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%	TITLE PAGE
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\title[]{Macro II} % The short title appears at the bottom of every slide, the full
% title is only on the title page

\author{Professor Griffy} % Your name
\institute[University at Albany, SUNY] % Your institution as it will appear on the bottom of
% every slide, may be shorthand to save space
{
UAlbany  \ % Your institution for the title page
}
\date{Spring 2024} % Date, can be changed to a custom date

\begin{document}

\begin{frame}
  \titlepage % Print the title page as the first slide
\end{frame}




% ----------------------------------------------------------------------------------------
%	PRESENTATION SLIDES
% ----------------------------------------------------------------------------------------

% ------------------------------------------------
\section{Introduction} % Sections can be created in order to organize your presentation into discrete blocks, all sections and subsections are automatically printed in the table of contents as an overview of the talk
% ------------------------------------------------

\begin{frame}
  \frametitle{Introduction}
  \begin{itemize}
  \item Midterm in a week!
  \item Homework next Tuesday.
  \item Today: Real Business Cycle Model
  \end{itemize}
\end{frame}

% ------------------------------------------------
\section{The RBC Model: Household's Dynamic Labor Supply Problem}

% \begin{itemize}
% \item Problem

% \begin{itemize}
% \item Owns and rents out capital

% \item Each member supplies up to 1 unit of labor

% \item Members receive utility from consumption and leisure

% \item Assume perfect foresight (for now)\pagebreak
% \end{itemize}

\begin{frame}\frametitle{Basic RBC Model}
\begin{itemize}
\item Household solves%
\begin{align}
\underset{\left\{ C_{t},I_{t},L_{t},K_{t+1}\right\} _{t=0}^{\infty }}{\max }%
& \sum_{t=0}^{\infty }\beta ^{t}N_{t}\left[ \ln \left( \frac{C_{t}}{N_{t}}%
\right) +\chi \frac{\left( 1-L_{t}/N_{t}\right) ^{1-\gamma }-1}{1-\gamma }%
\right]  \notag \\
s.t.\quad & C_{t}+I_{t}=r_{t}K_{t}+W_{t}L_{t},  \tag{BC} \\
& K_{t+1}=\left( 1-\delta \right) K_{t}+I_{t},  \tag{CA} \\
& L_{t}\in \left[ 0,N_{t}\right] ,  \notag \\
& K_{0}\;\text{given},\quad C_{t}\geq 0.  \notag
\end{align}
\item Parameter restrictions:$\hspace{0.3in}\chi >0$, $\gamma \geq 0$, $%
0<\beta <1$

\item $1-L_{t}/N_{t}$ is per capita leisure

\item Note that $K_{t}<0$ represents borrowing
\end{itemize}
\end{frame}

% \begin{frame}
% \begin{itemize}
% \item Combine (BC) and (CA) to get%
% \begin{align}
% C_{t}+K_{t+1}& =R_{t}K_{t}+W_{t}L_{t},  \tag{BC$^{\prime }$} \\
% R_{t}& \equiv r_{t}+\left( 1-\delta \right) .  \notag
% \end{align}

% \item Iterate on (BC$^{\prime }$) to get%
% \begin{eqnarray*}
% R_{t}K_{t} &=&C_{t}-W_{t}L_{t}+\frac{1}{R_{t+1}}R_{t+1}K_{t+1} \\
% &=&C_{t}-W_{t}L_{t} \\
% &&+R_{t+1}^{-1}\left[ C_{t+1}-W_{t+1}L_{t+1}+K_{t+2}\right] , \\
% &\vdots &
% \end{eqnarray*}%
% \end{itemize}
% \end{frame}

% \begin{frame}
% \begin{itemize}
% \item Iterating on (BC$^{\prime }$) will yield a bounded sum only if
% \begin{equation*}
% \underset{J\rightarrow \infty }{\lim }\left(
% \prod_{j=1}^{J-1}R_{t+j}^{-1}\right) K_{t+J}=0.
% \end{equation*}

