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67

Assume A = p. Finally, the disturbance follows a first-order autoregressive process: = -1 + e,, where -1 < < 1 and where the f/s are mean-zero, i.i.d. shocks.

(a) Find the first-order condition (Euler equation) relating C, and expectations of C(+i.

(il) Guess that consumption takes the form C, = a + pKt + yet. Given this guess, what is Xt+i as a function of Kt and e,7

(c) What values must the parameters a, /3, 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 e on the paths ofY,K, and C?

4.9. A simplified real-business-cycle model with taste shocks. (This follows Blanchard and Fischer. 1989, p. 361.) Consider the setup in Problem 4.8. Assume, however, that the technological disturbances (the es) are absent, and that the instantaneous utility function is u(C,) = (- ( (-nt ). The is are mean-zero, i.i.d. shocks.

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

ib) Guess that consumption takes the form Cf = a -i- pKt + yv,. Given this guess, what is Kt+\ as a function of K, and vt?

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

id) What are the effects of a one-time shock to v on the paths ofY.K, and ?

4.10. The balanced growth path of the model of Section 4.3. Consider the model of Section 4.3 without any shocks. Let y*, k*, c*, and G* denote the values of YIAL, IAL, IAL and G/AL on the balanced growth path; w* the value of w/A; £* the value of LIN; and r* the value of r.

(a) Use equations (4.1)-(4.4), (4.23), and (4.26) and the fact that y*, k*, c*, w*, *, and r are constant on the balanced growth path to find six equations in these six variables. (Hint: the fact that in (4.23) is consumption per person, IN, and c* is the balanced-growth-path value of consumption per unit of effective labor, IAL, implies that = c**A on the balanced growth path.)

(b) Consider the parameter values assumed in Section 4.7. What are the implied shares of consumption and investment in output on the balanced growth path? What is the implied ratio of capital to aimual output on the balanced growth path?

4.11. Solving a real-business-cycle model by finding the social optlmum. Consider the model of Section 4.5. Assume for simplicity that = g = A = N = 0. Let V(Kt,At), the value function, be the expected present value from the cur-

"This problem uses dynamic programming and the method of undetermined coefficients. These two methods are explained in Section 10.4 and Section 4.6, respectively.



*The calculation of /So is tedious and is therefore omitted.

rent period forward of lifetime utility of the representative individual as a function of the capital stock and technology.

(a) Explain intuitively why V(») must satisfy

V(Kt,A,} = max{[ln Cf + bind - )] + eff [V(Xt+i, Af+i)]}. c,.e,

This condition is known as the Bellman equation.

Given the log-linear structure of the model, let us guess that V(») takes theform V(Xt,A,) = j3o-i-In + /3 In , where the values ofthe/3sare to be determined. Substituttng this conjectured form and the facts that K,+i = ( - Cf and £f[lnAf+iJ = , into the Bellman equation yields

V(Kt,A,) = max{[ln Cf + fcln(l - /,)J + e "[fh + 1 ( - Cf) + 1 ,]}.

Ci.tt

(b) Find the first-order condition for Cf. Show that it impUes that Cf/ , does not depend on or A,.

(c) Find the first-order condition for £f Use this condition and the result in part (b) to show that does not depend on Kt or A,.

id) Substitute the production function and the results in parts (b) and (c) for the optimal Cf and St into the equation above for V(»), and show that the resulting expression has the form V(K,,At) = , -i- InKt + p InAf.

(e) What must fe and fe be so that pf = fi and /3 = fe?*

(f) What are the implied values of / and £7 Are they the same as those found in Section 4.5 for the case of = g = 0?

4.12. Suppose the behavior of techriology is described by some process other than (4.8)-(4.9). Do Sf = S and = f for all f continue to solve the model of Section 4.5? Why or why not?

4.13. Consider the model of Section 4.5. Suppose, however, that the instantaneous utility function, Ut, is given by = In Ct + b{l - {(/{1 - y), b > 0, > 0, rather than by (4.7) (see Problem 4.4).

(a) Find the first-order condition analogous to equation (4.26) 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?

4.14. (a) If the Afs are uniformly zero and if In Yt evolves according to (4.39), what

path does Iny, settle down to? (Hint: note that we can rewrite (4.39) as In Yt-{n + g)t = Q + a[ln 1 - in + g)(t 1)] -i- (1 - a)A where Q =

a Ins + (1 - a)[A -I- In I* + - a{n + g).)

(b) Defining y, as the difference between 1 , and the path found in (a), derive (4.40).



"One could express r* in terms of the discount rate p. Campbell (1994) argues, however, that it is easier to discuss the models impUcations in terms of r* instead of p.

4.15. The derivation of the log-linearized equation of motion for capital, (4.52).

Consider the equation of motion for capital, Kti = Kt + K,(AtL,)" - Q -Gt - SKf

(a) (i) Show that <?hi,+i/<?In, (holding At,L,,Ct, and Gt fixed) is (1 + rt+i){Kt/Kt+i).

(ii) Show that this implies that dlnKt+i/dlnKt evaluated at the balanced growth path is (1 + r*)/e"+0.

(il) Show that

f+i XiKt + + It) + A,Gf -I- (1 - Al - A2 - )0.

where Ai = (1 + r*)/e"+0,A2 = (1 - a)(r* + 8)/ae"+0, and = -ir* + S) (G / Y)*/ae"+e; and where (G / ¥)* denotes the ratio of G to F on the balanced growth path without shocks. (Hints: 1. Since the production function is Cobb-Douglas, Y* = (r*-t-6)X*/a;2.0nthebalancedgrowthpath, Kt+i = e+ex,, which implies that C* = * - G* - 6K* - (e"+0 - l)K*.)

(c) Use the result in (ii) and equations (4.43)-(4.44) to derive (4.52), where = Ai--A2aLK+(l-Al-A2-A3)acK, = A2(l-baiA)-i-(l-Ai-A2-A3)acA, and i>kg = Azlg -h -I- (1 - Al - A2 - ) -

4.16. A Monte Carlo experiment, and the source of bias in OLS estimates of equation (4.56). Suppose output growth is described simply by 1 = St, where the fs are independent, mean-zero disturbances. NormaUze the initial value of Iny, Inyo, to zero for simplicity. This problem asks you to consider what occurs in this situation if one estimates equation (4.56), 1 yt = a + blny, i + St, by ordinary least squares.

(a) Suppose the sample size is 3, and suppose each e is equal to 1 with probability I and -1 with probability . For each of the eight possible realizations of (f 52, f)((l, 1,1), (1,1,-1), and so on), what is the OLS estimate of b? What is the average of the estimates? Explain intuitively why the estimates differ systematically from the true value of b = 0.

ib) Suppose the sample size is 200, and suppose each e is normally distributed with a mean of 0 and a variance of 1. Using a random-number generator on a computer, generate 200 such fs; then generate In ys using In yt = ff and Inyo = 0; then estimate (4.56) by OLS; finally, record the estimate of b. Repeat this process 500 times. What is the average estimate of b? What fraction of the estimated bs are negative?



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