CHAPTER 11

PRICING PRACTICES

1. A firm produces two products (A and B) jointly. Every time a unit of A is produced, a unit of B is also produced as a byproduct. The demand functions for A and B are:

Q_A = 400 − 4P_A           Q_B = 100 − 2P_B

Assuming that disposal is costless, determine the number of units of A and B that the firm should produce, the number of units of A and B that the firm should sell, and the price that should be charged for each of the products if the firm's marginal cost of producing a unit of joint output is:

(i) MC = 100 + Q

(ii) MC = 0.125Q

Solution:

For levels of output where the marginal revenue for both products is nonnegative, the firm's marginal revenue function is obtained by vertically summing the two individual marginal revenue functions as follows:

P_A = 100 − 0.25Q_A so MR_A = 100 − 0.50Q_A

P_B = 50 − 0.50Q_B so MR_B = 50 − Q_B

MR_T = MR_A + MR_B = 100 − 0.50Q + 50 − Q = 150 − 1.50Q

For levels of output where the marginal revenue for one product is negative, the firm's marginal revenue function is equal to the nonnegative marginal revenue function. If output exceeds Q = 50, then MR_B is negative so the firm's marginal revenue function is equal to MR_A.

(i) MR_T = 150 − 1.50Q = 100 + Q = MC implies Q = 20. Since this is less than 50,

the firm should produce Q_A = Q_B = 50. This yields prices of P_A = 95 and P_B = 40.

(ii) MR_T = 150 − 1.50Q = 0.125Q = MC implies Q = 92.31. Since this is greater than 50, the firm should not produce this level of output. Instead, it should set MR_A = MC as follows:

MR_A = 100 − 0.50Q = 0.125Q = MC so Q_A = 160 and P_A = 60. Q_B must also be

equal to 160, but the firm should dispose of any output beyond that which results in a marginal revenue from sale of B equal to zero, that is, beyond Q_B = 50 and P_B = 25. The firm will dispose of 110 units of product B.

2. A firm produces two products (A and B) jointly. Every time a unit of A is produced, a unit of B is also produced as a byproduct. The demand functions for A and B are:

Q_A = 3000 − 20P_A            QB = 1000 − 10P_B

Assuming that disposal is costless, determine the number of units of A and B that the firm should produce, the number of units of A and B that the firm should sell, and the price that should be charged for each of the products if the firm's marginal cost of producing a unit of joint output is:

(i) MC = 70 + 0.30Q

(ii) MC = 10 + 0.04Q

Solution:

For levels of output where the marginal revenue for both products is nonnegative, the firm's marginal revenue function is obtained by vertically summing the two individual marginal revenue functions as follows:

P_A = 150 − 0.05Q_A so MR_A = 150 − 0.10Q_A

P_B = 100 − 0.10Q_B so MR_B = 100 − 0.20Q_B

MR_T = MR_A + MR_B = 150 − 0.10Q + 100 − 0.20Q = 250 − 0.30Q

For levels of output where the marginal revenue for one product is negative, the firm's marginal revenue function is equal to the nonnegative marginal revenue function. If output exceeds Q = 500, then MR_B is negative so the firm's marginal revenue function is equal to MR_A.

(i) MRT = 250 − 0.30Q = 70 + 0.30Q = MC implies Q = 300. Since this is less than

500, the firm should produce Q_A = Q_B = 300. This yields prices of P_A = 135 and

P_B = 70.

(ii) MR_T = 250 − 0.30Q = 10 + 0.04Q = MC implies Q = 705.88. Since this is greater than 500, the firm should not produce this level of output. Instead, it should set

MR_A = MC as follows:

MR_A = 150 − 0.10Q = 10 + 0.04Q = MC so Q_A = 1000 and P_A = 100. Q_B must also

be equal to 1000, but the firm should dispose of any output beyond that which

results in a marginal revenue from sale of B equal to zero, that is, beyond Q_B =

500 and P_B = 50. The firm will dispose of 500 units of product B.

3. A firm manufactures a product that is sold on two different markets (A and B) that have the following demand functions:

Q_A = 100 − 0.50P_A   Q_B = 60 − 0.50P_B

The firm has the following marginal cost function:

MC = 20 +0.80Q

If the firm is engaging in price discrimination, what prices should be charged on each market and how many units should be sold on each market?

Solution:

The horizontal sum of the marginal revenue functions (MRS) for the two markets is calculated as follows:

P_A = 200 − 2Q_A so MR_A = 200 − 4Q_A and Q_A = 50 − 0.25MR_A

P_B = 120 − 2Q_B so MR_B = 120 − 4Q_B and Q_B = 30 − 0.25MR_B

Q = Q_A + Q_B = 50 − 0.25MRS + 30 − 0.25MRS = 80 − 0.50MRS

MRS = 160 − 2Q

This horizontally summed marginal revenue function applies when marginal revenue is below 120. When marginal revenue is above 120, only the marginal revenue function for market A is relevant. Setting the firm's marginal cost function equal to MRS yields:

MRS = 160 − 2Q = 20 + 0.80Q = MC so Q = 50 and MRS = 60

Substituting MRS = 60 into the marginal revenue function for the two markets yields Q_A= 35 and Q_B = 15.

