\begin{equation} \ifnumber \label{eq:kBartp1i}
\bar{\mathbf{k}}_{t+1,i} =
\begin{cases}
         \mathbf{a} _{t,i}/\omega & \text{for period $t+1$ updaters}
\\  \bar{\mathbf{a}}_{t,i}/\omega & \text{for period $t+1$ nonupdaters}
\end{cases}
\end{equation}
\input eq/kBartp1i.tex



\subsection{Dynamic Stochastic General Equilibrium Model}

The PE/SOE model takes aggregate wages and interest rates
to be exogenous.  This section examines an alternative approach, which is to begin with a standard
representative agent dynamic stochastic general equilibrium model and
modify it to permit sticky expectations.  Idiosyncratic wage and
productivity shocks are suppressed for
tractability.\footnote{Developing a model with fully realistic
  idiosyncratic transitory and permanent shocks as well as
  frictionless expectations general equilibrium dynamics is a very
  formidable problem; see \cite{krusell&smith:hetero} for a version
  with simple idiosyncratic shocks.}

We assume a Cobb-Douglas aggregate production function; aggregate labor supply and aggregate output are
respectively
\begin{eqnarray}
    \mathbf{L}_{t} & = & \Theta_{t} {P}_{t}
\\ F(\mathbf{K}_{t},\mathbf{L}_{t}) & = & \mathbf{K}_{t}^{\varepsilon}\mathbf{L}_{t}^{1-\varepsilon}
%\\ & = & r_{t} \mathbf{K}_{t} + W_{t} \mathbf{L}_{t}
\end{eqnarray}

Using notation consistent with that above, market resources and the
transition are defined by
\begin{eqnarray}
    \mathbf{M}_{t}  & = & \mathbf{K}_{t}+F(\mathbf{K}_{t},\mathbf{L}_{t})
%\\ & = & R_{t} \mathbf{K}_{t} + W_{t} \mathbf{L}_{t}
\\  \mathbf{A}_{t}  & = & \mathbf{M}_{t}-\mathbf{C}_{t}
\\  \mathbf{K}_{t+1} & = & \daleth \mathbf{A}_{t}
\end{eqnarray}
where the only novelty is the introduction of a capital depreciation factor $\daleth$ (necessary
for calibration purposes).

\subsubsection{Frictionless Expectations}

Normalizing market resources by permanent productivity,
\begin{eqnarray}
   M_{t} & = & \mathbf{M}_{t}/{P}_{t}
\\       & = &(\mathbf{K}_{t}+\mathbf{K}_{t}^{\varepsilon}(\Theta_{t}{P}_{t})^{1-\varepsilon})/{P}_{t}
\\ & = & K_{t}+K_{t}^{\varepsilon}\Theta_{t}^{1-\varepsilon} \label{eq:trueM}
\\ & = & K_{t}+\Theta_{t}^{1-\varepsilon}(\varepsilon K_{t}^{\varepsilon-1} K_{t} + (1-\varepsilon) K_{t}^{\varepsilon})
\\ & = & \underbrace{R_{t}}_{\equiv 1+\varepsilon K_{t}^{\varepsilon-1}\Theta^{1-\varepsilon}_{t} } K_{t} + \underbrace{W_{t}}_{\equiv (1-\varepsilon)K_{t}^{\varepsilon}\Theta_{t}^{-\varepsilon}} \Theta_{t}
\end{eqnarray}
so the normalized frictionless representative agent's problem has a structure parallel to that of the frictionless
PE/SOE problem (cf. \eqref{eq:scaledv})
\begin{eqnarray}
  V(M_{t}) & = & \max_{C_{t}} \left\{ u(C_{t}) + \beta E_{t}[(\ifGG{G}{} \Psi_{t+1})^{1-\rho} V(M_{t+1})]\right\}
\\ & \mbox{s.t.} & \nonumber
\\ A_{t} & = & M_{t}-C_{t}
\\ {K}_{t+1} & = & A_{t} \daleth/\ifGG{G}{} \Psi_{t+1}
\\ M_{t+1} & = & R_{t+1} {K}_{t+1}+ W_{t+1} \Theta_{t+1}
\end{eqnarray}
and the process for consumption can be shown to satisfy the Euler equation
\begin{eqnarray}
  C_{t}^{-\rho} & = & \beta \mathbf{E}_{t}[\overbrace{\mathcal{R}_{t+1}}^{\equiv R_{t+1} \daleth} (\ifGG{G}{} \Psi_{t+1} C_{t+1})^{-\rho} ] \label{eq:DSGEuler}
\end{eqnarray}
%yielding a policy function $C(M_{t}).$

