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# 8108notes_2010 - Lecture Notes in Macroeconomic Theory 8108...

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Lecture Notes in Macroeconomic Theory 8108 Jos´ e-V´ ıctor R´ ıos-Rull University of Minnesota May 2, 2010 March 23th and March 25th, 2010 1 Introduction Equilibrium can be deﬁned as a prediction of what will happen and therefore it is a mapping from environments to allocations. One equilibrium concept that we will deal with is Competitive Equilibrium 1 . Lets deﬁne the CE for an economy without uncertainty : The commodity space is L = { ( l 1 ,l 2 3 ) : l i = ( l it ) t =0 l it R , X t | l it | < , i = 1 , 2 , 3 } The consumption possibility set is X ( k 0 ) = { x ∈ L : ( c t ,k t +1 ) t =0 such that t = 0 , 1 ,... c t t +1 0 x 1 t + (1 - δ ) k t = c t + k t +1 - k t +1 x 2 t 0 - 1 x 3 t 0 k 0 = k 0 } 1 Arrow-Debreu or Valuation Equilibrium 1

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The production possibility set is Y = Q t Y t where Y t = { ( y 1 t ,y 2 t 3 t ) R 3 : 0 y 1 t F ( - y 2 t , - y 3 t ) } Deﬁnition 1 An Arrow-Debreu equilibrium is ( x * * ) X × Y , and a continuous linear functional ν * such that i. x * arg max x X,ν * ( x ) 0 t =0 β t u ( c t ( x ) , - x 3 t ) ii. y * arg max y Y ν * ( y ) iii. x * = y * Now, let’s look at the one-sector growth model’s Social Planner’s Problem: max t =0 β t u ( c t , - x 3 t ) ( SPP ) s.t. c t + k t +1 - (1 - δk t = x 1 t 0 x 2 t k t 0 x 3 t 1 0 y 1 t F ( - y 2 t , - y 3 t ) x = y k 0 : given. Suppose we know that a solution in sequence form exists for (SPP) and is unique. Homework : Clearly stating suﬃcient assumptions on utility and production function, show that (SPP) has a unique solution. Two important theorems show the relationship between CE allocations and Pareto optimal allocations: Theorem 1 Suppose that for all x X there exists a sequence ( x k ) k =0 , such that for all k 0 , x k X and U ( x k ) > U ( x ) . If ( x * * * ) is an Arrow-Debreu equilibrium then ( x * * ) is Pareto eﬃcient allocation. Theorem 2 If X is convex, preferences are convex, U is continuous, Y is convex and has an interior point, then for any Pareto eﬃcient allocation ( x * * ) there exists a continuous linear functional ν such that ( x * * ) is a quasiequilibrium, that is (a) for all x X such that U ( x ) U ( x * ) it implies ν ( x ) ν ( x * ) and (b) for all y Y , ν ( y ) ν ( y * ) . 2
Note that at the very basis of CE deﬁnition and welfare theorems is implicit assumption of perfect commitment and perfect enforcement. Note also that the FWT implicitly as- sumes there is no externality or public goods (achieves this implicit assumption by deﬁning a consumer’s utility function only on his own consumption set but no other points in the commodity space). From the First Welfare Theorem, we know that if a Competitive Equilibrium exits, it is Pareto Optimal. Moreover, if the assumptions of the second welfare theorem is satisﬁed and if the SPP has unique solution then competitive equilibrium allocations is unique and they are same as PO allocations. Prices can be constructed using this allocations and ﬁrst order conditions.

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## This note was uploaded on 11/12/2010 for the course ECON 8108 taught by Professor Staff during the Spring '08 term at Minnesota.

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8108notes_2010 - Lecture Notes in Macroeconomic Theory 8108...

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