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\begin{document}
\begin{tabular}{lcr}
   Econ. 511b \phantom{mmmmmmmm} &amp;  Spring 1999 &amp;  \phantom{mmmmm} George Hall and Chris Sims
\end{tabular}
\bigskip

\begin{center}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
   \textbf{\Large The Simplest Optimizing Sticky Price Model?\\  New ISLM}\footnote{Copyright 1999 by Christopher A. Sims.  This
   document may be reproduced for educational and research purposes, so long as
   the copies contain this notice and are retained for personal use or
   distributed free.}
\end{center}
\section{Introduction}
Recently several macroeconomists have begun to use a stylized model, based on
dynamically optimizing behavior with sticky prices, that uses just two
equations to analyze the effects of monetary and fiscal policy.  The two
equations bear some relation to a traditional ISLM equation and a Phillips
curve.  Here we show the kind of equilibrium model that can lead to such
equations, and discuss the limitations of this kind of modeling.

\section{The consumer}
The representative consumer maximizes
\begin{equation}\label{eq:Cobj}
  E\left[\sum_{t=0}^{\infty}\beta^t\left(U(C_t,1-L_t)+\gamma\left(\frac{M_t}{P_t}\right)\right)\right]
\end{equation}
subject to
\begin{equation}\label{eq:Cconstraint}
  C_t+\frac{B_t+M_t}{P_t}
  =\frac{W_t}{P_t}L_t+\frac{R_{t-1}B_{t-1}+M_{t-1}}{P_t}+\pi_t-\tau_t\:.
\end{equation}
The consumer chooses time paths for $C$, $L$, $B$, and $M$, with the usual
dating convention that choice variables dated $t$ are allowed to depend on
non-choice variables dated $t$ and earlier only.  The variable names should be
largely self-explanatory.  The variable $\pi$ represents profit distributions
from the representative firm, which is owned by the consumer, but
non-tradeable; and $\tau$ is taxes.  The term involving $\gamma$ in the utility
function is meant to represent the transactions value of money.  It is a
drawback of this setup that it implies no increase in the transactions value of
money as the level of consumption increases, so long as real balances $M/P$
stay constant.  More natural would be to have transactions costs as a
proportion of output stay constant as long as the ratio of $M/P$ to $C$ stays
constant.  But this more realistic specification would keep us from getting to
the simple stylized model we are aiming at.

The government budget constraint is
\begin{equation}\label{eq:GBC}
  \frac{B_t+M_t}{P_t}=\frac{R_{t-1}B_{t-1}+M_{t-1}}{P_t}-\tau_t\:,
\end{equation}
so that in equilibrium
\begin{equation}\label{eq:SRCa}
  C_t = \frac{W_t}{P_t}L_t+\pi_t\:.
\end{equation}

The consumer Euler equations are
\begin{align}\label{eq:CDC}
  \partial C\tcol&amp;&amp;&amp; D_1U_t=\lambda_t \\
  \partial L\tcol &amp;&amp;&amp; D_2U_t=\lambda_t\frac{W_t}{P_t} \label{eq:CDL}\\
  \partial M\tcol &amp;&amp;&amp;
  \gamma'\left(\frac{M_t}{P_t}\right)=\frac{\lambda_t}{P_t}-\beta
  E_t\left[\frac{\lambda_{t+1}}{P_{t+1}}\right] \label{eq:DM}\\
  \partial B\tcol &amp;&amp;&amp; \frac{\lambda_t}{P_t}=R_t\beta
  E_t\left[\frac{\lambda_{t+1}}{P_{t+1}}\right]\label{eq:DB}\:.
\end{align}
\section{The firm}
The firm maximizes
\begin{equation}\label{eq:fobj}
  E\left[\sum_{t=0}^{\infty}\beta^t \Phi_t \pi_t\right]
\end{equation}
subject to
\begin{align}\label{eq:fcash}
\mu\tcol &amp;&amp;  &amp;\pi_t=C_t-\frac{W_t}{P_t}L_t\\
\psi\tcol &amp;&amp;&amp;
C_t=F(L_t)-\frac{\xi}{2}\left(\frac{P_t}{P_{t-1}}-1\right)^2\label{eq:ftech}\\
\nu\tcol &amp;&amp;&amp; \frac{C_t}{\bar C_t}=\left(\frac{P_t}{\bar
P_t}\right)^{-\theta}\label{eq:fdemand}\:,
\end{align}
where the Greek letters at the left are the Lagrange multipliers we will
associate with each constraint.  The firm chooses time paths for $\pi$, $C$,
$L$, and $P$. The stochastic discount factors $\Phi_t$ are taken by the firm as
given by the market.  It will turn out that in the simple, linearized special
case we consider below these discount factors do not matter, so long as they
are constant in steady state.  Equation \eqref{eq:fcash} defines profit
distributions as what's left over from production after wage payments. Equation
\eqref{eq:ftech} defines production as a function of labor (no capital) less a
charge for rapid proportional change in prices.  Equation \eqref{eq:fdemand}
defines the firm's demand curve, as derived from the Dixit-Stiglitz setup.  In
the firm's problem, $C_t$ is interpreted as the firm's own output of the
consumption good, while $\bar C_t$ is the aggregate output.  In equilibrium,
all firms will make the same choice and we will have $C_t\equiv \bar C_t$, but
individual firms perceive a possibility of deviating, and we must account for
this in solving their maximization problem.

