Estimating Elasticities of Substitution in African Crops
by
Fatima Zouhair
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Economics
Auburn, Alabama
May 7, 2012
Keywords: Allen elasticities of substitution, cross price elasticities,
Hicks elasticity of complementary
Copyright 2012 by Fatima Zouhair
Approved by
Henry Thompson, Chair, Professor of Economics
John D. Jackson, Professor of Economics
Norbert Wilson, Associate Professor of Agriculture Economics
ii
Abstract
This paper examines cross price elasticities of substitution for labor, capital, and fertilizer
for the agricultural sector in Africa with a translog production function. The data are from the
Food Agricultural Organization and World Development Indicators Database. The elastisities
are calculated directly from the estimated parameters of a production function. Those elastisities
include the Allen Substitution Elasticity (AES), factor price substitution elasticity, and Hicks
elasticity of complementary (HEC).
Allen elastiticies of substitution of all inputs are positive. Results show that only labor
and capital are complements among the inputs in HEC and factor price substitution elasticities.
Own price elasticities for labor, capital, and fertilizer are inelastic. Labor is the most sensitive of
the inputs to input prices. Furthermore, HEC shows that fertilizer demand is elastic with respect
to the capital price and inelastic with respect to the wage.
iii
Acknowledgments
First and foremost, I wish to appreciate my utmost assistance and valuable suggestions to
enable finalization of my research work to the committee chairman Dr. Henry Thompson at
Economic Department. I really cherish his kindness, guidance, advice, and patience throughout
the entire process.
I am also grateful to the committee members Dr. John D. Jackson and Dr. Norbert Wilson
for their valuable suggestions and helpful comments. Finally I would like to thank my family for
its maximum support.
iv
Table of Contents
Abstract ??????????????????????????????????.. ii
Acknowledgments ?????????????????????????????? iii
List of Table ????????????????????????????????.. vi
List of Figures ???????????????????????????????... vii
Chapter 1 Introduction ???????????????? ??????????? ....... 1
1.1 Background and Motivation ???????????????????.. 1
1.2 Objective of the Study ????????????????????? ... 2
1.3 Approach and Methods ?????????????????????.. 3
1.4 Organization of the Study ????????????????????.. 3
Chapter 2 Review of Literature ?????????????????????????.. 4
2.1 African Agriculture ?????????????????????? ... 4
2.2 Related Studies ????????????????????????.. 6
Chapter 3 Methodology ??????????????????????????? ..... 9
3.1 Production Theory ?????????????????????? ? 9
3.2 Assumptions ??????? ??????????????????. . 9
3.3 Empirical Model specification ??????????????????. 10
3.4 Data and Variables ?????????????????????? ... 15
Chapter 4 Results and Discussions ???????????????????????.. 16
v
4.1 Specification and Estimation of the Model ?????????????. 16
4.2 Hausman Test ????????????????????????.. 18
4.3 Heterskedasticity Test ?????????????????????. 18
4.4 Autocorrelation Test ????????????????????....... 19
4.5 Estimation and hypothesis Test ?????????????????.. 20
4.6 Results of the Empirical Estimation ???????????????... 21
Chapter 5 Conclusions ????????????????????????????. 25
References ????????????????????????????????? 27
vi
List of Tables
The Share of Agriculture in GDP Table 1 ?????????????????????32
Description of Variables Used in the Analysis Table 2 ???????????????... 33
Summary Statistics of the Variables Table 3 ???????????????????... 33
Hypothesis Tests Table 4 ???????????????????????????. 34
Parameter Estimation of Translog Production Table 5 ???????????????? 34
Separability Tests Table 6 ??????????????????????????? 35
Estimated Marginal Product and Output Elasticites for African Crops Table 7 ??????.. 35
Estimated Allen Substitution Elasticity Table 8 ??????????????????.. 36
Factor Price Substitution Elasticity Table 9 ????????????????????. 36
Hicks Elasticity of Complementary Table 10 ???????????????????.. 36
vii
List of Figures
Trend of fertilizer Consumption in Africa in Thousands Figure1 ???????????... 30
Trend of African Agricultural Labor in Thousands Figure2 ?????????????... 30
Capital Use in African Agriculture in Thousands Figure3 ??????????????.. 31
1
CHAPTER 1
INTRODUCTION
1.1 Background and Motivation
Agriculture dates back as far as 9500 BC. At that time farmers within Southwest Asia
commenced to cultivate special crops. By the middle ages, industry came and irrigation and crop
rotation started. Finally, by the middle of nineteen century and early twenty century, technology
was applied leading to agricultural production growth through increasing crop yields. The
technological improvements enabled farmers to meet the demand for food until the late 20th
century, particularly in Asia and Africa.
From 1973, the African continent has relied heavily on food imports and food aid. Even
though Africa possesses large portions of arable land, food insecurity and malnutrition affect
more than half of the population according to the International Food Policy Research Institute
(2008). One of the most critical challenges facing Africa today is how to increase agricultural
production to meet increasing domestic demand arising from increased population pressure
(Pretty, 1995). Despite these circumstances, the aim of agriculture production is to feed the
population, assure food security, and ease poverty in the region.
