Positive Mathematical Programming for Agricultural and Environmental Policy Analysis: Review and Practice
Application of PMP when more data are available
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- 5.2Constraints on input and output quantities
- 5.3Functional form
- 5.4Aggregation issues
- 6.Conclusions
- Acknowledgements
5.1Application of PMP when more data are availableSPEP (Paris, 2001) is an example of extension of PMP to a full sample of farms sharing the same technology. In that case, the amount of information is considerably higher than in the typical single farm case of PMP. Yet the method is designed for only one year of data – a cross section. It is a strong hypothesis to assume that differences in output prices across farms in a cross section do indeed reveal the supply curve. More likely, differences in prices reveal differences in products, possibly local marketing conditions or differences in quality. Figure # -5 shows a plot of price versus quantity produced across the year 2000 FADN sample of winter wheat producers in Belgium. If such a sample is used in SPEP to extract a cost function, and the marginal cost is set equal to the price, the supply curve slope is negative. 
Figure #-5. Price and output of wheat (including by-products) in Belgium (2000) and least-squares regression On the other hand, it is certainly true that when a producer expects the price of a certain crop to rise relatively to the other crops, he will increase production. In other words, observing the farm at different points in time seems important. This points out to panel data estimation, for which FADN data are suitable.
Some inputs are nevertheless truly quantity-constrained at the farm level, for example family labour3 for obvious reasons but also pesticide or fertilizer uses because they may be limited by law. Quotas, such as those existing in the milk and sugar sub-sectors, may also be binding at farm level, although it seems that in most EU countries, they can be traded. This means that, similarly to applications of PMP at regional level, there may be limiting resources that affect estimation of the parameters of the cost function at farm level. When there is no price for additional units of one resource, the marginal cost of producing one output is not anymore equal to that output price and the shadow price of the limiting resource is used to modify the marginal cost as in equation (6). Therefore, to extend PMP to a sample of farms, constraints on input quantities are relevant for issues such as pesticide use and manure production, but these topics have yet to be addressed. 5.3Functional formThe original quadratic cost function of PMP, although quite simple, allows for simultaneous production of several outputs. This is a necessity in agricultural modeling, where most farms supply more than one product. Following Mundlak (2001), such diversification may have four causes: interdependence in production, fixed inputs, savings due to vertical integration, and risk management. The simplicity of the PMP quadratic cost function is, however, obtained by suppressing all input prices from the cost function, leaving only the output quantities and some quantity-constrained inputs. All the inputs that are not quantity-constrained are implicitly used in fixed proportions to the quantity-constrained ones (most often: land). Regarding the PMP applications in the EU, some specificities of the FADN sample with respect to inputs are noteworthy. First, data on land use and land price are available per farm and per output, to some extent that is also true for fertilizers. For other inputs, such as pesticides, seeds and hired services, only the expenses per output are known not the quantity. Other inputs, such as capital, labor and machinery, are not allocated per output. Multi-product cost functions developed in the literature (e.g. Khumbakkar, 1994) are designed only for the last type of inputs. Because the FADN farm-level data holds much richer information, there is scope and need to develop a cost function (or equivalently a profit or production function) that exploits fully this information.
