs . s=1 X Next Class

Asset Prices and Equilibrium