Question

1.  Consider an industry in the U.S. facing aggregate (inverse) demand function:

p(y) = 1050 – 5y

The industry is currently in long run equilibrium. The market price is $225 and there are n = 11 firms producing. Each firm’s variable cost is:

c_v(y) = [1]  y³

a.  What is each firm’s fixed cost?

Answer 1

let the fixed cost be $F$

total cost = $TC(y) = F + \frac{1}{3}\times y^{3}$

marginal cost $= MC(y) = \frac{d TC(y)}{dy} = y^{2}$

average cost $= \frac{TC(y)}{y} = \frac{F}{y}+\frac{1}{3}\times y^{2}$

set $MC(y) =AC(y)$ to determine the output level at which $AC(y)$ is minimized

$y^{2} = \frac{F}{y}+\frac{1}{3}\times y^{2}$

$\frac{2}{3}y^{2} = \frac{F}{y}$

$y^{3} = \frac{3}{2}F$

$y= (1.5F)^{\frac{1}{3}}$

so $MC$ at this output is given by $MC(y*)$ $=(1.5F)^{\frac{2}{3}}$

set $MC(y*) =$ long run price

$(1.5F)^{\frac{2}{3}}= 225$

$1.5F=(225)^{\frac{3}{2}}=3375$

$F=\$2250$ (fixed cost)