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\begin{document}
\title{Ch. 6~ Nominal Rigidity: Part A (macro)}
\author[1]{Clara Jace}%
\affil[1]{George Mason University}%
\vspace{-1em}
\date{\today}
\begingroup
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\sloppy
A major limitation of real-business-cycle models is their omission of
any role for monetary changes in driving macroeconomic fluctuations. It
is therefore important to extend our analysis of fluctuations to
incorporate a role for such changes.~
For monetary disturbances to have real effects, there must be some type
of nominal rigidity or imperfection. Otherwise, a monetary change only
means proportional changes in all prices with no impact on real prices
or quantities. Thus introducing an important role for nominal
disturbances usually involves significant changes to the microeconomics
of the model. The goal of this chapter is to understand the effects of
nominal rigidity and to analyze the effects of various assumptions about
the specifics of the rigidity (i.e. sticky prices). The key question is
how barriers to nominal adjustment (almost certainly small) can lead to
substantial aggregate nominal rigidity.~
Below we have key macroeconomic variables:\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Screen-Shot-2017-10-23-at-7-59-19-PM/Screen-Shot-2017-10-23-at-7-59-19-PM}
\end{center}
\end{figure}
\section*{Part A~ Exogenous Nominal
Rigidity}
{\label{690267}}
\subsection*{6.1~ A Baseline Case: Fixed
Prices}
{\label{585467}}
We begin with a couple of assumptions. Time is discrete, and firms
produce output using labor as their only input. Aggregate output is
therefore
\begin{quote}
\(Y = F(L), F'(*) > 0, F''(*) < 0\)
\end{quote}
Government and international trade are absent from the model, thus,
aggregate consumption and aggregate output are equal. There is a fixed
number of infinitely lived households that obtain utility from
consumption and from holding real money balances and disutility from
working. We ignore population growth; this is the representative
household's objective function:
\begin{quote}
\(U = \sum^\infty _{t = 0} \beta ^t [U(c_t) + \Gamma (\frac{M_t}{P_t}) - V(L_t)], 0 < \beta < 1\)
\end{quote}
Where there is diminishing marginal utility of consumption and money
holdings and increasing marginal disutility from working such that the
first derivatives of the functions of~\(\Gamma, V, U >0 \) and the second
derivatives of the functions of~~\(\Gamma, V, U < 0\). We assume that
utility and money holdings take our usual
constant-relative-risk-aversion forms:
\begin{quote}
\(U(C_t) = \frac{C_t ^{1 - \theta}}{1 - \theta}, \theta > 0\)
\(\Gamma (\frac{M_t}{P_t}) = \frac{(\frac{M_t}{P_t})^{1 - v}}{1 - v} , v > 0\)
\end{quote}
The assumption that money is a direct source of utility is a shortcut
(as it only means that the household can purchase goods more easily, aka
convenience).~~
We allow for two assets: money, paying a nominal interest rate of zero
and bonds, paying an interest rate of~\(i_t\).
Let~\(A_t\) denote the household's wealth at the start of
period t. Its labor income is~\(W_t L_t\) and its consumption
expenditures will be~\(P_t C_t\). The quantity of bonds it holds
from t to t + 1 is therefore~\(A_t + W_t L_t - P_t C_t - M_t\) and thus wealth evolves
according to:
\begin{quote}
\(A_{t + 1} = M_t + (A_t + W_t L_t - P_t C_t - M_t)(1 + i_t)\)
\end{quote}
The household takes the paths of P, W, and i as given. It chooses the
paths of C and M to maximize its lifetime utility subject to its flow
budget constraint (and the no-Ponzi condition). Because we allow for
wage rigidity, we do not know whether the labor supply, L , is exogenous
or choice. The path of M is set by the central bank, though households
would see the path of i as given and the path of M as something chosen,
though this is the opposite of general equilibrium.~
\subsection*{Household Behavior}
{\label{404939}}
In period t, the household's choice variables
are~\(C_t\)and~\(M_t\). Consider the experiment
from Section 2.2 and 5.4 to find the Euler equation
relating~\(C_t\) and~\(C_{t + 1}\). This was done by
reducing~\(C_t\) by~\(dC\) and thus increasing
the household bond holdings by~\(P_t dC\). It then uses those
bonds and the interest on them to increase~~\(C_{t + 1}\)
by~\((1 + i_t) \frac{P_tdC}{P_{t + 1}}\). Thus it increases the next period consumption
by~\((1 + r_t)dC\) where r is the real interest rate. This yields:
\begin{quote}
\(C_t ^{-\theta} = (1 + r_t) \beta C_{t + 1} ^{-\theta}\)
\end{quote}
After taking logs, three things are done to make this more useful:
recalling that the only use of output is consumption (substituting Y for
C), and using the economists favorite rule (\(\ln(1 + r) \approx r\)), and
suppressing the constant term (\(-(\frac{1}{\theta}) \ln \beta\)) will give us:
\begin{quote}
\(\ln Y_t = \ln Y_{t + 1} - \frac{1}{\theta} r_t\)
\end{quote}
This is also known as the~\emph{new Keynesian IS curve,~}as it is
derived from microeconomic foundations. It is most important to
appreciate that this implies an inverse relationships
