Microeconomics Problems and Solutions - Optimization and Models

Problem 5.5: Increase in w1 and w2 with w1/w2 unchanged

5.5. Consider the problem investigated in (5.16) (5.21).

 

(a)Show that an increase in both w1 and w2 that leaves w1/w2 unchanged does not affect ?1 or ?2.

 

(b)Now assume that the household has initial wealth of amount Z > 0.

 

(i)Does (5.23) continue to hold? Why or why not?

 

(ii)Does the result in (a) continue to hold? Why or why not?

(a)Use an argument analogous to that used to derive equation (5.23) to show that household optimization requires b/(1 ? ?t) = e?? Et [wt (1 + rt +1)b/ wt +1(1 ? ?t +1)].

(b)Show that this condition is implied by (5.23) and (5.26). (Note that [5.26] must hold in every period.) 

 

5.8. A simplified real-business-cycle model with additive technology shocks. (This follows Blanchard and Fischer, 1989, pp. 329 331.) Consider an economy consisting of a constant population of infinitely lived individuals. The representative individual maximizes the expected value of ??t=0u(Ct)/(1+?)t,?>0. The instantaneous utility function, u(Ct), is u Assume that C is always in the range where u'(C) is positive. Output is linear in capital, plus an additive disturbance: Yt = AKt + et. There is no depreciation; thus Kt +1 = Kt + Yt ? Ct, and the interest rate is A. Assume A = ?. Finally, the disturbance follows a first-order autoregressive process: et = ?et ?1 + ?t, where ?1 < ? < 1 and where the ?t ’s are mean-zero, i.i.d. shocks.

(a)Find the first-order condition (Euler equation) relating Ct and expectations of Ct +1.

(b)Guess that consumption takes the form Ct = ? + ? Kt + ?et. Given this guess, what is Kt +1 as a function of Kt and et?

(c)What values must the parameters ?, ?, and ? have for the first-order condition in part (a) to be satisfied for all values of Kt and et?

(d)What are the effects of a one-time shock to ? on the paths of Y, K, and C?

5.9. A simplified real-business-cycle model with taste shocks. (This follows Blanchard and Fischer, 1989, p. 361.) Consider the setup in Problem

5.10. Assume, however, that the technological disturbances (the e’s) are absent and that the instantaneous utility function is u(Ct) = Ct ? ? (Ct + ?t)2. The ? ’s are mean-zero, i.i.d. shocks.

(a)Find the first-order condition (Euler equation) relating Ct and expectations of Ct +1.

(b)Guess that consumption takes the form Ct = ? + ?Kt + ? et. Given this guess, what is Kt +1 as a function of Kt and et?

(c)What values must the parameters ?, ?, and ? have for the first-order condition in (a) to be satisfied for all values of Kt and et?

(d)What are the effects of a one-time shock to ? on the paths of Y, K, and C?  
 
5.12.Suppose technology follows some process other than (5.8) (5.9). Do st = ? and ?t=ˆ? for all t continue to solve the model of Section 5.5? Why or why not?

5.13.Consider the model of Section 5.5. Suppose, however, that the instantaneous utility function, ut, is given by ut = ln ct + b (1 ? ?t)1??/(1 ? ?), b > 0, ? > 0, rather than by (5.7) (see Problem 5.4).

(a)Find the first-order condition analogous to equation (5.26) that relates current leisure and consumption, given the wage.

(b)With this change in the model, is the saving rate (s) still constant?

(c)Is leisure per person (1 ? ?) still constant?