Question

9. In the past few years, a few new "private garage and track" facilities have opened in various places across the U.S. For example, in Michigan, the M1 Concourse (Links to an external site.) facility opened recently opened.

At the basic level, these facilities charge consumers an annual membership fee and allow them to have a dedicated garage space to store their car. (Usually a hot rod or track car like a Corvette, Porsche, or Ferrari.) Members then also have access to a track where they can race their car.

Suppose that you are the pricing director at one of these facilities and you're trying to figure out how much to charge, both for the annual membership fee and for each hour of track time that a member uses.

Suppose you know that the market of potential members can be divided into two groups: die-hards and enthusiasts. Each "die-hard" consumer has an annual demand for track time that can be given by Q = 130 - P. Each "enthusiast" consumer has an annual demand for track time that can be given by Q = 100 - P. In both equations P is the price per hour of track time and Q is the number of hours of track time demanded per year.

The facility has fixed costs, and the marginal cost of each hour of track time used is only $10.

Suppose that you could distinguish which group each potential member was in, and that you could set (and maintain) a different pricing scheme for each consumer group. Your goal is to maximize the facility's annual profits.

• What price, per-hour, would you set for the die-hard consumers?
  $ 70 per hour
• What annual membership fee would you set for the die-hard consumers
• $ 4200 per year
• What price, per-hour, would you set for the enthusiast consumers?
  $ 55 per hour
• What annual membership fee would you set for the enthusiast consumers?
  $ 2475 per year

Now suppose that you can't tell the customer groups apart. That is, you don't know what "type" any potential member is and you have no way of charging them different prices. Suppose that you decide to set the per-hour track price at your marginal cost. Finally, suppose that you know that 50% of your potential customers are "die-hards" and 50% are "enthusiasts".

• To maximize profits, what should you set the annual membership fee to?
  (a) $_________ per year

What would you do if, instead, you knew that 60% of your potential customers are "die-hards" and only 40% are "enthusiasts"?

• (b) To maximize profits, what should you set the annual membership fee to?
  $_______________ per year

Answer 1

ANSWER

a) To maximise profits Q+P=100+130=230 UNITS

=230*40/100=230*.0.04=9.2 UNITS Per year

A firm maximizes profit by operating where marginal revenue equals marginal cost. In the short run, a change in fixed costs has no effect on the profit maximizing output or price. The firm merely treats short term fixed costs as sunk costs and continues to operate as before.There are several perspectives one can take on this problem. First, since profit equals revenue minus cost, one can plot graphically each of the variables revenue and cost as functions of the level of output and find the output level that maximizes the difference (or this can be done with a table of values instead of a graph). Second, if specific functional forms are known for revenue and cost in terms of output, one can use calculus to maximize profit with respect to the output level.

since 60% of the customers are die hard=60/100*130=0.06*130=7.8 units per year

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