\documentclass{article}
\input{1402-preamble}
\psetnum{1}
\date{2009/02/20}

\begin{document}
\begin{pset}
  \begin{problem}
    
    \begin{subproblem}
      \textbf{False.} The real GDP growth rate is
      $\frac{Y_{t}}{Y_{t-1}} -1$. $Y_t$ and $Y_{t-1}$ both depend on the
      choice of base year, in that the quantities of goods produced
      are weighted by their prices in the base year, so changing the
      base year could cause $Y_{t}$ to be greater relative to
      $Y_{t-1}$. If so, however, the inflation rate would
      decrease. Since the GDP deflator for year $t$ is
      $\frac{\$Y_t}{Y_t}$, and the nominal GDP $\$Y_t$ is unaffected
      by the base year, if the base year choice causes $Y_t$ to be
      higher, then it will cause the GDP deflator to be lower.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True.} The GDP deflator is the ratio of nominal to real
      GDP, so its change reflects the change in prices of goods
      weighted by the total amount produced in the economy, including
      goods not typically purchased by consumers. The CPI reflects the
      change in the cost of a basket of goods typically purchased by a
      consumer. Assuming that this basket actually is representative
      of consumer consumption patterns, then it will be a better
      indicator for the cost of living. (This might not be true if
      consumption patterns change over time, for example.)
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} Private savings is discretionary income minus
      consumption,
      \[S = Y^d - C = Y - T - C\]
      and so, using the equilibrium condition and definition of total
      demand,
      \[S = C + I + G - T - C = I + G - T\]
      so investment is only equal to private savings if public savings
      ($G -T$) is zero.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False. }When the value of a bank's assets falls, it must
      write down its capital accordingly. However, this increases its
      leverage. In order to keep its leverage constant (or reduce it,
      if procyclical), it must either raise new capital, or sell some
      assets to reduce debt.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    
    \begin{subproblem}
      \begin{subsubproblem}
        Nominal GDP, 2006 base
        \begin{align*}
          \$Y_{2006} &= P_{S,2006} Q_{S,2006} + P_{B,2006} Q_{B,2006} &=
          2 \cdot 1200 + 25 \cdot 500 &= 14900\\
          \$Y_{2007} &= P_{S,2007} Q_{S,2007} + P_{B,2007} Q_{B,2007} &=
          2.5 \cdot 1500 + 24 \cdot 400 &= 13350\\
          \$Y_{2008} &= P_{S,2008} Q_{S,2008} + P_{B,2008} Q_{B,2008} &=
          2.5 \cdot 1650 + 24 \cdot 440 &= 14685
        \end{align*}
      \end{subsubproblem}

      \begin{subsubproblem}
        Real GDP, 2006 base
        \begin{align*}
          Y_{2006} &= P_{S,2006} Q_{S,2006} + P_{B,2006} Q_{B,2006} &=
          2 \cdot 1200 + 25 \cdot 500 &= 14900\\
          Y_{2007} &= P_{S,2006} Q_{S,2007} + P_{B,2006} Q_{B,2007} &=
          2 \cdot 1500 + 25 \cdot 400 &= 13000\\
          Y_{2008} &= P_{S,2006} Q_{S,2008} + P_{B,2006} Q_{B,2008} &=
          2 \cdot 1650 + 25 \cdot 440 &= 14300
        \end{align*}        
      \end{subsubproblem}

      \begin{subsubproblem}
        GDP deflator, 2006 base
        \begin{align*}
          \text{GDP deflator}_{2006} &= \frac{\$Y_{2006}}{Y_{2006}}
          &= \frac{14900}{14900} &= 1\\
          \text{GDP deflator}_{2007} &= \frac{\$Y_{2007}}{Y_{2007}}
          &= \frac{13350}{13000} &\approx 1.027\\
          \text{GDP deflator}_{2008} &= \frac{\$Y_{2008}}{Y_{2008}}
          &= \frac{14685}{14300} &\approx 1.027
        \end{align*}
      \end{subsubproblem}

      \begin{subsubproblem}
        Real GDP growth rate, 2006 base
        \begin{align*}
          \text{Real GDP growth}_{2007} &= \frac{Y_{2007} -
            Y_{2006}}{Y_{2006}} &= \frac{13000-14900}{14900} &\approx -0.128\\
          \text{Real GDP growth}_{2008} &= \frac{Y_{2008} -
            Y_{2007}}{Y_{2007}} &= \frac{14300-13000}{13000} &= 0.1
          \end{align*}
      \end{subsubproblem}

