Dynamics of Factor Productivity Dispersions∗
Christian Bayer† , Ariel M. Mecikovsky‡, Matthias Meier‡
This version: February 16, 2013
Abstract
This paper documents a new set of stylized facts on the joint distribution of labor
and capital productivity in the cross section of firms differentiating between low
and high frequencies. We exploit panel data from Germany, Chile, Colombia and
Indonesia and show that the basic patterns are similar across the economies. First,
we show that across factors there are significant correlation patterns that differ
between low and high frequency. At high frequency, differences between a firm’s
and the average productivity are positively correlated across factors, while they are
negatively correlated across factors at low-frequency. Second, we show that the
bulk of the cross-sectional dispersion in factor productivities of firms is due to longlasting differences even after controlling for industry. We then develop a simple
model of technology choice and show that the model is capable to replicate the
basic patterns. Moreover, we show that such model economies that grow faster,
i.e. developing economies, will endogenously exhibit larger dispersions in factor
productivities. Finally, we discuss the welfare implications of our model.
Keywords: Productivity Dispersions, Technology Choice, Heterogeneous Firms.
JEL Classification Numbers: .
∗
The research leading to these results has received funding from the European Research Council
under the European Research Council under the European Union’s Seventh Framework Programme
(FTP/2007-2013) / ERC Grant agreement no. 282740. We would like to thank the Deutsche Bundesbank,
Instituto Nacional de Estadsticas de Chile, Badan Pusat Statistik and Devesh Raval, for providing us
access to the firm level data from Germany, Chile, Indonesia and Colombia respectively.
†
Department of Economics, Universität Bonn. Address: Adenauerallee 24-42, 53113 Bonn, Germany.
email: christian.bayer@uni-bonn.de.
‡
Bonn Graduate School of Economics, Department of Economics, Universität Bonn. Address: Adenauerallee 24-42, 53113 Bonn, Germany.
1
Introduction
In a frictionless economy with perfect competition marginal factor productivities
should be equal across firms as factors move to those firms where they are most productive. However, empirical results show that there are large difference in factor productivities even within the same sector such that factors are not allocated optimally.1
This holds true for both capital and labor. A fast growing literature has been studying
this phenomenon and its aggregate implications since the seminal papers by Foster et al.
(2001) and Hsieh and Klenow (2009) and has highlighted the large impact of these factor
mis-allocations to aggregate productivity.2
In this paper, we contribute to this empirical research by investigating the structure
and dynamics of factor productivity dispersions at low and high frequency using establishment/firm level data from Chile, Colombia, and Indonesia as developing countries
and from Germany as developed economy.3
First, we establish a novel set of stylized facts: we find across all economies that firmlevel labor and capital productivities show a negative cross-sectional correlation at low
frequencies, while they are positively correlated in the cross section at high frequencies.
Moreover, long-lasting differences in factor productivities explain the bulk of the total
deviations of firm-level factor productivities from the average. More than 48% (70%) of
the productivity variance of labor (of capital) is due to long-lasting deviations.
Second, we ask how much of these stylized facts we can explain with a simple model
of frictional technology choice. For this purpose, we develop and estimate a model, in
which monopolistically competitive firms choose the labor intensity of their production
subject to fixed costs in this choice. We consider an environment along the balanced
growth path, where aggregate capital intensity grows at a constant rate. Hence, firms
are continuously driven to readjust their technology. However, due to the fixed costs in
this choice, they do so in a lumpy way from time to time, which creates dispersion in
observed labor productivities. We show that such model can explain both the positive
correlation in productivities across factors in high-frequency and the negative correlation
in low-frequency.
In this second aspect our paper contributes also to the recently developing literature
that tries to model in detail the frictions that give rise to productivity dispersions as
1
See also Restuccia and Rogerson (2013) for an extensive analysis on the contribution of reallocation
to aggregate productivity growth.
2
See for instance Collard-Wexler et al. (2011), Jones (2011), Bartelsman et al. (2009), Buera et al.
(2011), Collard-Wexler et al. (2011), Gilchrist et al. (2010), Hsieh and Klenow (2012), Peters (2011) and
Yang (2012).
3
For a description of the datasets and a list of papers who have used them, see Apendix A.1.
2
endogenous objects4 . For example, Midrigan and Xu (2012) evaluates the effect of financial frictions in entry and technology adoption decisions and differences in the returns
to capital across firms. They conclude that financial frictions has only an important
role on entry and technology adoption decisions, as these include investment that will
have a gradual effect over time and is difficult to finance using internal funds from firms.
Compared to Midrigan and Xu (2012), who assume two sectors with different fixed production technologies, we assume that firms select from a menu of short-run Leontieff
production functions. Once they have committed to a particular one, they can only
change this by paying a fixed cost instead.
The model has two interesting implications. First, because of monopolistic competition the technology adoption is inefficient, as part of the higher marginal costs coming
from suboptimal technology choice are borne by the consumers through higher prices.
Hence, the basic findings of the IO literature on R&D effort apply. Firms wait too long
in technology adjustment and invest inefficiently too little in adopting their technology
to current relative prices. Second, because growth is larger in developing economies,
the downward drift in labor-intensity is stronger. Our model predicts that they will experience a larger dispersion in factor productivities as the drift drives capital intensity,
and consequently, labor and capital productivities, faster away from their optimal level
(return point). This shows that some of the higher dispersion of factor productivities in
developing economies might be an outcome of high growth and not only a cause of low
productivity levels.
In order to elaborate on the role of technology choice we also contrast our model to
alternative models that produce productivity dispersions: with frictions in hiring and
capital adjustment, and with financial frictions. We show that these models are not able
to match the time and correlation structure as our model of technology choice.
The remainder of this paper is organized as follows: Section 2 describes our data
sets, empirical model and methods, and provides our empirical results. Section 3 sets
up our model of technology choice, and Section 4 provides the model calibration and
an analyis of the model mechanism. Section 5 estimates the parameters describing our
adjustment probability function. We use a method of simulated moments and conduct
a counterfactual analysis using our estimated model. Next, in Section 6 we show that
alternative models are not able to match the main empirical findings from each country.
Finally, Section 7 concludes. An Appendix follows which provides more details on the
data and estimation procedure.
4
See also Bhattacharya et al. (2013), Buera et al. (2011) and Amaral and Quintin (2010).
