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

\begin{document}
\psetnum{5}
\date{2004/10/22}

\begin{pset}
  \begin{problem}
    \begin{subproblem}
      \textbf{False.}
      The short run supply curve is equal to the marginal cost curve
      (truncated below the minimum of the average variable cost). The
      marginal cost is the derivative of the total cost, so the fixed
      costs do not affect it, or the supply curve. In a perfectly
      competitive market, the individual firms cannot set the
      price. So the price is determined by the intersection of the
      demand curve with the industry supply curve, which is the sum of
      the individual firms' supply curves. These do not depend
      on fixed costs, so the price charged by a firm will not depend
      on the fixed costs.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True.} If the firm's average cost exceeds its minimum
      long run average cost, then it can change its fixed costs to
      decrease its marginal costs. So the firm will be able to produce
      more output for the same cost in the long run, and assuming the
      market price remains constant, will choose to increase its
      output.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True.} The consumer surplus is the difference between
      what consumers are willing to pay for a good and its cost. If
      demand is perfectly inelastic, then consumers are willing to pay
      infinite amounts for a good. Thus, regardless of the actual cost
      of the good (equilibrium cost), consumer surplus is infinite.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} Consider the following diagram:
      
      \includegraphics[scale=1.5]{ps5-1d}

      The initial price is $p_1$. The consumer surplus for the more
      elastic demand
      curve is $B+C$, and the consumer surplus for the less elastic
      demand curve is $A+B+C+D$. Then the price rises to $p_2$. The consumer
      surplus for the more elastic demand curve is now $B$, a decrease
      of $C$. The consumer surplus for the less elastic demand curve
      is now $A+B$, a decrease of $C+D$. So the less elastic demand
      curve caused a larger consumer surplus loss.
    \end{subproblem}
  \end{problem}
  
  \begin{problem}
    \begin{subproblem}
      \[MRTS = -\frac{MP_L}{MP_K} =
      -\frac{\frac{K^{\nicefrac{1}{2}}}{2L^{\nicefrac{1}{2}}}}{\frac{L^{\nicefrac{1}{2}}}{2K^{\nicefrac{1}{2}}}}
      = -\frac{K}{L} \]
      To minimize costs, the firm will choose a point where $MRTS =
      -\frac{w}{r}$
      \begin{align*}
        -\frac{K}{L} &= -\frac{w}{r} = -1 \\
        K &= L \\
      \end{align*}
      So $q = K = L = 10000$.
    \end{subproblem}

    \begin{subproblem}
      The firm's capital is fixed in the short run, so we can simplify
      the production function
      \[ q = \sqrt{10000 L} = 100 \sqrt{L} \]
      and thus
      \[ L = \left(\frac{q}{100}\right)^2 = \frac{q^2}{10000}\]
      The cost function is
      \[ C(q) = K r + w L(q) = 100000 + 10 \frac{q^2}{10000} = 100000
      + \frac{q^2}{1000} \]
      This is the total cost function.

      The fixed cost is
      \[FC = 100000\]

      The variable cost function is
      \[VC(q) = \frac{q^2}{1000}\]

      The average cost function is
      \[AC(q) = \frac{100000 + \frac{q^2}{1000}}{q} = \frac{100000}{q}
      + \frac{q}{1000}\]

      The average variable cost function is
      \[AVC(q) = \frac{\frac{q^2}{1000}}{q} = \frac{q}{1000}\]

      The marginal cost function is
      \[MC(q) = \partialder{C(q)}{q} = \frac{q}{500}\]
    \end{subproblem}

    \begin{subproblem}
      The firm's revenue will be $100q$, so its profit will be
      \[\pi(q) = 100q - C(q) = 100q - 100000
      - \frac{q^2}{1000} \]

      The firm will set its output where the marginal profit is zero
      \begin{align*}        
        \partialder{\pi(q)}{q} = 100 - \frac{q}{500} &= 0 \\
        100 &= \frac{q}{500} \\
        q &= 50000 \\
      \end{align*}

      The average variable cost at this output is $AVC(50000) =
      50$. This is less than the market price, so the firm does not
      down.


