Exact calibration of programming models of agricultural supply against exogenous supply elasticities

1. Introduction

Since the publication of Howitt's (1995b) ‘Positive Mathematical Programming’, programming models of agricultural supply have been widely used for policy analysis. Recently, conventional positive mathematical programming (PMP) models have come under scrutiny, in part due to their inability to reproduce robust and realistic supply responses.¹ The reason is that traditional specifications of these models typically use information on a single observation (the ‘cropping pattern’) to calibrate model parameters directly controlling supply responses, and they do so in an arbitrary way (Heckelei and Britz, 2005).

The case for incorporating prior information regarding the responsiveness of activities to price changes into PMP models that rely on one observation has been made repeatedly in the recent literature (Heckelei, 2002; Heckelei and Britz, 2005). The argument is two-fold. First, PMP models, in particular positive quadratic programming models, are typically under-identified. Additional information on supply elasticities can reduce, though not eliminate, the under-identification problem. For instance, it can be used to construct quadratic cost functions for marginal activities that typically had linear cost functions in the early PMP model of Howitt (1995b), leading to erratic supply responses (Heckelei and Britz, 2005).²

More fundamentally, the information content available in a single observation is not sufficient, in principle, to infer the value of model parameters that directly control the way the model responds to changes in price conditions, because a single-year observation on activity and input levels does not provide any information on second-order properties of the objective function (Heckelei and Britz, 2000, 2005).³ All information on how producers react to changing economic conditions must therefore come from prior exogenous information (Heckelei, 2002).⁴ Model parameters that directly control these reactions should then be calibrated so that the model reproduces behaviour consistent with the prior information.⁵

The use of exogenous information on supply elasticities should not be limited to calibration models, however. When information on more than one observation is available and the programming model is estimated through generalised maximum entropy (GME) rather than calibrated to a base-year allocation, the use of (correct) prior information on supply elasticities has been shown by Heckelei and Wolff (2003) to improve the convergence of the GME estimator towards true parameter values, particularly in small samples. Similarly, under-identified models estimated by GME can easily accommodate additional constraints on implied supply elasticities whenever prior information is available.

In spite of the above arguments, so far few studies have attempted to use prior information on supply elasticities for calibration of programming models of agricultural supply (one often-cited example is Helming et al. 2001). When they have done so, these studies have often relied on ‘myopic’ calibration procedures, that is, model parameters have been chosen to calibrate against exogenous supply elasticities holding the implicit price of constrained resources constant.⁶ The reason for using a myopic calibration procedure, rather than the exact one, is mainly that each activity can then be calibrated separately from all others, greatly simplifying the modeller's task. However, ‘myopic’ parameter values have been shown by Heckelei (2002) to yield erroneous model elasticities, because changes in crop prices induce changes in the shadow values of constrained resources that are ignored by myopic calibration. In contrast, prior information on supply elasticities typically comes from econometric estimates that implicitly take into account all limitations faced by farmers, notably the land constraint (Buysse et al., 2007). The extent of the distortion caused by using ‘myopic’ calibrated values rather than ‘exact’ ones, and its relationship to the base-year allocation and the set of exogenous elasticities, has yet to be elucidated. We intend to fill this gap for two popular programming models, the Leontief-quadratic model of Howitt (1995b) and the CES-quadratic model of Howitt (1995a).⁷

One reason why information on supply elasticities has not been widely or properly used, despite the long-standing popularity of PMP models – aside from, perhaps, the limited availability and/or reliability of such information – may be that the link between mathematical programming model parameters and the model's implied supply elasticities is often difficult to elucidate. As a result, analytical expressions that relate implied supply elasticities to unknown parameters to be calibrated against said elasticities are not widely available, except for the simple Leontief-type production function with quadratic adjustment cost Heckelei (2002). Heckelei (2002) and Heckelei and Wolff (2003) argue that for more flexible models of input allocation, closed-form expressions for the model elasticities may not even exist, while Jansson and Heckelei (2008) recognise that in the general case with continuous derivatives, the implicit function theorem may be used.

