\documentclass{article}
\input{1401-preamble}

\begin{document}
\psetnum{6}
\date{2004/10/29}

\begin{pset}
  \begin{problem}
    \begin{subproblem}
      The inverse demand function is
      \[ p = 10 - Q_D \]

      so the marginal revenue is
      \[ MR = p(Q) + Q\partialder{p}{Q} = (10 - Q) + Q(-1) = 10 - 2Q
      \]

      The firm maximizes profits at the quantity where marginal profit
      equals marginal costs, i.e.
      \[ MR = 10 - 2Q =  2 + Q = MC \]
      \[ Q = \frac{8}{3} \]
      At this quantity, the price is $10 - \frac{8}{3} = \frac{22}{3}
      \approx 7.333$.
    \end{subproblem}

    \begin{subproblem}
      In a competitive market, the marginal cost is equal to the
      inverse demand curve.
      \[ 10 - Q = 2 + Q \]
      \[ Q = 4 \]
      and so $p = 10 - 4 = 6$.

      In a competitive market, the consumer surplus is \[\int_0^{Q_e}
      p(Q_d) - p_e \; dQ_d = \int_0^4 10 - Q - 6 \; dQ = 8\] and the producer
      surplus is \[\int_0^{Q_e} p_e - MC(Q) \; dQ = \int_0^4 6 - (2 + Q)
      \; dQ = 8\] so the welfare is $16$.

      In the monopoly situation, the consumer surplus is \[\int_0^{Q_e}
      p(Q_d) - p_e \; dQ_d = \int_0^{\frac{8}{3}} 10 - Q -
      \frac{22}{3} \; dQ = \frac{32}{9}\] and the producer surplus is
      \[\int_0^{Q_e} p_e - MC(Q) \; dQ = \int_0^{\frac{8}{3}}
      \frac{22}{3} - (2 + Q) \; dQ = \frac{32}{3}\] so the welfare is
      $\frac{128}{9}$.

      So the deadweight loss is $16 - \frac{128}{9} = \frac{16}{9}
      \approx 1.777$.
    \end{subproblem}

    \begin{subproblem}
      To eliminate the deadweight loss, the government must provide a
      sufficient subsidy that the monopolist will be willing to
      produce the same quantity as would be produced in a competitive
      market equilibrium. This is $Q = 4$, $p = 6$.

      The monopolist's marginal revenue is now
      $MR' = MR + \mathscr{S}$, where $\mathscr{S}$ is the amount of
      the subsidy. At the profit-maximizing point, $MR = MC$, so
      \[ 10 - 2Q + \mathscr{S} = 2 + Q \]
      \[ 3 Q = 8 + \mathscr{S} \]
      \[ Q = \frac{8 + \mathscr{S}}{3} \]
      We wish this quantity to be equal to the competitive market
      quantity, so
      \[ \frac{8 + \mathscr{S}}{3} = 4 \]
      \[ \mathscr{S} = 4 \]
      The government must give a subsidy of $4$ per unit.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      \textbf{True}. At the profit-maximizing choice of quantity and
      price, the producer surplus (the monopolist's profit) is
      maximized by definition. Total welfare is not maximized, since
      the consumer surplus is lower than it could be and
      there is a deadweight loss. However, increasing the consumer
      surplus would require decreasing the producer surplus, so this
      choice of price and quantity is Pareto-efficient.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True}. Using the standard definition of social welfare
      as the sum of the consumer and producer surpluses, social
      welfare is the same. In a perfectly competitive market, the
      producer surplus is the area between the price line and the
      supply curve, and the consumer surplus is the area between the
      demand curve and the price line (both to the left of the
      equilibrium point). With a perfectly price-discriminating
      monopolist, the producer surplus is the area between the supply
      and demand curves (left of the equilibrium point). This is the
      same area. But even though the total welfare is the same, it has
      shifted all the surplus from the consumers to the producer,
      which is arguably detrimental to welfare.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True}. The marginal revenue can be written as
      \[ MR = \left(1 + \frac{1}{\epsilon}\right) \]
      For maximum profits, the marginal revenue equals the marginal
      cost, which is zero in this case. So $\frac{1}{\epsilon}  = -1$,
      or $\epsilon = -1$.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The demand curve for workers is

      \includegraphics{ps6-3a1}

      and the demand curve for students is

      \includegraphics{ps6-3a2}
      
     The total quantity demanded is the sum of that demanded by
     workers and students. The workers demand $Q_W = 6 - \nicefrac{1}{2}
     p$, and the students demand $Q_S = 10 - p$. So the total demand
     is \[Q = Q_w + Q_s = 6 - \nicefrac{1}{2} p + 10 - p = 16 -
     \nicefrac{3}{2} p\]

     The demand curve is thus
     \[ p = \frac{2}{3} \left(16 - Q\right) \]
     This is shown below.

