\documentclass{article}
\input{1401-preamble}
\begin{document}
\psetnum{4}
\date{2004/10/15}

\begin{pset}
  \begin{problem}
    \begin{subproblem}
      At the isoquant, $\overline{q}$ is fixed in $\overline{q} =
      K^\alpha L^\beta$. So
      \begin{align*}
        K^\alpha &= \frac{\overline{q}}{L^\beta}\\
        K &= \sqrt[\alpha]{\frac{\overline{q}}{L^\beta}}
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      The marginal rate of technical substitution is
      \begin{align*}
        MRTS &= -\frac{MP_L}{MP_K} =
        \frac{\partialder{q}{L}}{\partialder{q}{K}} = \frac{\beta
          K^\alpha L^{\beta - 1}}{\alpha L^\beta K^{\alpha -1}} =
        \frac{\beta K}{\alpha L}
      \end{align*}
      At the optimal combination of inputs, the MRTS equals the ratio
      of the costs of labor and capital:
      \[MRTS = \frac{\beta K}{\alpha L} = \frac{w}{r} \]
      \[ K = \frac{\alpha w L}{\beta r} \]
      We wish to produce output $q^*$, so
      \[ q^* = K^\alpha L^\beta = \left(\frac{\alpha w L}{\beta
          r}\right)^\alpha L^\beta \]
      \[ L^{\alpha + \beta} = q^* \left(\frac{\beta r}{\alpha
          w}\right)^\alpha\]
      \[L = {q^*}^{\frac{1}{\alpha + \beta}} \left(\frac{\beta
          r}{\alpha w}\right)^{\frac{\alpha}{\alpha+\beta}} \]
      and
      \[ K = \frac{\alpha w L}{\beta r} = \frac{\alpha w}{\beta r}
      {q^*}^{\frac{1}{\alpha + \beta}} \left(\frac{\beta
          r}{\alpha w}\right)^{\frac{\alpha}{\alpha+\beta}}\]
    \end{subproblem}

    \begin{subproblem}
      The cost $C(q)$ is the cost of the optimal amount of inputs
      required to produce quantity $q$:
      \[C(q) = wL + rK = wL + r\frac{\alpha w L}{\beta r} = \left(w +
        \frac{\alpha w}{\beta}\right) q^{\frac{1}{\alpha +
          \beta}} \left(\frac{\beta
          r}{\alpha w}\right)^{\frac{\alpha}{\alpha+\beta}}\]
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      \begin{enumerate}
      \item \[q(2K, 2L) = \left(2K\right)^{\nicefrac{3}{4}}
        (2L)^{\nicefrac{1}{4}} = 2 K^{\nicefrac{3}{4}}
        L^{\nicefrac{1}{4}} = 2q(K,L) \]
        Constant returns to scale.
      \item \[q(2K,2L) = 5(2K) + 3(2L) = 2(5K + 3L) = 2q(K,L)\]
        Constant returns to scale.
      \item \[q(2K, 2L) = \min(2(2K), 3(2L)) = 2 \min(2K, 3L) =
        2q(K,L)\]
        Constant returns to scale.
      \end{enumerate} 
    \end{subproblem}

    \begin{subproblem}
      \begin{enumerate}
      \item \[MP_K = \partialder{q}{K} = \frac{3}{4}
        L^{\nicefrac{1}{4}} K^{-\nicefrac{1}{4}} \]
        As $K$ increases, $MP_K$ decreases, so the production function
        exhibits diminishing $MP_K$.
      \item \[MP_K = \partialder{q}{K} = 5 \]
       $MP_K$ is independent of $K$, so the production function
       exhibits constant $MP_K$.
      \item \[MP_K = \partialder{q}{K} =
        \begin{cases}
          2 & 2K < 3L \\
          0 & 2K \ge 3L
        \end{cases}
        \]
        As $K$ increases, $MP_K$ can only decrease, so the production function
        exhibits diminishing $MP_K$.
      \end{enumerate}
    \end{subproblem}

