Answer to Question #240929 in Microeconomics for Kuds

Question

You are the manager of a monopoly. A typical consumer’s inverse demand function for your firm’s product is 𝑃 = 250 − 4𝑄 and your cost function is 𝐶(𝑄) = 10𝑄. a) Determine the profit-maximizing quantity, and price under standard monopoly pricing. b) Determine the optimal two-part pricing strategy. c) Suppose you want to use block pricing strategy. How many units should you put in a block and what price you should charge for the block? d) Compare your profits from parts (a), (b) and (c)

Answer 1

Solution:

a.). Profit maximizing quantity and price under monopoly is determined where MR = MC

First derive TR:

TR = P x Q

P = 250 – 4Q

TR = (250 – 4Q)$\times$ Q = 250Q – 4Q²

Derive MR:

MR = $\frac{\partial TR} {\partial Q}$ = 250 – 8Q

TC = 10Q

Derive MC:

MC = $\frac{\partial TC} {\partial Q}$ = 10

Set MR = MC

250 – 8Q = 10

250 – 10 = 8Q

240 = 8Q

Q = 30

Substitute to get price:

P = 250 – 4Q = 250 – 4(30) = 250 – 120 = 130

P = 130

Profit = TR – TC

Profit = (130 $\times$ 30) – (10 $\times$ 30) = 3,900 – 300 = 3,600

Profit under standard monopoly pricing = 3,600

 

b.). The optimal two-part pricing strategy:

Set the price equal to marginal cost and charge a fee equal to the consumer surplus at that price.

P = MC = 10

Therefore, the per-unit price will be $10, which equals marginal cost.

Substituting this into the demand equation:

10 = 250 – 4Q

4Q = 250 – 10

4Q = 240

Q = 60

Consumer surplus = 0.5 (250 – 10) (60) = 7,200

The monopoly will charge an additional fee of 7,200 under the two-part pricing strategy.

 

c.). Block-pricing strategy:

TR = P x Q

P = 250 – 4Q

TR = (250 – 4Q)$\times$ Q = 250Q – 4Q²

Derive MR:

MR = $\frac{\partial TR} {\partial Q}$ = 250 – 8Q

TC = 10Q

Derive MC:

MC = $\frac{\partial TC} {\partial Q}$ = 10

Substituting this into the demand equation:

10 = 250 – 4Q

4Q = 250 – 10

4Q = 240

Q = 60

Optimal quantity to package = 6 0

Derive the optimal price for the package:

= 0.5(250 – 10) (60) + (10 x 60) = 7,200 + 600 = 7,800

Profit under block pricing strategy = 0.5(250 – 10) (60) + (10 $\times$ 60) – (10 $\times$ 60) = 7,800 – 600 = 7,200

 

d.). The profits under the two-part pricing strategy and block pricing strategy is more profitable compared to standard monopoly pricing. Therefore, price discrimination is paramount under the two-part pricing and block pricing strategy.