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\begin{document}
\title{\textbf{Bargaining and Wage Rigidity in a Matching Model for the US}}
\date{This version: \today}
\maketitle
\begin{abstract}
This paper uses robust econometric methods to assess previous empirical
results for the \cite{Mortensen-Pissarides(1994):RES} matching model. Assuming
all wages are negotiated each period is inconsistent with the history
dependence in US wages, even allowing for heterogeneous match productivities,
time to build vacancies and credible bargaining. Flexible wages for job
changers, with rigid wages for job stayers, allows the model to capture this
history dependence and is not inconsistent with parameter calibrations in the
literature. Such wage rigidity affects only the timing of wage payments over
the duration of matches; conclusions about other characteristics are
unaffected by it. \bigskip\
\emph{Keywords: }Matching frictions, wage bargaining, history dependence, wage
rigidity, weak instruments
\emph{JEL classification: }E2, J3, J6
Word count: 10981
\end{abstract}
\renewcommand{\thefootnote}{\arabic{footnote}}
\section{Introduction}
The \cite{Mortensen-Pissarides(1994):RES} matching model is the basis for most
recent discussions of unemployment and vacancies at the macroeconomic level in
the US.\footnote{There is also a growing literature applying disaggregated
versions of the matching model with heterogeneous firms and employees to micro
data. For examples, see \cite{Cahuc-PostelVinay-Robin(2006):Ectra} and
\cite{Robin(2011):Ectra}.} The empirical implementations in
\cite{Cole-Rogerson(1999):IER}, \cite{Yashiv(2000):AER},
\cite{Shimer(2005):AER}, \cite{Yashiv(2006):EER} and
\cite{Hagedorn-Manovskii(2008):AER} assume Nash bargaining of wages in all
matches each period. \cite{Hall(2005):REStat}, \cite{Hall(2005):AER},
\cite{Hall-Milgrom(2008):AER} retain bargaining of wages in all matches each
period but replace Nash by other forms of bargaining. Others, such as
\cite{Shimer(2004):JEEA}, \cite{Gertler-Trigari(2009):JPE},
\cite{Pissarides09}, \cite{Rudanko(2009):JME}, \cite{Rudanko(2011):AER},
\cite{Haefke-Sonntag-Rens(2013):JME} \cite{Kudlyak(2014):JME} and
\cite{Gertler-Huckfeldt-Trigari(2015):WP}, argue that introducing some form of
wage rigidity enables the matching model to better capture the relationship
between unemployment, vacancies and wages in US data. There is also a
substantional literature on New Keynesian models with search and matching
frictions that has looked at wage persistence (Christoffel et al,
2009\nocite{Christoffel2009}, Krause and Lubik, 2007\nocite{KrauseLubik2007},
Krause et al 2008\nocite{KrauseLubikLopezSalido2008}, Trigari,
2009\nocite{Trigari2009}).
In most of these studies, empirical results are based on calibration or
full-information estimation, both of which require strong assumptions to pin
down the distribution of the data. The studies using calibration rely on
specific assumptions about the distribution of shocks. Those using
full-information estimation rely on other aspects of the full model being
correctly specified. In addition, all empirical results reported in the
literature rely on the assumption of strong identification, since none of the
papers use methods that are robust to weak identification or weak instruments.
Hence the rejection of the spot market model, and evidence on any of the
proposed extensions, is conditioned on those assumptions.
The contribution of this paper is to assess the robustness of results in
previous studies to dropping those strong assumptions. We use
limited-information analysis so that our results do not rely on correct
specification of the other aspects of the model.\footnote{This approach has
been used successfully in many other areas of macroeconomics. e.g., see Gali
and Gertler (1999) for the New Keynesian Phillips curve.} Moreover, the models
we estimate are forward-looking rational expectations models and this
characteristic provides straightforward criteria for determining valid
instruments. Furthermore, our estimation is based on the generalized method of
moments (GMM) making use of methods of inference that are robust to weak
instruments, see \cite{Stoc00}. This innovation relative to the literature on
matching in wage determination is important because weak instruments are known
to pose a very serious threat to empirical validity across many areas of
economics, see, e.g., Stock \textit{et al} (2002).
The paper establishes two main robust findings on the basis of these
estimation methods. First, none of the formulations in the aforementioned
literature with wages in all matches negotiated each period satisfactorily
captures the history dependence in wages. Second, there is a formulation of
wage rigidity that does so and, consistent with the micro evidence reported in
\cite{Pissarides09} and reiterated by \cite{Haefke-Sonntag-Rens(2013):JME},
applies only to continuing matches.
This second finding has important implications. If wage rigidity applies to
new, as well as continuing, matches, as in \cite{Gertler-Trigari(2009):JPE},
it affects job creation and hence vacancies and unemployment. But, as pointed
out by \cite[Section 4]{Malcomson(1999):Hndbk} and \cite{Pissarides09}, if it
applies only to continuing matches, it affects only the timing of wage
payments over the duration of a match. So conclusions for vacancies and
unemployment drawn from studies without wage rigidity continue to
apply.\footnote{\cite{Gertler-Huckfeldt-Trigari(2015):WP} criticizes this
conclusion but does not test for history-dependence in wages, the issue we
address in this paper.}
The underlying problem for capturing the history dependence of wages with
models in which wages in all matches are determined by the Nash bargain in
every period is illustrated in Figure \ref{fig: HM}. This compares the
variation in the wage predicted by the calibration in
\cite{Hagedorn-Manovskii(2008):AER} with the average wage in the data. The
residuals are the difference between these. It is apparent from the figure
that the residuals are highly persistent (as measured by the autocorrelation
function), which is contrary to what one would expect if the model captured
adequately the history dependence of wages in the data.
Subsequent contributions have extended the model with all wages negotiated
each period to address this problem. \cite{Hall-Milgrom(2008):AER} consider a
different model of bargaining, \emph{credible bargaining}.
\cite{Mortensen-Nagypal(2007):RevEcDy} allow for separation shocks in addition
to productivity shocks. \cite{Hagedorn-Manovskii(2011):IER} allow for
\textquotedblleft time to build\textquotedblright\ in vacancy creation.
\cite{Hagedorn-Manovskii(2013):AER} add heterogeneity to match productivities
and allow \textquotedblleft on the job\textquotedblright\ search by workers.
To these can be added the fixed costs (for training, negotiation or
administration) incurred \emph{after} matching suggested by
\cite{Pissarides09}. In this paper, we construct an empirically implementable
model that encompasses all these when all wages are negotiated each period and
show that, even with them all combined, the model still does not
satisfactorily capture the history dependence of wages in the US.
As an alternative to all wages being negotiated in each period, this paper
constructs an empirically implementable model with wage rigidity for
continuing matches but wages negotiated afresh for all new matches. The form
of wage rigidity is that developed in \cite{Gertler-Trigari(2009):JPE} but
applied only to continuing, not to new, matches. This formulation
satisfactorily captures the history dependence of wages in the US, despite
having essentially only one additional parameter. Alternative forms of wage
rigidity for continuing matches are developed by
\cite{Thomas-Worrall(1988):REStud}, \cite{Beaudry-DiNardo(1991):JPE},
\cite{Rudanko(2009):JME} and \cite{MacLeod-Malcomson(1993):AER}. The first
three, however, depend on workers being risk averse, which is not part the
standard matching model, and \cite{MacLeod-Malcomson(1993):AER} is harder to
implement empirically with aggregate data. In any case, the purpose here is
not to select between different models of wage rigidity for continuing matches
but to show that a wage equation with these general characteristics can
capture the history dependence of wages. The model we use is an empirically
tractable one that suffices for this purpose.
The paper is organized as follows. The next section sets out the theoretical
wage equations we use for econometric analysis. That is followed by sections
on empirical specifications, the data and estimation results. These are, in
turn, followed by a conclusion. \ref{Appendix: Model} sets out the full
details of the theoretical model from which the wage equations are derived.
\ref{Appendix: Derivations} contains further derivations of equations in
\ref{Appendix: Model}, \ref{Appendix: Data} additional information about the
data used, and \ref{Appendix: Additional empirical results} supplementary
empirical results and robustness checks.
\section{Theoretical wage equations}
\label{S: theoretical wage equations}
\subsection{Wage equations without wage rigidity}
\label{S: wage equations without wage rigidity}The basic framework used here
for wage equations without wage rigidity is the matching model of
\cite{Mortensen-Pissarides(1994):RES} as developed by
\cite{Hagedorn-Manovskii(2008):AER} for empirical analysis. The model consists
of five equations: (1) a value equation for a filled job, (2) a value equation
for an unfilled vacancy that is set to zero because free entry is assumed, (3)
a value equation for an employed worker, (4) a value equation for an
unemployed worker, and (5) a Nash bargaining equation that determines wages.
We adapt those equations to encompass the credible bargaining model of
\cite{Hall-Milgrom(2008):AER} as an alternative to the Nash bargaining model,
\textquotedblleft time to build\textquotedblright\ in vacancy creation as
suggested in \cite{Hagedorn-Manovskii(2011):IER}, heterogeneity in match
productivities as suggested in \cite{Hagedorn-Manovskii(2013):AER} and fixed
costs incurred only \emph{after} matching as suggested by \cite{Pissarides09},
together with some minor generalizations of inessential restrictions that
there is no reason to require the data to satisfy. The full model is set out
in \ref{Appendix: Model}. There we derive a wage equation that, when wages in
all matches are negotiated each period and with the notation in Table
\ref{T: notation}, takes the form%
\begin{equation}
w_{t}=\frac{z_{t}}{1+\lambda}+\frac{\lambda}{1+\lambda}\left[ p_{t}%
+c_{t}\frac{f_{t}}{\left( 1-f_{t}\kappa\right) q_{t}}\right] +\frac
{\lambda}{1-\lambda^{2}}\frac{\gamma_{t}-E_{t}\left( \delta_{t+1}^{\kappa
}\gamma_{t+1}\right) }{1-f_{t}\kappa}.
\label{wage bargained every period free entry}%
\end{equation}
This reduces to exactly the wage equation in
\cite{Hagedorn-Manovskii(2008):AER} when the parameters incorporating the
extensions to the model are set to appropriate values; specifically,
$\kappa=0$ (employment starts at $t+1$ for new matches at $t$) and $\gamma
_{t}=0$ for all $t$ (no cost to making offers so bargaining is Nash). Given
appropriate specifications for $z_{t},c_{t}$ and $\gamma_{t}$,
(\ref{wage bargained every period free entry}) is an equation for the average
wage that can be estimated from available data.