3
2
A new set of stylized facts
2.1
Data
To understand the dynamics and correlation structures of the dispersion of factor productivities we analyze plant/firm level data from four countries: USTAN (Germany),
ENIA (Chile), EAM (Colombia) and IBS (Indonesia). When cleaning and preparing
the data for our analysis, we make sure to treat the data in the most comparable way,
Appendix A.1 provides further information on each dataset, and the procedures used in
cleaning the data.5 Table 1 gives a short description from each dataset. Our main data
source USTAN, is large in size and has a broad coverage in terms of ownership, firm-size,
and industry.
From each dataset, we are able to get information on payroll, value added and capital
stocks. The latter, we re-compute by a perpetual inventory method from balance-sheet
data on capital stocks and investments. We exploit that in all cases we have information
for disaggregated capital by type of capital good. This makes our measure of the capital
stock robust to heterogeneity in capital portfolios, which may otherwise be driving our
results due to wrongly estimated depreciation rates.
Our measure of misallocation is based on factor shares in value added.
L
αit
:=
wit Lit
Pit Yit
K
αit
:=
(rit + δit )Kit
Pit Yit
(1)
where wit Lit is the wage bill of a firm, rit is the real interest rate. Both are divided by
the value added Pit Yit of a firm6 .
We remove 2-digit industry-year effects from our variables α to focus on differences across firms that are purely idiosyncratic. We thus eliminate differences in 2-digit
industry-specific responses to aggregate shocks as well as predictable heterogeneity between firms that comes from the industry structure. After proceeding with the cleaning
L and αK .
steps, we get the estimates for αit
it
For the simple case of Cobb-Douglas production functions with elasticities αL and αK
5
Even though we do not have information whether a plant is single-plant firm or a multi-plant firm
for the case of ENIA, EAM and IBS, the National Statistics from each country provide some estimations
that more than 90% of manufacturing establishment are single-plant firm. Therefore, every time we refer
to an entity as a firm, ignoring if they are part of a multi-plant firm.
6
As shown by Foster et al. (2008), the distinction between revenue and physical productivity is
important. Establishment-level prices are typically not observable and hence one has to rely on industrylevel prices in order to deflate revenue and to obtain output measures. Therefore, our measures are
revenue productivity and we recognize that they could also capture idiosyncratic demand shifts or market
power variation besides quality of production.
4
Table 1: Data Characteristics (before cleaning procedures) from each dataset
Total
Observations
Total
Firms
Avg. Number
of Years per Firm
Avg. Number
of Firms per Year
Period
Germany
854,105
72,853
11.7
32,850
1973-1998
Chile
70,217
10,939
6.4
5,410
1995-2007
Colombia
103,011
24,308
4.2
6,891
1977-1991
Indonesia
422,564
60,802
6.9
19,284
1988-2010
Dataset
and the absence of any other real frictions, one would obtain as conditions for productive
efficiency of an allocation
L
αL = αit
K
and αK = αit
,
(2)
which means intuitively that the dispersion of the expenditure shares around their means
indicates deviations from productive efficiency.
Our measures are not factor productivities from the narrow definition, e.g. ouput per
hour worked. Looking at factor shares instead of factor productivities has the advantage
that it controls for differences in quality of inputs and outputs insofar as they are reflected
K,L
in the market price. In fact, αit
are inversely related to αL,K and as we focus on the
second moments, we would expect similar results7 . For simplicity, we will use factor
productivity as synonym for factor expenditure share knowing the difference.
2.2
High and Low Frequency Deviations
In all four countries we observe large deviations of the log factor shares from their
averages. In order to give the data more structure, we create sequences of J = 5
(J = 11) year sub-panels from the data sets and calculate for any firm that we observe
in {t − (J − 1)/2, . . . , t − 1, t, t + 1, . . . , t + (J − 1)/2} the average log factor share in this
7
The dispersions and correlations were of the same sign and similar magnitude.
5
panel
(J−1)/2
K,L
ᾱit
=J
X
−1
K,L
αit+j
.
j=−(J−1)/2
This gives us for each firm in the sample a series of J-year moving averages which
approximate the low frequency movements in factor shares for that firm. We then
K,L
measure high frequency movements in factor shares, α̂it
by the difference between the
actual factor share and the moving average
K,L
K,L
K,L
α̂it
= αit
− ᾱit
.
This way we can decompose to what extent high and low frequency movements are
responsible for the observed dispersions in factor shares. Moreover, by comparing within
L and αK at both frequencies we can try to understand what drives the factor
firm αit
it
share dispersions at the different frequencies.
2.3
Results
Table 2 provides the results of this exercise for all four countries. Overall the dispersions in factor expenditure shares are smallest in the German sample, much in line with
previous literature, e.g. Foster et al. (2001) that finds smaller dispersions in developed
economies. Yet most surprising is the enormous persistence the dispersions show. For
Germany, the low frequency dispersions are 4 to 7 times higher than the high frequency
ones. For the developing economies, both low and high frequency dispersions are larger
but in particular the high frequency ones exceed the German ones by factor 2 to 3, while
the low frequency dispersions are only larger by at most 40% for labor and 25% for capital. Still this means that in terms of the overall variance of factor shares, low-frequency
movements explain the bulk in all countries. This finding does not change much wether
we take 5 or 11 year moving averages even though the longer moving averages bias the
samples towards larger firms that are observed in all 11 years.
Table 2 reveals another important finding. There is a significant difference in the
co-movement of labor and capital shares at low and high frequencies. At low frequencies
firms that have high labor expenditure shares have low capital expenditure shares – the
correlation is negative, whereas for high frequency deviations the correlation is positive.
Firms that have higher labor expenditure shares in the short run have also higher capital
expenditure shares in the short run. Again this finding holds robustly across all countries.
