Macro Ii Problem Set 3
Problem Set 3 ECON 973 Fall 2012 Fluctuations: Discrete Time Models
Ben Brewer 12/10/12
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
Ben Brewer 12/10/12
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