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Activity 10.7.5 While the quantity of a product demanded by consumers is often a function of the price of the product, the demand for a product may also depend on the price of other products. For instance, the demand for blue jeans at Old Navy may be affected not only by the price of the jeans themselves, but also by the price of khakis. Suppose we have two goods whose respective prices are p₁ and P2. The demand for these goods, 91 and 92, depend on the prices as 91= 150-2p1 - P2 92 = 200-P1-3p2. (10.7.1) (10.7.2) The seller would like to set the prices p1 and p2 in order to maximize revenue. We will assume that the seller meets the full demand for each product. Thus, if we let R be the revenue obtained by selling q₁ items of the first good at price pi per item and q2 items of the second good at price p2 per item, we have CIN R= P191 +P292- We can then write the revenue as a function of just the two variables pi and p2 by using Equations (10.7.1) and (10.7.2), giving us R(P1, P2) = P1(150-2p1-P2) + P2(200-p1-3p2) = 150p1 +200p2-2p1p2-2p1-3p². =

Activity 10.7.5 While the quantity of a product demanded by consumers is often a function of the price of the product, the demand for a product may also depend on the price of other products. For instance, the demand for blue jeans at Old Navy may be affected not only by the price of the jeans themselves, but also by the price of khakis. Suppose we have two goods whose respective prices are p₁ and P2. The demand for these goods, 91 and 92, depend on the prices as 91= 150-2p1 - P2 92 = 200-P1-3p2. (10.7.1) (10.7.2) The seller would like to set the prices p1 and p2 in order to maximize revenue. We will assume that the seller meets the full demand for each product. Thus, if we let R be the revenue obtained by selling q₁ items of the first good at price pi per item and q2 items of the second good at price p2 per item, we have CIN R= P191 +P292- We can then write the revenue as a function of just the two variables pi and p2 by using Equations (10.7.1) and (10.7.2), giving us R(P1, P2) = P1(150-2p1-P2) + P2(200-p1-3p2) = 150p1 +200p2-2p1p2-2p1-3p². =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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10. DERIVATIVES OF Activity 10.7.5 While the quantity of a product demanded by consumers is often a function of the price of the product, the demand for a product may also depend on the price of other products. For instance, the demand for blue jeans at Old Navy may be affected not only by the price of the jeans themselves, but also by the price of khakis. Suppose we have two goods whose respective prices are p₁ and Pp2. The demand for these goods, q1 and 92, depend on the prices as 91150-2p1 - P2 92 = 200-P1-3p2- The seller would like to set the prices p₁ and p2 in order to maximize revenue. We will assume that the seller meets the full demand for each product. Thus, if we let R be the revenue obtained by selling q₁ items of the first good at price pi per item and q2 items of the second good at price p2 per item, we have R= P191 +P292- We can then write the revenue as a function of just the two variables pi and p2 by using Equations (10.7.1) and (10.7.2), giving us R(P1, P2) = P1 (150-2p1 - p2)+p2(200-p1-3p2) = 150p1 +200p2-2p1p2-2p1-3p². A graph of R as a function of p₁ and p2 is shown in Figure 10.7.2. 40001 2000- 0- -2000- -4000- -6000- -8000- (10.7.1) (10.7.2) -10000 TTTT 10 20 30 40 50 60 P2 Figure 10.7.2 A revenue function. a. Find all critical points of the revenue function, R. (Hint: You should obtain a system of two equations in two unknowns which can be solved by elimination or substitution.) 60 50 40 30 20 10 0 P1
Transcribed Image Text:10. DERIVATIVES OF Activity 10.7.5 While the quantity of a product demanded by consumers is often a function of the price of the product, the demand for a product may also depend on the price of other products. For instance, the demand for blue jeans at Old Navy may be affected not only by the price of the jeans themselves, but also by the price of khakis. Suppose we have two goods whose respective prices are p₁ and Pp2. The demand for these goods, q1 and 92, depend on the prices as 91150-2p1 - P2 92 = 200-P1-3p2- The seller would like to set the prices p₁ and p2 in order to maximize revenue. We will assume that the seller meets the full demand for each product. Thus, if we let R be the revenue obtained by selling q₁ items of the first good at price pi per item and q2 items of the second good at price p2 per item, we have R= P191 +P292- We can then write the revenue as a function of just the two variables pi and p2 by using Equations (10.7.1) and (10.7.2), giving us R(P1, P2) = P1 (150-2p1 - p2)+p2(200-p1-3p2) = 150p1 +200p2-2p1p2-2p1-3p². A graph of R as a function of p₁ and p2 is shown in Figure 10.7.2. 40001 2000- 0- -2000- -4000- -6000- -8000- (10.7.1) (10.7.2) -10000 TTTT 10 20 30 40 50 60 P2 Figure 10.7.2 A revenue function. a. Find all critical points of the revenue function, R. (Hint: You should obtain a system of two equations in two unknowns which can be solved by elimination or substitution.) 60 50 40 30 20 10 0 P1
95 CHAPTER 10. DERIVATIVES OF MULTIVARIABLE FUNCTIONS b. Apply the Second Derivative Test to determine the type of any critical point (s). c. Where should the seller set the prices p₁ and p2 to maximize the revenue?
Transcribed Image Text:95 CHAPTER 10. DERIVATIVES OF MULTIVARIABLE FUNCTIONS b. Apply the Second Derivative Test to determine the type of any critical point (s). c. Where should the seller set the prices p₁ and p2 to maximize the revenue?
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