% \item Define%
% \begin{align*}
% c_{t}& =\frac{C_{t}}{A_{t}N_{t}},\quad \ell _{t}=\frac{L_{t}}{N_{t}}, \\
% k_{t}& =\frac{K_{t}}{A_{t}N_{t}},\quad w_{t}=\frac{W_{t}}{A_{t}}.
% \end{align*}%
% \end{itemize}
% \end{frame}

\begin{frame}\frametitle{Basic RBC Model II}
\begin{itemize}
\item Assume constant growth in population and productivity%
\begin{eqnarray*}
N_{t} &=&N_{0}N^{t},\quad N_{0},N>0,\quad \beta N<1, \\
A_{t} &=&A_{0}A^{t},\quad A_{0},A>0.
\end{eqnarray*}

\item The per-effective-worker problem becomes:%
\begin{align*}
\hspace{-0.025in}\underset{\left\{ c_{t},k_{t+1},\ell _{t}\right\}
_{t=0}^{\infty }}{\max }& \sum\nolimits_{t=0}^{\infty }\left( \beta N\right)
^{t}\left[ \ln \left( A_{t}c_{t}\right) +\chi \frac{\left( 1-\ell
_{t}\right) ^{1-\gamma }-1}{1-\gamma }\right] , \\
s.t.\quad & c_{t}+ANk_{t+1}=R_{t}k_{t}+w_{t}\ell _{t}, \\
& \ell _{t}\in \left[ 0,1\right] ;\quad k_{0}\;\text{given},\quad c_{t}\geq
0, \\
& \underset{J\rightarrow \infty }{\lim }\left(
\prod_{j=1}^{J-1}R_{t+j}^{-1}\right) A_{t+J}N_{t+J}k_{t+J}=0.
\end{align*}%
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Solution}
\begin{itemize}

\item The first order conditions are
\begin{align}
\frac{1}{c_{t}}& =\beta A^{-1}\frac{1}{c_{t+1}}R_{t+1},  \tag{EE} \\
u^{\prime }(A_{t}c_{t})A_{t}w_{t}& =v^{\prime }\left( 1-\ell _{t}\right)
\notag \\
\Leftrightarrow \frac{1}{c_{t}}w_{t}& =\chi \left( 1-\ell _{t}\right)
^{-\gamma }.  \tag{LL}
\end{align}%
\item Euler equation and ``portfolio allocation''
\end{itemize}
\end{frame}

% \begin{frame}\frametitle{Balanced growth path}
% \begin{itemize}
% \item Data suggests that per capita output, capital and consumption grow at
% the rate of growth of $A.$ Per capita labor is constant

% \item Does this result hold here?

% \item Suppose that per-effective worker consumption $\left( c\right) $ and
% the effective wate $\left( w\right) $ are constant

% \item Then (LL) implies that per capita labor $\left( \ell \right) $ and per
% capital leisure $\left( 1-\ell \right) $ are constant

% \item Many utility functions do not deliver this
% \end{itemize}
% \end{frame}

\begin{frame}\frametitle{Effect of interest rate changes on savings}
  \begin{align*}
    \frac{1}{c_{t}}& =\beta A^{-1}\frac{1}{c_{t+1}}R_{t+1}
  \end{align*}
\begin{itemize}
\item Substitution effect: Increasing $R_{t+1}$ lowers the price of future
consumption, inducing substitution into the cheaper good (future
consumption), inducing more saving

\item Income effect

\begin{itemize}
\item Positive assets: Increasing $R_{t+1}$ raises future income and
consumption, lowers future $MU_{C},$ inducing less savings

\item Negative assets: Increasing $R_{t+1}$ reduces future income and
consumption, raises future $MU_{C},$ inducing more savings
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Effect of interest rate changes on savings}
  \begin{align*}
    \frac{1}{c_{t}}& =\beta A^{-1}\frac{1}{c_{t+1}}R_{t+1}
  \end{align*}
\begin{itemize}

\item General (empirical) consensus

\begin{itemize}
\item Consumers are net savers: the aggregate income effect of higher
interest rates is to lower saving

