SUPPLY RESPONSE WITHIN THE FARMING SYSTEM CONTEXT
WEEK 3: DAY 1
ESTIMATING AGGREGATE SUPPLY RESPONSE FROM PRODUCTION AND
PROFIT FUNCTIONS
by Colin Thirtle and Yogi Khatri, University of Reading and London School of Economics
CONTENTS
1. DIRECT ESTIMATION OF SUPPLY RESPONSE FROM THE PRODUCTION
FUNCTION
1.1. The Determinants of Aggregate Supply
1.2. Estimation of Supply Parameters from the Production Function
2. THEORETICAL ADVANCES: FLEXIBLE FUNCTIONAL FORMS AND
DUALITY
2.1. Introduction
2.2. Flexible Functional Forms
2.3. Duality: Production, Cost and Profit Functions
3. A PROFIT FUNCTION APPROACH TO LAND REFORM IN ZIMBABWE
3.1. Overview
3.2. Introduction
3.3. The Dual Profit Function Approach
3.4. Data
3.5. Elasticity Results and Interpretation
3.6. Returns to Research
3.7. Shadow Prices
3.8. Conclusions
REFERENCES
LIST OF TABLES
Table 1.
Table 2.
Table 3.
Table 4.
Calculating Contributions to Supply Response
Estimated Elasticities
Shadow Prices of Capital and Buildings
Shadow Prices for the Fixed and Conditioning Factors
LIST OF FIGURES
Figure 1. The Structure of Duality Relationships
1
1. DIRECT ESTIMATION OF SUPPLY RESPONSE FROM THE PRODUCTION
FUNCTION
1.1. The Determinants of Aggregate Supply
The discussion in this chapter concentrates on the supply function because of its obvious
relevance to agricultural price policy. However, in the overall context of production, the
response of output to changes in the output price is only one of many relationships that may be
of interest. We noted that this issue was addressed by Binswanger (1990), who argued that
although individual crops respond strongly to price factors, this is at the expense of alternative
outputs. The short run aggregate response is very low, because the main inputs, land, labour and
capital, are fixed. To get a good aggregate response in the long run requires more resources
and/or better technology and infrastructure investments in roads, markets, irrigation, education
and health. We now proceed to incorporate some of these variables in our model.
1.2. Estimation of Supply Parameters from the Production Function
Supply response to price is determined by a combination of (1) the physical response of output
to a change in the level of input use and (2) the behavioural response of farmers in changing the
level of input use, in response to changes in the price of outputs (or inputs)1. The production
function (Figure 1, week 2, day 5) measures the physical response of output to inputs. In the
Figure, we considered only one variable input in order to be able to draw a two dimensional
diagram. Now, the production function can be stated with one output as a function of several
inputs, which may be variable or fixed:
Y = f ( X 1 , X 2 , X 3 , ...., X n )
1
where Y is output and X1, X2, ..., Xn are inputs such as land, labour, animal and mechanical
power, implements, fertiliser and other agricultural chemicals. The elasticity of output, Y, with
respect to an input, Xi, is defined as:
ε yx =
i
δY / Y
δY / δ X i
δLnY
=
=
Y / Xi
δ Xi / Xi
δ LnX i
2
which measures the physical response. The behavioural response of farmers to changes in the
output price is defined as:
εx
i
py
=
δ X i / δ Py
δ Xi / Xi
δ LnX i
=
=
δ Py / Py
δ LnP y
X i / Py
3
which is the elasticity of input demand for Xi with respect to the output price, Py.
1
Some care is needed here. The elasticity of output with respect to the input price was defined
in the last chapter in equation (11). The concept being introduced here is the elasticity of input
use with respect to the output price. If we are looking at the aggregate input, then the two will be
the same, but with opposite signs, provided that the supply function is homogenous of degree
zero.
2
Under fairly general assumptions these two elasticities can be combined to provide a measure of
supply response that is an alternative to the direct estimation of the supply function:
ε sp = Σi ε yx ε x p
i
i
y
4
Summing the product of the output elasticity and the input demand elasticity over all inputs, Xi,
gives the supply elasticity of good Y, for a change in the price of the output; the contribution of
an individual input, Xi, is just (εyxi)(εxipy).
It is possible to estimate the production function with data only on outputs and inputs, but a
functional form must be chosen. It is convenient to start with a function that is linear in
logarithms:
LnY = Ln ß 0 + ß 1 LnX 1 + ß 2 LnX 2 + ..., + ß n LnX n
5
This form is known as the Cobb Douglas. One attractive feature is that little data is required, and
another is ease of interpretation. It should not come as a surprise that the coefficients, ßi, are the
output elasticities. This has to be so, since from equation (5):
δLnY
δY / δ X i
= ßi
=
Y / Xi
δ LnX i
ε yx =
i
6
i.e. the coefficient is just the slope of the function, with both variables defined in logarithms.
This is convenient, but we can go further. Even if the function cannot be estimated, we can
calculate output elasticities, provided we are prepared to assume that the Cobb Douglas is
appropriate and that the system is in equilibrium. In equilibrium, the value marginal product of
an input will equal its price (section 1.1, yesterday), which we rearranged to give:
MPP x i =
δY
Px
=
δ Xi
Py
7
and for the Cobb Douglas, manipulation gives the MPP of Xi:
MPP x i =
δY
Y
= ßi
δ Xi
Xi
8
if the MPP is equal to the ratio of prices, as in (7), then multiplying both sides by Xi/Y gives the
output elasticity:
ε yx =
i
δY X i
Px X i
=
δ Xi Y
PyY
9
The right-hand term is the share of factor i in total output (or total cost), meaning that the output
elasticities for a (constant returns) Cobb Douglas are simply the factor shares, which can usually
be calculated directly from the data.