      \begin{subsubproblem}
        Inflation rate, 2006 base
        \begin{align*}
          \text{Inflation}_{2007} &= \frac{\text{GDP deflator}_{2007}
            - \text{GDP deflator}_{2006}}{\text{GDP deflator}_{2006}}
          &\approx \frac{1.027 - 1}{1} &\approx 0.027\\
          \text{Inflation}_{2008} &= \frac{\text{GDP deflator}_{2008}
            - \text{GDP deflator}_{2007}}{\text{GDP deflator}_{2007}}
          &\approx \frac{1.027 - 1.027}{1.027} &= 0
        \end{align*}
      \end{subsubproblem}
    \end{subproblem}

    \begin{subproblem}
      \begin{subsubproblem}
        Nominal GDP, 2007 base --- independent of base year, so same as
        2006 base
        \begin{align*}
          \$Y_{2006} &= P_{S,2006} Q_{S,2006} + P_{B,2006} Q_{B,2006} &=
          2 \cdot 1200 + 25 \cdot 500 &= 14900\\
          \$Y_{2007} &= P_{S,2007} Q_{S,2007} + P_{B,2007} Q_{B,2007} &=
          2.5 \cdot 1500 + 24 \cdot 400 &= 13350\\
          \$Y_{2008} &= P_{S,2008} Q_{S,2008} + P_{B,2008} Q_{B,2008} &=
          2.5 \cdot 1650 + 24 \cdot 440 &= 14685
        \end{align*}        
      \end{subsubproblem}

      \begin{subsubproblem}
        Real GDP, 2007 base
        \begin{align*}
          Y_{2006} &= P_{S,2007} Q_{S,2006} + P_{B,2007} Q_{B,2006} &=
          2.5 \cdot 1200 + 24 \cdot 500 &= 15000\\
          Y_{2007} &= P_{S,2007} Q_{S,2007} + P_{B,2007} Q_{B,2007} &=
          2.5 \cdot 1500 + 24 \cdot 400 &= 13350\\
          Y_{2008} &= P_{S,2007} Q_{S,2008} + P_{B,2007} Q_{B,2008} &=
          25 \cdot 1650 + 24 \cdot 440 &= 14685
        \end{align*}
      \end{subsubproblem}

      \begin{subsubproblem}
        GDP deflator, 2007 base
        \begin{align*}
          \text{GDP deflator}_{2006} &= \frac{\$Y_{2006}}{Y_{2006}}
          &= \frac{14900}{15000} &\approx 0.993\\
          \text{GDP deflator}_{2007} &= \frac{\$Y_{2007}}{Y_{2007}}
          &= \frac{13350}{13350} &= 1\\
          \text{GDP deflator}_{2008} &= \frac{\$Y_{2008}}{Y_{2008}}
          &= \frac{14685}{14685} &= 1
        \end{align*}        
      \end{subsubproblem}

      \begin{subsubproblem}
        Real GDP growth rate, 2007 base
        \begin{align*}
          \text{Real GDP growth}_{2007} &= \frac{Y_{2007} -
            Y_{2006}}{Y_{2006}} &= \frac{13350-15000}{15000} &= -0.11\\
          \text{Real GDP growth}_{2008} &= \frac{Y_{2008} -
            Y_{2007}}{Y_{2007}} &= \frac{14685-13350}{13350} &= 0.1
          \end{align*}
      \end{subsubproblem}
      
      \begin{subsubproblem}
        Inflation rate, 2007 base
        \begin{align*}
          \text{Inflation}_{2007} &= \frac{\text{GDP deflator}_{2007}
            - \text{GDP deflator}_{2006}}{\text{GDP deflator}_{2006}}
          &\approx \frac{1-0.993}{0.993} &\approx 0.007\\
          \text{Inflation}_{2008} &= \frac{\text{GDP deflator}_{2008}
            - \text{GDP deflator}_{2007}}{\text{GDP deflator}_{2007}}
          &\approx \frac{1-1}{1} &= 0
        \end{align*}
      \end{subsubproblem}
    \end{subproblem}

    \begin{subproblem}
      The inflation rate in 2007 was calculated as
      \[\text{Inflation}_{2007} = \frac{\text{GDP deflator}_{2007}
            - \text{GDP deflator}_{2006}}{\text{GDP deflator}_{2006}}
          = \frac{\text{GDP deflator}_{2007}}{\text{GDP deflator}_{2006}}
          - 1 \]    
      Using the definition of the GDP deflator,
      \[\text{Inflation}_{2007} =
      \frac{\frac{\$Y_{2007}}{Y_{2007}}}{\frac{\$Y_{2006}}{Y_{2006}}}
      - 1 = \frac{\$Y_{2007}}{\$Y_{2006}} \frac{Y_{2006}}{Y_{2007}}
      -1\]
      The nominal GDP is independent of the base year, so the only
      factor that can be affected by the base year is the
      $\frac{Y_{2006}}{Y_{2007}}$ factor.
      \[\frac{Y_{2006}}{Y_{2007}} = \frac{P_{S,\text{base}} Q_{S,2006}
        + P_{B,\text{base}} Q_{B,2006}}{P_{S,\text{base}} Q_{S,2007} +
        P_{B,\text{base}} Q_{B,2007}}\]
      Note that the quantities are weighted by their price in the base
      year. Because in 2006 books were more expensive relative to soda
      than they were in 2007, using a 2006 base means a difference in
      the quantity of books is weighed more heavily. Because the
      quantity of books dropped in 2007 and the quantity of soda
      increased, weighing books more heavily leads to a higher
      inflation rate for 2007 when 2006 is used as a base year than
      when 2007 is used as the base yearn.
    \end{subproblem}