This thesis will concentrate on the African continent. Table 1 depicts all the countries
included in this study. Agriculture is a crucial economic activity in this region. Agricultural
activities employs about 70% of the population and contributes to about 30% of the continent?s
2
gross domestic product, comprises of 50% of export value, brings a substantial amount of
exchange rate, generates 34% of income, and serves the basis for many industries.
The motivation of this study is to analyze and estimate the African agricultural
production function. There have only been a few studies on this matter. However, numerous
studies of production function and its elasticities have been done in different regions with a cross
sectional and time series data.
1.2 Objective of the Study
The objective of this study is to develop and analyze the structure of agricultural crops in
Africa from 1961 to 2007. This thesis will define, derive, and provide a comprehensive
understanding of the African agricultural production function. Investigation into the roles of
capital, labor, fertilizer, and output will be included so that the production function will be
estimated.
The results will be used to assess and highlight differences within substitution
possibilities between capital, labor, and fertilizer. In order to achieve good policy in this matter,
the elasticity of substitution is an appropriate method. It is necessary to estimate the elasticity of
substitution between inputs, and own and cross price elasticities of inputs. The empirical results
of the estimation of the production function and elasticities of substitution will be useful for
policy makers and future research.
3
1.3 Approach and Methods
In order to achieve the above objective, the thesis includes all crops in the region for 47
countries in an annual panel data from 1961-2007 to examine the production function with
capital, labor, and fertilizer as inputs.
1.4 Organization of the Study
The thesis has been divided into five chapters. The next chapter will provide an overview
of the literature and will contain a detailed discussion of the African continent and its agriculture.
Chapter 3 will set up the production function, define the various elasticities to be estimated, and
describe the methodological approach. The econometric procedures and empirical results from
the estimation of the model and its elasticities are presented in Chapter 4. Chapter 5 consists of
conclusions, highlights some remarks, and offers some suggestions to assist in creating better
growth in agricultural production.
4
CHAPTER 2
REVIEW OF LITERATURE
The African continent covers over 30 million square kilometers (11.58 square miles).
Africa is considered the second largest continent after Asia. In addition it surrounded by
Mediterranean Sea, Atlantic Ocean, Indian Ocean, and Red Sea. It climate is mostly tropical with
the majority (60%) of its land are deserts and dry land. The African continent is seen as a
complicated and diverse continent containing more than one billion of the world?s population
and more than 1000 different ethnic groups. (United States Department of State, 2009).
2.1 African Agriculture
In 1960 the independence of many African countries, the Continent was looked as self-
sufficient in its agricultural production. In contrast, the Asia continent began to endure food crisis
until the birth of the Green Revolution which came in the middle of the 1960s; a revolution that
primarily motivates the adaption of high-yielding seeds. Beginning in 1973, several African
nations began to depend on food imports. One of the most critical challenges facing Africa today
is how to scale up agricultural production to meet increasing domestic demand arising from
population pressure (Hunt, 2011).
Lack of rain fall has made agriculture difficult for many African nations. With areas that
get unreliable rainfall amounts, farmers are ambivalent of what kinds of crops to use. To survive
these severe conditions, farmers were obliged to swap their strategies frequently; on the other
5
hand, other areas are receiving too much rain that exceedingly destroying nourishing substances.
Another common issue is droughts. It is more frequent in the region than any elsewhere in the
world. As a result, crop yield is diminishing greatly and food insecurity is increasing
significantly. Climate variation is another impediment to African agriculture. Precipitation
sometimes delayed or advanced, leading to catastrophic consequences such as hunger, and
malnutrition. Another constraint to agriculture development is water shortage that is the
outcome from bad management of the environmental resources that are ruined regularly by wars
and mining usage. Water shortages in reservoirs cause lakes, rivers, and streams to dry out in the
warmer seasons which create water stress. Today many African countries have water shortages,
and they cannot meet their demands. Since water is a vital component to agriculture, soil has a
poor capability to keep or emit moisture. Soil depletion affects most areas in Africa with nutrient
losses estimated from 30kg to 60 kg per hectare each year (Henao and Baanante, 2006).
Irrigated area is less than 5 million hectares which represent 5 percent of the total cultivated land.
Further complication to African agriculture is Soil fertility. Soil is low in magnesium,
phosphorous, zinc, and nitrogen. The fast growing of the population has worsened the problem
because it decreases the length and the amount of fallow lands. Therefore, most of the countries
hinge on imported fertilizer in which most farmers can?t afford due to high price. Overall, the
average use of fertilizer in the African Continent still inferior than any other place.
Agriculture remains one of the most important issues in the region. Rural infrastructure
and change in policy is a must; a situation that urges full attention from all parties.
6
2.2 Related Studies
Over the last thirty years, many studies studied elasticity of substitution between capital
and labor. However, the reported estimates differ considerably depending on cross sectional and
time series studies.
Most of elasticity of substitution has been done in industry while there is only limited
amount of study has been conducted to agriculture in comparison to other sectors.
In a study of productivity and economic growth in Tunisian agriculture, Dhehibi (2006)
showed that labor and capital are substitutes. His elasticities show that an increase in labor
increases demand for capital by (1.63) while an increase in the capital increase demand for labor
by (0.23).
