Arrow Debreu Equilibrium. October 31, 2015

Transcription

1 Arrow Debreu Equilibrium October 31, 2015

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3 Θ 0 = {s 1,...s S } - the set of (unknown) states of the world assuming there are S unknown states. information is complete but imperfect n - number of consumers K- number of physical commodities

4 m number of firms this method proceeds by incorporating the state dependency directly in to the allocation - an allocation lies in X = R KnS a list of consumption plans for each individual in each state, AD is kind of Walrasian equilibrium each firm has a state contingent production set Y js R K

5 an allocation {x i } i=1,...n if there is a collection of state contingent production plans {y js } j=1,...,m,s=1,...s satisfying y js Y js for all j and s and n x is i=1 m j=1 y js for each s Preferences subjective expected utility - π i s - the probability with which consumer i thinks that state s will occur consumer i s utility for allocation x depends only on x i and is given by U i (x) is S s=1 πi su s i (x is )

6 auctioneer announces a price vector p R ks (one price for each commodity in each state) each consumer owns the share λ ij of firm j s shares and a state contingent endowment ω is firms maximize profits an Arrow-Debreu equilibrium is a Walrasian equilibrium for the problem above is a triple (p,x,y )

7 (profit maximization) each firm s state contingent production plan yj maximizes p y j subject to y j Y j (utility maximization) each consumer s consumption plan xi = {xis } s=1,...s maximizes subject to πsu i i (x is ) s psx is s=1 psω is + s=1 s=1 m j=1 λ ij p sy js

8 (feasibility) for each state s n i=1 x is n ω is + i=1 the preferences above satisfy the private value property, reasonable monotonicity restriction then ensure that the first Welfare Theorem holds so that an Arrow Debreu Equilibrium is Pareto Optimal which means that every Arrow Debreu equilibrium is interim efficient m j=1 y js

9 every Arrow-Debreu equilibrium allocation is ex post efficient provided π i s > 0 for each i and s. Proof: Let {x,y,p } be the Arrow Debreu equilibrium allocation. By the first welfare theorem, there does not exist an alternative feasible allocation {x, y, p} such that πsu i i s (x is ) s=1 s=1 π i su s i (x is) with strict inequality holding for some i.

10 Suppose that the allocation {x,y,p } is not ex post efficient. Then there is an alternative feasible allocation {x,y,p } such that u s i(x is) u s i(x is) with strict inequality holding for at least one i and s. But then it is immediate that s=1 π i su s i (x is) s=1 π i su s i (x is) with strict inequality holding for at least one i. This contradicts the first welfare theorem.

11 Radner Equilibrium now suppose that there are S forward markets for contingent delivery of good 1. z i s is the quantity of good 1 that consumer i buys on the forward market. Each such contract guarantees that the consumer will receive exactly 1 unit of commodity 1 if (and only if) state s occurs. this special kind of forward contract is referred to as an Arrow security let q = {q 1,...q S } denote the vector of S forward prices

12 once the state is realized deliveries occur as specified by the forward contracts (so consumer i get z i s units of good 1 in state s) - then trade in the other commodities occurs on a set of spot markets let {p 1s,...p Ks } be the prices that each consumer i expects to prevail on the spot markets for each of the K commodities in state s. the budget constraint faced by consumer i on the forward markets is just q s zs i 0 s=1 figure 1 gives a typical forward trade in the case where S = 1 and K = 1 in the figure, there are 2 states.

