hw-02-sol - Macroeconomic Theory I. Viktor Tsyrennikov...

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Homework 1 Macroeconomic Theory I. Viktor Tsyrennikov Question 1. An individual seeks to maximize his utility U ( c ) = s t =0 β t u ( d t ) , β (0 , 1) , where d t is a non-negative consumption level of a durable good in period t . Durable goods are accumulated according to: d t +1 = (1 - δ ) d t + c t , δ (0 , 1] , where c t g 0 is date t spending on durable goods. Assume that borrowing limits are such that they are never binding. Initial values a 0 , d 0 are given. a) Derive the durable-goods-consumption Euler equation. How does is it compare with the case when goods are perishable? Solution. The Lagrangian for this problem is L = s t =0 b β t u ( d t )+ λ t ( R t +1 ( a t + y t - c t ) - a t +1 )+ κ t ((1 - δ ) d t + c t - d t +1 )+ μ t +1 ( a t +1 + B t +1 ) B , (1) where λ t is the Lagrange multiplier on the agent’s date- t budget constraint, κ t is the lagrange multiplier on the evolution of durable goods equation and μ t +1 is the Lagrange multiplier on the borrowing constraint on the date t +1 assets. The FOCs for this problem for each date t are: a t +1 : - λ t + R t +2 λ t +1 + μ t +1 = 0 , d t +1 : - κ t + κ t +1 (1 - δ ) + β t +1 u ( d t +1 ) = 0 , c t : - R t +1 λ t + κ t = 0 . Combining the FOCs gives:
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hw-02-sol - Macroeconomic Theory I. Viktor Tsyrennikov...

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