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Unformatted text preview: Noah Williams Economics 714 Department of Economics Macroeconomic Theory University of Wisconsin Spring 2009 Problem Set 2 Solutions 1. An economy consists of two types of consumers indexed by i = 1 , 2. There is one non storable consumption good. Let ( e i t ,c i t ) be the endowment, consumption pair for consumer i in period t . Consumer 1 has endowment stream e 1 t = 1 for t = 0 , 1 ,..., 20, and e 1 t = 0 for t ≥ 21. Consumer 2 has endowment stream e 2 t = 0 for 0 ≤ t ≤ 20 and e 2 t = 1 for t ≥ 21. Both consumers have preferences ordered by ∞ X t =0 β t u ( c i t ) where u is increasing, twice differentiable, and strictly concave. (a) Let the Pareto weight on consumer 1 be λ ∈ (0 , 1) and the weight on consumer 2 be 1 λ . Compute the Pareto optimal allocation. Solution. The Pareto problem is max c 1 ,c 2 λU 1 ( c 1 ) + (1 λ ) U 2 ( c 2 ) s.t. c 1 t + c 2 t = e 1 t + e 2 t , ∀ t , where we know that e 1 t = 1 and e 2 t = 0 for ≤ t < 21 , and e 1 t = 0 and e 2 t = 1 for 21 ≤ t . As a result, the resource constraint can be rewritten as c 1 t + c 2 t = 1 , ∀ t . (1) The First Order Conditions are ∂L ∂c 1 t = 0 : λu ( c 1 t ) = μ t , ∀ t ∂L ∂c 2 t = 0 : (1 λ ) u ( c 2 t ) = μ t , ∀ t . Equating the above we have u ( c 1 t ) u ( c 2 t ) = 1 λ λ , ∀ t (2) and using in this the resource constraint we have u ( c 1 t ) u ( 1 c 1 t ) = 1 λ λ . (3) 1 Notice that from (1) we have that if c 1 t > c 1 t then it must be that c 2 t < c 2 t . But then u ( c 1 t ) u ( c 2 t ) < u ( c 1 t ) u ( c 2 t ) , which violates (2), so it must be that c 1 t = c 1 , ∀ t c 1 t = c 2 , ∀ t . The value of c 1 (and hence of c 2 ) is given by (3) and depends on λ and the utility function. (b) Define a competitive equilibrium for this economy (with trading at date 0). Solution. A competitive equilibrium is an allocation c 1 t ,c 2 t ∞ t =0 and a set of prices { p t } ∞ t =0 such that: i) Agent i ∈ { 1 , 2 } maximizes his utility, i.e., he solves the problem max { c i t } ∞ t =0 ∞ X t =0 β t u ( c i t ) s.t. ∞ X t =0 p t c i t = ∞ X t =0 p t e i t , (4) ii) Markets clear, i.e., c 1 t + c 2 t = 1 , ∀ t . (5) (c) Compute a competitive equilibrium. Interpret your results, relating them to the Pareto optima in part (a). Solution. The First Order Conditions of agent i ’s problem are ∂L ∂c i t = 0 : β t u ( c i t ) = κ i p t , ∀ t . (6) Normalizing by setting p = 1 we have κ i = u ( c i ) . Combining the F.O.C. for the two agents we have u ( c 1 t ) u ( c 2 t ) = u ( c 1 ) u ( c 2 ) ≡ ν . (7) Notice that from (5) we have that if c 1 t > c 1 t , then it must be that c 2 t < c 2 t . But then u ( c 1 t ) u ( c 2 t ) < u ( c 1 t ) u ( c 2 t ) , which violates (7), so it must be that c 1 t = c 1 , ∀ t (8) c 1 t = c 2 , ∀ t ....
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This note was uploaded on 06/19/2011 for the course ECON 714 taught by Professor Staff during the Spring '08 term at University of Wisconsin.
 Spring '08
 Staff
 Economics

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