The derivation of the log-linearized equation of motion for capital, (4.52). Consider the equation of motion for capital, K+1 K+K (AL)- - C- Gr-8Kt.

(a) (i) Show that In K+1/ln K, (holding A, L, C, and G, fixed) is (1 + Ft+1)(K/K+1). (ii) Show that this implies that a In K+1/8 In K, evaluated at the balanced growth path is (1+r)/en+9.47

(b) Show that K+1 =+( + L) + A + (1 - A - A2 - A3)t =2 where A = (1+r)/eng, A = (1 - a)(r* + 6)/ae"+, and A3 = -(r* + 8) (G/Y)*/ae"; and where (G/Y)* denotes the ratio of G to Y on the bal- anced growth path without shocks. (Hints: 1. Since the production func- tion is Cobb-Douglas, Y* = (*+8)K*/a; 2. On the balanced growth path, K+en+K, which implies that C*Y* - G-8K* - (en+9 - 1)K*.)

(c) Use the result in

(b) and equations (4.43)-(4.44) to derive (4.52), where bkk A1+A2aLK+(1-A1-A2-A3)ack, bKA A(1+aLA)+(1-A1-A2-3)aca, and bG A2aLG A3 + (1 A1 A2 - A3)acG.