To interpret (\ref{wage bargained every period free entry}), start with Nash
bargaining ($\gamma_{t}=0$ for all $t$). If employment lasted only a single
period, the worker would receive payoff $z_{t}$ and the firm $0$ (because, at
the bargaining stage, $c_{t}$ is a sunk cost) if they do not form a match at
$t$. If they form a match, the parties have productivity $p_{t}$ to share
between them. So Nash bargaining would result in $w_{t}=z_{t}+\beta\left(
p_{t}-z_{t}\right) $, where $\beta\in\left[ 0,1\right] $ is the bargaining
power of the worker. (To encompass the credible bargaining model, $\beta$ is
replaced by $\lambda/\left( 1+\lambda\right) $ in
(\ref{wage bargained every period free entry}).) With a continuing match,
there is also the future to consider. With Nash bargaining at $t+1$ as well as
at $t$, the worker's expected future gains are proportional to the firm's
expected future gains. Moreover, because of free entry, the firm's expected
gains from $t+1$ on equal the cost $c_{t}$ of posting a vacancy at $t$ less
the period $t$ gains. So, all future payoffs can be written explicitly in
terms of variables known at $t,$ including $c_{t}$ as in
(\ref{wage bargained every period free entry}). See \ref{Appendix: Model} for
the detailed derivation. The term in
(\ref{wage bargained every period free entry}) including $\gamma_{t}$ and
$\gamma_{t+1}$ needs to be added to incorporate credible bargaining. For that
model, $\lambda$ has a different interpretation, see Table \ref{T: notation},
but it still affects the other terms in the same way as with Nash bargaining.
when individual productivity is not observed
Equation (\ref{wage bargained every period free entry}) is also consistent
with \textquotedblleft on the job\textquotedblright\ search as modelled in
\cite{Hagedorn-Manovskii(2013):AER}, see \ref{Appendix: Model}. In their
paper, history dependence refers to the dependence of individual wages on past
labour market conditions when individual match productivity is not observed.
They show that such history dependence\ can arise through history dependence
in individual match productivity induced by \textquotedblleft on the
job\textquotedblright\ search.\ Because equation
(\ref{wage bargained every period free entry}) aggregates over all matches and
aggregate productivity is measurable, such history dependence is already
captured through our measure of productivity.
For reasons explained above, wage equation
(\ref{wage bargained every period free entry}) depends on the future gains
from the relationship being satisfactorily measured in terms of the cost
$c_{t}$ of creating a vacancy at $t$. It therefore depends on the value
equation for an unfilled vacancy being properly specified. If it is not,
(\ref{wage bargained every period free entry}) is mis-specified even if wages
in all matches are negotiated every period. In particular, there are some
specifications of the time to build a vacancy in
\cite{Hagedorn-Manovskii(2011):IER} that affect the value of an unfilled
vacancy but are not captured in the specification used to derive
(\ref{wage bargained every period free entry}). Thus, our subsequent finding
that (\ref{wage bargained every period free entry}) does not fit the data
could result purely from this mis-specification, not because some wages are
not negotiated every period.
To rule out this possibility, we derive in \ref{Appendix: Model} an
alternative wage equation when all wages are bargained each period that,
although making use of the free entry condition that the value of creating a
vacancy is zero, does not rely on the specification of the equation for that
value. This wage equation has the form%
\begin{multline}
\left( 1-f_{t}\kappa\right) \left( w_{t}-z_{t}\right) -\lambda\left(
p_{t}-w_{t}\right) \\
+E_{t}\sum_{n=1}^{\infty}\left[ \delta_{t,n}^{\kappa}-\delta_{t}%
\delta_{t+1,n-1}^{\kappa}\left( 1-s_{t+1}\right) \right] \left(
1-f_{t+n}\kappa\right) \left( w_{t+n}-z_{t+n}\right) \\
-\frac{\lambda}{1-\lambda}\left[ \gamma_{t}-\delta_{t}E_{t}\left(
1-s_{t+1}\right) \gamma_{t+1}\right] =0. \label{wage equation 4}%
\end{multline}
where $\delta_{t,n}^{\kappa}=\prod_{i=1}^{n}\delta_{t+i}^{\kappa},$\ with
$\delta_{t,0}^{\kappa}=1,$ and\ $\delta_{t}^{\kappa}=\delta_{t-1}\left(
1-s_{t}-f_{t-1}+\kappa s_{t}f_{t-1}\right) $. Instead of capitalising a
worker's future gains from forming a match in terms of the cost $c_{t}$ of
creating a vacancy, wage equation (\ref{wage equation 4}) spells out those
future gains explicitly in the terms $w_{t+n}-z_{t+n}$ for $n\geq1$. It is
thus robust to the specification of the equation for the value of creating a
vacancy and, in particular, remains valid for any length of time to build a
vacancy. It thus encompasses all the specifications in
\cite{Hagedorn-Manovskii(2011):IER}. Estimation of (\ref{wage equation 4})
requires the terms under the summation sign to be truncated at some finite
horizon.\footnote{This approach has been used by, for example,
\cite{RuddWhelan06} for studying the new Keynesian Phillips curve.} With
$\delta_{t}^{\kappa}$ strictly less than 1 as implied by the model, however,
the approximation error from truncation can be made arbitrarily small for a
sufficiently long horizon. Given appropriate specifications for $z_{t}$ and
$\gamma_{t}$, (\ref{wage equation 4}) is then an equation in current and
future average wages that can be estimated from available data.
\subsection{Wage equation with wage rigidity}
\label{S: wage equation with wage rigidity}\cite{Gertler-Trigari(2009):JPE}
use a form of wage rigidity that enables the matching model to account for the
cyclical behaviour of wages and labour market activity. Their form of wage
rigidity has a fixed probability that the wage for a match, whether new or
continuing, is bargained in any one period. But
\cite{Haefke-Sonntag-Rens(2013):JME} find little evidence of wage rigidity for
new hires at the microeconomic level, reinforcing the micro evidence surveyed
by \cite{Pissarides09}. Here, therefore, we use a model in which wages in
continuing matches are subject to wage rigidity of the type analysed by
\cite{Gertler-Trigari(2009):JPE} but those in new matches are all bargained.
In \ref{Appendix: Model}, we derive a wage equation for the average wage in
new matches, $w_{t}^{\ast}$, when the wage for continuing matches is
renegotiated with probability $1-\psi$. We allow for the possibility that
wages not renegotiated may be adjusted automatically to inflation by scaling
them by the factor $\pi_{t}^{-\mu}$, where $\pi_{t}$ is the ratio of prices at
$t$ to prices at $t-1$ and $\mu$ is a parameter to be estimated. Wages here
are measured in real terms. Thus, for $\mu=1$, the unrenegotiated wage is set
in nominal terms, for $\mu=0$ in real terms, with $\mu\in\left( 0,1\right) $
interpreted as the proportion of unrenegotiated wages set in nominal terms.
The wage equation then takes the form%
\begin{multline}
\left( 1-f_{t}\kappa\right) \left( w_{t}^{\ast}-z_{t}\right)
-\lambda\left[ \left( 1-f_{t}\kappa\right) \left( p_{t}-w_{t}^{\ast
}\right) +c_{t}\frac{f_{t}}{q_{t}}\right] \\
-\frac{\lambda}{1-\lambda}\left[ \gamma_{t}-E_{t}\left( \delta_{t+1}%
^{\kappa}\gamma_{t+1}\right) \right] +\psi\left( 1+\lambda\right)
E_{t}\left\{ \rule[-0.5cm]{0cm}{1cm}\delta_{t+1}^{\kappa}\left(
\rule[-0.3cm]{0cm}{0.6cm}\pi_{t+1}^{-\mu}w_{t}^{\ast}\right. \right. \\
\left. \left. -w_{t+1}^{\ast}\rule[-0.3cm]{0cm}{0.6cm}\right) \left[
\sum_{i=1}^{\infty}\prod_{j=2}^{i}\left( \delta_{t+j-1}\left( 1-s_{t+j}%
\right) \psi\pi_{t+j}^{-\mu}\right) \right] \rule[-0.5cm]{0cm}{1cm}%
\right\} =0, \label{eq: NCB}%
\end{multline}
where, as before, $\delta_{t}^{\kappa}=\delta_{t-1}\left( 1-s_{t}%
-f_{t-1}+\kappa s_{t}f_{t-1}\right) $. When $\psi=0,$ $w_{t}^{\ast}=w_{t}$
and wage equation (\ref{eq: NCB}) reduces to
(\ref{wage bargained every period free entry}). In
(\ref{wage bargained every period free entry}), with Nash bargaining only
contemporaneous variables appear. (With credible bargaining, the costs of
making offers at $t+1$ also appear.) Wage rigidity gives rise to
forward-looking behaviour because the wage currently negotiated may continue
to apply at future dates, which is captured by $\psi\neq0$. Thus, our
specification of wage rigidity involves relaxing the single restriction
$\psi=0,$ which turns out to make a big difference empirically.
The data contain the average wage $w_{t}$ but not the average wage for new
matches $w_{t}^{\ast}$. \ref{Appendix: Model} shows that, under the
assumptions of the model, the relationship between these is given by%
\begin{equation}
w_{t}^{\ast}=\frac{w_{t}-\pi_{t}^{-\mu}w_{t-1}}{1-\psi\left( 1-s_{t}\right)
j_{t-1}/j_{t}}+\pi_{t}^{-\mu}w_{t-1}. \label{starting wage}%
\end{equation}
As with (\ref{wage equation 4}), estimation of (\ref{eq: NCB}) requires the
terms under the summation sign on the right-hand side to be truncated at some
finite horizon. With $\delta_{t+j-1}\left( 1-s_{t+j}\right) \psi$ strictly
less than 1, the approximation error from truncation can be made arbitrarily
small for a sufficiently long horizon.