\item The substitution effect weakly dominates implying that savings
increases in interest rates
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Labor-leisure tradeoff}
  \begin{align*}
    \frac{1}{c_{t}}w_{t}& =\chi \left( 1-\ell _{t}\right)
^{-\gamma }
  \end{align*}
\begin{itemize}
\item $MU_{C}\times wage=MU_{L}$

\item Wealth effects: Holding $w_{t}$ constant, higher permanent income
raises current consumption, lowers marginal benefit of working

\begin{itemize}
\item Higher assets

\item Higher current or future non-labor income

\item Higher current or future labor income

\item Increasing non-labor component of permanent income lowers labor
supply
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Effects of increasing the current wage ($MU_{C}\times wage=MU_{L})$}
  \begin{align*}
    \frac{1}{c_{t}}w_{t}& =\chi \left( 1-\ell _{t}\right)
^{-\gamma }
  \end{align*}

\begin{itemize}
\item Substitution effect: holding $MU_{C}$ constant, and raising $w_{t}$
increases marginal benefit of working

\item Income effect: raising $w_{t}$ increases $y_{t}^{P},$ lowers $MU_{C}$
and marginal benefit of working
\end{itemize}
\end{frame}

\begin{frame}\frametitle{General (empirical) consensus}
  \begin{align*}
    \frac{1}{c_{t}}w_{t}& =\chi \left( 1-\ell _{t}\right)
^{-\gamma }
  \end{align*}

\begin{itemize}
\item Temporary wage increases generate more hours due to small income effect

\item Permanent wage increases generate no more hours because income and
substitution effects offset. Consistent with long-term data where wage rises
but labor hours do not

\item Our specification delivers this
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Labor supply curve}
\begin{itemize}
\item Rearrange (LL) to get%
\begin{equation*}
\ell _{t}=1-\left( c_{t}\chi \right) ^{1/\gamma }w_{t}^{-1/\gamma }.
\end{equation*}

\item Frisch supply curve%
\begin{equation*}
\ell _{t}=f\left( w_{t},MU_{C}\right) =f\left( w_{t},y_{t}^{P}\right) .
\end{equation*}

\begin{itemize}
\item Consider effects of changing wages with $MU_{C}$ held constant

\item Wealth effects ignored

\item Note: $MU_{C}$ can depend on things besides $y_{t}^{P},$ although it
does not here
\end{itemize}
\end{itemize}
\end{frame}


% \begin{frame}
% \begin{itemize}
% \item \textquotedblleft Regular\textquotedblright\ labor supply curve:
% Recognizes that $\Delta \text{wages}\Rightarrow \Delta \text{wealth}%
% \Rightarrow \Delta MU_{C}$

% % \begin{itemize}
% \item Persistent wage changes produce a steeper curve

% \item Both types of curves shifted by non-labor wealth\end{itemize}
% \end{frame}

\begin{frame}\frametitle{Intertemporal elasticity of substitution of labor ($IES_{L}$ or Frisch
elasticity)}

\begin{itemize}
\item Measures willingness to vary labor over time, holding $MU_{C}$
(wealth) constant%
\begin{equation*}
IES_{L}=\left. \frac{d\ln \left( \ell _{1}/\ell _{2}\right) }{d\ln \left(
w_{1}/w_{2}\right) }\right\vert _{MU_{C}}.
\end{equation*}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Derivation}
\begin{align*}
\frac{1}{c_{t}}& =\beta A^{-1}\frac{1}{c_{t+1}}R_{t+1},  \tag{EE} \\
\frac{1}{c_{t}}w_{t}& =\chi \left( 1-\ell _{t}\right)
^{-\gamma }.  \tag{LL}
\end{align*}%
\begin{itemize}
\item Combine (EE) and (LL)%
\begin{equation*}
\chi \frac{\left( 1-\ell _{1}\right) ^{-\gamma }}{w_{1}}=\beta A^{-1}\chi
\frac{\left( 1-\ell _{2}\right) ^{-\gamma }}{w_{2}}R_{2}.
\end{equation*}%
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Portfolio Allocation}
\begin{itemize}
\item Note that the household smooths leisure as well as consumption