between~\(r_t\) and~\(Y_t\). The demand for
goods has the same implication, and we will see that increases in the
real interest rate reduces the amount of investment firms want to
undertake. A rise in the country's interest rate would generally
increase demand for the country's assets, thus its exchange rate
appreciates.~~
Finding the first-order condition for households' money holdings (so as
to maximize) suppose the household raises~\(\frac{M_t}{P_t}\)
by~\(dm\) and lowers~\(C_t\)
by~\([\frac{i_t}{1 + i_t}]dm\) This change has no effect on the household's
wealth and if they are optimizing, this marginal change shouldn't affect
utility.~
The utility benefit of the change is~\(\Gamma ' (\frac{M_t}{P_t})dm\) and the utility
cost is~\(U'(C_t)[\frac{i_t}{1 + i_t}]dm\). The first-order condition for optimal money
holdings is therefore
\begin{quote}
\(\Gamma ' (\frac{M_t}{P_t})dm = U'(C_t)[\frac{i_t}{1 + i_t}]dm\)
\end{quote}
Since we have definitions for utility and gamma and consumption is equal
to output, this condition implies,~
\begin{quote}
\(\frac{M_t}{P_t} = Y_t ^{\frac{\theta}{v}} (\frac{1 + i_t}{i_t}) ^{\frac{1}{v}}\)
\end{quote}
Thus money demand is increasing in output and decreasing in the nominal
interest rate.~
\subsection*{Effects of Shocks with Fixed
Prices}
{\label{478898}}
Now we can see how a disturbance (like changes in the money supply)
impact households. With completely rigid prices, the nominal and real
interest rates are the same. The set of curves that will show the
inverse relationship between the interest rate and output that satisfy
the optimal money holdings equation are known as the IS and LM curves:
\par\null\par\null\par\null\par\null\par\null\par\null
Because of rigidity, monetary and governmental disturbances have real
effects (i.e. prices can't move up if government demands more goods).
\section*{6.2~ Price Rigidity, Wage Rigidity, and Departures from
Perfect}
{\label{881417}}
We've neglected an important question: Why do firms supply the
additional output (instead of not meeting additional demand)? One
important case in which this occurs is with labor-- workers won't supply
more unless the wage increases but employers won't hire more unless the
wage falls. Thus employment and output do not change when the money
supply increases; the rise in demand leads not to a rise in output but
to rationing in the goods market.~~
So we need to depart from perfect competition for monetary to affect the
product or labor markets.~ In all considered cases, incomplete nominal
adjustment is assumed, not derived, as the goal is to examine the
implications that different assumptions about nominal wage and price
rigidity and characteristics of the labor and goods market have for
unemployment, firms' pricing behavior and the behavior of the real wage
and the markup in response to demand flucuations.~
\subsection*{Case 1: Keynes's Model}
{\label{856908}}
The supply side of the model in Keynes's~\emph{General Theory} (1936)
has two key features. First, the nominal wage is completely unresponsive
to current-period developments (\(W = \bar W\)). Second, for
reasons Keynes failed to specify, the prevailing wage is above the level
that equates supply and demand.~
Keynes's assumptions concerning the goods market are conventional. Firms
are competitive, have flexible prices, and hire labor up to the point
where the marginal product of labor equals the real wage:
\begin{quote}
\(F'(L) = \frac{W}{P}\)
\end{quote}
Thus an increase in demand raises output through its impact on the real
wage. When money supply or another determinant of demand rises, goods
prices rise, so the real wage falls and employment rises. Thus there is
involuntary unemployment since the wage is above market-clearing level
and some workers who want to work cannot. Fluctuations in the demand for
goods lead to movements of employment and the real wage along the
downward-sloping labor demand curve:
\par\null\par\null\par\null\par\null\par\null\par\null\par\null
This view of the supply side of the economy therefore implies a
countercyclical real wage in response to aggregate demand shocks. This
prediction has consistently failed to find support.~
\subsection*{Case 2: Sticky Prices, Flexible Wages, and a Competitive
Labor
Market}
{\label{400552}}
Now the source of incomplete nominal adjustment is entirely in the goods
market. We assume that the flexible-price equilibrium involves prices
that exceed marginal costs (otherwise firms wouldn't meet additional
demand). Now the first-order condition for the optimal labor supply of
the utility function is~~
\begin{quote}
\(C^{ - \theta} \frac{W}{P} = V'(L)\)
\end{quote}
Thus in equilibrium where~\(C = Y = F(L)\) we have
\begin{quote}
\(\frac{W}{P} = [F(L)]^\theta V'(L)\)
\end{quote}
Thus labor is an increasing function of the real wage
\begin{quote}
\(L = L^s (\frac{W}{P}), L^{s'}(- ) > 0\)
\end{quote}
Finally, firms meet demand at the prevailing price so long as it doesn't
exceed the level where marginal cost equals price. Firms' demand for
labor is determine by their desire to meet the demand for their good.