An aggregate farm results from summing all the farms in a sample. This aggregate farm is always more diversified than any farm in the sample. Therefore, the cost function that can be calibrated from such an aggregate farm bears little resemblance to the cost function that is extracted from the individual farm. With farm-level data, there is a serious problem of heterogeneity in the sense that few farms produce the same products: this is the selection problem mentioned in section 3. Selection causes zero production for some products leading to two problems. First, the cost function must accommodate true zeroes. Second, it is necessary for simulation that the parameters of the cost function are estimated for all the outputs for all the farms in the sample. Hence, some hypothesis must be made regarding the homogeneity of the sample: can we use for some farm parameter values that have been estimated on the basis of the production of other farms? An additional aggregation issue is that in any sample, most farms are involved in a series of activities whose output levels are very limited. It is unclear whether those activities really belong to the core economic activities of the farm because they may be experimental or heavily regulated (such as tobacco). The question is whether to remove such activities from the farms or to aggregate them. The former option may seem dramatic, but the total farm area and income in fact virtually do not change. The later option may appear more cautious, but induces a strong heterogeneity. Generally speaking, aggregating within a farm causes heterogeneity in the sample because an output that is seemingly identical across farms may appear with widely different prices and technical characteristics. 6.ConclusionsPMP has renewed the interest in mathematical modelling for agricultural and environmental policies for several reasons. The main advantages of the PMP approach are the simplicity of the modelling of bio-economic constraints or policy instruments, the smoothness of the model responses to policy changes and the possibility to make use of very few data to model agricultural policies. In this review paper, the focus has been on farm-level data (Heckelei and Britz, 2005, supply additional insights for applications of PMP with regional data). The individual farm-based sector model SEPALE is an illustration of how PMP can be used with large farm-level samples. This model not only makes it possible to account for the individual farm structure, but also for the direct payment entitlement trade mechanisms. The results prove the relevance of the model for simulating possible alternatives to the implementation of the CAP Mid Term Review, but this example is certainly not limitative as other applications of the model (Henry de Frahan et al., 2003; Buysse et al., 2004) have already shown. The possibility of distinguishing the effects according to farm size or other criteria such as region or farm type is one of the main advantages of the individual farm-base modelling. It also opens avenues to model structural changes of the sector. Although already widely applied, as illustrated by the many references in this review and elsewhere (see Heckelei and Britz, 2005), PMP is still developing and each new application raises new questions and challenges. In section 5, some of the pending issues have been mentioned, but this is certainly not an exhaustive list. One can think, for example, about the inclusion of risk or other behavioural parameters in the model or about the extension of the model with environmental parameters. At the farm level, strong hypotheses must be maintained for PMP to be operational. The basic shortcoming, when considering large farm-level samples such as the FADN, is that PMP only makes use of a single data point and imposes considerable structure on the technology as embodied in the cost function. It disregards all the information that are present when considering several years of data (time series) or when the data on several farms can be pooled together. As reminded by Heckelei and Britz (2005), one observation of activity level on one farm is not enough to estimate how that farm could respond to changing economic conditions. Additionally the quadratic cost function used in standard PMP is not flexible and may constrain the farm behaviour in various ways. In particular, it could be “too smooth” with respect to reasonable expectations, as shown by Röhm and Dabber (2003). When large farm-level datasets are available, econometric estimation of general flexible functional form cost functions should solve these problems, but will pose others, especially regarding the regularity properties of those cost functions (see Wolff et al., 2004). The challenge is to maintain the flexibility of the PMP approach, in particular for the modelling of bio-economic constraints, in an econometric model that can better capture the information contained in large panel datasets.
