PS3.tex
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\begin{document}
\title{Problem Set 3}
\author{}
\maketitle
\be
\item Find the partial derivative, second partial derivatives, and cross
partial derivative of each function.
\begin{enumerate}
\item $f\left( x_{1},x_{2}\right) =12x_{1}^{4}-6x_{1}^{2}x_{2}+4x_{2}^{3}$
\item $f\left( x_{1},x_{2}\right) =\left( 3x_{1}^{2}+5x_{1}+1\right) \left(
x_{2}+4\right) $
\end{enumerate}
\item Find $\frac{dy}{dz}$ .
\begin{enumerate}
\item $y=f\left( x,w\right) =3x^{2}-2xw+w^{2}$ where $x=8z-18$ and $w=4z$
\item $y=f\left( x,z\right) =4x^{3}+\frac{1}{4}xz^{2}-2z$ where $x=z^{-2}$
\end{enumerate}
\item Find $\frac{\partial Z}{\partial u}$ and $\frac{\partial Z}{\partial v%
}$
\begin{enumerate}
\item $Z=f\left( x,y\right) =4x^{2}+2xy+y^{2}$ where $x=3u^{2}$ and $y=u-2v$
\item $Z=f\left( x,y\right) =ax^{3}-bx^{2}y+cy$ where $x=\gamma u+\theta v$
and $y=\theta u-\gamma v^{2}$
\end{enumerate}
\item Consider the Cobb-Douglas production function, $y=L^{\alpha
}K^{1-\alpha }$ where $y=$output, $L=$labor, $K=$capital, and $0<\alpha <1.$
\begin{enumerate}
\item Calculate $MP_{L}$ and $MP_{K}$
\item Find the rate of change of each due to changes in labor and \ capital.
\item Show that $f_{L}L+f_{K}K=y$
\end{enumerate}
\item Consider $U=x_{1}^{1/3}x_{2}^{2/3}.$ The demand curves associated
with this utility function are $x_{1}=\frac{M}{3p_{1}}$ and $x_{2}=\frac{2M}{%
3p_{2}}.$ Find the rate of change of $U$ with respect to changes in each
price and money income. Interpret.
\item Consider the production function $Q=K^{3/4}L^{3/4}$. Show that marginal
productivity of each factor is diminshing. Show, however, that for any
strictly positive input combination, if the input combination is doubled, then
output more than doubles.
\item Consider the following three utility functions: $U=x_{1}x_{2}$, $%
V=x_{1}^{2}x_{2}^{2}$, $W=\log x_{1}+\log x_{2}$
\begin{enumerate}
\item Find the MU of $x_{1}$ and $x_{2}$ for each utility function
\item Find the rates of change of MU of one good with respect to a change
in consumption of the other good for each utility function. Verify that, for
these functions, the change in the MU\ of one good due to a change in the
other good is the same, no matter which good is chosen first.
\item Find the MRS of $x_{1}$ for $x_{2}$ for each utility function
\item What do (b) and (c) imply for economic analysis?
\end{enumerate}
\item Let $U=f\left( x_{1},x_{2}\right) $ be a utility function and let $%
V\left( x_{1},x_{2}\right) =F\left( U\right) ,$ where $F^{\prime }\left(
U\right) >0$
\begin{enumerate}
\item Show that $\frac{V_{1}}{V_{2}}=\frac{U_{1}}{U_{2}}$
\item Find $V_{ij}$ in terms of $U_{ij}$ $i,j=1,2$. Show that in general $%
U_{ij}$ and $V_{ij}$ need not have the same sign.
\end{enumerate}
\ee
\end{document}
