Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

13th Edition

ISBN: 9780321869838

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Publisher: PEARSON

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Textbook Question

Chapter 7.2, Problem 85E

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:

x = f ( p , q ) Demand function for product A y = g ( p , q ) Demand function for product B

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
f_q(p, q) > and g_p(p, q) > 0	Competitive (substitute)
f_q(p, q) < and g_p(p, q) < 0	Complementary
f_q(p, q) ≥ and g_p(p, q) ≤ 0	Neither
f_q(p, q) ≤ and g_p(p, q) ≥ 0	Neither

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

93. Product demand. The monthly demand equations for the sale of skis and ski boots in a sporting goods store are

x = f ( p , q ) = 800 − 0.004 p 2 − 0.003 q 2 Skis y = g ( p , q ) = 600 − 0.003 p 2 − 0.002 q 2 Ski boots

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Chapter 7.2, Problem 84E

Chapter 7.2, Problem 86E

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Chapter 7 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

Ch. 7.1 - Prob. 1MP Ch. 7.1 - Matched Problem 2 Find f(2, 2, 3) for f in Example...Ch. 7.1 - Prob. 3MP Ch. 7.1 - Matched Problem 4 For the box in Example 4, find...Ch. 7.1 - Matched Problem 5 Refer to Example 5. Find the...Ch. 7.1 - Matched Problem 6 Find the coordinates of the...Ch. 7.1 - Matched Problem 7 (A) Describe the cross sections...Ch. 7.1 - Explore and Discuss 1 Imagine that you are facing...Ch. 7.1 - In Problems 18, find the indicated value of the...Ch. 7.1 - Prob. 2E

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Solution Summary: The author analyzes the monthly demand equations for the sale of skis and ski boots in a sporting goods.

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