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

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?

callculus42
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Kolten
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  • What is the demand equation? – saulspatz Jul 22 '18 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 '18 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 https://economics.stackexchange.com/ – saulspatz Jul 22 '18 at 13:49
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    You may want to include what you have tried in order to improve your chances of getting an answer – Green.H Jul 22 '18 at 13:54

2 Answers2

1

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

$$\mathcal L=X^{1/3}\cdot Y^{1/2}+\lambda (I-P_x\cdot X-P_y\cdot Y)$$

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

$\frac{\partial \mathcal L}{\partial X}=\frac13\cdot X^{-2/3}\cdot Y^{1/2}-P_X\lambda=0 \Rightarrow\frac13\cdot X^{-2/3}\cdot Y^{1/2}=P_x\lambda \quad (1)$

$\frac{\partial \mathcal L}{\partial Y}=\frac12\cdot X^{1/3}\cdot Y^{-1/2}-P_y\lambda=0 \Rightarrow\frac12\cdot X^{1/3}\cdot Y^{-1/2}=P_y\lambda \quad (2)$

$\frac{\partial \mathcal L}{\partial \lambda}=I-P_x\cdot X-P_y\cdot Y=0 \quad (3)$

Dividing (1) by (2)

$\frac23\frac{Y}{X}=\frac{P_x}{P_y}$

Solving for $P_y\cdot Y$

$P_y\cdot Y=\frac32P_x\cdot X$. The expression can be insert in (3).

$I-P_x\cdot X-\frac32P_x\cdot X=0$

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

callculus42
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0

You need to solve the following: $$\max_{X,Y}X^{1/3}Y^{1/2}$$ subject to $I \geq P_x X + P_y Y.$

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

FOC's yield: \begin{eqnarray*} \frac{1}{3}\left(\frac{Y^{1/2}}{X^{2/3}}\right) - \lambda P_x &=&0,\\ \frac{1}{2}\left(\frac{X^{1/3}}{Y^{1/2}}\right) - \lambda P_y &=&0,\\ I - (P_x X + P_y Y) &=&0. \end{eqnarray*} I will leave the algebra for you. You should get the following answer: $$X= \frac{2I}{5 P_x}.$$

Green.H
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