# Macro Ii Problem Set 3

1. a. Given this two-period problem of labor supply maxc1 ,n1 ,c2 ,n2 ln[c1 ] + ln[1 − n1 ] + βln[c2 ] + βln[1 − n2 ] subject to the intertemporal budget constraint c1 [1 + r] + c2 = w1 n1 [1 + r] + w2 n2 Dividing each side by [1+r] for convenience gives c1 + c2 w 2 n2 = w 1 n1 + 1+r 1+r

We can solve for consumption and labor supply in each period (c1 , c2 , n1 , n2 ) by ﬁrst setting up the Lagrangian for a constrained optimization problem as L = ln[c1 ] + ln[1 − n1 ] + βln[c2 ] + βln[1 − n2 ] − λ[n1 + Taking FOC’s with respect to c1 , c2 , n1 , n2 , λ) gives dL dc1 dL dn1 dL dc2 dL dn2 dL dλ = = = = = 1 +λ=0 c1 1 − − λw1 = 0 1 − n1 β λ =0 + c2 1+r β λw2 =0 − − 1 − n2 1+r c2 w2 n2 c1 + = w1 n1 + 1+r 1+r n2 w2 c2 c1 − ] 1+r 1+r

We can now rearrange some of these ﬁrst order conditions to help solve for the variables of interest in terms of just prices. Each of these FOC’s can now be arranged so that they are equal to λ − − 1 c1 = = = = λ λ λ λ

1 [1 − n1 ]w1 [1 + r]β − c2 [1 + r]β − [1 − n2 ]w2

There are probably several diﬀerent ways to proceed now but the ﬁrst one that jumped out to me was to set equal the equations involving c1 and c2 to get − 1 c1 = = − [1 + r]β c2 c2 1+r

βc1

We can now plug this arrangement into our constraint to get c1 + βc1 c1 [1 + β] = w 1 n1 + w 2 n2 1+r w 2 n2 = w 1 n1 + 1+r 2

To proceed we can now combine the FOC for c1 and n1 to get − 1 c1 c1 = = 1 [1 − n1 ]w1 [1 − n1 ]w1

w1 n1

= w1 − c1

Subbing this into our already modiﬁed budget constraint yields c1 [1 + β] c1 [2 + β] = w 1 − c1 + = w1 + w 2 n2 1+r

w2 n2 1+r

We can now combine the FOC’s for n1 and n2 to get − 1 [1 − n1 ]w1 w1 [1 − n1 ] and using the c1 = [1 − n1 ]w1 we get c1 − w 2 n2 1+r w 2 n2 1+r [1 − n2 ]w2 β[1 + r] w2 = βc1 − 1+r w2 = − βc1 1+r = = = − [1 + r]β [1 − n2 ]w2 [1 − n2 ]w2 β[1 + r]

and plugging this into our modiﬁed constraint yields c1 [2 + β] c1 [2 + 2β] w2 − βc1 1+r w2 = w1 + 1+r = w1 +

which is in terms of only prices like we wanted. This means that the optimal level of c1 is c1 = 1 w2 [w1 + ] [2 + 2β] 1+r (1)

We know from setting the FOC’s for c1 and c2 equal that c2 = [1 + r]βc1 which means that the optimal level of c2 is [1 + r]β w2 c2 = [w1 + ] (2) [2 + 2β] 1+r We also know that from setting the FOC’s for c1 and n1 equal and rearranging that n1 = 1 − will be 1 w2 n1 = 1 − [w1 + ] w1 [2 + 2β] 1+r c1 w1

which (3)

which is the optimal level of n1 . Finally we know from setting the FOC’s for c1 and n2 equal and β[1+r]c rearranging that n2 = 1 − w2 1 which will be n2 = 1 − which is the optimal level of n2 . β[1 + r] w2 [w1 + ] w2 [2 + 2β] 1+r (4)

3

b. Now that we have the optimal levels of labor supply we can determine the response of labor supply given an increase in the real interest rate (r). For simplicity we can expand n1 to get n1 = 1 − w2 [1 + r]−1 w1 − w1 [2 + 2β] w1 [2 + 2β]

and via standard comparative statics we have dn1 w2 >0 = dr w1 [2 + 2β][1 + r]2 (5)

which is positive since each of the parameters is positive. This says that an increase in the real interest rate r will lead to a higher level of labor supply in period 1. Intuitively, this makes sense because a higher interest rate will serve as an incentive to save more (as more can be earned via savings) so individuals will want to work more now so as to earn more this period and save a larger amount than otherwise to reap the beneﬁts from the higher interest rate. 2. a. Given a two-period world where consumers maximize their utility function U = ln[c1 ] + βln[c2 ] − subject to their budget constraint [w1 n1 − p1 c1 ][1 + i1 ] + w2 n2 − p2 c2 = 0 which can be rearranged to get the standard p 2 c2 w2 n2 p 1 c1 + = w 1 n1 + 1 + i1 1 + i1 We can carry out this problem by ﬁrst setting up the Lagrangian for constrained maximization as L = ln[c1 ] + βln[c2 ] − nγ w2 n2 p2 c2 nγ 1 − 2 − λ[[w1 n1 + − p1 c1 − ] γ γ...

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