\section{Empirical specifications}
\label{S: empirical specifications}Empirical implementation of
(\ref{wage bargained every period free entry})--(\ref{eq: NCB}) requires
specifications for $z_{t}$, $\gamma_{t}$ and $c_{t}$. In
\cite{Hagedorn-Manovskii(2008):AER}, productivity is detrended and $z_{t}$ is
a constant. The corresponding assumption here is that $z_{t}$ is proportional
to trend productivity, denoted $\bar{p}_{t}$.\footnote{Specifying $z_{t}%
=z^{0}\overline{p}_{t}+z^{1}p_{t}$ to make it procyclical as in
\cite{ChodorowReich-Karabarbounis(2016):JPE} would make $z_{t}$ collinear with
$p_{t}$, so it would not affect the fit of the model and the implications for
wage rigidity. It would, however, make $\beta$ unidentified unless additional
restrictions are imposed.} We make the same assumption for $\gamma_{t}$. These
give the specifications%
\begin{equation}
z_{t}=z\bar{p}_{t},\qquad\gamma_{t}=\gamma\bar{p}_{t},\qquad\text{where }%
z\geq0. \label{cu gamma empirical specification}%
\end{equation}
For the vacancy posting cost, \cite{Hagedorn-Manovskii(2008):AER} include two
components, capital and labour. Capital costs in period $t$ are proportional
to productivity in period $t$ and so can be written $c^{K}p_{t}$, where
$c^{K}$ is a non-negative
constant.\footnote{\cite{Hagedorn-Manovskii(2008):AER} actually assume that
capital costs are $c^{K}p_{t}/\bar{p}$ and normalize $\bar{p}$ to one, but
then HP\ filter the data, so this is equivalent to assuming that, after
detrending the cost by the productivity trend, it is proportional to the
business cycle variation in productivity.} Labour costs in period $t$ are
proportional to the cyclical component of productivity, $\tilde{p}_{t}%
=p_{t}/\bar{p}_{t}$, raised to the power $\xi\in\left[ 0,1\right] $, which
\cite{Hagedorn-Manovskii(2008):AER} interpret as the elasticity of the labour
cost of those engaged in hiring with respect to productivity, and so can be
written $c^{W}\tilde{p}_{t}^{\xi}\bar{p}_{t}$, where $c^{W}$ is a non-negative
constant. See \cite{Hagedorn-Manovskii(2008):AER} for a detailed discussion of
the motivation for these formulations. We add to these the post-matching fixed
costs suggested in \cite{Pissarides09}, denoted by $H_{t}$, which we allow to
have both capital and labour components specified in ways corresponding to the
vacancy posting costs in \cite{Hagedorn-Manovskii(2008):AER}, so%
\[
H_{t}=H^{K}p_{t}+H^{W}\tilde{p}_{t}^{\xi}\bar{p}_{t},\qquad H^{K},H^{W}\geq0.
\]
Because these costs are incurred only in the event of a match, they are
multiplied by the probability of matching $q_{t}$ in their impact on vacancy
creation. Combining all these components for $c_{t}$, we get the empirical
specification%
\begin{equation}
c_{t}=c^{K}p_{t}+c^{W}\tilde{p}_{t}^{\xi}\bar{p}_{t}+\left( H^{K}p_{t}%
+H^{W}\tilde{p}_{t}^{\xi}\bar{p}_{t}\right) q_{t},\qquad c^{K},c^{W}%
,H^{K},H^{W}\geq0,\xi\in\left[ 0,1\right] .
\label{psi empirical specification}%
\end{equation}
For estimation, we normalize the wage equations
(\ref{wage bargained every period free entry}), (\ref{wage equation 4}) and
(\ref{eq: NCB}) by trend productivity, which corresponds to the use of
detrended productivity in \cite{Hagedorn-Manovskii(2008):AER} and ensures all
variables are stationary.\footnote{We measure trend productivity using the HP
filter with parameter 1600, as in \cite{Hagedorn-Manovskii(2008):AER}. We
detrend wages and productivity by the same productivity trend. The results are
robust to the alternative of HP filtering each series separately, see Table
\ref{t: filter wages} in \ref{Appendix: Additional empirical results}.}
\section{Data}
\label{datatext}We use data on the nonfarm business sector of the USA, mainly
from the Bureau of Labor Statistics (BLS) and the OECD. The data are quarterly
and cover the period 1951q1 to 2011q4. Our baseline estimation results are for
the period up to 2004q4 for comparability with earlier studies. A fuller
description of the data is in \ref{Appendix: Data}.
We use this data to construct model-consistent data series. The number of new
matches at $t,$ $m_{t}$, is given by the total number of filled jobs at $t$,
$j_{t}$, less the number of continuing matches at $t$, $\left( 1-s_{t}%
\right) j_{t-1}$, so%
\begin{equation}
m_{t}=j_{t}-\left( 1-s_{t}\right) j_{t-1}. \label{eq: J}%
\end{equation}
The stock of vacancies at the end of period $t$, after matching takes place,
is denoted $v_{t}$. Hence, the total number of vacancies available to be
filled in period $t$ is $v_{t}+m_{t}$. The stock of unemployed workers seeking
matches in period $t$ consists of workers who were unemployed in the previous
period, $l_{t-1}-j_{t-1}$ (where $l_{t}$ is the labour force at $t$), workers
who were employed in the previous period but have lost their job,
$s_{t}j_{t-1}$, and net new entrants to the labour force, $\Delta l_{t}%
=l_{t}-l_{t-1}$, making $l_{t}-\left( 1-s_{t}\right) j_{t-1}$ in total.
Equivalently, this is given by the stock of unemployed workers at the end of
the period, $u_{t}$, plus the total matches during the period, $u_{t}+m_{t}.$
Thus, the probability of filling a vacancy in period $t$ is given by%
\begin{equation}
q_{t}=\frac{m_{t}}{v_{t}+m_{t}} \label{eq: q}%
\end{equation}
and the job-finding probability for unemployed workers by%
\begin{equation}
f_{t}=\frac{m_{t}}{u_{t}+m_{t}}. \label{eq: f}%
\end{equation}
Employment $j_{t}$ and unemployment $u_{t}$ are constructed and seasonally
adjusted by the BLS from the CPS. They correspond to the last month in the
quarter in accordance with the model used here. Employment consists of total
nonfarm dependent employment (excluding the self-employed). The labour force
is the sum of employed and unemployed.
We adopt the practice discussed by \cite{Blanchard-Diamond(1990):BPEA} of
constructing a series for separations from the number of short-term
unemployed, $u_{t}^{s},$ in our case (because we are using quarterly data)
those with spells shorter than 14 weeks. Moreover, if the increase in the
labour force all goes through the unemployment pool first, this increase
should be subtracted from the short-term unemployed before calculating the
separation rate. We adjusted the data for this, though the effect on the
calculated series for the separation rate $s_{t}$ is very small. We also
adjusted for direct job-to-job flows using the procedure suggested in
\cite{Shimer(2005):AER} based on the idea that, on average, a worker losing a
job has half a period to find a new one before being recorded as unemployed.
Thus, short-term unemployment satisfies%
\[
u_{t}^{s}=\left( 1-\frac{1}{2}f_{t}\right) \left( \Delta l_{t}+s_{t}%
j_{t-1}\right) .
\]
Use of (\ref{eq: J}) and (\ref{eq: f}) to express $f_{t}$ as $\left[
j_{t}-\left( 1-s_{t}\right) j_{t-1}\right] /\left[ l_{t}-\left(
1-s_{t}\right) j_{t-1}\right] $ enables us to solve for a series for $s_{t}$
that is consistent with the model.\footnote{Even with the adjustment suggested
by \cite{Shimer(2005):AER}, the measure of separations does not include
workers moving directly from jobs to self-employment or to leaving the labour
force but it is not clear how to allow for that.} The resulting series is
plotted in Figure \ref{fig: sep rate}. This series is higher than the monthly
separation rate series reported elsewhere (e.g., \cite[Figure 7]%
{Shimer(2005):AER}), but it matches the cyclical pattern of the (monthly)
series exactly. It illustrates the point made by
\cite{Mortensen-Nagypal(2007):RevEcDy} and by \cite{Shimer(2005):AER} that
separation rates have not been constant over this period.
Vacancy stocks $v_{t}$ are measured using the Conference Board Help-Wanted
Index (HWI), which is available in quarterly frequency from 1951 to
2008.\footnote{The HWI series based on printed newspaper advertising was
replaced by online advertising after 2008. The two series have coexisted since
2005. \cite{Barnichon(2010):EconLet} compiled a composite print and online HWI
index that extends to 2011. For comparability with \cite{Shimer(2005):AER} and
\cite{Hagedorn-Manovskii(2008):AER}, we use the original HWI\ series.
Robustness checks to alternative vacancy series are reported in
\ref{Appendix: Additional empirical results}.} The index is converted to total
units using the job-openings series from the Job Openings and Labor Turnover
Survey (JOLTS), which is available only since December 2000. The HWI\ is known
to contain low frequency fluctuations, such as those resulting from newspaper
consolidation in the 1960s and the internet revolution recently, that are
unrelated to labour market trends, see \cite{Shimer(2005):AER}. Following
\cite{Shimer(2005):AER}, we remove the effect of those trends using a low
frequency filter, see \ref{Appendix: Data} for details. The probability of
filling a vacancy $q_{t}$ is then calculated using data on employment and
vacancies via equations (\ref{eq: J}) and (\ref{eq: q}). The resulting series
is plotted in Figure \ref{fig: q}. We also plot on the same graph the
corresponding series for $v_{t}$ derived using the JOLTS data over the period
(2001 on) for which it is available. This shows that the two series match very
closely (their correlation is 0.9).
Wages and productivity are from the BLS, which provides a measure of the
labour share (including non-wage compensation) and output per person in the
nonfarm business sector. We adjust for the ratio of marginal to average
productivity using the scaling factor 0.679 computed by
\cite{Hagedorn-Manovskii(2008):AER}. Because we use quarterly data, we specify
the discount factor as $\delta_{t}=\frac{1}{1+r_{t}/4},$where $r_{t}$ is the
annualized gross real interest rate, which we measure as the quarterly average
of daily 3-month Treasury bill interest rates deflated using the implicit
price deflator for nonfarm business obtained from the BLS.
\section{Estimation results}
Estimation of equations (\ref{wage bargained every period free entry}),
(\ref{wage equation 4}) and (\ref{eq: NCB}) is performed with GMM
(\cite{Hansen(1982):Ectra}). For robustness to weak identification, we use the
continuously updated estimator (CUE) proposed by \cite{Hans96l}, and the S
test proposed by \cite{Stoc00}. Confidence sets based on the S test can be
empty if the identifying restrictions are rejected for all admissible values
of the parameters. To check whether this is the case, it suffices to compare
the minimum value of the S test statistic to its critical value. This
coincides with the $J$ statistic of \cite{Hansen(1982):Ectra} that tests the
validity of over-identifying restrictions, but unlike the standard Hansen
test, the use of a higher critical value makes it robust to weak
identification.\footnote{See, for example, \cite[Section A.2.6]%
{MavroeidisPlagborgMollerStock14}.}
Let $\phi_{t}\left( \theta\right) $ denote the expression on the left-hand
side of (\ref{wage bargained every period free entry}), (\ref{wage equation 4}%
) or (\ref{eq: NCB}), as appropriate, with the expectations operator removed,
where $\theta$ is the vector of model parameters. This is a parametric
function of observed variables, whose expectation conditional on variables
known at $t$ is zero if expectations are rational. The model's testable
implications can then be expressed in terms of orthogonality restrictions of
the form $E\left[ Z_{t}\phi_{t}\!\left( \theta\right) \right] =0,$ where
$Z_{t}$ is a vector of instruments at $t$. Rational expectations imply that
lagged values of variables are uncorrelated with current and future error
terms so, as standard in macroeconomic time-series models with rational
expectations, we use lags of the variables in the model as instruments. Given
the quarterly frequency of the sample, we use four lags.\footnote{Because
results may become unreliable when the number of instruments is large, see
\cite{AndrewsStock07}, we avoid using a larger number of instruments, but we
find that our results are robust to different sets of instruments, see
\ref{Appendix: Additional empirical results}.} For the estimation, parameter
values are constrained to be consistent with the model, specifically
$z,\kappa,\psi,\mu,\xi\in\left[ 0,1\right] $, and $\lambda,c^{K},c^{W}%
,H^{K},H^{W}\geq0$.
