Question

(30 marks) Consider a perfectly competitive market in the short run. Assume that market demand is P=1000 - Q. Denoting firm level quantity by q, assume TC=30q-10q²+q³ so that MC=30-20q+3q².

a) If there are 10 identical firm in the industry in the short run, what is the market equilibrium price and quantity?

b) Do firms make a profit or loss in the short run, and how much are these profits/losses?

c) What is the equilibrium price in the long run?

d) What will be equilibrium profit in the long run?

e) How many firms will there be in the long run?

Answer 1

a) In the short run P=MC:

$$1000-q=30-20q+3q^2,$$ $$3q^2-19q-970=0.$$

This quadratic equation has two roots: $q_1=21.4$ and $q_2=-15.1$. Since quantity can't be negative the correct answer $q_E=21.4.$ Then, we can find the equilibrium price:

$$P_E=1000-21.4=\$978.6$$

b) The profit can be found as follows:

$$\pi=TR-TC=P_Eq_E-30q+10q^2-q^3,$$ $$\pi=978.6\cdot21.4-30\cdot21.4+10\cdot(21.4)^2-(21.4)^3=\$15079.$$

Therefore, the firms make a profit of $15079.

(c) In the long run ATC=MC:

$$30-10q+q^2=30-20q+3q^2,$$ $$2q^2-10q=0,$$ $$q=5.$$

Then, we can find the equlibrium price in the long run:

$$P_E=1000-5=\$995.$$

(d) Let's find the equilibrium profit in the long run:

$$\pi=TR-TC=P_Eq_E-30q+10q^2-q^3,$$ $$\pi=995\cdot5-30\cdot5+10\cdot(5)^2-(5)^3=\$4950.$$

(e) Let's write the inverse demand function:

$$Q=1000-P.$$

Then, we can find the market quantity:

$$Q=1000-P=1000-\$995=5.$$

Finally, we can find the number of firms:

$$nq=Q,$$ $$n=\dfrac{Q}{q}=\dfrac{5}{5}=1.$$