PS4 - Problem 4

    \subsection*{Problem 4}
        \begin{enumerate}
            \item \
                \begin{center}
                $Q = 132 - P$\\
                thus;
                $P = 132 - Q$\\
                \end{center}

            \item \
                \begin{center}
                    $R_{(Q)} = P_{(Q)} * Q$\\
                    $MR_{(Q)} = \frac{dR_{(Q)}}{dQ} = P_{(Q)} + Q * \frac{dP_{(Q)}}{dQ}$\\
                    $R_{(Q)} = 132 - 2Q$
                \end{center}
            \item \
                \begin{center}
                    $MC_{(Q)} = \frac{dTC_{(Q)}}{dQ} = \frac{d(Q^2)}{dQ} = 2Q$
                \end{center}
            \item Bella Italia maximizes its profit when MR = MC. Thus;
                \begin{center}
                    $2Q = 132 - 2Q$\\
                    Therefore: $Q^* = 33$ and $P^* = 99$
                \end{center}
            \item \
            \begin{center}
                Profit = Revenue - Cost. Thus,\\
                $\pi = P^* * Q^* - AC_{(Q^*)}  * Q^*$\\
                $AC_{(Q^*)} = \frac{TC_{(Q^*)}}{Q^*} = 33$\\
                So, $\pi = 2178$.
            \end{center}
            \item Graph 4.6
            \item Consumer's Surplus will be:
                \begin{center}
                    $CS = Q^* * (\frac{1}{2}P_{(Q=0)} - P^*)$\\
                    $= 33 *\frac{1}{2} (132-99)$\\
                    So, $CS = 544.5$
                \end{center}
            \item Bella Italia's surplus will be:
                \begin{center}
                    $PS = P^* * Q^* - (\frac{1}{2} * Q^* * MR_{(Q^*)}$\\
                    $= 33 * 99 - (\frac{1}{2} * 33 * 66)$\\
                    So, $PS = 2178$
                \end{center}
            \item Total surplus = CS + PS. So,
                \begin{center}
                    $CS + PS = 2178 + 544.5 =$\\
                    $Total Surplus = 2722.5$
                \end{center}
            \item \
                \begin{center}
                    $MR = P = MC = \frac{dQ^2}{dQ} = 2Q$\\
                    So, $P_E = 2Q_E$ and $Q=132 - P$\\
                    Therefor, $P_E = 88$ and $Q_E = 44$
                \end{center}
            \item Quantity has been increased and price has been decreased comparing to the initial monopoly.
            \item Profits = Revenue - Costs. So,
                \begin{center}
                    $\pi = P_E * Q_E - {Q_E}^2 = 1936$
                \end{center}
            \item Graph 4.13
            \item \
                \begin{center}
                    $CS_1 = \frac{(Q^{**})*(132 - P^{**})}{2} = 968$
                \end{center}
            \item \
                \begin{center}
                    $PS_1 = \frac{(Q^{**})*(P^{**})}{2} = 1936$
                \end{center}
            \item \
                \begin{center}
                    $TS_1 = CS_1 + PS_1 = 2904$
                \end{center}
            \item \
                \begin{center}
                    $DL = TS_1 - TS = 2904 - 2722.5 = 181.5$
                \end{center}
            \item Graph 4.18
            \item Graph 4.19
            \item Let $P^c$ be Price ceiling. Then,
                \begin{center}
                    $MC = P^c$\\
                    Therefore, $P^c = 20$\\
                    So, $Q^{***} = \frac{P^c}{2}=32$ and $P^{***} = P^c = 64$
                \end{center}
            \item Graph 4.21
            \item \
                \begin{center}
                    $\pi_{(Q)} = R_{(Q)} - C_{(Q)}$\\
                    $\pi_{(Q)} = (P^{***}*Q^{***}) - (Q^{***})^2 = 1024$
                \end{center}
            \item There is an excess demand and the quantity demanded at that price is greater than the equilibrium quantity. Therefore not everyone can have that product.
            \item Let A be the correspondent point in the demand function to $Q^{***}$. So, $A = (P^{****},Q^{***})$\\
                \begin{center}
                    $P^{****} = 132 - Q^{***} = 100$\\
                    $CS_2 = \frac{(132-P^{****})(Q^{***})}{2} + (P^{****}-P^{***})(Q^{***}) = 1664$
                \end{center}
            \item \
                \begin{center}
                    $PS_2 = \frac{(P^{***})(Q^{***})}{2} = 1024$
                \end{center}
            \item \
                \begin{center}
                    $TS_2 = CS_2 + PS_2 = 2688$
                \end{center}
            \item Yes, because there is a DL since $P^c = P^{***}$ and it's not equal to the price corresponding to the optimal choice of perfect competition.
            \item $TS_1$ > $TS$ > $TS_2$. Therefore, TS with $P^c$ is the maximum, while $TS_1$ with monopoly is greater than $TS_2$ of perfect competition with the price ceiling. The DL between monopoly and perfect competition without price ceiling is less than the DL between perfect competition with price ceiling and perfect competition without price ceiling.
            \item Consumers prefer perfect competiotion with price ceiling since the conxumer surplus in this case is the greatest. the last preference is monopoly since most of CS is charged as PS or $CS_2$ > $CS_1$ > $CS$.
            \item Restaurants prefer monopoly since they can maximize PS in this case. The last preference is perfect competition with price ceiling since the consumers are much protected or we can say $PS$ > $PS_1$ > $PS_2$
            \item The best scenario for total welfare (TS) is perfect competition without price ceiling since $TS_1$ > $TS$ > $TS_2$. Perfect competition without price ceiling maximizes welfare and there are no losses (DL = 0).
        \end{enumerate}