\subsection*{Problem 1}
\begin{enumerate}

\item \
%Question 1
\begin{center}
$50=2L+3L$\\
$50=5L$\\
$L=10$
\end{center}

\item \
%Question 2
\begin{center}
$F=rK$\\
$F=4$
\end{center}


\item \
%Question 3
\begin{center}
$VC=wL$\\
$VC=4*\frac{q}{5}$\\
$VC=\frac{4q}{5}$
\end{center}



\item \
%Question 4
\begin{center}
$TC=FC+VC$\\
$TC=rK+wL$\\
$TC=4+\frac{4q}{5}$
\end{center}

\item \
%Question 5
\begin{center}
$TC=4+\frac{4*(50)}{5}$\\
$TC_{SR}=44$
\end{center}


\item \
%Question 6
\begin{center}
$min \ rK + wL$\\
$min\ 2K+3L$\\
$s.t.\ q=KL+3L$
\end{center}


\item \
%Question 7
\begin{center}
$L=\frac{q}{K+3}$\\
$TC=2K + 4*\frac{q}{k+3}$\\
$=2k+\frac{4q}{K+3}$\\
\end{center}

\begin{center}
$\frac{\partial {TC}}{K}=2-\frac{4q}{(K+3)^2}=0$
\end{center}

\begin{center}
$2(K+3)^2=4q$\\
$K+3 = \sqrt{2q}$\\
$K^*=\sqrt{2q}-3$
\end{center}

\begin{center}
$L=\frac{q}{\sqrt{2q}-3+3}$\\
$L^*=\sqrt{\frac{q}{2}}$
\end{center}


\item \
%Question 8
\begin{center}
$TC_{LR}=rK^*+wL^*$\\
$=2*(\sqrt{2q}-3)+4*\sqrt{\frac{q}{2}}$\\
$=2\sqrt{2q}-6+2\sqrt{2q}$\\
$TC_{LR}=4\sqrt{2q}-6$
\end{center}


\item \
%Question 9
\begin{center}
$K^*=\sqrt{2*50}-3$\\
$K^*=7$
\end{center}

\begin{center}
$L^*=\sqrt{\frac{50}{2}}$\\
$L^*=5$
\end{center}


\item \
%Question 10
\begin{center}
$TC_{LR}=2*7+4*5$\\
$TC_{LR}=34$
\end{center}
In the short run $TC=44$. Alhambra Guitarras is sacrificing $TC_{SR}-TC_{LR}=10$ euros.\\


\item \
%Question 11
\begin{center}
$\frac{\partial {TC}}{K}=2-\frac{16q}{(K+3)^2}=0$
\end{center}

\begin{center}
$2(K+3)^2=16q$\\
$K+3 = \sqrt{8q}$\\
$K^{**}=\sqrt{8q}-3$
\end{center}

\begin{center}
$L=\frac{q}{\sqrt{8q}-3+3}$\\
$L^{**}=\sqrt{\frac{q}{8}}$
\end{center}

\begin{center}
$K^{**}=\sqrt{8*50}-3$\\
$K^{**}=17$
\end{center}

\begin{center}
$L^{**}=\sqrt{\frac{50}{8}}$\\
$L^{**}=2.5$
\end{center}
When $w$ increases there will be more capital than before and less labor than before as labor has become relatively more expensive.\\

\item \
%Question 12
\begin{center}
$TC^*_{SR}=4+\frac{16*(50)}{5}$\\
$TC^*_{SR}=164$
\end{center}

\begin{center}
$TC^*_{LR}=2*(\sqrt{8q}-3)+16*\sqrt{\frac{q}{8}}$\\
$=2\sqrt{8q}-6+4\sqrt{2q}$\\
$=4\sqrt{2q}-6+4\sqrt{2q}$\\
$=8\sqrt{2q}-6$\\
$=8*\sqrt{2*50}-6$\\
$TC^*_{LR}=74$
\end{center}

\begin{center}
$TC_{SR}<TC^*_{SR}$\\
$TC_{LR}<TC^*_{LR}$
\end{center}
As the wage increases both in the short and long run total costs are higher than with the lower wage. Even when wages increase and you decrease the labor and increase the capital, the decrease of labour cannot compensate the extra costs.

