A Test for Factor Misallocation and Bounds on
Potential Output Gains
Thomas Schelkle∗
April 29, 2015
Abstract
The paper develops a novel test for factor misallocation which relies
only on production functions being homogeneous. These general assumptions reduce the risk of incorrectly rejecting an efficient allocation. Bounds
on potential output gains from reallocating factors are also derived. An
application finds misallocation of labor and capital between agriculture and
non-agriculture in 16 out of 24 analyzed countries in 1985 and a wide range
of potential output gains. Limiting output elasticities to a reasonable range
an efficient allocation is rejected in all countries. The bounds then indicate that potential output gains are negligible in developed countries and
substantial in developing ones. As a consequence factor misallocation between sectors contributes significantly to observed cross-country income differences.
JEL codes: O11, O13, O14, O47
Keywords: Misallocation, factor allocation, test, bounds, agriculture
University of Cologne; Center for Macroeconomic Research; Albertus-Magnus-Platz; 50923
Köln; Germany; Email: [email protected]
∗
1
1
Introduction
The allocation of scarce production factors to different production units is one of
the central topics in economics. An “efficient” factor allocation in the sense that
it maximizes the total value of output at current prices, requires the equalization
of marginal value products of factors across production units. Under certain ideal
conditions such a situation may arise in a market economy. However determining
whether an empirically observed factor allocation exhibits this property or involves
a misallocation of resources is extremely challenging. The main reason is that
marginal products are not directly observable. Instead only average products,
or more generally the value of output and inputs of each production unit, are
observed. In recent years there is a strong renewed interest in factor misallocation
following Hsieh and Klenow (2009) and Restuccia and Rogerson (2008) in order
to explain income and productivity differences between countries. But so far
drawing inferences from the observed data relies on strong and highly parametric
assumptions on the underlying production functions. This raises concerns that an
efficient factor allocation may often be incorrectly rejected simply because these
assumptions are too restrictive and misspecified. This is the Achilles’ heel of
any work trying to identify factor misallocation (in addition to ubiquitous data
problems).
The main contribution of this paper is that it addresses this concern by providing a novel test for factor misallocation which relies on much weaker assumptions. The approach only requires that production functions are homogeneous of
a known degree and satisfy standard regularity conditions. Under these assumptions the paper studies the allocation of two production factors, say capital and
labor, among two or more production units. The theoretical analysis establishes
that these general assumptions imply restrictions on the joint variation of capital
intensities and average products of labor across production units which may arise
in a situation of equalized marginal products. Thus one can point to observations
that are necessarily inconsistent with an efficient factor allocation. This allows to
test and possibly refute the null hypothesis that an observed factor allocation is
efficient. The generality of the underlying assumptions implies that a rejection of
an efficient allocation by this test should induce a relatively high level of confidence
that such a hypothesis is indeed wrong.
A natural measure of the economic significance of misallocation is by how
much total output could potentially be increased by eliminating the distortions
and reallocating resources efficiently. Computing this measure requires to impose
2
additional functional form assumptions as in previous work. However here the
contribution of the paper is to not depend on exact knowledge of the magnitude
of output elasticities, which are key quantities for these computations and the
identification of misallocation more generally. It is shown how one can then place
upper and lower bounds on the potential output gains from factor reallocation.
An extension allows to flexibly impose stronger assumptions on output elasticities,
which translate into a more demanding test procedure and tighter bounds on
potential output gains.
The developed framework can in principle be used to study misallocation in
many contexts and at various levels of aggregation, for example between firms, industries, sectors, regions or countries. This paper presents an empirical application
to the allocation of capital and labor between the agricultural and non-agricultural
sector in different countries. On the one hand this is motivated by the low productivity of agriculture relative to non-agriculture in poor countries combined with
the puzzling fact that most workers in these countries work in this low productivity sector. These observations have long been known and are empirically well
documented, cf. Gollin, Lagakos, and Waugh (2014) for the most recent measurement efforts. Thus one may suspect a misallocation of resources between these
sectors in poor countries. On the other hand even just casual observation suggests
that there are fundamental technological differences between countries and particularly in agriculture for example due to geographic and climatic differences. This
suggests caution to assume that these technologies, and more precisely the output
elasticities, are identical in all countries. Thus the generality and flexibility of the
presented approach is a key advantage in this context.
The empirical analysis considers a sample of 24 countries in 1985. For these
countries the labor input in both sectors can be corrected for observed differences
in schooling and hours of work based on the work of Gollin, Lagakos, and Waugh
(2014). This addresses an important data and measurement concern in prior
work on this topic. A first important result is that even for the most lenient
assumptions the test indicates factor misallocation between these sectors in 16
out of 24 countries. In contrast, the quantitative magnitude of potential output
gains from reallocation is far less clear and depends considerably on the exact
assumptions regarding output elasticities. For the most general assumptions the
lower bound of potential output gains is close to zero in all countries and never
exceeds 8%, while in many developing countries the upper bound takes an order of
100 − 250%. This range is wide and thus only informative on the overall possible
range and not on the exact magnitude of potential output gains. However I also
3
consider reasonable stronger assumptions on output elasticities, which are more
aligned with the prior literature and still considerably more general. In this case
the test rejects an efficient allocation in all countries and the bounds become
much tighter. One can then conclude that factor misallocation between these
sectors only plays a very minor role in developed countries. In contrast in the
least developed countries of the sample potential output gains are substantial and
fall in a range of 30% to 100%. All these numbers on potential output gains refer to
output evaluated at domestic prices, which is a natural measure when considering
each country in isolation.
I also investigate the implications of misallocation for the huge observed income
differences between countries. Here a contribution of the paper is to distinguish
carefully in the analysis between the allocative role of domestic prices and the accounting role of a common set of international prices which are used in standard
cross-country income comparisons. An interesting result is that in poor countries
the potential gains of output evaluated at common international prices are typically larger than those of output evaluated at domestic prices, while this difference
is small for rich countries. For instance for the stronger assumptions on output
elasticities the potential gains in internationally priced output fall between about
60% and 180% in the least developed countries of the sample. As a consequence in
the absence of intersectoral misallocation the variance of log income per worker in
international prices would be between 20% and 40% lower than the one observed in
this sample. This suggests a substantial explanatory role of factor misallocation
between the agricultural and non-agricultural sector for observed cross-country
income differences.
Finally I document that general equilibrium effects may also dampen the potential output gains from reallocating factors. While the results mentioned above
are all based on a small open economy assumption with constant prices, I also
consider a completely closed economy with demand elasticities calibrated to empirical estimates for each country. For the stronger assumptions on output elasticities, the potential gains in internationally priced output in the most agricultural
developing countries take a value between about 25% and 100% then. As a consequence the variance of log income per worker in international prices would only
be between 10% and 30% lower in the absence of misallocation compared to the
observed variance. General equilibrium effects and changing prices may therefore
be quantitatively important for potential output gains, but do not overturn the
main results.
The paper relates to various strands of the prior literature. First, it is related
4
to the literature on partial identification, which attempts to draw inferences from
data using only weak, but credible assumptions (Manski 2007). Such an approach
may help to establish a wide consensus among researchers on certain questions.
The paper applies this general idea to the topic of factor misallocation.
Second, the paper is related to an active recent literature on factor misallocation. A survey of recent work in this area is provided by Restuccia and Rogerson
(2013). The present paper fits into what Restuccia and Rogerson call the “indirect
approach” which attempts to measure the overall level of misallocation resulting
from the cumulative effect of all distortionary policies, institutions and market imperfections.1 The prior literature using the “indirect approach” has applied two
main methods to identify misallocation.
One set of papers like Hsieh and Klenow (2009) on the manufacturing sector in
China and India, and Bartelsman, Haltiwanger, and Scarpetta (2013) on a larger
sample of countries, or Vollrath (2009) on the allocation between agriculture and
non-agriculture assume Cobb-Douglas production functions and fix their parameters according to information from the United States. Under these assumptions
one can then draw conclusions from observed average products, or more generally
the value of output and inputs, to marginal products and misallocation in other
countries. The drawback is that these assumptions imply that all countries should
exhibit identical output elasticities and therefore the same average product dispersion in the absence of misallocation. These papers do not allow this dispersion
to be affected by any fundamental technological differences between countries. In
contrast the approach in this paper is robust to the presence of such effects.
Another strand of work using the “indirect approach” for identifying misallocation assumes that factor prices are equal to marginal products. Some papers then
take factor price differentials as a direct indication of misallocation. For instance
Banerjee and Duflo (2005) review extensive evidence on a great heterogeneity of
rates of return to the same factor in developing countries. Based on the same
basic assumption on factor prices, a related approach combines information on
factor income shares and average products to calculate marginal products. An
example is the analysis of Caselli and Feyrer (2007) of the cross-country capital
allocation. Furthermore, the approach mentioned above using Cobb-Douglas pro1
The paper is also related to a large literature that employs the “direct approach” and studies
the effects of specific imperfections and distortionary policies. Examples are financial frictions
(Buera, Kabowski, and Shin (2011), Caselli and Gennaioli (2013), Midrigan and Xu (2014), Moll
(2014)), frictional labor markets (Lagos 2006), size-dependent policies and regulation (Guner,
Ventura, and Xu (2008), Garcı́a-Santana and Pijoan-Mas (2014)) or imperfect output markets
(Peters 2013), among others. A more detailed survey of this literature is provided by Restuccia
and Rogerson (2013).
5
duction functions becomes equivalent to this approach if it fixes the parameters
of the Cobb-Douglas according to the factor income shares of each individual production unit instead of the ones of a reference country. In contrast the framework
developed in this paper is independent of any direct assumptions and data requirements concerning factor prices. This is an important advantage for two reasons.
One is the possible departure of factor prices from marginal value products, which
has traditionally played an important role in development economics as discussed
in the survey by Rosenzweig (1988). The other is that reported labor income shares
tend to underestimate true labor income shares because labor compensation of the
self-employed is often treated as capital income as argued by Gollin (2002). Such
a measurement problem may for example be particularly severe in agriculture in
poor countries, which involves a lot of small-scale subsistence farming.
Finally the empirical application in this paper is related to an enormous literature on the role of agriculture and structural transformation for economic development. There is no consensus on whether factor misallocation between the
agricultural and non-agricultural sector is a key feature in this context. Recent
studies that abstract from misallocation are for example Caselli (2005), Duarte
and Restuccia (2010), Gollin, Parente, and Rogerson (2002, 2004, 2007), Lagakos
and Waugh (2013) and Young (2013). Other papers emphasize the role of inefficient factor allocations such as Adamopoulos and Restuccia (2014), Caselli
and Coleman (2001), Chanda and Dalgaard (2008), Hayashi and Prescott (2008),
Restuccia, Yang, and Zhu (2008), Temple (2005), Temple and Wößmann (2006)
and Vollrath (2009). This paper contributes to this debate by providing evidence
on factor misallocation between agriculture and non-agriculture using very general
assumptions, which may help to reach a consensus on this issue.
The paper is structured as follows. The basic theoretical arguments underlying the test procedure and the bounds on potential output gains are developed in
section 2 and useful extensions are presented in section 3. The theoretical framework is applied to data on agriculture and non-agriculture in different countries in
section 4 and the implications for cross-country income differences are explored in
section 5. Section 6 investigates the role of possible general equilibrium effects and
price changes for the potential gains from factor reallocation. Section 7 concludes.
An appendix contains proofs and further results.
6
2
Basic Theoretical Framework
This section derives observable restrictions on factor allocations for given marginal
product differentials, which are valid for all well-behaved and homogenous production functions. These theoretical properties allow to test the hypothesis that
an observed factor allocation exhibits equalized marginal products and to place
bounds on the unobserved marginal product differentials. Under more stringent
assumptions one can also compute bounds on the potential output gains associated
with an elimination of misallocation. The section introduces the main substantive
ideas, but keeps the setup as simple as possible by considering a case with only
two production units and the most general assumptions. The following main section discusses extensions with stronger assumptions on production functions and
a general number of production units.
2.1
Observable Restrictions on Factor Allocations for given
Marginal Product Differentials
There are two production units called a and b, which could for example be two
firms, sectors or countries. The output of goods of these production units is denoted by Ya and Yb with associated given output prices pa and pb . The paper
concentrates on a situation where the production units use two common production factors which are called labor L and capital K here. The total amount of
factors that can be allocated between the two production units is exogenous and
for labor denoted by L and for capital by K. There may also be other factors of
production which are not directly part of the factor allocation problem, for example because they are only used by one of the two production units. The factor
allocation of labor and capital across the production units a and b is then given
by La , Lb , Ka and Kb . The two production units may differ in their production
functions. It is assumed that both production functions are “well-behaved” such
that they satisfy standard regularity conditions like continuity, differentiability
and are strictly increasing and concave in K and L. Furthermore the production
functions are assumed to be homogenous in K and L of degree 0 < λa ≤ 1 and
0 < λb ≤ 1, respectively.2
2
The theoretical properties of allocations with equalized marginal value products and the
test procedure derived below are in principle also valid for production functions with degrees of
homogeneity larger than one, i.e. for increasing returns to scale. But then an allocation with
equalized marginal value products is not necessarily a situation where the value of total output
is maximized. Accordingly equalization of marginal value products would not be desirable and
the developed test would not be very interesting then. This is the reason for restricting attention
7
In the following I allow for the presence of distortions that drive a wedge between the marginal products of the two production units. The labor and capital
wedge are denoted by dL and dK , respectively. These wedges are exogenous and
capture the cumulative effect of market imperfections, institutions and distortionary policies. For an interior solution the factor allocation is determined by
modified marginal value product equations that read as
∂Yb
∂Ya
= pb
∂La
∂Lb
∂Ya
∂Yb
d K pa
= pb
∂Ka
∂Kb
d L pa
(1)
(2)
and the resource constraints La + Lb = L and Ka + Kb = K. One may simply
view these equations as definitions of the marginal product differentials dL and
dK . Hence given known differentials dL and dK one can also determine the factor
allocation by these equations independently of how the factor allocation is determined in reality. A value of a wedge above one indicates that the marginal value
product is higher in production unit b than in a, and vice versa. If the wedges
dL and dK are not equal to one then marginal value products are not equalized
and accordingly total income is not maximized at this allocation. I refer to such a
situation as “factor misallocation” and the allocation being “inefficient”. In contrast an “efficient” allocation is one where marginal value products are equalized
(dL = dK = 1) such that the total value of output is maximized. The main aim
of this paper is to assess whether an observed factor allocation could or could not
be an “efficient” allocation in this total output maximizing sense.3
The main problem in identifying factor misallocation is that marginal products
are unobservable. Thus, I show first how marginal product differentials are related
to average product differentials and other observable variables. Dividing equation
(1) by (2) yields an equation involving marginal rates of technical substitution
to production functions which are at a maximum linearly homogeneous.
