Question

Robert has utility function u(c,l) = cl over consumption, c, and leisure, l. Robert is endowed with 16 hours of leisure. Let the price of consumption be p = 1. Robert can sell his time in the labor market at hourly wage, w. The equilibrium we will consider implies zero firm profits, so labor income is the only source of income for consumers. Thus, Robert’s budget line can be written by c + wl = 16w. Production of the consumption good is done by the Acme corporation according to production function Q = F(L) = AL, where A > 0 is some constant that captures firm productivity. There are no fixed costs. The acme corporation sells the output in the consumption market at the price p, and hires labor at wage w. We will in the following work out the general equilibrium.

a) For given wage w and consumption good price p, work out Robert’s optimal consumption and leisure choice (c*,l*). (Hint: Combine the budget line equation with MRS = p/w). What is Robert’s supply of hours to the labor market, h* = 16−l*. State the expressions as functions of the real wage, wˆ = w/p.

b) For a given wage and productivity A, determine Acme’s cost function, C(Q), and the associated input level for a given output requirement, L (Q).

c) Verify that Acme Corp’s marginal cost curve is flat. That is, Acme’s supply curve is flat. At what price level? Notice that Acme is willing to supply any level of output at this price level and that its profits are exactly zero regardless of what output level it chooses.

d) There are a total of 100 individuals in the economy all identical to Robert. State the market demand for consumption, QD as a function of p and w. Combine with the flat supply curve to determine the equilibrium price that equates market consumption demand with supply (it will be a function of w). This equilibrium price equation is effectively also an equilibrium condition on the real wage, wˆ = w/p. What is the equilibrium level of the real wage?

e) Set A = 2. State the equilibrium market demand of the consumption good, QD. Acme will be supplying exactly the same quantity. Verify that the labor market is also clearing.

f) In equilibrium, how much labor is Robert supplying to the market? How much is he consuming?

g) Acme has come up with a brilliant new production process which has increased the productivity parameter to A = 3. Determine the new general equilibrium. What is the equilibrium wage? How much labor is Robert supplying? How much is he consuming?

Answer 1