Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

The demand equations for two related products are given, where $p_{1}$ is the per unit price demanded for $x$ items of the first product and $p_{2}$ is the per unit price when $y$ units of the second product are demanded. Determine if the products are substitutes, complementary or neither.$$x=\frac{4}{p_{1} p_{2}}, \text { and } y=\frac{6}{p_{1} p_{2}}$$

Comp

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 5

Economic Applications

Partial Derivatives

Campbell University

Oregon State University

Harvey Mudd College

Baylor University

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

05:49

The demand and supply func…

01:05

Determine whether the dema…

01:04

here we have the demand and supply functions for two interdependent commodities that it's given by Q Sub D one equals 40 minus five. Piece of one minus piece of two. Q. D. Sub two equals 50 minus two. Piece of one minus four. Piece of two. Q s sub one equals negative three plus four piece of one and kill S. Sub two equals negative seven plus three piece of two. So where Cousteau beacuse of S and P. C. P denote the quantity demanded quantity supplied and price of the good respectively. We're going to determine the equilibrium price and quantity for this to commodity model. Then we're going to determine if the goods are substitute, herbal or complementary. So let's start by finding the equilibrium and equilibrium. Q D sub one equals Q S sub one and Q D sub two equals Q s substitute because quantity demanded equals quantity supplied in equilibrium. So then we get 40 minus five. Piece of one minus piece up to equals negative three plus four. Piece of one. And over here we're going to get 50 minus two. Piece of one minus four. Piece of two equals negative seven plus three p. Sub two. Let's go ahead and solve this equation here for peace up to I want to start by adding five piece of one And subtracting -40 from both sides. This gives me negative piece of two equals negative 43 plus nine P sub one. I'm going to divide everything by negative one or multiplied by -1. To get rid of this negative sign. Get peace of two equals 43 nine piece of one. Now that I have an expression here for peace up to. We're going to go ahead and substitute it in over here I get 50 -2. piece of 1 -4 times 43 -9. piece of one equals negative seven Plus three times 43 -9 piece of one. This gives me 50 -2. piece of one -4 times 43 is 172 plus 36. Piece of one equals negative seven plus three times 43 It's 129 -27 piece of one. In order to solve a piece of one. I need all my piece of buttons on one side and all of my numbers on the other side. Let's go ahead and combine like terms first. Before we start moving stuff around Here I have 50 and I have a negative 172. So 50 minus 1 72 I get negative 1, 22 minus two piece of one plus 36. Piece of one. So I get positive 34 piece of one equals negative seven Plus 129 I get 122 -27. Piece of one. I'm gonna go ahead and add 122 to both sides and we get 34 piece of one equals 244 minus 27. Piece of one. I'm going to add 27 piece of one to both sides as well. And that gives me 34 plus 27 I get 61. Piece of one equals 2, 44 Divide both sides by 61. And we find that piece of one is 2, 44, divided by 61 Is four. So it's $4. Piece of two equals 43 -9 times piece of one which is four. So piece of two equals 43 -36. We get a piece of two equals 7. So we found the equilibrium price it's four and 7. But we still need to find the quantity demanded or equilibrium quantity quantity demanded quantity supplied equally the same. Let's start with Q. S. Sub two which equals negative seven plus three P. Sub two quantity, supply two equals negative seven plus three times p. Sub two is seven. I get accused of S sub two equals negative seven plus 21. So the equilibrium quantity for product # two is 14. And for the quantity of the first one, Qs sub one equals negative three plus four P. One. This gives us negative three plus four times four. This gives us negative three plus 16 Which gives us an equilibrium quantity of 13. So we get a piece of one Is $4. The equilibrium price of good two is $7. Our equilibrium Quantity one Product one is 13 and our equilibrium quantity for product to is 14. Is this product a substitute, herbal or complementary good? This product is complementary, and the reason it is complementary is because as the price of either good goes up, the price of both falls, and we can see that that is demonstrated right here. In both of these, we have the same side, so when one goes up, it's affecting both.

View More Answers From This Book

Find Another Textbook

06:28

Given the function $f(x)=e^{-a x^{2}},$ where $a$ is a positive constant. Su…

01:13

Evaluate the given integral.$$\int x^{2} e^{4 x^{3}} d x$$

01:54

$S(R, r, n)=R\left(\frac{(1+r)^{n}-1}{r}\right)$ determine $S(10,000,0.0025,…

01:59

Use the method of Lagrange multipliers to optimize $f$ as indicated, subject…

03:03

Evaluate the given integral and check your answer.$$\int \frac{5 t^{7}-2…

01:28

Evaluate the given integral and check your answer.$$\int(3 x)^{3} d x$$<…

04:42

Sketch some of the level curves (contours) or the surface defined by $f$.

01:48

Evaluate the given integral and check your answer.$$\int \sqrt[3]{x} d x…

02:48

Find and classify, using the second partial derivative test, the critical po…

01:03

If $f^{\prime}(x)=3 / x,$ sketch the family of solutions.