Problem Set $1$

Consider the long$-$run average cost for production of computers is given by

$AC(Q)=(120-20Q+Q^2)(wr{)}^{\frac{1}{2}},$

where $Q$ is the annual output of a firm, $w$ is the wage rate for labor, and $r$ is the rental rate of capital which is equal to $1.$

a$)$ Determine the marginal cost.

b$)$ Determine the demand for labor for an individual firm.

c$)$ In a long$-$run competitive equilibrium, how much output will each firm produce?

d$)$ In the long$-$run competitive equilibrium, what will be the market price? This price will be a function of $w.$

e$)$ In the long$-$run competitive equilibrium, how much labor will each firm demand? This answer will also depend on $w.$

f$)$ Suppose that the market demand curve is given by

$D(P)=\frac{10,000}{P}.$

What is the market equilibrium quantity as a function of $w?$

g$)$ What is the long$-$run equilibrium number of firms as a function of $w?$

h$)$ Using your answers to e$)$ and g$),$ determine the overall demand for labor in this industry as a function of $w.$

i$)$ Suppose that supply curve for labor used in this industry is

$S(w)=50w.$

At what value of $w$ does the supply of labor equal the demand for labor?

j$)$ Using your answers to i$),$ go back through parts d$),$ f$),$ and g$)$ to determine the long$-$run

$1$

equilibrium price, market demand, and the number of firms in this industry.

k$)$ Repeat the analysis with demand curve now

$D(P)=\frac{20,000}{P}.$

Based on your answers to j$)$ and k$),$ draw the long$-$run industry supply curve.