     \includegraphics{ps6-3a3}
   \end{subproblem}

   \begin{subproblem}
     If the monopolist cannot distinguish the two types of consumers,
     it must charge them the same price, and demand is given by $Q_d =
     16 - \nicefrac{3}{2} p$. The marginal revenue is
     \[ MR = p(Q) + Q\partialder{p}{Q} = \frac{2}{3} \left(16 -
       Q\right) - \frac{2}{3}Q = \frac{32}{3} - \frac{4Q}{3} \]

     The marginal cost is fixed at $5$, so the profit maximizing
     choice has
     \[\frac{32}{3} - \frac{4Q}{3} = 5\]
     \[Q = \frac{17}{4} = 4.25\]
     \[p = \frac{2}{3} \left(16 - Q\right) = \frac{2}{3} \left(16 -
       \frac{17}{4}\right) = \frac{47}{6} \approx 7.833 \]

     This gives a profit
     \[ \pi = Q p - 5 Q = \frac{17}{4} \frac{47}{6} - 5 \frac{17}{4}=
     \frac{799}{24} - \frac{85}{4} = \frac{289}{24} \approx 12.04 \]

     The quantity consumed by full-time workers is $6 - \frac{1}{2}
     \frac{47}{6} = \frac{25}{12}$. The quantity consumed by students
     is $10 - \frac{47}{6} = \frac{13}{6}$.
     
     The consumer surplus for full-time workers is
     \[\int_0^{Q_e} p(Q_D) - p_e \; d Q_d = \int_0^{\frac{25}{12}} 12 -
     2 Q_D - \frac{47}{6} \; d Q_D = \frac{625}{144} \approx 4.34\]
     The consumer surplus for students is
     \[\int_0^{Q_e} p(Q_D) - p_e \; d Q_d = \int_0^{\frac{13}{6}} 10 -
     Q_D - \frac{47}{6} \; d Q_D = \frac{169}{72} \approx 2.35\]
   \end{subproblem}

   \begin{subproblem}
     The full time workers have higher opportunity cost of time, so
     they will be less likely to spend the time to apply for a
     card. So the advantage of the cards is that the monopolist will
     be able to charge a lower price to students (with the card) than
     to the full-time workers.
   \end{subproblem}

   \begin{subproblem}
     The price can now be set independently for students and full time
     workers. For card holders (students), the marginal revenue is
     \[ MR = p(Q) + Q\partialder{p}{Q} = 10 - Q + -1 Q = 10 - 2Q \]
     \[ MR = 10 - 2Q = 5  = MC \]
     The equilibrium quantity and price are
     \[ Q = 2.5 \]
     \[ p = 10 - Q_D = 7.5 \]
     The consumer surplus for students is
     \[ \int_0^{Q_e} p(Q_D) - p_e \; d Q_d = \int_0^{2.5} 10 - Q_D -
     7.5 = 3.125 \]
     and the profit from sales to students is
     \[ \pi = p Q - 5 Q = 6.25 \]

     For non-card holders (workers), the marginal revenue is
     \[ MR = p(Q) + Q\partialder{p}{Q} = 12 - 2Q + -2 Q = 12 - 4Q \]
     \[ MR = 12 - 4Q = 5 = MC \]
     \[ Q = 1.75 \]
     \[ p = 12 - 2Q_D = 8.5 \]
     The consumer surplus for workers is
     \[ \int_0^{Q_e} p(Q_D) - p_e \; d Q_d = \int_0^{1.75} 12 - 2 Q_D -
     8.5 = 3.0625 \]
     and the profit from sales to workers is
     \[ \pi = p Q - 5 Q = 6.126 \]

