Demand Function

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A consumer has the utility function over goods X and Y, $U(X; Y) = X^1/3cdot Y^1/2$

Let the price of good x be given by $P_x$, let the price of good $y$ be given by $P_y$, and let income be given by $I$.

(a) Derive the consumer’s generalized demand function for good $X$. This is simply the demand equation. $X$ is a function of Price and Income ($P_x$ and $I$).

Can someone help me get in the right direction for this?

economics

share|cite|improve this question

edited Jul 22 at 13:28

callculus

16.4k31427

asked Jul 22 at 13:20

Kolten

1

• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is the demand equation?
  – saulspatz
  Jul 22 at 13:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I am looking for the demand function. I do not have the demand equation. Set up the Lagrangean expression, L = X1/3Y1/2 + λ[I - PxX - PyY]:
  – Kolten
  Jul 22 at 13:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't know what is meant by the demand equation. This is a mathematics site. The question would be better asked on economics.stackexchange.com
  – saulspatz
  Jul 22 at 13:49
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  You may want to include what you have tried in order to improve your chances of getting an answer
  – Green.H
  Jul 22 at 13:54
  

  
  
  
  

add a comment | 

up vote
-2
down vote

favorite

A consumer has the utility function over goods X and Y, $U(X; Y) = X^1/3cdot Y^1/2$

Let the price of good x be given by $P_x$, let the price of good $y$ be given by $P_y$, and let income be given by $I$.

(a) Derive the consumer’s generalized demand function for good $X$. This is simply the demand equation. $X$ is a function of Price and Income ($P_x$ and $I$).

Can someone help me get in the right direction for this?

economics

share|cite|improve this question

edited Jul 22 at 13:28

callculus

16.4k31427

asked Jul 22 at 13:20

Kolten

1

• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is the demand equation?
  – saulspatz
  Jul 22 at 13:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I am looking for the demand function. I do not have the demand equation. Set up the Lagrangean expression, L = X1/3Y1/2 + λ[I - PxX - PyY]:
  – Kolten
  Jul 22 at 13:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't know what is meant by the demand equation. This is a mathematics site. The question would be better asked on economics.stackexchange.com
  – saulspatz
  Jul 22 at 13:49
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  You may want to include what you have tried in order to improve your chances of getting an answer
  – Green.H
  Jul 22 at 13:54
  

  
  
  
  

add a comment | 

up vote
-2
down vote

favorite

up vote
-2
down vote

favorite

A consumer has the utility function over goods X and Y, $U(X; Y) = X^1/3cdot Y^1/2$

Let the price of good x be given by $P_x$, let the price of good $y$ be given by $P_y$, and let income be given by $I$.

(a) Derive the consumer’s generalized demand function for good $X$. This is simply the demand equation. $X$ is a function of Price and Income ($P_x$ and $I$).

Can someone help me get in the right direction for this?

economics

share|cite|improve this question

edited Jul 22 at 13:28

callculus

16.4k31427

asked Jul 22 at 13:20

Kolten

1

A consumer has the utility function over goods X and Y, $U(X; Y) = X^1/3cdot Y^1/2$

Let the price of good x be given by $P_x$, let the price of good $y$ be given by $P_y$, and let income be given by $I$.

(a) Derive the consumer’s generalized demand function for good $X$. This is simply the demand equation. $X$ is a function of Price and Income ($P_x$ and $I$).

Can someone help me get in the right direction for this?

