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\begin{document}

\textbf{ECON 4113. OPTIONAL HOMEWORK PART 2. DUE MAY 2.}

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. Hint: graph the functions, this helps a lot.

\begin{enumerate}
\item For each value of the parameter $q\in [-1,1]$ solve the following problem
\begin{eqnarray*}
\max_{p\in\Real} &&qp^2-p\\
\mathrm{s.t.}&& 0\leq p\leq 1
\end{eqnarray*}
\item
\begin{eqnarray*}
\max_{x,y\in\Real} &&-x^2-2y^2-xy-2x\\
\mathrm{s.t.}&& x^2+y^2\leq 4\\
&& x+y\leq 2\\
&& x,y\geq -2
\end{eqnarray*}
\item
\begin{eqnarray*}
\max_{x,y\in\Real} &&e^{-(x+y)}\sin x\\
\mathrm{s.t.}&& y = |x|
\end{eqnarray*}
\item Dixit, Problem 3.2
\item
\begin{eqnarray*}
\max_{x,y\in\Real} &&x\\
\mathrm{s.t.}&& x^3+y^2=0
\end{eqnarray*}
\end{enumerate}
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