Question

In: Economics

Suppose that the demand for a monopolist's product is estimated to be Qd = 100 −...

Suppose that the demand for a monopolist's product is estimated to be Qd = 100 − 2P and its total costs are C(Q) = 10Q. Under first-degree price discrimination, the optimal price(s), number of total units exchanged, profit, and consumer surplus are:

Multiple Choice

  • 10 ≤ P ≤ 50; Q = 80, Π = $1,600; CS = $0.

  • 10 ≤ P ≤ 100; Q = 80; Π = $1,600; CS = $1,600.

  • P = $30; Q = 40, Π = $800; CS = $400.

  • P = $30; Q = 40, Π = $600; CS = $0.

Solutions

Expert Solution

Answer: P = $30; Q = 40, Π = $800; CS = $400.

The demand for a monopolist's product is estimated as;

Qd = 100 − 2P.........(1)

Its total costs are;

C(Q) = 10Q .........(2)

The monopolist produces the profit maximizing level of output at which marginal revenue(MR) and marginal cost(MC) are same, and then charges the price determined by the demand curve it faces in the market.

Now,we can write equation(1) as,

2P = 100 - Qd

Multiplying the above equation by Qd, we get,

2P * Qd = (100 - Qd) * Qd

Or, 2P * Qd = 100Qd - (Qd)2

Or, 2 TR = 100Qd - (Qd)2 , [TR = Total Revenue = Price * Quantity]

Or, TR = [100Qd - (Qd)2 ] / 2

Or, TR = 50Qd - (Qd)2 / 2

Now, differentiating the above equation with respect to Qd, we get,

d(TR) / d(Qd) = MR = 50 - Qd , [ MR = Change in TR for the change of an additional unit of output]

So, we get the monopolist's marginal revenue as;

MR = 50 - Qd

Now, let us find the marginal cost of the monopolist.

Differentiating equation(2) with respect to Q, we get,

d[C(Q)] / dQ = 10

Or, MC = 10

So we get the firm's marginal cost as,

MC = 10

Now at the profit-maximizing level of output, MR = MC.

Thus,

50 - Qd = 10

Or, - Qd = 10 - 50

Or, - Qd = - 40

Or, Qd = 40

__________________________________

Thus, the monopolist produces the profit-maximizing level of output 40 units.

Now, putting the value of Qd in equation(1), we get,

40 = 100 − 2P

Or, 2P = 100 - 40

Or, 2P = 60

Or, P = 60 / 2 = 30

Thus, the monopolist charges the price of $30.

_____________________________________

Now the profit (π) is the difference between the revenue and the cost, i.e.,

π = TR - TC......(3)

TR = Price * Quantity

Or, TR = $30 * 40

Or, TR = $1,200

Now, TC = C(Q) = 10Q

Or, TC = 10 * 40

Or, TC = $400

Now, putting the value of TR and TC in equation(3), we get,

π = $1,200 - $400

Or, π = $800

The monopolist's profit is $800.

_____________________________

From the given informations of the monopolist's demand function, and cost function, we have drawn the above figure, where curve 'D' is the demand curve , the monopolist faces. The curve 'MR' is the corresponding marginal revenue curve, and 'MC' is the marginal cost curve of the monopolist.

We can write the equation of the monopolist's inverse demand curve as;

Qd = 100 − 2P

2P = 100 - Qd

Or, P = 50 - Qd / 2

From the demand equation, we see that the maximum price, the consumer is willing to pay is $50, and the monopolist charges the price of $30.Thus we can calculate the consumer surplus(CS) as;

CS = 1/2 * 40 * ( $50 - $30)

Or, CS = 20 * ($20)

Or, CS = $400

Thus, the consumer surplus is $400

___________________________________________________


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