Question

A profit-maximizing monopolist operates in market with an inverse demand curve given by P = 20 − Q and an associated marginal revenue MR = 20 − 2Q. Their marginal cost of production is constant at MC = 4. Assume for now that they have to sell each unit of output for the same price.

a) Find the monopolist’s optimal choice of output and the socially efficient output.

b) Sketch demand, marginal revenue, and marginal cost. Indicate on your diagram the points you found in part a).

c) What is the amount of deadweight loss at the monopolist’s optimal choice? If a buyer is willing to pay more than marginal cost for the next unit beyond the monopolist’s choice of output, why doesn’t the monopolist choose to produce and sell that unit?

Answer 1

a) The inverse demand function and the marginal revenue function in the market are given as P=20-Q and MR=20-2Q respectively. The marginal cost of production of the monopolist is MC=4. The monopolist will maximize its overall or total profit by producing the output which corresponds to the equality between MR and MC.

Therefore, based on the profit-maximizing condition or principle of a monopolist firm, we can state:-

MR=MC

20-2Q=4

-2Q=-20+4

-2Q=-16

Q=-16/-2

Q=8

Therefore, the profit-maximizing output produced by the monopolist would be 8 units, in this case.

Now, the socially efficient output level refers to the equality between the P and MC.

Hence, based on the principle of socially efficient output production, we can state:-

P=MC

20-Q=4

-Q=-20+4

-Q=-16

Q=16

Thus, the socially efficient output in the market would be 16 units.

b) Figure-1 illustrates the MR, MC, and the demand curve in the monopoly market and the profit-maximizing output produced and the price per unit of output charged by the monopolist firm. Again MR, MC, and D curves in the figure denote the marginal revenue curve, marginal cost curve and the demand curve respectively in the monopoly market. The y-axis and the x-axis in figure-1 represent the cost/revenue/price and the units of output respectively. Note that the monopolist firm produces 8 units of output and charges 12 for selling each unit of output to the consumers or buyers which corresponds to the intersection of the MR and the constant MC curves. The socially efficient output level is indicated as 16 units which correspond to the intersection of the MC and the D curves and the socially efficient per-unit price level of output is 4, which is also equal to the constant MC as indicated in the figure.

c) The profit-maximizing output and output price of the monopolist firm are 8 units and 12 respectively and the socially efficient level of output and output price are 16 units and 4 respectively. Therefore, the deadweight loss at monopolist's desired or optimal choice=1/2*(16 units-8 units)*(12-4)=0.5*8 units*8=32

Hence, the deadweight loss in the market at the monopolist's optimal choice is 32.

The monopolist essentially maximizes its total profit by producing 8 units of output where the MR is equal to MC and charging a per-unit output price of 12 to sell its product. Now, referring to figure-1 note that as the monopolist decides to produce more than or beyond 8 units of output, the MC becomes greater than MR implying that by producing more the monopolist will incur an incremental or marginal loss for each unit of output produced. Hence, even if the consumer is willing to pay an output price higher than the MC, the monopoly will not be willing to produce beyond or above the profit-maximizing level as it will incur a marginal loss on each unit of output produced thereby reducing its total profit accordingly.