All calculations were performed using Ox, see \cite{Doornik07}.
\subsection{Results with wages bargained in all matches every period}
We first present estimates for the model with wages bargained in all matches
each period. Table \ref{t: Hagedorn-Manovskii} reports results for wage
equation (\ref{wage bargained every period free entry}).
Column 1 of Table \ref{t: Hagedorn-Manovskii} reports estimates for the
specification in \cite{Hagedorn-Manovskii(2008):AER} with $\kappa$, the
fraction of newly matched jobs that become active in less than one period, and
the cost parameters $H^{K}$ and $H^{W}$ set to 0. (The parameters $\gamma$ and
$\lambda$ do not appear in this specification.) A \textquotedblleft
period\textquotedblright\ in \cite{Hagedorn-Manovskii(2008):AER} is one week.
Here each period is a quarter, so this specification corresponds to a longer
lag between the decision to create a vacancy and the possibility of the job
becoming productive, as suggested by \cite{Hagedorn-Manovskii(2011):IER}. (A
shorter lag corresponds to $\kappa>0$, which we allow for in column 2.) The
point estimate of $z$, the value of non-work activity, is essentially
identical to the calibrated value 0.936 in \cite{Hagedorn-Manovskii(2011):IER}
for the corresponding productivity series and the other parameters have large
standard errors --- reassuring evidence that, when applied to the same
formulation, our estimation procedure yields results not inconsistent with
their calibration procedure. But the over-identifying restrictions of this
model (implied by errors uncorrelated with the lagged instruments we use) are
overwhelmingly rejected by the Hansen test, even that using the more
conservative projection $p$ value that is robust to weak identification.
Column 2 reports estimates with $\kappa$ restricted only to $\left[
0,1\right] $. The point estimate of $\kappa$ is zero and the value of the
GMM\ objective function indicates that this generalization of the timing does
not improve fit. Column 3 of Table \ref{t: Hagedorn-Manovskii} reports
estimates allowing for the fixed costs $H^{K}$ and $H^{W}$ in the spirit of
\cite{Pissarides09}. This generalization does not significantly improve the
fit of the model either. For both these extensions, the Hansen test continues
to overwhelmingly reject the over-identifying restrictions. The standard
errors are large, especially so for the cost parameters. But these standard
errors are unreliable for constructing confidence intervals because there is
no guarantee that the assumptions underlying standard $t$ tests are satisfied.
An alternative to Nash bargaining is the credible bargaining of
\cite{Hall-Milgrom(2008):AER}. Equation
(\ref{wage bargained every period free entry}) with $\gamma$ unrestricted and
$\lambda\in\lbrack0,1)$ corresponds to our formulation of credible bargaining
for application to quarterly data. Column 4 of Table
\ref{t: Hagedorn-Manovskii} reports estimates for this model. Allowing
$\gamma$ to be non-zero makes little difference to the fit, as measured by the
value of the GMM\ objective function. It also leaves the over-identifying
restrictions rejected just as overwhelmingly by the Hansen test.
Rejection by the Hansen test does not provide information about which aspects
of the model fail to fit the data. One way to assess this informally is by
regressing the residuals of the model on the instruments. This reveals that
the instrument that drives the rejection is $w_{t-1}$: its coefficient is
close to one, while the coefficients on all other instruments are very close
to zero. This supports the message from Figure \ref{fig: HM} that it is the
history dependence of wages that the model fails to capture.
Wage equation (\ref{wage bargained every period free entry}) is derived making
use of the equation for the value of a vacancy. Rejection of
(\ref{wage bargained every period free entry}) might, therefore, be the result
of misspecification of that equation and not a rejection of wage bargaining
for every job in every period. We check for this by estimating wage equation
(\ref{wage equation 4}), which is derived without any assumption about the
form of the equation for the value of a vacancy. It is consistent with any
length of time to build a vacancy of the type considered in
\cite{Hagedorn-Manovskii(2011):IER}. Table
\ref{t: Hagedorn-Manovskii no JC equation} reports the results of estimating that.
For the results in Table \ref{t: Hagedorn-Manovskii no JC equation}, the
infinite sum in (\ref{wage equation 4}) is truncated at 13 quarters, which
makes use of all the data available for the variables in that sum while
keeping the estimation sample the same as in Table \ref{t: Hagedorn-Manovskii}%
, for comparability with \cite{Hagedorn-Manovskii(2008):AER}. The point
estimates are, however, very insensitive to the truncation length. Column 1
gives results for Nash bargaining, column 2 for credible bargaining. Because
wage equation (\ref{wage equation 4}) does not include $c_{t}$, it does not
yield estimates of the cost parameters. But the important point here is that,
despite these specifications not depending on the form of the equation for the
value of a vacancy, both Nash and credible bargaining formulations continue to
be resoundingly rejected. The implication is that no alternative specification
of the value of creating a vacancy will enable either bargaining model with
wages for all jobs bargained in every period to capture the history dependence
of wages in the data.
The most important point from these results is that, for all the
specifications with wages negotiated in all matches in every period, the
Hansen tests resoundingly reject the over-identifying restrictions with a $p$
value of 1\% or lower even using the most conservative critical values robust
to weak identification, implying that the instruments are correlated with the
residuals. Another implication is that the confidence intervals discussed
above that are robust to weak identification are completely empty even at the
99\% level for every parameter, despite the large standard errors. There just
does not exist any set of economically feasible parameter values that enable
these models to capture in a statistically satisfactory way the pattern of
history dependence in the data. Because the formulation allows for
heterogeneous match productivity and on the job\ search as in
\cite{Hagedorn-Manovskii(2013):AER}, this is strong evidence that these
extensions of the basic matching model, either alone or together, are
insufficient to enable the model with the wage for each job bargained in every
period to capture the pattern of wages in the data. Moreover, the results in
Table \ref{t: Hagedorn-Manovskii no JC equation} imply that no alternative
formulation of the value of creating a vacancy, whether to incorporate time to
build as in \cite{Hagedorn-Manovskii(2011):IER} or anything else, can overcome this.
The implication is that something more is required to enable the model to fit
the history dependence of wages. It need not be wage rigidity. But in the next
section we show that the addition of wage rigidity of the form modelled in
Section \ref{S: wage equation with wage rigidity} enables the model to do so.
\subsection{Results with wage rigidity}
Wage equation (\ref{eq: NCB}) can be used to test the Nash and credible
bargaining specifications in the presence of wage rigidity of the form in
Section \ref{S: wage equation with wage rigidity}. The results of estimating
that specification, with $w_{t}^{\ast}$ specified by (\ref{starting wage}) and
the infinite sum truncated at 28 quarters, are reported in columns 1--5 of
Table \ref{t: NashCred}. (Truncation at 28 quarters makes use of all the data
available up to the end of 2011 for the variables in the infinite sum while
leaving the estimation sample at 1952q2-2004q4 as used by
\cite{Hagedorn-Manovskii(2008):AER}. Table \ref{t: sensitivity} in
\ref{Appendix: Additional empirical results} gives results for other
truncation lengths. Beyond 8 quarters, the point estimates are completely
insensitive to the truncation length.) The results in columns 1-4 are directly
comparable to the corresponding columns of Table \ref{t: Hagedorn-Manovskii}.
The most important finding is that, unlike the specifications without wage
rigidity, these specifications all comfortably pass the Hansen tests of
over-identifying restrictions at conventional levels of significance. Addition
of the single extra parameter $\psi$ is the reason for this. Thus for the
model with wage rigidity, unlike for that without wage rigidity, there exist
sets of parameter values that satisfactorily capture the history dependence of
wages in US data.
The cost parameters $c^{K},c^{W},\xi,H^{K}$ and $H^{W}$ are not precisely
estimated from the aggregate time-series data we use for columns 1-4 of Table
\ref{t: NashCred}. (This is shown formally by the confidence intervals robust
to weak identification that we report below). Calibration studies typically
use other data to determine values of cost parameters. In column 5, we report
estimates of $\beta$ and $z$ for the Nash bargaining model (together with the
wage rigidity parameters) when the cost parameters are calibrated following
the procedure in \cite{Hagedorn-Manovskii(2008):AER}. Fixing the cost
parameters in this way hardly affects the fit of the model (compare with
column 3, the unrestricted specification) and this specification still
comfortably passes the Hansen tests. Thus wage rigidity allows the model to
capture the history dependence of wages with values of the cost parameters
consistent with those commonly used in calibrated versions of the matching model.
Because standard errors are unreliable for constructing confidence intervals
in the present context, we construct confidence intervals that are robust to
weak identification using the $S$ test of \cite{Stoc00} in order to see how
large the set of statistically acceptable parameter values is. Unlike in the
models in which wages are bargained for every match in every period,
confidence intervals for standard significance levels constructed in this way
are not empty. Confidence intervals at the 95\% and 90\% level for the
specifications that correspond to columns 3 and 4 in Table \ref{t: NashCred}
are reported in the first two columns of Table \ref{t: S confidence sets},
respectively. (The final column of Table \ref{t: S confidence sets} is
discussed below). Confidence intervals for parameters not reported in Table
\ref{t: S confidence sets} comprise the entire parameter space; they are
completely uninformative. In the case of the cost parameters $c^{K},c^{W}%
,\xi,H^{K},H^{W},$ and $\gamma,$ this is explained by the fact that they are
unidentified when $\lambda=0$ because then these parameters drop out of
equation (\ref{eq: NCB}) and the restriction $\lambda=0$ (which also
corresponds to $\beta=0$) is acceptable at the 10\% level in both specifications.
It may help with interpreting the confidence intervals in Table
\ref{t: S confidence sets} to explain how they are constructed. Consider the
confidence interval for the parameter $\psi$ (the parameter that determines
the degree of wage rigidity). For each value of $\psi,$ say $\psi_{0},$ in the
range of economically feasible values (in this case $\left[ 0,1\right] $),
we check whether there are any values of the remaining parameters such that
the model's identifying restrictions are statistically acceptable at the
desired level of significance. Specifically, we compare the value of the S
statistic, minimized over all parameters subject to the restriction $\psi
=\psi_{0}$, to the appropriate 95\% or 90\% quantile of the $\chi^{2}$
distribution with degrees of freedom equal to the number of identifying
restrictions. The confidence interval contains all the values $\psi_{0}$ for
which this test accepts. This procedure is repeated over a grid of values from
0 to 1, with increment .01 (so the confidence intervals are correct to two
decimal places). Thus, the top row of confidence intervals for $\psi$ in Table
\ref{t: S confidence sets} shows that there is no set of values for the
parameters other than $\psi$ that enables the Nash bargaining specification to
pass the $S$ test at the appropriate significance level for a value of $\psi$
less than 0.15 or greater than 0.90. But there is a set that does so for each
intermediate value of $\psi$. For this parameter, all the confidence intervals
exclude zero, confirming the results of the previous section that, without
wage rigidity, the model is statistically unacceptable. They also exclude one,
that is, completely rigid wages in continuing matches. However, the confidence
intervals for $\mu$ cover the entire parameter space, indicating that this
parameter is not identified, so the data is not sufficiently informative to
distinguish between nominal and real wage rigidity.
