Positive Mathematical Programming: a Comparison of Different Specification Rules

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1 Positive Mathematical Programming: a Comarison of Different Secification Rules Fragoso R., Carvalho M.L. and Henriques P.D. 3 University of Évora/Management Deartment/ ICAM, rfragoso@uevora.t, Évora, Portugal University of Évora /Economics Deartment/ICAM/CEFAGE, leonor@uevora.t, Évora, Portugal 3 University of Évora versity/economics Deartment, damiao@uevora.t, Évora, Portugal Abstract In this aer, the rescritive caacity of different tyes of ositive mathematical rogramming models alied to the Alenteo agricultural sector is analysed. Model results are comared for and 4 agricultural rice and subsidies scenarios, regarding otimal combination of activities. Thus, it is tested, on one hand, models caacity to reroduce Alenteo agricultural sector behaviour, and by the other hand, their resonse and adustment caacities to changes in rices and in agricultural olicy. Keywords Positive mathematical rogramming, agricultural suly, Alenteo. I. INTRODUCTION Mathematical rogramming (MP) models have been largely utilized in the area of agricultural economics, because their structure can easily suit to the economic roduction theory. Based on an otimisation criterion, these models allow reresenting agricultural roduction conditions and the analysis of the adustments from technical, economic and institutional changes []. Early alications of MP to agricultural economics aimed to solve and to analyse roblems dealing with farm lanning [ and 3]. These models are simle to formulate and very useful for understanding reality, but have some limitations in suorting decision and evaluation of agricultural olicy and rural develoment measures. These limitations are rincially due to the need of detailed information to obtain suitable coefficients describing the roduction technologies, and to the deviations in otimal from observed values [4]. In order to aroimate the results of the MP models to the observed behaviour, it is usual to add arbitrary constraints which limit their analysis otential. In this contet, Positive Mathematical Programming (PMP) made u a feasible alternative that allows to automatically calibrating the models without additional constraints [5]. The resulting model is able to resond more smoothly to changes in arameters, so that it is more consistent with changes on observed behaviour. This technique can be understood as a comromise between econometric models and MP models, because arameterization is done based on observed behaviour, as for econometrics, and rimal solution ehibits an elicit secification of technology, as done in any MP model. Recently, the PMP methodology has been often used in the study of economic, social and environmental roblems, like those of modelling the Common Agricultural Policy. The obective of this aer is to evaluate the calibration and rescrition caacity of a suly resonse PMP model for the Alenteo region. The model will be calibrated for rices and agricultural subsidies of the base year ( scenario), using different secification rules of the cost function. Then, the model is utilized for the rescrition of the results for the scenario of rices and subsidies of 4. The aer is organised in more four sections regarding the PMP and cost function secification rules, the develoment of an agricultural model suly resonse for the Alenteo, results and finally conclusions. II. POSITIVE MATHEMATICAL PROGRAMMING AND COST FUNCTION SPECIFICATION RULES Even before its formal resentation [5], PMP had been emloyed in modelling economic roblems alied to the agricultural sector [6, 7, 8, 9]. After the article of Howitt [5], it was clear the interest with its use, and new develoments have intensified its interest [,,, 3, 4, 5, 6]. th Congress of the Euroean Association of Agricultural Economists EAAE 8

2 PMP uses the information contained in dual variables of the constraints of a rofit maimization LP roblem, which bound activities to observed levels. These dual variables are used to secify a non-linear obective function such that the otimal solution will reroduce the observed activity levels. The emirical rocedures of the PMP roblem consist of two hases, comrising the estimation of the calibration arameters (hase I), and the secification of a nonlinear obective function (hase II). In hase I the calibration constraints are used in order to force the LP model solution to the observed activity levels: ma Z = ' - c' s.t. A b [ λ] () o +ε ρ ( )[ ] Where: Z = obective function value reresenting the farm rofit; = (n) vector of roduct rices; c = (n) vector of variable costs er unit of activity; = (n) vector of roduction activity levels; A =(mn) matri of coefficients in resource constraints; b = (m) vector of available resources; λ = (m) vector of dual variables associated with the resource constraints; = (n) vector of observed activity levels; ε = (n) vector of a small ositive numbers to avoid a degenerate solution; ρ = dual variables associated with calibration constraints. The level of at least one of the activities in the LP model is not bounded by its calibration constraint, but for one of the fied resources constraint. In this way, the vector can be divided into a vector of referable activities ( ) bounded by the calibration constraints, and a vector of marginal activities ( m ), which are constrained by the resource constraints. The Kuhn- Tucker conditions are: ρ = - c - A λ () ρ m = [] (3) ( m - c m )( A m ) - λ = (4) Dual value of the calibration constrains for referable activities, for marginal activities and for resource constraints area given by the equations (), (3) e (4), resectively. In hase II, the dual values of the calibration constraints, ρ, are used to secify a non-linear obective function, such that the marginal cost of the referable activities are equal to the resective rice at the base year observed activity levels,. Given these conditions, the model should reroduce eactly the vector,. The quadratic cost function is often utilized for comutational simlicity and because it fits well to the hyothesis of decreasing returns in agricultural roduction: c '= d' + 'Q (5) Where d = (n) vector of arameters associated with the linear term; and Q = (nn) symmetric, ositive definite matri of arameters associated with the quadratic term. The linear marginal variable cost function is the sum of linear costs, c, and marginal costs, ρ: ( o ) v v C Cm = = d + Q o = c +ρ (6) Given d and Q, the non-linear rogramming roblem that reroduces the observed activity levels is: ma Z = ' - d' - 'Q s.t. A b [ λ] (7) The condition Cm = c + ρ imlies an undetermined system associated to an infinite resonse atterns. Trying to avoid arbitraries simulations on resonse behaviour, several methods for secification of the arameters d and Q of the variable cost function have been develoed [4]. A short overview of some of these methods is given. In the early utilizations of PMP, the secification roblem of the quadratic cost function was solved by doing d=c and setting equal to zero all off-diagonal elements of Q matri. In this aroach called standard secification the diagonal elements of Q, q i, were calculated as: ρ q = =,,...,n (8) Since ρ m =, the standard secification rule leads to a cost function which is linear in marginal activity. th Congress of the Euroean Association of Agricultural Economists EAAE 8