      For an arbitrary price $p$, we have 
      \begin{align*}        
        \partialder{\pi(q,p)}{q} = p - \frac{q}{500} &= 0 \\
        p &= \frac{q}{500} \\
        q &= 500p \\
      \end{align*}
      and so the $AVC$ is
      \[AVC(500p) = \frac{500p}{1000} = \frac{p}{2}\]
      Thus the average variable cost is always less than the market
      price, so the firm will not shut down at any non-zero price.
    \end{subproblem}

    \begin{subproblem}
      We assume there are five firms in the industry with the same
      costs and production function. From above, the short run
      marginal profit is zero at $q = 500p$; this is a single firm's
      supply curve. The supply curve for all firms is
      \[ q = 5(500p) = 2500p\]
    \end{subproblem}

    \begin{subproblem}
      Recall that in the long run, labor and capital are employed in
      equal proportion, so to produce $q$ with minimal costs, $q$
      units of labor and $q$ units of capital are required. Thus, the
      long run total cost is $wq + rk = 20q$. So the long run average
      cost is $\frac{20q}{q} = 20$ and the long run marginal cost is
      $\partialder{(20q)}{q} = 20$.

      We assume a perfectly competitive market. So the long run supply
      curve will be horizontal since firms will enter and leave the
      market. The value of the price is the minimum of the long run
      average cost, which is simply 20. So the supply curve is
      horizontal at $p=20$.

      Long run average cost is constant at 20, independent of the
      quantity produced, so constant returns to scale are exhibited.
    \end{subproblem}

    \begin{subproblem}
      The long run supply curve is horizontal at $p=20$, so the
      suppliers will supply any quantity at that price. So the
      equilibrium price will be 20. The quantity demanded will be
      \[Q_D(20) = 40 - 20 = 20\]
    \end{subproblem}

    \begin{subproblem}
      We cannot tell how many firms are in the market in the long run.
      This is the quantity demanded at equilibrium divided by the
      quantity supplied by each firm. However, we cannot determine the
      quantity supplied by each firm: since the average cost is
      constant, any quantity produced will be minimize average cost.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      \[ MC = \partialder{C(q)}{q} = 10 - 2q + q^2 \]
      \[ AVC = \frac{VC(q)}{q} = 10 -q + \frac{q^2}{3}\]
      \[ AC = \frac{C(q)}{q} = \frac{100}{q} + 10 -q + \frac{q^2}{3}\]
    \end{subproblem}

    \begin{subproblem}
      The firm will shut down in the short run if the market price is
      less than the minimum of the average variable cost. To find the
      minimum of the AVC, we set the derivative to zero:
      \[ -1 + \frac{2q}{3} = 0 \]
      \[ q = \frac{3}{2} \]
      At this point, the AVC is
      \[AVC(q) = 10 - \frac{3}{2} +
      \frac{\left(\frac{3}{2}\right)^2}{3} = 10 - \frac{3}{2} +
      \frac{9}{12} = 9.25\]
      So the firm will shut down if the price is below $9.25$.

      The firm will supply the profit-maximizing quantity (i.e. where
      the marginal profit is zero) when the price is above $9.25$.
     
      \[\pi(q) = pq - C(q) = pq - 100 + 10q - q^2 + \frac{q^3}{3}\]

      \[\partialder{\pi(q)}{q} = p - 10 + 2q - q^2 = 0 \]
      \[q^2 - 2q + (10 -p) = 0\]
      \[S(p) =
      \begin{cases}
        \sqrt{x - 9} + 1 \;\;\; & p > 9.25 \\
        0 & p \le 9.25 \\
      \end{cases}
      \]
    \end{subproblem}

    \begin{subproblem}
      We assume the long-run costs are the same as the short run
      costs. Since the model is perfectly competitive, the long run
      market supply curve is horizontal at the minimum of the long-run
      minimum average cost of a firm.