economics

share|cite|improve this question

edited Jul 22 at 13:28

callculus

16.4k31427

asked Jul 22 at 13:20

Kolten

1

share|cite|improve this question

share|cite|improve this question

edited Jul 22 at 13:28

callculus

16.4k31427

edited Jul 22 at 13:28

callculus

16.4k31427

edited Jul 22 at 13:28

callculus

16.4k31427

16.4k31427

asked Jul 22 at 13:20

Kolten

1

asked Jul 22 at 13:20

Kolten

1

asked Jul 22 at 13:20

Kolten

1

1

• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is the demand equation?
  – saulspatz
  Jul 22 at 13:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I am looking for the demand function. I do not have the demand equation. Set up the Lagrangean expression, L = X1/3Y1/2 + λ[I - PxX - PyY]:
  – Kolten
  Jul 22 at 13:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't know what is meant by the demand equation. This is a mathematics site. The question would be better asked on economics.stackexchange.com
  – saulspatz
  Jul 22 at 13:49
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  You may want to include what you have tried in order to improve your chances of getting an answer
  – Green.H
  Jul 22 at 13:54
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is the demand equation?
  – saulspatz
  Jul 22 at 13:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I am looking for the demand function. I do not have the demand equation. Set up the Lagrangean expression, L = X1/3Y1/2 + λ[I - PxX - PyY]:
  – Kolten
  Jul 22 at 13:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't know what is meant by the demand equation. This is a mathematics site. The question would be better asked on economics.stackexchange.com
  – saulspatz
  Jul 22 at 13:49
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  You may want to include what you have tried in order to improve your chances of getting an answer
  – Green.H
  Jul 22 at 13:54
  

  
  
  
  

What is the demand equation?
– saulspatz
Jul 22 at 13:29

What is the demand equation?
– saulspatz
Jul 22 at 13:29

I am looking for the demand function. I do not have the demand equation. Set up the Lagrangean expression, L = X1/3Y1/2 + λ[I - PxX - PyY]:
– Kolten
Jul 22 at 13:42

I am looking for the demand function. I do not have the demand equation. Set up the Lagrangean expression, L = X1/3Y1/2 + λ[I - PxX - PyY]:
– Kolten
Jul 22 at 13:42

I don't know what is meant by the demand equation. This is a mathematics site. The question would be better asked on economics.stackexchange.com
– saulspatz
Jul 22 at 13:49

I don't know what is meant by the demand equation. This is a mathematics site. The question would be better asked on economics.stackexchange.com
– saulspatz
Jul 22 at 13:49

1

1

You may want to include what you have tried in order to improve your chances of getting an answer
– Green.H
Jul 22 at 13:54

You may want to include what you have tried in order to improve your chances of getting an answer
– Green.H
Jul 22 at 13:54

add a comment | 

2 Answers
2

active

oldest

votes

up vote
0
down vote

You can solve the problem by using the method of lagrange multiplier as you have already written in the comment.

$$mathcal L=X^1/3cdot Y^1/2+lambda (I-P_xcdot X-P_ycdot Y)$$

Then you have to calculate the partial derivatives w.r.t $X,Y$ and $lambda$.

$fracpartial mathcal Lpartial X=frac13cdot X^-2/3cdot Y^1/2-P_Xlambda=0 Rightarrowfrac13cdot X^-2/3cdot Y^1/2=P_xlambda quad (1)$

$fracpartial mathcal Lpartial Y=frac12cdot X^1/3cdot Y^-1/2-P_ylambda=0 Rightarrowfrac12cdot X^1/3cdot Y^-1/2=P_ylambda quad (2)$

$fracpartial mathcal Lpartial lambda=I-P_xcdot X-P_ycdot Y=0 quad (3)$

Dividing (1) by (2)

$frac23fracYX=fracP_xP_y$

Solving for $P_ycdot Y$

$P_ycdot Y=frac32P_xcdot X$. The expression can be insert in (3).

$I-P_xcdot X-frac32P_xcdot X=0$

What is left is to solve the equation for $X$. Can you finish?

share|cite|improve this answer

answered Jul 22 at 13:59

callculus

16.4k31427

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I will try! So solving for X will create the demand function?
  – Kolten
  Jul 22 at 14:01
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, that´s right. I hope you can comprehend the steps before.
  – callculus
  Jul 22 at 14:02
  
  

  
  
  
  

add a comment | 

up vote
0
down vote

You need to solve the following:
$$max_X,YX^1/3Y^1/2$$
subject to $I geq P_x X + P_y Y.$

Set up Lagrangian
$$L = X^1/3Y^1/2 + lambda (I -(P_x X + P_y Y)).$$

FOC's yield:
begineqnarray*
frac13left(fracY^1/2X^2/3right) - lambda P_x &=&0,\
frac12left(fracX^1/3Y^1/2right) - lambda P_y &=&0,\
I - (P_x X + P_y Y) &=&0.
endeqnarray*
I will leave the algebra for you. You should get the following answer:
$$X= frac2I5 P_x.$$

share|cite|improve this answer

edited Jul 22 at 14:55

answered Jul 22 at 13:53

Green.H

1,046216

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Your production function is wrong. Also pay attention on the signs $L = X^1/3Y^2/3 + lambda (I -(P_x X colorred+ P_y Y)).$
  – callculus
  Jul 22 at 14:08
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @callculus Thanks, edited
  – Green.H
  Jul 22 at 14:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Now there is no contradiction between our answers anymore.
  – callculus
  Jul 22 at 15:05
  
