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\begin{document}

\textbf{ECON 4113. OPTIONAL HOMEWORK PART 1. DUE APRIL 27.}

Solve the following problems by all means necessary. If you use Lagrange Theorem do the Constraint Qualification properly. There is no need to check second order conditions.

\begin{enumerate}
\item 
\begin{eqnarray*}
\max_{x,y,z\in\Real} &&zx+y^2\\
\mathrm{s.t.}&& (y^2-1+2zx)^4\leq 0\\
&& x,y,z\geq 0
\end{eqnarray*}
\item
\begin{eqnarray*}
\max_{x,y\in\Real} &&9y+2xy-2x^2-2y^2\\
\mathrm{s.t.}&& 4x+3y\leq 10\\
&& y-4x^2\geq -2\\
&& x,y\geq 0
\end{eqnarray*}
\item
\begin{eqnarray*}
\max_{x,y\in\Real} &&x\\
\mathrm{s.t.}&& y-x^4\leq 0\\
&& x^3-y\leq 0\\
&& x\leq \frac12
\end{eqnarray*}
\item Consider the problem of maximizing the utility of a consumer in an exchange economy, keeping the utility of the other consumer no less than some fixed level $\bar u_2$:
\begin{eqnarray*}
\max_{x_1,y_1,x_2,y_2\in\Real} &&u_1(x_1,y_1)\\
\mathrm{s.t.}&& u_2(x_2,y_2)\geq \bar u_2\\
&& x_1+x_2\leq w_x\\
&& y_1+y_2\leq w_y\\
&& x_1,y_1,x_2,y_2\geq 0
\end{eqnarray*}
Suppose that $u_1$ and $u_2$ are strictly increasing in all arguments. Show that at the solution point $(x_1^*,y_1^*,x_2^*,y_2^*)$ the first three constraints bind. Suppose for simplicity that $x_1^*,y_1^*,x_2^*,y_2^*>0$. Find the conditions that $x_1^*,y_1^*,x_2^*,y_2^*$ should satisfy.
\end{enumerate}
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