    \begin{subproblem}
      \begin{center}
        \includegraphics{ps4-2a}
        \includegraphics{ps4-2b}
        \includegraphics{ps4-2c}
      \end{center}
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      \textbf{False.} Consider the Leontief production function
      \[f(L,K) = \min \set{L,K}\] A cost-minimizing firm will always
      consume labor and capital in a unit ratio, regardless of their
      relative prices.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True.} If we take the total derivative of the production
      function, we find
      \[ dq = \partialder{f}{L} dL = \partialder{f}{K} dK + \cdots \]
      To find the returns to scale, we use positive values of $dL, dK,
      \ldots$. If the firm exhibits decreasing marginal utility in
      each input, $\partialder{f}{L}, \partialder{f}{K}, \ldots$ will
      be negative. So $dq$ will be negative as well. Thus, decreasing
      marginal utility in every input means that the returns to scale
      cannot be increasing.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} In an increasing-cost industry, the long-run supply is
      everywhere upwards-sloping, and costs increase as the quantity
      supplied increases. If demand increases at every price, then
      more goods will be supplied at the new equilibrium point. So the
      costs to the firm will increase. This means that they may make a
      lower profit. 
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} Consider the production function $f(L,K) =
      \sqrt{L}$, which is independent of the amount of
      capital. It exhibits decreasing marginal returns with respect to
      labor, but increasing the amount of capital clearly does not
      affect the output.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      To produce $Y_0$, we have
      \[Y_0 = L^b\]
      \[L = \sqrt[b]{Y_0}\]
    \end{subproblem}

    \begin{subproblem}
      A firm's cost function gives the cost of producing a certain
      quantity of goods using the cost-minimizing bundle of inputs, as
      a function of the quantity desired and the prices of the
      inputs.
    \end{subproblem}

    \begin{subproblem}
      Since there is only one input, the cost function simply
      represents the cost of labor:
      \[C(Y_0, w) = w L = w \sqrt[b]{Y_0} \]
    \end{subproblem}

    \begin{subproblem}
      Suppose the desired production quantity increases by a factor of
      $k$ (with $k > 1$). Then the cost is
      \[C(k Y, w) = w \sqrt[b]{k Y} = w k^{\nicefrac{1}{b}}
      Y^{\nicefrac{1}{b}} = k^{\nicefrac{1}{b}} C(Y,w)\]
      So if $b < 1$, $C(kY,w) > k C(Y,w)$, i.e. producing $k$ times as
      many goods costs more than $k$ times as much. The firm exhibits
      diminishing returns to scale.

      If $b = 1$, $C(kY,w) = k C(Y,w)$, i.e. producing $k$ times as
      many goods costs exactly $k$ times as much. The firm exhibits
      constant returns to scale.

      Finally, if $b > 1$, $C(kY,w) < k C(Y,w)$, i.e. producing $k$
      times as many goods costs less than $k$ times as much. The firm
      exhibits increasing returns to scale.
    \end{subproblem}

    \begin{subproblem}
      A firm's profit function gives the amount of profit a firm will
      make by producing a specified quantity of goods, as a function
      of the quantity, the price, the cost of inputs, and possibly
      other variables.
    \end{subproblem}

    \begin{subproblem}
      With output $Y_0$, price $p_Y$, and wage $w$, the profit is
      \[ \pi(Y_0, w, p_Y) = R(Y_0, p_Y) - C(Y_0, w) = p_Y Y_0 - w
      Y_0^{\nicefrac{1}{b}} \]
    \end{subproblem}

    \begin{subproblem}
      At the profit-maximizing output level, the marginal profits will
      be zero:

      \begin{align*}
        \partialder{\pi}{q} = p_Y - \frac{w q^{\nicefrac{1}{b}
            -1}}{b} &= 0 \\
        \frac{b p_Y}{w} &= q^{\frac{1 - b}{b}} \\
        q &= \left(\frac{b p_Y}{w}\right)^{\frac{b}{1-b}}
      \end{align*}
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The fixed cost is $A$, and the variable cost is $Bq+Dq^2$.