Chandharry, Khan, and Nqair (1998) in a study on ?estimates of farm output supply and
input demand elasticities? in Pakistan agriculture, found that an increase in capital increases
demand for labor and fertilizer by 0.69 and 0.64 respectively.
The study of Lianos (1971) on production function in United States from 1950-1989
found that the short run wages elasticity was -1.04 while the long run was -3.51.
The study of Vincent (1979) of factor substitution in Australian agriculture from 1920 to
1970 used land, labor, and capital as factors inputs and the value added as output using both
Cobb Douglas and constant elasticity of substitution production functions. The estimation was
done through full information maximum likelihood and the major finding was that there is very
low elasticity of substitution between factor inputs.
Binswanger (1973) conducted a study on the measurement of technical change biases
with many factors of production for the years, 1949, 1954, 1959, and 1964 including 39 states of
United States. He used translog cost function with five inputs land, labor, machinery, fertilizer,
7
and all others. His results showed elasticity of substitution between land-labor, land-machinery,
labor-machinery, land-fertilizer were substitutes while labor-fertilizer and machinery-fertilizer
were complements.
Thrisk (1973) examined factor substitution in Colombian agriculture using two
estimation methods: ordinary and generalized least squares estimation techniques. He used cross
sectional data over major crops in the region. The crops included in his study were rice, cotton,
corn, wheat, barley, sesame, soybeans, and sorghum. He distinguished land, labor, and
machinery as factors inputs. The results of elasticity of substitution estimates were similar in
both approaches with 1.5 for labor-machinery.
Mupondwa (2005) in a study called induced technological change in Canadian
agriculture field crops-canola and wheat: 1962-2003, he used constant elasticity of substitution
production function with four inputs: land, machinery, labor, and fertilizer. The major finding
was that the long run elasticity of substitution in the wheat for fertilizer-land and machinery-
labor was -2.08 and -.346 respectively while canola had an elasticity of substitution for fertilizer-
land and machinery-land of -1.06 and -1.53 respectively.
Arrow and Solow (1961) analyzed capital-labor substitution and economic efficiency for
the years 1949-1955 for manufacturing. They included the following countries in their study:
United States, Canada, New Zealand, Australia, Denmark, Norway, Puerto Rico, United
Kingdom, Colombia, Ireland, Mexico, Argentina, Japan, El Salvador, Brazil, S. Rhodesia,
Ceylon, India, and Iraq. Their results showed that the bilateral elasticity of substitution between
capital and labor were less than unity.
In a study of the translog cost function analysis of U.S agriculture from 1939-77, Ray
(1982) based his analysis on neoclassical duality theory to derive the elasticity of substitution.
8
The model was based on the estimation of crops and livestocks as two separate outputs while his
inputs variables were limited to hired labor, farm capital, fertilizer, and miscellaneous input. The
results suggest a decline in substitutability between labor and capital while price elasticity
increased in all inputs.
9
CHAPTER 3
METHODOLOGY
3.1 Production Theory
The production function is a focal point in the economy. It deals with the maximum
amount of output from optimal inputs. The objective is to develop a production function for
African crops. The study focuses on three important inputs in African agriculture.
The production function will be:
Q = Q (K, L, F)
where
Q is the crop output (all crops are measured in quantity tons)
K is the capital (combine harvesters, threshers, and agriculture tractors in thousands)
L is labor (agricultural labor force in thousands)
F is fertilizer (fertilizer consumption in metric tons of nutrient in thousands)
3.2 Assumptions
Certain assumptions are made of production functions to assure technical validity and viability of
an economic optimum involving the following restrictions. (Chambers, 1988):
? The production is monoperiodic meaning that the production is autonomous from the
previous and subsequent period.
10
? The production function is homogeneous of degree one.
? The production is monotonic in X. that is, an increase in X (ceteris paribus) can never
diminish output Y.
? The production is quasi-concave (convexity of the isoquant and diminishing marginal
rate of technical substitution).
? The production is twice continuously differentiable.
3.3 Empirical Model Specification
Production can take numerous forms such as: linear functional, polynomial functional,
Cobb Douglas, translog, and CES. Various studies have been conducted with Cobb Douglas
production function due to its linearity in logarithms; however, its elasticity is constant and the
elasticity of substitution is unity.
This thesis uses translog production function to study productive behavior. The translog
function is more flexible in that it imposes few assumptions on the function and its elasticities.
The transcendental logarithmic function was introduced by Christensen and Lau (1973) and has
been used to analyze factors inputs and their substitution. The model is specified as follows:
= + + + + + + + (1)
+ +
where
, , and represent the first-order parameters.
11
, , , , , and represent second-order parameters. The variable
t is a time trend perhaps capturing technical change and are errors which are assumed to be
independently and identically distributed and have N (0, ) distribution.
Assuming perfect competition, the monotonocity condition necessitates positive marginal
physical product (MP) which can be derived from translog production. Mathematically,
Diminishing marginal productivity entails:
Using these conditions, the factor share for each input will be:
12
Young?s theorem imposes the following constraints:
= ; = ; =
The output elasticity can be calculated using equation (6).
