Question

The inverse demand equation for the output of a monopolist is P = 50 − 2Q. The monopolist's total cost equation is C(Q) = 100 + 2Q + Q2. What is the deadweight loss at the profit-maximizing output and price?

• $48

• $64

• $16

• $32

• None of the options.

Answer 1

Answer: $64

The monopolist produces the profit-maximizing level of output, where marginal revenue(MR) equals the marginal cost(MC), and then charges the price according to the market demand curve it faces.

A monopolist charges the price which is greater than its marginal cost(MC) of production, and produces less then the socially optimal level of output(output, where price = MC). Thus the deadweight loss arises in monopoly market due to inefficient production.

Now, here we see that the inverse demand equation for the output of a monopolist is,

P = 50 − 2Q...........(1)

The monopolist's total cost equation is,

C(Q) = 100 + 2Q + Q²........(2)

Multiplying equation(1) by Q , we get,

PQ = 50Q - 2Q²

Or , Total Revenue(TR) = 50Q - 2Q²

Now, MR = d(TR) / dQ

Differentiating TR above with respect to Q, we get,

d(TR) / dQ = d(50Q - 2Q²) / dQ

Or, MR = 50 - 4Q

Now, differentiating equation(2) with respect to Q, we get,

C(Q) = 100 + 2Q + Q²

d[C(Q)] / dQ = d(100 + 2Q + Q²)

Or, MC = 2 + 2Q , [d[C(Q)] / dQ = d(TC) / dQ = MC]

Now, the monopolist produces the profit-maximizing output, where MR = MC,

So, 50 - 4Q = 2 + 2Q

Or, -4Q - 2Q = 2 - 50

Or, -6Q = - 48

Or, Q = -48 / -6 = 8

Or, Q = 8

Putting the value of Q in equation(1), we get,

P = 50 − 2 * 8

Or, P = 50 - 16 = 34

Or, P = $34

Thus, the monopolist produces 8 units of output for which it charges the price $34.

Now, we know that, MC = 2 + 2Q

Let us make P = MC, we get,

34 = 2 + 2Q

Or, 2 + 2Q = 34

Or, 2Q = 34 -2 = 32

Or, Q = 32 / 2 = 16

When Q = 16,

P = 50 - 2 * 16 = 50 - 32 = 18

Or, P = $18

Thus, at the socially optimal level of production, the price is $18 , and the output is 16 units.

Therefore, the monopolist is producing 16 - 8 = 8 units less; and charging $34 - $18 = $16 more.

So, the deadweight loss(DWL) = 1/2 ( 16 - 8) ( $34 - $18)

Or, DWL = 1/2 * 8 * ($16) = 4 * ($16) = $64

Or, DWL = $64

The deadweight loss(DWL) at the profit-maximizing output and price is $64.

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