For $z$, the value of non-work activity as a proportion of productivity, the
confidence intervals in the first two columns of Table
\ref{t: S confidence sets} are sufficiently wide to contain the values
reported in earlier studies that used calibration. In the Nash bargaining
specification, the confidence intervals for $\beta$ are also wide, including
values from zero to more than 0.8. The corresponding parameter $\lambda$ in
the credible bargaining specification has the confidence intervals that
contain all the values from 0 to 0.999 that we searched over. (For $\lambda
=1$, the model is not defined.) In particular, they contain the calibrated
value 0.995 in \cite{Hall-Milgrom(2008):AER}. There is thus very considerable
latitude to choose parameter values based on other data sources used to
calibrate matching models that will enable the model to capture the history
dependence of wages if allowance is made for wage rigidity.
The column \textquotedblleft Nash Barg calibr.\textquotedblright\ reports
confidence intervals for $\psi,\beta$ and $z$ in the Nash bargaining
specification with wage rigidity when $\mu=0$ (real wage rigidity) and all
other parameters are fixed at the calibrated values in column 5 of Table
\ref{t: NashCred}. The confidence intervals for these parameters are very much
smaller than in the other columns, so the use of additional sources of
information to calibrate the cost parameters substantially reduces the
uncertainty surrounding the point estimates. These confidence intervals have
$\beta$ below 0.25, and the value of non-market activity $z$ no lower than
0.88, and neither inconsistent with their values in
\cite{Hagedorn-Manovskii(2008):AER} and \cite{Hagedorn-Manovskii(2011):IER}.
Such values of non-market activity are, however, higher than
\cite{Mortensen-Nagypal(2007):RevEcDy} and \cite{Hall-Milgrom(2008):AER}
regard as plausible (\cite{Hall-Milgrom(2008):AER} suggest a calibrated value
of 0.71), so this remains an important puzzle that cannot be addressed just by
allowing for wage rigidity of the form used here.
The key message from these results is that there is a form of wage rigidity
that enables the model to capture in a statistically satisfactory way the
history dependence of wages in US data. The time series data we use are not
themselves sufficiently informative to tie down the parameters of the model
tightly. But there is plenty of scope for determining parameter values from
other data sources that will capture that history dependence when wage
rigidity is included in the model. That is in contrast to the model with wages
for all matches bargained every period, for which the time series data are
sufficiently informative to rule out \emph{any} set of economically feasible
parameter values that does this. These conclusions remain robust to several
variations in the data and specification, see
\ref{Appendix: Additional empirical results}.
\subsection{Implications of the results}
One implication of our results is that matching models with fully flexible
wages of the type Hagedorn and Manovskii argue in a series of papers captures
the behaviour of vacancies and unemployment do not capture the history
dependence of wages. The alternative proposed by
\cite{Gertler-Trigari(2009):JPE} with the same degree of wage rigidity in all
matches has been criticised by \cite{Pissarides09} as inconsistent with the
micro evidence that wages of job changers (new matches) are substantially more
flexible than those of job stayers (continuing matches). A second implication
of our results is that a wage equation consistent with this micro evidence,
with wage rigidity only in continuing matches and flexible wages for all new
matches, can capture the history dependence of wages.
This second implication is important. As \cite[Section 4]%
{Malcomson(1999):Hndbk} and \cite{Pissarides09} point out, wage rigidity that
applies only to continuing matches, unlike wage rigidity that applies to all
matches as in \cite{Gertler-Trigari(2009):JPE}, has implications for
unemployment and vacancies no different from fully flexible wages, provided it
does not result in inefficient separations. Wage negotiation for new matches
takes account of the wage rigidity and sets an initial wage such that the
expected present value of wages over the duration of the match is unaffected
by that wage rigidity. The wage rigidity thus results merely in an
intertemporal redistribution of that expected present value and so has no
effect on the incentives for vacancy creation. Inefficient separations do not
occur in the model used here because all separations are assumed to be for
exogenous reasons, though even with endogenous separations they can be avoided
by renegotiation. There is thus no need to use wage equations that have
implications for unemployment and vacancies different from fully flexible
wages to capture the history dependence of wages.
Other messages that come across strongly from our results are the following.
The credible bargaining model of \cite{Hall-Milgrom(2008):AER} with all wage
negotiated each period fares no better than Nash bargaining in capturing the
history dependence of wages. With our formulation of wage rigidity that
applies only to continuing matches, with the wage negotiated for all new
matches, the lower bound of the 95\% confidence interval for the proportion of
wages not negotiated each quarter is comfortably above zero, and this
conclusion is robust to weak identification. The time-series data we use are,
however, not sufficiently informative to enable us to identify whether the
rigidity should be modelled in nominal or in real wages.
Moreover, among the sets of parameter values that capture the history
dependence, there is plenty of scope for selecting a set that is consistent
with the other empirical evidence typically used for calibration of matching
models. In particular, those sets include the calibrated values in
\cite{Hagedorn-Manovskii(2008):AER}, \cite{Hall-Milgrom(2008):AER},
\cite{Pissarides09}, \cite{Hagedorn-Manovskii(2011):IER},
\cite{Haefke-Sonntag-Rens(2013):JME} and many other papers in the literature.
Thus our results are not inconsistent with the findings of other studies
concerning those parameters, as long as allowance is made for wage rigidity in
continuing matches. Disagreements between those studies about appropriate
values for the parameters need to be settled by other evidence.
\section{Conclusion}
In this paper, we have investigated econometric wage equations for a matching
model of the US that are robust both to dropping the strong assumptions
typically required for calibration and for full-information estimation methods
and to weak instruments. We establish two main findings. First, we show that
none of the formulations in the literature with wages in all matches
negotiated each period robustly satisfies the natural criterion of adequately
capturing the history dependence of wages. Second, we provide a formulation of
wage rigidity that does so and, consistent with the micro evidence reported in
\cite{Pissarides09} and \cite{Haefke-Sonntag-Rens(2013):JME}, applies only to
continuing matches, with wages negotiated for all new matches.
We reach this conclusion by nesting the Nash bargaining model and the credible
bargaining model of \cite{Hall-Milgrom(2008):AER} within a common over-arching
framework of which each is a special case. The framework allows for
heterogeneous match productivities and \textquotedblleft on the job"
search\ as modelled in \cite{Hagedorn-Manovskii(2013):AER} and gives rise to a
wage equation that can be estimated in a way that allows for the time to build
in vacancy creation in \cite{Hagedorn-Manovskii(2011):IER} of any length. It
enables us to apply statistical tests that are robust to weak instruments to
investigate which models are statistically acceptable restrictions of the
over-arching framework and, in particular, capture the history dependence of
wages. Our statistically acceptable specification includes a parameter that
allows for wage rigidity in continuing, but not new, matches. Only with this
parameter strictly positive can the model capture the history dependence of
wages. But this form of wage rigidity has implications for unemployment and
vacancies no different from fully flexible wages. There is no need to use a
wage equation with wage rigidity for new matches, with its markedly different
implications for unemployment and vacancies, to capture the history dependence
of wages.
\bibliographystyle{agsm}
\bibliography{labour}
\newpage%
%TCIMACRO{\FRAME{fphFU}{5.4215in}{3.6175in}{0pt}{\Qcb{Top panel: real wages and
%corresponding fitted values for the calibration in
%\cite{Hagedorn-Manovskii(2008):AER}. Middle panel: residuals = differences
%between data and model. Bottom panel: autocorrelation function\ of residuals.
