Suppose a monopoly's inverse demand curve is p = 13 — Q, and its cost function is C(Q) = 25 + Q + 0.5Q2. (a) Use profit maximization to determine the profit maximizing quantity, Q*, for the monopoly. Show all steps for the profit maximization process. (b) At Q*, what is the price in the market and what is the monopoly's profit? (c) In the short run, should the monopoly operate or shut down? Why?

(a) Q* = 6

(b) The price in the market is $7 and the monopoly's loss is also 7.

(c) The monopoly should operate because the price of 7 is greater than the AVC of 4.

Explanation:

(a) Use profit maximization to determine the profit maximizing quantity, Q*, for the monopoly. Show all steps for the profit maximization process.

Given ............................ (1)

MC = C'(Q) = 1 + Q ............................................... (2)

p = 13 — Q ................................................................(3)

Profit is maximized when P = MC, Therefore, we equate equations (2) and (3) and have:

1 + Q = 13 - Q

Q + Q = 13 - 1

2Q = 12

Q* = 12/2

Q* = 6 <-------------- Profit maximizing Q*

(b) At Q*, what is the price in the market and what is the monopoly's profit?

Substituting 6 for Q in equation (3), we have:

p = 13 — 6

p = 7 <-------------- The price in the market

Revenue = Q*P = 6 * 7 = 42

To obtain cost, substitute 6 for Q in equation as follows:

Cost =

        = 31 + (0.5 * 36)

        = 31 + 18

Cost = 49

Profit (loss) = Revenue - Cost = 42 - 49 = - 7 loss

(c) In the short run, should the monopoly operate or shut down? Why?

From equation (1), we have:

Fixed cost = 25

Variable cost (VC) =

Average variable cost (AVC) = VC/Q = 24/6 = 4

p > AVC => 7 > 4

The monopoly should operate because price of 7 is greater than the AVC of 4. Continuing operation will therefore enable the monopoly to cover variable costs fully and some part of the fixed cost. This way, loss will be minimized.

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