Chapter 7.2, Problem 94E

Problems 91–94 refer to the following: If a decrease in demand for one product results in an increase in demand for another product, the two products are said to be competitive, or substitute,

products. (Real whipping cream and imitation whipping cream are examples of competitive, or substitute, products.) If a decrease in demand for one product results in a decrease in demand for another product, the two products are said to be complementary products. (Fishing boats and outboard motors are examples of complementary products.) Partial derivatives can be used to test whether two products are competitive, complementary, or neither. We start with demand functions for two products such that the demand for either depends on the prices for both:

     $\begin{array}{cc}x=f(p,q)& \text{Demand}\text{function}\text{for}\text{product}A\\ y=g(p,q)& \text{Demand}\text{function}\text{for}\text{product}B\end{array}$

The variables x and y represent the number of units demanded of products A and B, respectively, at a price p for 1 unit of product A and a price q for I unit of product B. Normally, if the price of A increases while the price of B is held constant, then the demand for A will decrease; that is, f_p(p, q) < 0. Then, if A and B are competitive products, the demand for B will increase; that is, g_r(p, q) > 0. Similarly, if the price of B increases while the price of A is held constant, the demand for B will decrease; that is, g_q(p, q) < 0. Then, if A and B are competitive products, the demand for A will increase; that is, f_q(p, q) > 0. Reasoning similarly for complementary products, we arrive at the following test:

Test for Competitive and Complementary Products

Partial Derivatives	Products A and B
fq(p, q) > and gp(p, q) > 0	Competitive (substitute)
fq(p, q) < and gp(p, q) < 0	Complementary
fq(p, q) ≥ and gp(p, q) ≤ 0	Neither
fq(p, q) ≤ and gp(p, q) ≥ 0	Neither

Use this test in Problems 91-94 to determine whether the indicated products are competitive, complementary, or neither.

94.    Product demand. The monthly demand equations for the sale of tennis rackets and tennis balls in a sporting goods store are

$\begin{array}{cc}x=f\left(p,q\right)=500-0.5p-q^2& \text{Tennis}\text{rackets}\\ y=g\left(p,q\right)=10,000-8p-100q^2& \text{Tennis}\text{balls}(cans)\end{array}$