3
The analysis so far covers only the estimation of the physical parameters of the production
function, that is, the output elasticities with respect to the inputs. To allow estimation of the
elasticity of output with respect to the output price, equation (4) indicates that we need also to
estimate the elasticity of input demand with respect to the output price (equation (3)). This
requires estimation of either the supply function or the input demand function. The link between
the two is made explicit by Tweeten (1989, Ch.5). Suppose, the input demand and output supply
functions are assumed to be linear in logarithms, and we stick to the case of one aggregate input.
We can specify an input demand function much like the output supply equation, (equation (14),
yesterday), but we ignore competing products for simplicity and drop the technology term
because it should not be important in input demand. Then the input demand function for the
aggregate input could be written as:
Py
X = ß0 ç ÷
Px
ß1
I ß2
10
where X is the aggregate input, Py is the output price, Px is the input price index, and I is a vector
of relatively fixed infrastructure variables. Then, suppose technology is measured by an index of
output quantity to input quantities, T = Y/X. Multiply both sides of the input demand function in
equation (10) by this index to get:
æ Pyö
æYö
X ç ÷ = Y = ß0 ç ÷
èX
è Px
ß1
I ß2 T ß3
11
which is the output supply function, provided that ß3 is approximately unity, which it should be.
Taking logarithms of equations (10) and (11) gives almost identical equations for input demand
(equation (12)) and output supply (equation (13)):
Ln X = Ln ß 0 + ß 1 Ln P y - ß 1 Ln P x i + ß 2 Ln I
12
Ln Y = Ln ß 0 + ß 1 Ln P y - ß 1 Ln P x i + ß 2 Ln I + ß 3 Ln T
13
and
Note that ß1 appears twice in both equations. It can be interpreted as the short run supply
response2 to the output price. But, the output supply elasticity and the input demand elasticity
with respect to the output price are both equal to ß1:
ε sp =
δLnY
= ε xp
δ LnP y
y
=
δLnX
= ß1
δ LnP y
14
and the elasticity of output with respect to the input price is equal to the elasticity of input
demand with respect to the input price, where these elasticities are just the same as those above,
but with the sign changed:
δLnY
δLnX
15
= ε xp =
= - ß1
ε yp =
δ
δ
2
LnP
LnP
x
x
In the last chapter, we considered the partial adjustment model, used to estimate short and
long-run response.
x
x
4
Thus, ß1 can be estimated from the input demand function, which is the statistically preferred
method. Or, it may be estimated from the output supply function, with a technology index. If
this is not available, it must be estimated from the output supply function, with a time trend as a
proxy for technology, but this can introduce considerable specification error.
So, there is a variety of alternative means of generating a supply elasticity. Before leaving the
subject, note that we have not yet used equation (4) to exploit any output elasticities that may
have been calculated from the production function estimation. This has happened because we
have so far looked only at a single aggregate input. The result is that Σiεyxi in equation (4) is
approximately equal to unity3, leaving;
ε sp =
δLnY
= ε xp
δ LnP y
y
=
δLnX
= ß1
δ LnP y
16
However, if output elasticities for the separate inputs were estimated from equation (5) and
elasticities of input demand with respect to their own prices had been estimated, using the
logarithmic version of last chapter’s equation (13), the contributions of different inputs to the
supply response can be calculated. When the inputs are fertiliser, land, labour and irrigation,
Table 1 gives some hypothetical results and calculations. The table shows that because the
contribution to supply response is the product of the two elasticities, the greatest effect will be
for price changes in an input where both are reasonably large. Reducing land prices will be
ineffective, because although the physical relationship to output is important, there is no
behavioural response of land use to price changes. Machinery prices are an effective policy tool,
because the output elasticity and the input demand elasticity are relatively large, giving this
variable the largest share in the total supply response.
Table 1: Calculating Contributions to Supply Response
Input
Output Elasticity
Elasticity of Input
Demand
Contribution to
Supply Response
Fertiliser
0.1
0.9
0.09
Land
0.3
0.0
0.0
Labour
0.3
0.2
0.06
Machinery
0.2
0.5
0.10
Irrigation
0.1
0.3
0.03
1.0
1.9
0.28
Total
By now the reader should be convinced that some knowledge of theory is an asset in the
estimation of supply elasticities. We leave this topic with two further examples of the uses of
theory to overcome data inadequacies. Firstly, there is a result associated with work by Mundlak
and others, that shows how reasonable estimates of short run supply responses at the industry
level can be obtained from the output elasticities alone. Suppose that the output elasticities were
3
A value of unity would indicate constant returns to scale.
5
as in Table 1 and that in the period being considered, land and labour should be viewed as fixed,
while fertiliser, machinery and irrigation as variable. The extent to which the short run supply
can change in response to price is shown to be the sum of the variable elasticities divided by the
sum of those that are fixed. So, in this hypothetical example, εp = 0.4/0.6 = 0.66.
The other example is attributable to Tweeten (1989, Ch.5) who shows that all the output price
elasticities in a complete system can be calculated from the input elasticities. Alternatively, all of
the cross-price elasticities for outputs and the input elasticities can be calculated from the ownprice output elasticities alone. A more complete treatment of this area requires duality theory.
This is a literature which has developed rapidly in the last two decades. Tweeten (1989, Ch.5)
provides an introduction and some references to duality theory which will be addressed in the
next section.
