Question

Suppose a firm owns a gym and is trying to determine the best pricing structure to use. The gym’s cost structure is C(Q) = 10, 000 + 15Q. There are two types of users: Heavy Gym users and Light Gym users. The individual demand curve for a Heavy user is qDH(P) = 400?10P and the individual demand for a Light user is qDL(P) = 125?5P. Suppose in the market there are 10 Heavy users and 5 Light users.

(a) Suppose the firm can not tell the two types apart and offers a single price. What quantity will be offered and at what price? Would the gym have both types attending? Graph the solution.

(b) How would this change if the gym decided to offer a two-part pricing strategy (assuming the gym cannot distinguish between the two groups)? Would the gym have both types attending (i.e. does the gym make higher profits excluding one group)? Would the gym prefer the pricing structure here or in part (a)?

(c) Suppose the gym can tell the two types apart. What would be the price and profits from group discrimination? Does the gym prefer this outcome to the previous parts? Why?

(d) Show the two-part pricing solution when the gym can distinguish between the two types. How do profits compare with part (b) and part (c)?

Answer 1

1. we need to use the information that we have in order to calculate the quantity and price of the market. We need to remember that in the market equilibrium we have the next equivalence

, where p is price, Mc is marginal cost and Mr is marginal revenue, which is the equilibrium condition.

We don’t have a price equation, but we can obtain it inverting the demand for each consumer. In our case, first we need to multiply each demand equations by 15 and 5 respectively and to sum the two demands in order to get a market demand (global demand).

The next step in the procedure is to invert the demand equation and get the price equation for the market, and then use the identities above to obtain the optimal quantity and price.

b) For this case, we can calculate our price and demand using just the demand equation for low and high users separately and then compare each profit with the combined profit in a) to see which is the best option for the Gym

Comparing the profits we can see that the best option is excluding low users and centering in the high users to get a higher profit thant a), and also, just the high users would attend to the gym.

c) the price and quantity would be the ones obtained in b) for high users and low users. the gym would prefer the price discrimination in order to get higher profit than if it chooses to get the two types of consumers attending to the gym.

d) the solution is merely identical that the one showed in b), because the procedure to get the prices and quantities would be the same.