Elasticities
This study will focus on Allen elasticity of substitution (AES), cross price elasticities,
and the Hicks elasticity of complementary (HEC).
AES is prevalent in the literature especially for production function; it measures the
changes in relative input as their relative price changes. Furthermore, the bigger the value of the
elasticity, the higher the substitution between the inputs. Cross price elasticity measures the
percentage changes in an input in response to a price change. HEC, on contrast, measures the
variation in the price as the quantity of the input change. The advantage of HEC is that it can be
applied even when the input price are falsified or are missing while the elasticity of substitution
is pertinent only when the price and the quantity are clearly stated. HEC involves specification
and estimation of production function when the quantity is exogenous and the price is
endogenous. (Kim, 2000)
A positive sign of the elasticity means that the inputs are substitutes while a negative sign
means that they are complements. The Allen elasticity can be calculated from
where
are the inputs (L, K, and F)
are inputs levels
13
is marginal product of inputs
is the cofactor in
is the determinant of the bordered Hessian matrix
And and are as defined in the equations (4) and (5)
The necessary and sufficient condition for the convexity of the isoquant requires the
bordered Hessian matrix to be negative semi-definite. That is, the determinants of all principal
minors of ( must alternate in sign starting with negative.
The price elasticity can be calculated now using (6)
The Hicks elasticity of complementary (HEC) is as follow:
Separability
Separability is a standard assumption in empirical work and it is necessary for checking
economic aggregation. They are two types of separability:
14
?Weak separabiliy: it must be satisfied that the marginal rate of substitution between two
goods from a group is independent of the quantities of the two goods which do not belong to that
group.?
?Strong separability: it is satisfied when the marginal rate of substitution between two
goods belonging to different groups does not depend on the quantities of the goods which do not
depend belong to any those groups? (Millan, 2003 p.11)
Berndt and Christensen (1973) argue that in translog function with three inputs
( are separable from the input when which can hold only
if .
Therefore separability in this translog production requires satisfaction of the following
conditions:
= 0 (
= 0 ( )
= 0 (
The above equations represent weak separability. The test will be implemented to linear
and non-linear separability. Starting with linear separability, if one of the above conditions is
satisfied, no further tests needed. If the test is rejected, an inquiry of one of the others types is
required until the test is accepted by any of them. If this hypothesis is rejected as well, then none
of these separability conditions is holds for this particular translog production function. The first
equation in the linear separability above will test if the capital and labor inputs are separable
from the fertilizer input. The second equation will test if the fertilizer and capital inputs are
separable from the labor input. The last equation will test if the fertilizer and labor inputs are
15
separable from the capital input. The purpose of testing separability is to see if any input in
production can be aggregated.
Consequently, the estimation of the translog production function will go under certain
tests. First, a test for symmetry conditions is obligatory by economic theory to assure that the
estimated model meets the restriction. Second, monotonicity necessitates that the marginal
product is positive. Finally, the concavity condition of the translog production function has to be
checked. The Hessian determinant built on the parameters estimation must be negative definite.
3.4 Data and Variables
There are abundant studies of translog production function conducted by cross-sectional
or time series but very limited ones that treat panel data. This study used annual panel data from
1961-2007 to estimate the parameters of the translog production function. The study focuses on
Africa and covers 47 countries while the other missing countries are excluded due to data
unavailability. The list of the countries is provided earlier in Table 1. All tables and figures are
presented in the Appendix.
Since the price data (wage of labor, rent of capital, and cost of fertilizer) in African
countries is unobtainable, physical quantity is utilized and the estimation comes from the
production function. Therefore, data include all African crops as agricultural output and labor,
capital, and fertilizer as agricultural inputs. In addition, trend variable (t) is employed as proxy
to catch technical change.
Table 2 presents a detailed explanation of the variables used in this research. Summary
statistics and other details of the data are reported in table 3.
16
The average labor in agricultural sector is close to 2.5 million which is superior than any
other inputs. Figure 2 shows the trend of agricultural labor. The average of fertilizer use is
almost 61 thousands metrics tons per year while the capital has the smallest portion of
approximately of 12 thousands which implies that labor and animal are the most performers of
agricultural work (Figure 3 and 4).
The entire data are divided by the mean before transformed to logarithms following
Friedlaender and Spady (1980).
Finally, different sources of data are used for this study. Output and input production
data conducted by the Food and Agricultural Organization?s online database and World
Development Indicators Database.
16
CHAPTER 4
RESULTS AND DISCUSSION
This chapter is divided into six sections. The model discussed in the previous chapter is
estimated with fixed effect regression. The first section starts by introducing the specification
and estimation of the model followed by several tests and its discussions. Further the chapter
reports the estimation results of the translog production function and its elasiticies with their
interpretation in the last section.
4.1 Specification and Estimation of the Model
Fixed effect
In the fixed effect (FE), the intercept can vary across individuals but the individuals
intercept terms cannot differ over the time and the coefficients of the independents variable does
not allow fluctuating across individuals or over time. (Gujarati, 2004).