3
This way of defining “efficiency” is motivated by a macroeconomic perspective focussed on
income comparisons and explaining income differences. However a maximization of the total
value of output at current prices is also related to traditional theoretical concepts like productive efficiency and pareto-optimality. Under the standard assumptions on production functions
maintained here, the allocation being total output maximizing at current prices is sufficient for
productive efficiency and the allocation being on the production possibility frontier. But it is
not necessary because the allocation could be on the production possibility frontier and only be
total output maximizing for a different set of prices. If one also makes standard assumptions
on households, which imply that their marginal rates of substitution are equal to the current
price ratio, then being total output maximizing at current prices is necessary and sufficient for
pareto-optimality of the allocation.
8
given by
dL
dK
∂Ya
∂La
∂Ya
∂Ka
=
∂Yb
∂Lb
∂Yb
∂Kb
.
(3)
It is more convenient to work with equations (1) and (3), which together with the
resource constraints also determine the factor allocation. The key step now is to
apply two simple “multiply and divide” tricks to equations (1) and (3). These
read as
Yb ∂Yb Lb
Ya ∂Ya La
= pb
La ∂La Ya
Lb ∂Lb Yb
∂Yb Lb
∂Ya La
Kb ∂L
dL Ka ∂La Ya
b Yb
.
=
∂Y
∂Y
a Ka
b
b
dK La ∂Ka Ya
Lb ∂Kb K
Yb
d L pa
(4)
(5)
Rearranging and conveniently renaming terms yields
yb
εLa
=
dL
ya
εLb
εLa εKb dL
kb
=
ka
εLb εKa dK
where
yb
ya
≡
pb Yb /Lb
pa Ya /La
(6)
(7)
is the ratio of average value products of labor between the
a
b
and ka = K
two production units.4 kb = K
are the capital intensities and εLa =
Lb
La
∂Yb Lb
∂Yb Kb
∂Ya La
∂Ya Ka
are the output elasticities of
, εLb = ∂Lb Yb , εKa = ∂Ka Ya and εKb = ∂K
∂La Ya
b Yb
labor and capital in the two production units.
Equations (6) and (7) are the key equations of the paper and provide a number
of insights. First, they show that there is indeed a meaningful relationship between
the average product ratio and the marginal product ratio.5 However one can
only draw direct conclusions from one to the other if one also knows the ratio of
output elasticities. Note that in general the output elasticities of a production
unit fully depend on the amount of factors used and are not constant. This shows
that using Cobb-Douglas functions, where output elasticities are constant, can
be restrictive in this context. An example is the question of factor misallocation
between the agricultural and non-agricultural sector in different countries that I
will use as an application in section 4. If one assumes Cobb-Douglas production
functions for these two sectors with common parameters in all countries then one
4
In the literature the average product of labor ratio is also often referred to as “relative labor
productivity” (RLP).
5
Instead of working with equation (7) one could also work with the equivalent of equation (6)
pb Yb /Kb
εKa
b
for capital, i.e. with ppabYYab /K
/Ka = εKb dK where pa Ya /Ka is the ratio of the average value product
of capital between the production units.
9
will automatically attribute all of the huge differences in yyab across countries to
differences in dL . However part of the yyab variation and possibly even all of it
may simply be caused by variation in output elasticities across countries. Such
a variation in output elasticities may result from different functional forms of
production functions or from different amounts of used factors. A second insight
from equations (6) and (7) is that in principle suitable values of dL and dK can
rationalize any observed allocation characterized by yyab and kkab .
So far the derivation is without much loss in generality. Now I impose the
assumption that the production functions of the two production units are homogenous in K and L of degree 0 < λa ≤ 1 and 0 < λb ≤ 1, respectively. By
Euler’s theorem this has implications for the sum of output elasticities reading as
εLa + εKa = λa
(8)
εLb + εKb = λb .
(9)
Note that given a factor allocation with an observed combination ( yyab , kkab ) and
assuming one knows the values of dL and dK , one can use equations (6), (7), (8)
and (9) to solve for the value of output elasticities εLa , εLb , εKa and εKb at this
allocation. However the standard regularity conditions of production functions
mentioned above require marginal products of K and L to be positive. Hence
output elasticities need to be positive as well and together with equations (8) and
(9) this implies bounds on output elasticities given by
εLa ∈ (0, λa ),
εKa ∈ (0, λa ),
εLb ∈ (0, λb),
εKb ∈ (0, λb).
This means that if one views equations (6), (7), (8) and (9) as a system in output
elasticities, this system is overidentified now. Conversely these equations together
with the bounds on output elasticities and given values of dL and dK imply restrictions on the observable quantities ( yyab , kkab ). These restrictions are stated by
the following proposition (all proofs are relegated to appendix A).
Proposition 1. If two production units a and b operate with well-behaved and
homogenous production functions of degree λa and λb and the factor allocation
exhibits marginal product differentials of labor dL and capital dK between the production units then the average product of labor ratio
10
yb
ya
and capital intensity ratio
kb
ka
at this allocation satisfy either
dL
kb
>
ka
dK
kb
dL
<
ka
dK
dL
kb
=
ka
dK
λa
yb
λa k b
dL <
<
dK ,
λb
ya
λb k a
λa k b
yb
λa
dK <
< dL ,
λb k a
ya
λb
yb
λa
= dL .
ya
λb
and
and
and
or
or
Proposition 1 shows that the maintained assumptions on production functions
and given values of λa , λb , dL and dK imply that the observable quantities yyab
and kkab fall within a certain set. This set of possible combinations of ( yyab , kkab ) is
illustrated as the shaded area in figure 1. Though this set is in principle large, the
key result here is that “not anything goes”. There are combinations of ( yyab , kkab )
which can never occur for given values of dL and dK . Thus hypotheses about
specific values of dL and dK can be refuted because one can point to observations
of ( yyab , kkab ) combinations which are inconsistent with such hypotheses. This forms
the basis for a hypothesis test proposed in the next subsection.
Figure 1: Illustration of Proposition 1
yb
ya
λa
d kb
λb K ka
λa
d
λb L
dL
dK
kb
ka
Notes: The shaded area represents the set of ( yyab , kkab ) combinations which may be consistent
with the basic assumptions and given specific values of λa , λb , dL and dK . The remaining area
represents ( yyab , kkab ) combinations which can never arise under these assumptions and parameter
values.
11
2.2
The Test for an Efficient Factor Allocation
The previous theoretical results can be used to test the null hypothesis that an
observed factor allocation is efficient, i.e. that marginal value products are equalized such that dL = 1 and dK = 1. The alternative hypothesis is that at least one
marginal product differential deviates from 1. The test relies on the assumptions
on production functions of the previous section. First one needs to pick specific
values for the degree of homogeneity λa and λb of the two production functions.
For example in many applications there may be good reasons to assume constant
returns to scale and accordingly choose a value of 1. In other applications the researcher may want to use a value below 1 because a fixed factor other than capital
or labor like land or managerial skills is also key for production and there are only
constant returns to scale to all factors.
The test requires observations on the average product of labor ratio yyab and
capital intensity ratio kkab between the production units a and b at the current
allocation. Given the assumed values λa and λb the test of the null hypothesis
then simply consist in checking whether the observed combination ( yyab , kkab ) satisfies
the conditions of proposition 1 for dL = dK = 1, which are either
kb
>1
ka
kb
<1
ka
kb
=1
ka
and
and
and
λa
yb
λa k b
<
<
,
λb
ya
λb k a
λa k b
yb
λa
<
< ,
λb k a
ya
λb
yb
λa
= .
ya
λb
or
or
In other words one checks whether the observed ( yyab , kkab ) combination is an
element of the shaded non-rejection region of figure 2, which is the equivalent to
figure 1 for the specific values dL = dK = 1. The figure also contains four examples
A-D of possible observations. If the observed ( yyab , kkab ) combination does not satisfy
the test conditions like observations A and C, then one rejects the null hypothesis.
Under the maintained assumptions these factor allocations cannot possibly be
efficient and necessarily involve a misallocation of resources. In contrast, if the
above conditions are satisfied for the observed allocation like for observations B
and D, then one cannot reject the null hypothesis. In this case the factor allocation
may be efficient. As in all tests a failure to reject the null hypothesis does not
imply that the null is correct. Here this means that even if the factor allocation
satisfies the above conditions, it may still be inefficient.
The pros and cons of the proposed test can then be explained using analogies
12
Figure 2: The Test for an Efficient Factor Allocation (dL = 1, dK = 1)
yb
ya
λa kb
λb ka
A
B
λa
λb
D
C
1
kb
ka
Notes: The shaded area represents the non-rejection region and the rest the rejection region. If
an observed ( yyab , kkab ) combination is an element of the rejection region then one rejects the null
hypothesis of an efficient factor allocation (dL = 1, dK = 1). If it falls into the non-rejection
region, one does not reject the null hypothesis. Points A-D refer to hypothetical examples that
one may observe.
from statistical hypothesis testing. The main advantage of the test is that it
relies on relatively weak assumptions. Accordingly the probability of making a
type 1 error (rejecting a correct null) due to a misspecification of the underlying
production model is small. This means that being able to reject the null hypothesis
with this test is very informative and should induce a high confidence that the
null hypothesis is indeed false. The flip side of this advantage is that there is a
higher probability of making a type 2 error (failing to reject a false null) because
of the weak assumptions on production functions. Accordingly, the power of the
test (the probability of not committing a type 2 error) may be small. In other
words one needs to be aware that failing to reject the null is not necessarily very
informative. The following subsection provides more details on these power issues
and shows what degree of misallocation may still be present in cases where one
cannot reject the null hypothesis. An extension to the framework presented in
section 3.1 allows to tighten the basic assumptions and to trade off the probability
of making these two types of errors.
It may also be helpful to directly compare the test developed here with the
implicit rejection and non-rejection region of the approach in the previous litera-
13
ture that assumes Cobb-Douglas production functions with some given values for
the elasticity parameters. In that case the non-rejection region, where one does
not reject an efficient factor allocation, shrinks to one single point in this graph.
This shows that the approach of the prior literature will basically always reject
an efficient factor allocation and many of these rejections may be incorrect unless
the used parameter values are exactly equal to the unknown true elasticities.
Finally, note that instead of testing for an efficient allocation (dL = 1 and
dK = 1) one can of course also test other hypotheses on specific values of dL and
dK . In this case one checks whether the observed ( yyab , kkab ) combination satisfies the
conditions of proposition 1 given the specific values of dL and dK that one wishes
to test.
2.3
Bounds on Marginal Product Differentials
This section provides bounds on the magnitude of marginal product differentials
dL and dK that are consistent with a factor allocation characterized by a specific
observed ( yyab , kkab ) combination. This aim can be achieved without adding any
further assumptions. The form of these bounds is described by the following
corollary which follows directly from proposition 1.
Corollary 1. If two production units a and b operate with well-behaved and homogenous production functions of degree λa and λb and the factor allocation involves an average product of labor ratio yyab and a capital intensity ratio kkab between
the production units then the marginal product differentials of labor dL and capital
dK between the production units are either
dL > deL
dL < deL
where
dL = deL
deL ≡
dK < deK ,
dK > deK ,
and
and
or
dK = deK ,
and
yb
ya
λa
λb
or
and
deK ≡
yb
ya
λa kb
λb ka
.
The intuition behind the corollary can be understood using graphical arguments and figures 1 and 2. Consider for example observation A in figure 2. This
observation is not part of the shaded area in this graph and thus inconsistent with
a situation of equalized marginal products (dL = 1 and dK = 1). One can then
ask which values of dL and dK would be consistent with this observation or in
14
other words how does the shaded area need to be shifted such that observation A
becomes part of it? The answer is provided by noting how changes to dL and dK
shift the straight lines in figure 1. For example a sufficiently high dL keeping dK
at a value of 1 or a sufficiently high dK keeping dL equal to 1 will shift the shaded
area such that it encompasses observation A. In fact many different combinations
of dL and dK achieve this aim. The exact set of possible (dL ,dK ) combinations is
provided by corollary 1 and characterized by the two boundary terms deL and deK .
Graphically these values for dL and dK are those that yield an intersection of the
horizontal and the upward-sloping line in figure 1 exactly at the observed ( yyab , kkab )
combination.
Figure 3 shows the possible marginal product differentials as described by corollary 1 for each of the examples A-D considered in figure 2. These graphs illustrate
that each ( yyab , kkab ) combination can in principle be generated by many different
(dL ,dK ) combinations. It is not possible to place a bound on each marginal product differential separately. Instead one can only place bounds on combinations of
dL and dK .
First consider observations A and C, for which the test rejects an efficient
factor allocation. The point dL = 1 and dK = 1 (marked by “+”), where marginal
products are equalized, is not part of the shaded area for these observations. This
is just a different way to state that one was able to reject dL = 1 and dK = 1
here. Also observe that for observation A (C) at least one of the marginal product
differentials needs to be larger (smaller) than 1.
In contrast for observations B and D, where the test does not reject the null
hypothesis of an efficient factor allocation, the point dL = 1 and dK = 1 is part of
the shaded area. However these observations are in principle also consistent with
marginal product differentials substantially different from 1, even though the test
could not reject the null hypothesis of an efficient factor allocation. This illustrates
the point raised earlier that not being able to reject the null hypothesis does not
imply that the allocation is necessarily efficient.
2.4
Bounds on Potential Output Gains
Factor misallocation implies that the economy could produce more output in total
with the given factor endowments. This section develops simple lower and upper
bounds on the potential output gain associated with moving from the current
allocation to an output maximizing allocation. Such an output gain G expressed
15
Figure 3: Illustration of Corollary 1
(a) Observation A
(b) Observation B
dK
dK
deK
+
1
1
1
e
dK
deL
1
dL
(c) Observation C
dK
1
deK
+
deL
1
deL
dL
(d) Observation D
dK
deK
+
1
+
deL 1
dL
dL
Notes: Each graph refers to one of the hypothetical observations A-D in figure 2. The shaded
area in each graph represents the marginal product differentials which may underly the respective
observation ( yyab , kkab ) under the basic assumptions and given specific values of λa and λb . The
remaining area represents marginal product differentials which are inconsistent with the observed
( yyab , kkab ) combination and these assumptions and values.
as a fraction of current output reads as
G=
Y∗−Y
Y
(10)
where Y = pa Ya +pb Yb denotes total output across the two production units for the
actually observed allocation and Y ∗ = pa Ya∗ + pb Yb∗ for the hypothetical output
maximizing allocation. Here and below a variable without a star refers to the
observed current allocation and a variable with a star to the hypothetical output
maximizing allocation. Note that in order to calculate a “real” total output gain,
one needs to evaluate both Y and Y ∗ at a common set of prices.
