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By David Romer

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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 .

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