FOC's for the firm are
\begin{align}
   \partial \pi\tcol &amp;&amp;&amp; \Phi_t = \mu_t\label{eq:FDpi}\\
   \partial C\tcol &amp;&amp;&amp; -\mu_t+\psi_t + \frac{\nu_t}{\bar C_t} = 0
   \label{eq:FDC}\\
   \partial L\tcol &amp;&amp;&amp; \mu_t\frac{W_t}{P_t}=\psi_t F'_t \label{eq:FDL}\\
   \partial P\tcol &amp;&amp;&amp;
   -\mu_t\frac{W_tL_t}{P_t^2}+\nu_t\theta\frac{P_t^{-\theta-1}}{\bar P_t^{-\theta}}
   +\psi_t\xi\left(\frac{P_t}{P_{t-1}}-1\right)\frac{1}{P_{t-1}}\notag\\
   &amp;&amp;&amp;\quad
   -\beta \xi E_t\left[\psi_{t+1}\left(\frac{P_{t+1}}{P_t}-1\right)\frac{P_{t+1}}{P_t^2}\right]=0
   \label{eq:FDP}\:.
\end{align}

Note that we use the same letters for $C$ and $L$ for the firm and consumer
problems, as a way of implicitly imposing market clearing.

\section{Analysis of the system of equations}
We can solve to get rid of Lagrange multipliers in both consumer and firm
problems.  For the consumer, we get from \eqref{eq:CDC}-\eqref{eq:DB}
\begin{align}
   \frac{D_2 U_t}{D_1 U_t}&amp;=\frac{W_t}{P_t} \label{eq:CRW}\\
   \frac{D_1 U_t}{P_t}&amp;=\beta R_t
   E_t\left[\frac{D_1U_{t+1}}{P_{t+1}}\right]\label{eq:Cret}\\
   \frac{\gamma'_t}{P_t}&amp;=\frac{D_1U_t}{P_t}-\beta
   E_t\left[\frac{D_1U_{t+1}}{P_{t+1}}\right]\label{eq:Cmval}\:.
\end{align}

For the firm, by using \eqref{eq:FDpi}-\eqref{eq:FDL} in \eqref{eq:FDP} and
multiplying the result through by $F'_tP_t^2/(W_t\Phi_t)$, we arrive at the single equation
\begin{multline}\label{eq:FDPf}
  -F'_tL_t+C_t\left(\frac{P_tF'_t}{W_t}-1\right)\theta\\
  +\xi\left(\frac{P_t}{P_{t-1}}-1\right)\frac{P_t}{P_{t-1}}-\beta\xi
  E_t\left[\left(\frac{P_{t+1}}{P_t}-1\right)\frac{W_{t+1}F'_t\Phi_{t+1}}{W_tF'_{t+1}\Phi_t}
  \right]\:,
\end{multline}
where we have used the fact that in equilibrium $P_t=\bar P_t$ and $C_t = \bar
C_t$.
In order to allow some further simplification, we will consider the special
case of linear production function, so that $F(L_t)=F'_t\cdot L_t$.  This lets
us conclude from \eqref{eq:FDPf} that in steady-state we will have (letting
barred variables without subscripts indicate steady-state values)
\begin{equation}\label{eq:ssmarkup}
  \frac{\bar W }{\bar P \bar F'} = \frac{\theta}{1+\theta} \:,
\end{equation}
i.e. a markup of marginal product of labor over real wage that depends on the
elasticity of demand.  Also, we will have $\bar C = \bar F' \bar L$, as the
penalty term on price changes in \eqref{eq:ftech} vanishes in deterministic
steady state.  (Note that this means we are considering only
zero-inflation steady states.)

Log-linearizing \eqref{eq:FDPf} about deterministic steady state leads then
(using lower case to indicate the log of a variable)
\begin{equation}\label{eq:logDP}
   (1+\theta)\bar C\cdot(dp_t +df'_t -dw_t) + \xi (dp_t-dp_{t-1}) - \beta \xi
   (dp_{t+1}-dp_t) = \eta_p(t+1) \:,
\end{equation}
where $\eta_p(t)$ is an endogenous error term with the usual property for such
terms that $E_t\eta_{t+1}=0$.  Note that the $dc_t$ term that might have been
expected drops out.