  

  
  
  
  

add a comment | 

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote

You can solve the problem by using the method of lagrange multiplier as you have already written in the comment.

$$mathcal L=X^1/3cdot Y^1/2+lambda (I-P_xcdot X-P_ycdot Y)$$

Then you have to calculate the partial derivatives w.r.t $X,Y$ and $lambda$.

$fracpartial mathcal Lpartial X=frac13cdot X^-2/3cdot Y^1/2-P_Xlambda=0 Rightarrowfrac13cdot X^-2/3cdot Y^1/2=P_xlambda quad (1)$

$fracpartial mathcal Lpartial Y=frac12cdot X^1/3cdot Y^-1/2-P_ylambda=0 Rightarrowfrac12cdot X^1/3cdot Y^-1/2=P_ylambda quad (2)$

$fracpartial mathcal Lpartial lambda=I-P_xcdot X-P_ycdot Y=0 quad (3)$

Dividing (1) by (2)

$frac23fracYX=fracP_xP_y$

Solving for $P_ycdot Y$

$P_ycdot Y=frac32P_xcdot X$. The expression can be insert in (3).

$I-P_xcdot X-frac32P_xcdot X=0$

What is left is to solve the equation for $X$. Can you finish?

share|cite|improve this answer

answered Jul 22 at 13:59

callculus

16.4k31427

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I will try! So solving for X will create the demand function?
  – Kolten
  Jul 22 at 14:01
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, that´s right. I hope you can comprehend the steps before.
  – callculus
  Jul 22 at 14:02
  
  

  
  
  
  

add a comment | 

up vote
0
down vote

You can solve the problem by using the method of lagrange multiplier as you have already written in the comment.

$$mathcal L=X^1/3cdot Y^1/2+lambda (I-P_xcdot X-P_ycdot Y)$$

Then you have to calculate the partial derivatives w.r.t $X,Y$ and $lambda$.

$fracpartial mathcal Lpartial X=frac13cdot X^-2/3cdot Y^1/2-P_Xlambda=0 Rightarrowfrac13cdot X^-2/3cdot Y^1/2=P_xlambda quad (1)$

$fracpartial mathcal Lpartial Y=frac12cdot X^1/3cdot Y^-1/2-P_ylambda=0 Rightarrowfrac12cdot X^1/3cdot Y^-1/2=P_ylambda quad (2)$

$fracpartial mathcal Lpartial lambda=I-P_xcdot X-P_ycdot Y=0 quad (3)$

Dividing (1) by (2)

$frac23fracYX=fracP_xP_y$

Solving for $P_ycdot Y$

$P_ycdot Y=frac32P_xcdot X$. The expression can be insert in (3).

$I-P_xcdot X-frac32P_xcdot X=0$

What is left is to solve the equation for $X$. Can you finish?

share|cite|improve this answer

answered Jul 22 at 13:59

callculus

16.4k31427

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I will try! So solving for X will create the demand function?
  – Kolten
  Jul 22 at 14:01
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, that´s right. I hope you can comprehend the steps before.
  – callculus
  Jul 22 at 14:02
  
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

You can solve the problem by using the method of lagrange multiplier as you have already written in the comment.

$$mathcal L=X^1/3cdot Y^1/2+lambda (I-P_xcdot X-P_ycdot Y)$$

Then you have to calculate the partial derivatives w.r.t $X,Y$ and $lambda$.

$fracpartial mathcal Lpartial X=frac13cdot X^-2/3cdot Y^1/2-P_Xlambda=0 Rightarrowfrac13cdot X^-2/3cdot Y^1/2=P_xlambda quad (1)$

$fracpartial mathcal Lpartial Y=frac12cdot X^1/3cdot Y^-1/2-P_ylambda=0 Rightarrowfrac12cdot X^1/3cdot Y^-1/2=P_ylambda quad (2)$

$fracpartial mathcal Lpartial lambda=I-P_xcdot X-P_ycdot Y=0 quad (3)$

Dividing (1) by (2)

$frac23fracYX=fracP_xP_y$

Solving for $P_ycdot Y$

$P_ycdot Y=frac32P_xcdot X$. The expression can be insert in (3).