16
Determining the counterfactual output maximizing allocation Y ∗ , or the underlying values of Ya∗ and Yb∗ , requires further assumptions in two respects. First
one needs to specify how prices react when one lifts the distortions and hypothetically moves to the optimal allocation. Here I rely on the simplest possible option
and assume that prices remain constant at their current level, which corresponds
to a small open economy assumption. Second one now needs to make more specific
functional form assumptions on production functions. Thus I use the standard assumption of Cobb-Douglas production functions with degrees of homogeneity λa
and λb . These two assumptions yield a parsimonious specification and are a useful
benchmark also with respect to the literature. However I also explore the role of
potential changes in domestic prices in section 6.
The production functions are then given by
Ya = Aa Kaαa Laλa −αa
(11)
Ab Kbαb Lbλb −αb
(12)
Yb =
where αa and αb are parameters governing the respective output elasticities of
capital in the two production units. Aa and Ab represent total factor productivity
of each production unit, which includes for example the effect of all factors of
production other than capital and labor. It is assumed that Aa and Ab are fixed
and remain unchanged when moving to the optimal allocation.
Though I rely on Cobb-Douglas functional forms here, the magnitude of output
elasticities and hence the parameters αa and αb are not fixed to some specific
values. Accordingly one can only provide bounds on the potential output gains.
However in order to derive these bounds step by step it is easier to first derive the
potential output gain as a function of some given parameters αa and αb , which
will be denoted as G(αa , αb ). This in turn requires to characterize the output
maximizing allocation Y ∗ for some given values of αa and αb , which is given by
max
L∗a ,L∗b ,Ka∗ ,Kb∗
pa Aa (Ka∗ )αa (L∗a )λa −αa + pb Ab (Kb∗ )αb (L∗b )λb −αb
s. t. Ka∗ + Kb∗ = K,
L∗a + L∗b = L.
Now define the share of the total capital stock employed in production unit
b by e
kb ≡ KKb ∈ [0, 1] and the share of total labor employed in production unit b
by ℓb ≡
Lb
L
∈ [0, 1]. Total output at the actual allocation can then be written in
17
terms of these quantities as
αa
Y = pa Aa K L
λa −αa
αb λb −αb αb λb −αb
e
k b ℓb
(1 − e
kb )αa (1 − ℓb )λa −αa + pb Ab K L
e∗ and ℓ∗ .
and similarly Y ∗ can be expressed in terms of k
b
b
Using these definitions the potential output gain G(αa , αb ) can be written as
(1 − e
kb∗ )αa (1 − ℓ∗b )λa −αa + pb (e
kb∗ )αb (ℓ∗b )λb −αb
G(αa , αb ) =
−1
k αb ℓλb −αb
(1 − e
kb )αa (1 − ℓb )λa −αa + pb e
b
(13)
b
where the output maximizing values e
kb∗ and ℓ∗b are given by
λa −αa
λb −αb
kb∗∗ )αa (1 − ℓ∗∗
+ pb (e
kb∗∗ )αb (ℓ∗∗
[e
kb∗ , ℓ∗b ] = arg max (1 − e
b )
b )
(14)
e
kb∗∗ ∈[0,1]
ℓ∗∗
b ∈[0,1]
and pb ≡
α
λ −α
p b Ab K b L b b
α
λ −α
p a Aa K a L a a
is a relative “price” that captures how the Cobb-Douglas
aggregate of fractions of total labor and capital relatively map into the value of
output in the two production units. pb is directly implied by the observed factor
allocation and given values of αa and αb because
αb λb −αb
e
pb Y b
kbαb ℓλb b −αb
pb Ab K L
=
αa λa −αa
pa Y a
(1 − e
kb )αa (1 − ℓb )λa −αa
pa Aa K L
yb Lb (1 − e
kb )αa (1 − ℓb )λa −αa
⇐⇒ pb =
.
e
ya La
kbαb ℓλb b −αb
Note that one needs to observe
Lb
La
(or equivalently ℓa =
La
L
(15)
or ℓb =
Lb
)
L
to
compute these potential output gains. This imposes a mild additional requirement
on the data.
The derivation up to now involves the potential output gain for a given observed
( yyab , kkab , ℓb ) combination and specific given values of αa and αb . However I continue
to assume that the specific value of output elasticities, and thus αa and αb , are
not known. Instead only the ranges of possible output elasticities are known,
which are αa ∈ (0, λa ) and αb ∈ (0, λb). The bounds on the potential output gain
for an observed ( yyab , kkab , ℓb ) combination consist of the lowest and highest output
gains G(αa , αb ) when searching over these admissible values of αa ∈ (0, λa ) and
αb ∈ (0, λb ). Note that for a given ( yyab , kkab , ℓb ) combination each of these possible
pairs of αa and αb is associated with a specific combination of marginal product
differentials dL and dK . Furthermore the values of αa and αb affect how these
18
marginal product differentials map into potential output gains. Formally the lower
bound on the output gain G from moving to an efficient allocation is then given
by
G=
inf
G(αa , αb )
(16)
sup
G(αa , αb ).
(17)
αa ∈(0,λa )
αb ∈(0,λb )
and the upper bound G by
G=
αa ∈(0,λa )
αb ∈(0,λb )
The lower and upper bounds on the potential output gains can thus be obtained
by solving the inf-max and sup-max problems consisting of equations (13) and
(14), and (16) and (17), respectively. In the application I solve these problems
computationally using simple numerical methods. For given values of αa and αb
the inner maximization is solved for e
kb∗ and ℓ∗b by a grid search algorithm. The
outer infimum and supremum problems are solved for the optimal values αa and
αb by a simplex algorithm.
The computed bounds allow to gauge the degree of “uncertainty” about the
magnitude of the potential output gains from eliminating misallocation. This
uncertainty derives from the fact that as before the framework does not assume
specific values for the output elasticities of the different factors.
3
Extensions to the Theoretical Framework
This section presents two useful extensions to the theoretical framework. Section
3.1 shows how one can flexibly impose stronger assumptions on output elasticities,
and thereby tighten the test procedure and the bounds on potential output gains.
An extension to a general number of production units is briefly discussed in section
3.2.
3.1
Stronger Assumptions on Output Elasticities
The framework presented in the previous section relies on the fact that mild assumptions on production functions already imply bounds on output elasticities.
These bounds on output elasticities in turn generate restrictions on the ratios of average products of labor and capital intensities one may observe for given marginal
product ratios. However in many potential applications the researcher may want
19
to make stronger assumptions on output elasticities. This section extends the
framework to allow imposing such assumptions and explains how this tightens
the set of allocations that may be consistent with equalized marginal products.
Though the general logic of the theoretical framework remains unchanged, there
are some modifications to the details.
Specifically, instead of requiring that output elasticities are only larger than
zero I now introduce general lower bounds for each elasticity given by
εLa > θLa ,
εKa > θKa ,
εLb > θLb ,
εKb > θKb
where the parameters θLa , θKa , θLb and θKb represent lower bounds on the output
elasticities of the respective factors. These lower bounds need to be consistent
with the degree of homogeneity λa and λb and equations (8) and (9). This implies
that the parameters θLa , θKa , θLb and θKb need to satisfy
θLa ∈ [0, λa − θKa ),
θKa ∈ [0, λa − θLa )
θLb ∈ [0, λb − θKb ),
θKb ∈ [0, λb − θLb )
in order to ensure that θLa + θKa < λa and θLb + θKb < λb . The parameters
θLa , θKa , θLb and θKb need to be set by the researcher based on prior information
about output elasticities. Together with equations (8) and (9) the bounds on
output elasticities are then given by
εLa ∈ (θLa , λa − θKa ),
εKa ∈ (θKa , λa − θLa )
εLb ∈ (θLb , λb − θKb ),
εKb ∈ (θKb , λb − θLb ).
The resulting specification nests the case considered in section 2 when all parameters θLa , θKa , θLb and θKb are set to zero, and allows to flexibly tighten the
test procedure. For any specified values of θLa , θKa , θLb and θKb and specific values
of the marginal product differentials dL and dK one can then again characterize
the set of ( yyab , kkab ) combinations which are in principle consistent with such a situation. This is formalized in the following proposition, which is the equivalent
to proposition 1 for the more general formulation with lower bounds on output
elasticities.
Proposition 2. If two production units a and b operate with output elasticities
of labor and capital bounded from below by θLa , θKa , θLb and θKb respectively
and their production functions are homogenous of degree λa and λb , and the factor
20
allocation exhibits marginal product differentials of labor dL and capital dK between
the production units then the average product of labor ratio yyab and capital intensity
ratio kkab at this allocation satisfy either
dL
kb
>
ka
dK
kb
dL
<
ka
dK
dL
kb
=
ka
dK
and
and
and
yb
< min{φ, ψ},
ya
yb
max{φ, ψ} <
< min{φ, ψ},
ya
yb
λa
= dL
ya
λb
max{φ, ψ} <
or
or
where
θKa kb
λa − θKa
dL +
dK ,
λb
λb k a
λa − θLa kb
θLa
dL +
dK ,
φ =
λb
λb
ka
λa kkab dK
,
ψ =
θKb + (λb − θKb ) kkab ddKL
φ =
ψ =
λa kkab dK
λb − θLb + θLb kkab ddKL
.
Note that proposition 2 is identical to proposition 1 when all parameters θLa ,
θKa , θLb and θKb are set to zero.
Given the assumed values of λa , λb , θLa , θKa , θLb and θKb the test for an
efficient factor allocation then consists in checking whether an observed ( yyab , kkab )
combination satisfies the conditions of proposition 2 for dL = 1 and dK = 1.
Depending on whether at least one of the parameters θLa , θKa , θLb or θKb is
unequal to zero the set of ( yyab , kkab ) combinations that are consistent with an efficient
allocation changes. Figure 4 illustrates this by presenting four different cases
where in each graph only one of these parameters is set to a positive value and all
others are kept at a level of zero. Unsurprisingly introducing the lower bounds on
output elasticities shrinks the set of observations that may be consistent with an
efficient factor allocation (the shaded region). However the resulting shape of the
admissible region differs depending on which parameter is set to a positive value
and accordingly on which output elasticity one sets a positive lower bound. In
practice one can of course set several of the lower bounds on output elasticities to
positive values. Such a case is presented in the empirical application of section 4.
The set of marginal product differentials dL and dK that are consistent with
21
Figure 4: The Test for an Efficient Factor Allocation (dL = 1, dK = 1) with Lower
Bounds on Output Elasticities
(a) Only θLa > 0
yb
ya
(b) Only θLb > 0
yb
ya
ψ
φ
φ
ψ
λa
λb
λa
λb
φ=ψ
1
1
kb
ka
(c) Only θKa > 0
yb
ya
φ=ψ
kb
ka
(d) Only θKb > 0
yb
ya
φ=ψ
φ=ψ
φ
λa
λb
ψ
1
ψ
λa
λb
φ
1
kb
ka
kb
ka
Notes: Each of the graphs considers a situation where only one of the lower bounds on output
elasticities is set to a positive value and all other lower bounds on elasticities are kept at a level
of zero. The shaded area represents the non-rejection region of the test, where one cannot reject
an efficient factor allocation, in each of these situations.
an observed ( yyab , kkab ) combination takes a more complicated shape in this case with
lower bounds on output elasticities. Thus I do not provide the equivalent to corollary 1 here. However it is very simple to solve for these sets computationally. In
the empirical application this is done by specifying a grid consisting of combinations of dL and dK . I then check which of these grid points satisfy the conditions
of proposition 2 for the observed ( yyab , kkab ) combination. With densely spaced grid
22
points this provides a good approximation to the boundaries of the true set.
The bounds on the potential output gains from eliminating misallocation can
then be computed as explained in section 2.4 with a small modification. The
admissible values of αa and αb for the infimum in equation (16) and the supremum
in equation (17) are now given by αa ∈ (θKa , λa − θLa ) and αb ∈ (θKb , λb − θLb ).
3.2
A General Number of Production Units
The discussion up to now has focussed on a situation with only two production
units in order to present the main ideas in the simplest possible setting. Furthermore the application to the agricultural and non-agricultural sector presented in
section 4 of this paper only requires the theoretical results for the case of two production units. However an extension of the framework to a general number of production units may be very useful for future work on misallocation between many
plants, firms or industries. Such a generalization of the theoretical framework is
straightforward. The notation becomes a bit more cumbersome, but conceptually
one simply needs to repeat the steps of the case with two units for a larger number
of pairs of production units in the general case. Therefore I relegate a detailed
derivation to appendix B. An application to a data set with many production
units is left for future research.
4
Empirical Application to Agriculture and NonAgriculture in Several Countries
This section presents an application of the theoretical framework to study the
allocation of labor and capital between the agricultural and non-agricultural sector
within different countries. First the main variables and data sources, as well as
the basic assumptions are presented. The test results and bounds on the potential
output gains for each country are then reported for two cases. The “general
case” refers to the situation covered in section 2 where output elasticities are just
required to be positive. In contrast the case with “stronger assumptions” imposes
strictly positive lower bounds on output elasticities as explained in section 3.1 to
align them more closely with assumptions in the previous literature. This whole
section analyzes the implications of misallocation for each individual country. The
following main section, section 5, evaluates the role of misallocation for crosscountry income comparisons and explaining the observed huge income differences
between countries.
23
4.1
Data
The data set contains information on the factor allocation between the agricultural
and non-agricultural sector in 24 countries observed in 1985. In order to relate the
data to the theoretical framework think of the agricultural sector as production
unit a and the non-agricultural sector as unit b. The data set contains the necessary information to conduct the procedures developed in this paper. These are
b
, the capital intenobservations on the average product of labor ratio yyab = ppabYYab /L
/La
Kb /Lb
and the share of the labor force employed in non-agriculture
sity ratio kkab = K
a /La
ℓb = LLb for each country. Data on these variables is derived from the following
sources.