Making the usual log approximations, defining $\mathbf{E}_{t}[r_{t+1}]
\equiv \mathbf{E}_{t}[\mathcal{R}_{t+1}-1]$, and implicitly defining
the time preference rate from $\beta = 1/(1+\tau)$, \eqref{eq:DSGEuler} can
be approximated by
\begin{eqnarray}
  \Delta \log \mathbf{C}_{t+1} & \approx & \rho^{-1}(\mathbf{E}_{t}[r_{t+1}]-\tau) + \epsilon_{t+1}, \label{eq:DSGERandom}
\end{eqnarray}
so that the random walk proposition in this context translates to a claim that
aggregate consumption growth is unpredictable using any variable aside
from expected real interest rates.

\subsubsection{Sticky Expectations}

The sticky-expectations representative agent updates his perceptions
of transitory and permanent labor productivity according to the same
equations that characterize the evolution of the average perceptions in the PE/SOE model:
\begin{eqnarray}
   \bar{{P}}_{t+1} & = & \gamma {P}_{t+1} + (1-\gamma) \bar{{P}}_{t} \ifGG{G}{}
\\ \bar{\Theta}_{t+1} & = & 1 + \gamma (\Theta_{t} -1)
\end{eqnarray}

Perceived aggregate capital dynamics are also the same (cf.\ \eqref{eq:barKtp1SOE}), except that depreciation must be incorporated:
\begin{eqnarray}
  \label{eq:barKtp1DSGE}
  \bar{\mathbf{K}}_{t+1} & = & \left(\gamma \mathbf{A}_{t} + (1-\gamma) \bar{\mathbf{A}}_{t}\right)\daleth.
\end{eqnarray}




Defining normalized perceived variables as before (e.g.\ $\bar{M}_{t+1}=\bar{\mathbf{M}}_{t+1}/\bar{{P}}_{t+1}$),
we assume that the representative agent's perception of $\bar{M}$ reflects
a correct understanding of the production function \eqref{eq:trueM},
\begin{eqnarray}
    \bar{M}_{t} & = & \bar{K}_{t}+\bar{K}_{t}^{\varepsilon}\bar{\Theta}_{t}^{1-\varepsilon}
\end{eqnarray}
and, as before, the assumption is that the representative agent
behaves according to the decision rule associated with the
frictionless solution, so that
\begin{eqnarray}
   \mathbf{C}_{t} & = & C(\bar{M}_{t})\bar{{P}}_{t},
\end{eqnarray}
yielding
\begin{eqnarray}
  \bar{\mathbf{A}}_{t} & = & \bar{\mathbf{M}}_{t}-\mathbf{C}_{t}.
\end{eqnarray}

\begin{comment}
It is tempting to interpret this as merely reflecting stickiness in
real wages, and indeed wages are sticky here in the sense that
they do not react immediately to productivity.  However, a model with
exogenous real wage stickiness would have quite different implications
from the model considered here; in that framework the
representative agent would instantly recognize an improvement in
permanent productivity, and consumption would instantly move most of
the way toward its new permanent level.  Here, instead, consumption
reacts sluggishly to the shock because the shock is not immediately
perceived.
\end{comment}


\section{Calibration and Equilibrium}

One insight from the recent precautionary saving literature is that
theoretical conclusions about importance of idiosyncratic risk depend
partly on model calibration.  In models calibrated so that the typical
consumer holds a large amount of wealth, behavior closely resembles
behavior of a representative agent economy calibrated to match an
aggregate wealth to income ratio~(\cite{aiyagari:ge};
\cite{krusell&smith:hetero}; henceforth, the `macro' literature).  If
the model is calibrated to match micro data (and richer
parameterizations of the income process that match micro data are
used), behavior diverges in quantitatively and qualitatively important
ways from the predictions of perfect foresight
models~(\cite{zeldes:qje}; \cite{carroll:brookings};
\cite{gourinchas&parker:lifecycle}; henceforth, the `micro'
literature).

We will begin by calibrating the DSGE model in a traditional way,
then specify the additional parameters necessary for solving the micro
model, with the aim of showing that the model's main results are robust
to alternative calibration choices.

