ps4_sol - ECO 475 PS4 Solution Yu LIU November 18, 2010...

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Unformatted text preview: ECO 475 PS4 Solution Yu LIU November 18, 2010 Question 1 De ne aggregate capital: K t , aggregate money holding: M ; for agents born at period t , individual savings: s y,t and s o,t , individual money holding: m y,t and m o,t . Since agents live two periods and supply labor only when young, and from log utility, agents want to smooth their consumption, then ( x y,t x o,t = 0 , x = s,m . (a) Young generation's problem: Given w t , r t +1 , p t and p t +1 , max { c y,t ,c o,t ,s y,t ,m y,t } log c y,t + log c o,t subject to c y,t + s y,t + m y,t p t = w t (1) c o,t = (1 + r t +1 ) s y,t + m y,t p t +1 (2) s y,t m y,t (b) Suppose constraints x y,t , x = s,m are not binding. Lagrangian: L = log c y,t + log c o,t + y,t w t- c y,t- s y,t- m y,t p t + o,t (1 + r t +1 ) s y,t + m y,t p t +1- c o,t FOC wrt c y,t : 1 c y,t- y,t = 0 (3) FOC wrt c o,t : c o,t- o,t = 0 (4) FOC wrt s y,t :- y,t + (1 + r t +1 ) o,t = 0 (5) FOC wrt m y,t :- y,t p t + o,t p t +1 = 0 (6) Combine equations (5) and (6), we can get 1 p t +1 p t = 1 1 + r t +1 (7) Firm's problem: Assume the production function F ( K t ,A t N t ) satis es standard assumptions, e.g: F is increasing, concave, and continuously di erentiable; F ( k, 0) = F (0 ,n ) = 0 for any k and n ; F is homogeneous of degree 1 (CRS). Especially, since there is no productivity or population growth, A t = 1 , N t = N . Let k t = K t N be capital per capita. De ne f ( k t ) F ( k t , 1) , then F ( K t ,N ) = Nf ( k t ) , F K ( K t ,N ) = f ( k t ) , F N ( K t ,N ) = f ( k t )- f ( k t ) k t . So for the rm, r t +1 = F K ( K t +1 ,N )- = f ( k t +1 )- (8) w t = F N ( K t ,N ) = f ( k t )- f ( k t ) k t (9) Substitute (8) into (7), we can get p t +1 p t = 1 f ( k t +1 ) + 1- (10) Intuition: In ation rate ( p t +1 p t ) gives the inverse of the marginal return to money holding. The marginal product of capital ( f ( k t +1 ) ) plus the fraction of capital which is not depreciated ( 1- ) gives the marginal return to savings. If constraints x y,t , x = s,m are not binding, both channels are utilized. Under no arbitrage condition, the marginal returns for the two channels should be equal....
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ps4_sol - ECO 475 PS4 Solution Yu LIU November 18, 2010...

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