%Data: \cite{Hagedorn-Manovskii(2008):AER}.}}{\Qlb{fig: HM}}{hm.eps}%
%{\special{ language "Scientific Word"; type "GRAPHIC";
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\begin{figure}[ph]%
\centering
\includegraphics[
natheight=3.617500in,
natwidth=5.421500in,
height=3.6175in,
width=5.4215in
]%
{/document/graphics/HM__1.pdf}%
\caption{Top panel: real wages and corresponding fitted values for the
calibration in \cite{Hagedorn-Manovskii(2008):AER}. Middle panel: residuals =
differences between data and model. Bottom panel: autocorrelation function\ of
residuals. Data: \cite{Hagedorn-Manovskii(2008):AER}.}%
\label{fig: HM}%
\end{figure}
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\newpage
%
%TCIMACRO{\FRAME{fphFU}{3.6296in}{2.4241in}{0pt}{\Qcb{The separation rate
%$s_{t}$ computed from $u_{t}^{s}=\left( 1-\frac{1}{2}\frac{j_{t}-\left(
%1-s_{t}\right) j_{t-1}}{l_{t}-\left( 1-s_{t}\right) j_{t-1}}\right)
%\left( \Delta l_{t}+s_{t}l_{t-1}\right) $ using employment, unemployment and
%short-term unemployment data from the BLS.}}{\Qlb{fig: sep rate}}%
%{sep.eps}{\special{ language "Scientific Word"; type "GRAPHIC";
%maintain-aspect-ratio TRUE; display "USEDEF"; valid_file "F";
%width 3.6296in; height 2.4241in; depth 0pt; original-width 6.0027in;
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natheight=2.424100in,
natwidth=3.629600in,
height=2.4241in,
width=3.6296in
]%
{/document/graphics/sep__2.pdf}%
\caption{The separation rate $s_{t}$ computed from $u_{t}^{s}=\left(
1-\frac{1}{2}\frac{j_{t}-\left( 1-s_{t}\right) j_{t-1}}{l_{t}-\left(
1-s_{t}\right) j_{t-1}}\right) \left( \Delta l_{t}+s_{t}l_{t-1}\right) $
using employment, unemployment and short-term unemployment data from the BLS.}%
\label{fig: sep rate}%
\end{figure}
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\newpage%
%TCIMACRO{\FRAME{fhFU}{3.6296in}{2.4241in}{0pt}{\Qcb{The vacancy-filling
%probability $q_{t}$ using employment data from the BLS and vacancy data from
%the Conference Board HWI and from JOLTS.}}{\Qlb{fig: q}}{jolts_hwi.eps}%
%{\special{ language "Scientific Word"; type "GRAPHIC";
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%cropbottom "0"; filename 'RES/jolts_hwi.eps';file-properties "XNPEU";}} }%
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\begin{figure}[h]%
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natheight=2.424100in,
natwidth=3.629600in,
height=2.4241in,
width=3.6296in
]%
{/document/graphics/jolts_hwi__3.pdf}%
\caption{The vacancy-filling probability $q_{t}$ using employment data from
the BLS and vacancy data from the Conference Board HWI and from JOLTS.}%
\label{fig: q}%
\end{figure}
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\begin{tabular}
[c]{|l|l|}\hline
\textbf{Notation} & \textbf{Description}\\
\multicolumn{1}{|c|}{$p_{t}$} & average productivity in matches at $t$\\
\multicolumn{1}{|c|}{$w_{t}$} & average wage in matches at $t$\\
\multicolumn{1}{|c|}{$\delta_{t}$} & discount factor applied to $t+1$ at $t$\\
\multicolumn{1}{|c|}{$s_{t}$} & match separation probability at $t$\\
\multicolumn{1}{|c|}{$c_{t}$} & vacancy posting cost at $t$\\
\multicolumn{1}{|c|}{$\gamma_{t}$} & difference in cost to firm and worker of
making offers at $t$ with credible bargaining\\
\multicolumn{1}{|c|}{$q_{t}$} & probability of filling a vacancy at $t$\\
\multicolumn{1}{|c|}{$f_{t}$} & probability of an unemployed worker finding
employment at $t$\\
\multicolumn{1}{|c|}{$z_{t}$} & value of non-work activity at $t$\\
\multicolumn{1}{|c|}{$\lambda$} & Nash bargain: $\beta/\left( 1-\beta\right)
$, for $\beta$ the bargaining power of a worker\\
\multicolumn{1}{|c|}{} & credible bargain: probability of negotiation
breakdown\\
\multicolumn{1}{|c|}{$\kappa$} & probability that employment starts at $t$ for
new match at $t$\\
\multicolumn{1}{|c|}{$\delta_{t}^{\kappa}$} & $\delta_{t-1}\left(
1-s_{t}-f_{t-1}+\kappa s_{t}f_{t-1}\right) $\\\hline
\end{tabular}
\caption{Notation for all wages bargained each period}\label{T: notation}%
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\begin{tabular}
[c]{c|cccc}\hline\hline
Parameter & 1 & 2 & 3 & 4\\\hline
$\beta$ & $\underset{\left( 0.891\right) }{0.036}$ & $\underset{\left(
0.843\right) }{0.036}$ & $\underset{\left( 2.568\right) }{0.033}$ &
$\underset{\left( -\right) }{-}$\\
$z$ & $\underset{\left( 0.060\right) }{0.937}$ & $\underset{\left(
0.056\right) }{0.937}$ & $\underset{\left( 0.308\right) }{0.887}$ &
$\underset{\left( 0.411\right) }{0.880}$\\
$c^{K}$ & $\underset{\left( 1803\right) }{0.440}$ & $\underset{\left(
953\right) }{0.440}$ & $\underset{\left( 751\right) }{0.006}$ &
$\underset{\left( 579\right) }{0.201}$\\
$c^{W}$ & $\underset{\left( 1817\right) }{0.267}$ & $\underset{\left(
966\right) }{0.270}$ & $\underset{\left( 777\right) }{0.399}$ &
$\underset{\left( 627\right) }{0.254}$\\
$\xi$ & $\underset{\left( 4292\right) }{0.382}$ & $\underset{\left(
2290\right) }{0.383}$ & $\underset{\left( 1891\right) }{0.006}$ &
$\underset{\left( 2388\right) }{0.000}$\\
$\kappa$ & $\underset{\left( -\right) }{-}$ & $\underset{\left(
0.390\right) }{0.000}$ & $\underset{\left( 0.382\right) }{0.069}$ &
$\underset{\left( 0.425\right) }{0.046}$\\
$H^{K}$ & $\underset{\left( -\right) }{-}$ & $\underset{\left( -\right)
}{-}$ & $\underset{\left( 245\right) }{2.279}$ & $\underset{\left(
1100\right) }{1.753}$\\
$H^{W}$ & $\underset{\left( -\right) }{-}$ & $\underset{\left( -\right)
}{-}$ & $\underset{\left( 243\right) }{0.157}$ & $\underset{\left(
1313\right) }{0.599}$\\
$\gamma$ & $\underset{\left( -\right) }{-}$ & $\underset{\left( -\right)
}{-}$ & $\underset{\left( -\right) }{-}$ & $\underset{\left( 86.3\right)
}{0.716}$\\\hline
$\lambda$ & $\underset{\left( -\right) }{-}$ & $\underset{\left( -\right)
}{-}$ & $\underset{\left( -\right) }{-}$ & $\underset{\left( 3.632\right)
}{0.031}$\\\hline
GMM objective & $34.459$ & $34.457$ & $33.395$ & $33.315$\\
Hansen test $p$ value & $0.001$ & $0.001$ & $0.000$ & $0.000$\\
Hansen test proj. $p$ value & $0.007$ & $0.007$ & $0.010$ & $0.010$%
\\\hline\hline
\end{tabular}
\caption{Estimates of models without wage rigidity, eq. (\ref{wage bargained every period free entry}).
\footnotesize{\protect{\newline \textbf{Notes:} In all models, $z$ is the value of non-work activity,
$c^K,c^W$ are capital and labour vacancy posting costs,
$\xi$ is the elasticity of the labour cost of those engaged in hiring with respect to productivity,
$\kappa$ is the fraction of matched jobs that become active within the quarter,
and $H^K,H^W$ are Pissarides (2009) fixed costs.
For the Nash bargaining model (columns 1--3), $\beta$ is the workers' bargaining weight.
For the credible bargaining model (column 4), $\gamma$ is the difference betwen firms' and
workers' costs of making offers, and
$\lambda$ is the probability negotiations do not break down between offers.
Estimation method is CUE-GMM with Newey-West weight matrix with prewhitening over the sample
1952q2-2004q4, with a constant and four lags of $w_t$, $p_t$, $f_t$ and $q_t$ as instruments.
Standard errors in parentheses. The Hansen test $p$ value is computed from a $\chi^2(N-\nu)$ distribution
where $N$ is the number of instruments and $\nu$ is the number of parameters estimated in the interior of
the parameter space. The Hansen test proj. $p$ value is derived from a $\chi^2(N)$ distribution.}}}\label{t: Hagedorn-Manovskii}%
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\begin{tabular}
[c]{c|cc}\hline\hline
Parameter & 1 & 2\\\hline
$\beta$ & $\underset{\left( 0.030\right) }{0.111}$ & $-$\\
$z$ & $\underset{\left( 0.018\right) }{0.865}$ & $\underset{\left(
0.019\right) }{0.933}$\\
$\kappa$ & $\underset{\left( 0.356\right) }{0.000}$ & $\underset{\left(
0.242\right) }{1.000}$\\
$\gamma$ & $\underset{\left( 0.000\right) }{0.000}$ & $\underset{\left(
0.219\right) }{-0.926}$\\\hline
$\lambda$ & $-$ & $\underset{\left( 0.023\right) }{0.092}$\\\hline
GMM objective & 37.899 & 36.486\\
Hansen test $p$ value & 0.001 & 0.001\\
Hansen test proj. $p$ value & 0.003 & 0.004\\\hline\hline
\end{tabular}
\caption{Estimates of models without wage rigidity independent of the value of creating a vacancy,
eq. (\ref{wage equation 4}), with the infinite sum truncated at 13 quarters.
\footnotesize{\protect{\newline \textbf{Notes:} In both models, $z$ is the value of non-work activity
and $\kappa$ is the fraction of matched jobs that become active within the quarter.
For the Nash bargaining model (column 1), $\beta$ is the workers' bargaining weight.
For the credible bargaining model (column 2), $\gamma$ is the difference betwen firms' and
workers' costs of making offers, and
$\lambda$ is the probability negotiations do not break down between offers.
Estimation method is CUE-GMM with Newey-West weight matrix with prewhitening over the sample
1952q2-2004q4, with a constant and four lags of $w_t$, $p_t$, $f_t$ and $q_t$ as instruments.
Standard errors in parentheses. The Hansen test $p$ value is computed from a $\chi^2(N-\nu)$ distribution
where $N$ is the number of instruments and $\nu$ is the number of parameters estimated in the interior of
the parameter space. The Hansen test proj. $p$ value is derived from a $\chi^2(N)$ distribution.}}}\label{t: Hagedorn-Manovskii no JC equation}%
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\begin{tabular}
[c]{c|ccccc}\hline\hline
Parameter & 1 & 2 & 3 & 4 & 5\\\hline
$\psi$ & $\underset{\left( 0.027\right) }{0.743}$ & $\underset{\left(
0.069\right) }{0.606}$ & $\underset{\left( 0.073\right) }{0.606}$ &
$\underset{\left( 0.084\right) }{0.631}$ & $\underset{\left( 0.026\right)
}{0.742}$\\
$\mu$ & $\underset{\left( 0.228\right) }{0.123}$ & $\underset{\left(
0.314\right) }{0.023}$ & $\underset{\left( 0.366\right) }{0.023}$ &
$\underset{\left( 0.454\right) }{0.000}$ & $\underset{\left( 0.218\right)
}{0.130}$\\
$\beta$ & $\underset{\left( 3.303\right) }{0.090}$ & $\underset{\left(
2.230\right) }{0.077}$ & $\underset{\left( 5.154\right) }{0.077}$ &
$\underset{\left( -\right) }{-}$ & $\underset{\left( 0.045\right)
}{0.060}$\\
$z$ & $\underset{\left( 0.198\right) }{0.946}$ & $\underset{\left(
0.137\right) }{0.943}$ & $\underset{\left( 0.340\right) }{0.943}$ &
$\underset{\left( 0.125\right) }{1.000}$ & $\underset{\left( 0.013\right)
}{0.947}$\\
$c^{K}$ & $\underset{\left( 1931.862\right) }{0.001}$ & $\underset{\left(
606.350\right) }{0.007}$ & $\underset{\left( 876.154\right) }{0.007}$ &
$\underset{\left( 121.400\right) }{0.000}$ & $\underset{\left( -\right)
}{0.194}$\\
$c^{W}$ & $\underset{\left( 1935.586\right) }{0.149}$ & $\underset{\left(
607.306\right) }{0.046}$ & $\underset{\left( 879.118\right) }{0.046}$ &
$\underset{\left( 130.336\right) }{0.083}$ & $\underset{\left( -\right)
}{0.045}$\\
$\xi$ & $\underset{\left( 9394.965\right) }{0.293}$ & $\underset{\left(
12360.664\right) }{0.085}$ & $\underset{\left( 17529.835\right) }{0.085}$ &
$\underset{\left( 1169.534\right) }{0.000}$ & $\underset{\left( -\right)
}{0.449}$\\
$\kappa$ & $\underset{\left( -\right) }{-}$ & $\underset{\left(
0.334\right) }{1.000}$ & $\underset{\left( 0.359\right) }{1.000}$ &
$\underset{\left( 0.395\right) }{1.000}$ & $\underset{\left( -\right)
}{0.000}$\\
$H^{K}$ & $\underset{\left( -\right) }{-}$ & $\underset{\left( -\right)
}{-}$ & $\underset{\left( 101.850\right) }{0.000}$ & $\underset{\left(
1200.852\right) }{0.011}$ & $\underset{\left( -\right) }{0.000}$\\
$H^{W}$ & $\underset{\left( -\right) }{-}$ & $\underset{\left( -\right)
}{-}$ & $\underset{\left( 102.460\right) }{0.000}$ & $\underset{\left(
1317.720\right) }{1.145}$ & $\underset{\left( -\right) }{0.000}$\\
$\gamma$ & $\underset{\left( -\right) }{-}$ & $\underset{\left( -\right)
}{-}$ & $\underset{\left( -\right) }{-}$ & $\underset{\left(
176.249\right) }{-1.376}$ & $\underset{\left( -\right) }{0.000}$\\\hline
$\lambda$ & $\underset{\left( -\right) }{-}$ & $\underset{\left( -\right)
}{-}$ & $\underset{\left( -\right) }{-}$ & $\underset{\left( 16.2\right)
}{0.131}$ & $\underset{\left( -\right) }{-}$\\\hline
GMM objective & 10.594 & 10.485 & 10.485 & 8.923 & 10.615\\
Hansen test $p$ value & 0.390 & 0.399 & 0.399 & 0.629 & 0.643\\
Hansen test proj. $p$ value & 0.877 & 0.882 & 0.882 & 0.943 &
0.876\\\hline\hline
\end{tabular}
\caption{Estimates of models with wage rigidity,
eq. (\ref{eq: NCB}), with $w_{t}^{*}$ specified by (\ref{starting wage}) and the infinite sum truncated at 28 quarters.
