Consider an economy with a representative worker, a representative firm, and perfectly competitive labour, capital, and goods markets. The worker is endowed with T > 0 hours of time and nonlabour income V ≥ 0. They divide their time between hours h ≥ 0 worked for the hourly wage w > 0 and hours L ≥ 0 consumed as leisure. They spend r all their labour and nonlabour income on the consumption of goods C, a unit of which sells at price p > 0. Their preferences over the consumption of goods C and leisure L can be represented by the utility function1

U(C,L)=γlogC+(1−γ)logL, 0<γ<1.

The firm rents capital K at a rental rate r > 0 per unit and hires labour E at the hourly wage w to produce output with the production function
f(K,E) = KαE1−α, 0 < α < 1,

which it sells at price p. Let V = 0. Consider the short run, i.e., K is fixed at K = K ̄ > 0.

1. Find the worker’s labour supply. What is the slope of the labour supply curve and why?

2. Find the firm’s short-run labour demand. Find and explain the effect of higher capital K ̄ .

3. Illustrate labour supply, labour demand, and the short-run equilibrium in a qualitatively accurate graph. Find expressions for the short-run equilibrium wage and employment.

4. Let T = 111, α = γ = 1/3, K ̄ = p = 1. Suppose that the labour market is in equilibrium when the government imposes a minimum wage w ̄ that affects all workers and that all employers comply with. Find and explain the effect of the minimum wage if w ̄1 = 1/6? How much labour is displaced? What is the unemployment rate? What if the minimum wage is w ̄2 = 1/4?