handoutcompletemkts - Pricing Assets with Complete Markets...

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Pricing Assets with Complete Markets This handout follows Ljungqvist and Sargent, Ch 8 “Equilibrium with Com- plete Markets". 1 Environment In f nite Horizon, Stochastic Pure Exchange Environment In each period, there is a realization of a stochastic event s t S which is observable at date t 0 . The history of events up to (and including t ) is denoted s t =( s 0 ,s 1 , ..., s t 1 t ) S t +1 which can be represented as a tree diagram. Such a representation will be useful in understanding decentralization. The unconditional probability of observing history s t is denoted π t ( s t ) and the probability of observing s t conditional upon realization s τ for t>τ is denoted π t ( s t | s τ ) . Each agent i is endowed with y i t ( s t ) of the time t history s t nonstoreable consumption good. We will assume y i t ( s t ) > 0 . Household i orders possibly history dependent consumption streams U ( c i )= T X t =0 X s t S t +1 β t π t ( s t ) u ( c i t ( s t )) (1) which is an increasing, twice di f erentiable strictly concave function which satis f es the Inada condition lim c 0 u 0 ( c . In most of what follows we will take T = . Note that while preferences are identical among agents, incomes are not (so a planner or the market may treat people di f erently). 2 Planner’s Problem Before moving onto the decentralized competitive equilibrium, we solve the planner’s problem for this economy. Since not all agents are assumed to have identical income realization paths, the planner solves max { c i } I i =1 I X i =1 λ i U ( c i ) 1
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subject to the resource feasibility constraint s.t. X i c i t ( s t ) X i y i t ( s t ) , t, s t . (2) Assigning multipliers θ t ( s t ) for each resource constraint, the langrangian for the plannner’s problem is X t =0 X s t S t +1 ( I X i =1 λ i β t π t ( s t ) u ( c i t ( s t )) + θ t ( s t ) I X i =1 £ y i t ( s t ) c i t ( s t ) ¤ ) The foc for any c i t ( s t ) is β t π t ( s t ) u 0 ( c i t ( s t )) = θ t ( s t ) λ i , i, t, s t . (3) Taking ratios of (3) for consumers i and WLOG 1 gives u 0 ( c i t ( s t )) u 0 ( c 1 t ( s t )) = λ 1 λ i ⇐⇒ c i t ( s t )= u 0 1 μ λ 1 u 0 ( c 1 t ( s t )) λ i , t, s t . (4) Plugging this expression into the feasibility constraint (2) yields X i u 0 1 μ λ 1 u 0 ( c 1 t ( s t )) λ i = X i y i t ( s t ) Y ( s t ) , t, s t (5) which is one equation in one unknown c 1 t ( s t ) for each t. For the example where u ( c )=log( c ) , then (4) is just c i t ( s t λ i λ 1 c 1 t ( s t ) so agent i ’s consumption is linear in agent 1 ’s and the more the planner weights i , the more he gives i. Further (5) is just c 1 t ( s t Y ( s t ) ³ 1+ P i λ i λ 1 ´ . Since all the variables on the rhs are numbers, the lhs is a number. Prop CC1. Given a realization of aggregate output Y ( s t ) , every agent’s consumption is a function of the aggregate endowment in an e cient allo- cation and does not depend on the realization of individual endowments.
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This note was uploaded on 08/06/2008 for the course ECON 387 taught by Professor Corbae during the Spring '07 term at University of Texas at Austin.

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handoutcompletemkts - Pricing Assets with Complete Markets...

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