Question

1. A monopolist knows that there are two types of consumers, “high demand” (H) and low demand (L) types. Inverse demand for each consumer of the two types is p = 50 − q_L and p = 100 − q_H . 60% of consumers are of the L type. Marginal cost is zero.

   a) Find the optimal price if the firm can only set a single price. (One way to do this is to pretend that there are 6 consumers of the L type and 4 of the H type, then construct a total demand curve, then calculate.
   b) Suppose that the firm can set a fee for purchasing any amount, plus a price per unit. Explain, using a diagram, how the firm can earn more from this strategy than from using a single price (without a fee).

Answer 1

It is given that 60% of the consumers are of L type, so the remaining (40%) will be of H type. For the sake of simplicity, we can assume that there are 6L consumers and 4H consumers as mentioned in the question.

It is also given that

Low demand is or

High demand is or

Now that we got their individual demands, their total demands are

(Aggregate inverse demand)

Now lets answer the questions now that we got our euqations

a) The first question asks the optimal price of the firm if it sets only a single price. If thats the case, Marginal revenue must be euqal to Marginal cost. And given that Marginal cost or MC = 0, the proft will be at its max when MR is 0

Total Revenue (TR) =

Marginal Revenue (MR) =

Profit is maximimum when that is 0

When q = 350

P = 35

Therefore, profit = PQ = 35*350 = 12250

b) If the monopolist implements a two part tariff (fee + per unit fee), then to maximize profit, MC = 0

Now the firm could go two ways. He can set fee equal to consumer surplus of low demand consumer, then the total fee will be area of the triangle AOB which is half of 50*50 or 1250. If he does this, total buyers will be 10 (6L + 4H) and total profit is 1250*10 = 12500

He can set fee equal to consumer surplus of high demand consumer, then the total fee will be area of the triangle A'OB' which is half of 100*100 or 5000. If he does this, total buyers will be 4 (4H) and total profit is 5000*4= 20000

We can clearly see that the profit is maximum in the latter one. Thus that will be implmented where only high demand consumers wil buy and total profits is 5000 per head or 20000