This makes something of a vertical line for the labor demand curve,
described by the term \emph{effective labor demand}.~
\par\null\par\null\par\null\par\null\par\null\par\null\par\null
The real wage is the intersection of the supply and demand curves. This
implies a procyclical real wage in the face of demand fluctuations,
since a rise in demand leads to a rise in effective labor demand and
increase in real wage. Finally, this model implies a countercyclical
markup (ratio of price to marginal cost). This means that a rise in
demand leads to a rise in costs since both the wage rises and the
marginal product of labor declines as output rises. Since prices are
fixed, the ratio of price to marginal cost falls.~
Empirical work has largely reached a consensus that countercyclical
markups are prevalent in reality. This model is important because it
uses the natural starting point of nominal stickiness for prices and
shows there is no necessary connection between nominal rigidity and
unemployment.~
\subsection*{Case 3: Sticky Prices, Flexible Wages, and Real Labor Market
Imperfections}
{\label{684822}}
Since fluctuations in output appear to be associated with fluctuations
in unemployment, it is natural to ask whether movements in the demand
for goods can lead to changes in unemployment when nominal prices adjust
sluggishly. We assume some non-Walrasian feature of the labor market
that causes the real wage to remain above the level that equates demand
and supply. Thus we write a ``real-wage function''
\begin{quote}
\(\frac{W}{P} = w(L)\)
\end{quote}
This would look like:
\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null
Just as the last assumption, increases in demand raise output up to the
point where marginal cost equals the exogenous price level.~
\subsection*{Case 4: Sticky Wages, Flexible Prices, and Imperfect
Competition}
{\label{171438}}
Now there is a markup function since imperfect competition causes a
price markup over marginal cost:
\begin{quote}
\(P = \mu (L) \frac{W}{F'(L)}\)
\end{quote}
Where~\(\frac{W}{F'(L)}\) is marginal cost and mu is the markup. If the
markup is countercyclical-- lower in booms than in recoveries-- the real
wage can be acyclical or procyclical even if the nominal rigidity is
entirely in the labor market. The firgures below demonstrate this:
\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null
\section*{6.3 Empirical Application: The Cyclical Behavior of the Real
Wage}
{\label{532342}}
Some results find that the real wage is roughly twice as procyclical at
the individual level as in the aggregate. Thus a fall in the
unemployment rate of 1 percentage point is associated with a rise in the
typical work's real wage of about 1.2 percent. More importantly, some
economists find that short-run wage movements are far from purely
temporary, making an explanation based on movements along the labor
supply function problematic.~
\section*{6.4 Toward a Usable Model with Exogenous Nominal
Rigidity}
{\label{404439}}
\subsection*{A Permanent Output-Inflation
Tradeoff?}
{\label{931889}}
To build a model usable in practice, we need to relax the assumption
that nominal prices or wages never change. Thus we can imagine that the
level of current prices and wages were determined by the previous
period, adjusted to make up for the previous period's inflation:
\begin{quote}
\(W_t = AP_{t - 1}, A >0\)
\end{quote}
Recall that employment is determine by the marginal product of labor
equation, thus:
\begin{quote}
\(F'(L_t) = \frac{AP_{t - 1}}{P_t}\)
\(= \frac{A}{1 + \pi _t}\)
\end{quote}
where~\(\pi _t\) is the inflation rate. This equation implies a
stable, upward-sloping relationship between employment and inflation.
This would mean there is a permanent output-inflation tradeoff:
policymakers can permanently raise output only by accepting inflation.