The share of agriculture in nominal value added is taken from the World Development Indicators. Denoting the agricultural value added share by va one can then
a
. Using domestic prices
compute the output ratio ppab YYba in domestic prices by 1−v
va
instead of some common international prices is appropriate because the factor allocation in a country is determined by domestic prices. Moreover evaluating the
potential output gains at domestic prices is the natural choice when considering
each single country in isolation. However I also present potential gains of output
at international prices in section 5 when I investigate the role of misallocation for
cross-country income differences.
The Food and Agriculture Organization (FAO) provides data on the number of
economically active persons in agriculture and in the total economy. This data is
used to compute the number of workers in agriculture Na and in non-agriculture
Nb . In order to accurately measure the effective labor input I take differences
between sectors in human capital and hours of work into account. Adjusting data
on the number of workers for these two differences is an important advantage
over the previous literature. Here I build on the recent measurement efforts of
Gollin, Lagakos, and Waugh (2014) who collected information on average levels
of schooling and hours of work in these two sectors in different countries from
micro data.6 Schooling levels si of sector i = a, b are transformed into human
6
I thank David Lagakos for sending me their data. The corrections of the labor input based
on this data have the effect that the true labor input is higher in non-agriculture and lower in
agriculture in developing countries relative to what the raw number of workers suggests. The
surveys on which this data is based were typically conducted at the end of the 1990s or beginning
of 2000s, i.e. later than 1985. Though using the data for these corrections to the labor input is
clearly advantageous, a possible concern about these timing differences is that the true schooling
and hours differences between sectors may have been even larger in 1985. This would imply that
the correction may not fully account for the differences in 1985. On the other hand this should
not be a major concern to the extend that the sectoral differences in education and hours only
change slowly over time.
24
capital levels by a Mincerian method such that hi = exp(ξsi ) where ξ is the
return to schooling. The ratio of average human capital levels is then given by
hb
= exp(ξ[sb −sa ]). I assume a return to schooling of 10% such that ξ = 0.1. This
ha
construction of human capital stocks follows Gollin, Lagakos, and Waugh (2014).7
Denoting the sectoral average hours of work by ni the labor force in units of human
capital adjusted hours in sector i is then given by Li = ni hi Ni . Accordingly, the
Nb h b n b
human capital and hours adjusted labor force ratio is computed as LLab = N
.
a h a na
This ratio implies the share of the human capital and hours adjusted labor force
employed in non-agriculture ℓb because ℓb = LLab /(1 + LLab ).
Data on sectoral physical capital stocks in fixed 1990 US-Dollars is obtained
from Crego, Larson, Butzer, and Mundlak (1998). For the capital stock in agriculture Ka I use their series of total agricultural capital which includes fixed capital,
livestock and tree capital. The non-agricultural capital stock Kb is calculated by
subtracting agricultural fixed capital from total economy-wide fixed capital. These
Kb
.
series are then used to calculate the capital ratio between sectors K
a
yb
ya
From these observations I finally compute the average product of labor ratio
K b Lb
as yyab = ppab YYab / LLab and the capital intensity ratio kkab as kkab = K
/ .
a La
4.2
Basic Assumptions
The procedures developed in this paper require to first specify the degree of homogeneity with respect to capital and labor of the production functions in agriculture
λa and non-agriculture λb . I choose parameter values which are very similar to
the ones in the previous literature. For the non-agricultural sector I follow the
bulk of the literature and assume constant returns to scale such that λb is set to
1. In agriculture land is likely to be a key factor of production in addition to
capital and labor. Most of the prior literature assumes constant returns to scale
in agriculture only to all three factors. Accordingly I assume decreasing returns to
scale in agricultural production to capital and labor alone. Specifically, I set λa to
a value of 0.8. This is motivated by an income share of land in agriculture in the
United States of about 18% as found by Valentinyi and Herrendorf (2008), and
calibrations by Caselli and Coleman (2001), Caselli (2005), Vollrath (2009) and
others who use a share of land of 19%.8 There are a few prior studies which also
7
Using a step function in schooling levels for constructing human capital stocks as specified
by Hall and Jones (1999) and Caselli (2005) gives very similar results to the procedure used
here, cf. appendix C.
8
For this calibration of λa I am implicitly relying on the marginal product of land being equal
to its rental price.
25
allow a role of land in non-agriculture such as Caselli and Coleman (2001) and
Valentinyi and Herrendorf (2008). But the factor income share or output elasticity
of land in non-agriculture takes a very small value of only about 5 − 6% in these
studies such that abstracting from this feature is not a major concern. However I
also conduct a series of sensitivity checks on the chosen values for λa and λb which
yield mostly very similar results, cf. appendix C for details. The parameters λa
and λb are held constant across all analyzed countries and scenarios concerning
the lower bounds on output elasticities.
4.3
Results for the General Case
First I conduct the test procedure under the most general assumptions, which
only require output elasticities to be positive. This corresponds to the framework
presented in section 2 or equivalently the one of section 3.1 with all lower bounds
on elasticities set to zero. In the following I present most results graphically, but
detailed results for this and the subsequent subsection and the basic data are also
presented in table 1 below.
The ( yyab , kkab ) combinations that may be consistent with an efficient allocation
are displayed as the shaded region of figure 5 along with the actual observations
for the different countries. For 16 out of 24 countries the test rejects an efficient
factor allocation. This means the factor allocation of about two thirds of all
countries in this sample cannot possibly be characterized by marginal product
equalization. This is an important result because it is based on very general
assumptions. Accordingly, one can be relatively confident that there is indeed
factor misallocation between agriculture and non-agriculture in the majority of
countries. On the other hand there are also 8 countries for which one cannot
reject an efficient factor allocation without more restrictive assumptions. Many
of these are developing countries and feature substantial average product of labor
differences between sectors. However these may be efficient factor allocations
characterized by substantial differences in capital intensities between sectors. Such
allocations can in principle be rationalized by a relatively high output elasticity
of capital in non-agriculture and a low one in agriculture, and a relatively high
output elasticity of labor in agriculture and a low one in non-agriculture.
Figure 6 presents the range of potential output gains associated with moving
to an efficient allocation in percent of current output. Here I have ordered and
grouped countries by their (human capital and hours adjusted) employment share
in agriculture, which can be thought of as a measure of economic development.
26
Figure 5: Test for an Efficient Factor Allocation (General Case)
5.5
ZWE
KEN
5
4.5
MWI
4
3.5
b
y /y
a
IDN
3
2.5
2
1.5
1
ZAF
HND TUR
CHL
ITA PAK
EGY
LKA
CAN
VEN
GBR
FRA
USA
PHL
PRT
SYR
NLD
AUS
SWE KOR
0.5
0
0
2
4
6
kb / ka
8
10
12
Notes: The shaded area represents the non-rejection region. The graph plots the observed
( yyab , kkab ) combination between the agricultural and non-agricultural sector in each country. If an
observed ( yyab , kkab ) combination is not an element of the shaded region then one rejects the null
hypothesis of an efficient factor allocation (dL = 1, dK = 1) in that country.
Note that the horizontal axis is therefore not to scale. For each country the
bottom and top end of the vertical bar represent the lower and upper bound
of these potential output gains. It turns out that without stronger assumptions
the range of potential output gains is very wide and in particular in developing
countries. One noteworthy result is that the output losses from misallocation are
not necessarily very high even in those countries where an efficient factor allocation
can be rejected. The lower bound of potential output gains does not exceed 8% of
current output in any country. On the other hand output losses from misallocation
may also be of a very substantial magnitude as shown by the upper bound of
potential output gains. There are nine countries where the upper bound exceeds
100%, and for three of those it even exceeds 200%. These results are informative
on the overall possible range in which the true potential output gains may fall.
However the general assumptions also allow combinations of output elasticities
which may seem unreasonable or unlikely. Thus the following subsection explores
the effect of imposing stronger assumptions on output elasticities that are more
aligned with standard assumptions in the prior literature.
27
Table 1: Data and Results
Data
General Case
28
Country
Code
yb
ya
kb
ka
ℓa (%)
Australia
Canada
Chile
Egypt
France
Honduras
Indonesia
Italy
Kenya
Malawi
Netherlands
Pakistan
Philippines
Portugal
South Africa
South Korea
Sri Lanka
Sweden
Syria
Turkey
United Kingdom
United States
Venezuela
Zimbabwe
AUS
CAN
CHL
EGY
FRA
HND
IDN
ITA
KEN
MWI
NLD
PAK
PHL
PRT
ZAF
KOR
LKA
SWE
SYR
TUR
GBR
USA
VEN
ZWE
1.05
1.51
2.01
2.30
1.38
2.21
3.16
1.97
4.98
4.49
1.13
1.99
1.32
1.22
2.55
1.12
1.88
1.04
1.16
2.24
1.43
1.37
1.45
5.02
0.72
0.66
0.75
3.81
1.06
1.77
11.29
1.03
4.31
6.42
0.76
1.65
2.35
1.96
1.54
1.83
5.00
0.99
1.96
2.58
0.69
0.75
1.43
4.56
5.9
5.1
14.3
36.5
6.2
38.2
48.9
8.8
70.7
77.1
4.7
44.3
30.0
17.5
12.3
14.9
41.9
4.9
24.6
36.3
2.5
3.1
8.9
59.5
Reject
Y
Y
Y
N
Y
Y
N
Y
Y
N
Y
Y
N
N
Y
N
N
Y
N
Y
Y
Y
Y
Y
Stronger Assumptions
G (%)
G (%)
Reject
0.7
1.8
7.8
0.1
1.5
6.2
0.0
4.3
6.6
0.0
0.7
7.0
0.0
0.0
3.5
0.0
0.0
0.5
0.0
0.3
0.8
0.9
0.6
3.6
56.4
41.3
48.8
126.5
45.2
90.0
227.4
37.1
202.2
248.6
49.7
97.5
116.2
97.5
45.3
95.9
168.6
52.2
106.9
103.7
35.0
38.4
59.0
159.9
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
G (%)
G (%)
1.0
2.3
8.8
5.3
1.7
13.7
6.7
4.4
43.9
33.3
1.0
14.8
0.2
0.2
5.3
0.0
1.2
0.5
0.1
8.2
1.0
1.1
1.4
31.4
3.6
3.7
11.0
19.7
2.2
22.4
36.2
4.6
96.6
102.6
2.6
24.3
9.6
5.4
7.2
4.6
20.3
2.2
7.2
19.8
1.6
1.8
2.8
68.7
“General Case”: Setting of subsection 4.3. “Stronger Assumptions”: Setting of subsection 4.4. “Reject”: Does the test reject an efficient factor allocation
(Y=Yes, N=No)? G (G): Lower (upper) bound on potential output gains (in %).
Figure 6: Range of Potential Output Gains (General Case)
300
MWI
250
Potential Output Gain (in %)
IDN
KEN
200
LKA
ZWE
150
100
50
0
EGY
PHL
SYR
TUR
KORPRT
HND
PAK
VEN
AUS
NLDSWE
FRA
ZAF CHL
CAN
ITA
GBRUSA
10
30
Share of Labor Force in Agriculture (in %, not to scale)
50
Notes: For each country the bar represents the upper and lower bound of potential output gains
for the considered set of underlying output elasticities. Here all output elasticities are only
required to be positive. Countries are ordered by their (human capital and hours adjusted)
employment share in agriculture on the horizontal axis, which is not drawn to scale.
4.4
Results with Stronger Assumptions
The previous subsection shows that even under the most general assumptions one
can already reject an efficient factor allocation in the majority of countries. In
contrast the potential output gains from eliminating misallocation may fall in a
very wide range. In other words their magnitude depends considerably on the
exact magnitude of current output elasticities. The wide range may be caused by
a possibly too agnostic attitude that also allows for implausible values of output
elasticities. Thus this section explores how reasonable stronger assumptions on
output elasticities affect the results of the test and tighten the bounds on output
gains. The aim here is to strike a compromise between generality on the one hand
and prior information or beliefs on the rough magnitude of output elasticities on
the other. This is done by setting suitable lower bounds on output elasticities
represented by the parameters θLa , θKa , θLb and θKb as explained in section 3.1.
In principle there are many ways how one can set these parameters. Ideally I
would like to base this choice on direct observations of output elasticities of the
agricultural and non-agricultural sector in a variety of contexts such as different
countries, time periods and levels of development. However in practice I have to
29
rely on the available estimates and calibrations in the prior literature. These are
usually for the United States and identify output elasticities from factor income
shares, which is problematic in the first place as argued in the introduction. With
these important caveats in mind, I set the parameters θLa , θKa , θLb and θKb such
that the considered range of elasticities contains the values of various recent studies. For this purpose I select all studies on agriculture and non-agriculture cited
in the introduction that consider a specification with labor, capital and land in
agriculture, and at least labor and capital in non-agriculture. These papers form
five different groups of studies which share the same parameter values: first Caselli
(2005) and Vollrath (2009), second Caselli and Coleman (2001) which is very similar to the first group, third the observations of factor income shares by Valentinyi
and Herrendorf (2008), which are also used by Adamopoulos and Restuccia (2014)
and Lagakos and Waugh (2013), fourth Gollin, Parente, and Rogerson (2007) and
fifth Hayashi and Prescott (2008).9 There is substantive variation in the value of
output elasticities between these studies. I then set θLa = 0.4, θKa = 0.1, θLb = 0.4
and θKb = 0.2. These parameter values generate intervals of elasticities that contain the ones of this prior literature as illustrated by figure 7.10 Of course labor
and capital elasticities in each sector still need to sum to the assumed values λa
and λb , respectively. The intervals considered in this subsection are more aligned
with the prior literature than the more general case presented in the previous
subsection. At the same time these assumptions are clearly more general than the
point values used by any single previous study.
The factor misallocation test for this scenario is displayed graphically in figure
8 and detailed results for each country are again reported in table 1. The set of
( yyab , kkab ) combinations, which may be consistent with marginal product equalization, shrinks strongly relative to the general benchmark of the previous subsection.
Even though the assumptions on output elasticities are still fairly general, they
imply that only a very limited amount of average product of labor dispersion can
be rationalized as an efficient allocation. As a consequence the test now rejects
an efficient factor allocation for all countries.
The bounds on potential output gains for the different countries are reported in
figure 9. Under the more restrictive assumptions for all countries the upper bound
9
For the study of Hayashi and Prescott (2008) their factor income shares in value added are
used. For Gollin, Parente, and Rogerson (2007) I consider their modern agricultural technology,
which is the only one meeting the selection criteria.
10
I allow a bit of extra room at the end points of the intervals. Fitting the bounds on output
elasticities as tightly as possible around the values from the previous studies tightens the test
and the bounds on potential output gains even more. But the results are broadly similar, cf.
appendix C.
