Stochastic Capital Depreciation

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Stochastic Capital Depreciation and the
Hours-Productivity Correlation
Michael J. Dueker Federal Reserve Bank of
St. Louis Andreas M. Fischer Swiss National
Bank and CEPR Robert Dittmar Federal
Reserve Bank of St. Louis
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Introduction

• Numerous DSGE models allow for variation in the
• depreciation rate of capital
• endogenous variable
• function of capital utilization
• Greenwood, Hercowitz, and Huffman (1988)
• Burnside, Eichenbaum, and Rebelo (1996)
• King and Rebelo (2000)
• or maintenance and repair
• Licandro and Puch (2000), Collard and Kollintzas
  (2000)
• McGrattan and Schmitz (1999)

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Introduction
Capital utilization procyclical amplification -
reduces the dependence of having to work with
technology shocks that have a large variance.
The probability of technical regress is
reduced. Maintenance and Repair depreciation is
a function of MR technology shocks have an
income(-) and substitution () effect on
depreciation rate. Positive productivity shock
causes goods production to be more efficient than
maintenance. Labor input is freed up for
alternative functions such as maintenance.
Endogenous depreciation does not allow for
random changes in depreciation as an independent
source of economic fluctuations.
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Introduction
Alternatively, the depreciation rate can
be stochastic This puts depreciation shocks on
the same level as technology shocks as the
driving force behind macroeconomic
fluctuations short-hand approach to complicated
endogenous decisions Ambler and Paquet
(1994) first to introduce stochastic depreciation
as a white noise process similar to technology.
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Introduction
Explain hours-productivity correlation is
low high depreciation rates cannot be the answer
4-14. Motivation of the paper is demonstrate
that stochastic fluctuation of the depreciation
rate within a narrow band at a moderate level is
by itself able to generate a low
hours-productivity correlation, without the
occurrence of high rates of depreciation.
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Introduction
Stochastic depreciation as a 2-state Markov
process Nonlinear decision rules Gong (1995)
calculate linear decision rules for each Markov
state and weight them by their probabilities. An
dolfatto and Gomme (2001) Markov switching for
money growth using a grid-based approximation to
the nonlinear decision rule functions. We use
Judd's (1998) Projection Method, which can handle
up to 10 ten state variables.
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Outline of the paper
The Program 1. Difficulties in Calibrating
Time-Varying Depreciation 2. Model Structure and
Calibration 3. Empirical Implementation 4.
Empirical Results Sensitivity Results Impulse
Responses 5. Concluding Remarks
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1. Difficulties in Calibrating T-V Depreciation
Scant Empirical Evidence on Time-Varying
Depreciation Big differences between say
construction and software and office eq.
Moore's Law - technology cycle of
microprocessors is independent of the business
cycle. (Service sector) bidding rounds
military contracts weekend prem.
Movies medical trials, new environmental or
safety standards Capital is destroyed through
obsolescence

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Sectoral data show considerable
variance Stylized facts The depreciation
rates differ across countries Annual BEA data
for the US 1947-2001 1) strong
persistence 2) they fluctuate in a relatively
narrow band 3) they do not appear to be
strongly procyclical Markov Switching should
be able to handle these properties.
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3. Calibration
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3. Calibration
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3. Calibration
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3. Calibration
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3. Calibration
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3. Calibration
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3. Calibration
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3. Calibration
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3. Model Description
Standard RBC Model Features indivisible time
spent at work Hansen/Rogerson capital stock
equation kt1 1 - ?(st )kt it
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3. Model Description
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4. Empirical Implementation

Markov Switching in the DSGE model has two
problematic features for traditional solution
techniques 1) No model steady state to serve
as a center of approximation (switching
parameters are crucial for determining the
decisions of agents) Cannot be simply turned
off. 2) Agent does not have full information
about depreciation rate and needs to
generate beliefs - states have very
nonlinear transitions through time. Used method
Judd (1998) 'Projection Method' approximate the
agent's decision rule by polynominals that
nearly solve the optimization problem.
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5. Sensitivity Analysis

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5. Sensitivity Analysis

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5. Sensitivity Analysis

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5. Impulse Responses
Impulse Responses

No standard procedure in the context of Markov
switching Need to define 'what is a shock' run
parallel simulations of the DSGE model of
'switch' and 'no switch' scenarios. Simulations
share a common technology shock and a
depreciation rate 0.02 that randomly follows
the Markov process until 25 periods before the
end of the sample of length 200.
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Empirical Setup Simulation 1 one period in
the low, four in the high, and one in the
low.The difference between the paths of
variables with and without the four-period
sojourn defines the impulse responseThe
reported impulse is the average response from 200
simulations of the model.
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5. Empirical Results

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6. Summary and Conclusions
Concluding Remarks 1) nonlinear decision
rules - Projection Method 2) Application to
Stochastic capital depreciation -
modeled as time varying
Stylized findings Level of switching and
persistence is inversely related to the
hours-productivity correlation Markov
switching allows the level of persistence,
variance to fluctuate that are consistent
with sectoral data.