A monopolist faces the following demand curve:

P=50-0.5Q

where Q is the quantity demanded and P is the price.

The monopolist's total cost function is given by:

TC=50+4Q where TC is the total cost and Q is the quantity produced.

What is the value of the deadweight loss created by this monopoly?

Question

A monopolist faces the following demand curve:

P=50-0.5Q

where Q is the quantity demanded and P is the price.

The monopolist's total cost function is given by:

TC=50+4Q where TC is the total cost and Q is the quantity produced.

What is the value of the deadweight loss created by this monopoly?

BlackTom AI · Accepted Answer

We start by laying out the given relationships: the demand is P = 50 − 0.5Q, and the total cost is TC = 50 + 4Q, so marginal cost is MC = dTC/dQ = 4.

First, find the monopoly quantity by equating marginal revenue to marginal cost. The total revenue is TR = P·Q = (50 − 0.5Q)Q = 50Q − 0.5Q^2, so marginal revenue is MR = d(TR)/dQ = 50 − Q. Set MR = MC: 50 − Q = 4, which gives Qm = 46.

Next, determine the price the monopolist charges at Qm using the demand curve: Pm = 50 − 0.5(46) = 50 − 23 = 27.

Now identify the competitive (socially efficient) quantity by setting price equal to marginal cost (P = MC): 4 = 50 − 0.5Qc, which yields 0.5Qc = 46 and thus Qc = 92. At this point, the price under competition would be Pc = MC = 4.

The deadweight loss (DWL) is the triangular area between the monopoly quantity Qm and the competitive quantity Qc, bounded by the demand curve. The base of this triangle is (Qc − Qm) = 92 − 46 = 46, and the height is the difference between the competitive price and the monopoly price, i.e., Pc − Pm = 4 − 27 = −23 (taking absolute value gives 23).

Compute DWL = (1/2) × base × height = (1/2) × 46 × 23 = 23 × 23 = 529.

Therefore, the deadweight loss created by this monopoly is 529.

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