30
Figure 7: Considered Set of Elasticities Compared to Other Studies
0
0.4
0 0.1
0.4
0
0.4
0
0.7
εLa
1
εKa
1
0.8
0.2
εLb
1
0.6
εKb
1
Notes: The shaded regions represent the interval in which each of the elasticities is assumed
to fall given the specified elasticity bounds. The dots refer to estimates or calibrations of each
elasticity in the prior literature cited in the main text. Some studies use the same values such
that there are less dots than studies.
Figure 8: Test for an Efficient Factor Allocation (Stronger Assumptions)
5.5
ZWE
KEN
5
4.5
MWI
4
3.5
b
y /y
a
IDN
3
2.5
2
1.5
1
ZAF
HND TUR
CHL
ITA PAK
EGY
LKA
CAN
VEN
GBR
FRA
USA
PHL
PRT
SYR
NLD
AUS
SWE KOR
0.5
0
0
2
4
6
k /k
b
8
10
12
a
Notes: The shaded area represents the non-rejection region. The graph plots the observed
( yyab , kkab ) combination between the agricultural and non-agricultural sector in each country. If an
observed ( yyab , kkab ) combination is not an element of the shaded region then one rejects the null
hypothesis of an efficient factor allocation (dL = 1, dK = 1) in that country.
is lower and the lower bound higher compared to the benchmark case. Thus the
possible range of output gains shrinks and this effect is quite strong for most countries. Based on the stronger assumptions on output elasticities maintained here
the computed bounds reveal the following interesting results. There is a general
31
pattern that the potential output gains are higher in more agricultural countries.
For the most developed countries with less than ten percent of their labor force in
agriculture, the upper bound of potential output gains never exceeds 5% of current
output such that this type of misallocation necessarily plays a minor role for these
countries. On the other hand for the most heavily agricultural developing countries with more than half of their labor force in agriculture, the lower bound on
potential output gains exceeds 30%. Misallocation between the agricultural and
non-agriculture sector is thus a phenomenon of first-order importance for these
countries. However in these cases the potential output gains may still fall in a
wide range, but the upper bound indicates that eliminating misallocation cannot
yield more than a doubling of output. The remaining countries are somewhat in
between these two groups.
Figure 9: Range of Potential Output Gains (Stronger Assumptions)
110
MWI
100
KEN
Potential Output Gain (in %)
90
80
ZWE
70
60
50
40
IDN
30
HND
LKA
TUREGY
20
PAK
CHL
10
0
ITA
CANAUS
VEN
FRA
GBRUSANLDSWE
ZAF
PHL
SYR
KORPRT
10
30
Share of Labor Force in Agriculture (in %, not to scale)
50
Notes: For each country the bar represents the upper and lower bound of potential output gains
for the considered set of underlying output elasticities. Here output elasticities are required to
satisfy the more stringent requirements described in the main text. Countries are ordered by
their (human capital and hours adjusted) employment share in agriculture on the horizontal
axis, which is not drawn to scale.
Finally, I turn to the set of marginal product differentials dL and dK which
may underly the observed factor allocations of each country. This set was only
derived formally for the general case in corollary 1. However for the more complicated case with stronger assumptions I simply solve for this set computationally
as explained in section 3.1. Figure 10 presents for each country the set of marginal
32
Figure 10: Possible Marginal Product Differentials
Canada
7
7
7
6
6
6
5
5
5
4
dK
8
4
3
3
2
2
2
1
1
2
3
4
d
5
6
7
8
1
1
2
3
L
6
7
8
1
7
6
6
5
5
5
dK
7
6
4
4
3
3
2
2
2
1
1
1
5
6
7
8
1
2
3
Indonesia
4
dL
5
6
7
8
1
7
7
6
6
6
5
5
5
dK
7
dK
8
4
3
3
2
2
2
1
1
3
4
d
5
6
7
8
2
3
4
d
5
6
7
8
Malawi
1
6
5
5
5
dK
7
6
dK
7
6
4
3
3
2
2
2
1
1
1
6
7
8
1
2
3
Philippines
4
dL
5
6
7
8
1
7
7
6
6
6
5
5
5
dK
7
dK
8
4
3
3
2
2
2
1
1
3
4
dL
5
6
7
8
8
3
4
d
5
6
7
8
3
4
dL
5
6
7
8
6
7
8
4
3
2
7
South Africa
8
1
2
Portugal
8
4
6
4
3
5
5
Pakistan
7
4
dL
2
Netherlands
8
3
4
dL
L
8
2
3
L
8
1
8
1
1
L
4
7
4
3
2
6
Kenya
8
1
2
Italy
8
4
5
4
3
4
dL
4
d
Honduras
7
3
3
France
8
2
2
L
8
dK
dK
5
8
1
dK
4
d
L
Egypt
dK
4
3
1
dK
Chile
8
dK
dK
Australia
8
1
1
2
3
4
dL
33
5
6
7
8
1
2
3
4
dL
5
Figure 10 (Continued)
Sri Lanka
7
7
7
6
6
6
5
5
5
4
dK
8
4
3
3
2
2
2
1
1
1
2
3
4
dL
5
6
7
8
1
2
3
Syria
4
dL
5
6
7
8
1
7
7
7
6
6
6
5
5
5
dK
8
4
4
3
3
2
2
2
1
1
3
4
dL
5
6
7
8
2
3
United States
4
dL
5
6
7
8
1
7
6
6
5
5
5
dK
7
6
dK
7
4
3
3
2
2
2
1
1
1
4
dL
5
6
7
8
7
8
3
1
2
3
4
dL
4
dL
5
6
7
8
5
6
7
8
4
3
3
6
Zimbabwe
8
2
2
Venezuela
8
1
5
1
1
8
4
4
dL
4
3
2
3
United Kingdom
8
1
2
Turkey
8
dK
dK
4
3
1
dK
Sweden
8
dK
dK
South Korea
8
5
6
7
8
1
2
3
4
dL
Notes: The graphs report the set of marginal product differentials which may underly the observed factor allocation. The case with general assumptions is represented by both shaded areas
together and the case with stronger assumptions is represented by the darker area only. The
point of equalized marginal products (dL = 1 and dK = 1) is marked by “+”.
product differentials of labor and capital between agriculture and non-agriculture
which may underly the observed factor allocation. The graphs report this for the
general case of the previous subsection (both shaded areas together) and the case
with stronger assumptions of this subsection (only the darker area). Imposing the
stronger assumptions on output elasticities shrinks this set considerably. Under
these assumptions the results for some countries indicate unambiguously whether
the allocation of a factor is distorted towards agriculture or non-agriculture. For
example in many countries the labor allocation is distorted in favor of agriculture such that the marginal product of labor is higher in non-agriculture than in
agriculture.
34
5
Misallocation and Cross-Country Income Differences
This section investigates the role of misallocation between the two sectors considered in the previous section for explaining the enormous observed aggregate income
differences between countries. For this purpose I compare observed differences in
aggregate income per worker to the differences prevailing in a hypothetical world
where all countries operate with efficient allocations of factors between agriculture
and non-agriculture. If income differences are significantly smaller in the world
with efficient allocations then misallocation between these sectors is an important
explanation of observed cross-country income differences.
The standard approach in the literature for conducting cross-country income
comparisons is to evaluate output of each country by a common set of international
prices. This is supposed to control for the fact that the same goods typically have
lower prices in poor countries, so I follow this approach. For observed aggregate
income per worker in international prices y int this is straightforward and here I
simply use data for the year 1985 from the Penn World Tables (PWT) version
8.11 However obtaining maximized income per worker in international prices y int∗
requires an additional analysis and careful attention to the following issues. The
potential output gains derived in the theoretical framework and computed in the
previous subsections refer to potential gains of domestically priced output G and
not to potential gains of internationally priced output Gint . One needs the latter
quantity to calculate maximized income per worker in international prices y int∗ by
(1 + Gint )y int for each country. Domestic prices are the ones which are allocative
and by which the efficiency or inefficiency of allocations should be judged. Instead
international prices do not determine the allocation of factors within countries, but
are only used for accounting purposes when aggregating and weighting the produced goods. This differential role of the two sets of prices needs to be incorporated
when computing the potential internationally priced output gains. Furthermore,
if the relative price of agricultural and non-agricultural goods at common international prices differs from the relative domestic price, then potential gains of
internationally priced output can differ markedly from the corresponding gains in
domestically priced output. It turns out that this is important in this application
and implies that potential gains in internationally priced output will on average
11
Income per worker is obtained by dividing the variable cgdpo, which is Output-side real GDP
at current PPPs (in millions of 2005 US-$), by the variable emp, which refers to the number of
persons engaged (in millions).
35
be larger than the gains in domestically priced output.
Taking these considerations on the differential role of domestic and international prices into account, the potential gain of internationally priced output Gint
from eliminating misallocation is given by
Gint
=
i
∗
int ∗
pint
a Y a + pb Y b
int
pint
a Y a + pb Y b
where pint
and pint
are international prices, and Ya∗ and Yb∗ are the goods outa
b
puts that maximize the value of total output at domestic prices pa and pb . This
formulation accounts for the allocative role of domestic prices pa and pb and the
int
accounting role of international prices pint
a and pb .
Using the same basic assumptions and functional forms and similar steps as
in the previous potential output gain calculations, the potential internationally
priced output gain Gint for given parameters αa and αb can then be written as
int
G (αa , αb ) =
(1 − e
kb∗ )αa (1 − ℓ∗b )λa −αa +
int
pint
b /pa
pb /pa
(1 − e
kb )αa (1 − ℓb )λa −αa +
pb (e
kb∗ )αb (ℓ∗b )λb −αb
int
pint
b /pa
pb /pa
pb e
kbαb ℓλb b −αb
−1
(18)
where the values of e
kb∗ and ℓ∗b which are output maximizing at domestic prices
continue to be calculated as
λa −αa
λb −αb
kb∗∗ )αa (1 − ℓ∗∗
+ pb (e
kb∗∗ )αb (ℓ∗∗
[e
kb∗ , ℓ∗b ] = arg max (1 − e
b )
b )
(19)
e
kb∗∗ ∈[0,1]
ℓ∗∗
b ∈[0,1]
and pb ≡
p b Ab K
p a Aa K
αb λb −αb
L
αa
L
λa −αa
can be determined as explained before in equation (15).
Note that domestic prices are allocative because they determine the hypothetical
factor allocation e
kb∗ and ℓ∗b when the distortions are lifted. In contrast when
weighting the actual and hypothetical amount of each of the produced goods to
calculate the potential output gains only international prices show up there and
domestic prices cancel out.
As before I do not assume exact knowledge of output elasticities and only the
range in which they need to fall is known. For the case with general lower bounds
on elasticities the parameters αa and αb need to satisfy αa ∈ (θKa , λa − θLa ) and
αb ∈ (θKb , λb − θLb ). Thus I again can only compute bounds on the potential gain
36
of internationally priced output. The lower bound Gint is then given by
Gint =
and the upper bound G
int
G
inf
Gint (αa , αb )
(20)
sup
Gint (αa , αb ).
(21)
αa ∈(θKa ,λa −θLa )
αb ∈(θKb ,λb −θLb )
by
int
=
αa ∈(θKa ,λa −θLa )
αb ∈(θKb ,λb −θLb )
The computation of these bounds on potential internationally priced output
gains can proceed similarly to the domestically priced ones. However the derivations show that there is an additional data requirement because one needs to observe the ratio between the international relative price and the domestic relative
int
pint /pint
(pint
pint /pint
b Yb )/(pa Ya )
,
price bpb /paa . Here I calculate this ratio by noting that bpb /paa = (p
Y
)/(p
Ya )
a
b b
where pa Ya and pb Yb are domestically priced output of the two sectors and pint
a Ya
and pint
b Yb refer to sectoral output in international prices. The domestically priced
output ratio ppab YYab is already part of the data set described in section 4.1. The
internationally priced output ratio
pint
b Yb
pint
a Ya
is calculated by combining data for 1985
on agricultural GDP at international prices from the FAO (Rao 1993) and data
on aggregate GDP at international prices from the Penn World Tables. Due
to methodological differences between the FAO and PWT data I first apply the
rescaling procedure suggested by Caselli (2005) to the FAO agricultural GDP
data to make it comparable to the Penn World Table data on aggregate GDP.12
Non-agricultural GDP at international prices is then calculated as the difference
between aggregate GDP and rescaled agricultural GDP. The internationally priced
output ratio is calculated as the ratio of the resulting non-agricultural GDP and
rescaled agricultural GDP. Finally dividing the internationally priced output ratio
by the domestically priced output ratio gives the ratio between the international
pint /pint
and domestic relative prices bpb /paa .
As noted by Caselli (2005) these estimates suggest that for almost all countries
in my sample the international non-agricultural relative price is higher than the
domestic one, or vice versa that the international agricultural relative price is lower
than the domestic one. This implies that when the elimination of misallocation
leads to a relatively stronger increase of non-agricultural production than agricultural production, both measured in goods units, then the gain in internationally
12
The rescaling procedure also corrects automatically for the fact that the FAO data is expressed in 1985 Dollars and the PWT data in 2005 Dollars.
37
priced total output will be larger than the gain in domestically priced total output.
On the other hand in a case where the move to an efficient allocation involves a
relatively stronger increase in agricultural production than non-agricultural production the internationally priced output gain will be lower than the domestically
priced output gain. These effects are substantial as shown in figure 11, which
reports the bounds on the potential internationally priced output gains. Comparing these results with the earlier ones on domestically priced output gains, both
described directions are possible depending on the country and the considered assumptions. For the general assumptions it is even possible that in some countries
an elimination of misallocation leads to a reduction in total output evaluated at
international prices. However in particular for the most agricultural countries in
the case of stronger assumptions the potential gains in internationally priced output may well be much larger than the gains in domestically priced output. For
instance for the three most agricultural developing countries the potential gains in
internationally priced output take a value between about 60% and 180% compared
to a range between 30% and 100% for domestic prices reported earlier.
Detailed results on these bounds in output gains for each individual country
are provided in table 2. The table also reports information on observed income
differences to the United States, as well as bounds on maximized income per
worker y int∗ implied by the bounds on output gains relative to the observed income
of the United States. This is informative on how the income difference to the
United States is affected if a country eliminates misallocation and income in the
US remains at its observed level. This shows that an elimination of misallocation
may allow some countries to catch-up considerably to the current income level of
the United States. However catch-up is not complete and one should be aware
that even if misallocation is eliminated considerable differences to the current US
income level will continue to exist.