    \begin{subproblem}
      For both the inflation rate and the growth rate, the only factor
      dependent on the base year is
      \[\frac{Y_{2007}}{Y_{2008}} = \frac{P_{S,\text{base}} Q_{S,2007}
        + P_{B,\text{base}} Q_{B,2007}}{P_{S,\text{base}} Q_{S,2008} +
        P_{B,\text{base}} Q_{B,2008}}\]
      as shown above. In 2008, the quantities produced of books and
      soda both grew by the same relative amount, 10\%. So, although
      the prices from the base year affect their relative weighting,
      this has no impact because both quantities increased by the same
      percentage.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    
    \begin{subproblem}
      The endogenous variables are $C$ (determined by the first equation
      given), $Z$ (determined by the demand equation $Z = C + I + G$), $Y$
      (determined by the equilibrium condition $Y = Z$), and $Y^d$
      (determined by its definition $Y^d = Y - T$).      
    \end{subproblem}

    \begin{subproblem}
      \[Z = C + I + G\]
      \[Y = Z\]
    \end{subproblem}

    \begin{subproblem}
      \begin{align*}
        Y = Z &= C + I + G\\
        Y &= (c_0 + c_1 Y^d) + I + G\\
        Y &= c_0 + c_1 (Y - T) + I + G\\
        Y - c_1 Y &= c_0 + I + G - c_1 T\\
        Y &= \frac{c_0 + I + G - c_1 T}{1 - c_1}\\
        Y &= \frac{200 + 400 + 150 - 0.6 \cdot 150}{1 - 0.6} =
        \frac{660}{0.4} = 1650\\
        C &= Y - I - G = 1650 - 400 - 150 = 1100
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      \begin{align*}
        Y' &= \frac{c_0 + I + G - c_1 T}{1 - c_1}\\
        Y' &= \frac{140 + 400 + 150 - 0.6 \cdot 150}{1 - 0.6} =
        \frac{600}{0.4} = 1500\\
        C' &= Y' - I - G = 1500 - 400 - 150 = 950
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      \begin{align*}
        Y - c_1 Y &= c_0 + I + G' - c_1 T\\
        G' &= (1 - c_1) Y - c_0 - I + c_1 T\\
         &= (1 - 0.6) 1650 - 140 - 400 + 0.6 \cdot 150 = 660 - 140 -
         400 + 90 = 210
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      \begin{align*}
        Y - c_1 Y &= c_0 + I + G - c_1 T'\\
        c_1 T' &= c_0 + I + G - (1 - c_1) Y\\
        T' &= \frac{c_0 + I + G - (1 - c_1) Y}{c_1}
        &= \frac{140 + 400 + 150 - (1 - 0.6) 1650}{0.6} =
        \frac{30}{0.6} = 50
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      Changing $G$ gives a budget deficit $G' - T = 210 - 150 =
      60$. Changing $T$ gives a budget deficit $G - T' = 150 - 50 =
      100$. The budget deficit is larger for the tax cut, because the
      tax cut required to increase total demand back to its original
      value is larger than the increase in government spending that
      has the same effect. This is because only $c_1 = 60\%$ of the
      money returned to consumers as a tax cut contributes to total
      demand, while the full increase in government spending
      contributes to total demand.
    \end{subproblem}

    \begin{subproblem}
      Changing $G$ gives consumption level $C = Y - I - G' = 1650 -
      400 - 210 = 1040$. Changing $T$ gives consumption level $C = Y -
      I - G = 1650 - 400 - 150 = 1100$. The tax cut gives a higher
      consumption level, which is no surprise because both the tax cut
      and government spending increase were chosen to give the same
      total demand, and the tax cut achieves this total demand by
      increasing only consumption $C$ ($I$ and $G$ remain unchanged),
      whereas the spending increase also increases $G$, requiring less
      increase in consumption.
    \end{subproblem}
  \end{problem}
\end{pset}
\end{document}
