Question

A monopolist operates in a market of demand q = 10−p with a total cost of C(Q, e) = (3/2)e^2 +(5−e)Q, where e represents effort.

a. Calculate the price, effort, quantity, and welfare that results from an unregulated monopoly.

b. A regulator establishes that price must equal marginal cost. The monopolist is free to select the level of effort. Calculate price, effort, quantity, and welfare in this situation.

c. A regulator decides to force price equal to marginal cost and mandates that monopolist must choose the level of effort that minimizes costs for a given level of quantity. Calculate price, effort, quantity, and welfare in this new situation.

d. Based on your answer to the previous parts, should a regulator focus on allocative efficiency and ignore productive efficiency?

Answer 1

a. The market demand function faced by the monopolist is given as q=10-p where q is the overall or total quantity of the concerned product in the market and p represents the per-unit price of the product in the market. Thus, the inverse market demand function of the product would be p=10-q. The total cost function of the monopolist is C(Q,e)=3/2e^2+(5-e)Q=1.5e^2+(5-e)Q where e is the effort of the monopolist. The marginal cost function of the monopolist or MC would be=dC/dQ=1.5e^2+5-e=0.5e^2+5

The total revenue of the monopolist or TR=p.q=(10-q)*q=10q-q^2. Therefore, the marginal revenue function of the monopolist or MR=dTR/dq=10-2q. Now, based on the profit-maximzing condition of the monopoly, a monopolist firm would maximize its profit by producing the total output which corresponds to the equality between the marginal revenue and marginal cost of production.

Therefore, based on the profit-maximizing condition of a monopolist firm, we can state, in this case:-

MR=MC

10-2q=0.5e^2+5

-2q=-10+5+0.5e^2

-2q=0.5e^2-5

q=(0.5e^2-5)/-2

q=2.5-0.25e^2

q-2.5=-0.25e^2

10-q/0.25=e^2

=e

Hence, the profit-maximizing output and the level of effort of the monopolist would be 2.5-0.25e^2 and respectively.

Now, plugging the profitmaxizing quantity of the monopolist into the inverse market demand function, we get:-

p=10-q

p=10-(2.5-0.25e^2)

p=10-2.5+0.25e^2

p=7.5+0.25e^2

Hence, the profit-maximizing per-unit price charged by the monopolist for the product would be 7.5+0.25e^2

Now, based on the inverse demand function, when q is 0 the p is 10. The consumer surplus, in this case=0.5*(10-7.5-0.25e^2)*(2.5-0.25e^2)=0.5*(2.5-0.25e^2)^2

b. The price is derived as p=10-q from the market demand function and the MC of the monopolist=0.5e^2+5

Hence, based on the mandate by the regulator, it can be stated:-

p=MC

10-q=0.5e^2+5

-q=0.5e^2+5-10

-q=0.5e^2-5

q=5-0.5e^2

q-5=-0.5e^2

(q-5)/-0.5=e^2

10-q/0.5=e^2

=e

Hence, the quantity and the level of effort by the monopolist, in this case would be 5-0.5e^2 and respectively.

Now, plugging the value of the quantity into the inverse demand function, we obtain:-

p=10-q

p=10-(5-0.5e^2)

p=10-5+0.5e^2

p=5+0.5e^2

Therefore, the price, in this case would be 5+0.5e^2

The consumer surplus in the market would be=0.5*(10-5-0.5e^2)*(5-0.5e^2)=0.5*(5-0.5e^2)*(5-0.5e^2)=0.5*(5-0.5e^2)^2

c. In this case, minimization of the total cost of the monopolist or C construe that the MC would be equal to 0. The regulator mandates that the price has to be equal to the MC, in this case as well.

Therefore, based on the total cost-minimization condition, it can be stated:-

MC=0

0.5e^2+5=0

0.5e^2=-5

e^2=-5/0.5

e^2=-10

e=

Thus, the level of effort by the monopolist, in this case would be

Now, based on the regulatory mandate and plugging the value of e^2 derived in the above equation into the MC function, we get

p=MC

10-q=0.5e^2+5

10-q=0.5*(-10)+5

10-q=-5+5

10-q=0

-q=-10

q=10

Hence, the output quantity in this case is 10 units.

Now, plugging the value of the output quantity into the inverse demand function, we get:-

p=10-q

p=10-10

p-0

Thus, the per-unit price charged by the monopolist would be 0, in this case.

The consumer surplus, in those case would be=0.5*(10-0)*10=0.5*10*10=50

d. Based on the comparison between the consumer surplus in part b. and part c., the consumer surplus in part b. would be expectedly and theoretically higher than that of in part c. as the regulatory condition or the mandate as mentioned in part b. of the question signifies the socially efficient and optimal market condition which would maximize the overall market welfare and efficiency by raising the consumer surplus. On the other hand, as the monopolist would try to minimize its total cost in part c. even if the market price of the product is equal to the marginal cost of the monopolist, the market welfare would not be maximized as the monopoly would attempt to minimize its total or overall cost.