\footnotesize{\protect{\newline \textbf{Notes:} In all models, $\psi$ is the proportion of wages not negotiated in a quarter,
$\mu$ is the proportion of wages not negotiated that are set in nominal terms,
$z$ is the value of non-work activity,
$c^K,c^W$ are capital and labour vacancy posting costs,
$\xi$ is the elasticity of the labour cost of those engaged in hiring with respect to productivity,
$\kappa$ is the fraction of matched jobs that become active within the quarter, and
$H^K,H^W$ are Pissarides (2009) fixed costs.
For the Nash bargaining model (columns 1--3 and 5), $\beta$ is the workers' bargaining weight.
For the credible bargaining model (column 4), $\gamma$ is the difference between firms' and
workers' costs of making offers, and
$\lambda$ is the probability negotiations do not break down between offers.
In column 5, the cost parameters are calibrated using the approach of Hagedorn and Manovskii (2008).
Estimation method is CUE-GMM with Newey-West weight matrix with prewhitening
over the sample 1952q2-2004q4, with a constant and four lags of $w_t$, $p_t$, $f_t$ and $q_t$ as instruments.
Standard errors in parentheses. The Hansen test $p$ value is computed from a $\chi^2(N-\nu)$ distribution
where $N$ is the number of instruments and $\nu$ is the number of parameters estimated in the interior of
the parameter space. The Hansen test proj. $p$ value is derived from a $\chi^2(N)$ distribution.}}}\label{t: NashCred}%
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\begin{tabular}
[c]{c}%
\begin{tabular}
[c]{c|cccc}\hline\hline
Parameter & & Nash Barg. & Cred. Barg. & Nash Barg. calibr.\\\hline
$\psi$ & 95\% & $\left[ 0.15,0.90\right] $ & $\left[ 0.25,0.90\right] $ &
$\left[ 0.62,0.90\right] $\\
& 90\% & $\left[ 0.21,0.88\right] $ & $\left[ 0.25,0.88\right] $ &
$\left[ 0.64,0.87\right] $\\
$\beta$ & 95\% & $\left[ 0.00,0.94\right] $ & $-$ & $\left[
0.00,0.24\right] $\\
& 90\% & $\left[ 0.00,0.84\right] $ & $-$ & $\left[ 0.00,0.22\right] $\\
$z$ & 95\% & $\left[ 0.25,1.00\right] $ & $\left[ 0.11,1.00\right] $ &
$\left[ 0.88,0.97\right] $\\
& 90\% & $\left[ 0.65,1.00\right] $ & $\left[ 0.20,1.00\right] $ &
$\left[ 0.89,0.97\right] $\\
$\lambda$ & 95\% & $-$ & $\left[ 0.00,0.999\right] $ & \\
& 90\% & $-$ & $\left[ 0.00,0.999\right] $ & \\\hline\hline
\end{tabular}
\end{tabular}
\caption{Confidence intervals based on the $S$ test of Stock and Wright (2000)
for the specifications in columns 3, 4 and 5 of Table \ref{t: NashCred}.
\footnotesize{\protect{\newline \textbf{Notes:} Confidence intervals reported only for
parameters for which they do not comprise the entire admissible parameter range. In the first two columns
all the remaining parameters are unrestricted. In the column ``Nash Barg. calibr.'', $\mu=0$ and the rest of the
parameters are fixed at the calibrated values given in
column 5 of Table \ref{t: NashCred}.}}}\label{t: S confidence sets}%
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\appendix%
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\begin{center}
{\LARGE For Online Publication}
\end{center}
\section{Full theoretical model}
\label{Appendix: Model}%
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\subsection{Basic framework}
\label{S: framework}The path of output in a match is determined by a random
draw at the time the match is formed but may change over time (because, for
example, of a general increase in productivity) at a rate common to all
matches. The distribution of match productivity is such that it is always
worthwhile to form a match when a vacant job and an unemployed worker meet.
Match productivity in match $k$ at time $t$ is denoted $p_{t}^{k}$. In the
basic model, separations occur only for exogenous reasons and at the same rate
for all matches --- there is no on the job\ search. Thus the distribution of
productivity in actual matches is the same as the distribution of the
productivity of potential matches. Denote by $J_{\tau,t}^{k}$ the expected
present value of current and future profits at $t$ to a firm from having a
filled job in match $k$ whose wage was most recently negotiated at $\tau\leq
t$. This equals (i) output $p_{t}^{k}$ net of wage costs $w_{\tau,t}^{k}$ for
period $t$, plus (ii) the expected present value of profits $\tilde{J}%
_{\tau,t+1}^{k}$ from period $t+1$ on (when taking account of the possibility
that the wage is renegotiated at $t+1$), discounted by the discount factor
$\delta_{t}$ and the probability $\left( 1-s_{t+1}\right) $ that the
relationship is not ended before production at $t+1$ because the match is
destroyed for exogenous reasons, plus (iii) the expected payoff $V_{t+1}$ (if
non-negative) of going back into the market for another employee if the match
is destroyed. (A new match results in a new productivity draw and negotiation
of a new wage, so $V_{t+1}$ does not depend on $k$.) Thus%
\begin{equation}
J_{\tau,t}^{k}=p_{t}^{k}-w_{\tau,t}^{k}+\delta_{t}E_{t}\left\{ \left(
1-s_{t+1}\right) \tilde{J}_{\tau,t+1}^{k}+s_{t+1}\max\left[ 0,V_{t+1}%
\right] \right\} ,\text{ for all }k,t\geq\tau, \label{eq: Pi}%
\end{equation}
where $E_{t}$ is the expectation operator conditional on information available
at $t$. \cite{Hagedorn-Manovskii(2008):AER} and \cite{Hall-Milgrom(2008):AER}
assume $s_{t+1}$ constant for all $t$. Here we allow for separation shocks in
view of the importance \cite{Mortensen-Nagypal(2007):RevEcDy} attribute to these.
In \cite{Hagedorn-Manovskii(2008):AER}, a new match at $t$ results in
employment starting at $t+1$ and thus expected future profit $\delta_{t}%
E_{t}\tilde{J}_{t,t+1}$, where $\tilde{J}_{\tau,t}$ (with no superscript and
$t>\tau$) is the average, over the distribution of productivities of matches
that negotiated the wage most recently at $\tau\tau$) is the average, over the distribution of
productivities of matches that negotiated wages most recently at $\tau\tau$) is the average, over the distribution of productivities of matches
that negotiated the wage most recently at $\tau0$.)
If negotiations break down, the parties search for alternative matches.
In \cite{Hall-Milgrom(2008):AER}, the parties alternate in making offers,
starting with the firm, with at most one offer made each period.
\cite{Hall-Milgrom(2008):AER} envisage each period as corresponding to a day.
With the data available, we are constrained to having each period correspond
to a quarter, so the assumption of at most one offer per period seems
implausible. For this reason, we generalize the model to allow offers at fixed
intervals that may be less than a whole period. Consider an offer from the
firm in match $k$ at time $\eta$ ($0\leq\eta<1$) between $t$ and $t+1$ that
would yield the worker present value payoff $W_{t+\eta,t+\eta}^{k}$. The
worker will accept that offer if $W_{t+\eta,t+\eta}^{k}$ is at least as great
as the payoff from rejecting the offer, having negotiations break down with
probability $\alpha$ and receiving payoff $U_{t+\eta}$ of seeking an
alternative match, but otherwise making a counter-offer resulting in expected
present value payoff denoted $\hat{W}_{t+\eta}^{k}$. Recognizing this, the
firm will make the lowest offer satisfying that requirement, which gives the
indifference condition%
\begin{equation}
W_{t+\eta,t+\eta}^{k}=\alpha U_{t+\eta}+\left( 1-\alpha\right) \hat
{W}_{t+\eta}^{k},\quad\eta\in\left[ 0,1\right) . \label{CBWI}%
\end{equation}
Symmetrically, the firm will accept an offer with present value payoff
$J_{t+\eta}^{\prime k}$ made by the worker at time $\eta$ ($0\leq\eta<1$)
between $t$ and $t+1$ if $J_{t+\eta}^{\prime k}$ is at least as great as the
payoff from rejecting the offer, having negotiations break down with
probability $\alpha$ and receiving payoff $V_{t+\eta}$ of seeking an
alternative match, but otherwise making a counter-offer resulting in expected
present value payoff denoted $\hat{J}_{t+\eta}^{k}$. Recognizing this, the
worker will make the lowest offer satisfying that requirement, which gives the
indifference condition%
\begin{equation}
J_{t+\eta}^{\prime k}=\alpha V_{t+\eta}+\left( 1-\alpha\right) \hat
{J}_{t+\eta}^{k},\quad\eta\in\left[ 0,1\right) . \label{CBFI}%
\end{equation}
In \cite{Hall-Milgrom(2008):AER}, the firm makes the first offer and in
equilibrium that offer is always accepted, so the bargained outcome
corresponds to $W_{t+\eta,t+\eta}^{k}$ for $\eta=0$.
