Assignment:
Question 1. Consider a representative consumer with preferences over consump¬tion (C) and leisure (1) given by the utility function U (C,l) =1n C + b . 1, where parameter b > 0. The consumer receives labor income from working equal to to wNs and receives div¬idends from the firm equal to π. The consumer was all income to purchase consumption goods.
The firm hires labor and produces output according to the Cobb-Douglas production function Y = z Ka (Nd)1-a where a E (0,1) . Note that the capital stock is fixed.
The government in this economy finances spending C, 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
max c, l,N {ln C + b • l}
subject to: C= wNs+ π
l + Ns = h
The firm maximizes Profits
Max N { (1-t) [zKa(Nd)1-a - wNd]}
(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.