Finally I provide a summary measure on the overall effect of eliminating misallocation in all countries on cross-country income differences. For this purpose
I compare the variance of observed log income per worker Var[log(yiint)] to the
variance of log maximized income per worker Var[log(yiint∗ )]. Since I have only
int
derived the range of potential output gains [Gint
i , Gi ] for each country, I can also
only place bounds on the latter statistic. The lower bound on Var[log(yiint∗ )] is
given by
int
int
int
min Var[log((1 + Gint
∈ [Gint
i , Gi ],
i )yi )] s.t. Gi
M
{Gint
i }i=1
38
i = 1, ..., M
(22)
Table 2: Results on Internationally Priced Potential Output Gains and Observed and Hypothetical Income Differences to the
United States
Data
39
Country
Code
int
yiint /yUS
Australia
Canada
Chile
Egypt
France
Honduras
Indonesia
Italy
Kenya
Malawi
Netherlands
Pakistan
Philippines
Portugal
South Africa
South Korea
Sri Lanka
Sweden
Syria
Turkey
United Kingdom
United States
Venezuela
Zimbabwe
AUS
CAN
CHL
EGY
FRA
HND
IDN
ITA
KEN
MWI
NLD
PAK
PHL
PRT
ZAF
KOR
LKA
SWE
SYR
TUR
GBR
USA
VEN
ZWE
0.83
0.88
0.28
0.09
0.74
0.12
0.09
0.75
0.07
0.04
0.75
0.11
0.12
0.35
0.37
0.29
0.14
0.60
0.10
0.38
0.65
1.00
0.45
0.12
General Case
(%)
Gint
i
0.3
2.7
10.5
-14.1
2.3
15.9
-53.8
4.8
30.0
-26.3
1.0
25.5
-40.5
-25.2
3.9
-26.0
-62.8
-4.2
-4.8
5.4
0.9
0.9
-1.6
17.1
int
Gi
(%)
60.9
31.2
40.1
100.8
36.9
77.9
145.8
29.0
224.6
295.4
45.8
86.1
78.5
35.8
42.7
32.9
103.1
23.7
94.6
68.5
33.0
38.4
32.1
150.7
Stronger Assumptions
/yUS
y int∗
i
yint∗
/yUS
i
0.83
0.90
0.31
0.07
0.76
0.14
0.04
0.79
0.09
0.03
0.76
0.14
0.07
0.26
0.38
0.21
0.05
0.58
0.09
0.40
0.66
1.01
0.45
0.14
1.34
1.15
0.40
0.17
1.02
0.22
0.23
0.97
0.23
0.14
1.09
0.21
0.21
0.47
0.53
0.38
0.28
0.74
0.19
0.64
0.87
1.38
0.60
0.31
(%)
Gint
i
0.6
3.4
11.6
12.6
2.5
25.3
15.9
5.8
85.2
83.7
1.3
37.3
3.1
-0.5
5.7
-3.5
3.9
-0.3
0.3
24.4
1.1
1.1
3.2
56.0
int
Gi
(%)
3.8
4.5
13.8
29.0
2.7
36.0
58.0
6.0
154.3
184.8
2.5
52.5
18.5
12.5
7.6
9.0
42.6
2.5
6.6
41.0
1.6
1.8
5.7
100.8
/yUS
yint∗
i
yint∗
/yUS
i
0.84
0.90
0.32
0.10
0.76
0.15
0.11
0.79
0.13
0.06
0.76
0.15
0.12
0.35
0.39
0.28
0.14
0.60
0.10
0.47
0.66
1.01
0.47
0.19
0.86
0.91
0.32
0.11
0.76
0.17
0.15
0.79
0.18
0.10
0.77
0.17
0.14
0.39
0.40
0.31
0.19
0.62
0.11
0.53
0.66
1.02
0.48
0.25
int
“General Case”: Setting of subsection 4.3. “Stronger Assumptions”: Setting of subsection 4.4. yiint /yUS
: Ratio between observed income per worker at
int∗
int∗
int
int
international prices in country i relative to the one observed in the United States. y i /yUS (y i /yUS
): Lower (upper) bound on ratio between the
maximized income per worker at international prices in country i and the observed one in the United States.
Figure 11: Range of Potential Internationally Priced Output Gains
(a) General Case
MWI
300
250
Potential Output Gain (in %)
KEN
200
IDNZWE
150
PHL
TUR
AUS
50
EGY
SYR
100
NLD
USA
GBR
CAN
SWE
FRA
ITA VEN
LKA
HND
PAK
ZAF CHL
KORPRT
0
−50
10
30
Share of Labor Force in Agriculture (in %, not to scale)
50
(b) Stronger Assumptions
200
MWI
180
Potential Output Gain (in %)
160
KEN
140
120
ZWE
100
80
IDN
60
PAK
LKA
TUR
HND
EGY
40
20
CANAUSFRA
GBRUSANLDSWE
ITA VENZAF
PHL
CHL
PRT
KOR
SYR
0
10
30
Share of Labor Force in Agriculture (in %, not to scale)
50
Notes: For each country the bar represents the upper and lower bound of potential gains in output
at international prices. “General Case”: Assumptions on output elasticities as in subsection 4.3.
“Stronger Assumptions”: Assumptions on output elasticities as in subsection 4.4. Countries are
ordered by their (human capital and hours adjusted) employment share in agriculture on the
horizontal axis, which is not drawn to scale.
40
and the upper bound by
int
int
int
max Var[log((1 + Gint
i )yi )] s.t. Gi ∈ [Gi , Gi ],
M
{Gint
i }i=1
i = 1, ..., M
(23)
where M is the number of countries in the sample. These bounds take all the
int
possible combinations of output gains Gint
∈ [Gint
i , Gi ] between countries into
i
account. I compute these bounds numerically.
Table 3 reports the results of this exercise for each of the two scenarios. The
first column contains the variance of observed log income per worker Var[log(yiint )]
in this sample, which takes a value of 0.96. The next two columns present the
lower and upper bound on the variance of log maximized income per worker
Var[log(yiint∗ )]. The last two columns report the corresponding percentage change
when moving from the observed variance to the hypothetical variance bounds.
Table 3: Cross-Country Income Differences for the Observed and the OutputMaximizing Allocation
Scenario
General Case
Stronger Assumptions
Var[log(yiint )]
0.96
0.96
Var[log(yiint∗ )]
Change (in %)
Min
Max
Min
Max
0.35
0.58
1.57
0.76
-63.3
-39.2
63.7
-20.5
Notes:Var[log(yiint )]: Variance of logarithm of observed income per worker. Var[log(yiint∗ )]:
Variance of logarithm of maximized income per worker. Change: Percentage change in variance
of logarithm of income per worker when moving from observed to maximized income, i.e. from
Var[log(yiint )] to Var[log(yiint∗ )]. Min (Max): Lower (upper) bound on respective variable.
For the stronger assumptions on output elasticities which are in line with the
prior literature, eliminating misallocation causes a reduction of the variance of
log income between about 20% and 40%. Though this is a wide range, this still
indicates a substantial explanatory role of this type of misallocation for observed
cross-country income differences. The results for the more general assumptions
indicate that in a world without misallocation the variance of income may be even
lower with a maximum reduction of about 60%. On the other hand it is theoretically possible that potential output gains are larger in rich than in poor countries.
In this case without misallocation the variance of log income per worker across
countries could be up to about 60% higher. However this theoretical possibility
is ruled out if output elasticities satisfy the stronger assumptions, which are more
aligned with the prior literature and still more general.
41
6
Role of Price Changes and the Demand Side
When computing potential gains in total output from lifting the distortions I have
assumed that domestic prices remain unchanged, which corresponds to a small
open economy assumption. This assumption seems reasonable because there is
substantial international trade in agricultural and non-agricultural products and
many of the considered countries are “small” relative to the world economy. On
the other hand there are also good reasons to be suspicious of this assumption for
instance because non-agricultural production contains non-tradeable goods or because of protectionist policies that exist in many countries. Therefore this section
analyzes the case of a completely closed economy where the local demand structure
plays a role in determining how much prices may change in general equilibrium.
It seems hard to judge whether a completely closed or a small open economy are
more realistic assumptions. Therefore the aim of this section is to understand how
sensitive the computed potential output gains are to these assumptions on price
responses.
For this purpose I use empirical estimates of income and price elasticities of
demand in different countries and assume that the true demand functions are
locally well approximated by a constant elasticity demand function. The demand
for agricultural goods Ca is then given by
Ca = ζ(pa )ε (pb )−(ε+η) (M)η
(24)
where M is income, η is the income elasticity, ε is the own price elasticity,
−(ε + η) is the cross-price elasticity in order to satisfy homogeneity of demand
functions and ζ is a scaling parameter. All these demand parameters are countryspecific, but I omit a country index here for convenience. Accordingly, at the
new hypothetical allocation the demand for agricultural goods Ca∗ satisfies Ca∗ =
ζ(p∗a )ε (p∗b )−(ε+η) (M ∗ )η . Combining these equations implies that the ratio of agricultural consumption levels in the hypothetical and actual situation reads as
Ca∗
=
Ca
p∗b /p∗a
pb /pa
−(ε+η) M ∗ /p∗a
M/pa
η
.
(25)
Imposing market-clearing for agricultural goods such that Ca = Ya and Ca∗ = Ya∗ ,
noting that total income is just given by the value of production such that M =
pa Ya + pb Yb and M ∗ = p∗a Ya∗ + p∗b Yb∗ , and substituting in the production functions
42
yields
(1 − e
kb∗ )αa (1 − ℓ∗b )λa −αa
= pe−(ε+η)
eb )αa (1 − ℓb )λa −αa
(1 − k
p∗ /p∗
!η
(1 − e
kb∗ )αa (1 − ℓ∗b )λa −αa + pe pb (e
kb∗ )αb (ℓ∗b )λb −αb
kbαb ℓλb b −αb
(1 − e
kb )αa (1 − ℓb )λa −αa + pb e
(26)
where pe = pbb /paa denotes the ratio between new and old relative prices.
The values of e
kb∗ and ℓ∗b which are output maximizing at the new domestic
prices, and the ratio of new to old relative prices pe are then determined by solving
simultaneously
λa −αa
λb −αb
kb∗∗ )αa (1 − ℓ∗∗
(27)
+ pe pb (e
kb∗∗ )αb (ℓ∗∗
[e
kb∗ , ℓ∗b ] = arg max (1 − e
b )
b )
e
kb∗∗ ∈[0,1]
ℓ∗∗
b ∈[0,1]
and equation (26). The resulting e
kb∗ and ℓ∗b are then substituted into equation (13)
for the gain in total output at domestic prices or into equation (18) for the gain
at international prices, respectively. The search for the lower and upper bound on
these gains proceeds as before.
The income and own-price elasticity parameters η and ε of agricultural demand
for each country of the sample are taken from the empirical estimates of Seale,
Regmi, and Bernstein (2003) for the category “food, beverages and tobacco” for
the year 1996.13 These estimates feature an income elasticity η below one for
all countries, which is decreasing in the level of development from a value of
about 0.772 for Malawi to 0.103 for the United States. The estimates of ownprice elasticities ε are negative and decreasing in absolute value in the level of
development from about −0.796 in Malawi to −0.098 for the United States. The
cross-price elasticities are close to zero in all countries.
Figure 12 reports the results for the bounds on potential output gains at domestic and international prices for the case with the stronger assumptions on output
elasticities. The black line refers to the results with changing prices and the gray
line plots the respective benchmark results with constant prices to facilitate the
comparison. One observes that the range of potential output gains is generally
lower for the changing price scenario compared to constant prices. This dampening effect is negligible for rich countries, but of a substantial magnitude for poor
countries. For instance for the three most agricultural developing countries the
potential gains in internationally priced output take a value between about 25%
13
Their paper does not contain information for Honduras and South Africa. Therefore I impute
the elasticities of these countries by using the values of the country with the most similar GDP
per worker.
43
Figure 12: Range of Potential Output Gains with Changing Domestic Prices
(Stronger Assumptions)
(a) Domestically Priced Output
120
MWI
Potential Output Gain (in %)
100
KEN
80
ZWE
60
40
IDN
HND
LKA
TUREGY
20
PAK
CHL
CANAUSFRA
GBRUSANLDSWE
ITA
ZAF
VEN
PHL
SYR
KORPRT
0
10
30
Share of Labor Force in Agriculture (in %, not to scale)
50
(b) Internationally Priced Output
200
MWI
180
Potential Output Gain (in %)
160
KEN
140
120
ZWE
100
80
IDN
60
PAK
LKA
HND
EGY
TUR
40
20
ZAF
CANAUSFRA ITA VEN
GBRUSANLDSWE
PHL
CHL
PRT
KOR
SYR
0
10
30
Share of Labor Force in Agriculture (in %, not to scale)
50
Notes: For each country the bar represents the upper and lower bound of potential gains in
output at (a) domestic prices and (b) international prices for the “Stronger Assumptions”, i.e.
assumptions on output elasticities as in subsection 4.4. The black line refers to results with
domestic price changes and the gray line to the benchmark with constant prices. Countries are
ordered by their (human capital and hours adjusted) employment share in agriculture on the
horizontal axis, which is not drawn to scale.
44
and 100% with changing domestic prices, compared to a range of 60% to 180%
with constant domestic prices. I omit the results for the general assumptions
on output elasticities for brevity and because these are less sensitive to changing
prices.
These results have consequences for how much lower the cross-country variance
of log income per worker at international prices will be if the distortions are lifted.
Table 4 reports these results for the situation with domestic price changes. For
the stronger assumptions on output elasticities, eliminating misallocation causes
a reduction of the variance of log income per worker between about 10% and
30% then, compared to a range of 20% to 40% in the benchmark with constant
domestic prices.
Table 4: Cross-Country Income Differences for the Observed and the OutputMaximizing Allocation with Changing Domestic Prices
Scenario
General Case
Stronger Assumptions
Var[log(yiint )]
0.96
0.96
Var[log(yiint∗ )]
Change (in %)
Min
Max
Min
Max
0.35
0.68
1.20
0.87
-63.7
-28.7
25.8
-8.8
Notes:Var[log(yiint )]: Variance of logarithm of observed income per worker. Var[log(yiint∗ )]:
Variance of logarithm of maximized income per worker. Change: Percentage change in variance
of logarithm of income per worker when moving from observed to maximized income, i.e. from
Var[log(yiint )] to Var[log(yiint∗ )]. Min (Max): Lower (upper) bound on respective variable.