$I-P_xcdot X-frac32P_xcdot X=0$

What is left is to solve the equation for $X$. Can you finish?

share|cite|improve this answer

answered Jul 22 at 13:59

callculus

16.4k31427

You can solve the problem by using the method of lagrange multiplier as you have already written in the comment.

$$mathcal L=X^1/3cdot Y^1/2+lambda (I-P_xcdot X-P_ycdot Y)$$

Then you have to calculate the partial derivatives w.r.t $X,Y$ and $lambda$.

$fracpartial mathcal Lpartial X=frac13cdot X^-2/3cdot Y^1/2-P_Xlambda=0 Rightarrowfrac13cdot X^-2/3cdot Y^1/2=P_xlambda quad (1)$

$fracpartial mathcal Lpartial Y=frac12cdot X^1/3cdot Y^-1/2-P_ylambda=0 Rightarrowfrac12cdot X^1/3cdot Y^-1/2=P_ylambda quad (2)$

$fracpartial mathcal Lpartial lambda=I-P_xcdot X-P_ycdot Y=0 quad (3)$

Dividing (1) by (2)

$frac23fracYX=fracP_xP_y$

Solving for $P_ycdot Y$

$P_ycdot Y=frac32P_xcdot X$. The expression can be insert in (3).

$I-P_xcdot X-frac32P_xcdot X=0$

What is left is to solve the equation for $X$. Can you finish?

share|cite|improve this answer

answered Jul 22 at 13:59

callculus

16.4k31427

share|cite|improve this answer

share|cite|improve this answer

answered Jul 22 at 13:59

callculus

16.4k31427

answered Jul 22 at 13:59

callculus

16.4k31427

answered Jul 22 at 13:59

callculus

16.4k31427

16.4k31427

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I will try! So solving for X will create the demand function?
  – Kolten
  Jul 22 at 14:01
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, that´s right. I hope you can comprehend the steps before.
  – callculus
  Jul 22 at 14:02
  
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I will try! So solving for X will create the demand function?
  – Kolten
  Jul 22 at 14:01
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, that´s right. I hope you can comprehend the steps before.
  – callculus
  Jul 22 at 14:02
  
  

  
  
  
  

I will try! So solving for X will create the demand function?
– Kolten
Jul 22 at 14:01

I will try! So solving for X will create the demand function?
– Kolten
Jul 22 at 14:01

Yes, that´s right. I hope you can comprehend the steps before.
– callculus
Jul 22 at 14:02

Yes, that´s right. I hope you can comprehend the steps before.
– callculus
Jul 22 at 14:02

add a comment | 

up vote
0
down vote

You need to solve the following:
$$max_X,YX^1/3Y^1/2$$
subject to $I geq P_x X + P_y Y.$

Set up Lagrangian
$$L = X^1/3Y^1/2 + lambda (I -(P_x X + P_y Y)).$$

FOC's yield:
begineqnarray*
frac13left(fracY^1/2X^2/3right) - lambda P_x &=&0,\
frac12left(fracX^1/3Y^1/2right) - lambda P_y &=&0,\
I - (P_x X + P_y Y) &=&0.
endeqnarray*
I will leave the algebra for you. You should get the following answer:
$$X= frac2I5 P_x.$$

share|cite|improve this answer

edited Jul 22 at 14:55

answered Jul 22 at 13:53

Green.H

1,046216

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Your production function is wrong. Also pay attention on the signs $L = X^1/3Y^2/3 + lambda (I -(P_x X colorred+ P_y Y)).$
  – callculus
  Jul 22 at 14:08
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @callculus Thanks, edited
  – Green.H
  Jul 22 at 14:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Now there is no contradiction between our answers anymore.
  – callculus
  Jul 22 at 15:05
  
  

  
  
  
  

add a comment | 

up vote
0
down vote

You need to solve the following:
$$max_X,YX^1/3Y^1/2$$
subject to $I geq P_x X + P_y Y.$

