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 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)=logC+(1)logL, 0<<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) = KE1, 0 < < 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 workers labour supply. What is the slope of the labour supply curve and why?

2. Find the firms 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, = = 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?