Production Functions

ECO 310: Empirical Industrial Organization
Lecture 7 - Production Functions (I)
Dimitri Dimitropoulos
Fall 2014
UToronto
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References
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ABBP Section 2
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Olley and Pakes (1996). "The Dynamics of Productivity in the Telecommunications Equipment Industry." Econometrica, Vol. 64(6), pp.
1263-1297.
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Estimation of Production Functions: Motivation
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A …rm’s production technology is describe by its Production Function (PF)
I
A production function Q = F (L; K ) gives the maximum about of output Q
that can be produced using the inputs L and K
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Production functions are important components in many economic models.
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Estimating production functions is useful for, among many others:
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Quantifying economies of scale and economies of scope
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Measuring productivity growth
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Evaluating the e¤ects of new technologies
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Quantifying production externalities
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Estimation of Production Functions: Motivation
I
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There are some important econometric issues in the estimation of PF
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Data Problems - measurement error in outputs and inputs
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Speci…cation Problems - functional form assumptions
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Multicolinearily - inputs are highly correlated.
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Selection - …rm entry and exit from a sample is not exogenous
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Simultaneity - observed inputs may be correlated with productivity shocks
In this course, we will focus on the simultaneity problem, and examine the
approaches taken by the Empirical IO literature to overcome them.
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The Model
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Suppose we have a panel of J …rms observed over T years.
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For each …rm j at time t, we have information on output, labor and capital
fQjt ; Ljt ; Kjt : j = 1; 2; :::J and t = 1; 2; :::T g
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A …rm’s production technology is describe by its production funciton, which
gives the maximum about of Q that can be produced using L and K
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We’re interested in the estimation of a Cobb-Douglass Production Function
Qjt = Ajt Ljtl Kjt k
where
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Ajt represents the Hicksian Neutral e¢ ciency level of …rm j at time t.
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We take
ln Ajt =
I
I
0
+
jt
0
is the mean e¢ ciency level across all …rms
jt
is the deviation from mean e¢ ciency for …rm j at time t
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The Model
I
Suppose we have a panel of J …rms observed over T years.
I
For each …rm j at time t, we have information on output, labor and capital
fQjt ; Ljt ; Kjt : j = 1; 2; :::J and t = 1; 2; :::T g
I
A …rm’s production technology is describe by its production funciton, which
gives the maximum about of Q that can be produced using L and K
I
We’re interested in the estimation of a Cobb-Douglass Production Function
Qjt = Ajt Ljtl Kjt k
where
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l
is the elasticity of output with respect to labor
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k
is the elasticity of output with respect to capital
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A …rm’s returns to scale are
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I
l
increasing returns to scale if
decreasing returns to scale if
+
l
k
+
l +
k
k
>1
<1
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The Model
I
Taking logs results in a linear equation for the production
qjt =
0
+
l ljt
+
k kjt
+
jt
where lower case letters denote natural logs, e.g. qjt = ln Qjt
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We have known since Marshack and Andrews (1944) that OLS estimation of this
equation is problematic
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Speci…cally, the regressor variables, labor and capital, are the choices of a …rm.
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If a …rm has knowledge of its
choices will be correlated with
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jt
when making these input choices, then these
jt
As a result both ljt and kjt are endogenous, and OLS estimation of the PF is bias
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The Simultaneity Problem
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For example, suppose that …rm j purchases its labor and capital from
competitive markets, at the wage rate Wjt and capital rental rate Rjt :
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The cost-minimization problem of the …rm is
min Wjt Ljt + Rjt Kjt subject to Qjt = Ajt Ljt l Kjt k
L jt ; K jt
and it is straight forward to show that the optimal choices of the …rm are
ljt = constant + c1 (rjt
wjt ) + c3 (qjt
jt )
kjt = constant + c2 (wjt
rjt ) + c3 (qjt
jt )
where lower case denotes logs, c1 =
I
l
Since both ljt and kjt depend directly on
cov (ljt ;
jt )
c3 ;; c2 =
jt ;
k
c3 ; c3 = 1=(
l
+
k
)
thus
6= 0 and cov (kjt ;
and OLS will result in bias coe¢ cient estimates
jt )
6= 0
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The Simultaneity Problem
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To be precise about where exactly the endogeneity problems will come from,
we divide jt into two components
jt
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= !jt + "jt
The "productivity shock" !jt
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factors of production that are unobserved by the researcher, but are observed
by the …rm, e.g. variables like managerial ability, input quality, materials, etc.