Set up Lagrangian
$$L = X^1/3Y^1/2 + lambda (I -(P_x X + P_y Y)).$$

FOC's yield:
begineqnarray*
frac13left(fracY^1/2X^2/3right) - lambda P_x &=&0,\
frac12left(fracX^1/3Y^1/2right) - lambda P_y &=&0,\
I - (P_x X + P_y Y) &=&0.
endeqnarray*
I will leave the algebra for you. You should get the following answer:
$$X= frac2I5 P_x.$$

share|cite|improve this answer

edited Jul 22 at 14:55

answered Jul 22 at 13:53

Green.H

1,046216

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Your production function is wrong. Also pay attention on the signs $L = X^1/3Y^2/3 + lambda (I -(P_x X colorred+ P_y Y)).$
  – callculus
  Jul 22 at 14:08
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @callculus Thanks, edited
  – Green.H
  Jul 22 at 14:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Now there is no contradiction between our answers anymore.
  – callculus
  Jul 22 at 15:05
  
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

You need to solve the following:
$$max_X,YX^1/3Y^1/2$$
subject to $I geq P_x X + P_y Y.$

Set up Lagrangian
$$L = X^1/3Y^1/2 + lambda (I -(P_x X + P_y Y)).$$

FOC's yield:
begineqnarray*
frac13left(fracY^1/2X^2/3right) - lambda P_x &=&0,\
frac12left(fracX^1/3Y^1/2right) - lambda P_y &=&0,\
I - (P_x X + P_y Y) &=&0.
endeqnarray*
I will leave the algebra for you. You should get the following answer:
$$X= frac2I5 P_x.$$

share|cite|improve this answer

edited Jul 22 at 14:55

answered Jul 22 at 13:53

Green.H

1,046216

You need to solve the following:
$$max_X,YX^1/3Y^1/2$$
subject to $I geq P_x X + P_y Y.$

Set up Lagrangian
$$L = X^1/3Y^1/2 + lambda (I -(P_x X + P_y Y)).$$

FOC's yield:
begineqnarray*
frac13left(fracY^1/2X^2/3right) - lambda P_x &=&0,\
frac12left(fracX^1/3Y^1/2right) - lambda P_y &=&0,\
I - (P_x X + P_y Y) &=&0.
endeqnarray*
I will leave the algebra for you. You should get the following answer:
$$X= frac2I5 P_x.$$

share|cite|improve this answer

edited Jul 22 at 14:55

answered Jul 22 at 13:53

Green.H

1,046216

share|cite|improve this answer

share|cite|improve this answer

edited Jul 22 at 14:55

edited Jul 22 at 14:55

edited Jul 22 at 14:55

answered Jul 22 at 13:53

Green.H

1,046216

answered Jul 22 at 13:53

Green.H

1,046216

answered Jul 22 at 13:53

Green.H

1,046216

1,046216

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  Your production function is wrong. Also pay attention on the signs $L = X^1/3Y^2/3 + lambda (I -(P_x X colorred+ P_y Y)).$
  – callculus
  Jul 22 at 14:08
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @callculus Thanks, edited
  – Green.H
  Jul 22 at 14:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Now there is no contradiction between our answers anymore.
  – callculus
  Jul 22 at 15:05
  
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Your production function is wrong. Also pay attention on the signs $L = X^1/3Y^2/3 + lambda (I -(P_x X colorred+ P_y Y)).$
  – callculus
  Jul 22 at 14:08
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @callculus Thanks, edited
  – Green.H
  Jul 22 at 14:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Now there is no contradiction between our answers anymore.
  – callculus
  Jul 22 at 15:05
  
  

  
  
  
  

Your production function is wrong. Also pay attention on the signs $L = X^1/3Y^2/3 + lambda (I -(P_x X colorred+ P_y Y)).$
– callculus
Jul 22 at 14:08

Your production function is wrong. Also pay attention on the signs $L = X^1/3Y^2/3 + lambda (I -(P_x X colorred+ P_y Y)).$
– callculus
Jul 22 at 14:08

@callculus Thanks, edited
– Green.H
Jul 22 at 14:59

@callculus Thanks, edited
– Green.H
Jul 22 at 14:59

Now there is no contradiction between our answers anymore.
– callculus
Jul 22 at 15:05

Now there is no contradiction between our answers anymore.
– callculus
Jul 22 at 15:05

add a comment | 

 

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