\item For example, interest rates affect labor supply

\item Rearrange the previous equation%
\begin{eqnarray*}
\beta A^{-1}R_{2}\left( \frac{w_{1}}{w_{2}}\right) &=&\frac{\left( 1-\ell
_{1}\right) ^{-\gamma }}{\left( 1-\ell _{2}\right) ^{-\gamma }}, \\
\ln \left( \beta A^{-1}R_{2}\right) +\ln \left( \frac{w_{1}}{w_{2}}\right)
&=&-\gamma \ln \left( 1-\ell _{1}\right) +\gamma \ln \left( 1-\ell
_{2}\right) , \\
&=&-\gamma \big[\ln \left( 1-\exp \left( \ln \ell _{1}\right) \right) \\
&&\quad \quad -\ln \left( 1-\exp \left( \ln \ell _{2}\right) \right) \big].
\end{eqnarray*}%
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item Implicitly differentiate:%
\begin{eqnarray*}
d\ln \left( \frac{w_{1}}{w_{2}}\right) &=&\gamma \frac{\exp \left( \ln
\left( \ell _{1}\right) \right) }{1-\exp \left( \ln \left( \ell _{1}\right)
\right) }d\ln \left( \ell _{1}\right) \\
&&-\gamma \frac{\exp \left( \ln \left( \ell _{2}\right) \right) }{1-\exp
\left( \ln \left( \ell _{2}\right) \right) }d\ln \left( \ell _{2}\right) .
\end{eqnarray*}

\item Now assume that $\ell _{1}=\ell _{2}=\ell $
\begin{eqnarray*}
d\ln \left( \frac{w_{1}}{w_{2}}\right) &=&\gamma \frac{\ell }{1-\ell }d\ln
\left( \ell _{1}\right) -\gamma \frac{\ell }{1-\ell }d\ln \left( \ell
_{2}\right) \\
&=&\gamma \frac{\ell }{1-\ell }\left[ d\ln \left( \ell _{1}\right) -d\ln
\left( \ell _{2}\right) \right] \\
&=&\gamma \frac{\ell }{1-\ell }d\ln \left( \frac{\ell _{1}}{\ell _{2}}%
\right) .
\end{eqnarray*}%
\end{itemize}
\end{frame}


\begin{frame}
\begin{itemize}
\item Finally, we get%
\begin{eqnarray*}
IES_{L} &=&\left. \frac{d\ln \left( \ell _{1}/\ell _{2}\right) }{d\ln \left(
w_{1}/w_{2}\right) }\right\vert _{MU_{C}} \\
&=&\frac{1}{\gamma }\left( \frac{1-\ell }{\ell }\right) .
\end{eqnarray*}

\item Tip: if $\gamma =0$ such that utility is linear in leisure, then $%
IES_{L}$ is infinite
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Non-Separable Preferences (Low, 2005)}

\begin{itemize}
\item Household solves%
\begin{align*}
\hspace{-0.025in}\underset{\left\{ c_{t},k_{t+1},\ell _{t}\right\}
_{t=0}^{\infty }}{\max }& E_{0}\left( \sum\nolimits_{t=0}^{\infty }\left(
\beta N\right) ^{t}u\left( A_{t}c_{t},1-\ell _{t}\right) \right) , \\
s.t.\quad & c_{t}+ANk_{t+1}=R_{t}k_{t}+w_{t}\ell _{t}, \\
& \ell _{t}\in \left[ 0,1\right] ,
\end{align*}

\item and the other usual constraints

\item The first-order conditions are%
\begin{eqnarray*}
u_{Ac}\left( A_{t}c_{t},1-\ell _{t}\right) A_{t} &=&\lambda _{t}, \\
u_{1-\ell }\left( A_{t}c_{t},1-\ell _{t}\right) &=&\lambda _{t}w_{t}, \\
\lambda _{t} &=&\beta A^{-1}E_{t}\left( R_{t+1}\lambda _{t+1}\right) .
\end{eqnarray*}%
where $\lambda _{t}$ is the multiplier on the budget constraint
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}

