homework9 2010 solutions

homework9 2010 solutions - ECN/APEC 7240 Spring 2010...

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Unformatted text preview: ECN/APEC 7240 Spring 2010 Homework 9 Solutions 1) Consider an economy with a single consumer. There is one good in the economy, which arrives in the form of an exogenous endowment obeying , 1 1 t t t y y + + = λ where y t is the endowment at time t and and λ t +1 is governed by a 2-state Markov chain with transition matrix [ ] , 1 1 ] | Pr[ 22 22 11 11 1 -- = = = = + p p p p p ij i t j t λ λ λ λ and initial distribution Pr[ λ = λ i ] = π i where π 1 = π and π 2 = 1 - π . We calibrate λ 1 = 0.98 and λ 2 = 1.03. Assume the histories y t and λ t are observed at t . The consumer has preferences , ) ~ ( ∑ ∞ = t t t c u E β where β ∈ (0, 1) and γ γ- =- 1 ) ( 1 c c u for γ = 1. a) Let λ t = ( λ , . . . , λ t ). We define a consumption plan c t ( λ t ) to specify consumption at time t if the history of growth rates is λ t . Let us define , ) ( 1 ∏ = = t s s t t y y λ λ where y ( λ ) = y . Let 2 2 1 1 ) ( ) ( π λ π π λ π = = and given π t ( λ t ) define ). , ( ) , , ( 1 1 1 i t t ij j i t t p λ λ π λ λ λ π-- + = ECN/APEC 7240 Spring 2010 Let L = { λ 1 , λ 2 }. A competitive equilibrium consists of a consumption plan c t ( λ t ) and a price function q t ( λ t ) such that (1) given q t ( λ t ) the consumption plan solves the problem ∑ ∑ ∞ = ∈ )) ( ( ) ( max t L t t...
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homework9 2010 solutions - ECN/APEC 7240 Spring 2010...

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