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





















  • 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














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





















  • 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












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













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












share|cite|improve this question




share|cite|improve this question








edited Jul 22 at 13:28









callculus

16.4k31427




16.4k31427









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
















  • 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










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





















  • 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


















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























  • 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











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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





















  • 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















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





















  • 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













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













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













share|cite|improve this answer



share|cite|improve this answer











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

















  • 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











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























  • 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















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























  • 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













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















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















share|cite|improve this answer



share|cite|improve this answer








edited Jul 22 at 14:55


























answered Jul 22 at 13:53









Green.H

1,046216




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

















  • 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













 

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