1. (45 points) Consider the closed-economy one-period macroeconomic model developed in class. The consumer is endowed with h units of time, and chooses consumption C and leisure ` to maximize U = log(C) + θlog(`), subject to the budget constraint C = wNs + π. Production is described by Y = zNd . Government spending G is financed with a proportional revenue tax (tax rate τ ) on the firm.
(a) (10) Find the firm’s optimal demand for labor Nd , as a function of w and τ .
(b) (10) Find the consumer’s optimal levels of consumption, leisure, and labor, given the real wage w and firm profits π.
(c) (15) Solve for the equilibrium allocation (C, `, Y , N), equilibrium wage w, and equilibrium tax rate τ using the conditions that must be satisfied by an equilibrium: i) (C, `, N) must be optimal for the consumer, given w and π; ii) N must maximize the firm’s profits, given w; iii) the government budget constraint is satisfied; and iv) market clearing (Y = C + G, and Nd = Ns ). Note: you should be solving for C, `, Y , N, τ , and w as functions of exogenous variables and parameters.
(d) (10) Now solve for the Pareto Optimal allocation. That is, find the bundle (C, `, Y , N) that satisfies the Pareto Optimality conditions: 1) MRSlc = MRTlc; and 2) Y = C + G. Is the equilibrium of this economy Pareto Optimal?
2. (55 points) Consider the Solow growth model. Output is produced every period according to the following constant returns to scale technology: Y = Kα (AN) 1−α . The capital stock accumulates according to K0 = (1 − δ)K + sY , where 0
(a) (10) Write down an equation that characterizes the evolution of capital per effective units of labour. Derive the steady-state level of this variable. Derive the steady-state level of capital per capita, and describe (with words or a diagram) how it evolves over time.
(b) (10) Derive an expression for the steady-state level of consumption per capita. Find the savings rate that maximizes steady-state consumption per capita.
(c) (15) Suppose the economy is in steady state when a natural disaster destroys half of the existing capital stock. Illustrate how consumption per capita evolves before, at the time of, and after the disaster. 2 (d) (20) Now assume the production function for this economy is Y = zKα (AN) 1−α , where A and N behave as above, and z is constant. Derive an expression for consumption per capital in steady state. Suppose there are two otherwise identical countries with different z’s (z R > zP ). Derive an expression for the factor difference in per-capita consumption between the two countries. Now suppose that in addition, g R