Lecture 11 – The Shimer Puzzle
Does the DMP Model Fit the Business Cycle
Facts?
Econ 471 - Spring 2022
Jim Albrecht
Does the DMP Model Fit the Business Cycle Facts?
I
The Diamond-Mortensen-Pissarides model is the workhorse
model in macro labor. It is analytically tractable, it gives clear
and intuitive comparative steady-state results, and these
results have been applied to a variety of important labor
market policy issues.
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A natural question to ask is whether we can use this model to
understand how unemployment, vacancies and wages vary
over the business cycle.
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In an in‡uential paper, Rob Shimer (AER 2005) argues that
the DMP model does a very bad job of “…tting the business
cycle facts.”
The Shimer Puzzle
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Shimer compares the predictions of a calibrated business-cycle
version of DMP with the corresponding patterns observed in
quarterly US data over the period 1951-2003.
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The cyclical version of the model assumes that labor
productivity and/or the job separation rate (both of which are
observable in the data) vary exogenously over time.
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The question is whether the response of unemployment,
vacancies and wages to these exogenous shocks in the
calibrated model lines up with the response that we see in the
data. Shimer’s answer is a clear “No.”
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In the U.S. data, Shimer documents:
(a) strong counter-cyclicality of u
(b) strong pro-cyclicality of v
(c) strong pro-cyclicality of θ = v /u
(d) strong pro-cyclicality of the job-…nding rate, m (θ )
(e) labor productivity, y , and the separation rate, λ, relatively
stable with mild ‡uctuations around trend
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The question is whether the DMP model can translate mildly
procyclical shocks in y or mildly countercyclical shocks in λ
into the strongly procylical movement that we see in θ.
Job Finding Rate
Labor productivity
Results
Aside – The Hodrick-Prescott Filter
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Let yt , t = 1, ..., T , be a time series, typically measured in
logs. For business cycle analysis, we want to focus on
deviations from trend, taking into account that the trend may
not be constant through time.
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Let yt = τ t + ct , where τ t is the trend component and ct is
the cyclical component of the time series.
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Once we choose a sequence fτ t gTt=1 , we can “…lter out” the
trend component. The HP …lter is a particular method for
…tting the trend sequence. Speci…cally, for a given “smoothing
parameter,” λ, the sequence fτ t gTt=1 is chosen to minimize
T
∑ (yt
t =1
τ t )2 + λ
T
1
∑
t =2
((τ t +1
τt )
(τ t
τt
1 ))
2
Model and Shocks
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Shimer uses the simplest version of the DMP model for this
exercise. Speci…cally, he endogenizes the job-…nding rate but
treats the separation rate as exogenous.
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The model is set in continuous time. Labor productivity, y ,
and the job separation rate, λ, are assumed to follow a
…rst-order Markov process, i.e., the probability distribution for
y (t + dt ) is conditioned on y (t ), and similarly for λ.
Speci…cally, an “extrinsic shock variable” evolves according to
=
γxdt + σdb
y = z + e x (y
z)
x
λ = e λ
dx
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Notation and technical point – Shimer uses z to denote the
‡ow value of unemployment, while b is a “Brownian motion
component” – essentially a random shock drawn from a
standard normal distribution.
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Worker and job values take into account the assumption that
y and λ vary through time in a less-than-completely
predictable way.
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v can be adjusted instantaneously; u requires time to adjust:
u = (1
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u )λ
m ( θ )u
The objective of the calibration/simulation exercise is to see
how well the properties of the simulated time series for
u, v , w , y , etc. match the corresponding properties in the
data.
Calibration
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The basic idea is to use steady-state averages of the
endogenous variables to choose values for the DMP model
parameters.
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Functional form assumption – Cobb-Douglas matching
function:
M (u, v ) = Au β v 1
β
or m (θ ) = Aθ 1
β
,
where β is the worker share parameter in the Nash bargain
(Hosios condition).
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The parameters γ and σ are chosen to match the time series
properties of productivity and of the separation rate. In one
set of simulations, λ is held constant, and only y varies
through time; in a second set of simulations, y is held
constant and only λ varies through time.
Calibration
Parameter
y
λ
r
z
m(θ )
β
c
σ
γ
Productivity
Separation
stochastic
0.1
0.012
0.4
1.355θ 0.28
0.72
0.213
0.0165
0.004
1
stochastic
0.012
0.4
1.355θ 0.28
0.72
0.213
0.0570
0.220
Results
Results
Results
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In the stochastic version of the DMP model, separation
shocks lead to a positive correlation between u and v , while
productivity shocks induce only small movements along a
negatively sloped Beveridge curve.
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Why does the model do so poorly in terms of matching the
business cycle facts? What can be done to make the model
work better in this dimension? These questions have spawned
a large macro literature.
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Shimer’s calibration has been questioned (especially his choice
of values for z and β), as has his assumption that only
‡uctuations in productivity drive labor market outcomes.
Solving the Shimer Puzzle
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One way to help the model do a better job of …tting the
business cycle facts is to make wages less ‡exible. In the data,
a positive productivity shock leads to a large increase in θ. In
the model, an increase in y leads to a large increase in w ,
which chokes o¤ the increase in θ.
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How can we “…x” this? One approach is to change the
calibration. One approach (Hagedorn and Manovskii, AER
2008) is to radically increase z and decrease β. Speci…cally,
Hagedorn and Manovskii set z = 0.955.and β = 0.052. This
“works” because under this calibration, wages are more or less
anchored at z. The problem with this approach is that it has
other, counterfactual, implications. In particular, a small
increase in unemployment compensation, i.e., a small change
in z, would have huge employment e¤ects under this
parameterization.
More on Wage Rigidity
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Another way to make wages more rigid is to change the
assumed wage-setting mechanism. Hall and Milgrom (AER
2008) achieve this by using a “strategic bargaining” rather
than a Nash bargaining approach to wage determination.
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Finally, one could simply “assume” some wage rigidity as is
sometimes done in the macro literature; i.e., assume that
wages can only be adjusted infrequently (Gertler and Trigari,
JPE 2009).
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However, are wages really rigid? The evidence (summarized in
Pissarides, Econometrica 2009) is that wages in ongoing
matches are fairly rigid but that wages in new matches are
quite ‡exible. There are wage-setting mechanisms that will
deliver this result, but the resulting model isn’t as tractable as
DMP, essentially because it becomes necessary to keep track
of distributions of wages through time.
Other Approaches
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An alternative approach emphasizes feedbacks between the
labor market and other markets. For example, in Kaplan and
Menzio (JPE 2016) a negative productivity shock increases
unemployment, as in the basic business-cycle version of the
DMP model. However, they argue further that the increase in
unemployment acts like a further negative shock to the goods
market (i.e., demand falls), which further increases
unemployment, etc. Models with feedbacks generate
“multiplier” e¤ects.There is also a literature that emphasizes
feedbacks between …nancial markets and the labor market.
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Bottom line: The DMP model works well for comparative
steady-state analysis but (in its basic form) does a bad job of
matching the business cycle facts. The question of how to
modify DMP (or to construct a new model) to do a good job
both for steady-state and business cycle analysis – and to do
so in a tractable way – is very much at the frontier of macro
labor research.