This new stylized fact – positive correlation between factor productivities at high
6
Table 2: Five and Eleven Year Moving Average Block Estimates
L)
std(αit
K)
std(αit
L , αK )
corr(αit
it
Deviation from 5 year firm average
L)
std(αit
K)
std(αit
L , αK )
corr(αit
it
5 year firm averages
DE
0.067
(0.002)
0.119
(0.006)
0.363
(0.036)
0.234
(0.006)
0.716
(0.047)
-0.169
(0.019)
CL
0.176
(0.003)
0.217
(0.003)
0.557
(0.011)
0.284
(0.002)
0.894
(0.003)
-0.280
(0.007)
CO
0.147
(0.001)
0.169
(0.002)
0.558
(0.007)
0.304
(0.001)
0.759
(0.001)
-0.174
(0.003)
ID
0.230
(0.002)
0.336
(0.003)
0.419
(0.006)
0.318
(0.002)
0.752
(0.002)
-0.211
(0.004)
L)
std(αit
K)
std(αit
L , αK )
corr(αit
it
L)
std(αit
K)
std(αit
L , αK )
corr(αit
it
Deviation from 11 year firm average
11 year firm averages
DE
0.079
(0.002)
0.151
(0.011)
0.352
(0.026)
0.206
(0.002)
0.603
(0.033)
-0.165
(0.002)
CL
0.201
(0.007)
0.263
(0.009)
0.463
(0.028)
0.220
(0.003)
0.741
(0.005)
-0.335
(0.015)
CO
0.168
(0.003)
0.21
(0.004)
0.501
(0.013)
0.271
(0.002)
0.685
(0.003)
-0.217
(0.008)
ID
0.233
(0.006)
0.401
(0.009)
0.360
(0.021)
0.260
(0.003)
0.678
(0.007)
-0.284
(0.013)
All standard errors from block bootstrap.
7
frequencies and negative at low frequencies – suggests that the long-lived deviations in
factor shares are results of some labor-capital trade-off in the long-run. Moreover, as
the bulk of differences in factor shares is fairly persistent a model that tries to explain
dispersions in factor shares overall will need a mechanism that generates persistent differences in factor productivities that increase the factor productivity in one factor at the
expense of the other.
However, the results also tell us that a model which tries to explain the differences
in factor productivity dispersion between developed and developing economies should
focus to some extent on the high-frequency deviations8 . What is more, our empirical
findings suggest that while the correlation at low-frequency should capture a capitallabor tradeoff this trade-off should be absent at high frequency.
An alternative way to phrase the results from Table 2 is to say that at high frequencies
firms operate at inefficient scales while at low frequency firms operate at inefficient
capital-intensities. For a given scale of operations, they employ inefficiently too much
or too little labor relative to capital.
In this paper we focus on the latter aspect of the data and relate our findings in
the next Section to a literature that goes back at least to Samuelson (1947) and Frisch
(1965), where firms employ different production technologies in the short and long run.
We spell out this short- vs. long-run in a dynamic model of technology choice that
features this capital-labor trade-off in the long-run while fixing a firms technology in the
short-run. We see this real frictions approach as complementary to other approaches that
try to micro-found the factor-share or productivity dispersions across firms for example
by financial frictions. However, the correlation findings imply that financial frictions, if
they explain the dispersions, should either directly bias the size of the firm and not the
(shadow) user costs of capital if the frictions are short lived or they need to explain why
financial frictions distort the factor mix a firm is using for extended periods of time.
An initial approximation on explaining this last effect is provided by Midrigan and Xu
(2012), where constrained producers are not able to adjust to the optimal technology,
and given that adoption decisions embodies investments that pay off gradually along the
time, then the existent friction has a long-run effect in the economy.
8
It is important to recognize that the real difference in factor productivity dispersion at high frequency might be obscured due to measurement error that our data presents even after proceeding with
our cleaning steps. Nevertheless, the effect of the measurement error should cancel out at low frequencies,
so this potential bias might appear only at high frequency.
8
3
A Simple Model of Technology Choice
3.1
Model setup
We model a firm that produces output yit using homogeneous capital Kit and labor
Nit under a Cobb-Douglas constant returns production function
α
yit = Kitα Nit1−α = kit
Nit ,
where kit =
Kit
Nit
(3)
is the capital intensity. The firm faces an iso-elastic demand function
for its differentiated product, such that price pit and revenues pit yit are given by
pit = zit
−ξ
yit
,
1−ξ
pit yit = zit
α N )1−ξ
(kit
it
,
1−ξ
where 1/(1−ξ) is the markup of the monopolistic firm and zit is a profitability shock, i.e.
a demand shifter/measure of idiosyncratic productivity, for firm i at time t, z reflects
that the distribution of consumer tastes across products is stochastic over time or timevarying production efficiency at the firm level.
Profits are given by
πit =
α N )1−ξ
zit (kit
it
− wt Nit − rt kit Nit ,
1−ξ
(4)
where wt is the wage-rate and rt are the user costs of capital. We can set wt = 1 without
loss of generality and denote the model in labor units.
While firms can in principle choose to produce output with factor mixes according
to the production function (3), they are bound to the capital intensity they have chosen
in the past for short-run variations in output, i.e. they operate a Leontieff technology.
Whenever a firm actively adjusts the capital intensity, it has to pay some adjustment
cost κ. Following Costain and Nkov (2009), we assume that κ is distributed according
to the cdf
P (κ < x) =
λ0
λ0 + (1 − λ0 ) (λ1 /x)λ2
(5)
This distribution function nests as special cases e.g. Calvo adjustment costs for λ2 = 0,
or non-stochastic adjustment costs at level λ1 for λ2 → ∞.
We formulate the model around a steady state growth trend, where capital, output
and most importantly capital intensity grow by a constant annual rate. This implies, that
in periods of no technology adjustment, the firm’s technology ages relative to the steady
9
state and becomes less capital intense in relative terms, such that absent adjustment:
kit = (1 − γ)kit−1 ,
where γ is a constant annual rate growth rate of aggregate capital intensity.
We assume that the firm commits to its production scale N with a time to build of
one period. Moreover, we assume that κ is i.i.d. while productivity z and the user cost
of capital r follow a first order Markov chain. This yields for the firms dynamic planning
problem
V (k, z, r, N, κ) = π(k, z, r, N ) + max V0 (k, z, r, N ) , Vadj (k, z, r, N ) − κ
V0 (k, z, r, N ) = β max
Ez 0 ,r0 ,κ0 V (k(1 − γ), z 0 , r0 , N 0 , κ0 )|z, r
0
N
0 0 0
0 0
0 ,r 0 ,κ0 V (k , z , r , N , κ )|z, r
Vadj (k, z, r, N ) = β max
E
z
0 0
(6)
N ,k
where 0 denotes next period variables. V is a firm’s value when behaving optimally in all
future periods. V0 is the continuation value obtained when only adjusting the scale of
production, Vadj is the continuation value obtained from adjusting both scale and capital
intensity.