Heckelei (2002) has derived a closed-form expression for the implied model elasticities in the case of the simple Leontief production function with quadratic adjustment cost. His expression was subsequently used by several authors, including Heckelei and Wolff (2003), Jansson (2007) and Jansson and Heckelei (2008). However, to the extent of our knowledge, no one has attempted to derive the general conditions under which calibration of the Leontief-quadratic model against exogenous elasticities is in fact feasible. In Section 2, we show that for a given model specification and a given base-year allocation, not all sets of supply elasticities can be reproduced. In essence, this implies that the information contained in one single observation on activity levels, though it cannot by itself determine the value of supply elasticities, does put restrictions on the set of supply elasticities that can be reproduced by positive quadratic programming models.⁸

The question that arises is then: how to characterise the sets of supply elasticities that are reproducible, given a base-year allocation? Focusing on the case where (i) one resource constraint is binding, (ii) prior information consists of values for the own-price supply elasticities, and (iii) the matrix of quadratic coefficients is diagonal, we formally derive the necessary and sufficient condition under which the specified model can, indeed, be calibrated against the supply elasticities.⁹

This condition, which we refer to as the ‘no dominant response’ rule, can be summarised as follows: no activity can have a desired acreage response that is higher than (or equal to) the sum of all others. Here the acreage response is defined as the derivative of crop acreage with respect to the gross margin per acre. The criterion explicitly relates the information contained in the base-year allocation to the prior information on supply elasticities and ensures that these two information sources are ‘compatible’.

Extending the calibration criterion to the case of more than one binding constraints is beyond the scope of this article. Nonetheless, we show at the end of Section 2 that a necessary condition to calibrate quadratic models with K binding constraints is that the number of positive activities in the base-year allocation be large enough, a condition we refer to as the ‘number of crops’ rule. In addition we provide, for the case of K binding constraints, a rule of thumb to assess whether the model elasticities implied by a myopically calibrated model are likely to differ much from the prior information.

The second contribution of the paper (Section 3) consists in proposing a novel procedure to obtain closed-form implied elasticity expressions for models where a closed-form solution to the first-order conditions of the optimisation programme does not exist, for instance, the models of input allocation proposed by Howitt (1995a) and Heckelei and Wolff (2003). The procedure is applicable to any model with crop-specific production functions. It is implemented on the CES-quadratic model of Howitt (1995a) with land constraint. Interestingly, the ‘no dominant response’ rule derived for the fixed-proportion case still applies, with a straightforward generalisation of the notion of acreage response.

In the rest of this article, the following notation is used. There are I non-zero crop activities and L allocated inputs, indexed from 1 to L. Input 1 represents land. There are K linear binding constraints, and we assume that I > K. The constraints may reflect the limited availability of inputs (land, water), technological relationships between activities, or policy constraints. The base-year conditions are described by an output price vector (p₁, … , p_I) and input prices (c₁, … , c_L). The base-year allocation is described by a vector of input and output quantities for each crop i, and a shadow price for binding constraints, ⁠.¹⁰

2. The Leontief-quadratic model

In this section, the production technology for each activity is assumed to be of the Leontief type. For crop i, define the per acre observed cost ⁠, the constant yield and the gross margin ⁠. The symbol denotes the acreage allocated to crop i, where the input index is dropped to simplify notation.

2.1. Calibration procedure

The problem then consists of choosing a set of positive coefficients so that the implied model elasticities coincide with an exogenous set of elasticities ⁠.¹¹

Equation (4) shows that the ability of the model to calibrate against the supply elasticities depends on the base-year allocation only through the information contained in the parameters ⁠, that is, the outputs and the acreages ⁠. The values assigned to the shadow prices are irrelevant to calibration of the model's supply response.

While system (4) may have multiple solutions over the unrestricted set of real-valued diagonal matrices Γ, we are only interested in solutions with positive coefficients, to ensure that program (1) has a strictly concave objective. By way of definition, we will say that the calibration problem has a positive solution if there exists a diagonal matrix Γ with positive entries that solves system (4).

In what follows, we seek to identify restrictions on the set of elasticities that ensure that a positive solution to the calibration problem exists. The non-existence of a positive solution can be interpreted as a sign that the set of exogenous supply elasticities is not consistent with the base-year observation and the model specification taken as a whole. Potential remedies include choosing a different model specification or modifying the set of supply elasticities so as to allow for calibration, while minimising the departure from the elasticity prior.

2.2. Necessary and sufficient condition for exact calibration when K = 1

In this section, we restrict our attention to the case where there is only one binding constraint, interpreted as a land constraint: ∑_i=1^Ix_i = b.¹³ The following proposition provides the necessary and sufficient condition under which calibration of model (1) against the elasticities is feasible.

Proposition 1.

Suppose that K = 1. Then, system (4) has a positive solution if and only if

(5)

When it exists, this solution is unique. It satisfiesfor alli = 1, … , I.

Proof.

Necessity. Suppose that there exists a positive solution to system (4). With K = 1, the system can be written

(6)

Therefore, we have, for each i = 1, … , I:

and

Since for all i = 1, … , I, it is apparent from these expressions that ⁠.