\end{enumerate}






\subsection*{Problem 5}

\begin{enumerate}
\item \
%Question 1
\begin{center}
$P_P = 400-2Q$\\
$MR=400-4Q$\\
$MC=120$\\
$MR=MC$\\
$400-4Q=120$\\
$Q^*_P=70$
\end{center}
\begin{center}
    $P_P=400-2*(70)$\\
    $P^*_P=260$
\end{center}

\item \
%Question 2
\begin{center}
$\Pi_P=(P^*_P-MC)*Q^*_P$\\
$=(260-120)*70$\\
$\Pi_P=9,800$
\end{center}


\item \
%Question 3
\begin{center}
$P_i = 200-2Q$\\
$MR=200-4Q$\\
$MC=120$\\
$MR=MC$\\
$200-4Q=120$\\
$Q^*_i=20$
\end{center}
\begin{center}
    $P_i=200-2*(20)$\\
    $P^*_i=160$
\end{center}



\item \
%Question 4
\begin{center}
$\Pi_i=(P^*_i-MC)*Q^*_i$\\
$=(160-120)*20$\\
$\Pi_i=800$
\end{center}


\item \
%Question 5
\begin{center}
$\varepsilon_P=\frac{1}{2}*\frac{260}{70}$\\
$\varepsilon_P=1.86$\\
\end{center}

\begin{center}
$\varepsilon_i=\frac{1}{2}*\frac{160}{20}$\\
$\varepsilon_i=4$\\
\end{center}

Price discrimination is possible by charging passionate customers more as they are less elastic than almost indifferent customers at point of equilibrium.



\item \
%Question 6
\includegraphics[scale=0.65]{Q56.PNG}


\item \
%Question 7
\includegraphics[scale=0.65]{Q57.PNG}

\end{enumerate}

\subsection*{Problem 5 Part 2}
\begin{enumerate}
\item \
%Question 1.2
\begin{center}
$Q^D=Q^D_P + Q^D_i$\\
$q^D_i$ has a limit of $P=200$
\end{center}

\begin{center}
    $\begin{cases}
    $$Q^D = 200 - \frac{P}{2} \ \ \ \ \text{if} \ \  P>200$$\\
    $$Q^D = 300 - P \ \ \ \ \text{if} \ \  P\leq200$$\\
    \end{cases}$\\
\end{center}


\item \
%Question 2.2
\begin{center}
    $\begin{cases}
    $$P = 400 - 2Q \ \ \ \ \text{if} \ \  P < 100$$\\
    $$P = 300 - Q \ \ \ \ \text{if} \ \  P\geq100$$\\
    \end{cases}$\\
\end{center}


\item \
%Question 3.2
\begin{center}
    $\begin{cases}
    $$MR = 400 - 4Q \ \ \ \ \text{if} \ \  P < 100$$\\
    $$MR = 300 - 2Q \ \ \ \ \text{if} \ \  P\geq100$$\\
    \end{cases}$\\
\end{center}


\item \
%Question 4.2
\begin{center}
    $Q<100$\\
    $MR=MC$\\
    $400-4Q=120$\\
    $Q=70$\\
    $P=260$
\end{center}

\begin{center}
    $Q\geq100$\\
    $MR=MC$\\
    $300-2Q=120$\\
    $Q=90$\\
    This contradicts the condition $Q\geq100$, therefore it is not optimal
\end{center}

\begin{center}
    Optimal:\\
    $Q^*=70$\\
    $P^*=260$
\end{center}

\item \
%Question 5.2
$\Pi^*=(260-120)*70$\\
$\Pi^*=9,800$


\item \
%Question 6.2
Bella Italia will sell menus to passionate customers only as the optimal price is too high for almost indifferent customers.


\end{enumerate}