Question

Suppose you own a sports gym. You identify two segments of consumers who want to use your equipment: High volume: P = 8 – 0.05Q Low volume: P = 9 – 0.2Q where P is the price in $ per visit to the gym, and Q is the number of visits per year.

Suppose you can charge both an annual membership fee and a per visit fee to your customers. Suppose you are able to identify and target the low demand consumers using (non-transferable) coupons for both annual membership fees as well as for per visit charges, so that you can effectively set separate membership and per visit fees for the two segments. Suppose the marginal cost per visit per person is $1. Then, to maximize profits, you should:

Answer 1

To maximize profits for different segments of customers at a sports gym, the owner should use a two-part pricing strategy and price discrimination, setting optimal membership and per visit fees based on the marginal cost of $1.

Step-by-step explanation:

To maximize profits for a sports gym, the owner should use price discrimination to set different prices for high volume and low-volume customers using a two-part pricing strategy, which includes an annual membership fee and a per-visit fee. Since we know the marginal cost per visit is $1, the firm would want to set marginal revenue equal to marginal cost to determine the optimal per visit charge for each group of consumers.

For high volume customers, the profit-maximizing quantity (Q) is where MR = MC, and this is found by differentiating the total revenue (TR) function derived from the demand function and equating it to $1. After finding Q, we plug it back into the original demand function to find the per-visit price. We would follow the same process for low volume customers but taking into account their specific demand function.

Since the firm can perfectly segment the market, the maximization problem involves maximizing profit from each group independently. The gym can set a higher per-visit charge and a lower membership fee for low volume customers to extract more consumer surplus. High-volume customers might receive a lower per-visit charge and a higher membership fee in order to maximize profits.

The annual membership fee would be set at the level where the total fee captures the consumer surplus from each type of customer. It is important to note that setting these fees requires careful consideration of consumer surplus for each segment, which is the area under the demand curve above the price line up to the demand quantity.