Let consider the model below:
(11)
where and is the dependent variable; are independents variables; is the
error; and are ?random individual-specific effects?. (Cameron and Trivedi, 2005). The are
allowed to be correlated with and is uncorrelated with .
The estimation of fixed effect can be obtained by mean differencing of (7), in which it
removes the
17
(12)
Since the have been excluded, OLS estimates will be consistent. Kumbhakar and
Lovell (2000) establish that fixed effect models are simple to implement and its consistency does
not rely on the independence or distribution of the regressors.
Random effect
In the random effect (RE) model, the are random and uncorrelated with the
independent variables. The estimation can be conducted with feasible generalized least square
(FGLS) or transformed OLS.
Assuming and are independently and identically distributed with variance of
and respectively. Since then variance ( and
covariance ( , ; so the correlation is
Then the random effect estimator can be calculated as follow:
(13)
Where
The advantage of the random effect model is that it produces estimates for all coefficients
including the time invariant ones and uses few degrees of freedom.
18
4.2 Hausman test
The Hausman test (1978) chooses the appropriate model between FE and RE. The null
hypothesis is that no correlation exists (RE) meaning that there is no differences between the
estimators since both of them are consistent. While the alternative suggests that both estimators
are dissimilar. If the null hypothesis is rejected then the FE is the appropriate model.
The Hausman Test (H) is
where
are the vector of RE and FE estimates.
4.3 Heteroskedasticity Test
Since this analysis is panel data, several problems arise. Heteroskedasticity arises when
the variance of the errors are not constant. However, it does not generate bias but the standard
errors estimated are underestimated leading to high t values. Hypotheses testing may lead to
reject the right null and accept the wrong null, leading to non-reliable results. Thus, this is a
strong incentive to check heteroskedasticity. The modified Wald test is one among others.
Greene (2000) defines it as
where
is the estimated variance.
19
The solution can be by correcting the bias of the standard errors, or using generalized least
squares.
4.4 Autocorrelation Test
The second common issue that panel data encounter is autocorrelation. It is defined as a
correlation between successive values and statistically as the error in one period correlated with
the error in the next period.
The consequences of autocorrelation are the same as heteroskedasticity. Autocorrelation
does not generate bias however the standard errors are underestimated leading to high t values.
Thus hypotheses testing may lead to the failure to reject the null when it is false. This is a strong
incentive to check if the errors are related.
However, detecting autocorrelation is not an easy task because the error cannot be
observed. The model must be estimated to get the residuals. This latter reflects the pattern in the
error term. A number of tests can diagnose this problem but the most recent one is the
Wooldridge test (2002). This test is simple and imposes few assumptions.
From equation (12)
The test is based on residuals . Wooldridge (2002) examined first If they are not
correlated, then the correlation of the first differences ( of and equal to - 0.5. Using
20
this technique, Wooldridge proceed with the regression of on and test of whether the
coefficient is equal to -0.5 or not to determine if serial correlation exists.
The problem of autocorrelation can be solved by either correcting the bias in the
standard errors, or transforming the data and using generalized least squares or feasible
generalized least squares.
4.5 Estimation and Hypothesis Test
Panel data analysis will be implemented in this study. One step is to choose the
appropriate functional form. Many factors are taken in consideration when making that decision
to choose the functional form that obeys certain restrictions such as concavity and non-
negativity of the input and output. Cobb Douglas, translog, and quadratic functional forms are
most widely used in agricultural production functions. Translog production function is
implemented, evaluated, and tested in this analysis.
Before the estimation of equation (1), a test for the functional form of the production
models must be conducted using the below Cobb Douglas and translog equations.
= + + +
= + + + + + + +
+ +
where The variables and the parameters are as defined before in equation (1).
The estimation of the Cobb Douglas and translog function were conducted with the Wald
test estimator using STATA. A hypothesis test was performed to choose the best functional
form. The results are presented in Table 4.
21
The first test is that the null hypothesis is a Cobb Douglas function which strongly
rejected implying that the translog is the adequate function for this analysis. The second test is
that there is no technical change in African agriculture over time. This test is also rejected.
4.6 Results of the Empirical Estimation
After the functional form has been chosen, the next step is to apply the adequate
econometric estimation to get the parameters for the model. The estimation of the empirical
model is based on fixed effects since the Hausman test rejects random effects regression. The
results are presented in Table 5.
Before proceeding with any conclusions to the analysis, a few tests are applied in order to
ensure reliable results. As mentioned in the last section, heteroskedasticity, autocorrelation, and
cross sectional dependence are common in panel data. The test used for heteroskedasticity is
Wald test indicating its existence. The test for serial correlation is Lagrangian-multiplier test
which also confirms its existence. A further test of cross-sectional dependence is specified in
this analysis using Breusch-Pagan Lagranger Multiplier, the results shows no correlation of
residuals.
The monotocity test is performed (Table7) showing that the marginal products of the
three inputs are positive. A concavity test is also adapted in this translog production function. To
make sure that this condition holds as imposed by economic theory, the bordered Hessian matrix
is checked and confirmed that it is negative semi definite. Another test of symmetry is
performed, the results from the model shows that the p value was 0.23 meaning that the null
hypothesis of symmetry is not rejected.
