<p>First, we want to find the optimal good baskets, <em>ie</em> how much to buy of goods $x$,$y$ and $z$ to get the maximum value of U out of it.</p>

<p>Let's write down the $MRS$ for $x$ and $y$ against $z$. For this we need to differentiate the utility function for each of the variables:
$d_xU(x,y,z)=a/x$ , $d_yU(x,y,z)=b/y$,$d_zU(z,y,z)=1$. We hence deduce the rates of substitution to be $-1a/x$ and $-b/y$ respectively.</p>

<p>So we have:
$p_x/p_z=a/x$ and $p_y/p_z=b/y$</p>

<p>The budget constraint is $p_xx+p_yy+p_zz=I$ which we rewrite
$p_x(ap_z/p_x)+p_y(bp_z/p_y)+p_zz=I$, which simplfies in
$ap_z+bp_z+zp_z=I$</p>

<p>$z^*=(I/p_z)-a-b$</p>

<p>$(x^*,y^*,z^*)=(ap_z/p_x, bp_z/p_y,(I/p_z)-a-b)$</p>

<p>Good news, you were right :).</p>

<p>The cross price effect is the rise of demand in good $x$ following a rise in the price of good $z$. Just calculate the difference between $x^*$ for $p_z$ and $x^*$ for $p_z'=p_z+\epsilon$ (using $x^*=ap_z/p_x$).</p>

<p>Finally, the six curves asked are how demand change when price change holding income ($I$) constant, or utility ($U$) constant. Draw same by taking two prices fixed for each situation.</p>

CC BY-SA 3.0
2015-05-15T23:50:34.020Z
2015-05-15T23:50:34.020Z
VicAche
<p>Hi don't forget that there is a possibility of corner solution as consumer's problem is:
\begin{matrix}
max_{x,y,z}U(x,y,z)=aln(x)+bln(y)+z &amp;\text{subject to}  &amp; xp_x+yp_y+zp_z=I, &amp; x,y&gt;0, &amp; z\geq 0 
\end{matrix}.
With FOCs,
\begin{matrix}
\frac{ay}{bx}=\frac{p_x}{p_y}, &amp;\frac{a}{x}=\frac{p_x}{p_z},  &amp;\frac{b}{y} =\frac{p_y}{p_z}.
\end{matrix}
Now this condition is valid iff $$\frac{I}{p_z}\geq (a+b)$$ otherwise the consumer"s problem reduces to optimal choice between good <strong>x</strong> and good <strong>y</strong> i.e.
 \begin{matrix} max_{x,y}U(x,y)=aln(x)+bln(y) &amp;\text{subject to}  &amp; xp_x+yp_y=I, &amp; x,y&gt;0 \end{matrix}.
With FOCs,
$$\frac{ay}{bx}=\frac{p_x}{p_y}$$
By the above info the demand function (Uncompensated demand function) of this consumer is:
$$(x,y,z)(p_x,p_y,p_z,I)=\begin{cases}
\left ( \frac{ap_z}{p_x},\frac{bp_z}{p_y},\frac{I}{p_z}-a-b \right ), &amp; \text{ if } \frac{I}{p_z}\geq (a+b),  \\ 
 \left ( \frac{Ia}{(a+b)p_x},\frac{Ib}{(a+b)p_y},0 \right )&amp; \text{ otherwise. } 
\end{cases}$$</p>

CC BY-SA 3.0
2015-05-21T12:59:52.327Z
2015-05-21T12:59:52.327Z
Dreyfus
<p>I'm working on a problem set for my intermediate microeconomics course, but I'm having trouble deriving the compensated and uncompensated demand functions. This is the utility function:</p>

<p>$U(x, y, z) = aln(x) + bln(y) + z$, with goods $x$, $y$, $z$ and income $I$. I found the following optimal values:</p>

<p>$x = (P_z/P_x)a$, $y = (P_z/P_y)b$, $z = (I/P_z) - a - b$</p>

<p>Even though I followed the steps, I'm not completely sure this is right. Moreover, I have to find cross price effects and both the compensated and uncompensated demand functions, but I'm having some serious trouble solving this problem.</p>