We can simplify the model by rewriting the model and optimizing out the scale N .
We define v := V − π, v 0 := V0 − π, and v adj := Vadj − π. Further we define optimal
next period’s profits after optimizing for labor demand by
π ∗ (k 0 , z, r) := β max
{E(π(k 0 , z 0 , r0 , N 0 )|z, r)}.
0
N
Then, our firm model is now given by
v(k, z, r, κ) = max v 0 (k, z, r) , v adj (k, z, r) − κ
v 0 (k, z, r)
v adj (k, z, r)
∗
(7)
0
0
0
= π (k(1 − γ), z, r) + βEv(k(1 − γ), z , r , φ |z, r)
= max
{π ∗ (k 0 , z, r) + βEv(k 0 , z 0 , r0 , φ0 |z, r)}.
0
k
The latter variant of the dynamic discrete choice model is for obvious reasons computationally more efficient and in Appendix B.1 we show that a unique solution to this
problem exists.
To close the model, we need to specify laws of motion for the two random variables
10
zit and rt . We assume them to follow a lognormal AR(1) processes
log(zit ) = ρz log(zit−1 ) + it
log(rt ) = ρr log(rt−1 ) + ηt
, where ρz < 1 and it ∼ N (µz , σz2 )
(8)
, where ρr < 1 and ηt ∼ N (µr , σr2 ),
(9)
and we assume z to be conditionally independent across i and t and r conditionally
independent across t but perfectly correlated across firms.
The latter implies that there is some commonality in the capital intensity choice
across all firms that adjust in a given point in time. If for example the user cost of
capital is particularly high relative to the wage rate, then all firms adjusting their capital
intensity choose small levels of the capital intensity. On the contrary if r is low firms
will tend to choose high levels of capital intensity. This together with the drift γ, creates
a differences in capital intensities between firms depending on the time of their last
adjustment.
3.2
Implications for High and Low Frequency Factor Share Dispersions
The previous argument shows how our model should not only be able to generate
expenditure share dispersions, but also that the adjustment/non-adjustment in capital
intensities should generate a negative correlation between capital and labor expenditure
shares. In order to understand better the mechanics of our model in generating factor share dispersions at different frequencies it is helpful to go through the first order
conditions that determine optimal production scale N ∗ .
Differentiating E(π 0 ), equation (4), w.r.t. N 0 we obtain as the first-order condition:
E(z 0 |z)N 0−ξ k 0α(1−ξ) = 1 + E(r0 |r)k 0 ,
(10)
which expresses optimal labor supply tomorrow as a function of capital intensity tomorrow and the stochastic states today. Making use of z 0 = E(z 0 |z) exp( − σ2 /2), we can
use this expression to rewrite labor and capital shares (using w = 1)
log α
log α
L
K
:= log
:= log
w0 N 0
p0 y 0
r0 K 0
p0 y 0
= log
= log
1−ξ
0α(1−ξ)
0
zk
N 0−ξ
K 0 (1 − ξ)
z 0 k 0α(1−ξ) N 01−ξ
= log
exp(−0 + σ2 /2)(1 − ξ)
1 + E(r0 |r)k 0
= log
exp(−0 + σ2 /2)k 0 (1 − ξ)
1 + E(r0 |r)k 0
,
.
Since profitability shocks have no serial correlation, is responsible for the high frequency dispersions in factor shares. On the other hand, k is persistent if technology
11
adjustment cost are non negligible, such that differences in the capital intensity k are
responsible for the low frequency differences.
Importantly, z and k imply different co-movement of factor shares. While innovations
in z move both the capital and the labor shares in the same direction, an increase in the
capital intensity decreases labor share and increases capital share. We can use this to
decompose the covariance of factor shares:
Cov log αL , log αK
= Cov − 0 − log[1 + E(r0 |r)k 0 ], −0 − log[1 + E(r0 |r)k 0 ] + log(k 0 )
= σ2 + Cov − log[1 + E(r0 |r)k 0 ], − log[1 + E(r0 |r)k 0 ] + log(k 0 )
≈ σ2 + Cov − E(r0 |r)k 0 , −E(r0 |r)k 0 + k 0 − 1 = σ2 − E(r0 |r)[1 − E(r0 |r)]V ar(k 0 ).
The latter shows formally, what we described intuitively in the preceding paragraph. The
covariance of the two factor shares is driven by profitability shocks that lead to positive
comovement and differences in capital intensity leading to negative comovement. The
first term is a high frequency difference while the latter is a low frequency difference.
Next we will ask to what extent a realistically calibrated version of our model is able to
capture the quantitative structures, we found in the data.
4
Calibration and model mechanics
In this section, we present the general calibration strategy. In particular, we discuss
the calibration of adjustment costs. Assuming Calvo adjustment costs, we show an upper
boundary to the model’s dispersions.
Table 3 contains all model parameters except for the calibration of adjustment costs.
We rely on standard choices for the labor elasticity, α = 1/3, and aggregate depreciation
rates, δ = 0.1. We differentiate between countries for average user cost of capital,
choosing a two percentage point differential, which reflects differences in the development
of financial sectors9 . The discount factor is determined in accordance with our balanced
1+γ
growth environment β = Et 1+r
, where γ is the average growth rate of per capital output.
t
We calibrate the stochastic process of the profitability shock z following Bachmann and
Bayer (2011). We set the persistency to 0.95 and σz = 0.29.
Additionally, we construct time series of hourly wages, interest rates and depreciation
rates to get relative factor prices10 . We then estimate the lognormal AR(1) process
9
10
We refer to Bustos et al. (2003) for estimates of user costs and depreciation in Chile.
We obtained the hourly wage series from Statistisches Bundesamt, Beiheft zur Fachserie 18, Reihe
12
specified in the setup of our model. Finally, we calibrate the demand elasticity by
1 − ξ = ᾱL + ᾱK .
Table 3: Parameter values
Germany
Chile
Colombia
Indonesia
α
δ
uc
¯
ρz
σz
0.33
0.10
0.15
0.95
0.29
0.33
0.10
0.17
0.95
0.29
0.33
0.10
0.17
0.95
0.29
0.33
0.10
0.17
0.95
0.29
ρr
σr
γ
β
ξ
0.66
0.11
0.02
0.97
0.32
0.88
0.30
0.03
0.97
0.38
0.80
0.34
0.02
0.95
0.21
0.60
0.37
0.04
0.96
0.07
Assigned parameters
Labor elasticity
Aggregate capital depreciation
Average user cost of capital
Persistence of demand shifter
Std. of demand shifter
Estimated outside the model
Persistence of relative factor price
Std. of relative factor price
Steady state growth rate
Discount factor
Elasticity of demand
An important component of our model is the specification of adjustment costs11 .