      The marginal cost is $\partialder{C}{q} = B + 2Dq$. The average
      cost is $\frac{A + Bq + Dq^2}{q} = \frac{A}{q} + B + Dq$. The
      cost function exhibits increasing returns to scale when the
      marginal cost is less than the average cost:
      \begin{align*}
      B + 2Dq &< \frac{A}{q} + B + Dq \\
      Dq &< \frac{A}{q} \\
      q^2 &< \frac{A}{D} \\
      \end{align*}
      When equality holds, we have constant returns to scale, and when
      the inequality is reversed, we have diminishing returns to
      scale. So for $q < \sqrt{\frac{A}{D}}$, the returns are
      increasing; for $q = \sqrt{\frac{A}{D}}$ the returns to scale
      are constant, and for $q > \sqrt{\frac{A}{D}}$ the returns are
      decreasing.
    \end{subproblem}

    \begin{subproblem}
      The total cost function is
      \[ TC(q) = A + Bq + Dq^2 = 400 + 2q + 0.04 q^2 \]
      \begin{center}
        \includegraphics{ps4-5b-1}
      \end{center}
      The average cost function is
      \[ AC(q) = \frac{TC(q)}{q} = \frac{A + Bq + Dq^2}{q} =
      \frac{A}{q} + B + Dq = \frac{400}{q} + 2 + 0.04q \]
      The average variable cost function is
      \[ AVC(q) = \frac{VC(q)}{q} = \frac{Bq + Dq^2}{q} = B + Dq = 2 +
      0.04q \]
      The marginal cost function is
      \[ MC(q) = \partialder{C}{q} = B + 2Dq = 2 + 0.08q\] 
      \begin{center}
        \includegraphics{ps4-5b-2}
      \end{center}
    \end{subproblem}

    \begin{subproblem}
      Average costs are minimized in the short run when the marginal
      cost equals the average cost, i.e. when \[q = \sqrt{\frac{A}{D}}
      = \sqrt{\frac{400}{0.04}} = \sqrt{10000} = 100\] At $q = 100$,
      the average cost is \[AC(100) = \frac{400}{100} + 2 + 0.04(100)
      = 4 + 2 + 4 = 10\] So in the short run, the minimum price at
      which the widget holders can be produced is \$10. The firm will
      not be willing to produce any output below this price in the
      short run. In the long run, changing the variable costs may make
      it possible for the firm to lower its costs.
    \end{subproblem}

    \begin{subproblem}
      The revenue function is
      \[R(q) = p_W q = \$1 q\]
      The profit is the revenue minus the costs:
      \[\pi(q) = R(q) - C(q) = p_W q - \left(A + Bq + Dq^2\right) = q
      - 400 - 2q - 0.04 q^2\]
    \end{subproblem}

    \begin{subproblem}
      AWH will choose its output so that the marginal profits are
      zero:
      \begin{align*}
        MP(q) = \partialder{\pi}{q} &= 0 \\
        p_W - B - 2Dq &= 0 \\
        q &= \frac{p_W - B}{2D}
      \end{align*}

      If the price $p_W = \$10$, then
      \[ q = \frac{p_W - B}{2D} = \frac{10 - 2}{2(0.04)} = 100 \]
    \end{subproblem}

    \begin{subproblem}
      The supply curve is the amount of product produced at any price
      (above the minimum). Since the firm is profit-maximizing, the
      points on the supply curve have zero marginal profit. With the
      formula from above,
      \[ S(p_W) = \frac{p_W - B}{2D} = \frac{p_W}{0.08} -
      \frac{1}{0.04} \text{ for } p_W > 10\]

      If twenty firms use the same technology, they will make the same
      profit-maximizing decision. So the supply curve is simply twenty
      times as large:
      \[ S(p_W) = 20 \frac{p_W - B}{2D} = 20 \frac{p_W}{0.08} -
      \frac{20}{0.04} \text{ for } p_W > 10\]
      
    \end{subproblem}
  \end{problem}
\end{pset}
\end{document}