Sufficiency. Suppose that ⁠. In system (6), consider the change of variables ⁠. We can rewrite the system as

(7)

or, equivalently

(8)

Now consider the following function ϕ, defined on the set

as

with ⁠. Given the premise that and since for all j, it is easy to see that indeed lies in the interval ⁠, so that Δ is stable by ϕ. In addition, Δ is a compact subset of ⁠, and ϕ is a continuous function. By Brouwer's fixed point theorem, there exists a vector r in Δ such that ϕ (r) = r. For such r, we have r_i ≥ 1 and therefore ⁠.

Uniqueness. See Supplementary material, Appendix.

Proposition 1 provides the rule to determine ex ante whether, for a given base-year allocation, the quadratic programming model can be calibrated to the set of elasticities ⁠. It explicitly relates the observed allocation to the exogenous information through the quantities and ensures that the two information sources are compatible. The idea is that no single crop can have an acreage response that is larger than the sum of all others, hence the ‘no dominant response’ rule. An equivalent way of writing (5) is

so that the analyst only needs to check that the condition holds for the crop with the highest desired acreage response ⁠.

Finally, note that condition (5) implies that the number of positive activities in the base-year allocation must be larger than 2, that is, I ≥ 3.

2.3. Necessary condition for exact calibration when K ≥ 2

Ideally, the analyst should have conditions to verify ex ante whether a (unique) solution to the general calibration problem (4) exists. While it is beyond the scope of the present article to extend the results of the previous section to the case of more than one constraint, we nonetheless point out an important condition under which calibration will not be feasible. This condition relates to the number of positive activities in the base-year allocation.

To avoid cumbersome notation, we write K for the set {1, … , K}, and I_−i for the set ⁠. The following lemmas are proved in Supplementary material, Appendix. 

Lemma 1

where denotes the K × K matrix obtained by deleting the columns from matrix A, and denotes the K × K matrix obtained by deleting the columns and rows with indices from matrix Γ. The notation under the summation sign indicates that the indices must all be different.

Denote ⁠, with typical element ⁠. System (4) can then be written as

(9)

Lemma 2

Lemmas 1 and 2 imply the following proposition.

Proof.

When I = K + 1, Lemma 1 implies that ⁠, while Lemma 2 implies that ⁠. These equalities together imply that

The matrices A_−i and are square and we have

The determinant is equal to ⁠. Therefore, we can write

This expression shows that the unknown parameters enter each of the equations of system (9) solely through the quantity ⁠. Therefore, unless the quantities are all equal to each other, the system does not have a solution.

Although the model can certainly be calibrated with K + 2 activities when K = 1 (provided the necessary and sufficient condition derived in Section 2.2 holds),¹⁴ interestingly when there is more than one binding constraint K + 2 activities are no longer sufficient to consider calibration. A proof of the following proposition is provided in Supplementary material, Appendix.

2.4. Myopic vs. exact calibration

It is apparent that this system is much easier to solve than (4) since now each coefficient γ_i can be calibrated independently of all others. The implied model elasticities will differ from the prior ⁠, however, as shown in Heckelei (2002).

The following lemmas are used below to compare and ⁠.

Lemma 3

Proof.

Lemma 4

Proof.

Since for all i = 1, … , I, we write with ⁠. Lemma 1 then implies that

(11)

all of the determinants on the right-hand side being positive since the matrices are all square. Lemma 2 implies that¹⁵

(12)

Q.E.D.

Proposition 4.

If the myopic coefficientsare used, then the implied elasticitiesare smaller than the exogenous elasticitiesfor all crops.

This result is not surprising in light of Heiner (1982) and Braulke (1984).¹⁶ Reinterpreting the activities 1, … , I as individual firm outputs and the constrained resources 1, … , K as Braulke's inputs for which supply to the industry is less than perfectly elastic, his results imply that the industry's response when resources are perfectly elastic must be larger than that when resource levels are fixed in aggregate (the latter being itself larger than when resources are fixed for each activity individually). The result is also in accordance with the intuitive result that the presence of fixed factors of production at the industry level should dampen the overall industry response to output price changes, compared with the case where all productive factors are perfectly elastic.

Expression (10) shows that the error made by using the myopic values will be negligible whenever

(13)

This leads to the following proposition.

Proposition 5.

If, for all i = 1, … , I, ⁠, then the supply elasticities implied by using the myopic valuesare close to the exogenous elasticities⁠.

Proposition 5 constitutes a useful rule of thumb, because it can be used ex ante to determine the merits of using the exact calibration procedure rather than the simpler myopic calibration. Equation (10) in fact implies that represents the relative difference between the exogenous elasticity and the (smaller) model elasticity when the myopic parameters are used. For instance, if ⁠, then the value differs from by exactly 10%.