Both the magnitude and variation of adjustment costs drive our model results. For
large costs, the firm will adjust less often or never its technology. On the other side, if
these costs fluctuate across time, the firm may find it optimal to wait for a lower future
realisation of costs. Moreover, the tails of the distribution may influence the dispersion
of capital and labor productivities. All moments of the distribution are given by the
three parameters of our cdf in equation 5.
We start our model analysis using a simple specification of Calvo adjustment costs
under the ad hoc assumption of 5% adjustment probability, which implies technology
3 (Germany), Instituto Nacional de Estadistica, INE (Chile), Departamento Nacional de Estadistica,
DANE (Colombia) and Badan Pusat Statistik, BPS (Indonesia). We further proceed indexing the series
using the consumer price index from each country. The depreciation rates are obtained from from VGR
(Germany), Henriquez (2008) (Chile and Colombia) and van der Eng (2008) (Indonesia). As for interest
rates, we take bank overdraft real interest rate (Germany) and the average indexed lending interest rate
of the financial system for the rest of the countries. For more information, see Appendix A.1.
11
See Gourio and Kashyap (2007) for an extensive discussion on the subject.
13
adjustment every 20 years on average. Our general specification of adjustment probability in (5) nests this case under λ2 = 0. To be precise, with probability λ0 , the firm faces
zero adjustment costs and else, it cannot adjust at all. Given the particular shape of this
distribution, some firms will deviate strongly from their optimal technology. Conditional
on the adjustment frequency, this will impose an upper bound on the dispersion. The
inefficiency imposed by the choice of technology in this model disappears under Calvo
adjustment costs. Table 4 reports the simulated moments using the parametrization of
Table 3.
Our model generates the right signs of correlations at low and high frequencies. Given
an adjustment frequency of 5%, our model is able to explain more than half of the low
and high frequency dispersions. What is more, our simple specification does explain
some of the differences between countries based on the growth rate of capital intensity
and relative factor prices.
Table 4: Calvo adjustment costs (λ0 = 5)
Germany
Data Model
Chile
Data Model
Colombia
Data Model
Indonesia
Data Model
HF Corr(αL , αK )
HF Std(αL )
HF Std(αK )
0.36
0.07
0.12
0.58
0.09
0.12
0.56
0.18
0.22
0.34
0.10
0.19
0.56
0.15
0.17
0.53
0.10
0.15
0.42
0.23
0.34
0.20
0.11
0.27
LF Corr(αL , αK )
LF Std(αL )
LF Std(αK )
-0.17
0.23
0.72
-0.81
0.11
0.32
-0.28
0.28
0.89
-0.83
0.13
0.48
-0.17
0.30
0.76
-0.82
0.13
0.31
-0.21
0.32
0.75
-0.85
0.17
0.42
Mean(αL )
Mean(αK )
0.53
0.16
0.47
0.21
0.45
0.16
0.45
0.17
0.50
0.29
0.53
0.28
0.58
0.35
0.59
0.34
LF/HF: Low/High Frequency.
All second moments are based on a 5 year moving average block procedure.
14
5
Model estimation
We estimate the adjustment probability function proposed in (5), i.e. λ0 , λ1 , and
λ2 , in order to match the high and low frequency dispersions and correlations. Given
that we have 3 free parameters, this should allows us to explain the relevant moments
for each country individually.
In order to analyze the importance of structural differences between developing and
developed countries, we change one of the estimated parameters of Chile, Colombia and
Indonesia to the level of the estimator for Germany. For example, a developed country
typically exhibits less volatile economic environments, which implies lower variances of
the demand shifter. By adjusting to a lower level of volatility idiosyncratic demand in a
developing country, we should expect to see lower dispersion at high frequencies.
6
Alternative models
6.1
Factor Adjustment Frictions
to be added
6.2
Financial Frictions
to be added
6.3
Matching Frictions
to be added
7
Conclusion
This paper contributes to the literature on dispersions in factor productivities, which
reflect some form of misallocation of factors. We first establish two new sets of stylized
facts by differentiating between high and low frequency aspects of the dispersions in factor productivities. First, persistent differences in factor productivities explain the bulk
of overall factor-productivity dispersions, which shows the importance to understand the
dynamic structure of factor productivity dispersions. When we decompose the deviations
into a long-lived and a short-lived, more than 48% (70%) of the productivity variance of
labor (of capital) is due to long-lived deviations. Second, we find that these persistent
15
deviations from average factor productivities show negative correlation across factors,
while transitory differences from the average are positively correlated across factors.
We then relate the observed patterns of productivity dispersions to some real friction
in technology choice, by assuming that firms have to select the capital intensity of their
production and then operate a Leontieff production function until they pay fixed costs to
change the capital intensity of their technology. We show that such model is in principle
able to rationalize the observed patterns and can predict at least some of the differences
across countries.
16
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19
A
Data analysis
A.1
Data Sources
German Firm Data: USTAN
Our main data source, USTAN, is large in size and has a broad coverage in terms of
ownership, firm-size, and industry.
USTAN is itself a byproduct of the Bundesbank’s rediscounting and lending activity.
The Bundesbank had to assess the creditworthiness of all parties backing promissory
notes or bills of exchange put up for rediscounting (i.e. as collateral for overnight lending). It implemented this regulation by requiring balance sheet data of all parties involved, which were then archived and collected, see Bachmann and Bayer (2011), Stoess
(2001), and von Kalckreuth (2003) for details.
We remove observations from East German firms to avoid a break of the series in
1990. This leaves us with a sample of 854,105 firm-year observations, which corresponds
to observations on 72,853 firms. The resulting sample covers roughly 70% of the WestGerman real gross value added in the private non-financial business sector.12
Chilean Plant Data: ENIA
We use the Annual Census of Manufacturing (Encuesta Nacional Industrial Anual,
ENIA) conducted by the National Institute of Statistics (Instituto Nacional de Estadsticas, INE) from 1995 to 2007 to get plant level data from Chile.
