Assignment: Macroeconomic Theory and Policy
Answer all questions
1. Consider the Solow's growth model with standard assumptions of constant growth of labor(n), technology (g) and depreciation (δ), and that the production function is given by Y=K^α (AL)^(1-α).
(i) Determine the steady-state values of capital stock(k^*), output (y^*), and consumption(c^*), all in per unit of effective labor, as functions of the parameters of themodel, s, n, δ, g, and α.
(ii) Determine the golden-rule value of capital stock(k_GR)?
2. (i) In Solow's growth model, assuming population and technology growing at the constant rates of n and g respectively, derive the elasticity of the balanced-growth path level of output (y^*)with respect to the saving rate (s). If α_k (k^* )=1/3, what is the impact of a 10 per cent increase in saving rate on the output per worker in the long run?
Assuming Cobb-Douglas production function in the Solow's growth model, derive the marginal product of capital (rate of return to capital). In the context of cross-country growth differences, assuming that the elasticity of output with respect to capital is one-third ("i.e." α_k=1⁄3), what is the magnitude of difference in the rates of return to capital that the model predicts for a tenfold difference in output per worker between two countries?
3. Consider an infinitely horizon Ramsey-Cass-Koopmans economy with agents' utility function given by u(t)=C(t)^(1-θ)/(1-θ).
Set up the optimization problem, derive the optimal consumption plan.
Using the capital accumulation equation, derive and describe the dynamics of the model in (c,k) plane. Determine the rate of steady state growth in output per unit of effective labor.
Consider the economy to be on its balanced growth path. What is the impact of a fall in the discount rate?
Assume that the government buys output at rate G(t) per unit of effective labour per unit time. What are the effects of a permanent increase in government purchases in this economy?
The Ramsey-Cass-Koopmans model assumes that government purchases do not affect utility from private consumption. However, the opposite view is that government purchases and private consumption are perfect substitutes. Suppose that the modified utility function in the Ramsey-Cass-Koopmans model is given by
U=B∫_(-∞)^(+∞)e^(-βt [c(t)+G(t)]^(1-θ)/(1-θ)) dt.
Where B≡A(0)^(1-θ) L(0)/H and β≡ρ-n-(1-θ)g.
If households' preferences are given by U and if the economy is initially on its balanced growth pat, what are the effects of a temporary increase in government purchases on the paths of consumption, capital, and the interest rate?