Substituting these quantities into the demand functions yields P_A = 130 and P_B = 90.

4. A firm manufactures a product that is sold on two different markets (A and B) that have the following demand functions:

Q_A = 400 − 2P_A                 Q_B = 240 − 2P_B

The firm has the following marginal cost function:

MC = 20 + 0.20Q

If the firm is engaging in price discrimination, what prices should be charged on each market and how many units should be sold on each market?

Solution:

The horizontal sum of the marginal revenue functions (MRS) for the two markets is calculated as follows:

P_A = 200 − 0.50Q_A so MR_A = 200 − Q_A and Q_A = 200 − MR_A

P_B = 120 − 0.50Q_B so MR_B = 120 − Q_B and Q_B = 120 − MR_B

Q = Q_A + Q_B = 200 − MRS + 120 − MRS = 320 − 2MRS               

MRS = 160 − 0.50Q

This horizontally summed marginal revenue function applies when marginal revenue is below 120. When marginal revenue is above 120, only the marginal revenue function for market A is relevant. Setting the firm's marginal cost function equal to MRS yields:

MRS = 160 − 0.5Q = 20 + 0.20Q = MC so Q = 200 and MRS = 60

Substituting MRS = 60 into the marginal revenue function for the two markets yields Q_A= 140 and Q_B = 60.

Substituting these quantities into the demand functions yields P_A = 130 and P_B = 90.

5. A firm has two semiautonomous divisions: production and marketing. The production division manufactures a product that is purchased and then resold by the marketing division. The marginal cost functions for the production division and for the value added by the marketing division are defined below.

MC_P = 100 + 6Q                       MC_M = 4Q

The demand function for the product is:

QD = 120 − 0.20P

(i) Assume that there is no external market for the output of the production division. How many units should be produced and what transfer price should be paid to the production division by the marketing division?

(ii) Assume that the external market for the output of the production division is

perfectly competitive and that the market price is $292. How many units should be produced by the production division, how many should be purchased by the marketing division, what transfer price should be paid to the production division by the marketing division, and what price should be charged for the product by the marketing division?

Solution:

(i) In the absence of an external market, production is determined by vertically

summing the two marginal cost functions and setting the sum (MCS) equal to

marginal revenue:

P = 600 − 5Q so MR = 600 − 10Q

MCS = MC_P + MC_M = 100 + 6Q + 4Q = 100 + 10Q

MR = 600 − 10Q = 100 + 10Q = MCS so Q = 25 at P = 475

The transfer price is set equal to the marginal cost of manufacturing the optimal level of output:

P_T = MC_P = 100 + (6)(25) = 250

(ii) The production division should manufacture the quantity that sets its marginal

cost equal to the competitive price:

P = 292 = 100 + 6Q so Q_P = 32

The transfer price is equal to the competitive market price and so:

MCS = P_T + MC_M = 292 + 4Q

Finally, MCS is set equal to marginal revenue to determine the number of units

that will be purchased by the marketing division and the price at which they will be sold:

MR = 600 − 10Q = 292 + 4Q = MCS and Q_M = 22 at P = 490

6. A firm has two semiautonomous divisions: production and marketing. The production division manufactures a product that is purchased and then resold by the marketing division. The marginal cost functions for the production division and for the value added by the marketing division are defined below.

MC_P = 2Q                   MC_M = Q

The demand function for the product is:

QD = 100 − P

(i) Assume that there is no external market for the output of the production division. How many units should be produced and what transfer price should be paid to the production division by the marketing division?

(ii) Assume that the external market for the output of the production division is

perfectly competitive and that the market price is $52. How many units should be produced by the production division, how many should be purchased by the

marketing division, what transfer price should be paid to the production division by the marketing division, and what price should be charged for the product by the marketing division?

Solution:

(i) In the absence of an external market, production is determined by vertically

summing the two marginal cost functions and setting the sum (MCS) equal to

marginal revenue:

P = 100 − Q so MR = 100 − 2Q

MCS = MC_P + MC_M = 2Q + Q = 3Q

MR = 100 − 2Q = 3Q = MCS so Q = 20 at P = 80

The transfer price is set equal to the marginal cost of manufacturing the optimal level of output:

P_T = MC_P = (2)(20) = 40

(ii) The production division should manufacture the quantity that sets its marginal cost equal to the competitive price:

P = 52 = 2Q so Q_P = 26

The transfer price is equal to the competitive market price and so

MCS = P_T + MC_M = 52 + Q

Finally, MCS is set equal to marginal revenue to determine the number of units

that will be purchased by the marketing division and the price at which they will be sold:

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MR = 100 − 2Q = 52 + Q = MCS and Q_M = 16 at P = 84