Lemma 3 further implies that a necessary condition for myopic calibration to yield model elasticities that are close to is that the number of positive activities I be large enough, and that, for given I, a larger number of binding constraints K will make the requirement of Proposition 5 more difficult to satisfy.

It is worth trying to understand how the base-year observation and the elasticities influence the magnitude of the terms and thereby the need for a more thorough calibration procedure. To gain intuition in this regard, let us consider the case where K = 1, that is, there is only one binding constraint. In this case, condition (13) simplifies to

where denotes the desired acreage response of crop j from a unit increase in the gross margin P_j. Clearly, for this condition to be satisfied for all i = 1, … , I, there needs to be a sufficient number of positive activities. Therefore, in systems with a limited number of crops, we expect the supply elasticities implied by myopic calibration to differ significantly from the desired elasticities, at least for a subset of crops.

But a large number of crops is not sufficient to ensure that condition (13) is satisfied. In addition, the quantity must be very small relative to ⁠. The criterion thus requires that the ‘contribution’ of crop i to the sum be small, this contribution being defined as the product of ⁠, a measure of activity i's per acre use of the limited resource, and ⁠, a measure of the desired acreage response of activity i to changes in its gross margin. For instance, suppose that land is the only binding constraint. Then a₁,_i = 1 for all i, and condition (13) requires that for all i, a condition that will be satisfied if (i) I is large and (ii) the number of s that are relatively large is large. This is intuitive: if one crop has a much larger desired acreage response than all others, the anticipated effect of a rise in its price on the shadow price of land will be large, and therefore the values ⁠, that by construction ignore the change in this shadow price, will lead to supply elasticities that differ significantly from the desired ones. Note however that crops for which condition (13) holds will be calibrated closely (albeit not exactly).

3. Calibration of models of input allocation

In this section, we introduce a procedure to obtain closed-form implied elasticity equations in models of input allocation, for which a closed-form solution for the optimal input allocation does not always exist. The procedure is then applied to the CES-quadratic model of Howitt (1995a) with land constraint. For this model, we explicitly derive the necessary and sufficient conditions under which the model can be calibrated against an exogenous set of supply elasticities. Interestingly, the ‘no dominant response’ rule derived in Section 2 for the Leontief-quadratic model generalises to the case of variable proportions.

3.1. Elasticity equation for general production functions

Heckelei and Wolff (2003) indicate that for models with input allocation, there may be no closed-form solution to the elasticity equation. Heckelei (2002: 77) is able to incorporate prior information on supply elasticities in such a model, but does not derive a closed-form solution for the elasticity either. Instead, he introduces I sets of ‘duplicate’ first-order conditions, each of them with a small disturbance in the price of a given crop, which force the model parameters to be consistent with the expected supply response.

Here we propose a simpler and exact procedure that allows calibration against the elasticities without the use of artificial sets of first-order conditions. The procedure overcomes the non-solvability of the model's optimal allocation functions by making use of the implicit function theorem, twice in a row, to derive an explicit expression for the supply elasticity. The derivation of elasticity equations is relevant for both calibration problems and estimation problems that seek to incorporate prior information on supply elasticities through additional constraints to the GME programme, as proposed by Heckelei and Wolff (2003). An application to popular model specifications shows that the elasticity expressions are very tractable.¹⁷

The first-order necessary conditions for profit maximisation with respect to x_jl, l = 1, … , L implicitly define the above relationships and we can apply the implicit function theorem to them to obtain the derivatives and ⁠, l ≥ 2. (Only the first of these derivatives is used subsequently.)

Fourth, the first-order conditions for profit maximisation with respect to x_il, l = 1, … , L define a system of L equations in the L variables λ₁, x_i2 , … , x_iL and the parameter p_i. One can apply the implicit function theorem to this system to obtain the derivatives and ⁠, l ≥ 2. These derivatives will include the derivative ⁠, which from the previous step has a closed-form expression when evaluated at the base-year allocation.

All the derivatives appearing in equation (14) are evaluated at the base-year allocation, have closed-form expressions, and are sole functions of the model parameters and the base-year data.

