?Detailed Question: Assignment if for a first year graduate course in macro theory. Based on the Ramsey Cass Koopman model?
Problem 1:
Assume that the instantaneous utility function u(C) is ln C. Consider the problem of a household maximizing lifetime utility subject to its budget constraint. Find an expression for C at each time as a function of initial wealth plus the present value of labor income, the path of r(t), and the parameters of the utility function.
Problem 2:
Consider a Ramsey-Cass-Koopmans economy that is on its balanced growth path, and suppose there is a permanent (unanticipated) fall in g.
a) How, if at all, does this affect the k'=0 curve?
b) How, if at all, does this affect the c'=0 curve?
c) What happens to c at the time of the change?
d) Find an expression for the impact of a marginal change in g on the fraction of output that is saved on the balanced growth path. (i.e. ds/dg, in a way) Can one tell whether this expression is positive or negative?
e) For the case where the production function is Cobb-Douglas with parameter alpha, rewrite your answer to part d in terms of p, n, g, theta, and alpha. (Hint: Use the fact that f'(k*)=p+theta*g.)
Problem 3:
Suppose that the lifetime utility function in the Ramsey-Cass-Koopmans model is given by
U = B.t=0∫∞ e-βt.[c(t) + G(t)]1-θ /(1-θ) dt
a) What does this imply about the relationship between the utility from private consumption and the utility from government spending?
b) If the economy is initially on its balanced growth path and if households' preferences are given by U, what are the effects of a permanent increase in government purchases on the paths of consumption, capital, and the interest rate?
Problem 4:
In class, we discussed the intuition behind the speed to adjustment to a balanced growth path in the Ramsey- Cass-Koopmans model versus in the Solow model. In addition, your textbook has a section entitled "The Speed of Adjustment" in chapter 2 that relates to the Ramsey-Cass-Koopmans model. (It will likely also be helpful to review the previous section.) For this problem, you can assume a Cobb-Douglas production function with parameter alpha.
Assume alpha = 1/3, p = 2%, n = 1%, g = 3%, and theta = 1/2.
a) What is the real interest rate along the balanced growth path?
b) What is the effective saving rate along the balanced growth path?
c) What is the implied rate of adjustment in the Ramsey-Cass-Koopmans model?
d) What is the analogous rate of adjustment in the Solow model?
e) Does this comparison match with the intuition that we discussed in class?