This relationship is known as the \emph{Phillips curve}, with
theoretical and empirical support.~
\subsection*{The Natural Rate}
{\label{568533}}
The case for a stable tradeoff was shattered in 1960-70s. Friedman and
Phelps attacked the theory and formulated their~\emph{natural-rate
hypothesis,~}stating the~idea that nominal variables (money supply,
inflation) could permanently affect real variables (output,
unemployment) is unreasonable -- in the long run, these behaviors would
be determined by real forces.~
They argued that a shift to an expansionary policy changes the way that
prices/wages are set. The progression would be: expansion -- permanent
increase output/employment -- permanent reduction real wage. If there
are forces driving towards equilibrium without inflation, they will
still be there with inflation, essentially meaning that there is some
natural rate of unemployment. This would be ``the level that would be
ground out by the Walrasian system of general equilibrium
equations\ldots{}'' in Friedman's famous definitions.~
One of the failures in the Phillips curve is mundane: that disturbances
to supply are even predict inflation and unemployment by our former
models. Still, they don't explain what happened in the 1960-70s. This
leads to the \textbf{Expectations Augmented Phillips Curve.~}
\subsection*{Aggregate Demand, Aggregate Supply, and the AS-AD
Diagram}
{\label{461189}}
The demand side of the economy had two elements, the new Keynesian IS
curve:
\begin{quote}
\(\ln Y_t = E[\ln Y_{t + 1}] - \frac{1}{\theta} r_t\)
\end{quote}
And the LM curve:
\begin{quote}
\(\frac{M_t}{P_t} = Y_t ^{\theta /v} [\frac{(1 + i_t) }{i_t}]^{1/v}\)
\end{quote}
Coupled with the assumption that~\(M_t\) was set exogenously
by the central bank, these equations led to the IS-LM diagram.~
The IS curve captures much useful information: increases in the real
interest rate reduce the current demand for goods relative to future
demand and increases in expected future income raise current demand. But
the LM curve is problematic when it comes to prices being fixed and the
money supply. An approach to avoid these difficulties is to assume that
the central bank conducts policy in terms of a rule for the interest
rate. So the real interest rate is an increasing function of the gap
between actual and potential output and of inflation:
\begin{quote}
\(r_t = r( \ln Y_t - \ln \bar Y_t, \pi _t), r_1(*) > 0, r_2(*) >0\)
\end{quote}
This implies an upward-sloping relationship between output and the real
interest rate, known as the MP curve. In combination with the old IS
curve, the AD curve is produced. The AS curve follows directly from~
\begin{quote}
\(\pi _t = \pi _t ^* + \lambda(\ln Y_t - \ln \bar Y_t) + \epsilon _t ^S\)
\end{quote}
Consider a rise in inflation. The IS curve is unaffected but since the
monetary rule includes this (pi)~ the rise in inflation increases the
real interest rate that the central bank sets for a given level of
output. Thus the MP curve shifts up and r and Y falls:
\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null
\subsection*{Example: IS Shocks}
{\label{475928}}
A common approach to obtaining a model with the three equations (IS, MP,
and AS) is assuming that core inflation,~\(\pi _t ^*\), is given
by lagged inflation,~\(\pi _{t - 1}\) and the MP curve is linear.
This is difficult unless we drop the~\(E_t [\ln Y_{t + 1}]\) to make it a
traditional IS curve so output depends negatively on the real interest
rate (and not any other endogenous variable). Assuming that
monetary-policy rule depends only on output, the only shocks are to the
IS curve and that~\(\ln \bar Y_t = 0\), we have the system:
\begin{quote}
AS curve:~\(\pi _t = \pi _{t - 1} + \lambda y_t, \lambda > 0\)
MP curve:~\(r_t = by_t , b> 0\)
IS curve:~\(y_t = E_t [y_{t + 1}] - \frac{1}{\theta} r_t + u_t ^{IS}, \theta > 0\)
IS shocks:~\(u_t ^{IS} = \rho _{IS} u^{IS} _{t - 1} + e_t ^{IS} , -1 \rho _{IS} < 1\)
\end{quote}
where~\(e_t ^{IS}\) is white noise. Now combining the middle two
equations for output we have:
\begin{quote}
\(y_t \equiv \phi E_t[y_{t + 1}] + \phi u_t ^{IS}\)
\end{quote}
where~\(\phi = \frac{\theta}{\theta + b}\). Since there is still the expectation of the
future value (E term) it's not clear what happens with a shock. Using
the \emph{law of iterated projections}, we can know that the current
expectation of a future expectation o fa variable equals the current
expectation of the variable. Iterating forward will eventually give us
this expression:
\begin{quote}
\(y_t = \frac{\theta}{\theta + b - \theta \rho _{IS}} u_t ^{IS}\)
\end{quote}
which finally shows how various forces influence how shocks to demand
affect output. For example, an aggressive monetary-policy response to
output movements (a higher b) dampens the effects of shocks.~
This equation coupled with the AS equation imply that inflation is given
by
\begin{quote}
\(\pi _t = \pi _{t - 1} + \frac{\theta \lambda}{\theta + b - \theta \rho_{IS}} u_t ^{IS}\)
\end{quote}
Since there is no feedback from inflation to the real interest rate,
there is no force acting to stabilize inflation. Through other examples,
we can see that if agents' expectation of output rises, they will act in
ways that make their expectations correct. The economy has multiple
equilibria and the spontaneous, self-fulfilling output movement is known
as \emph{sunspot solutions}. If it is not, it's called the
\emph{fundamental solution}.~ ~
\selectlanguage{english}
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