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the …rm doesn’t choose !jt , but (im)perfectly observes it when making
production decisions – as such, !jt is correlated with ljt and kjt
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the researcher has no information about the value !jt , and thus also includes
it as an econometric error
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The Simultaneity Problem
I
To be precise about where exactly the endogeneity problems will come from,
we divide jt into two components
jt
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= !jt + "jt
The latent error "jt
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factors of production that are unobserved by the …rm (and hence the
researcher), e.g. measurement error, labor problems, machine breakdowns
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the …rm doesn’t know the value of "jt at the time it makes its production
decisions, and thus "jt has no in‡uence on ljt or kjt
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likewise, the researcher has no information about the value !jt , and thus also
includes it as an econometric error
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Solutions to the Endogeneity Problem
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Thus, our model
qjt =
0
+
l ljt
+
k kjt
+
qjt =
0
+
l ljt
+
k kjt
+ !jt + "jt
jt
or
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The composite econometric error in our model is
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We’ve consolidated the endogeneity problems into !it ;and "it is not a concern.
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Various approaches have been taken by the literature to address the simulataneity
problem. We will study:
jt
1. Traditional Solutions
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Instrumental Variables
Fixed E¤ects
2. Control Function Approach (Olley & Pakes)
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Traditional Solutions – Instrumental Variables
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For the production function
qjt =
0
+
l ljt
+
k kjt
+
jt
IV appraches rely on …nding appropriate instruments – variables that:
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are correlated with the endogenous variables ljt and kjt ; but
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do not appear in the production function, and thus are uncorrelated with
jt
Producer theory suggests a set of inuitive instruments – input prices.
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Input prices are correlated with input demands
cov (ljt ; wjt ) 6= 0
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and
cov (kjt ; rjt ) 6= 0
Input prices are determined in competitive factor markets, and not by the
production decisions of any particular …rm
cov ( jt ; wjt ) 6= 0
and
cov ( jt ; rjt ) 6= 0
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Traditional Solutions – Instrumental Variables
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IV is probably the best method if at all possible, as it relies on fewer
assumptions than the other structural approaches we sill study.
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However, for a variety of reasons, IV solutions have not have not been
broadly used in practice
1. Data on input prices is not always available
2. Even if available:
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To use input prices as instruments requires econometrically helpful variation in
these variables, i.e. enough shifts to trace out input demands.
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However, if there is only one competitive input market in the population under
study then all …rms will face the same input prices.
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Then, there will be no cross-sectional variation in input prices; and, time-series
variation is not enough for identi…cation.
3. IV relies on the assumption that …rm do not have any in‡uence on the
realization of !jt (beyond the scope of this course).
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Traditional Solutions – Fixed E¤ects
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A 2nd approach to dealing with the production function endogeneity
prodblem is …xed e¤ects estimation.
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In fact, …xed e¤ects estimators were introduced to economics in the context
of production function estimation by Mundlak (1961)
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Mundalak (1961) studies the estimation of farm production functions
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Mundlak assumes that the productivity shock takes the form of
time-invariant unobserved hetereogeneity – i.e. !jt is constant over time
jt
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Then, the production function takes the form of
qjt =
I
= !j + "jt
0
+
l ljt
+
k kjt
+ !j + "jt
This is the Fixed E¤ects Model
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Traditional Solutions – Fixed E¤ects
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This is the Fixed E¤ects Model
qjt =
0
+
l ljt
+
k kjt
+ !j + "jt
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Since we observe each …rm j at multiple points in time we can de…ne a series of
dummy variables, one for each …rm, to indirectly serve as controls for the !j ’s
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De…ne the …rm i dummy by:
Djti =
1
0
if observation jt is on …rm i
otherwise
then
qjt =
0
+
l ljt
+
k kjt
+
P
i
i !j Djt
+ "jt
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Formally, we’re treating the !j ’s as parameters (on the …rm dummies) to estimate.
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In e¤ect, we are allowing each …rm to have its own intercept term
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Traditional Solutions – Fixed E¤ects
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NB: the availability of panel data (observe each …rm at multiple points in time) is
key, else we do not have enough degrees of freedom to identify the parameters
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In particular, the number of parameters in the model
qjt =
is equal to 3 + (J
0
+
l ljt
1) – namely
+
0;
k kjt
l;
+
k;
P
i
i !j Djt
and the (J
+ "jt
1) !j ’s
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With a panel of J …rms over T years, the number of observations is J
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For T
2 we have J T > 3 + J 1; and thus there are more than enough
degrees of freedom to estimate the model
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However, if he had cross-sectional data, and only observe each …rm once, then the
number of observations in the data is J
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And, since J < 3 + J
T
1;we cannot estimate all the parameters
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Traditional Solutions – Fixed E¤ects
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In practice the …xed e¤ects estimator has not been judged to be very sucessful at
solving endogeneity problems in production functions
1. Appropriateness of time-invariant !jt
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Mundlak’s assumption that the productivity shock !jt takes the form of
time-invariant unobserved hetereogeneity may be plausible for the
estimation of agricultural production function
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However, it is very unrealistic for manufacturing …rms, or other modern
industrial organizations.
2. Attenuation Bias
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If we allow of each …rm to have its own intercept, we must rely on
variation in inputs around this intercept to identify l and k
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However, there is often not enough such variation in the output of a …rm
– the signal-to-noise ratio of the data around each …rm’s intercept is low
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