2. THEORETICAL ADVANCES: FLEXIBLE FUNCTIONAL FORMS AND
DUALITY
2.1. Introduction
Three basic methodologies are used to tackle the measurement and explanation of agricultural
efficiency and productivity. They are, econometric estimation of the production, cost or profit
function; the accounting approach, using index number theory and non-parametric programming
techniques, sometimes called data envelopment analysis4. For the purposes of analysing supply
response, we have so far used the production function and will now concentrate on the dual
profit function, since it provides a unifying framework, from which the output supply and input
demand functions can be derived.
All the approaches and representations start form the basic notion of a relationship between
outputs, Yi and inputs, Xj. The simplest form is the single output production function, Y = F(Xj),
which was the starting point for week 2, day 5’s analysis. This is a purely technical relationship
and economics is only introduced when an economic problem is stated, such as maximising
profits with the production function as the technological constraint. The three major recent
advances in this area are the development of flexible functional forms, duality theory, and the
application of time series techniques associated with the concept of cointegration. We covered
cointegration yesterday and will now introduce new functional forms and duality.
2.2. Flexible Functional Forms
The Cobb Douglas production function is unduly restrictive. In the 1970's and 1980's, flexible
functional forms were developed, such as the translog. These functions differ from old fashioned
functional forms by incorporating enough estimated parameters to take account of the
interactions between variables and to allow for non-linearity in the parameters. Thus, a typical
specification is the translog:
LnY = Ln α 0 +
α i LnX i +
4
1
2
β ij LnX i LnX
j
17
The input-output approach, which is particularly useful for examining sectoral interactions,
is not discussed here.
6
where Y is aggregate output, the Xis are inputs and all the αis and ßijs are coefficients.
Without the last term, this is the familiar Cobb Douglas that was discussed in week 2, day 5.
This last term allows for interactions between inputs, when i≠j. These terms allow the elasticities
of substitution between each pair of inputs to be estimated from the data, whereas the Cobb
Douglas imposes substitution elasticities of unity for all pairs of inputs.5 Thus, if two inputs,
such as R&D and extension expenditures, are complements rather than substitutes, this will be
taken into account. When i = j, the last term adds squared terms for each input, which allows for
non-linearity and estimation of a system of simultaneous equations consisting of the production
function itself and all but one of the input share equations. The internal rate of return to R&D
can be estimated in this framework, since the coefficient(s) of the R&D term relate changes in
R&D expenditures to changes in output.
2.3. Duality: Production, Cost and Profit Functions
The second major advance is the use of dual forms, instead of direct estimation of the production
function. Duality is not a new concept and now even undergraduate textbooks point out that the
average and marginal cost curves are simply the average and marginal product curves inverted.
This is because they are derived from the familiar S shaped total product and total cost curves,
which are also the inverse of each other. Alternatively, in the programming literature, reversing
the objective function and the constraint is known to give the same solution. Thus, maximising
output subject to a cost constraint (moving along a given isocost line until the tangency with the
highest possible isoquant is found) is the same as minimising costs subject to an output
constraint (moving along a given isoquant until the tangency with the highest possible isocost
line is found).
Figure 1 shows how modern duality theory has generalised these fairly obvious concepts. The
most intuitively appealing progression is the derivation of the dual cost function on the left hand
side of the figure. We begin by stating the problem, (1), which is to minimise costs, subject to
the production function constraint, with a particular level of output, Q*. The Xjs are variable
inputs, the Zks are fixed inputs, the Rs are input prices, C is total cost and F represents some
functional form. Formally, the problem is solved by setting up the Lagrangian and taking
derivatives with respect to all the variables. These are the first order conditions, (2), which are
not shown, with each derivative set equal to zero, to find maximum or minimum values. The
system is solved with the variable inputs as the dependent variables. This gives the input
demand functions (3), for all the Xjs as a function of the input prices, the levels of the fixed
inputs and the level of output. The last term indicates that these are the output constrained input
demand functions, which are the production equivalent of the Hicksian compensated demand
functions in consumer theory. Substituting the expressions for the Xjs into the objective function
(4) gives the dual cost function, (5), which expresses costs, C, as a function of the input prices,
the levels of the fixed inputs and the level of output.
[Figure 1: The Structure of Duality Relationships]
5
This follows from the fact that the Cobb Douglas output elasticities are also the factor
shares (when constant returns is imposed) and they remain constant. The only way this can
happen is if input price changes are exactly compensated by quantity changes, which requires
unitary elasticities of substitution.
7
The point of all this is not the derivation itself, but its implications. The duality relationship
indicated between the constrained cost minimisation problem and the dual cost function means
that the dual cost function contains all the information in the constrained minimisation problem.
Thus, we no longer need to solve these complicated problems, but can start instead by defining a
functional form for the dual cost function. Then, Shephard's Lemma proves that all we need to
do to retrieve the input demand functions is to take the derivatives of the cost function with
respect to the input prices. This enormously simplifies the process of generating the estimating
equations needed to model agricultural production with the behavioural assumption of cost
minimisation incorporated.
The limitation of the cost function is that like the production function, it is restricted to a single
output.6 This is unfortunate, since most farms are multiple output enterprises. The transition to
multiple outputs is made by stating the more general problem of profit maximisation, shown in
(8), on the right hand side of the figure. The notation is the same as for the cost function, with
the addition of Pis, which are output prices. Now, the output mix can be varied, so profit is
maximised as the sum of value of all the outputs, (PiQi), minus the costs of the variable and
fixed inputs, subject to the constraint of the transformation function which is included to allow
for crop switching.
