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\begin{document}

\title{\textbf{4113: Math Econ\\
Midterm}}%
\author{Instructor: Sasha Vostroknutov}%
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\textbf{YOUR NAME:...................................................................}

\vspace{1in}

\center{
\textbf{YOU HAVE 1 HOUR 15 MINUTES. THERE ARE 6 PAGES AND 2 QUESTIONS.}

\vspace{0.5in}
\textbf{DO NOT FORGET TO CHECK CONSTRAINT QUALIFICATION.}

\vspace{0.5in}
\textbf{THERE IS A LIST OF PROPERTIES OF PREFERENCE RELATIONS ON THE LAST PAGE.}


\vspace{0.5in}
\textbf{THE MAXIMIZATION PROBLEM DOES HAVE SIMPLE SOLUTION.\\
IF YOUR FIRST ORDER CONDITIONS GET TOO COMPLICATED THINK!}


\vspace{0.5in}
\textbf{GOOD LUCK!}}


\break
\begin{enumerate}
\item Solve the following problem
\begin{eqnarray*}
\max_{x,y,z\in\Real} &&e^zxy+x+y\\
\mathrm{s.t.}&& x+y+e^z \leq 2\\
&& z(x+3y)\geq 0\\
&& x,y,z\geq 0
\end{eqnarray*}

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\phantom{qwerty}
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\item Consider the preference relation $\gtrdot$ over $\Real^2$ defined like this:
$$(x,y)\gtrdot (x',y')\ \mathrm{if\ and\ only\ if}\ x > x'\ \mathrm{and}\ y > y'$$
\begin{description}
\item[a)] Is $\gtrdot$ complete? Prove your claim.
\vspace{1.5in}
\item[b)] Is $\gtrdot$ transitive? Prove your claim.
\vspace{1.5in}
\item[c)] Is $\gtrdot$ reflexive or irreflexive? Prove your claim.
\vspace{1.5in}
\item[d)] Is $\gtrdot$ symmetric or asymmetric? Prove your claim.
\end{description}

\break
Suppose we have a binary relation $R$ (you can use any other symbol, for example $\succsim$) on the set of alternatives $X$. Remember that $R\subseteq X^2$ and for $x,y\in X$ we say that $xRy$ is true if $(x,y)\in R$. Here are some definitions. You should read the statements below as follows: $R$ is \textbf{reflexive} if for all $x\in X$ it is true that $xRx$.

\begin{itemize}
\item[] \textbf{reflexive} if $xRx$
\item[] \textbf{irreflexive} if $\neg (xRx)$
\item[] \textbf{transitive} if $xRy\ and\ yRz\Rightarrow xRz$
\item[] \textbf{complete} if $xRy\ or\ yRx\ or\ both$
\item[] \textbf{symmetric} if $xRy\Rightarrow yRx$
\item[] \textbf{asymmetric} if $xRy\Rightarrow \neg yRx$
\end{itemize}

\end{enumerate}

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