<p>You're right in that <span class="math-container">$\frac{1+y}{p_x}=\frac{x}{p_y}$</span>. This implies that <span class="math-container">$yp_y=xp_x-p_y$</span>, and using this in the budget constraint <span class="math-container">$I=xp_x+yp_y$</span> yields
<span class="math-container">$ I =xp_x+xp_x-p_y=2xp_x-p_y$</span> or</p>

<p><span class="math-container">$$x=\frac{I+p_y}{2p_x}.$$</span></p>

<p>In order to determine whether these goods are gross substitutes (or complements), you need to compute <span class="math-container">$\frac{\partial x}{\partial p_y}$</span> and observe that this is </p>

<p><span class="math-container">$$\frac{\partial x}{\partial p_y}=\frac{1}{2p_x}&gt;0.$$</span></p>

<p>This means that increases in the price of <span class="math-container">$y$</span> are associated with increases in the consumption of <span class="math-container">$x$</span>, a behaviour that corresponds to gross substitutes.</p>

<p>As for whether <span class="math-container">$x$</span> is a normal good, you need to determine if increases in income are associated with increases or decreases in its consumption, so we calculate</p>

<p><span class="math-container">$$\frac{\partial x}{\partial I}=\frac{1}{2p_x}&gt;0.$$</span></p>

<p>As consumption of <span class="math-container">$x$</span> increases with <span class="math-container">$I$</span>, we say that <span class="math-container">$x$</span> is a normal good (has positive income elasticity).</p>

CC BY-SA 4.0
2019-02-06T10:16:19.243Z
2019-02-06T10:16:19.243Z
Patricio
<p>Utility function: U=X+XY and I=XPx+YPy
Good Y is a composite good, so Py= 1</p>

<p>What would be the demand function for good X?</p>

<p>I got Mux=1+y and Muy=x and since Mux/Px=Muy/Py</p>

<p>1+y/Px=x/py</p>

<p>so that I=XPx+(Xpx-1)Py---> X=(I+Py)/(Px(1+Py)</p>

<p>Is this correct? Or am I supposed to set Py= to 1 in the calculations so that it would instead be X= (I+1)/2Px?</p>

<p>Also, would it be correct to say that these are substitutes? And good X is a normal good?</p>

<microeconomics>
Microeconomics, deriving demand equations from utility unction
Jason
# Microeconomics, deriving demand equations from utility unction

(Asked by: username_0 on 2019-02-06)

Utility function: U=X+XY and I=XPx+YPy
Good Y is a composite good, so Py= 1

What would be the demand function for good X?

I got Mux=1+y and Muy=x and since Mux/Px=Muy/Py

1+y/Px=x/py

so that I=XPx+(Xpx-1)Py---> X=(I+Py)/(Px(1+Py)

Is this correct? Or am I supposed to set Py= to 1 in the calculations so that it would instead be X= (I+1)/2Px?

Also, would it be correct to say that these are substitutes? And good X is a normal good?

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2019-02-06. Score: 1

You're right in that $\frac{1+y}{p\_x}=\frac{x}{p\_y}$. This implies that $yp\_y=xp\_x-p\_y$, and using this in the budget constraint $I=xp\_x+yp\_y$ yields
$ I =xp\_x+xp\_x-p\_y=2xp\_x-p\_y$ or

$$x=\frac{I+p\_y}{2p\_x}.$$

In order to determine whether these goods are gross substitutes (or complements), you need to compute $\frac{\partial x}{\partial p\_y}$ and observe that this is

$$\frac{\partial x}{\partial p\_y}=\frac{1}{2p\_x}>0.$$

This means that increases in the price of $y$ are associated with increases in the consumption of $x$, a behaviour that corresponds to gross substitutes.

As for whether $x$ is a normal good, you need to determine if increases in income are associated with increases or decreases in its consumption, so we calculate

$$\frac{\partial x}{\partial I}=\frac{1}{2p\_x}>0.$$

As consumption of $x$ increases with $I$, we say that $x$ is a normal good (has positive income elasticity).


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