Before we proceed to linearize what this literature labels the ``IS" curve,
equation \eqref{eq:Cret}, we introduce one further simplification that makes
$L$ drop out of that equation:  we assume $U$ is additively separable in $C$
and $1-L$.  So log-linearized \eqref{eq:Cret} becomes
\begin{equation}\label{eq:logCret}
  dc_t=dc_{t+1}-\sigma^{-1}(dr_t-dp_{t+1}+dp_t)+\eta_c(t+1)\:,
\end{equation}
where $\sigma=-D_{11}\bar U\cdot \bar C/\bar D_1U$ is the rate of relative risk aversion
at steady state consumption.

Log-linearizing \eqref{eq:CRW} leads to
\begin{equation}\label{eq:logRW}
  \sigma dc_t + \alpha d\ell_t = dw_t - dp_t\:,
\end{equation}
where $\alpha$ is a positive constant.  Using also the linearized technology
constraint \eqref{eq:ftech},
\begin{equation}\label{eq:logftech}
  dc_t=df'_t+d\ell_t\:,
\end{equation}
we can express \eqref{eq:logDP}
entirely in terms of $dp$, $dc$, $dr$, the exogenous technology disturbance
$df'$, and the endogenous error term:
\begin{multline}
   (1+\theta)\bar C\cdot(-(\sigma+\alpha) dc_t \\+ (1+\alpha)df'_t) +\xi (dp_t-dp_{t-1}) - \beta \xi
   (dp_{t+1}-dp_t) = \eta_p(t+1)\label{eq:logFDPff}
   \:.
\end{multline}

Note that (contrary to what I said in class Tuesday 4/6) this equation is
written entirely in terms of inflation, with no terms in the price level.
It may be helpful to intuition to note that \emph{if} the inflation rate does
not explode at the rate $\beta^{-1}$ or faster, then the equation can be solved
forward to express current inflation as a weighted sum of current and future
$dc$, with positive weights, and technology shocks $df'$, with negative weights.
However, it is also important to recognize that $dc$ and $dp$ are jointly
determined endogenous variables, so no conclusions about existence and uniqueness
of stable solutions are possible by examining \eqref{eq:logFDPff} in isolation.

\section{Completing the system:  Monetary policy}
To keep the system down to two dimensions, let us postulate a monetary policy
rule, in terms of deviations from steady state, of the form
\begin{equation}
   dr_t = \delta_0 dq_t + \delta_1 dc_t + \eps_m(t)\:.\label{eq:mpol}
\end{equation}
Here we are introducing the notation $dq_t=dp_t-dp_{t-1}$ for the deviation of
inflation from steady state.  By omitting lags (an important omission for a
serious study of the effects of monetary policy) we can just substitute for
$dr_t$ in \eqref{eq:logCret} and arrive at the system, in matrix format,
\begin{multline}\label{eq:matsys}
  \begin{bmatrix}
   1 &amp; \sigma^{-1}\\
   0 &amp; \beta
\end{bmatrix}\begin{bmatrix}
   dc_{t+1}\\dq_{t+1}
\end{bmatrix}\\=\begin{bmatrix}
   1+\sigma^{-1}\delta_1 &amp; \sigma^{-1}\delta_0 \\
   -(\sigma+\alpha)\bar C &amp; 1
\end{bmatrix}\begin{bmatrix}
   dc_t \\ dq_t
\end{bmatrix}+ \begin{bmatrix}
   0 &amp; \sigma^{-1}\\\bar C\frac{1+\alpha}{\xi} &amp; 0
\end{bmatrix} \begin{bmatrix} df'_t \\ \eps_m(t) \end{bmatrix}
+ \begin{bmatrix}
   \eta_c(t+1)\\ \eta_p(t+1)
\end{bmatrix}\:.
\end{multline}

Using the standard notation of previous lectures and notes on linear rational
expectations models, the matrices of coefficients in this equation are labeled
$\Gamma_0$, $\Gamma_1$, and $\Psi$, with what is called $\Pi$ in the notes (the
coefficient of the $\eta$'s) not appearing because it is the identity.
We can expect to solve this system forward, eliminating the indeterminacy due
to the presence of the two $\eta$'s, if $A=\Gamma_0^{-1}\Gamma_1$ has all its
eigenvalues greater than one in absolute value.  A computational trick that
helps limit the algebra here is to note that the $x$'s that solve
$\abs{Ix-\Gamma_0^{-1}\Gamma_1}=0$ are the same as those that solve
$\abs{\Gamma_0x-\Gamma_1}=0$.  (Though inverting $\Gamma_0$ is not a major
task,here.)