22
The last test is the separability test (Table 6), a complete test is performed, and the null
hypothesis is rejected at 5% level indicating continuing testing until one of the conditions is
accepted. As results, investigating linear separability, the null hypothesis is rejected also at 5
percent level for (LK-F) and (LF-K) indicating that the optimum ratio of (LK) and (LF) is
influenced by the level and the price of F and K respectively. On the other hand, the null
hypothesis is accepted for (KF-L) meaning that capital and fertilizer are independent from the
level and the price of labor. Since one of the separability tests is accepted no further tests (non-
linear separability) can hold.
After the tests are implemented to the model and the corrections are made when
necessary, the standards errors are robust for analysis. Table 5 shows that most of the
coefficients are statistically significant supporting the relationship between the inputs/output and
the first order parameters are well behaved since their estimate fall between zero and one. The
three inputs are significantly different from zero. The output elasticities are calculated for each
input at the mean level. The output elasticities with respect to labor, fertilizer, and capital, are
(0.19), (0.10), and (0.07).
Labor appears to be the most important factors in African agricultural growth. Fertilizer
has low effect due to the average intensity of its use has been stagnant since 1980s and then
declined sharply (50%) in 1990s. Africa remains the lowest of any developing countries in using
fertilizer (8 kilograms (kg) per hectare (h)) compared to 86 kg/h in Latin America and 123kg/h in
Asia (FAO, 2002). The low estimate of capital maybe related to the use of archaic cultivation
techniques. Human and animal powers are still cultivating the majority of the total area.
The sum of the elasticities is less than one, indicating that African agricultural production
is experiencing decreasing returns to scale at the sample mean (Table 6). The coefficient of time
23
is 0.014 suggests that the output has been increasing by 1.4 % per annum due to technical change
while the estimate of time squared means that technical progress rate increases by 0.2 %. These
estimates are statistically significant at the 1 and10 percent level.
The elasticities are calculated directly from the parameter estimation of the production
function. The AES are reported in Table 8 (Appendix). From the results, the Allen elastiticies of
substitution of all inputs are positive. Capital appears to be more substitutable with respect to
labor than to fertilizer. On the other hand, fertilizer appears to be more substitutable to labor than
to capital.
Factor price substitution is presented in Table 9. The own price elasticities are negative
as expected with labor is more elastic (-0.82) followed by capital (-0.80) while the fertilizer input
has the smallest own elasticity (-0.64).
The sign of the estimated cross price elasticities for labor and capital are negative
indicating that they are complements while the cross price elasticities of capital and fertilizer and
fertilizer and labor are positive indicating that they are substitutes. The demand for fertilizer is
sensitive to the price change of capital (0.12) and the demand for labor is sensitive to the price
change of fertilizer by the same amount. The demand for capital and labor is not as sensitive to
price change of labor and capital with (-0.02) and (-0.05) respectively. Furthermore, a 10 percent
increase in the price of labor will have positive effect on demand for fertilizer of about 7 percent
and an increase of 10 percent in the price of fertilizer will have increase demand for capital of
about 8 percent.
The HEC are reported in Table 10. The results show that only labor and capital are
complements as factor price substitution elasticities found earlier. A rise in capital will lead the
24
wage of labor to fall but the impact is opposite on the fertilizer. A rise in the fertilizer will raise
the price of both the capital and labor.
25
CHAPTER 5
CONCLUSION
The objective of this study is to estimate cross price elasticities of substitution for labor,
capital, and fertilizer for African agriculture. To conduct such an analysis, all crops in the region
are estimated with annual panel data from 1961-2007. The model is specified as a translog
production function estimated with a fixed effect regression. In order to get reliable results,
several tests and corrections are implemented. The symmetry condition is imposed on the
model.
To gain insights of the relation between the production function and its inputs, the Allen
substitution elasticity, factor price substitution elasticity, and Hicks elasticity of complementary
are computed. The Allen elastiticies of substitution of all inputs are positive. The results suggest
that the substitution is low for capital and fertilizer. Capital appears to be more substitutable with
respect to labor than to fertilizer. On the other hand, fertilizer appears to be more substitutable
for labor than capital.
Own price elasticities for labor, capital, and fertilizer are less than one, implying inelastic
demand with the labor is the most inelastic. It is reasonable to expect this result knowing that
developing countries in Africa rely heavily on labor input in agricultural production.
The own price elasticity for labor and capital are sensitive to the price change to almost
26
the same degree. Fertilizer is the least sensitive of the three inputs. Elasticity of substitution
results of factor price elasticity and HEC suggests that labor and capital are the only
complements among the inputs. Government should promote investment in agricultural
mechanization. Fertilizer use needs to be encouraged also by different motivations. Such as
efforts will help African agriculture.
27
REFERENCES
Alene, A.D. (2010. ?Productivity Growth and the Effects of R&D in African Agriculture.?
Agricultural Economics 41: 223-238.
Arrow, K.J. Chenery, H.B. Minhas, B.S, and Sollow R.M. (1961). ?capital-labor substitution and
economic efficiency?. The Review of Economics and Statistics, Volume 43. P 225-250
Baum, F. Christopher (2001). ?Residual Diagnostics for Cross Section- Time Series Regression
Models.? The Stata Journal. 1, n.1. p 101-103.
