A perfectly competitive industry is characterized by the cost function for indiv

Question

A perfectly competitive industry is characterized by the cost function

for individual firms: TC(q) = 0.01q2 + 100

and by demand function: D(p) = 10, 000 100p

Compute the long-run equilibrium price, quantity, and number of
firms in the market.

[Hint: Competition drives down the price
to the minimum point on the average cost curve which is char-
acterized by AC(q) = MC(q). Compute the long-run equilib-
rium quantity q* using this equality. Once you have q*, calculate
p* by using the competitive firm’s profit maximization condition
p = MC(q). Then get the number of firms using the demand
equation.]

Explanation / Answer

MC = 0.02q <--- this is the derivative of the TC curve.

AC = TC/q
AC = 0.01q + 100/q

0.01q + 100/q = 0.02q
-0.01q -0.01q

100/q = 0.01q
100 = 0.01q^2
10000 = q^2
q* = 100 <--- this is the long-run equilibrium q

Plug this into MC to get a number.

MC = 0.02(100) = 2

Now, because p = MC in a perfectly competitive market, and we just found MC = 2, we can substitute to get p* = 2

Now plug this into the demand function:

D = 10,000 - 100(2) = 9,800

Because each firm is producing 100, we can divide the total demand by the number each firm is producing to get the number of firms as such:

9800/100 = 98 firms

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