\include{EasyBeamer}
\newcommand{\StanFig}{3.5}

\m{By How Much Does GDP Rise If the Government Buys More Output?}
{\begin{center}
Robert E. Hall\\
\vspace{0.5in}
Hoover Institution and Department of Economics, Stanford\\

\vspace{0.5in}

October 13, 2009
\end{center}}

\blackslide

\xltab{regs}{Estimates of Output and Consumption Multipliers for Military Spending}{1.8}

\fig{weights}{Weights Implicit in Regression Estimates of Output and Consumption Multipliers}{\StanFig}

\xltab{VARs}{Estimates of Multipliers from Vector Autoregressions}{5}

\ee{Consumption and labor supply}
{\frac{c^{1-1/\sigma}}{1-1/\sigma}-\gamma \frac{h^{1+1/\psi}}{1+1/\psi}}
{wc^{-1/\sigma}=\gamma h^{1/\psi}}

\eee{Simple static model}
{y = h^{\alpha}}
{w = \alpha h^{-(1-\alpha)}}
{(y-g)^{-1/\sigma} = \frac{\gamma}{\alpha}y^{\frac{1+1/\psi}{\alpha}-1}}

\eee{Output multiplier}
{\mbox{Normalize so }y=1, \mbox{ wlog}}
{\gamma = \alpha (1-g)^{-1/\sigma}}
{m_y = \frac{dy}{dg} = \frac{\alpha}{\alpha + \sigma(1-g)(1-\alpha+1/\psi)}}

\mmm{Properties of the static model}
{ $\alpha \le 1$ and $\psi > 0$, so the \emph{output multiplier cannot exceed one.}}
 {Multiplier increases with $\psi$, increases with $\alpha$, and decreases with $\sigma$}
 {Multiplier is close to one if
  \begin{itemize}
  \item $\psi$ large \emph{and} $\alpha$ close to one, OR
  \item $\sigma$ close to zero, OR
  \item $g$ close to one
  \end{itemize}}

\m{Consumption multiplier}
{ \emph{The consumption multiplier is never positive.}}

\mmmm{Parameter values}
{ $\alpha$, is widely believed to be around 0.7}
 {Estimates of the Frisch  elasticity of labor supply, $\psi$,  range from 0.2 to 1.0 }
 { $\sigma$ at the fairly standard value of 0.5}
  {$g$ = 0.2}

  \mm{Implied multipliers}
 {Output multiplier is about 0.4, at the low end of the range of empirical findings}
 {Consumption multiplier is $-0.6$, out of line with all of the empirical evidence}

 \mmmm{Boosting the multipliers}
  {Variations in the markup of price over cost}
     {Unemployment}
       {Complementarity of consumption and hours of work}
           {Negative response of investment to government purchases---dynamic model}

 \eee{Markup ratio declining with output}
 {\mu(y) = y^{-\omega}}
 {w =\frac{1}{y^{-\omega}} \alpha h^{-(1-\alpha)}}
{m_y = \frac{dy}{dg} = \frac{\alpha}{\alpha + \sigma(1-g)[1-(1+\omega)\alpha+1/\psi]}}

\mmm{Implications}
{Condition for output multiplier above one: $\omega > \frac{1-\alpha + 1/\psi}{\alpha}$}
{If $\psi = 0.5$, the markup elasticity $\omega$ needed to deliver an output multiplier of 1 is 3.3, far above the plausible range.}
 {With $\omega=0.5$, the output multiplier is 0.5 and the consumption multiplier is $-0.5$.}

\mmmm{Employment rate and hours}
{Framework in my 2009 JPE paper}
{Frisch labor supply function $h(w)$ with elasticity 0.7}
{Employment rate function $n(w)$ with elasticity 1.2}
{Elasticity of total hours of work $h(w)n(w)$ is 1.9}

\m{Implications}
{Output multiplier becomes 0.8 and the consumption multiplier $-0.2$, an important step toward realism}

\mmm{Complementarity}
{\[\frac{c^{1-1/\sigma}}{1-1/\sigma}-\chi c^{1-1/\sigma}h^{1+1/\psi} - \gamma \frac{h^{1+1/\psi}}{1+1/\psi}\]}
{Parameter values: $\sigma=0.4$, $\psi=1.54$, $\chi = 0.334$ and $\gamma=1.1$}
{Frisch elasticities:
\begin{itemize}
\item Own-price elasticity of consumption: $-0.5$
\item Wage elasticity of hours of work: $1.9$
\item Elasticity of consumption with respect to wage: $0.4$
\end{itemize}}

\m{Implications}
{With the negative of the elasticity of the markup, $\omega$, at 0.5, the output multiplier is 0.97 and the consumption multiplier is $-0.03$, figures easily consistent with the empirical evidence.}

\eeee{Dynamic model with investment}
{q_t = \kappa \frac{k_t - k_{t-1}}{k_{t-1}} + 1}
{b_t=q_{t-1}(r_t+\delta)-\Delta q_t.}
{(1 - \alpha) \frac{y_t}{\mu b_t}=k_{t-1}}
{k_t + \frac{\kappa}{2}\frac{(k_t-k_{t-1})^2}{k_{t-1}} = (1-\delta)k_{t-1} +y_t-c_t-g_t.}

\ee{Discounting and Euler equation}
{m_{t,t+1} = \beta \frac{c_{t+1}^{-1/\sigma}}{c_t^{-1/\sigma}}\mbox{ } \frac{1-\chi(1-1/\sigma)h_{t+1}^{1+1/\psi}} {1-\chi(1-1/\sigma)h_t^{1+1/\psi}}}
{(1+r_{t+1})m_{t,t+1}=1}

\e{Path of government purchases}
{g_t = \bar{g}+g\phi^t.}

\m{Adjustment cost}
{I take the parameter $\kappa$ to be 8 at a quarterly rate, corresponding to 2 at an annual rate, a representative value from the literature on this subject.}

\m{Implications}
{Output multiplier is 0.9 and consumption multiplier is zero, in line with the evidence.}

\xltab{params}{Parameter values}{5}

 \xltab{IRFs}{Impulse responses}{5}

 \m{Conclusion}
{The key to a substantial effect of government purchases on output is a combination of a markup that declines when output expands and elastic labor supply, possibly associated with a sticky wage.}

\m{Zero bound on nominal interest rate}
{At the zero bound, the output multiplier rises from 0.95 to 1.72 and the consumption multiplier from -0.07 to 0.26.}

\xltab{NKresults}{Markup response in an NK model}{4.5}

\eeee{Measuring the markup elasticity}
{\mbox{Marginal cost } =\frac{w}{\partial Y/\partial L}}
{\mu = \frac{p}{\frac{w}{\partial Y/\partial L}}}
 {= \frac{pY}{wL}\frac{L}{Y}\frac{\partial Y}{\partial L}}
 {=\frac{\alpha}{s}}

 \fig{CDmarkup}{Cobb-Douglas Index of Markup Ratio and Employment Rate, 1948-2009}{4}

\xltab{stim}{Effects of Stimulus Measure of February 2009 and of an Alternative}{3}

\end{document} 