 ## Problem Set 3

### Module 2-2: The Neoclassical Growth Model in Continuous Time  留学生宏观经济代写

1.Ramsey-Cass-Koopmans model. Suppose the planner seeks to maximize subject to resource constraint

and feasible consumption c 0. Consider the optimal growth problem with isoelastic utility function and the Cobb-Douglas production function F (K, L) = AKαL 1α , where α (0, 1). Suppose for simplicity that the level of productivity A > 0 is constant. Let ct , kt , yt etc. denote consumption, capital, output etc in per worker units. Assume n > 0, δ > 0 .

### (a) Write down the complete optimal growth problem. Solve this problem using the Hamil-tonian method and derive two difffferential equations in the variables (c, k).  留学生宏观经济代写

(b) Assume for now that labor force remains constant, n = 0. Our baseline economy transitions from an initial level of capital k0 which is below the steady state level k.Now,consider a modifified economy which is identical to the original economy except for the initial level of capital which is set to k0/2. Draw the time paths of k and  for both the baseline and the modifified economies. How do they compare? Use a phase diagram motivate your answer.

(c) Assume n > 0, δ > 0 . Consider an economy at the balanced growth path. At time 0, it experiences a sudden increase in productivity A> A. Draw the time paths of k and c following this event. Proceed using the following steps:

(i) draw the phase diagram for the baseline case, and suppose that k0 is equal to k for this case;

(ii) draw the modifified phase diagram, indicating what has changed with the higher value of A;

(iii) set k0 is equal to baseline k and draw the modifified time paths of k and c, indicating how they compare with the baseline time paths.

What effffect does the increase in A have? Pay special attention to the value of consumption at t = 0; is it higher or lower in the modifified case? Why? Give an intuitive answer.