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ps5_ans_sp08_part2 - Econ 387L Macro II Spring 2008...

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Econ 387L: Macro II Spring 2008, University of Texas Instructor: Dean Corbae Answers Problem Set #5 - Part 2 I. This question explores the equivalence between time 0 Arrow-Debreu forward markets and Arrow’s sequential representation in a nonstochastic exchange economy. There are I households and the horizon is in fi nite. Goods are nonstorable and household i has an endowment of goods y i > 0 each period. Household i ’s utility function is given by X t =0 ¡ β i ¢ t u i ( c i t ) where u i is strictly increasing and concave, continuously differentiable. 1. Write household i ’s budget constraint in a forward markets economy where p t is the forward price of the consumption good delivered at time t set at time 0 . Answer. X t =0 p t c i t X t =0 p t y i . (1) 2. Write household i ’s budget constraint in a sequential markets economy where q t is the spot price of the consumption good in period t, a i t +1 are loans (if positive) or borrowings (if negative) by household i between time t and t + 1 , and r t is the interest paid at t on assets chosen at t 1 . Assume that a 0 = 0 . Answer. q t c i t + a i t +1 q t y i + (1 + r t ) a i t (2) 3. Show under what conditions the two constraints equivalent. Would a borrowing constraint be suf fi cient to generate the equivalence? Answer. Dividing (2) by (1 + r 0 )(1 + r 1 ) · · (1 + r t ) yields q t (1 + r 0 )(1 + r 1 ) · · (1 + r t ) c i t + 1 (1 + r 0 )(1 + r 1 ) · · (1 + r t ) a i t +1 (3) q t (1 + r 0 )(1 + r 1 ) · · (1 + r t ) y i + 1 (1 + r 0 )(1 + r 1 ) · · (1 + r t 1 ) a i t Adding up the fi rst T constraints yields T X t =0 b p t c i t + 1 (1 + r 0 )(1 + r 1 ) · · (1 + r T ) a i T +1 T X t =0 b p t y i . (4) where b p t q t (1 + r 0 )(1 + r 1 ) · · (1 + r t ) To see this, notice that the t 1 version of (3) is q t 1 (1 + r 0 )(1 + r 1 ) · · (1 + r t 1 ) c i t 1 + 1 (1 + r 0 )(1 + r 1 ) · · (1 + r t 1 ) a i t (5) q t 1 (1 + r 0 )(1 + r 1 ) · · (1 + r t 1 ) y i + 1 (1 + r 0 )(1 + r 1 ) · · (1 + r t 2 ) a i t
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