Question

The average consumer at a firm with market power has an inverse demand function of P = 10 – Q, and the firm's cost function is TC = 2Q. Assume the firm engages in two-part pricing.
a. What is the optimal fixed fee to charge each consumer?
b. What is the optimal price to charge a consumer for each unit purchased?
c. How do the firm’s profits arising from two-part pricing compare to profits arising from the optimal single price?

Answer 1

Profit Maximizing two part pricing strategy

In order to maximize Profit a firm should Produce that quantity at which P = MC and also for each good it should charge Price equal to its Marginal Cost and whatever Consumer Surplus consumer receives he should charge Fixed Fee Equal to that Consumer Surplus.

(a) TC = 2Q

=> MC = dTC/dQ = 2

Hence P = MC => 10 - Q = 2 => Q = 8

Now, Consumer Surplus is the Area above Price (P = 2) and Below Demand Curve :

Hence, Consumer Surplus = (1/2)(10 - 2)8 = 32

Hence, As discussed above,

Fixed Fee = Consumer surplus = 32

Hence He should Charge Fixed Fee = 32

(b)

TC = 2Q

=> MC = dTC/dQ = 2

Hence As Discussed above,  for each good it should charge Price equal to its Marginal Cost

Hence  for each good it should charge Price = 2

(c)

Under 2 part Pricing

Profit = Total Revenue - Total Cost

Total Revenue = Price*Quantity + Fixed Fee = 2*8 + 32 = 48

Total Cost = 2Q = 2*8 = 16

Hence Profit = 48 - 16 = 32

Single Pricing

In order to maximize profit Monopoly should Produce that quantity at which MR = MC

As calculated above MC = 2

MR = dTR/dQ = d(PQ)/dQ = d((10 - Q)Q)/dQ = 10 - 2Q

=> MR = MC => 10 - 2Q = 2 => Q = 4

Hence P = 10 - 4 = 6

Hence Profit = PQ - 2Q = 6*4 - 2*4 = 16

Hence (Profit under 2 part pricing) = 32 is greater than (Profit under single Pricing) = 16