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Consider the closed-economy one-period macroeconomic model. The consumer is endowed with h units of time, and...

Consider the closed-economy one-period macroeconomic model. 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.

Find the firm’s optimal demand for labor Nd , as a function of w and τ .

(b) Find the consumer’s optimal levels of consumption, leisure, and labor, given the real wage w and firm profits π.

(c) 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) 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?

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Answering the first part of the question: Apply Walras Law:

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