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?
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