The ENIA provides information for all manufacturing plants with total employment
of at least ten employees13 . According to INE statistics, ENIA covers about 50% of
total manufacturing employment. The data was used for example by Pavcnik (2002)
and Doms and Bartelsman (2000).
Colombian Plant Data: EAM
For Colombian estimates, we use the plant-level data from the Colombian Manufacturing Survey (Encuesta Anual Manufacturera, EAM) made available through the
National Institute of Statistics (Departamento Administrativo Nacional de Estaditicas,
DANE) for the period 1977 to 1991.
The survey covers information on plants with more than 10 employees or annual
12
We refer to the sectors Agriculture, Energy and Mining, Manufacturing, Construction, and Trade
together as the private non-financial business sector throughout this paper.
13
Even though we do not have information whether a plant is single-plant firm or a multi-plant firm,
Pavcnik (2002) states that more than 90% of Chilean manufacturing firms are single-plant firm.
20
production above 60 million pesos in 1997 prices14 . See, for example, Das et al. (2007)
and Roberts and Tybout (1997) who have utilized this dataset.
Indonesian Plant Data: IBS
The Indonesian estimation is based on the Manufacturing Survey of Large and
Medium establishments (Survei Tahunan Perusahaan Industri Pengolahan, IBS), provided by the National Institute of Statistics (Badan Pusat Statistik, BPS).
The survey covers all entities with 20 or more employees. It consist of plant level
data on capital stocks, labor costs, output value, intermediate inputs and value added
among other variables
15 .
Given that we have information on capital stocks at plant-level from 1988, we estimate our model using the information from the period 1988 to 2010.
For the industry census in 1996 and 2006 we only have information for the aggregate
stock of capital and investment. To construct the dissagregated capital by each source
in those years, we multiply the aggregate stock of capital in that year with the average
of the specific capital share based on the year before and after the industry census.
The Indonesian data was employed, for example, by Blalock and Gertler (2009),
Peters (2011) and Yang (2012).
A.2
Data processing and correction for outliers
Specific cleaning steps in Indonesia
The firm level data provided by the National Institute of Statistics of Indonesia is an
extensive dataset that allows us to examine our hypothesis. However, before proceeding
with the general cleaning steps we needed to implement some specific corrections. For
these corrections, we are based on the methodology applied by Blalock and Gertler
(2009):
• We have to correct from mistakes due to data keypunching. If the sum of the
capital categories is a multiple of 10n (with n being an integer) of the total capital,
we replace the latter with the sum of the categories.
• For some years, we have found duplicate observations (i.e. observations which have
the same values for all the variables but differ in their identification number). In
14
Equivalent to US$ 52,500. This threshold is subject to changes every year.
As in the case of Colombia and Chile, we do not have data at firm level, however, Blalock and
Gertler (2009) explains that, based on BPS estimation, less than 5% of plants belong to multi-plant
firms.
15
21
those cases we drop those observations.
• In order to avoid for sudden changes in total employees in the plant, we drop for
those observations where the change in total workers with respect to previous year
is higher (lower) than 150% (−150%).
• We have corrected for those cases where value added is not consistent with the
formula provided by BPS.
• The industry survey have changed their industry classification from ISIC Rev. 2
in 1998 to ISIC Rev. 3 in 1999. Given that we only need a classification to the two
digit level for our empirical analysis, we were able to deal with this issue without
major problems using the United Nations concordance table.
In addition, we have missing values in capital reported for some firms who reported
in previous and/or subsequent years. In order to overcome with this issue, we follow Vial
(2006) and we estimate the capital category using the following regression by two-digit
sectoral level:
ln Kit = β0 + β1 ln Kit−1 + θ ln Xit−1 + µi + it
(11)
where Kit is the capital category, µi is the individual fixed effect and Xit−1 are a
set of explanatory variables (total output, input, employees and wages). Given that we
have the lagged capital category as explanatory variable, we will only impute the capital
for those cases where the plant have reported a value for the previous period. For the
rest of the cases, we do not impute it.
Finally, to proceed with the PIM, we need have a measure of the depreciation from
each capital source for the initial period. Nevertheless, the survey stop providing a
question regarding depreciation after 2001. For the subsequent periods, we estimate the
log of depreciation as a function of log of capital (by each capital category).
General cleaning steps all countries
We proceed with data-cleaning steps in order to avoid estimation with outliers likely
caused by measurement error. First, using as a threshold 3 standard deviations, we do
not consider firm-year observations at the top-bottom of the percent changes of labor,
capital and value added, of the levels of labor and capital productivities, and finally,
L + αK , we
of the changes in labor and capital productivities. Second, for the value αit
it
do not consider those observations that are below 1/3 and above 3. The rationale from
22
the last cleaning step relates to the fact that we do not take into account firms whose
operation costs are significatively low/high relative to their value added, and we infer
from this values that are subject to measurement errors.
From each dataset, we select those firms/plants that report information on payroll,
gross value added (before depreciation) and capital stocks and for which we have at least
five consecutive observations16 .
A.3
Construct average factor productivities
For our empirical analysis we first need to construct firm-level average labor productivity and average capital productivity. Our panel contains all required variables,
however we should be careful with potential bias induced by the balance sheet nature of
our data. In order to control for measurement errors, we employ the perpetual inventory method (PIM). In particular, this methodology ensures that aggregated movements
in firm-level capital match investment and aggregate yearly depreciation rates, see Caballero et al. (1995). To be more precise, we first separate capital by each source and
apply PIM separately to them for the reason that both types of capital are different in
depreciation behavior. We then determine our new measure of capital merely by the
capital accumulation equation Kt+1,i = Kt,i (1 − δte ) + It,i and initialize using balance
sheet capital. We adjust PIM capital by multiplying with a sector-wise correction factor
and dropping the observation with a correction factor in the bottom 0.1 or top 99.9
percentile. The correction factor is equal to the industry mean of the ratio between the
capital deflated and the value of the capital deflated provided by the capital accumulation equation. We repeat this procedure until convergence of the mean correction factors
to near one or until the procedure exceeds 4 iterations.
The procedure described above is applied for all countries except of Indonesia. For
this country we have information on the current price of capital, so we do not need to
proceed with the correction factor described above17 . Therefore, we create our capital
measure using the capital accumulation equation Kt+1,i = Kt,i (1 − δte ) + It,i , using as
initialization the current price capital.
