Problem
A closed-form solution of the Ramsey model. (This follows Smith, 2006.) Consider the Ramsey model with Cobb-Douglas production, y(t) = k (t) α, and with the coefficient of relative risk aversion (θ) and capital's share (α) assumed to be equal.
(a) What is k on the balanced growth path (k∗)?
(b) What is c on the balanced growth path (c∗)?
(c) Let z(t) denote the capital-output ratio, k (t)/y(t), and x (t) denote the consumption-capital ratio, c (t)/k(t). Find expressions for z?(t) and x?(t)/x (t) in terms of z, x, and the parameters of the model.
(d) Tentatively conjecture that x is constant along the saddle path. Given this conjecture:
(i) Find the path of z given its initial value, z(0).
(ii) Find the path of y given the initial value of k, k (0). Is the speed of convergence to the balanced growth path, d ln[y(t) - y∗]/dt, constant as the economy moves along the saddle path?
(e) In the conjectured solution, are the equations of motion for c and k, (2.24) and (2.25), satisfied?
The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.