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\title[Real Business Cycles]{MA Advanced Macroeconomics: \\ Introduction to Real Business Cycles}
\author{Karl Whelan}
\institute[UCD]{School of Economics, UCD}
\date{February 10, 2011}
\begin{document}
\frame{\titlepage}



\frame{
\frametitle{Working Through A DSGE Model}
\small{
\begin{itemize}

\item  We have described methods for solving and simulating linear models with lags, leads
and rational expectations.

\item Now it is time to go through a particular model to see how these methods get combined
with economic theory.

\item Specifically, we will work through a version of the Real Business Cycle (RBC)\ model---introduced
in a famous 1982 paper by Finn Kydland and Edward Prescott---is the original
DSGE\ model.\footnote{%
``Time to Build and Aggregate Fluctuations,'' \textit{Econometrica},
November 1982, Volume 50, pages 1345-1370. This paper was cited in the 2004
Nobel prize award given to Kydland and Prescott.} 

\item We will set out a basic RBC model and discuss how the
model's first-order conditions can be turned into a system of linear
difference equations of the form we know how to solve.  

\item This will require explaining another
new technique, known as \textit{log-linearization}.

\item While many now question the specific assumptions underlying the early RBC
models, the \textit{methodology} has endured.


\end{itemize}
}
}


\part{Introduction to the RBC Model}
\frame{\partpage}

\frame{
\frametitle{An RBC Model}
\small{
\begin{itemize}

\item The basic RBC\ model assume perfectly functioning competitive markets, so the outcomes generated by decentralized decisions by firms and households can be replicated as the solution to a social planner
problem. 

\item The social planner wants to maximize 
\begin{equation*}
E_t\left[ \sum_{i=0}^\infty \beta ^i\left( U(C_{t+i}) - V (N_{t+i}) \right) \right]
\end{equation*}
where $C_t$ is consumption, $N_t$ is hours worked, and $\beta $ is the
representative household's rate of time preference. 

\item The economy faces constraints described by 
\begin{eqnarray*}
Y_t &=&C_t+I_t=A_tK_{t-1}^\alpha N_t^{1-\alpha } \\
K_t &=&I_t+\left( 1-\delta \right) K_{t-1}
\end{eqnarray*}
and a process for the technology term $A_t$, usually a log-linear AR(1):
\begin{equation*}
\log A_t=\left( 1-\rho \right) \log A^{*}+\rho \log A_{t-1}+\epsilon _t
\end{equation*}

\end{itemize}
}
}




\frame{
\frametitle{Discussion of the RBC Approach: 1}
\small{
\begin{itemize}

\item The RBC approach is often criticised but it is worth understanding the reasons why
it is a useful baseline model.

\item \textit{Criticism: Perfect Markets and Rational Expectations:}
\begin{itemize}
	\item Can the economy really be characterized as a perfectly competitive market equilibrium
solution describing the behaviour of a set of completely optimizing rational agents? Surely we know that markets are not
always competitive and surely people are not always completely rational in their economic decisions?
	
\end{itemize}

	\item  The RBC model should be seen as a \textit{benchmark} against which
more complicated models can be assessed. If a model with optimizing agents
and instantaneous market clearing can explain the business cycle, then can
market imperfections such as sticky prices really be seen as crucial to
understanding macroeconomic fluctuations? 

	\item Imperfect competition means the decentralized market outcome cannot be characterized as the outcome of a
social planner problem. So, one needs to derive the FOCs from separate
modelling of the decisions of firms and households. But this is easily done.
\end{itemize}
}
}




\frame{
\frametitle{Discussion of the RBC Approach: 2}
\small{
\begin{itemize}

\item \textit{Criticism: Monetary and Fiscal Policy}:

\begin{itemize}
	\item RBC models exhibit complete
monetary neutrality, so there is no role at all for monetary policy,
something which many people think is crucial to understanding the
macroeconomy. We haven't put government spending in this model, but if we
did, the model would exhibit Ricardian equivalence, so the effects of fiscal
policy would be different from what most people imagine them to be.
\end{itemize}

	\item However, most modern models that build on the RBC approach have introduced mechanisms that allow monetary and fiscal policy to 	have Keynesian effects.  For instance, most of the DSGE models used these days feature various forms of sticky prices and sticky wages, which lead to real effects for monetary policy.
\end{itemize}
}
}



\frame{
\frametitle{Discussion of the RBC Approach: 3}
\small{
\begin{itemize}

\item \textit{Criticism: Skepticism about Technology Shocks}:

\begin{itemize}
	\item RBC models give primacy
to technology shocks  as the source of economic
fluctuations (all variables apart from $A_t$ are deterministic). But what are these shocks? 