We furthermore adjust firm-level depreciation rates to match the aggregate economic
depreciation rate on average. In the case of Germany, we are able to get it from the
National Accounts data (vermgenswirtschaftliche Gesamtrechnung, VGR), however for
16
We need to include as a restriction at least 5 consecutive periods for a firm because our empirical
model, specially the 5 year moving average block, requires it.
17
In the specific case of buildings in 1989, the current price is equal to the book value. We correct
from that problem by multiplying the value of buildings in that year with the two-digit industry mean
of the ratio of current and book values from all years.
23
the rest of the countries we are not able to get it from National Accounts. For Chile, we
use a report from Henriquez (2008) which provides an estimate of gross and consumption
of fixed capital by each source for the period 1985-2005, and allows us to derive from
these estimates the depreciation by each capital source. For the case of Colombia, we
use as an indicator for the economic depreciation the average of the derived depreciation
in Chile for the available period. Finally, for Indonesia we use the values from van der
Eng (2008) who estimated the stock of capital in the country for the period 1950-2007.
The average economic depreciation are presented in Table 5.
Table 5: Economic Depreciation
Germany
Chile
Colombia
Indonesia
Machinery
15.1%
6.8%
6.6%
6.3%
Buildings
3.3%
2.6%
2.7%
3.3%
As a time series of risk-free annual interest rates we take bank overdraft real interest
rate for Germany, and the average lending interest rate for 90 days to 1 year (indexed
or adjusted with inflation) for the rest of the countries. Given the fact that agents make
decisions based on the expected value of the real interest rate, we estimate an AR(2)
process of the interest rate, and we use this final value for the construction of αK .
Finally, we deflate all variables in order to adjust from price changes. For capital,
we take an index price of capital (by each capital category when available) using the
information of gross fixed capital formation at current and constant prices from National
Accounts. And for value added and labor expenses, we use either the CPI index or the
GDP deflator.
A.4
Robustness
In order to test if the results are still consistent when looking only at continuing
firms, we provide estimates of the model keeping only the firms that have been in all
years in the dataset. Given that for the longer datasets, such as the IBS, restricting the
sample for all the period would leave us with a significatively low amount of observations,
we have consider that the threshold we should impose for all datasets is equivalent to
24
the total years available from the Chilean firm level data (12 years18 ). The difference
between our empirical and our theorical models consist in that for the latter we do not
consider firm entry or exit, while in the former case, the results could be driven by this
effect. Hence, we keep only firms that have been in all the panel to test if estimates are
still robust when we only look at this subsample.
Additionally, the Chilean plant level data provides a variable called correccion monetaria (monetary correction), which is a measure for price capital adjustments based on
the changes in the purchasing power of the local currency or changes in the exchange
rate. We present the estimation for our econometrical models, using the capital measure
adjusted with the monetary correction factor.
We were also able to test if the financial crisis in Indonesia during the period 19971998 affected our results, by splitting the sample before the crisis (before 1997) and after
the crisis (from 1999)19 .
The results from Tables 6 are similar to the estimates presented in Section 2. The
correlation between labor productivity and capital productivity is positive at high frequencies while at low frequencies is negative.
Summarizing, our estimations are robust when only looking at continuing firms and
when we use a different specification to construct the capital measure.
18
The results are still robust when using different thresholds for the amount of years. The estimates
for Germany are not yet available. We will provide them for a newer version of the paper.
19
For the sample before the crisis we were not able to compute the 11 year moving average given that
we do not have at least 11 consecutive firm observations.
25
Table 6: Other specifications: Moving Average Block estimates
L)
std(αit
K)
std(αit
L , αK )
corr(αit
it
Deviation from 5 year firm average
L)
std(αit
K)
std(αit
L , αK )
corr(αit
it
5 year firm averages
CL - Monetary
correction
0.172
(0.002)
0.264
(0.003)
0.439
(0.012)
0.253
(0.002)
0.905
(0.003)
-0.276
(0.007)
CL - Continuing
firms
0.161
(0.004)
0.200
(0.004)
0.558
(0.018)
0.248
(0.004)
0.746
(0.006)
-0.261
(0.015)
CO - Continuing
firms
0.139
(0.002)
0.160
(0.003)
0.557
(0.010)
0.293
(0.002)
0.709
(0.003)
-0.177
(0.008)
ID - Continuing
firms
0.202
(0.003)
0.331
(0.006)
0.403
(0.014)
0.293
(0.003)
0.706
(0.007)
-0.211
(0.011)
ID - Before
crisis
0.221
(0.002)
0.309
(0.003)
0.429
(0.011)
0.334
(0.002)
0.737
(0.003)
-0.212
(0.006)
ID - After
crisis
0.243
(0.003)
0.378
(0.006)
0.385
(0.015)
0.256
(0.003)
0.780
(0.005)
-0.164
(0.009)
L)
std(αit
K)
std(αit
L , αK )
corr(αit
it
L)
std(αit
K)
std(αit
L , αK )
corr(αit
it
Deviation from 11 year firm average
11 year firm averages
CL - Monetary
correction
0.196
(0.006)
0.333
(0.011)
0.359
(0.035)
0.198
(0.003)
0.755
(0.006)
-0.347
(0.018)
CL - Continuing
firms
0.196
(0.007)
0.268
(0.008)
0.473
(0.033)
0.224
(0.004)
0.726
(0.006)
-0.331
(0.017)
CO - Continuing
firms
0.167
(0.003)
0.214
(0.004)
0.504
(0.016)
0.270
(0.002)
0.683
(0.003)
-0.225
(0.008)
ID - Continuing
firms
0.230
(0.005)
0.404
(0.008)
0.367
(0.023)
0.258
(0.003)
0.680
(0.008)
-0.288
(0.015)
ID - After
crisis
0.230
(0.020)
0.374
(0.041)
0.415
(0.086)
0.250
(0.009)
0.820
(0.018)
-0.164
(0.038)
All standard errors from block bootstrap
26
B
Technology choice models
B.1
Dynamic Optimization
Let us show existence and uniqueness of the transformed model of equation 7. We
can first rewrite the nested equations in a more compact way:
= max{π ∗ (k, z, r) + βEt v(k, z 0 , r0 , φ0 ),
v(k, z, r, φ)
(12)
max
{π ∗ (k 0 , z, r) − φ + βEt v(k 0 , z 0 , r0 , φ0 )}}.
