Question

Suppose a pharmaceutical firm just invents a new drug to treat mad cow disease

and this is the only effective drug available in the market. The demand and total

cost functions of the firm are:

Demand: P = 36 – Q

Total cost: C = 24 + 2Q2

where P is the price of the drug,

Q is the quantity of the drug and

C is the total cost of the firm.

a. Find the marginal cost and marginal revenue functions of the firm. Show

your steps. (4 marks)

b. Find the monopoly output, price and profit of the firm. Show your steps.

(5 marks)

c. Draw a diagram to show the price, output and profit of the firm. Label

your diagram clearly. (6 marks)

d. Suppose the pharmaceutical firm needs to spend a lump sum of $M to

apply for the patent of the drug. Explain whether this cost of patent

affects the price, output and profit of the firm. (5 marks

Answer 1

Answer

The pharmaceutical firm's demand and total cost function are as follows;

Demand: P = 36 – Q ....... (1)

Total cost: C = 24 + 2Q².......(2)

a. Marginal revenue(MR) is the change in total revenue(TR) for the change in an additional unit of output(Q).

TR = Price(P) * Output

Now, multiplying equation(1) by Q, we get,

P * Q = Q(36 -Q)

Or, TR = 36Q - Q²

Now differentiating TR with respect to Q, we get,

d(TR) / dQ = 36 - 2Q

Or, MR = 36 - 2Q

Therefore, the firm's marginal revenue function is, MR = 36 - 2Q

The marginal cost(MC) is the change in total cost(TC) for the change in one additional unit of output.

Now, differentiating equation(2), 'C = 24 + 2Q²' with respect to Q, we get,

dC / dQ = d(24 + 2Q²) / dQ

Or, dC / dQ = 4Q

Or, MC = 4Q

Therefore, the firm's marginal cost function is, MC = 4Q

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b. A monopolist produces the profit-maximizing level of output at the point, where marginal revenue equals the monopolist's marginal cost of production, and then sets the price according to the market demand curve it faces.

We know the MC and MR of the firm.

Now, MC = MR means,

4Q = 36 - 2Q

Or, 4Q + 2Q = 36

Or, 6Q = 36

Or, Q = 36 / 6 = 6

Therefore, the firm's profit-maximizing level of output(Q) is 6 units.

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

P = 36 – 6 = 30

Therefore, the firm's profit-maximizing level of price is 30 or $30.

Profit = Total Revenue(TR) - Total Cost(TC)

Total Revenue(TR) = Price * Output = 30 * 6 = $180

So, the firm's total revenue is $180

Now, putting the value of Q in equation(2), we get the TC as,

C = 24 + 2Q²

Or, C = 24 + 2 * (6)²

Or, C = 24 + 2* 36

Or, C = 24 + 72

Or, C = 96

So, the firm's total cost is $96

The firm's average total cost(ATC) = TC / Q = $96 / 6 = $16

Firm's MC = 4 *6 = 24 = $24

Therefore, the firm's profit is, .

Profit = $180 - $96 = $84.

So, the firm's profit is $84

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c.

The firm's average total cost(ATC) = TC / Q = $96 / 6 = $16

Firm's MC = 4 *6 = 24 = $24

In the above figure, we are measuring price(P), costs(C) , and revenues(R) on the vertical axis; and the quantity on the horizontal axis. The curve 'D' is the monopolist's demand curve, 'MR' is the monopolist's marginal revenue curve. The 'MC', and 'ATC' are the marginal cost and average total cost curves of the monopolist.The monopolist produces the profit-maximizing output 6 units for which the price it charges is $30. The monopolist's ATC is $16. The green shaded rectangle shows the monopolist's profit.

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d. If the pharmaceutical firm needs to spend a lump sum of $M to apply for the patent of the drug, its total cost will rise by $M. So the total cost will be,

C = 24 + 2Q² + $M

Or, C = (24 + $M) + 2Q²

(24 + $M) is the firm's new fixed cost.

The rise in fixed cost won't affect the MC. So the MC will remain constant. The firm's market demand curve is also given. So the MR will not change. Thus the firm's price will not be affected for the rise in fixed cost for the application of patent. The firm's profit maximizing quantity will also remain same.

The rise in fixed cost will only affect the firm's profit, as the total cost rises by $M.

The new profit will be,

Profit New = $180 - ($96 + $M)

So, If the pharmaceutical firm needs to spend a lump sum of $M to apply for the patent of the drug, it's price, and output will remain same ; only profit will decrease by $M.

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