8108notes_2010

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

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