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

Solution :-

(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.

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 ]

So, The maximum profit is :-

A = TR - TC

= P x Q - MC x Q

= ( P - MC) x Q

= ( 110 - 20) x 450

= 90 x 450

A = $40500

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.

Consider the above fig, here under the quantity discounting the associated MR is step function, the profit maximizing production if Q = 600.

So, The consumer will buy 600 units in total.

(C) :-

Computing the profit if the quantity discount is implemented,

P1 = $120

P2 = $80

MC = $20

Q1 = 400

Q2 = 600

A = ( P1 - MC) x Q1 + ( P2 - MC) x ( Q2 - Q1)

= ( 120 - 20) x 400 + ( 80 - 20) x ( 600 - 400)

= 100 x 400 + 60 x 200

= 40000 + 12000

= $52000.

The profit of the producer is = $52000.

(D) :-

​​​​​​P = 200 - 0.2Q

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

So, the total quantity sold will be :-

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 x ( 200 - MC) x Q

= 0.5 x ( 200 - 20) x 900

= 0.5 x 180 x 900

= $81000

The total revenue is given by :-

TR = Total fixed fee + Variable fee

= 81000 + MC x Q

= 81000 + 20 x 900

= 81000 + 18000

= $99000.

Total revenue is = $99000.

So, The total Variable cost is :-

TVC = MC x Q

= 20 x 900

= $18000

The total Variable cost is = $18000.