In the specification in \cite{Hall-Milgrom(2008):AER} with only one offer per
period,%
\begin{equation}
\hat{J}_{t}^{k}=\delta_{t}E_{t}\left( -\gamma_{t+1}^{f}+J_{t+1,t+1}%
^{k}\right) ;\quad\hat{W}_{t}^{k}=z_{t}+\delta_{t}E_{t}\left( -\gamma
_{t+1}^{w}+W_{t+1}^{\prime k}\right) ,
\label{CB payoffs with 1 offer per period}%
\end{equation}
where $W_{t+1}^{\prime k}$ is the payoff to the worker from making an offer at
$t+1$. The alternative we consider here, which seems more appropriate with
periods of a quarter, is to let the time interval between offers go to zero.
Then%
\begin{equation}
\hat{J}_{t}^{k}=-\gamma_{t}^{f}+J_{t,t}^{k};\quad\hat{W}_{t}^{k}=-\gamma
_{t}^{w}+W_{t}^{\prime k}. \label{CB payoffs with 0 time between offers}%
\end{equation}
In that case, the indifference conditions (\ref{CBWI}) and (\ref{CBFI}) with
$\eta=0$ can be solved to give the following sharing rule as an alternative to
(\ref{eq: bargaining outcome}):%
\begin{equation}
W_{t,t}^{k}-U_{t}=\left( 1-\alpha\right) \left( J_{t,t}^{k}-V_{t}\right)
+\frac{1-\alpha}{\alpha}\gamma_{t}, \label{eq: bargaining outcome CB}%
\end{equation}
where $\gamma_{t}=\left( 1-\alpha\right) \gamma_{t}^{f}-\gamma_{t}^{w}$.
Note that $\gamma_{t}^{f}$ and $\gamma_{t}^{w}$ cannot be separately
identified from (\ref{eq: bargaining outcome CB}). But permitting $\gamma
_{t}^{w}>0$ allows the model to be consistent with an estimated $\gamma_{t}<0$.
\subsubsection{Nesting Nash and credible bargaining}
\label{S: nesting}The Nash and credible bargaining outcomes
(\ref{eq: bargaining outcome}) and (\ref{eq: bargaining outcome CB}) are
special cases of the more general formulation%
\begin{equation}
W_{t,t}^{k}-U_{t}=\lambda\left( J_{t,t}^{k}-V_{t}\right) +\frac{\lambda
}{1-\lambda}\gamma_{t}, \label{general sharing rule}%
\end{equation}
with the models satisfying the restrictions%
\begin{align}
\text{Nash bargaining (\ref{eq: bargaining outcome})} & :\lambda=\frac
{\beta}{1-\beta}\in\left[ 0,\infty\right) ;\gamma_{t}=0;\nonumber\\
\text{Credible bargaining (\ref{eq: bargaining outcome CB})} &
:\lambda=1-\alpha\in\left[ 0,1\right) . \label{restrictions for NB and CB}%
\end{align}
Averaged over all match productivities, (\ref{general sharing rule}) becomes%
\begin{equation}
W_{t,t}-U_{t}=\lambda\left( J_{t,t}-V_{t}\right) +\frac{\lambda}{1-\lambda
}\gamma_{t}, \label{general sharing rule averaged}%
\end{equation}
where we have made use of the linearity property of
expectations.\footnote{This follows by Tonelli's theorem, see \cite[Theorem
18.3]{Billingsley95}.}
\subsubsection{Wages bargained every period}
With all wages bargained every period, the average wage $w_{t}$ is just the
average of the wages for each individual match $k$ given by
(\ref{general sharing rule}). Because, from (\ref{eq: Pi}) and (\ref{eq: W}),
$J_{t,t}^{k}$ and $W_{t,t}^{k}$ are linear in $w_{t,t}^{k}$, the average wage
is given by (\ref{general sharing rule averaged}). In this case,
(\ref{general sharing rule averaged}) can be combined with (\ref{eq: Pibar}),
(\ref{eq: free entry}) and (\ref{eq: Wbar}) to yield wage equation
(\ref{wage bargained every period free entry}) for the average wage. For Nash
bargaining, $\lambda=\beta/\left( 1-\beta\right) $ and $\gamma_{t}=0$ so
(\ref{wage bargained every period free entry}) can be written%
\begin{equation}
w_{t}=\beta p_{t}+\left( 1-\beta\right) z_{t}+\beta c_{t}\frac{f_{t}}%
{q_{t}\left( 1-f_{t}\kappa\right) }. \label{HM wage equation}%
\end{equation}
When $\kappa=0$, (\ref{HM wage equation}) is exactly the wage equation in
\cite{Hagedorn-Manovskii(2008):AER}.
\subsubsection{Wage equation independent of vacancy creation equation}
A wage equation that is independent of the vacancy creation equation can be
derived as follows. From the free entry condition (\ref{eq: free entry}),
$V_{t}=0$ for all $t$. With the wage for every job bargained in every period,
$\tilde{J}_{\tau,t}=J_{t,t}$ for $\tau\leq t$. With those specifications,
(\ref{eq: Pi}) averaged over all match productivities becomes%
\begin{equation}
J_{t,t}=p_{t}-w_{t}+\delta_{t}E_{t}\left( 1-s_{t+1}\right) J_{t+1,t+1}%
,\text{ for all }t. \label{Pi for Vt = 0}%
\end{equation}
Also, with the wage for every job bargained in every period, $\tilde{W}%
_{\tau,t}=W_{t,t}$ for $\tau\leq t$. Then (\ref{eq: W}) averaged over all
match productivities and (\ref{eq: Wbar}) can be jointly solved forward to
write%
\begin{equation}
W_{t,t}-U_{t}=E_{t}\sum_{n=0}^{\infty}\delta_{t,n}^{\kappa}\left(
1-f_{t+n}\kappa\right) \left( w_{t+n}-z_{t+n}\right) ,
\label{W - Wbar solved}%
\end{equation}
where $\delta_{t,n}^{\kappa}=\prod_{i=1}^{n}\delta_{t+i}^{\kappa},$\ with
$\delta_{t,0}^{\kappa}=1,$ and\ $\delta_{t}^{\kappa}=\delta_{t-1}\left(
1-s_{t}-f_{t-1}+\kappa s_{t}f_{t-1}\right) $. These two conditions can be
used with (\ref{general sharing rule averaged}) to yield the wage equation
(\ref{wage equation 4}).
\subsection{Extensions to the basic model}
\label{S: model extensions}Two important generalizations of the basic model in
the literature are to on the job\ search and to wages that are not negotiated
every period.
\subsubsection{On the job\ search}
\label{S: on-the-job search}In the \cite{Hagedorn-Manovskii(2013):AER} model
of on the job\ search, wage determination in match $k$ takes, up to a
log-linear approximation, the form (see their equation (1))%
\begin{equation}
w_{t,t}^{k}=\left( p_{t}^{k}\right) ^{\zeta}\left( \vartheta_{t}\right)
^{\rho},\qquad\text{for all }k,t, \label{HM10 wage equation}%
\end{equation}
where $\vartheta_{t}$ is a business cycle indicator that incorporates labour
market tightness. For empirical purposes, \cite[eq. (35)]%
{Hagedorn-Manovskii(2013):AER} normalize $\zeta=1$ because it is not
identified separately from the standard deviation of the distribution of
productivities. With $\zeta=1$ and averaging over $k,$
(\ref{HM10 wage equation}) corresponds to (\ref{HM wage equation}) with%
\[
\left( \vartheta_{t}\right) ^{\rho}=\left( 1-\beta\right) \frac{z_{t}%
}{p_{t}}+\beta\left[ 1+\frac{c_{t}}{p_{t}}\frac{f_{t}}{q_{t}\left(
1-f_{t}\kappa\right) }\right] .
\]
\subsubsection{Wage rigidity}
\label{S: wage rigidity}Following \cite{Pissarides09}, we model wage rigidity
as applying only to continuing matches, with wages for new matches all
negotiated. The form of wage rigidity is that developed by
\cite{Gertler-Trigari(2009):JPE} but applied only to continuing, not new,
matches. Persistence takes the form of a fixed probability $1-\psi$ that a
firm renegotiates its wage in any period. In the absence of such
renegotiation, the wage remains the same as in the previous period. Thus the
wage at $t$ for a match with wage most recently negotiated at $\tau\leq t-1$
is%
\begin{equation}%
\begin{array}
[c]{ll}%
w_{t,t}^{k}, & \text{with probability }1-\psi,\\
\pi_{t}^{-\mu}w_{\tau,t-1}^{k}, & \text{with probability }\psi,
\end{array}
,\quad\psi,\mu\in\left[ 0,1\right] , \label{wage rigidity}%
\end{equation}
where $\pi_{t}$ is the ratio of prices at $t$ to prices at $t-1$, which we
incorporate to allow for the possibility that the previous period's wage may
be adjusted automatically in response to inflation, with $\mu$ a parameter to
be estimated. With wage rigidity of the form in (\ref{wage rigidity}),%
\begin{align}
\tilde{J}_{\tau,t}^{k} & =\left( 1-\psi\right) J_{t,t}^{k}+\psi J_{\tau
,t}^{k}=J_{t,t}^{k}-\psi\left( J_{t,t}^{k}-J_{\tau,t}^{k}\right)
\label{J tilde}\\
\tilde{W}_{\tau,t}^{k} & =\left( 1-\psi\right) W_{t,t}^{k}+\psi W_{\tau
,t}^{k}=W_{t,t}^{k}-\psi\left( W_{t,t}^{k}-W_{\tau,t}^{k}\right) .
\label{W tilde}%
\end{align}
(Recall that $\tilde{J}_{\tau,t}^{k}$ and $\tilde{W}_{\tau,t}^{k}$ refer to
matches with wage negotiated at $\tau$ but not renegotiated before $t$.)
With wages negotiated in all new matches, (\ref{eq: Pi}), (\ref{eq: Pibar}),
(\ref{eq: W}) and (\ref{eq: Wbar}) continue to apply. Manipulation of these
conditions gives%
\begin{align}
& J_{t+1,t+1}^{k}-J_{t,t+1}^{k}\nonumber\\
& \quad=-\left( W_{t+1,t+1}^{k}-W_{t,t+1}^{k}\right)
\label{wage rigidity payoff deviations}\\
& \quad=\left( \pi_{t+1}^{-\mu}w_{t,t}^{k}-w_{t+1,t+1}^{k}\right)
E_{t+1}\left[ \sum_{i=1}^{\infty}\prod_{j=2}^{i}\left( \delta_{t+j-1}\left(
1-s_{t+j}\right) \psi\pi_{t+j}^{-\mu}\right) \right] ,
\label{Jt+1, t+1 - Jt, t+1 alt}%
\end{align}
with the convention that the product term equals $1$ for $i