\subsection{Calibration}

If there were no aggregate shocks, the DSGE model would have a deterministic
steady state satisfying the Euler equation with $C_{t+1}=C_{t}$:
\begin{eqnarray}
    \mathcal{R} \beta \ifGG{{G}^{-\rho}}{} & = & 1
\ifGG{\\  \beta & = & {G}^{\rho}/\mathcal{R}}{}.
\end{eqnarray}

A standard calibration for DSGE models is to choose a discount parameter
so that the perfect foresight steady state model's ratio of capital
to annual output is around 3 (which corresponds to 12 in our quarterly
model), implying
\begin{eqnarray}
  K/K^{\varepsilon} & = & 12
\\ K & = & 12^{1/(1-\varepsilon)}
\end{eqnarray}
which implies a net-of-depreciation interest rate of about 2 percent
annually.\footnote{$[0.975 (1+0.36/12)]^{4} \approx 1.02$}



% Calibration
We fix the aggregate interest factor to $R=1.02^{1/4}$, which is close
to the value implied by the general equilibrium model considered
below.  The term $\beta G^{-\rho} \mathbf{E}[(\Psi \psi)^{-\rho}]$
plays a role similar to that of the pure time preference factor in
a model with no growth or uncertainty.  We somewhat arbitrarily
calibrate the model so that this term has a value of 0.99 per quarter,
corresponding to a common calibration for the pure time preference
factor in the perfect foresight model.


% Intuitively, this criterion merely requires that the average wealth-to-income ratio not head to infinity; the condition can therefore be characterized as an `impatience' requirement.

The paper therefore considers two parameterizations of
the model, one designed to produce a large aggregate wealth-to-income
ratio typical of the `macro' literature, and one designed
to generate a wealth-to-income ratio at the lower range of those
matched in the `micro' literature.

For simplicity, the difference in the calibrations is confined to a
single parameter, the time preference factor; however, unreported
experiments confirm that similar conclusions emerge if the
differing impatience were generated by alternative values of $R$ or
$G$.

Table~\ref{table:calib} reports the baseline parameter values used
for both calibrations, along with some basic statistics about the
ergodic distributions of wealth produced in steady state by the two
economies.  The patient calibration produces a wealth ratio ratio of
about 12 (recall that this is a quarterly model, so this corresponds
to a ratio of wealth to annual income of about 4).  The impatient
calibration produces a wealth ratio of about 2, which is low even by
the standards of the micro literature.

\begin{table}
\caption{Steady State Statistics}
\label{table:calib}
\end{table}


\subsection{Utility Costs of Sticky Expectations}

[Show that utility costs of updating epidemiologically are small]


\section{Simulated Empirical Tests}

\subsection{Macro Benchmarks}

A large literature in the 1980s attempted to test the CRRA-utility-revised
random walk proposition \eqref{eq:DSGERandom} by estimating aggregate regressions of the form
\begin{eqnarray}
  \Delta \log \mathbf{C}_{t+1} & = & \alpha + \vartheta \mathbf{E}_{t}[r_{t+1}] + \zeta {X}_{t-1} + \epsilon_{t+1}
\end{eqnarray}
where the random walk proposition implies that $\zeta=0$ for any
possible instrument ${X}$ (and in principle the intertemporal
elasticity of substitution should be captured by $\alpha_{1}$,
\cite{hall:substitution}).\footnote{$X$ is lagged twice relative to
  $\Delta \log \mathbf{C}_{t+1}$ to control for potential time aggregation problems
  \cite{working:timeagg}.}
Unfortunately for the random walk hypothesis, many instruments ${X}$
were found with statistically reliable predictive power for
consumption growth.  (The inability to obtain robust and sensible
estimates for the intertemporal elasticity $\alpha_{1}$ has been viewed as a
less serious problem for the model for a variety of reasons).




The simplest alternative to the random walk model, influentially
advocated by \cite{campbell&mankiw:c-mmodel} (henceforth CM), is the
presence some `rule-of-thumb' consumers who set consumption equal to
income in every period.  CM argue that when an equation of the form
\begin{eqnarray}
  \label{eq:CM}
  \Delta \log \mathbf{C}_{t+1} & \approx & \varsigma + \eta \mathbf{E}_{t}[\Delta \log \mathbf{Y}_{t+1}] +\epsilon_{t+1}
\end{eqnarray}
is estimated, the coefficient $\eta$ should reveal
the proportion of aggregate income received by rule-of-thumb
consumers (and $\eta=0$ if the pure random walk model is correct); they argue that this coefficient is robustly
around 0.5, both for the U.S. and for many other countries
(\cite{campbell&mankiw:intl}).