13 Consumption/Delivery in State 2 b a z 2 ω 2 c z 1 0 ω 1 State 1

14 The forward trade is given by the line segment 0a which has the trader selling z 1 units of good 1 forward (there is only one good) in state 1 and buying z 2 units of good 1 forward in state 2. the trader s state contingent consumption is given by adding the forward trades to the trader s endowment (ω 1,ω 2 ) in the first figure, the endowment is at point c while the forward trades take the consumer to point b the second figure shows some on the interesting things that can occur when there are multiple goods

15 Good 2 b d ω a ω 1 z 0 ω 2 cω 1 +z Good 1 Figure: 2 Physical Goods, 2 States

16 the trader s endowment is the same in both state 1 and state 2 and is given by the point ω the trader buys z units of good 1 forward in state 1. If the state is realized this takes the trade to the budget line ab and he trades along this to the tangency to pay for these forward deliveries, the trader sells z units of good 1 forward in state 2 moving to the budget line cd when the state occurs notice that the forward trade in good 1 is larger than the consumers total endowment of good 1 in state 2 - this is called selling short. It means that the consumer commits himself to buy good 1 in state 2 to meet his forward commitments.

17 Radner Equilibrium with Arrow Securities describing firms preferences in Radner equilibrium is complicated, so to start, focus on an exchange economy in which there are no firms consumers trade arrow securities - for each state s there is an arrow security that pays 1 unit of good 1 if and only if state s occurs. No one has any endowment of arrow securities an allocation rule is, as always X = R nks consumer s preferences over any allocation rule x areu i (x i ) = S s=1 πi sui s (x is ) - consumers only care about their own consumption

18 to describe the institutions, each consumer chooses a trade vector z i R s describing the quantity of arrow securities bought and sold there is a common belief {p 1,...,p s } describing the spot price vectors that every consumer believes will prevail in each state s there is a price vector q R S + describing the prices of the Arrow Securities.

19 A Radner Equilibrium is a set (x,z,q,p ) of consumption plans, forward trades, securities prices and spot prices such that 1. (feasibility) m i=1 z i = 0; and for all s S m i=1 x is = m i=1 ωi s 2. for each i, {{xis },z i } maximizes πsu i i (x s) s=1 subject to the constraints for each s = 1,...S; and p x p ω i s +p 1sz s q z 0

20 If an allocation x to consumers is supported as (or is part of) a Radner equilibrium with exactly S arrow securities, it can also be supported as an Arrow Debreu equilibrium. part of the implication of this is that Radner equilibrium with S arrow securities inherits all the properties of AD, i.e, is is pareto optimal, etc.

21 Proof: Basically we are going to show that the Arrow Debreu and Radner equilibrium price functions are linear transformations of one another that preserve every consumer s budget set. That means that anything that a consumer could have purchased in AD is also available in Radner, and conversely. Radner equilibrium delivers asset prices q and spot prices p s in state s. Define ξ s = q s p 1s and let ξ = {ξ 1,...ξ s }. Lets use the notation ρ for Arrow Debreu, so ρ s is the vector of K commodity prices in state s in the Arrow Debreu equilibrium. Define state contingent prices to be ρ s = p s ξ s.

22 step 1 - {x s} is feasible in a Radner equilibrium with asset prices q and spot prices p s then it must be feasible with direct trade at state contingent prices ρ s = p s ξ s. Radner feasibility implies that there is some vector z of asset trades such that qz = 0 and p s x s = p s ω i s +p 1s z s Since the budget set is homogenous of degree zero this implies that p s ξ s x s = p s ξ s ω i s +p 1s ξ s z s which implies by definition that p s ξ s x s = p s ξ s ω i s +q s z s

23 Sum both sides over s to get which says that the consumption plan is feasible at state contingent prices ρ = {p s λ s } s=1,s p s ξ s x s = p s ξ s ωs i + q s z s s=1 s=1 s=1 which by the feasibility of the asset trade z s implies that p s ξ s x s s=1 p s ξ s ωs i s=1

24 step 2 the consumption plan x s is feasible at state contingent prices p s ξ s. Then ρx = ρω i implies p s ξ s x s = s=1 Define the asset trade p s ξ s ωs i s=1 z s = p sx s p s ω i s p s1

25 This asset trade trivially makes the consumption x s affordable at the Radner equilibrium spot prices. Further q s z s = s=1 s=1 q s p s x s p s ω i s p s1 = ( ξ s ps x s p s ωs) i = 0 s=1