Solving the first order conditions (9) gives both the input demand functions and the output
supply functions (10), with output prices, input prices and the levels of the fixed factors as the
independent variables. Rather than constructing ad hoc supply functions, as in the analysis
presented in week 2, day 5, we now have a complete system of supply functions and will
generate a complete set of own and cross price elasticities. Furthermore, estimation of the
system of output supply and input demand functions also produces shadow values for the fixed
inputs. These shadow prices are the marginal revenue products of these inputs and can be used
to calculate rates of return to the investments in capital items, infrastructure and technology
variables. Thus, the profit function provides a sound theoretical basis for the study of long run
aggregate supply response, which takes technological change and infrastructure into account.
Again, the derivation is only useful, because it shows the relationships involved. The point is
that substituting the output supply and input demand functions into the objective function, gives
the dual profit function (12). Hotelling's Lemma (13) shows that taking derivatives with respect
to output prices and input prices directly gives a system of output supply and input demand
functions (14), which can be estimated. Thus, we have no further need of the constrained
maximisation problem and can begin our analysis by specifying a flexible functional form for
the dual profit function. This we do in the next section, which is an application of the dual profit
function to commercial agriculture in Zimbabwe.
3. A PROFIT FUNCTION APPROACH TO LAND REFORM IN ZIMBABWE
3.1. Overview
6
In fact, it is possible to generalise the cost function to the multiple output case, but for our
purposes the profit function is preferable.
8
The purchase of commercial farm land in Zimbabwe for resettlement has been a major factor in
government policy since independence in 1980, but from 1980 to 1989 only 52,000 families
were relocated. The Land Acquisition Bill of 1992 made compulsory purchase easier, and at
present the government has announced its intention to considerably increase the rate of
resettlement. But, Zimbabwe has a serious food security problem and the output effects of land
redistribution are a matter of dispute. The World Bank estimate that 3,000,000 hectares of
commercial farmland are under-utilised is contested by the Commercial Farmer's Union. Fitting
a normalised residual profit function to the data for the commercial sector, allows estimation of
the shadow price of commercial farm land. We find that the model suggests that the World Bank
is correct, in that the marginal value product of land is negative, meaning that there is underutilisation. However, negative values of capital assets are common when real interest rates are
negative, so the result should be treated with some caution. Also, the problem of identifying the
under-utilised land is not trivial and redistributing intra-marginal land would have output effects.
3.2. Introduction
"Zimbabwe's one million communal farm households are restricted to half the total area suited
for agricultural production. The other half is occupied by 4,500 large-scale commercial farmers,
most of whom are white. To compound this inequality, the communal lands have a much lower
agricultural potential; 74% of the communal lands is in natural regions IV and V, and 51% of
the commercial farming area is in natural regions I-III (CSO, 1989). This grossly unequal land
distribution is the most fundamental and least tractable of all Zimbabwe's problems. It is also a
significant cause of food insecurity in the rural areas." (Christensen and Stack, 1992).
There is also, in theory, an efficiency argument for land redistribution, since in any dual
economy, output can be increased by redistributing resources until their marginal products are
equal in the two sectors. But, it is widely accepted that the communal farmers cannot produce at
the same level as the commercial farmers without considerable support, and the government is
already under extreme pressure to cut expenditures. Without considerable investment7, the
expectation is that food production would decrease, exacerbating the food security problem.
Christensen and Stack (1992) estimate that 420,000 rural and 125,000 urban households are
suffering from chronic food insecurity. In this respect, the land reform issue in Zimbabwe is
quite different from the South African situation, where output exceeds consumption by a wide
margin8 and food grains are exported at below cost. Thus, South Africa can afford to redistribute
land, even if the result is a substantial decline in output, but Zimbabwe cannot ignore the
possibility that land reform could result in even greater food security problems.
Whereas many of the arguments over land reform are complex, the value of marginal land in the
commercial sector can be estimated quite simply. One legacy of the colonial past is that
Zimbabwe has a statistical system not much different from the UK, which has collected
agricultural statistics for the national income accounts that can be used for the estimation of
7
The cost of resettling 52,000 families, from 1980-89 has been about US $112 million
(Bratton, 1991)
8
Self-sufficiency indices for South Africa, with 100 meaning sufficiency, show grain
production at 150, horticultural products at 132 and livestock production at 98 (van Zyl et al
(1993).
9
production relationships. The data for the commercial sector is qualitatively not much different
from the information available in European countries (indeed, better than some). These data
were used for the Total Factor Productivity estimates in Thirtle et al (1993), but direct
comparison of the two sectors was deliberately avoided, on the grounds that they are too
dissimilar. However, by fitting production, cost, or profit functions to the two sectors separately,
estimates of variables such as marginal products and shadow prices of inputs can be derived.
These indicate relative factor scarcities, allowing quantification of the costs and benefits of
reallocating resources between the two sectors.
The next section briefly explains the profit function approach used in this study, which was
more fully explained in section 2. This is followed by the elasticity results, shadow prices and
the calculation of the returns to research. The final section concludes by considering the policy
implications.
3.3. The Dual Profit Function Approach
The profit function provides estimates of a full range of economic variables, whereas the
production function and the TFP index concentrate only on the physical relationships between
inputs and outputs. The commercial sector is treated as single production units to which the
restricted or variable profit function (Lau 1972, 1976) is applied. Consider a multiple output
technology producing outputs yi, (i = 1, ..., m), with the respective expected output prices pi,
using n variable inputs xj, (j = 1, ..., n) with prices wj. We then define variable expected profits
as:
π =
pi y i -
wj xj
18
j
i
which is simply the value of output minus variable costs.
