\documentclass{article}
\input{1401-preamble}
\begin{document}
\psetnum{3}
\date{2004/10/01}

\begin{pset}
  \begin{problem}
    \begin{subproblem}
      The marginal utility of butter is 1 and the marginal utility of
      margarine is 5, so
      \[MRS_{bm} = - \frac{MU_b}{MU_m} = - \frac{1}{5}\] The
      indifference curves are parallel lines with slope of
      $-\nicefrac{1}{5}$, with butter on the $x$ axis and margarine on
      the $y$ axis.
    \end{subproblem}
    
    \begin{subproblem}
      The two goods are perfect substitutes, with five sticks of
      butter equal to one stick of margarine, so at equal prices a
      consumer will only consume margarine.
      \begin{align*}
        M &= \frac{\$45.00}{\$1.50} = 30 \\
        B &= 0 \\
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      The prices are still equal, so the consumer will spend his or
      her entire income on margarine.
      \begin{align*}
        M(I) &= \frac{I}{\$1.50} \\
        B(I) &= 0 \\
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      Let $p$ be the ratio of the price of margarine to the price of
      butter: $p = \frac{p_M}{p_B}$. If $p$ is less than 5, the entire
      income will be spent on margarine as this maximizes utility; if
      $p$ is greater than 5, then the entire income will be spent on
      butter. If $p$ equals exactly 5, then nothing can be said about
      the relative amounts of butter and margarine purchased except
      that the total cost is equal to the income $I$:

      \begin{align*}
        M(p,I) &=
        \begin{cases}
          \frac{I}{p_M} & \;:\; p < 5 \\
          0 & \;:\; p > 5 \\
          \frac{I - p_B B}{p_M} & \;:\; p = 5 \\
        \end{cases} \\
        B(p,I) &=
        \begin{cases}
          0 & \;:\; p < 5 \\
          \frac{I}{p_B} & \;:\; p > 5 \\
          \frac{I - p_M M}{p_B} & \;:\; p = 5 \\
        \end{cases} \\
      \end{align*}
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The two goods are perfect substitutes at a 1:1 equivalence
      ratio, and the price of movies is less.
      \begin{align*}
        m &= \frac{\$200}{\$1} = 200 \\
        b &= 0 \\
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      The two goods are still perfect substitutes, but now books are
      cheaper.
      \begin{align*}
        m &= 0 \\
        b &= \frac{\$200}{\$0.50} = 400 \\
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      To maximize utility, one unit of books must be purchased per
      unit of movies. So
      \[ m = b = \frac{\$200}{\$2 + \$1} = \frac{200}{3} \]
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The marginal utility of clothes is
      \[MU_C = \partialder{U}{C} = \frac{3}{4C}\]
      and the marginal utility of food is
      \[MU_F = \partialder{U}{F} = \frac{1}{4F}\]
      so the marginal rate of substitution is
      \[MRS = -\frac{MU_C}{MU_F} = - \frac{12F}{4C} = \frac{3F}{C}\]
      At the equilibrium point, the MRS is equal to the MRT, so
      \begin{align*}
        \frac{3F}{C} &= \frac{p_C}{p_F} = \frac{20}{5} \\
        C &= \frac{3}{4}F \\
      \end{align*}
      We apply the budget constraint $20C + 5F = 2000$ and find
      \[ (F,C) = (100,75) \]
    \end{subproblem}

    \begin{subproblem}
      The cost of the same bundle is now
      \[100 p_f^a + 75 p^a_c = 1000
      + 1875 = 2875 \]
    \end{subproblem}

    \begin{subproblem}
      The MRS is the same as before, but the MRT is different.
      \begin{align*}
        \frac{3F}{C} &= \frac{p^a_C}{p^a_F} = \frac{25}{10} \\
        C &= \frac{6}{5}F \\
      \end{align*}
      The new budget constraint is $25 C + 10F = 2875$, so
      \begin{align*}
        25\frac{6}{5}F + 10F &= 2875 \\
        F &= \frac{2875}{40} = 71.875 \\
      \end{align*}
      and hence
      \[C = \frac{2875 - 718.75}{25} = 86.25 \]
    \end{subproblem}

    \begin{subproblem}
      Her utility in Cambridge is
      \[ U(75,100) = \frac{1}{4} \ln 100 + \frac{3}{4} \ln 75 \approx
      4.3894 \]
      and her utility in Palo Alto is
      \[ U(86.25,71.875) = \frac{1}{4} \ln 71.875 + \frac{3}{4} \ln
      86.25 \approx 4.4117\]
      So her utility is greater in Palo Alto.
    \end{subproblem}

    \begin{subproblem}
      The budget lines for Cambridge and Palo Alto and relevant
      indifference curves are shown below:
      
      The budget line in Cambridge is
      \[ 20C + 5F = 2000 \]
      and the budget line in Palo Alto is
      \[ 25C + 10F  = 2875 \]
      The indifference curve in Cambridge is
      \[ \frac{1}{4} \ln F + \frac{3}{4} \ln C \approx 4.3894 \]
      and in Palo Alto
      \[ \frac{1}{4} \ln F + \frac{3}{4} \ln C \approx 4.4117 \]
      \begin{center}
        \includegraphics[width=5in]{ps3-3d}
      \end{center}
      The budget line for Palo Alto intersects a higher indifference
      curve than the budget line for Cambridge, so utility is greater
      in Palo Alto.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The marginal utility of coffee is
      \[MU_C = \partialder{U}{C} = \frac{D^{\nicefrac{1}{2}}}{2
        C^{\nicefrac{1}{2}}} \]
      and the marginal utility of donuts is
      \[MU_D = \partialder{U}{D} = \frac{C^{\nicefrac{1}{2}}}{2
        D^{\nicefrac{1}{2}}} \]
      so the marginal rate of substitution is
      \[MRS = -\frac{MU_C}{MU_D} = -\frac{D}{C} \]
      At the equilibrium point,
      \[MRS = -\frac{D}{C} = -\frac{p_C}{p_D} = -\frac{\$1}{\$0.5}\]
      \[D = 2C\]
      So, applying the budget constraint $0.5D + C = I$,
      \[ (D,C) = (I, \frac{I}{2}) = (100, 50) \] 
    \end{subproblem}

