A closed-form solution of the Ramsey model. (This follows Smith, 2006.) Consider the Ramsey model with Cobb-Douglas production, y(t ) = k (t )A closed-form solution of the Ramsey model. (This follows Smith, 2006.)
Consider the Ramsey model with Cobb-Douglas production, y(t ) = k (t )a, and with the coefficient of relative risk aversion (?) and capital's share (a) 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?a, and with the coefficient of relative risk aversion (?)