3.2. The CES-quadratic model

Here, we apply the above procedure to the CES-quadratic model of Howitt (1995a) with one binding constraint, which shall be interpreted as a land constraint. We call this model the CES-quadratic model because the specified production relationship between inputs and output is of the constant-elasticity-of-substitution form (with constant returns to scale), and calibration against the base-year allocation is achieved through the addition of a quadratic cost term for each activity. In this model, the quadratic term typically involves only one input (land) and there are no quadratic interaction terms between inputs. The choice of land as the quadratic term in the objective function can be justified heuristically by the heterogeneity of land quality (Howitt, 1995a), although calibration against supply elasticities can also be achieved, subject to the restrictions discussed below, using quadratic terms in alternate inputs.¹⁹

The calibration problem consists of choosing the set of unknown parameters so that the model exactly reproduces the base-year allocation ⁠, and the implied supply elasticity for each crop coincides with the prior value ⁠.

3.2.1. Myopic calibration

Equation (19) further shows that for a given value of σ_i, myopic calibration against will be possible whenever the base-year expenditure on inputs l ≥ 2 is small enough relative to the base-year net expenditure on input 1, defined as the difference ⁠. From (16), it is apparent that this net expenditure includes not only the average cost ⁠, but also the implicit expenditure ⁠. This suggests a simple rule for choosing ex ante which of the three inputs to use in the quadratic cost term, particularly if the ratio is low: namely the input l for which the net expenditure in the reference allocation is the largest. Of course, there is no guarantee that condition (19) will be satisfied even when one chooses the calibrating input optimally.

3.2.2. Exact calibration

Proposition 6.

In the CES-quadratic model (15), the implied acreage response of crop i, defined as the derivative of x_i1 with respect to p_i divided by the base-year yield, is equal to

The relationship in Proposition 6 is in fact true for any quadratic model with crop-specific production functions displaying constant returns to scale.²⁰ From the results of Section 2, we can readily derive the following propositions.

Proposition 7.

Suppose that I ≥ 3. If the myopic coefficients are used, the implied supply elasticities are smaller than the desired elasticities ⁠. In addition, if, for all i = 1, … , I,

then the supply elasticities implied by using the myopic valuesare close to the exogenous elasticities⁠.

Note that here, unlike in the Leontief-quadratic model, the quantity overstates the relative difference ⁠, because

Proposition 8.

4. Conclusion

In this article, we derived necessary and sufficient conditions to calibrate certain quadratic programming models of agricultural supply against exogenous sets of (own-price) supply elasticities. Focusing on models with diagonal matrix of quadratic coefficients, we showed that the number of positive activities in the reference allocation needs to be large enough for the calibration problem to have a solution. The ‘number of crops’ rule is that if only one linear constraint is binding, there should be at least three crops in the base-year allocation. If there are K ≥ 2 constraints, the minimum number of crops rises to (at least) K + 3.

Even when the ‘number of crops’ rule is satisfied, calibration may not be feasible while maintaining the strict concavity of the objective function. We derived necessary and sufficient conditions for exact calibration of two popular models, the Leontief-quadratic and the CES-quadratic, for the case K = 1. Calibration is feasible whenever there is no ‘dominant response’, in the sense that no crop has a desired acreage response larger than the sum of the desired acreage responses of all other crops. For the general case of K constraints, the use of myopic values for calibrating parameters is defendable under the conditions derived in Section 2.4. All these conditions can be routinely incorporated into PMP algorithms prior to attempting exact calibration against supply elasticities.

The second contribution of this article was to propose a general procedure to obtain closed-form expressions for the model elasticities when there are several allocated inputs. The derivation of explicit elasticity equations is relevant for both calibration models and econometric programming models where it is deemed appropriate to incorporate priors on supply elasticities through additional constraints to the GME program.

The calibration criteria derived in Propositions 1 and 8 cover only a subset of mathematical programming models. One limitation of our approach is that we do not control for cross-price elasticities. Yet, estimates of cross-price elasticities are not always available, and in that case the inability to control for their magnitude should not be seen as a fundamental flaw.²²

One important generalisation of our results would consist of deriving conditions for the calibration of models with a richer set of constraints. We hope that the methodologies presented here will serve as a template for further work in this area. Nonetheless, the case where one resource constraint – e.g. land – is binding should not, in our view, be dismissed as a mere scholarly example. The reason is that not all constraints typically introduced in programming models of agricultural supply are relevant for calibration against supply elasticities: their inclusion should depend on the nature of the econometric estimates available to the analyst. Notably, if the estimates of supply elasticities reflect, as they should, the underlying scarcity of productive factors and the limitations imposed by technology, rather than government policies, there is no reason to include policy constraints into the calibration phase.²³

Acknowledgements

The authors thank Richard Howitt, Thomas Heckelei, and three anonymous referees for useful comments on this article. All errors are ours. Financial support from CHEVRON is greatly acknowledged.

References

Author notes