\item Benchmark utility specification is isoelastic Cobb-Douglas%
\begin{equation*}
u\left( A_{t}c_{t},1-\ell _{t}\right) =\frac{1}{1-\gamma }\left(
(A_{t}c_{t})^{\chi }(1-\ell _{t})^{1-\chi }\right) ^{1-\gamma }
\end{equation*}

\item The derivatives of this function are%
\begin{eqnarray*}
u_{Ac} &=&\chi (1-\gamma )\frac{1}{A_{t}c_{t}}u\left( A_{t}c_{t},1-\ell
_{t}\right) , \\
u_{1-\ell } &=&(1-\chi )(1-\gamma )\frac{1}{1-\ell _{t}}u\left(
A_{t}c_{t},1-\ell _{t}\right) , \\
u_{1-\ell ,Ac} &=&\frac{\chi (1-\chi )(1-\gamma )^{2}}{(1-\ell
_{t})A_{t}c_{t}}u\left( A_{t}c_{t},1-\ell _{t}\right) .
\end{eqnarray*}%
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}

\item Key issue: Is consumption at time-$t$ a substitute or a complement for
leisure at time-$t$?

\begin{itemize}
\item This depends on the sign of the cross-partial derivative $u_{Ac,1-\ell
}(\cdot )$: $u_{Ac,1-\ell }>0$ implies complements

\item For the benchmark specification%
\begin{align*}
u_{1-\ell ,Ac}=\chi & (1-\chi )(1-\gamma ) \\
& \times (A_{t}c_{t})^{\chi (1-\gamma )-1}(1-\ell _{t})^{(1-\chi )(1-\gamma
)-1}.
\end{align*}

\item This term will be negative if $\gamma >1$

\item Baseline assumption: $\gamma =2.2,$ implying consumption and leisure
are substitutes
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}

\item Combining first-order conditions yields%
\begin{equation*}
u_{1-\ell }\left( A_{t}c_{t},1-\ell _{t}\right) =u_{Ac}\left(
A_{t}c_{t},1-\ell _{t}\right) A_{t}w_{t}.
\end{equation*}

\item With the baseline preferences, this becomes%
\begin{align*}
\frac{1-\chi }{1-\ell _{t}}& =\chi \frac{w_{t}}{c_{t}}, \\
\Rightarrow \ell _{t}& =1-\left( \frac{1-\chi }{\chi }\right) \frac{c_{t}}{%
w_{t}}.
\end{align*}

\item This specification produces constant hours along a balanced growth path

% \begin{itemize}
\item King et al (1989) provide a general set of conditions\end{itemize}
\end{frame}

\begin{frame}\frametitle{Data Puzzle 1}

\begin{itemize}
\item Consumption tracks income over the life-cycle: Inconsistent with
consumption smoothing

\item If consumption and leisure are substitutes, people working more hours
will consume more implying that consumption tracks income (Heckman, 1974)
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Data Puzzle 2}

\begin{itemize}
\item There is a discrete drop in consumption immediately after retirement
which is inconsistent with consumption smoothing

\item If consumption and leisure are substitutes, then consumption will drop
at retirement (French 2005, Aguiar and Hurst 2005)
\end{itemize}
\end{frame}


\begin{frame}\frametitle{Data Puzzle 3}
\begin{itemize}

\item Low-wage young people work many hours; high-wage old people work fewer
hours: Inconsistent with the intertemporal substitution of labor

\item Young people work long hours to fund precautionary saving

\item This precautionary saving builds up assets and reduces the need to
work when old

\item This result does not require non-separable preferences

\item It does require life-cycle (not infinite-horizon) framework with low
initial wealth
\end{itemize}
\end{frame}

% --------------------------------------------------------------

\begin{frame}
  \frametitle{Conclusion}
  \begin{itemize}
  \item Nice weather!!!!! Woooooo!
  \item Midterm in one week!
  \item Homework in one week!
  \end{itemize}
\end{frame}


\end{document}