    \begin{subproblem}
      The income elasticity of demand for donuts is
      \[ \xi_D = \partialder{Q_D}{I} \frac{I}{Q_D} = 1 \frac{I}{I} = 1\]
      and the income elasticity of demand for coffee is
      \[ \xi_C = \partialder{Q_C}{I} \frac{I}{Q_D} = \frac{1}{2}
      \frac{I}{\nicefrac{I}{2}} = 1\]

      The income elasticities are both
      positive, so he purchases more of both goods as his income
      rises. The income elasticities are both 1, so the increase in
      his purchases of each good is proportional to the increase in
      his income.
    \end{subproblem}

    \begin{subproblem}
      If the price of coffee increases to \$2, 
      \[MRS = -\frac{D}{C} = -\frac{p_C}{p_D} = -\frac{\$2}{\$0.5}\]
      \[D = 4C\]
      and the budget constraint becomes $0.5D + 2C = I$,
      \[ (D,C) = (I, \frac{I}{4}) = (100, 25) \] 
    \end{subproblem}

    \begin{subproblem}
      The bundle purchased was $(D,C) = (I, \frac{I}{2})$, which now
      equals $(200, 100)$.
    \end{subproblem}

    \begin{subproblem}
      The utility at the original equilibrium point is
      \[ U(50,100) = 50^{\nicefrac{1}{2}} 100^{\nicefrac{1}{2}} =
      50\sqrt{2} \]
      So the original indifference curve is $U(C,D) = 50\sqrt{2}$. The
      substitution effect point $e^*$ is the point at which this
      indifference is tangent to the budget line parallel to the one
      after the shift. So we are tasked with finding the point at
      which $U(C,D) = 50\sqrt{2}$ and the imaginary budget line is
      parallel to the indifference curve.
      \[MRS = -\frac{D}{C} = -\frac{p_C'}{p_D'} = -\frac{\$2}{\$0.5}\]
      \[D=4C\]
      So
      \[U(C,D) = C^{\nicefrac{1}{2}} D^{\nicefrac{1}{2}} =
       C^{\nicefrac{1}{2}} (4C)^{\nicefrac{1}{2}} = 50 \sqrt{2}\]
      \[C = 25\sqrt{2}\]
      and
      \[D = 100\sqrt{2}\]
      So the substitution effect decreases the quantity of coffee
      consumed from 50 to $25\sqrt{2}$ (a decrease by a factor of
      $\sqrt{2}$), and decreases the quantity of donuts consumed from
      100 to $50\sqrt(2)$, again a decrease by a factor of $\sqrt{2}$.

      The income effect corresponds to a shift perpendicular to the
      budget line, from $(C,D) = (25\sqrt{2},50\sqrt{2})$ to $(25,100)$.
  
    \end{subproblem}

    \begin{subproblem}
        The income and substitution effects are shown graphically
        below. The substitution effect moves the demand from point
        $e_1$ to $e_*$, and the income effect moves the demand from
        point $e_*$ to $e_2$.
      \begin{center}
        \includegraphics{ps3-4f}
      \end{center}
    \end{subproblem}
  \end{problem}

  \begin{problem}
    Let $a$ and $b$ be goods for which an agent's preferences are
    given by $U(a,b) = 3a + 2b$. We show that neither good can be a
    Giffen good. To do so, we show that $a$ is not a Giffen good; the
    argument for $b$ is analogous. Note from the utility function that
    $a$ and $b$ are perfect substitutes. Thus, the quantity purchased
    of $a$ is zero if $\frac{p_a}{p_b} > \frac{3}{2}$, and the full
    income is spent on $\frac{y}{p_a}$ units of $a$ if
    $\frac{p_a}{p_b} < \frac{3}{2}$. Suppose $a$ were a Giffen good.
    Then at some point, as the price $p_a$ increased with all
    variables held constant, the quantity $a$ purchased of $a$ would
    increase. But this cannot happen. Consider first the case when the
    initial ratio of prices $\frac{p_a}{p_b} > \frac{3}{2}$. Then $a$
    is zero. If $p_a$ increases, the consumer will still not purchase
    any of good $a$ as their utility is maximized by purchasing only
    $b$. Alternatively, the initial ratio of prices could satisfy
    $\frac{p_a}{p_b} < \frac{3}{2}$. In this case, $a =
    \frac{y}{p_a}$. If $p_a$ increases, it will be impossible to
    purchase this many or more units of $a$; it would violate the
    budget constraint. So $a$ is a non-Giffen good. The same argument
    can be made for $b$. So neither good is a Giffen good.
  \end{problem}

  \begin{problem}
    The initial bundle consumed is
    \[(a,b) = (6,14)\]
    When the price of $p_b$ is reduced to 5, and the consumer's income
    is reduced to maintain the same indifference curve, the bundle
    consumed is
    \[(a,b) = (3,18)\]
    So the change due to substitution effects is $(3-6,18-14) =
    (-3,4)$.
    
    And since the bundle consumed after the price change with the
    original income level is
    \[(a,b) = (5,30)\]
    the change due income effects is $(5-3,30-18) = (2,12)$, or a
    scaling by a factor of $\nicefrac{5}{3}$.
  \end{problem}
\end{pset}
  
\end{document}