0
k
Assumption 1: α ≤
1
1+ξ .
Lemma 1: π ∗∗ = maxk0 π ∗ (k 0 , z, r) is bounded from above and below ∀z, r.
Proof: Since we know that π ∗ (k 0 , z, r) is a continuous function, it is sufficient to
show that limk0 →∞ |π ∗ (k 0 , z, r)| < ∞. If this is the case, then π ∗ (k 0 , z, r) is bounded
for k 0 → ∞ and by the Weierstrass extreme value theorem, it is bounded for any finite
interval [0, c] ∀c ∈ < and hence bounded everywhere.
Making use of equation 10, we can express optimal labor demand as a function of
capital intensity
"
E(z)k α(1−ξ)
N ∗ (k, z, r) =
1 + (E(r) + δ)k
#1
ξ
,
which we can then use to maximize out labor and express profits as a function of the
choice variable k:
∗
π (k, z, r)
=
zE(z)
1−ξ
ξ
(1 + (E(r) + δ)k)k
α(1−ξ2 )
ξ
(13)
1
(1 − ξ)(1 + (E(r) + δ)k) ξ
1
−
(1 − ξ)E(z) ξ k
α(1−ξ)
ξ
− (1 − ξ)(r + δ)k
1
α(1−ξ)+ξ
ξ
.
(1 − ξ)(1 + (E(r) + δ)k) ξ
When taking the limit for k approaching infinity, only the largest exponents matters, so
27
for covenience we rewrite above equation as
π̃(k) =
Ak
α(1−ξ2 )+ξ
ξ
− Bk
α(1−ξ)
ξ
− Ck
α(1−ξ)+ξ
ξ
1
,
Dk ξ
where A, B, C, D ∈ <. As α, ξ ∈ (0, 1) it is straightforward that as for the numerator,
the largest exponent is the one attached to the first term. For π ∗ (k, z, r) to exist as k
approaches infinity, the exponent of the denominator must be at least as large as the
one attached to the first term, which yields us the following condition
1
α(1 − ξ 2 ) + ξ
1
≥
⇔ α≤
.
ξ
ξ
1+ξ
By Assumption 1, limk0 →∞ |π ∗ (k 0 , z, r)| < ∞.
Lemma 2: v(k, z, r, φ) is constant in N.
Proof: It follows directly that when the return function π ∗ −φI{k0 6=k} is not a function
in N, then v(k, z, r, φ) is not a function in N.
Lemma 3: Let us define the operator T by posing (T v)(k, z, r, φ) equal to the righthand-side of equation 12. This operator is defined on the set B of all real-valued, bounded
and almost everywhere (a.e.) continuous functions with domain <+ ×<++ ×<++ ×[0, φ].
Then T (i) preserves boundedness, (ii) preserves continuity, and (iii) satisfies Blackwell’s
sufficient conditions.
Proof: Our standard neoclassical, monopolisitic competition profit function when
abstracting from adjustment costs is continuous, concave and bounded on the set of
state variables. Including adjustment costs implies a.e. continuous functions.
(i) Consider u ∈ B bounded below by u and bounded above by u. Then (T u)(k, z, r, φ)
is bounded from above since
(T u)(k, z, r, φ)
{π ∗ (k 0 , z, r) − φ}}
≤βu + max{π ∗ (k, z, r), max
0
(14)
≤βu + max
{π ∗ (k 0 , z, r)},
0
(15)
k
k
where the first inequality follows from boundedness of u and the second one from dropping adjustment costs. The second term is bounded above as shown by Lemma 1. To be
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precise, the value function is bounded from above by the maximum continuation value
plus the maximum static profit. Similarly, (T u)(k, z, r, φ) is bounded from below since
(T u)(k, z, r, φ)
≥βu + max{π ∗ (k, z, r), max
{π ∗ (k 0 , z, r) − φ}}
0
(16)
≥βu − φ.
(17)
k
(ii) To show that the function (T u) is continuous for u ∈ B, we note that from
equation 12 it follows that (T u)(k, z, r, φ) is the maximum of two functions. The first
inherits continuity from u(k, z, r, φ) and the second one is constant. It follows that (T u)
is continuous.
(iii) We want to show that (T u) satisfies monotonicity and discounting. For the
former, we note that if u1 , u2 ∈ B and u1 (k, z, r, φ) < u2 (k, z, r, φ) for all k, z and r,
then integrating over the distribution of the stochastic variables (z,r) preserves the above
inequality. Thus
(T u1 )(k, z, r, φ) < (T u2 )(k, z, r, φ).
(18)
As for discounting, we can show that, for any u ∈ B and any constant a:
(T [u + a])(k, z, r, φ) = (T u)(k, z, r, φ) + βa.
(19)
Hence the second Blackwell condition holds as well.
Propositon 1: Equation 12 has exactly one solution (in the metric space B).
Proof: We know from Lemma 3 that T defines a contraction mapping on the metric
space B with modulus β. Existence and uniqueness then follow from the Contraction
Mapping Theorem.
B.2
Technology adjustment costs
We assume adjustment costs to be of fixed nature, which can be justified by planning
costs. A firm’s CEO has to invest time to find out which is the optimal technology
and in more practical terms how to implement it. Assuming that a CEO’s wage is
drawn from the same distribution whether the firm is small or large, justifies fixed costs.
Furthermore, fixed costs imply that adjustment is relatively cheaper for larger firms and
hence large firms should adjust more frequently. Supportively, when applying the split
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sample to the rotation model, table ?? and ??, we find that the dispersion of x1 , which
captures mark-ups, is larger for small firms than for large firms. This points at large
firms adjusting more frequently than small firms.
On the other side, scale-dependent adjustment costs can be interpreted as expertisebased cost or reorganization cost. The former captures losses of technology-specific
human capital losses, whereas the latter could reflect production halts while adopting
a new technology. Besides, replacing fixed by proportional adjustment costs has only
small quantitative and no qualitative effects on the statistics relevant for our study.
Another explanation of technology adjustment costs could lie in consumer preferences. For a traditional shoemaker whose product are handmade leather shoes, consumers’ demand is tied to the specific production technology used. If the shoemaker
changed her technology, she would not only incur planning costs but also loose consumers who are demanding handmade leather shoes only.
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