26 Trading in Firms Shares suppose there are J assets that pay off in units of good 1. Each asset j = 1,...J generates a vector of returns r j R S ++ r js denotes the amount of good 1 delivered in state s Arrow security r s = {0,0,...,1,...0} un-contingent delivery r j = {1,1,...1}

27 option to purchase security r j = {r j1,r j2,...r js } at a price c gives security r o j = {0,0...r js c,r js+1 c,...r js c} define the matrix R to be equal to r 11 r r J1 r 12.. r J r 1S r 2S... r JS

28 Now we ll assume there are exactly S firms Define the return matrix R as follows p 1 y 11 p 1 y m1 = R p S y 1S p S y ms

29 each consumer i has an endowment λ ij of firm j s shares. These shares trade in the securities market at price q j a portfolio now consists of a trade vector z i R J+m, as before z ij is interpreted as the amount of asset j (i.e., firm j) that i buys or sells. In particular if i decides to sell his own shares we just subtract this from his initial endowment an allocation is a collection {x,y,z}in R KnS R KS2 R Sn while a price system is a pair {p,q} R KS R S

30 as before, we view the firms share prices q as something that traders observe in the first period, while the spot price system is something that traders believe will occur in the second period a consumption plan x i for i is feasible if there is a portfolio z i of firms shares satisfying p s x is p s ω is + n λ ij p s y js + j=1 j=1 z ijp s y js for each s = 1,...,S and zjq i j = 0 j=1 (the latter constraint could be written qz i = 0)

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32 a Radner equilibrium is a collection {x,y,z,p,q } satisfying four conditions n i=1 z i = 0 for each i, x i is at least as good as any other consumption plan that is feasible for i at prices p and q for each j and s p sy j p sy y Y js n i=1 x is = n i=1 ω is + S j=1 y js in each state s

33 Theorem: A Radner equilibrium outcome can also be supported as an Arrow Debreu equilibrium provided the returns matrix R has rank S.

34 Proof: The proof mimics the technique outlined above - we explicitly calculate the Arrow Debreu equilibrium prices that support the same allocation. Begin with the Radner equilibrium spot prices p and asset prices q. These define the returns matrix R. Since it has rank S by assumption, it has an inverse. Define ξ = qr 1 defined this way ξ is a row vector consisting of S elements. Define AD prices ρ s ξ s p s. note - it is possible to show that ξ has only non-negative components (another time)

35 step 1 - if {x s} is feasible in a Radner equilibrium with asset prices q and spot prices p s nd production vectors y js then it must be feasible with direct trade at state contingent prices ρ s = ξ s p s. Radner feasibility implies that there is some vector z of asset trades such that qz 0 and p s x s = p s ω is + which immediately gives λ ij p s y js + j=1 z jp s y js j=1 ξ s p s x s = ξ s p s ω is + λ ij ξ s p s y js + j=1 z jξ s p s y js j=1

36 Sum both sides over s to get ξ s p s x s = s=1 ξ s p s ω is + λ ij ξ s p s y js + z jξ s p s y js s=1 s=1 j=1 s=1 j=1

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38 notice that the only reason that this isn t the same as the AD budget constraint is the last term ξrz j s=1 z jξ s p s y js j=1 Recall ξ is a row vector (with uninterpretable components except that there are S of them, so rewrite this as ξ s p s y js z j s=1 and notice that the sum over j is the same as taking the inner product of a row in the return matrix R and z j this inner product is then multiplied by ξ s then summed over s which, means the overall is given by j=1

39 since ξ = qr 1 this reduces to qz j which is zero by feasibility.

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41 finally, note that y j maximizes p y j over j s state s production set, so (provided ξ s > 0) it also maximizes ξ s psy js s=1 overall. So the radner equilibrium produces a feasible allocation that is at least as good for both firms and workers as anything feasible for them at arrow debreu prices ξ s p s - therefore the same allocation is an arrow debreu equilibrium.