Normalising the profit function with respect to an output or input price has the practical
advantages of ensuring that the homogeneity requirement is met and reduces the number of
parameters to be estimated. Also let pi represent input prices as well as output prices, to keep the
notation compact. Thus, if:
*
pi =
pi
p0
19
pi* is a vector of normalised output and input prices and the functional form for the generalised
quadratic profit function is defined as:
π* =
π
= α0 +
p0
*
α i pi
+
i
1
2
β ij p *i p *j +
i, j
where β ij = β
γ
i, k
ik
p *i z k
20
ji
where π* is normalised profits and zk is a vector of fixed or quasi-fixed inputs and conditioning
factors, such as R&D and patents. The vector α and matrices β, γ contain the parameter
10
coefficients to be estimated. Applying Hotelling's lemma, we derive the optimal levels of output
supply and input demand respectively:
β ij p j +
y *i = α i +
j
β ij w j +
j
β ij p j +
- x *i = α i +
β ij w j +
j
j
γ ik z k
21
γ ik z k
22
k
k
where we have again distinguished between output and input prices, so that (21) are the output
supply functions and (22) are the input demand functions.
Then, the price elasticities of outputs and inputs for the non-numeraire cases are:
*
ηij = β ij
pj
yi
*
23
wj
ηij = - β ij
xi
Convexity of the profit function implies that the own-price elasticities should be positive for an
output and negative for an input. The cross-price elasticities for pairs of inputs are negative for
complementary inputs and positive for substitutes. For pairs of outputs, positive cross-price
elasticities imply complementarity in supply and output substitutes are indicated by negative
cross-products.
If the elements of z are treated as short-run constraints on production, we can derive the effects
of relaxing the z variable constraints on the output and variable input levels. We can derive these
effects in elasticity form by logarithmic differentiation of (21) and (22) with respect to the
elements of z:
ε ik
z
= k γ ik
yi
24
zk
γ jk
ε jk = xj
Shadow prices for the variables in the z vector can be derived as partial derivatives of the profit
function (Diewert, 1974, Huffman,1987) with respect to the z variables. The derived shadow
values can be interpreted equivalently as (i) the marginal change in profits for an increment in a
particular element of z, (ii) as the imputed rental value for an additional unit of that factor or (iii)
the effects on expected profit of relaxing the particular constraint represented by each z variable.
The shadow value of land (treated as fixed) provides the implicit value in production as opposed
to the market price. The difference between the market price and shadow value indicates
11
whether land is over, under or optimally utilised. The shadow prices of the other conditioning
factors (such as R&D) can be used to assess their effectiveness.
3.4. Data
The data was compiled by Thirtle et al (1992) and a detailed description of the basic data can be
found therein. Here we define the compilation of the quasi-fixed inputs and conditioning factors
and discuss the aggregation of the output and variable input groups.
The outputs are Divisia aggregated into three groups: food crops (Y1), industrial crops (Y2) and
livestock and livestock products (Y3). The first group contains largely maize and other grains.
The second group contains largely tobacco, coffee and other export crops. The third aggregate is
of animals and animal related outputs.
The variable inputs are Divisia aggregated into four groups. These are, hired labour (XL),
livestock related inputs (feed, veterinary costs, purchases from the communal sector, etc) (XV),
chemical/crop related inputs (fertiliser, other chemicals and packing) (XC) and running costs
(vehicle maintenance, transport, sundries, services and licences, etc) (XO). Vehicles and fixed
capital in the form of buildings and other fixed improvements are assumed to be quasi-fixed.
Foreign exchange constraints support the fixity of farm capital.
Two capital categories were constructed. These are farm vehicles (CAP) and buildings (BLD).
The number of tractors and combine harvesters was used together with their purchase prices to
construct a value of agricultural machinery stock (at purchase prices). Assuming, on average,
that each vehicle is worth half of its purchase price, we divided the value derived by two.
Deflating by the FAO machinery price index for Zimbabwe, we derive a stock of machinery.
No details on gross fixed capital formation could be found for the period, so to construct a series
for buildings, we assumed that the building maintenance and repairs expenditure was equal to
the depreciation on buildings and that the average life-span of a building or fixed improvement
was 30 years. Thus, we can construct an implicit stock of buildings. Subtracting net own account
capital formation9 and deflating by an index of building material prices, we derive the stock of
building and fixed improvements.
Total area of land (LAND) in the commercial sector is included as a fixed input. Other fixed,
exogenous or conditioning factors included are, Research and Extension (RES), Rainfall (RAIN)
and world patents pertaining to agricultural machinery and chemicals (PAT).
The research and extension expenditures were aggregated into an agricultural knowledge stock.
Patents are used to capture transferable technology (and thus international spillovers). The
patents are simply the total number of agriculture-related patents registered in the US by all
countries. This number is the straight aggregate of the number of mechanical and chemical
patents. A stock variable of internationally produced and available knowledge was constructed
from the patent numbers.
9
Own account capital formation is both an output and an input. Thus adopting the net output
approach (USDA, 1980), we subtract fixed capital formation net of materials (ie value added)
from the output and input side of the account.
