7. Consider a Diamond economy in which C, and C2, represent the consumption in period t of young and old individuals, respectively. The utility, U, of an individual born at time / depends on Cir + C2; and that its utility function takes the form: U, = 1 1-0 1+ p(1-0) 41-8 , 0>0, p>-1 The second-period consumption of the individual born at time t is specified as: C2141 = (1+ 1;4)(W,A, -Cy). Capital stock in period + 1 is the amount saved by the individual in period t so that: K, HI = S(1,4) L, A,w, [Hints: L, = (1+n)L, ; A, = (1+g)A,;r, = f'(k); and w, = f(k,)-kf'(k,) . The variables are defined as follows: A, = knowledge; w, = wage rate; r= interest rate; p= discount rate; and 0 = coefficient of relative risk aversion; L, = labour; L, A, = effective labour; and k, = capital per unit of effective labour]. (a) Derive the budget constraint (b) Set up the Lagrangian and derive the Euler equation (c) Express C), in terms of labour income and interest rate (d) Given your answer in (c), derive the fraction of income saved, and interpret your result (e) Show how k,+, implicitly depends on K,8. Consider a Ramsey-Cass economy in which the representative household maximizes its utility function specified as: I U=8[" et D 4 = 1-g Subject to the budget constraint that takes the form: A(0)L(0) +r. X O(7) A(0)L(0)e" =" it H +=0 AQLOE o H - H r e "e(t) t=0 _ A0)""L(0) Where B= H and ,3 =p0R {l = 9}3 -2 1 8 [Hints: L =total number of Lt workers; H = total number of households; %= the number of workers in each household at time t; A()L(t) = effective labour; k(0)= initial capital holdings per unit of effective labour; w= wage I rate per unit of effective labour; p = the houschold's time preference; R(f) = I 5 (r)dr ; A(D)L(t) = A0)L(0)e " 1. The campus police are baffled: is it murder or is it suicide? Assist them in solving the case by explaining the questions below: (a) Derive the Euler equation for the maximisation problem. You are required to interpret your result; (b) Present the phase diagram for the dynamics of consumption () and capital stock (k) per unit of effective labour, and explain the quadrant where both the values of cand k are increasing; (c) Explain why in the phase diagram in (ii), & (the level of k that implies =0) is set below the golden-rule value of k (the value of k associated with the peak of the k = 0locus), what is the implication of this on the model