ReferencesBritz W., Heckelei, T., and Wolff, H., 2003, Symmetric Positive Equilibrium Problem: A Framework for rationalizing Economic Behavior with Limited Information: Comment, American Journal of Agricultural Economics, 85(4): 1078-81. Buysse, J., Fernagut, B., Harmignie, O., Henry de Frahan, B., Lauwers, L., Polomé, P., Van Huylenbroeck, G., and Van Meensel, J., 2004, Modelling the impact of sugar reform on Belgian Agriculture, Selected paper presented at the International Conference on Policy Modelling, Paris, 30 June - 2 July, 2004. Gohin, A., and Chantreuil, F.,.1999, La programmation mathématique positive dans le modèles d'exploitation agricole: principes et importance du calibrage, Cahiers d'Economie et Sociologie Rurales, 52: 59-78. Golan, A., Judge, G., and Miller, D., 1996, Maximum Entropy Econometrics., Wiley, Chichester UK. Graindorge C., Henry de Frahan, B., and Howitt, R., 2001, Analysing the Effects of Agenda 2000 Using a CES Calibrated Model of Belgian Agriculture, in Heckelei T., Witzke, H.P., and Henrichsmeyer, W. (eds.) Agricultural Sector Modelling and Policy Information Systems, Vauk Verlag, Kiel.. Heckelei, T., and Britz, W., 2000, Positive mathematical programming with multiple data points: a cross-sectional estimation procedure, Cahiers d'Economie et Sociologie Rurales, 57: 28-50. Heckelei, T., and Britz., W., 2001, Concept and Explorative Application of an EU-wide, Regional Agricultural Sector Model, in Heckelei T., Witzke, H.P., and Henrichsmeyer, W. (eds.) Agricultural Sector Modelling and Policy Information Systems, Vauk Verlag, Kiel. Heckelei, T., and Britz, W., 2005, Models Based on Positive Mathematical Programming: State of the Art and Further Extensions, Plenary paper presented at the 89th EAAE Seminar – 3-5 February 2005 – Parma. Paper available at www.unipr.it/arpa/dipseq/EAAE/PR/Homepage.htm Heckelei, T., and Wolff, H., 2003, Estimation of constrained optimisation models for agricultural supply analysis based on generalised Maximum Entropy, European Review of Agricultural Economics, 30(1): 27-50. Helming, J.F.M., Peeters, L., and Veendendaal. P.J.J., 2001, Assessing the Consequances of Environmental Policy Scenarios in Flemish Agriculture, in Heckelei T., Witzke, H.P., and Henrichsmeyer, W. (eds.) Agricultural Sector Modelling and Policy Information Systems, Vauk Verlag, Kiel.. Henry de Frahan, B., Elouhichi, K., Harmignie, O., Polomé, P., Van Huylenbroeck, G., Buysse, J., Lauwers L., and Fermagut, B., 2004, La Politique Agricole Commune : une analyse de la réforme récente, Regards économiques, 19: 2-13. Howitt, R.E., 1995a, Positive mathematical programming, American Journal of Agricultural Economics, 77(2): 329-342. Howitt, R.E 1995b. A Calibration Method for Agricultural Economic Production Models, Journal of Agricultural Economics, 46(2): 147-159. Just, R.E., and Pope, R.D., 2001, The Agricultural Producer: Theory and Statistical Measurement, in Gardner, B.L., and Rausser, G.C., (eds.), Handbook of Agricultural Economics Volume 1A - Agricultural Production, Elsevier, Amsterdam. Judez, L., Chaya, C., Martinez, S., and Gonsalez, A.A., 2001, Effects of the measures envisaged in 'Agenda 2000' on arable crop producers and beef and veal producers: an application of Positive Mathematical Programming to representative farms of a Spanish region" Agricultural Systems, 67: 121-138. Kumbhakar, S. C., 1994, A Multiproduct Symmetric Generalized McFadden Cost Function., Journal of Productivity Analysis, 5: 349-57. Mittelhammer, R.C., Judge, G.G., and Miller, D.J., 2000, Econometric Foundation., Cambridge University Press. Mundlak, Y., 2001, Production and supply, in Gardner, B.L., and Rausser, G.C., (eds.), Handbook of Agricultural Economics Volume 1A - Agricultural Production, Elsevier, Amsterdam. Paris, Q., 1997, CAPRI Meeting Follow-up, CAPRI Workshop in Reggio Emilia, Italy, May 1997. Paris, Q., 2001, Symmetric Positive Equilibrium Problem: A framework for rationalizing economic behavior with limited information, American Journal of Agricultural Economics, 83(4): 1049-1061. Paris, Q., and Arfini, F., 2000, Frontier Cost Functions, Self-Selection, Price Risk, PMP and Agenda 2000, Eurotools Working Papers Series, No. 20, May 2000. Paris, Q., and Howitt, R.E., 1998, An analysis of ill-posed production problems using Maximum Entropy, American Journal of Agricultural Economics, 80(1): 124-138. Röhm, O., and Dabbert, S., 2003, Integrating Agri-Environmental Programs into Regional Production Models: An extension of positive mathematical programming, American Journal of Agricultural Economics, 85(1): 254-65. Wolff, H., Heckelei, T., and Mittelhammer, R.C., 2004, Imposing Monotonicity and Curvature on Flexible Functional Forms, Selected Paper prepared for presentation at the American Agricultural Economics Association Annual Meeting, Denver, Colorado, August 1-4, 2004. 1 In short, the maximum entropy approach consists in estimating parameters regarded as expected values of associated probability distributions defined over a set of a priori discrete supports (Golan et al., 1996). 2 In other words, the ‘estimated’ value of the dual vector cannot converge to the true dual vector as more observations are added because PMP always selects the highest possible value for the dual vector . 3 Family labor must be considered separately from hired labor because it is immune to moral hazard. 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