      The long-run average cost of the firm is $\frac{100}{q}
      + 10 -q + \frac{q^2}{3}$. The derivative is $\frac{2q}{3} -
      \frac{100}{q^2} -1$; this equals zero at the minimum: $q_0 = 5.863$. At this
      quantity, the average cost is approximately 32.65.

      So the long run market supply curve will be horizontal at $p_0
      \approx 32.65$.
    \end{subproblem}

    \begin{subproblem}
      The market demand is $Q(p) = 1000-p$. Since the supply curve is
      horizontal at $p_0$, the equilibrium price is $p_0$. The
      quantity demanded is the quantity supplied, $Q(p_0) = 1000 -
      p_0$. Each firm supplies $q_0$ units (where $AC(q_0) = p_0$), so
      the quantity supplied by $n$ firms is $q_0 n$.
      \begin{align*}
      Q(p_0) &= q_0 n \\
      1000 - p_0 &= q_0 n \\
      n &= \frac{1000 - p_0}{q_0} \\
      \end{align*}

      For the values of $p_0$ and $q_0$ above,
      \[ n = \ceil{\frac{1000-p_0}{q_0}} =
      \ceil{\frac{1000-32.65}{5.863}}  = \ceil{164.99} = 165\]
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The cost function is $C(q) = 5q^2$, so the average cost and
      average variable cost functions are both $5q$, and the marginal
      cost function is $10q$. The minimum of the average variable cost
      function is zero ($AVC(0)=0$), so the firm will never shut
      down. Thus, the supply curve for a single firm is given by
      \[MC(q) = 10q = p \]
      \[q = \frac{p}{10}\]
      With ten identical firms, the market supply is ten times this
      quantity:
      \[S(p) = p\]
      At the equilibrium point, supply equals demand
      \begin{align*}
        D(p) &= S(p) \\
        10-p &= p \\
        p = 5 \\
      \end{align*}
      so the equilibrium price is $5$ and the equilibrium quantity is
      $S(5) = 5$.

      The consumer and producer surpluses are shown in the following
      figure:
      
      \includegraphics[scale=1.5]{ps5-4a}

      The consumer surplus is region $A$, and the producer surplus is
      region $B$. Both regions have area $\nicefrac{1}{2}(5)(5) =
      12.5$, so this is the value of both the producer and consumer
      surpluses.
    \end{subproblem}

    \begin{subproblem}
      This decreases welfare, as seen in the graph below:

      \includegraphics[scale=1.5]{ps5-4b}
      
      Before the government intervention, the consumer surplus is $A +
      B$. After the government intervention, it is $A + B + C + D +
      E$. Before the price change, the producer surplus is $C + F$.
      After the government intervention, the producer surplus is $F -
      D - E - G - H$.

      So the consumer surplus increases but the producer surplus
      decreases. The total welfare was originally $A + B + C + F$, but
      it changes to $A + B + C + D + E + F - D - E - G - H = A + B + F -
      G - H$. Thus, the change in total welfare is $-G - H$. By inspection
      from the graph, the area $G + H$ is $\nicefrac{1}{2}(2)(1) = 1$. So total
      welfare decreases by $1$ due to the government
      intervention.
    \end{subproblem}

    \begin{subproblem}
      The quantity supplied at any price after the tax is equal to the
      quantity supplied before at the price minus the tax:
      \[ Q_S'(p) = Q_S(p-2) = p-2 \]
      so the new equilibrium point has
      \[ 10-p = p-2 \]
      or
      \[ p = 6 \]
      and the equilibrium quantity is $4$.

      To see the effects on the surplus, see the following graph.

      \includegraphics[scale=1.5]{ps5-4c}
      
      Initially, the consumer surplus is $A + B+ C +D$ and the
      producer surplus is $E + F + G + H + I$; there is no tax
      revenue. After the tax, the consumer surplus falls to $A$ and
      the producer surplus falls to $H + I$. The tax revenue is $B + E
      + C + F$. So society suffers a deadweight loss of $D + G$. The
      value of this loss is $\nicefrac{1}{2}(2)(1) = 1$. The amount of
      the government revenue is $2(4) = 8$. 
    \end{subproblem}
  \end{problem}
\end{pset}
\end{document}