Beattie, Bruce R., Taylor, Robert. C, and Watts, Myles. J (2009). The Economics of Production
book. Second edition. P 1-8
Behar Alberto (2008). ?Does Training Benefit those Who Do not Get Any? Elasticities of
Complementarity and Factor Price in South Africa?. Working Paper. N.73. p 7-15
Berndt, E., and L. Christensen (1973b). "The Translog Function and the Substitution of
Equipment, Structures and Labor in U.S. Manufacturing 1929-68." Journal of Econometrics 1
Cameron, Collin. A and Trivedi, Pravin. K (2009). Microeconometrics Using Stata book. P 156-
162
Chambers, Robert. G (1988). Applied Production Analysis: A dual approach. P 6-15
Chandhary, Muhammed. Ali, Khan, Mushtaq. Ahmed and Naqvi Hassan (1998). ?Estimates of
Farm Output Supply and Input Demand Elasticities: The Translog Profit Function Approach? P
1-7
Christensen, L., D. Jorgenson, and L. Lau (1973). "Transcendental Logarithmic Production
Frontiers." Review of Economics and Statistics 55, no.1.
Coelli, T. Rahman, S., Thirtle, C.( 2003). ?A Stochastic Frontier Approach to Total Factor
Productivity Measurment in Bangladesh Crop Agriculture, 1961-92.? Journal of
International Development 15: 321-333.
28
Dhehibi Boubaker (2006). ?productivity and economic growth in Tunisian agriculture: An
Empirical Evidence.? Presented Paper at International Association of Agricultural Economists
Conference, Australia. P 1-9
Drukker, David (2003). ?Testing for Serial Correlation in Linear Panel Data Models?. The Stata
Journal. 3, n.2. p 168-177.
Edmund K. Mupondwa (2005). ?Induced technological change in Canadian agriculture field
crops-canola and wheat: 1962-2003?. Paper presented at American Agricultural Economics
Association Annual Meeting. P 1-22
FAO (2008). Agricultural Mechanization in Africa ?.Time for Action. Available at:
http://www.unido.org/fileadmin/user_media/Publications/Pub_free/agricultural_mechanization_i
n_Africa.pdf, (accessed 2008).
FAOSTAT database. Available at: http://apps.fao.org/. (Accessed to 2010).
Friedlaender, A.F and R. H. Spady (1980). ? General Spexification of Cost and Technology:
Freight Transport Regulation?. Cambridge MA, MIT Press.
Greene, W. (2000). Econometric Analysis. Upper Saddle River, NJ: Prentice-Hall.
Gujarati, Domador. N. Basic Econometrics. Fourth edition. P 792- 822
Hans P. Binswanger (1973). ?The measurement of technical changes biases with many factors of
production? Staff Paper. P 1-28
Henao Julio and Baanante Carlos. 2006. Agricultural Production and Soil Nutrient Mining in
Africa. An International Center for Soil Fertility and Agricultural Development. Forthcoming
Technical Bulletin.
Hausman, J.A., (1978). ?Specification tests in econometrics?, Econometrica 46, 1251-1272.
Hunt Diana. (2011). ?Green Revolutions for Africa.? Programme Paper: AFP 2011/01.
Khalil, M. Ali. ?A Cross Section Estimate of Translog Production Function: Jordanian
Manufacturing Industry?. Paper presented at Al-Ahiyya Amman University.
Kibaara Betty (2005). ?Technical Efficiency in Kenyan?s Maize Production: An Application of
the Stochastic Frontier Approach?. Masters of Science Thesis, Colorado State University.
Kim, H. Youn (2000). ?The Antonelli Versus Hicks Elasticity of Complementarity and Inverse
Input Demand Systems?. Australian Economic Papers.Volume 39, Issue 2,P 245-258
29
Kumbhakar, S. C and C.A.K Lovell (2000). ?Stochastic Frontier Analysis? Cambridge:
Cambridge University pres.
Millan, Pablo. Goto (2003). Utility and Production Theory and application.Second Edition. P 11
and 171-175
Nicholson Walter (2005). Microeconometric Theory: Basic Principles and Extensions. Ninth
edition. P 181-200
Pretty, J.N. (1995). Regenerating Agriculture. Earthscan Publications Limited, London.
Ray, Subhash. C (1982): A Translog Cost Function Analysis of U.S. Agriculture, 1939- 1977.