A separate literature, with an original contribution by
\cite{campbell&deaton:smooth} and more recent work by
\cite{rotemberg&woodford:macroannual}, \cite{fuhrer:habits}, and
others, has argued that the key problem for the frictionless model is
that aggregate consumption is excessively smooth; an extensive
literature now argues that habit formation explains this excess
smoothness (along with a variety of asset pricing puzzles).

The
degree to which consumption is excessively smooth can be captured by
the coefficient $\chi$ in a regression of the form
\begin{eqnarray}
  \label{eq:chiEst}
 \Delta \log \mathbf{C}_{t+1} & = & \varsigma + \chi \mathbf{E}_{t-1}[\Delta  \log \mathbf{C}_{t}] + \epsilon_{t+1}.
\end{eqnarray}

In the intertemporally separable model, the random walk proposition implies
$\chi = 0$; \cite{dynan:habits} shows that in a simple habit formation model,
$\chi$ will is an estimate of the magnitude of the strength of the habits
parameter.  \cite{sommer:habits} estimates such a model on U.S.
aggregate data and obtains a point estimate of $\chi \approx
0.7$, an estimate which is consonant with findings by \cite{fuhrer:habits}
and others.

\subsection{Micro Benchmarks}

The extensive theoretical literature on precautionary saving, starting
with \cite{zeldes:qje}, has emphasized that precautionary motives can
introduce a correlation between predictable income growth and
consumption growth even in the absence of liquidity constraints or
rule-of-thumb consumers; the most extreme version of the argument can
be found in \cite{carroll:death}, who shows that if a microeconomic
version of \eqref{eq:CM} is estimated
and the instruments used for constructing $\mathbf{E}_{t,i}[\Delta \log
\mathbf{y}_{t+1,i}]$ capture predictable cross-household differences
in permanent income growth, theory predicts that $\eta \approx 1$
even if there are no rule of thumb consumers.

Another insight from the precautionary saving literature is that
consumption growth should be forecastable, to some extent, from
the consumer's level of current resources.  Consumers with assets
below a target value will be saving in order to build up their buffer
stocks, and consequently their consumption growth will be faster than
that of consumers who happen to have experienced a run of good luck
and therefore currently hold more than the target stock of
precautionary assets.


A simple way to capture this effect is to include in the consumption
growth regression a dummy variable that indicates whether the consumer
is a ``low-wealth'' consumer in period $t$.  This corresponds, for
example, to the empirical approach of \cite{zeldes:jpe}, who splits
his sample based on a measure of liquid wealth and finds faster
consumption growth for the low-wealth consumers.  While Zeldes
interprets this as evidence for liquidity constraints, it could also
indicate the presence of a precautionary saving motive.

Finally, the previously-cited paper by \cite{dynan:habits} estimates
\eqref{eq:chiEst} using household level data but, rather than finding
$\chi \approx 0.7$ as in the macro literature, instead estimates $\chi \approx 0$.

In sum, the micro literature suggests estimating an equation of the form
\begin{eqnarray}
\Delta \log \mathbf{c}_{t+1,i} & \approx & \varsigma + \chi \mathbf{E}_{t,i}[\Delta \log \mathbf{c}_{t,i}] + \eta \mathbf{E}_{t,i}[\Delta \log \mathbf{y}_{t+1,i}]+\alpha a_{t,i} + \epsilon_{t+1,i}
\end{eqnarray}




\subsection{Combined Benchmark}

In the

Taken together, these arguments suggest estimating equations of the
form of \eqref{eq:CGrowBench} which also incorporates a lag of
consumption growth to detect any serial correlation not explained by
the precautionary and CM terms.

\begin{verbatimwrite}{eq/CGrowBench.tex}
   \newcommand{\CGrowBench}{\Delta \log       \mathbf{c} _{t+1,i}  &=&  \varsigma  + \chi       \Delta \log      \mathbf{c} _{t,i} + \eta           {\mathbf{E}}_{t,i}[\Delta \log       \mathbf{y} _{t+1,i}]+ \alpha {     a }_{t,i}}
