Demand
- Symbolic Analysis -

Situation:

From past experience, a publisher knows that a particular type of book will sell 14,000 copies at a price of $20 each. Market research further indicates that for every one-dollar increase in price, sales will fall by 400 copies. The publisher asks for your advice in deciding the suggested retail price of the next book published.¹

Symbolic Analysis:

In this situation, the number of books sold will change if the publisher decides to change the suggested retail price of the book. Since the number of books sold depends on the number of times the publisher decides to increase the price, one may use the variable x to represent the number of increases.

The suggested retail price may then be thought of as the usual price ($20) plus the amount that the publisher decides to increase the price of the book. Since the increases come in $1 dollar increments, the total increase may be written as 1x ($1 per increase times the number of increases). Of course, 1x is usually written as just x. Therefore, the suggested retail price can be written as 20 + x.

The number of books expected to be sold may be thought of as the usual number (14,000) less the number not sold as a result of raising the asking price of the book. The number not sold will be 400 for each increase in price. Therefore, the number not sold may be written 400x. The expected number of books sold may be represented by 14000 - 400x.

The revenue from book sales is calculated by multiplying the number of books sold by the price charged for each book. In previous times this could be determined by multiplying 14000 books by $20 per book for a total revenue of $280,000. If the publisher changes the suggested retail price of the book, the total revenue will be the number of books expected to be sold times suggested retail price. Using the analysis above this would be written as (14000 - 400x)(20 + x).

Conclusion:

A function useful in determining the anticipated revenue could be represented by:

R(x) = (14000 - 400x)(20 + x)

Further analysis of this situation may be accomplished by graphing the function listed above or by examining projected sales data.

1	Problems of this type may be found in Mathematical Analysis, 3rd. Ed. By Arya and Lardner, published by Prentice Hall, 1989, page 93.