12
3.5. Elasticity Results and Interpretation
The system of output and variable input equations were estimated using an iterative Zellner
procedure, which provides maximum likelihood parameter estimates on convergence of the
weighted error-covariance matrix. Convergence was achieved for the normalised system with
symmetry imposed. The symmetry restrictions could not be tested due to a limited number of
degrees of freedom, but were imposed to ensure consistency with the continuity property of the
profit function. The system provides a majority of significant parameter estimates (at the 1%
level) and the 'goodness of fit' measure represented by R2s of the estimated supply and demand
system equations vary between 0.77 and 0.99, which is high, even for a time series model. The
parameter estimates themselves have limited economic interpretation, and are thus relegated to
the workshop presentation. The parameters are used to construct measures of elasticities and
shadow prices for the quasi-fixed and fixed inputs.
13
Table 2: Estimated Elasticities*
Dependant Variable
Regressors
Y1
Y2
Y3
XL
XV
XC
XO**
P1
0.8
(4.4)
-0.08
(-.99)
-0.10
(-1.2)
-0.15
(3.8)
-0.44
(-4.4)
0.36
(2.92)
0.33
P2
-0.19
(-0.98)
-0.31
(-1.6)
0.47
(2.8)
0.33
(4.1)
0.68
(3.6)
-0.29
(-1.8)
0.37
P3
-0.14
(-1.2)
0.28
(2.8)
0.83
(3.4)
-0.44
(-3.9)
-0.21
(-0.74)
-0.53
(-3.9)
0.49
WL
-0.24
(-3.8)
0.22
(4.1)
-0.5
(-3.9)
-0.11
(-1.4)
0.33
(1.95)
0.27
(2.97)
0.06
WV
-0.34
(-4.4)
0.22
(3.6)
0.11
(0.74)
0.16
(1.95)
0.33
(0.97)
0.3
(2.76)
-0.84
WC
0.34
(2.9)
-0.11
(-1.8)
-0.35
(-3.9)
0.16
(2.97)
0.36
(2.76)
-0.4
(-3.8)
0.32
WO
-0.3
-0.15
-0.34
0.03
-1.1
0.33
-0.84
CAP
-1.29
(-1.8)
0.33
(.78)
1.4
(3.6)
0.34
(1.8)
2.1
(4.5)
-0.46
(-0.9)
12.1
BLD
-0.14
(-0.2)
-0.69
(-1.8)
-1.88
(-4.4)
.07
(0.37)
-0.74
(-1.54)
0.99
(2.0)
-6.3
LAND
-0.24
(-0.49)
-0.54
(-2.0)
1.0
(3.4)
1.1
(7.8)
0.6
(1.7)
0.14
(0.4)
4.9
RES
0.9
(2.1)
-0.12
(-0.5)
-0.16
(-0.64)
-0.04
(-0.39)
0.5
(1.8)
0.96
(3.2)
-1.6
PAT
0.58
(1.7)
-0.07
(-0.4)
-0.24
(-1.4)
-0.27
(-3.4)
0.06
(0.3)
0.48
(2.1)
-2.7
* t-values are in parentheses; the critical value is taken to be 2.26
** t-values are not computed for numeraire input elasticities as the numeraire input and the derived elasticities are
gained residually from (6).
Table 2 summarises the short-run elasticities of supply and variable input demand with
respect to prices, quasi-fixed inputs and conditioning factors at the variable means. The
significant own-price supply and demand elasticities (on the diagonals of the upper shaded
blocks)10 have the expected sign, and are of plausible magnitudes. Note that at this level of
aggregation, we get supply elasticities for food crops and livestock of about 0.8, whereas the
highest elasticity for aggregate output in Table 3 in the last chapter was 0.34. The own-price
elasticity of the industrial crop aggregate has the wrong sign, but the t-statistic indicates that
the elasticity is not significantly different from zero. The input demand elasticities form the
other upper shaded block. Apart from livestock related inputs (not significant), the variable
input own-price elasticities have the expected signs. However, the hired labour elasticity is of
low significance and for running costs, which was the numeraire, we get no t-statistic. Thus,
10
The elasticities that have meaningful economic interpretations are in the shaded areas of
table 2; the others are not discussed.
14
only the crop input own-price elasticity is significantly different from zero at high confidence
levels. Indeed, all the own-price output supply and input demands are inelastic.
For the outputs, complementarity (substitutability) is indicated by a positive (negative) crossprice elasticity. Thus, industrial crops and livestock are complements and food crops are not
related to industrial crops or livestock, due to the low t values. Input complementarity
(substitutability) is indicated by a negative (positive) cross-price elasticity. Thus, livestock
inputs may be complementary to running costs (no t statistic for the numeraire) and labour,
livestock inputs and crop inputs are all substitutes for one another.
If we consider the quasi-fixed, fixed and conditioning factors as constraints in production, the
long-run output and variable input elasticities with respect to these factors can be regarded as
the responses to relaxing these constraints. The quasi-fixed inputs are stock variables that are
endogenous in the long-run, but changing their levels requires investment. Thus, in the short
run, the costs of adjusting these stock levels may be considered in terms of forgone
production. The levels of the conditioning variables are assumed to be beyond the control of
farmers and the costs of adjustment are not considered to be incurred by farmers. Thus, since
the reported elasticities are for the short-run, we might predict net negative output elasticities
with respect to fixed and quasi-fixed factors and positive output elasticities with respect to the
conditioning factors representing technology. However, the effect on individual outputs
cannot easily be predicted, as changing capital stock levels, or technology levels may favour
certain outputs and also affect the variable input levels, which in turn affects output.
These elasticities are reported in the shaded lower part of the table. The food crop output
elasticities follow the predicted pattern; with respect to machinery, buildings and land the
elasticities are negative (but insignificant) and positive and significant with respect to
research and international technology spillovers. All the elasticities for the industrial crops
are insignificant and for livestock, machinery and land appear to increase output, even in the
short run.
