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Unformatted text preview: Homework 1. Solution Macroeconomic Theory I. Viktor Tsyrennikov Question 1. An individual seeks to maximize his utility U ( c 1 , c 2 ) = ln ( c 1 ) + ln ( c 2 ) , (0 , 1) , where c 1 and c 2 are non-negative consumption levels of a perishable good in period 1 and 2 respectively. The individual receives income y 1 and y 2 in the two periods that he lives. He can save at interest rate r s > 0 and borrow at rate r b > r s . a) Write down the individuals budget constraint for the case when he is a saver and when he is a borrower. Solution. 1 c 1 + c 2 / (1 + r s ) = y 1 + y 2 / (1 + r s ) , if saver ( y 1 greaterorequalslant c 1 ) , c 1 + c 2 / (1 + r b ) = y 1 + y 2 / (1 + r b ) , if borrower ( y 1 lessorequalslant c 1 ) . b) Solve for the optimal consumption plan. Solution. Lets s 1 greaterorequalslant 0 and b 1 greaterorequalslant 0 denote the individuals savings and bor- rowing decisions in period 1. So we can write the unified budget constraint as c 1 + s 1 = y 1 + b 1 , c 2 = y 2 + (1 + r s ) s 1 (1 + r b ) b 1 . These hold both for savers and borrowers. The Lagrangian for the agents problem is: L = ln ( c 1 ) + ln ( c 2 ) + 1 ( y 1 + b c 1 s ) + 2 ( y 2 + (1 + r s ) s (1 + r b ) b c 2 ) + s s + b b. 1 Note: In my solution first period income includes a . That is if a negationslash = 0 then add a to y 1 in every formula....
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- Spring '11