\item Is it really credible that all economic fluctuations, including recessions, are just an optimal response to
technology shocks? If we were
experiencing these big shocks wouldn't we be reading about them in the
newspapers? Wouldn't recessions have to be periods in which their is
outright technological regress, so that (for some reason)\ firms are using
less efficient technologies than previously---is this credible?
	\end{itemize}
	
	
\item  Technology shocks are probably more important than one might
think at first. Growth theory teaches us that increases in TFP are the
ultimate source of long-run growth in output per hour. Is there any
particular reason to think that these increases have to take place in a
steady trend-like manner? Put this way, random fluctuations in TFP growth
doesn't seem so strange. Also, at least in theory, RBC models could generate
recessions without outright declines in technology. This can happen if the
elasticity of output with respect to technology is greater than one.
\end{itemize}
}
}
	


\part{Solving the RBC Model}
\frame{\partpage}
	

\frame{	
\frametitle{A Technical Note}
\small{
\begin{itemize}

\item Our problem involves maximizing
\begin{equation*}
E_t\left[ \sum_{i=0}^\infty \beta ^i\left( U(C_{t+i}) - V (N_{t+i}) \right) \right]
\end{equation*}
where the uncertainty here relates to not knowing future values of $A_t$.


\item Technically, the best way to solve these problems is using \textit{stochastic
dynamic programming} but I don't have time to teach that.  Instead, I will effectively treat
it as a deterministic problem and then substitute $E_t X_{t+i}$ for $X_{t+i}$.

\item Justification?  Suppose
\begin{equation*}
G\left( x\right) = \sum_{k=1}^Np_kF\left( a_k,x\right)
\end{equation*}

\item This is maximized by setting 
\begin{equation*}
G^{\prime }\left( x\right) = \sum_{k=1}^Np_kF^{\prime }\left( a_k,x\right)
=E_tF^{\prime }\left( x\right) =0
\end{equation*}
So, the FOCs for for maximizing $E_tF\left( x\right) $ are just $E_tF^{\prime
}\left( x\right) =0$.

\end{itemize}
}
}



\frame{
\frametitle{Formulating the Social Planner's Problem}
\small{
\begin{itemize}

\item Remember our two constraints
\begin{eqnarray*}
Y_t &=&C_t+I_t=A_tK_{t-1}^\alpha N_t^{1-\alpha } \\
K_t &=&I_t+\left( 1-\delta \right) K_{t-1}
\end{eqnarray*}

\item We can simplify the problem by combining them into one equation:
\[
A_{t}K_{t-1}^\alpha N_{t}^{1-\alpha} = C_t + K_t - \left( 1-\delta \right) K_{t-1} 
\]

\item We can then formulate the social planner's problem as a Lagrangian problem involving 
picking a series of values for consumption
and labour, subject to satisfying a series of constraints of the form just described:
\begin{eqnarray*}
L & = & E_t \sum_{i=0}^\infty \beta ^i  \left[U(C_{t+i}) - V (N_{t+i}) \right] \\
  &  & + E_t \sum_{i=0}^\infty \beta ^i \lambda _{t+i} \left[   A_{t+i}K_{t+i-1}^\alpha N_{t+i}^{1-\alpha
}+\left( 1-\delta \right) K_{t+i-1}-C_{t+i}-K_{t+i} \right]
\end{eqnarray*}

\end{itemize}
}
}



\frame{
\frametitle{How to Get the First-Order Conditions}
\small{
\begin{itemize}

\item This equation might look a bit
intimidating. It involves an infinite sum, so technically there is an
infinite number of first-order conditions for current and expected future
values of $C_t,K_t$ and $N_t$. 

\item But the problem is less hard than
this makes it sound, Note that the time-$t$ variables appear in this sum as 
\begin{eqnarray*}
U(C_t) - V(N_t) +\lambda _t\left( A_tK_{t-1}^\alpha
N_t^{1-\alpha }-C_t-K_t +\left( 1-\delta \right) K_{t-1} \right)  \\ 
+\beta E_t\left[ \lambda_{t+1} \left( A_{t+1}K_t^\alpha N_{t+1}^{1-\alpha }+\left( 1-\delta \right)
K_t \right) \right]  
\end{eqnarray*}


\item After that, the time-$t$ variables don't ever appear again. \ So, the FOCs
for the time-$t$ variables consist of differentiating this equation 
with respect to these variables and setting the derivatives equal to
zero. 

\item Then, the time $t+n$ variables appear exactly as the time $t$
variables do, except that they are in expectation form and they are
multiplied by the discount rate $\beta ^n$. But this means the FOCs for the time $t+n$
variables will be identical to those for the time $t$ variables. So
differentiating this equation gives us the equations for the optimal dynamics at all times.


