Question

Suppose the demand equation facing a firm is Q = 1000 – 5P, MR = 200 – 0.4 Q, and MC = $20.

1. Compute the maximum profit the firm can earn.
2. Suppose the firm is considering a quantity discount. It offers the first 400 units at a price of $120, and further units at a price of $80. How many units will the consumer buy in total?
3. Compute the profit if the quantity discount is implemented.
4. If the firm implemented a two-part pricing strategy, what would be the fixed fee, variable fee, total revenue, and the total variable cost?

Answer 1

(a) Suppose the demand equation facing a firm is Q = 1000 – 5P, MR = 200 – 0.4 Q, and MC = $20.

The maximum profit the firm can earn is:
Q = 1000 - 5P
5P = 1000 - Q
P = 200 - 0.2Q

At Equilibrium: MR = MC
200 - 0.4Q = 20
200 - 20 = 0.4Q
180 = 0.4Q
Q = 180/0.4
Q = 450

The profit maximizing price is:
P = 200 - 0.2Q
= 200 - 0.2 x 450
= 200 - 90
P = $110

Thus, The maximum profit is:
A = TR - TC
= P x Q - MC x Q
= ( P - MC) x Q
= ( 110 - 20) x 450
= 90 * 450
= $40500

Therefore, the maximum profit the firm can earn = $40500.

(b) Suppose the firm is considering a quantity discount. It offers the first 400 units at a price of P1 = $120, and further units at a price of $80.

As per the graph above, here under the quantity discounting the associated MR is step function, the profit maximizing production if Q = 600.
Therefore, the consumer will buy 600 units in total.

(c) Compute the profit if the quantity discount is implemented:
P1 = $120 ;P2 = $80
MC = $20
Q1 = 400; Q2 = 600
A = ( P1 - MC) * Q1 + ( P2 - MC) * ( Q2 - Q1)
= ( 120 - 20) * 400 + ( 80 - 20) * ( 600 - 400)
= 100 * 400 + 60 * 200
= 40000 + 12000
= $52000.

Therefore, the profit if the quantity discount is implemented is $52000.

(d) If the firm implemented a two-part pricing strategy, what would be the fixed fee, variable fee, total revenue, and the total variable cost?

​​​​​​P = 200 - 0.2Q

If the firm implemented a two-part pricing strategy, then the variable fee should be exactly equal to MC = $20.

Thus, the total quantity sold is:
P = MC
200 - 0.2Q = 20
200 - 20 = 0.2Q
180 = 0.2Q
Q = 180/0.2
Q = 900

The fixed fee will be the consumer surplus.

Consumer surplus:
= 0.5 ( 200 - MC) * Q
= 0.5 ( 200 - 20) * 900
= 0.5 * 180 * 900
= $81000

The total revenue is:
TR = Total fixed fee + Variable fee
= 81000 + MC * Q
= 81000 + 20 * 900
= 81000 + 18000
= $99000.

Thus,
Total Fixed Fee is $81000.
Total Variable Fee is $18000.
Total Revenue is $99000.

The Total Variable Cost is:
TVC = MC * Q
= 20 * 900
= $18000

Total Variable Cost is $18000.