Consider a monopolistically competitive market with N firms. Each firm's business opportunities are described by the following equations:

Demand: Q=100/N−P

Marginal Revenue: MR=100/N−2Q

Total cost: TC=50+Q2

Marginal Cost: MC=2Q.

(a.) How does N, the number of firms in the market, affect each firms demand curve? Why.

(b.) How many units does each firm produce? (The answer to this and the next two questions depend on N.)

(c.) What price does each firm charge?

(d.) How much profit does each firm make?

(e.) In the long run, how many firms will exist in this market?

(a.) The number of firms in the industry can both shift the position of an individual firm's demand curve and affect its slope. If excess (deficient) profits prevail in the industry, more firms will enter (exit) and the firm's demand curve will shift inward (outward). Also, the more close substitutes there are for a firm's product, the more horizontal (elastic) its demand curve may become as firms enter and exit the industry in response to changing profit opportunities in the long run.

(b.) The number of units each firm produces is determined by the point where the firm's marginal revenue and marginal cost curves intersect. Since MR depends in part on how many firms are in the industry, we assume initially there are 6 firms.

MR=MCMR=MC

(100/N)−2Q=2Q(100/N)−2Q=2Q

(100/6)−2Q=2Q(100/6)−2Q=2Q

4Q=16.674Q=16.67

Q=4.17.Q=4.17.

So, each firm initially produces 4.17 units.

(c.) The price each firm charges in the short-run is determined from its demand curve:

Q=(100/N)−PQ=(100/N)−P

4.17=(100/6)−P4.17=(100/6)−P

P=(100/6)−4.17=12.50P=(100/6)−4.17=12.50

So, initially each firm charges $12.50

(d.) The profit each firm makes is equal to total revenue minus total cost:

Profit=TR−TC=(P)(Q)−(50+Q2)=(12.50)(4.17)−(50+4.172)=52.13−67.39=−15.26.Profit=TR−TC=(P)(Q)−(50+Q2)=(12.50)(4.17)−(50+4.172)=52.13−67.39=−15.26.

So, each firm is actually taking a loss equal to $15.26.

(e.) The long-run equilibrium will be achieved by firms exiting the industry until the price charged just equals the firm's average total cost (ATC). The ATC curve will be tangent to the demand curve at that point so that price equals ATC and profit is zero.

Equate MC with MR to solve for N in terms of Q:

MR=MCMR=MC

(100/N)−2Q=2Q(100/N)−2Q=2Q

(100/N)=4Q(100/N)=4Q

4QN=1004QN=100

N=100/4QN=100/4Q

N=25/Q.N=25/Q.

Express the demand equation with P as the dependent variable:

Q=(100/N)−PQ=(100/N)−P

P=(100//N)−Q.P=(100//N)−Q.

Substitute for N in the demand equation:

P=100/(25/Q)−Q=4Q−Q=3Q.P=100/(25/Q)−Q=4Q−Q=3Q.

Set profit, which is equal to total revenue minus total cost, equal to zero and solve for Q:

TR−TC=0TR−TC=0

PQ−(50+Q2)=0PQ−(50+Q2)=0

(3Q)Q−50−Q2=0(3Q)Q−50−Q2=0

3Q2−50−Q2=03Q2−50−Q2=0

2Q2=502Q2=50

Q2=25Q2=25

Q=5.Q=5.

From the earlier expression for N, we can determine the long-run equilibrium number of firms:

N=25/Q=5.N=25/Q=5.

So, 1 firm will be forced to exit the industry.

Note as well that P and average total cost (ATC) will now be equal at $15:

P=100/5−5=15P=100/5−5=15

ATC=TC/Q=(50+Q2)/Q=50/5+5=15.