The effects of changes in the fixed inputs and technology variables on the variable inputs are
mostly insignificant, but increasing machinery increases livestock inputs and running costs.
Increasing buildings reduces running costs and increasing land raises both labour inputs and
running costs. For the technology variables, R&D increases crop inputs (which is reasonable,
since improved varieties use more fertiliser and pesticide), but technology spillovers reduce
both labour inputs and running costs. This is entirely sensible, since the majority of patents
are for machinery.
3.6. Shadow Prices
The shadow prices for the quasi-fixed, fixed and conditioning factors provide measures of the
implicit value in production of additional units of the factors. In equilibrium, the shadow
prices of a quasi-fixed factor should equal its opportunity cost, or rental value. Excess
capacity or under-utilisation of a quasi-fixed input would be indicated by an estimated
shadow prices less than the opportunity cost. Similarly, under-investment is indicated by a
shadow value greater than the opportunity cost, indicating that revenue can potentially be
increased by increasing the stock of the quasi-fixed factor until the shadow price equals the
opportunity cost (Berndt and Fuss, 1986; Morrison, 1986).
15
The economic reasoning behind these propositions is sound enough, but does not take good
account of economies with persistent high inflation and negative interest rates. In such
circumstances, the opportunity cost of capital investment is negative, but rental rates are not.
For Zimbabwe, if the opportunity cost is taken to be the real return on bank deposits, the rate
has been negative since 1976. For long term investments, such as 25 year government stock,
the average real rate of interest has been negative over the same period. This implies that a
rational farmer should invest in capital up to the point where the return is negative. The
Zimbabwe case is further complicated by rationing and allocation of farm machinery, which
would suggest that the supply is inadequate (at these prices). These factors should be taken
into account in interpreting Table 3, below, which reports the values of the estimated shadow
prices of capital, buildings, land and the technology variables, and the opportunity costs. The
opportunity cost of machinery capital is taken to be the real rate of return on bank deposits
and that for buildings the real rate of interest on 25 year stock (Bouchet, 1987).
Considering firstly farm capital (which we took to be harvesters and tractors), we see that
there existed an excess capacity until the mid-1970's. This appears consistent with the general
encouragement of the government then, to maintain growth in agricultural output. Towards
independence, inflation grew and real interest rates became increasingly negative. The over
capitalisation, thus became under capitalisation by the opportunity cost criteria. That is,
financial capital invested in machinery depreciated in real terms less quickly than the
alternative of bank deposits. The assumption that bank deposits are the main alternative to
capital investment is strengthened by the existence of foreign exchange controls and
restrictions on foreign investment.
As we see from the table, the shadow value of capital in the post independence period tends
toward the opportunity cost of capital. Thus, although the implication of a negative shadow
price, is a negative marginal product (implying an economically inefficient allocation of that
factor), the policy restrictions on alternative forms of investment imply that, post
independence investment in machinery capital stock is consistent with the opportunity cost
criteria.
Investment in buildings and other fixed capital is a longer-term investment decision, and is
therefore, subject to a greater degree of uncertainty. The opportunity cost of such long-term
investments would be related to longer term financial assets, and thus the yield on long bonds
for example, might be used as a measure of the opportunity cost of investment in buildings
and fixed capital. We use the yield on 25 year government stock together with the CPI to
derive the opportunity cost of investment in buildings in a similar manner to that derived for
capital.
Before suggesting the implications of comparing the shadow price to the opportunity cost of
building stocks, we note the compilation of the buildings stock is far from ideal (simply a
multiple of the costs of maintenance and repairs) and the summary measure (at variable
means) is not significantly different from zero, and therefore, subject to assumption that the
measure of building stocks is reliable.
Table 3: Shadow Prices of Capital and Buildings
Year
Shadow price of
Capital
Opportunity Cost of
Capital
16
Shadow Price of
Buildings
Opportunity Cost of
Buildings
1971
-0.08
1.46
0.06
3.40
1972
-0.07
1.46
0.06
3.40
1973
-0.09
1.46
0.07
3.40
1974
-0.08
1.46
0.07
3.40
1975
-0.08
1.29
0.07
4.20
1976
-0.09
-0.10
0.07
2.77
1977
-0.08
-2.58
0.08
0.22
1978
-0.09
-3.42
0.09
0.51
1979
-0.21
-4.15
0.12
-0.25
1980
-0.32
-5.84
0.11
-2.01
1981
-0.45
-1.27
0.06
2.35
1982
-0.84
-1.44
-0.02
2.18
1983
-1.47
-1.87
-0.10
1.73
1984
-1.74
-4.45
-0.14
-0.95
1985
-2.61
-4.91
-0.25
-1.42
1986
-2.99
-5.60
-0.38
-1.96
1987
-4.06
-5.60
-0.56
-1.36
1988
-4.45
-5.81
-0.72
-1.49
1989
-4.69
-3.17
-0.96
1.27
At Var.
means*
-0.83
(-3.36)
-2.27
0.0153
(0.143)
1.02
* t-values in parentheses
The figures in the table imply that there was under-investment in buildings and fixed capital
before independence and a movement toward the negative rate after independence. The
possible under-investment in the pre-independence period might be attributable to the
expectation of independence and the related uncertainty. In the post independence period, the
shadow price of buildings is still largely less than, but possibly moving towards the trend in,
the opportunity cost. This still indicates under-investment, and as the investment decisions
relating to building are very long-term, this may indicate continued uncertainty of the largely
white commercial farmers with respect to their future positions, in the light of past and
current land reform policy.
