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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.

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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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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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