\end{itemize}
}
}




\frame{
\frametitle{The First-Order Conditions}
\small{
\begin{itemize}

\item Differentiating
\begin{eqnarray*}
U(C_t) - V(N_t) +\lambda _t\left( A_tK_{t-1}^\alpha
N_t^{1-\alpha }-C_t-K_t +\left( 1-\delta \right) K_{t-1} \right)  \\ 
+\beta E_t\left[ \lambda_{t+1} \left( A_{t+1}K_t^\alpha N_{t+1}^{1-\alpha }+\left( 1-\delta \right)
K_t \right) \right]  
\end{eqnarray*}

\item We get following first-order conditions:
\begin{eqnarray*}
\frac{\partial L}{\partial C_t} &:& U'\left(C_t \right) -\lambda _t=0   \label{cfoc} \\
\frac{\partial L}{\partial K_t} &:& -\lambda _t + \beta E_t\left[ \lambda _{t+1} \left( \alpha \frac{Y_{t+1}}{K_t}%
+ 1-\delta \right) \right] =0
\label{kfoc} \\
\frac{\partial L}{\partial N_t} &:&- V' \left(N_t \right) +\left( 1-\alpha \right) \lambda _t%
\frac{Y_t}{N_t}=0 \\
\frac{\partial L}{\partial \lambda _t} &:&A_tK_{t-1}^\alpha N_t^{1-\alpha
}-C_t-K_t +\left( 1-\delta \right) K_{t-1}=0
\end{eqnarray*}


\end{itemize}
}
}




\frame{
\frametitle{The Keynes-Ramsey Condition}
\small{
\begin{itemize}

\item Define the marginal value of an additional unit of capital next year as 
\begin{equation*}
R_{t+1}=\alpha \frac{Y_{t+1}}{K_t}+1-\delta
\end{equation*}

\item Then the FOC for capital can be written as 
\begin{equation*}
\lambda _t=\beta E_t\left( \lambda _{t+1}R_{t+1}\right)
\end{equation*}

\item This can then be combined with the FOC for consumption to give 
\begin{equation*}
U'(C_t) =\beta E_t\left[ U'(C_{t+1}) R_{t+1}\right]
\end{equation*}

\item Interpretation:
\begin{itemize}
	\item  Decrease consumption by $\Delta$ today, at a loss of $U'(C_t) \Delta$ in utility.
	\item Invest to get $R_{t+1}\Delta$ tomorrow.
	\item Worth $\beta E_t\left[ U'(C_{t+1}) R_{t+1} \Delta \right]$ in terms of today's utility. 
	\item Along an optimal path, must be indifferent.
\end{itemize}

\end{itemize}
}
}




\frame{
\frametitle{CRRA Consumption and Separable Consumption-Leisure}
\small{
\begin{itemize}

\item We are going to work with a utility function of the form:
\[
U(C_t) - V(N_t) = \frac{C_t^{1-\eta }}{1-\eta }-aN_t
\]

\item This formulation of the Constant Relative Risk Aversion (CRRA) utility from consumption and
separate disutility from labour turns out to be necessary for the model to have a stable growth
path solution.

\item The Keynes-Ramsey condition becomes
\[
C_t^{-\eta }=\beta E_t\left( C_{t+1}^{-\eta }R_{t+1}\right)
\]

\item And the condition for optimal hours worked becomes
\[
-a+\left( 1-\alpha \right) C_t^{-\eta } \frac{Y_t}{N_t}=0
\]

\end{itemize}
}
}



\frame{
\frametitle{The Full Set of Model Equations}
\small{
The RBC model can then be defined by the following six equations (three
identities describing resource constraints, one a definition, and two FOCs describing optimal behaviour) 
\begin{eqnarray*}
Y_t &=&C_t+I_t  \label{yci} \\
Y_t &=&A_tK_{t-1}^\alpha N_t^{1-\alpha }  \label{prodfunc} \\
K_t &=&I_t+\left( 1-\delta \right) K_{t-1} \\
R_t &=&\alpha \frac{Y_t}{K_{t-1}}+1-\delta  \label{rdef} \\
C_t^{-\eta } & = & \beta E_t\left( C_{t+1}^{-\eta }R_{t+1}\right) \\
\frac{Y_t}{N_t} &=&\frac a{1-\alpha }C_t^\eta 
\end{eqnarray*}
and the process for the technology variable
\begin{equation*}
\log A_t=\left( 1-\rho \right) \log A^{*}+\rho \log A_{t-1}+\epsilon _t
\end{equation*}

}
}
\end{document}