Overall in a completely closed economy the general equilibrium effects and
changing domestic prices dampen the potential output gains considerably compared to a small open economy in this application. However the price changes do
not completely overturn the benchmark results. As argued above the truth probably lies somewhere between the two polar cases considered here. In the general
spirit of this paper one could therefore also view the combination of the black
and gray lines in figure 12 as bounds on potential output gains from the combined
“uncertainty” on output elasticities and the strength of general equilibrium effects.
7
Conclusions
This paper develops a theoretical framework to test for the equalization of marginal
products of labor and capital across production units under very general assumptions. The procedure relies in its most general form only on production functions
being homogenous of a known degree and satisfying standard regularity condi45
tions. The advantage of a test based on more general assumptions than the prior
literature is that it faces a lower risk of incorrectly rejecting an efficient factor allocation simply because production functions have been misspecified. In addition
bounds on the potential output gains from factor reallocation are derived, which
require more parametric assumptions. However these bounds are still more general than calculations of the previous literature because they do not require exact
knowledge of output elasticities, which are key parameters in this context. An
extension allows to tighten the test and the bounds on potential output gains by
imposing stronger assumptions on output elasticities. The developed procedures
can in principle be used to analyze misallocation in many contexts and at various
levels of aggregation.
As an illustration the framework is applied to study whether capital and labor are allocated efficiently between the agricultural and non-agricultural sector
in different countries, which is a long-standing and controversial question in development economics. I analyze a sample of 24 countries observed in 1985, for
which important measurement concerns can be addressed by correcting the sectoral labor data for differences in schooling and hours of work. The test rejects an
efficient factor allocation in about two thirds of all countries even under the most
general assumptions. In contrast the ranges resulting from the bounds on potential output gains are wide. However under reasonable stronger assumptions on
output elasticities which are more aligned with the prior literature and still more
general, the test rejects equalized marginal products in all countries. In this case
the bounds on potential output gains are considerably tighter and indicate large
potential output gains in the most heavily agricultural developing countries and
only small gains in developed countries. However I also note that general equilibrium effects may dampen the magnitude of potential output gains. Distinguishing
carefully between the allocative role of domestic prices and the accounting role
of international prices, I find that in the absence of misallocation cross-country
income differences would be substantially smaller than they are in reality. Accordingly, factor misallocation between agriculture and non-agriculture is an important
explanation for the vast observed income differences between countries.
When thinking about policy implications of these findings one needs to keep
in mind that the decisions of people in poor countries are probably optimal given
the institutions and frictions they face. The results of this study therefore do
not support deliberately moving people out of agriculture. Instead determining
the specific frictions, policies or institutions responsible for misallocation is an
important topic for future research, which could then inform policy reforms in
46
poor countries.
Future work could extend and improve the theoretical framework of this paper.
One could for example modify the basic assumptions on production functions or
the demand specifications underlying possible price changes. It would also be possible to search over a whole set of such specifications when computing the bounds
on potential output gains. Finally, the theoretical framework in its present or a
modified form could also be applied to many other important settings including
studies of misallocation between a larger number of production units.
47
References
Adamopoulos, T. and D. Restuccia (2014). The Size of Farms and International
Productivity Differences. American Economic Review 104 (6), 1667–1697.
Banerjee, A. and E. Duflo (2005). Growth Theory through the Lens of Development Economics. In P. Aghion and S. N. Durlauf (Eds.), Handbook of
Economic Growth, pp. 473–552. Amsterdam: Elsevier.
Bartelsman, E., J. Haltiwanger, and S. Scarpetta (2013). Cross-Country Differences in Productivity: The Role of Allocation and Selection. American
Economic Review 103 (1), 305–334.
Buera, F. J., J. P. Kabowski, and Y. Shin (2011). Finance and Development:
A Tale of Two Sectors. American Economic Review 101 (5), 1964–2002.
Caselli, F. (2005). Accounting for Cross-Country Income Differences. In
P. Aghion and S. N. Durlauf (Eds.), Handbook of Economic Growth, pp.
679–741. Amsterdam: Elsevier.
Caselli, F. and W. J. Coleman (2001). The U.S. Structural Transformation
and Regional Convergence: A Reinterpretation. Journal of Political Economy 109 (3), 584–616.
Caselli, F. and J. Feyrer (2007). The Marginal Product of Capital. Quarterly
Journal of Economics 122 (2), 535–568.
Caselli, F. and N. Gennaioli (2013). Dynastic Management. Economic Inquiry 51 (1), 971–996.
Chanda, A. and C.-J. Dalgaard (2008). Dual Economies and International Total
Factor Productivity Differences: Channeling the Impact from Institutions,
Trade and Geography. Economica 75, 629–661.
Crego, A., D. Larson, R. Butzer, and Y. Mundlak (1998). A New Database
on Investment and Capital for Agriculture and Manufacturing. World Bank
Policy Research Working Paper 2013.
Duarte, M. and D. Restuccia (2010). The Role of the Structural Transformation
in Aggregate Productivity. Quarterly Journal of Economics 125 (1), 129–
173.
Garcı́a-Santana, M. and J. Pijoan-Mas (2014). The Researvation Laws in India and the Misallocation of Production Factors. Journal of Monetary Economics 66, 193–209.
48
Gollin, D. (2002). Getting Income Shares Right. Journal of Political Economy 110 (2), 458–474.
Gollin, D., D. Lagakos, and M. E. Waugh (2014). The Agricultural Productivity
Gap. Quarterly Journal of Economics 129 (2), 939–993.
Gollin, D., S. L. Parente, and R. Rogerson (2002). The Role of Agriculture
in Development. American Economic Review Paper and Proceedings 92 (2),
160–164.
Gollin, D., S. L. Parente, and R. Rogerson (2004). Farm Work, Home Work and
International Productivity Differences. Review of Economic Dynamics 7 (4),
827–850.
Gollin, D., S. L. Parente, and R. Rogerson (2007). The Food Problem and
the Evolution of International Income Levels. Journal of Monetary Economics 54 (4), 1230–1255.
Guner, N., G. Ventura, and Y. Xu (2008). Macroeconomic Implications of SizeDependent Policies. Review of Economic Dynamics 11 (4), 721–744.
Hall, R. E. and C. I. Jones (1999). Why do some Countries produce so
much more Output per Worker than Others? Quarterly Journal of Economics 114 (1), 83–116.
Hayashi, F. and E. C. Prescott (2008). The Depressing Effect of Agricultural
Institutions on the Prewar Japanese Economy. Journal of Political Economy 116 (4), 573–632.
Hsieh, C.-T. and P. J. Klenow (2009). Misallocation and Manufacturing TFP
in China and India. Quarterly Journal of Economics 124 (4), 1403–1448.
Lagakos, D. and M. E. Waugh (2013). Selection, Agriculture, and Cross-Country
Productivity Differences. American Economic Review 103 (2), 948–980.
Lagos, R. (2006). A Model of TFP. Review of Economic Studies 73 (4), 983–
1007.
Manski, C. F. (2007). Identification for Prediction and Decision. Harvard University Press.
Midrigan, V. and D. Y. Xu (2014). Finance and Misallocation: Evidence from
Plant-Level Data. American Economic Review 104 (2), 422–458.
Moll, B. (2014). Productivity Losses from Financial Frictions:
Can
Self-Financing Undo Capital Misallocation? American Economic Review 104 (10), 3186–3221.
49
Peters, M. (2013). Heterogeneous Mark-Ups, Growth and Endogenous Misallocation. Working Paper .
Rao, D. P. (1993). Intercountry Comparisons of Agricultural Output and Productivity. FAO Economic and Social Development Paper 112.
Restuccia, D. and R. Rogerson (2008). Policy Distortions and Aggregate Productivity with Heterogeneous Establishments. Review of Economic Dynamics 11 (4), 707–720.
Restuccia, D. and R. Rogerson (2013). Misallocation and Productivity. Review
of Economic Dynamics 16 (1), 1–10.
Restuccia, D., D. T. Yang, and X. Zhu (2008). Aggriculture and Aggregate
Productivity: A Quantitative Cross-Country Analysis. Journal of Monetary
Economics 55 (2), 234–250.
Rosenzweig, M. R. (1988). Labor Markets in Low-Income Countries. In H. Chenery and T. N. Srinivasan (Eds.), Handbook of Development Economics,
Volume I, pp. 713–762. Amsterdam: North-Holland.
Seale, J., A. Regmi, and J. Bernstein (2003). International Evidence on Food
Consumption Patterns. United States Department of Agriculture Technical
Bulletin 1904.
Temple, J. (2005). Dual Economy Models: A Primer for Growth Economists.
The Manchester School 73 (4), 435–478.
Temple, J. and L. Wößmann (2006). Dualism and Cross-Country Growth Regressions. Journal of Economic Growth 11 (3), 187–228.
Valentinyi, A. and B. Herrendorf (2008). Measuring Factor Income Shares at
the Sectoral Level. Review of Economic Dynamics 11 (4), 820–835.
Vollrath, D. (2009). How Important are Dual Economy Effects for Aggregate
Productivity? Journal of Development Economics 88 (2), 325–334.
Young, A. (2013). Inequality, the Urban-Rural Gap, and Migration. Quarterly
Journal of Economics 128 (4), 1727–1785.
50
Appendix
A
Proofs
A.1
Proof of Proposition 1
The general strategy of the proof is to show that equations (6), (7), (8) and (9)
together with the bounds on output elasticities and given values of dL and dK
imply the restrictions on the quantities ( yyab , kkab ) stated in the proposition.
yb
yb
= dyLa and εLa = dyLa εLb . Using these
First note that equation (6) implies εεLa
Lb
expressions and equations (8) and (9) one can then substitute for εLa , εKa and
εKb in equation (7) such that it reads as
yb
kb
λb − εLb dL
y
= a
y
ka
dL λ − yab ε dK
a
dL Lb
#
y
"
yb
kb d K
ya
λa − εLb
⇐⇒
ka d L
dL
b
=
ya
dL
[λb − εLb ]
yb
yb kb d K
kb d K
ya
ya
⇐⇒
λa − λb =
− 1 εLb
ka d L
dL
d L ka d L
If
kb
ka
=
dL
dK
then equation (28) directly implies
of proposition 1.
However if kkab 6=
dL
dK
yb
ya
=
λa
d ,
λb L
(28)
which is the last line
then equation (28) can be solved for εLb as
εLb =
λa kkab dyKb − λb
ya
kb dK
ka dL
−1
(29)
Given the value of εLb the other elasticities εLa , εKa and εKb are implied by equations (6), (8) and (9).
Now impose the restrictions εLa ∈ (0, λa ), εKa ∈ (0, λa ), εLb ∈ (0, λb) and
εKb ∈ (0, λb ). First note that if εLa ∈ (0, λa ) and εLb ∈ (0, λb ) then this directly
implies εKa ∈ (0, λa ) and εKb ∈ (0, λb) due to equations (8) and (9). Thus it
suffices to impose the restrictions εLa ∈ (0, λa ) and εLb ∈ (0, λb ).
First consider the case kkab > ddKL . Note that the denominator of the RHS of
equation (29) is positive in this case. The restrictions in the first line of equations
of proposition 1 are the collection of the following restrictions:
• εLb > 0 requires that the numerator of the RHS of equation (29) is positive
51
which implies
yb
ya
<
λa kb
d .
λb ka K
• εLb < λb requires that the RHS of equation (29) is smaller than λb which
implies
kb d K
kb d K
λa
− λb < λb
−1
ka yyab
ka d L
and hence
yb
ya
>
λa
d .
λb L
• εLa > 0 does not generate further
constraints because it is always satisfied
y
when εLb > 0 because εLa =
b
ya
ε .
dL Lb
• εLa < λa requires due to εLa =
yb
ya
dL
and hence
yb
ya
>
λa
d .
λb L
yb
ya
ε
dL Lb
that
"
#
kb d K
kb d K
λa
−1
− λb < λa
ka yyab
ka d L
This is the same constraint as imposed by εLb < λb .
Second consider the case kkab < ddKL . Note that the denominator of the RHS
of equation (29) is negative in this case. The restrictions in the second line of
equations of proposition 1 are the collection of the following restrictions:
• εLb > 0 requires that the numerator of the RHS of equation (29) is negative
which implies
yb
ya
>
λa kb
d .
λb ka K
• εLb < λb requires that the RHS of equation (29) is smaller than λb which
implies
kb d K
kb d K
−1
λa
− λb > λb
ka yyab
ka d L
and hence
yb
ya
<
λa
d .
λb L
• εLa > 0 does not generate further
constraints because it is always satisfied
y
when εLb > 0 because εLa =
b
ya
ε .
dL Lb
• εLa < λa requires due to εLa =
yb
ya
dL
and hence
yb
ya
<
λa
d .
λb L
yb
ya
ε
dL Lb
that
"
#
kb d K
kb d K
λa
−1
− λb > λa
ka yyab
ka d L
This is the same constraint as imposed by εLb < λb .
52
A.2
Proof of Corollary 1
Corollary 1 follows from proposition 1. The strategy to prove corollary 1 is to
show that the (dL ,dK ) combinations stated in the corollary are consistent with
proposition 1, but all other (dL ,dK ) combinations lead to a contradiction with
proposition 1.
As a preliminary step, note that by definition of deL and deK it holds that
e
dL
= kkab .
deK
First consider the case of dL = deL :
• It directly follows from the definition of deL that
as stated in the proposition then
dL
dK
=
last equation of proposition 1.
deL
deK
=
kb
.
ka
yb
ya
=
λa
d .
λb L
If also dK = deK
This is consistent with the
• Now confirm that any other dK 6= deK does not satisfy proposition 1. Note
that for the case yyab = λλab dL proposition 1 requires kkab = ddKL which implies
deL
dL
kb
=
=
=
ka
dK
dK
Thus
kb
ka
=
dL
dK
yb
ya
λa
d
λb K
⇐⇒ dK =
yb
ya
λa kb
λb ka
≡ deK .
can only be satisfied for dK = deK . Instead any dK 6= deK leads
to a contradiction with proposition 1.
Second consider the case of dL > deL :
• If as stated in the proposition dK < deK then
kb
ka
=
deL
deK
<
dL
dK
and
λa k b e
yb
λa
λa
λa k b
dK <
dK =
= deL < dL
λb k a
λb k a
ya
λb
λb
which is consistent with the second equation of proposition 1.
