4)] In this question we use the cash-in-advance (CIA) constraint and endogenize the labour supply decision. Less than fully confid- ent readers may want to attempt this question after studying the Ramsey model of Chapter 13. The infinitely-lived representative household has a lifetime utility func- tion of the form: A= = Su (C (1),1-L (1) e-pt dt, (Q11.16) where C (t) is consumption at time t, L (t) is labour supply (1 - (t) is leisure), p is the rate of time preference (p > 0), and U) is the felicity function. It has the usual features: 0, U-L = >0, ac a (1-L) 22u 22u Ucc = 0, where UC,1-L = du The household budget identity is given by: aca(1-2) (t) = r(t) A (t) + w (t) L (t) + T (t) - C(t) [r (t) + n(t)] m(t), (Q11.17) where A (t) is the stock of tangible assets, r(t) is the real interest rate, w (t) is the real wage rate, T (t) is transfers received from the government, it (t) is the inflation rate (T (t) > 0), and m(t) is the real stock of money balances. Obviously, r(t) + (t) is the nominal interest rate. As usual, we have that (t) = dA (t)/dt. Total wealth consists of real money balances plus the value of the capital stock: A (t) = V(t) + m(t). (Q11.18) The cash-in-advance (CIA) constraint is given by: m(t) > C($). (Q11.19) The household is blessed with perfect foresight and takes the path of government transfers as given. The perfectly competitive firm maximizes the value of the firm, V(0) = S[F(K (1),L(t) w(t)L(t) 1 (h)]e=R68 dt, (Q11.20) where R(t) = Sor(s) ds is the cumulative interest factor, and I (t) is gross invest- ment, and F.) features constant returns to scale. The firm faces the capital accu- mulation identity, K (t) = I(t) - 8K (t), and takes as given the initial capital stock, K(0). The choice variables are I (t) and L (t) (and thus K (1)). (a) Solve the household's optimization problem by means of optimal control meth- ods. (b) Solve the firm's optimization problem by means of optimal control methods. Prove that V (0) = K (0) in the optimum