By David Romer
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Extra info for Advanced Macroeconomics Solution Manual
From equation (3), (1 rt 1 )S t C 2,t 1 (1 n)T . Solving for St yields C 2 ,t 1 (1 n) T. (4) S t 1 rt 1 (1 rt 1) Now substitute equation (4) into equation (2): C 2 ,t 1 (1 n) (5) C1,t T. Aw t T 1 rt 1 (1 rt 1) Rearranging, we get the intertemporal budget constraint: C 2 ,t 1 ( rt 1 n) Aw t T. (6) C1,t (1 rt 1 ) 1 rt 1 We know that with logarithmic utility, the individual will consume fraction (1 + )/(2 + ) of her lifetime wealth in the first period.
Thus the kt+1 function shifts down over this range of kt 's. Finally, right at lnkt = 1/(1 - ), the old and new kt+1 functions intersect. 15 (a) We need to find an expression for kt+1 as a function of kt. Next period's capital stock is equal to this period's capital stock, plus any investment done this period, less any depreciation that occurs. Thus (1) Kt+1 = Kt + sYt - Kt . To convert this into units of effective labor, divide both sides of equation (1) by At+1Lt+1 : K t 1 K t (1 ) sYt K t (1 ) sYt k t (1 ) sf ( k t ) , (2) A t 1L t 1 A t 1L t 1 (1 n)(1 g)A t L t (1 n)(1 g) which simplifies to s 1 (3) k t 1 kt f (k t ).
We can now derive the intertemporal budget constraint. From equation (4), (19) S t C 2,t 1 1 rt 1 T . Substituting equation (19) into equation (2) yields C 2,t 1 (20) C1,t Aw t T T , 1 rt 1 or simply C 2,t 1 Aw t . (21) C1,t 1 rt 1 This is just the usual intertemporal budget constraint in the Diamond model. Solving the individual's maximization problem yields the usual Euler equation: (22) C 2 ,t 1 1 1 (1 rt 1 ) C1,t Substituting this into the budget constraint, equation (21), yields (23) C1,t 1 2 Aw t .