We now move on to consider the fixed factors in the model, shown in Table 4.
Table 4: Shadow Prices for the Fixed and Conditioning Factors
Year
Shadow Price of Land
Shadow Price of Research
Shadow Price of Patents
1971
0.023
0.0009
0.0279
1972
-0.003
0.0009
0.0283
17
1973
-0.040
0.0010
0.0339
1974
-0.038
0.0011
0.0377
1975
-0.040
0.0011
0.0435
1976
-0.037
0.0011
0.0463
1977
-0.040
0.0013
0.0532
1978
-0.040
0.0015
0.0617
1979
-0.051
0.0018
0.0846
1980
-0.050
0.0018
0.0928
1981
-0.051
0.0019
0.0999
1982
-0.066
0.0019
0.1092
1983
-0.091
0.0014
0.1458
1984
-0.095
0.0002
0.1521
1985
-0.117
0.0000
0.1681
1986
-0.136
-0.0005
0.1711
1987
-0.188
-0.0014
0.2150
1988
-0.232
-0.0034
0.2347
1989
-0.354
-0.0076
0.2424
At Var.
Means*
-0.0738
(-4.6)
0.00148
(.43)
.109
(2.05)
* t-values in parentheses
The table indicates a negative shadow price for land in the commercial sector for the period
1972-1989 and shadow prices evaluated at the variable means is negative and highly
significant. A negative shadow value for land implies that land area is not an effective
constraint to production in the commercial sector. The shadow values become a larger
negative over the period, even after the policy of land redistribution from the commercial
sector to the communal areas. Possible reasons for this include, the adoption of new chemical
and biological technologies that are effectively land substitutes (less land area required to
produce a given quantity of a given crop). This does seem to be supported with respect to the
food crop and industrial crop outputs by the negative elasticities of these outputs with respect
to land area. It is also possible that the land redistribution has largely been restricted to underutilised or low quality land in the commercial sector. Furthermore, even with the 15% or so
of land that has already been redistributed, there is still sufficient land in the commercial
sector such that it still does not represent an effective constraint to production. This result
appears to support the land reforms programme in Zimbabwe.
The shadow prices of the research knowledge stock (RKS) and the patent knowledge stock
(PKS) indicate the addition to profit for a unit increase in the stock variables. The shadow
price of research peaks in the early 1980's and diminishes thereafter, becoming negative in
the last four years. This implies the counter-intuitive result that additions to the RKS in these
years reduced profitability. This is possible if funds for research compete with other
technology or infrastructure projects, thus research funded at the cost of other potential
18
projects might show a net negative marginal product. We note, however, that the shadow
value of RKS evaluated at the variable means is insignificant.
The shadow value of the PKS at the variable means is large relative to that of the RKS but
they are not strictly comparable as the RKS relates to a stock value and the PKS relates to
patent numbers. The shadow value at the variable means is significant and represents the
addition to profits attributable largely to the research systems of other countries (i.e., spillover
effects). Some developmental research expenditure may be required to customise the
available stock of international technology for local use. Thus, the PKS may be taking some
of the credit for developmental research, resulting in a declining or even negative shadow
price for the RKS.
3.7. Returns to Research
We found above that research augmented food crop production and that there is some a priori
expectation as well as empirical evidence that research is neutral with respect to other
outputs. Assuming that research relates only to food crops, we can derive the marginal
product of the RKS from the supply equation of food crops as the partial derivative of food
crops with respect to the RKS. The marginal product of the RKS in food crop production is
found to be 2.18 (significant at the 95% level). We can use the formula in Ito (1991, p.7) to
estimate an internal rate of return (IRR), given the compilation of the RKS and its marginal
product, as:
∞
exp(r, L)= ç
0
∂Y
÷ exp(-rt)dt
∂(RKS)
25
where r is the IRR, L is the diffusion lag and the partial derivative gives the marginal product
of the RKS. Assuming a diffusion lag of 5 years (Thirtle et al, 1993) we derive an estimated
IRR to public sector research of 36%.
Thirtle et al (1993) derive an IRR of 43% using the same RKS in a primal (translog
production function) model. However, no account was taken of international spillovers in that
model, implying that some upward bias existed in the estimated IRR due to the omission of a
variable representing internationally available technology. Lastly, the shadow price of patents
is positive and significant, indicating that international spillovers are important. This cannot
be easily quantified because the series is the number of patents registered, which has no
obvious connotations, in terms of financial magnitudes.
3.8. Conclusions
The model generates supply elasticities of 0.8 for two of the three enterprise groups, which is
considerably higher than the results reported in the previous chapter. The model suggests that
the World Bank is correct, in that the marginal value product of land is negative, meaning
that there is under-utilisation. However, negative values of capital assets are common when
real interest rates are negative, so the result should be treated with some caution. The extent
of the distortions of macroeconomic variables, such as the interest rate, must have a
considerable effect on the efficiency of resource allocation in the agricultural sector. The
combination of the over-valued exchange rate and negative real interest rates would lead to
undue substitution of capital for labour. With unemployment estimated at about one million,
19
minimising employment in agriculture makes no sense at all, and has only been restricted by
the shortage of foreign exchange. Lastly, the returns to research appear to have fallen at the
end of the period and Zimbabwe is now more dependent on international spillovers of
agricultural technology.
20
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by Colin Thirtle and Yogi Khatri
21
University of Reading
Department of Agricultural Economics and Management
4 Earley Gate, Whiteknights Road
P.O. Box 237
Reading, Berks., RG6 2AR
United Kingdom
22