<microeconomics><consumer-theory>
Struggling with uncompensated/compensated demand
Econ123
# Struggling with uncompensated/compensated demand

(Asked by: username_0 on 2015-05-14)

I'm working on a problem set for my intermediate microeconomics course, but I'm having trouble deriving the compensated and uncompensated demand functions. This is the utility function:

$U(x, y, z) = aln(x) + bln(y) + z$, with goods $x$, $y$, $z$ and income $I$. I found the following optimal values:

$x = (P\_z/P\_x)a$, $y = (P\_z/P\_y)b$, $z = (I/P\_z) - a - b$

Even though I followed the steps, I'm not completely sure this is right. Moreover, I have to find cross price effects and both the compensated and uncompensated demand functions, but I'm having some serious trouble solving this problem.

**Tags:** microeconomics, consumer-theory


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## 2 answers

### Answer by username_1 on 2015-05-15. Score: 1

First, we want to find the optimal good baskets, *ie* how much to buy of goods $x$,$y$ and $z$ to get the maximum value of U out of it.

Let's write down the $MRS$ for $x$ and $y$ against $z$. For this we need to differentiate the utility function for each of the variables:
$d\_xU(x,y,z)=a/x$ , $d\_yU(x,y,z)=b/y$,$d\_zU(z,y,z)=1$. We hence deduce the rates of substitution to be $-1a/x$ and $-b/y$ respectively.

So we have:
$p\_x/p\_z=a/x$ and $p\_y/p\_z=b/y$

The budget constraint is $p\_xx+p\_yy+p\_zz=I$ which we rewrite
$p\_x(ap\_z/p\_x)+p\_y(bp\_z/p\_y)+p\_zz=I$, which simplfies in
$ap\_z+bp\_z+zp\_z=I$

$z^\*=(I/p\_z)-a-b$

$(x^\*,y^\*,z^\*)=(ap\_z/p\_x, bp\_z/p\_y,(I/p\_z)-a-b)$

Good news, you were right :).

The cross price effect is the rise of demand in good $x$ following a rise in the price of good $z$. Just calculate the difference between $x^\*$ for $p\_z$ and $x^\*$ for $p\_z'=p\_z+\epsilon$ (using $x^\*=ap\_z/p\_x$).

Finally, the six curves asked are how demand change when price change holding income ($I$) constant, or utility ($U$) constant. Draw same by taking two prices fixed for each situation.


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### Answer by username_2 on 2015-05-21. Score: 0

Hi don't forget that there is a possibility of corner solution as consumer's problem is:
\begin{matrix}
max\_{x,y,z}U(x,y,z)=aln(x)+bln(y)+z &\text{subject to} & xp\_x+yp\_y+zp\_z=I, & x,y>0, & z\geq 0
\end{matrix}.
With FOCs,
\begin{matrix}
\frac{ay}{bx}=\frac{p\_x}{p\_y}, &\frac{a}{x}=\frac{p\_x}{p\_z}, &\frac{b}{y} =\frac{p\_y}{p\_z}.
\end{matrix}
Now this condition is valid iff $$\frac{I}{p\_z}\geq (a+b)$$ otherwise the consumer"s problem reduces to optimal choice between good **x** and good **y** i.e.
\begin{matrix} max\_{x,y}U(x,y)=aln(x)+bln(y) &\text{subject to} & xp\_x+yp\_y=I, & x,y>0 \end{matrix}.
With FOCs,
$$\frac{ay}{bx}=\frac{p\_x}{p\_y}$$
By the above info the demand function (Uncompensated demand function) of this consumer is:
$$(x,y,z)(p\_x,p\_y,p\_z,I)=\begin{cases}
\left ( \frac{ap\_z}{p\_x},\frac{bp\_z}{p\_y},\frac{I}{p\_z}-a-b \right ), & \text{ if } \frac{I}{p\_z}\geq (a+b), \\
\left ( \frac{Ia}{(a+b)p\_x},\frac{Ib}{(a+b)p\_y},0 \right )& \text{ otherwise. }
\end{cases}$$


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