American Journal of Agricultural Economics. P 1-10
Salazar, Jorge. Ibarra (1994). ?Price Elasticities and Substitution Elasticities among Productive
Factors in the Border Mexican Maquiladora Industry? P 1-20
Tchale Hardwick and Sauer Johannes (2007). ?The Efficiency of Maize Farming in Malawi: A
Bootstrapped Transkog Frontier?. Cahiers d??conomie et sociologie rurales, n: 82-83
Tzouvelekas Evaggelos (2000). ?Approximation Properties and Estimation of the Translog
Production Function with Panel Data?. Agricultural Economics Review, Vol.1, n. 1 pp. 33-45
Valle Ikerene, Astorkiza Inmaculada, and Astorkiza Kepa. ?Estimating the Elasticity of
Substitution among Inputs Making up Fishing Effort: An Application to VIII Division European
Anchovy Fishery?. Paper presented in the EAFE 2000 meeting celebrated in Esberg, Denmark
Vincent, D.P. (1977). ?Factor substitution in Australian agriculture?. Australian journal of
Agricultural Economics, Vol. 21, No.2 pp. 119-129
Wayne Thirsk (1974). ?Factor substitution in Colombian agriculture?. American Journal of
Agricultural Economics. P 1-11
Wooldridge, J. M. 2002. Econometric Analysis of Cross Section and Panel Data. Cambridge,
MA: MIT Press.
30
Figure1. Trend of fertilizer Consumption in Africa in Thousands
Figure2. Trend of African Agricultural Labor in Thousands
0
20
40
60
80
100
120
1950 1960 1970 1980 1990 2000 2010
Fertilizer
Fertilizer
31
Figure3. Capital Use in African Agriculture in Thousands
32
Table1. The Share of Agriculture in GDP
Country Share of Agriculture in GDP (%) Country Share of Agriculture in GDP
Algeria 8 Madagascar 26
Angola 8 Malawi 30
Benin 33 Mali 37
Botswana 2 Mauritania 19
Burkina Faso 33 Mauritius 5
Burundi 35 Morocco 14
Cameroon 9 Mozambique 28
Cape Verde 9 Namibia 9
Central African 54 Niger 40
Chad 13 Nigeria 33
Comoros 45 Reunion -
Cote d?Ivoire 24 Rwanda 36
Democratic Congo 42 Senegal 13
Congo 4 Sierra Le 50
Egypt 14 Somalia -
Gabon 5 South Africa 3
Gambia 29 Sudan 28
Ghana 29 Swaziland 7
Guinea 25 Togo 44
G-B 62 Tunisia 30
Kenya 20 Uganda 24
Lesotho 8 Zambia 22
Liberia 55 Zimbabwe 23
Libya 2
Source: 2011 World Bank Indicator database
33
Table2: Description of Variables Used in the Analysis
Variable Description
Production frontier variables:
Output includes aggregate detailed crops measured as quantity in millions
tons
Inputs
Labor agricultural labor force in thousands, farmers and employers
Capital combines harvesters, threshers, and agriculture tractors
Fertilizer measured as consumption of nitrogen, phosphate, and potash in metric tons
Table3. Summary Statistics of the Variables
Variable Means Std. Dev Min Max
Output 7389242 1.42e+0.7 30809 1.61e+0.8
Labor 2430078 2747816 8000 1.45e+07
Capital 11562.81 45358.09 1 1095478
Fertilizer 60518.55 1742523 0 1824243
34
Table4. Hypothesis Tests
Null Hypothesis F statistic Critical Decision at 5%
168.26 Reject
122.08 Reject
Note: The critical values come from table 4 (Griffiths, Hill, and Judge, 1993).
Table5. Parameter Estimation of Translog Production Function
Serial no Parameters Coefficients Estimates T-ratio
0 Constant 0.967*** -16.28
1 ln L 0.107** 2.40
2 ln K 0.056*** 2.88
3 ln F 0.134** 7.49
4 T 0.014*** -0.08
5 ln L * ln L -0.001 2.27
6 ln K * ln K 0.009** 6.06
7 ln F * ln F 0.026*** -2.25
8 ln K * ln L -0.016** -1.40
9 ln K * ln F 0.001 0.36
10 ln F * ln L -0.007 7.56
11 ln L * T 0.001*** 2.58
12 ln K *T 0.001** 2.94
13 ln F * T 0.001** 2.08
14 TT 0.002 2.05
0.72
N of obs 2209
*** is significant at 1%
** is significant at 5 %
*is significant at 10%
35
Table 6: Separability Tests
Type
Restrictions Number of
restriction
Decision
Symmetry
=
=
=
3 Fail to reject
Complete
Separability
= 0
= 0
= 0
3 Fail to reject
Linear
LK- F separability
LF- K separability
KF-L separability
= 0 = 0
= 0 = 0
= 0 = 0
2
2
2
Rejected
Rejected
Fail to reject
Table 7: Estimated marginal product and output elasticites for African crops
Inputs Output Elasticities Marginal Products
Labor
Capital
Fertilizer
0.187
0.068
0.102
0.568
43.379
12.504
36
Table8. Estimated Allen Substitution Elasticity
Allen Elasticities of
Substitution
Elasticies Estimates
1.078
1.032
0.852
Table9. Cross Price Substitution Elasticity
Factor Price Elasticities
Elasticies Estimates Elasticies Estimates Elasticies Estimates
-0.818
-0.048
0.118
-0.018
-0.800
0.078
0.065
0.117
-0.643
Table10. Hicks Elasticity of Complementary (HEC)
Hicks Elasticity of Complementary (HEC)
Elasticies Estimates Elasticies Estimates Elasticies Estimates
-4.291
-0.203
0.632
-0.203
-11.449
1.143
0.632
1.143
-6.400