A contour plot of the absolute value of the minimum root as a function of $\delta_0$ and
$\delta_1$ appears in the figure below.  These were computed with $\beta=.95$, $\alpha=\sigma=2$,
and $\bar C=.5$.
\vspace{.2in}

\begin{center}
\includegraphics[width=.8\textwidth]{minroots.eps}
\end{center}

Clearly the minimum root is larger than one in absolute value when $\delta_0$
is above one.  Large positive values of $\delta_1$ make it possible for
$\delta_0$ to go just slightly below one.  That $\delta_0&gt;1$ should be required
makes sense, as this is the condition that the monetary authority responds to
inflation with a strong enough rise in the nominal interest rate to make the real rate
increase.

Assuming the $df'_t$ shocks are i.i.d., we can easily derive the solution for
$dc_t$ and $dp_t$ in terms of the exogenous disturbances $df'_t$ and
$\eps_m(t)$, by solving forward.  Using the same parameter settings as for the
graph above, with $\delta_0=1.1$ and $\delta_1=.5$, we obtain as the matrix of
impacts of exogenous disturbances
\begin{equation}\label{eq:impact}
  \begin{bmatrix}
 0.3511 &amp;  -0.2128\\
   -0.7979 &amp;  -0.4255
\end{bmatrix}
\end{equation}
This means that a surprise money contraction ($\eps_m$ increase) shrinks both
consumption and inflation, while a surprise favorable productivity shock
($df'_t$ increase) produces a rise in consumption and a fall in prices.  This
accords nicely with our informal ideas about how ``demand" and ``supply" shocks
should affect the economy.  But note that if $df'_t$ and $\eps_m(t)$ are i.i.d., the
forward-solved model implies i.i.d. $dq$ and $dc$,  despite the price
``stickiness".  This does not match very well what we know about the data.

\section{limitations of the model}
If the model had increasing costs in the investment goods industry or had
adjustment costs on capital, internal to the firms, then we would find it
necessary to include more lagged variables in the linearization.  This would
create more persistence in response to shocks.

The Keynesian econometric modeling tradition thought of wages as sticky, due to
the institutional characteristics of wage bargaining, and of prices as
responding in a relatively simple way to wages via markup rules.  Statistical
evidence for this view that stickiness in some sense originates in the labor
market, with little delay in the reaction of prices to wages, is weak.
However, there is certainly no strong support for the opposite view, that
stickiness originates in pricing, with wages responding quickly to prices.  The
model discussed in these notes attributes all stickiness to price menu costs
and has real wages adjusting quickly to clear markets.  The implications of
this model cannot easily be made to match the data in certain respects,
particularly the cyclical behavior of real wages.

One could set up the model with stickiness in wages, just by having individual consumers
face Dixit-Stiglitz demand curves for their particular types of labor and a
menu cost for wage changes.  The main reason this has not often been done is
probably that there is no intuitively appealing fable, like the restaurant
owner reprinting his menus, that helps make us comfortable with the idea of a
monopolistically competitive worker facing wage-change menu costs.  But that
wages are indeed sticky is certainly plausible, and the idea that a worker, via length of
search, faces a tradeoff between the level of wages he seeks and the fraction
of time he will be working, is also plausible.  A few papers, e.g. the thesis
of Jinill Kim and my paper ``Stickiness" have combined wage and price
stickiness, with monopolistic menu costs for both.

The model as it stands makes no use of the government budget constraint.
Results we have obtained on existence and uniqueness of equilibrium are
incomplete, therefore.  Cases where the equilibrium looks indeterminate in the
system as we have studied it here can turn out to be determinate under certain
conditions on fiscal policy.  Cases that appear to have a unique, determinate
price level can turn out to imply non-existence of equilibrium under other
conditions on fiscal policy.  This leads us to ask more systematically how
monetary and fiscal policy are related, the topic of the next section of the
course.

\section{Exercise}

Consider the linearized model we discussed above, with the policy equation
modified to the form
\begin{equation}\label{eq:exMpol}
   dr_t = \delta_0 dq_t + \delta_1 dc_t + \rho dr_{t-1} \eps_m(t)
\end{equation}
Use the same for parameters in other equations as were used in deriving the
numerical results above, but now consider $\rho=.1$, $\rho=.9$, and $\rho=1.1$.
Is the conclusion still that there is a unique price level when $\delta_0&gt;1$
and not when $\delta_0$ is much below 1?  Is there a systematic dependence of
the critical value of $\delta_0$ on $\rho$?
\end{document}
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