Consider an economy with a representative consumer whose preferences are represented by the utility function:

U(c, l) = log c + η log l

Consumer faces the budget constraint:

c = wNs + π

and the time constraint:

l + Ns = h

where h > 0 is time endowment.

There is also a representative firm that uses technology: Y = zNd

(a) Write the consumer and firm’s problem. Don’t solve them yet.

(b) Define the competitive equilibrium.

(c) Solve consumer and firm’s problem and find the competitive equilibrium. Check to make sure that all conditions required for a competitive equilibrium are satisfied. Now introduce a government that spends an amount G of consumption good financed through taxes.

(d) How does the definition of competitive equilibrium change?

(e) Suppose the government imposes lump-sum tax on the consumer. Solve the consumer and firm’s problem and find the competitive equilibrium allocations and prices in this case.

(f) Formulate the planner’s problem and solve for Pareto optimal allocations.

(g) Now suppose that the government imposes labor income tax to finance the government expenditure instead. In particular, the consumer’s budget constraint changes to c = (1 − t)wNs + π Find competitive equilibrium allocations and prices in this case.

(h) If the government cared for the consumer’s welfare, which tax regime would it choose? Why?

(i) Suppose the government chooses distortionary tax to finance G. Using the competitive equilibrium value for labor supply, write the revenue function for the government. Solve for tax rates that can finance G. If there are multiple tax rates what is the effect of using one vs. the other?