Assignment:
Question 1. Consider a representative consumer with preferences over consumption (C) and leisure (l) given by the utility function U (C, I) =1n C + b . 1, where parameter b > 0. The consumer receives labor income from working equal to wNs and receives dividends from the firm equal to π. The consumer uses all income to purchase consumption goods.
The firm hires labor and produces output according to the Cobb-Douglas production function Y = zKa (Nd)1-a where a (0,1) . Note that the capital stock is fixed.
The government in this economy finances spending G, by imposing a proportional tax t on the firm's profits. As a result, after-tax profits for the firm are (1 - t) [zka (Nd)1-a - wNd] .
The consumer's problem and the firm's problem are given below.
Consumer maximizes utility subject to constraints The firm maximizes profits
max C,l,N{ ln C + b . 1} max N {(1 - t) [zKa (Nd)1-a - wNd]
subject to:
C=wNs + π
I + Ns= h
(a) Write an expression for the amount of revenues collected by the government.
(b) Define a competitive equilibrium in this environment. Be specific about which vari-ables are endogenous and exogenous; who solves what problem; write down everyone's prob-lem explicitly; specify what is taken as given by each agent; and which markets clear.
(c) Solve for the equilibrium levels of C, l, Nd, NS, Y, π, w. Make sure your solution only depends on exogenous variables! Otherwise it is not a solution!
(d) How does the equilibrium labor N change when the tax t increases? Explain why.