• Now confirm that any other dK ≥ deK does not satisfy proposition 1. Note
that dL > deL implies yyab < λλab dL . In this case proposition 1 requires yyab >
λa kb
d
λb ka K
which can be written as
yb dK
yb
λa k b
λa deL
dK =
⇐⇒ dK < deK .
>
dK =
e
ya
λb k a
λb d K
ya deK
Thus yyab > λλab kkab dK can only be satisfied if dK < deK . Instead any dK ≥ deK
leads to a contradiction with proposition 1.
53
Third consider the case of dL < deL :
• If as stated in the proposition dK > deK then
kb
ka
=
deL
deK
>
dL
dK
and
λa
yb
λa k b e
λa k b
λa
dL < deL =
=
dK <
dK
λb
λb
ya
λb k a
λb k a
which is consistent with the first equation of proposition 1.
• Now confirm that any other dK ≤ deK does not satisfy proposition 1. Note
that dL < deL implies yyab > λλab dL . In this case proposition 1 requires yyab <
λa kb
d which can be written as
λb ka K
Thus
yb
ya
<
λa k b
λa deL
yb
yb dK
<
dK =
dK =
⇐⇒ dK > deK .
ya
λb k a
λb deK
ya deK
λa kb
d
λb ka K
can only be satisfied if dK > deK . Instead any dK ≤ deK
leads to a contradiction with proposition 1.
A.3
Proof of Proposition 2
The initial steps and the general setup of the proof are identical to the one in
section A.1. Again kkab = ddKL directly implies yyab = λλab dL because of equation
(28). For the case of kkab 6= ddKL one now needs to impose the restrictions εLa ∈
(θLa , λa − θKa ) and εLb ∈ (θLb , λb − θKb ) on equation (29).
First consider the case kkab > ddKL . Note that the denominator of the RHS of
equation (29) is positive in this case. The restrictions in the first line of equations
of proposition 2 are the collection of the following restrictions:
• εLb > θLb requires that
kb d K
kb d K
−1
λa
− λb > θLb
ka yyab
ka d L
λa kkab dK
yb
⇐⇒
<
ya
λb − θLb + θLb kkab ddKL
• εLb < λb − θKb requires that
kb d K
kb d K
− λb < (λb − θKb )
−1
λa
ka yyab
ka d L
⇐⇒
λa kkab dK
yb
>
ya
θKb + (λb − θKb ) kkab ddKL
54
• εLa > θLa requires that
yb
ya
dL
#
kb d K
kb d K
− λb > θLa
λa
−1
ka yyab
ka d L
"
⇐⇒
θLa
λa − θLa kb
yb
<
dL +
dK
ya
λb
λb
ka
• εLa < λa − θKa requires that
yb
ya
dL
#
kb d K
kb d K
− λb > (λa − θKa )
λa
−1
ka yyab
ka d L
"
⇐⇒
λa − θKa
θKa kb
yb
<
dL +
dK
ya
λb
λb k a
Second consider the case kkab < ddKL . Note that the denominator of the RHS of
equation (29) is negative in this case. When imposing the restrictions εLb > θLb ,
εLb < λb −θKb , εLa > θLa and εLa < λa −θKa , all inequalities are reversed compared
to the previous case. This generates the restrictions in the second line of equations
of proposition 2.
B
Details on the Extension to a General Number of Production Units
All basic assumptions and most of the notation are equivalent to the ones presented
in the main text except that they apply to N ≥ 2 different production units now.
In order to facilitate the comparison to the case with only two units, take any
one of these N production units and call it unit a. Then let b now be an index
running from 1 to N − 1 that refers to the remaining N − 1 production units. The
marginal value product equations for labor and capital then read as
∂Ya
∂Yb
= pb
, b = 1, ..., N − 1
∂La
∂Lb
∂Yb
∂Ya
= pb
, b = 1, ..., N − 1
dbK pa
∂Ka
∂Kb
dbL pa
(30)
(31)
PN −1
PN −1
and the resource constraints are La + b=1
Lb = L and Ka + b=1
Kb = K. The
difference to before is that there are N − 1 such marginal product equations for
each pair consisting of unit a and one of the remaining N − 1 units indexed by
N −1
N −1
b. {dbL }b=1
and {dbK }b=1
refer to the marginal product differentials of labor and
55
capital for each of these pairs.
Applying the same tricks as before yields the key equations
εLa b
yb
=
d ,
ya
εLb L
kb
εLa εKb dbL
=
,
ka
εLb εKa dbK
b = 1, ..., N − 1
(32)
b = 1, ..., N − 1
(33)
involving the average product of labor and capital intensity ratios for the pairs
consisting of a and b where b = 1, ..., N − 1.
Again the production functions are assumed to be homogenous in K and L of
N −1
degree λa and {λb }b=1
, where the value of λb may of course differ between each of
the units b = 1, ..., N −1. By Euler’s theorem the sum of output elasticities in each
production unit satisfies εLa + εKa = λa and εLb + εKb = λb for all b = 1, ..., N − 1.
As in section 3.1 I cover here the case with general lower bounds on output
N −1
N −1
elasticities given by θLa , θKa , {θLb }b=1
and {θKb }b=1
, which need to be set by the
researcher. These parameters need to satisfy θLa ∈ [0, λa − θKa ), θKa ∈ [0, λa − θLa
and θLb ∈ [0, λb − θKb ) and θKb ∈ [0, λb − θLb ) for all b = 1, ..., N − 1.
These assumptions then imply bounds on output elasticities given by εLa ∈
(θLa , λa − θKa ), εKa ∈ (θKa , λa − θLa ) and εLb ∈ (θLb , λb − θKb ) and εKb ∈ (θKb , λb −
θLb ) for all b = 1, ..., N − 1.
For a general number of production units and specific values of marginal prodN −1
N −1
uct differentials {dbL }b=1
and {dbK }b=1
the combinations of average product of
N −1
which may be consistent with
labor ratios and capital intensity ratios { yyab , kkab }b=1
such a situation are characterized by the following proposition.
Proposition 3. If N ≥ 2 production units a and b = 1, ..., N − 1 operate with
N −1
output elasticities of labor and capital bounded from below by θLa , θKa , {θLb }b=1
N −1
and {θKb }b=1
, and their production functions are homogenous of degree λa and
N −1
{λb }b=1
, and the factor allocation exhibits marginal product differentials of labor
N
−1
N −1
{dbL }b=1 and capital {dbK }b=1
between production unit a and each of the production
units b = 1, ..., N −1 then the average product of labor ratio yyab and capital intensity
ratio kkab for all b = 1, ..., N − 1 at this allocation satisfy either
db
kb
> bL
ka
dK
kb
db
< bL
ka
dK
kb
db
= bL
ka
dK
and
and
and
yb
< min{φb , ψb },
ya
yb
max{φb , ψb } <
< min{φb , ψb },
ya
λa
yb
= dbL
ya
λb
max{φb , ψb } <
56
or
or
where
λa − θKa b
θKa kb b
dL +
d ,
λb
λb k a K
θLa b
λa − θLa kb b
=
dL +
d ,
λb
λb
ka K
λa kkab dbK
,
=
db
θKb + (λb − θKb ) kkab dKb
φb =
φb
ψb
L
ψb =
λa kkab dbK
λb − θLb + θLb kkab
dbK
dbL
.
The proof follows the exactly same logic and steps as the proof of proposition
2, which is presented in section A.3. The only difference is that the earlier proof
did this for only one pair of production units consisting of a and b, and here this
is done for N − 1 pairs consisting of a and each of the units b = 1, ..., N − 1. Thus
these steps are not repeated here.
N −1
N −1
N −1
Given the assumed values of λa , {λb }b=1
, θLa , θKa , {θLb }b=1
and {θKb }b=1
the
test for an efficient factor allocation then consists in checking whether an observed
N −1
satisfies the conditions of proposition 3 when one
factor allocation { yyab , kkab }b=1
sets dbL = 1 and dbK = 1 for all b = 1, ..., N − 1. In terms of the earlier graphical
illustration of the test, this means to consider a separate graph of the test region for
each pair of a and b = 1, ..., N − 1 in terms of the respective ( yyab , kkab ) combination.
The test rejects an efficient allocation if the ( yyab , kkab ) observation on at least one of
the N −1 graphs is not an element of the respective non-rejection region. However
if one has set the parameters λb , θLb and θKb to the same respective value for all
production units b = 1, ..., N − 1, then the test region on all these N − 1 graphs
is identical. One can then just use one graph and check whether all observations
N −1
of the whole collection { yyab , kkab }b=1
are an element of the non-rejection region.
The bounds on potential output gains from eliminating factor misallocation
are derived under the same basic assumptions as before. Now total output is
PN −1
calculated by summing over N production units such that Y = pa Ya + b=1
pb Y b .
One also needs to adjust the number of optimizers in the respective max, inf and
sup optimization steps. The potential output gain G(αa , αb ) for specific values of
αa and αb then reads as
G(αa , αb ) =
PN −1 ∗ λa −αa PN −1
PN −1 e∗ αa
kb∗ )αb (ℓ∗b )λb −αb
+ b=1 pbb (e
b=1 ℓb )
b=1 kb ) (1 −
−1
PN −1 eαb λb −αb
PN −1 λ −α
PN −1 e α
pbb kb ℓb
ℓb ) a a + b=1
kb ) a (1 − b=1
(1 − b=1
(34)
(1 −
57
where
i
h
N −1
N −1
=
{e
kb∗ }b=1
, {ℓ∗b }b=1
1−
N
−1
X
b=1
and pbb ≡
α
arg max
e∗∗ ∈[0,1]}N−1 s.t. PN−1 e
kb∗∗ ≤1
{k
b
b=1
Pb=1
N−1
N−1 ∗∗
∈[0,1]}
s.t.
{ℓ∗∗
b
b=1
b=1 ℓb ≤1
e
kb∗∗
λ −α
p b Ab K b L b b
α
λ −α
p a Aa K a L a a
!αa
1−
N
−1
X
ℓ∗∗
b
b=1
!λa −αa
(
+
N
−1
X
b=1
(35)
λb −αb
pbb (e
kb∗∗ )αb (ℓ∗∗
b )
)
(36)
can be determined by
yb Lb (1 −
pbb =
ya La
for all b = 1, ..., N − 1.
PN −1 e αa
PN −1 λa −αa
j=1 kj ) (1 −
j=1 ℓj )
α
λ
−α
e
k bℓ b b
b
(37)
b
The lower bound G is then given by
G=
inf
G(αa , αb )
(38)
sup
G(αa , αb ).
(39)
αa ∈(θKa ,λa −θLa )
{αb ∈(θKb ,λb −θLb )}N−1
b=1
and the upper bound G by
G=
αa ∈(θKa ,λa −θLa )
{αb ∈(θKb ,λb −θLb )}N−1
b=1
Computing these bounds may be more challenging in practice, but conceptually
there is no major difference to the case with two production units.
C
Robustness Checks for Application
This section describes a series of robustness checks on the basic assumptions. Each
of the following alternative specifications varies one parameter or feature only and
keeps all other parameters as in the benchmark. Exact results of course vary for
each country (available upon request), but I provide some summary measures for
each alternative specification and each of the two cases considered in the main text
in table 5. The first column contains the number of countries for which the test
rejects an efficient factor allocation out of a total of 24 countries. The next two
columns present the lower and upper bound on the variance of log internationally
priced maximized income per worker Var[log(yi∗)]. The last two columns report
58
the corresponding percentage change when moving from the observed variance of
0.96 to the hypothetical variance bounds.
First I scrutinize the role of the assumed degree of homogeneity in agriculture
λa and non-agriculture λb . I consider a smaller role for the factor land in agriculture relative to the benchmark with an associated larger degree of homogeneity
to labor and capital of λa = 0.9 and also the reverse situation with λa = 0.7. The
latter is consistent with a land share in agriculture of 30% as assumed by Gollin,
Parente, and Rogerson (2007). In the benchmark I abstract from a role of land in
non-agriculture, but I also consider an alternative situation here with λb = 0.95
which is consistent with a 5% land share as found by Valentinyi and Herrendorf
(2008). These specifications yield broadly similar results to the benchmark. The
only exception is that for the case with a lower land share in agriculture (λa = 0.9)
one cannot rule out that for the stronger assumptions the variance in the hypothetical situation is not significantly lower than the observed one. The reason is
that there are more countries in this situation than in the benchmark for whom
an elimination of misallocation may lead to a reduction in internationally priced
output.
Second for the case with stronger assumptions I tighten the bounds on output
elasticities further such that they are as tight as possible around the values from
the previous studies and the resulting intervals still contain all of these values.
This results in θLa = 0.44, θKa = 0.1, θLb = 0.5 and θKb = 0.28. This tightens the
bounds on output gains and the variance of log maximized income a bit more, but
the effect is not huge.
Finally, instead of assuming a constant return to schooling of 10% in the calibration of human capital stocks I use a step function following Hall and Jones
(1999) and Caselli (2005). Here the returns to schooling are 13.4% for the first
four years, 10.1% for the next four years and 6.8% for all additional years. The
results are very similar to the benchmark.
59
Table 5: Results for Robustness Checks
A. General Case
Check
Reject
Benchmark
λa = 0.9
λa = 0.7
λb = 0.95
Human Capital Step Function
16
15
16
16
16
Var[log(yiint∗ )]
Change (in %)
Min
Max
Min
Max
0.35
0.33
0.37
0.37
0.35
1.57
1.93
1.35
1.52
1.56
-63.3
-66.0
-60.9
-61.8
-63.4
63.7
101.8
40.6
58.6
63.3
B. Stronger Assumptions
Check
Benchmark
λa = 0.9
λa = 0.7
λb = 0.95
Tighter Elasticity Bounds
Human Capital Step Function
Reject
24
20
24
22
24
24
Var[log(yiint∗ )]
Change (in %)
Min
Max
Min
Max
0.58
0.58
0.58
0.60
0.61
0.58
0.76
0.95
0.75
0.77
0.71
0.76
-39.2
-39.9
-38.9
-36.9
-36.1
-39.6
-20.5
-0.7
-21.8
-19.3
-26.1
-20.8
Notes: “Reject”: Number of countries (out of 24) for which an efficient factor allocation is
rejected. Var[log(yiint∗ )]: Variance of logarithm of maximized income per worker. Change:
Percentage change in variance of logarithm of income per worker when moving from observed to
maximized income, i.e. from Var[log(yiint )] to Var[log(yiint∗ )]. Min (Max): Lower (upper) bound
on respective variable.
60