     So the total profit is $6.25 + 6.125 = 12.375$.
   \end{subproblem}

   \begin{subproblem}
     Star Market makes a greater profit with the card (12.375 vs
     12.04), so they are better off. The total consumer surplus
     without the cards is $4.34 + 2.35 = 6.69$, and with the cards it
     is $3.125 + 3.0625 = 6.1875$, which is less, so consumers are
     worse off.
   \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The inverse demand function is $p= 30 - Q$, and the marginal
      revenue for the monopolist is
      \[ MR = p(Q) + Q\partialder{p}{Q} = 30 - Q + -1 Q = 30 - 2Q \]
      The marginal cost is
      \[ MC = \partialder{C}{q} = q \]
      so
      \[ 30 - 2Q = Q\]
      \[ Q = 10\]
      \[ p = 30 - Q = 20 \]
      This is the profit-maximizing monopoly price. The profit to the
      monopolist is
      \[ \pi = p Q - C(q) = 200 - \frac{10^2}{2} = 150\]
    \end{subproblem}

    \begin{subproblem}
      The socially optimal price has $MC = p$, so
      \[ Q = 30 - Q \]
      \[ Q = 15\]
      \[ p = 30 - Q = 15\]
      The profit to the monopolist (or producer surplus) is
      \[ \pi = p Q - C(q) = 225 - \frac{15^2}{2} = 112.5 \]
      The consumer surplus at the socially optimal price is
      \[ \int_0^{Q_e} p(Q_D) - p_e \; d Q_d = \int_0^{15} 30 - Q_D -
      15 = 112.5 \]
      So the social welfare is 225.

      At the monopoly price the consumer surplus is
      \[ \int_0^{Q_e} p(Q_D) - p_e \; d Q_d = \int_0^{10} 30 - Q_D -
      20 = 50 \]
      So the social welfare in the monopoly situation is 200. The
      deadweight loss is 25.
    \end{subproblem}

    \begin{subproblem}
      Suppose a price ceiling is set at $p = 18$.  The demand at this
      price is $30 - 18 = 12$. So the marginal revenue is as before
      for quantities above 12, and fixed at 18 for quantities
      below 12:
      \[ MR =
      \begin{cases}
        30 - 2Q & Q > 12 \\
        18 & Q < 12 \\
      \end{cases}
      \]

      The monopoly will maximize its profits by setting the price
      equal to the price ceiling. Then $p = 18$ and $Q = 12$. The
      profit of the monopolist is $pQ- C(Q) = 18(12) - \frac{12^2}{2}
      = 144$. The consumer surplus is
      \[ \int_0^{Q_e} p(Q_D) - p_e \; d Q_d = \int_0^{12} 30 - Q_D -
      18 = 72 \]
      So the total welfare is 216, which gives a deadweight loss of 9.
    \end{subproblem}

    \begin{subproblem}
      To maximize total welfare, the government should set a price
      ceiling at the competitive price, 15. Then the monopolist will
      maximize profits by producing 15 units, at a profit of
      112.5. This is the same outcome as in a competitive market, so
      the consumer and producer surpluses are both 112.5 (from part
      b), and there is no deadweight loss.
    \end{subproblem}

    \begin{subproblem}
      The business fee is a fixed cost, as it is independent of the
      quantity produced. So the marginal cost and marginal revenue are
      the same, and the profit maximizing outcome for the monopoly
      will be the same as in part a: $q = 10, p = 20$. Before the
      business fee, the profit is $150$; the business fee reduces the
      profit to $20$. But this is still a positive profit so the
      monopolist will not shut down.

      With the price ceiling from part d, the profit is 112.5. This is
      positive, so the monopoly will choose to stay in business.
      \footnote{I'm not entirely clear on what the problem was asking
        here. I'm assuming 4e is asking whether the monopoly will stay
        in business if the price ceiling is imposed with no business
        fee, and 4f is asking whether it'll stay in business with both
        the price ceiling and the business fee.}
    \end{subproblem}

    \begin{subproblem}
      With the price ceiling from part d, the profit is 112.5. This is
      less than 130, so when the business fee is subtracted, the
      monopolist will be losing money. So it will choose to shut down
      and avoid the business fee instead.      
    \end{subproblem}
  \end{problem}
\end{pset}

\end{document}
