Macro Prelim Q1 -answers (by K. Fukushima)
نویسنده
چکیده
1. A competitive equilibrium is a sequence (p * t , c * t , s * t+1) T t=0 such that (i) (c * t , s * t+1) T t=0 solves max (ct,s t+1) T t=0 T t=0 β t u(c t), subject to (1) c t + p * t s t+1 ≤ (p * t + y t)s t , ∀t c t , s t ≥ 0, ∀t s 0 = 1 and (ii) markets clear: c * t = y t , ∀t s * t = 1, ∀t 2. The equation is: p * t = β u (y t+1) u (y t) (p * t+1 + y t+1) Proof: Let (p * t , c * t , s * t+1) T t=0 be a competitive equilibrium. Then (c * t , s * t+1) T t=0 is an interior solution to the agent's problem (1). (Interiority follows from c * t = y t > 0 and s * t+1 = 1 > 0.) Thus there are Lagrange multipliers (λ * t) T t=0 such that the Kuhn-Tucker conditions hold: β t u (c * t) = λ * t , ∀t = 0, 1, ..., T (2) λ * t p * t = λ * t+1 (p * t+1 + y t+1), ∀t = 0, 1, ..., T − 1 (3) λ * T p * T = 0. The result follows from (2), (3), and the market clearing condition c * t = y t. 3. The nal period security price is p * T = 0. To see this, combine (2) for t = T and (4) to get β T u (c * T)p * T = 0. Since β T u (c * T) > 0 we have p * T = 0.
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تاریخ انتشار 2011