3 3 This imlies that a rice change of a referable activity only leads to a substitution of the marginal activity. The advantages of this method are basically on the simlicity of the secification and on ease comutational, mainly, when available information is shortened. Paris [7] used an alternative secification rule (Paris standard) where the arameter d of the cost function is equal to zero and the elements of the Q matri are calculated as a function of the observed elicit costs in the base year, c, and of the dual values of the calibration constraints, ρ. d = q = c +ρ =,,...,n (9) Diagonal elements of Q for marginal activities are all ositive. So, a change of a referable activity is done not at the eense of the marginal activities, but of the other referable activities. Other secification of the cost function, named by average cost, assumes that the observed vector of the accounting cost er activity unit in the base year, c, is equal to the average cost of quadratic variable cost function: ρ q = =,,...,n () d = c ρ =,,...,n In this aroach, the diagonal elements of Q are larger than those obtained from the standard rule in (8), what imlies smaller imlicit elasticities, but the roblem of the marginal activities with constant returns remains. Another aroach that allows the incororation of rior information is the eogenous suly elasticities. Being / equal to q -, then rice elasticity for activity is calculated by: ε = q =,,...,n The arameters q and d of the cost function are determined as: q = ε d = c +ρ =,,...,n q =,,...,n () III. REGIONAL MODEL OF AGRICULTURE SUPPLY FOR ALENTEJO REGION In order to analyse the rescrition caacity of the considered secification rules for the cost function, a PMP model adated to the regional characteristics of the Alenteo region was develoed. The simlified formulation of this model is Ma Z = X + a X + i Y i + a i Y i i i - c - Y X - qx - ciyi qii i () i i - ht - i E Subect To e f Y i X f (3) i X s *, X set (4) s X b s (5) h X + h i Y i b t + T (6) i c X + c i Y i b c +E (7) i Where: X and Y i are the decision variables concerning the area of cro activities in hectares (ha) and the size of livestock activities i in livestock units; T and E are the overtime working units and the additional oeration caital units;, a, c, and h are, resectively, the outut value, subsidies, variable costs and work needs er unit of activity end i; h and i are the hour cost of T and the annual loan interest rate of E; e if are the livestock stocking rates; and bs, bt and bc are the fied resources land, work and caital availability. The obective function () maimizes the gross margin in euros and it is calculated by the difference between revenue and total variable costs. The revenue th Congress of the Euroean Association of Agricultural Economists EAAE 8

4 4 includes agricultural outut value and the direct subsidies. The variable costs comrehend short time linear inut costs (c and ci), costs with overtime working (h) and oerating caital (i) and also marginal costs coefficients of activities (q and qii). Decision variables in the model include eighteen agricultural activities of the Alenteo Region between cros and livestock activities. Cro activities comrise cereals and oil seeds, horticulture and fruit culture, fruit trees, vineyards, olive tree, ermanent astures, forage, comulsory set-aside, fallow and an activity regarding land occuied by forests. Livestock activities comrehend beef cattle, shee and etensive swine. Permanent astures and forages are intermediate activities because they are not sold but are an inut for livestock activities. So, these activities only have costs, being their rofits indirectly obtained from animal activities. The rofit transfer between activities is done essentially by equation (3), which defines the balance between forage areas (X f ) and the total number of animals. Equation (4) models the set-aside (X set ) imosed by CAP. This equation states that % of the cro area (X s ) has to be retired from roduction and ut in setaside. Equations (5) to (7) stay for the use of land, labour and caital. These equations state that the resource demand is less than or equal to their availability. In site of the obective function reresent the return to land, labour and caital, model solution is limited only by land availability in (5). Labour (6) and caital (7) demand can eceed their availabilities by urchasing additional hours of labour at an hour cost of 3.5 and additional units of caital at an annual loan interest rate of 7%. IV. RESULTS The results of PMP model of agriculture suly of the Alenteo is obtained for each one of the secification rules of the cost function. First, the PMP model is calibrated for the base year (). Then, rices and subsidies vectors are changed and the model is used for rescrition of results for 4 scenario. In both scenarios, results are comared to available data for the Alenteo region, concerning cro areas and the number of livestock units. For the base year the model reroduces eactly the observed level of the activities, whatever the secification rule of the cost function used. The different secification aroach of the cost function give the same results, because the condition Cm = c + ρ constitutes an undetermined system. So, there are an infinite number of values for the arameters q e q ii satisfying the conditions of the PMP roblem. Table resents the absolute deviation to activities observed levels in 4. This table also resents the total weighted absolute deviation, which have in account the relative weight of each cro on the total land and of each animal activity on the total livestock unit. Table Absolute deviation on the activity levels for 4 (%) Cro activities Standard Paris Average Eogen. Standard Cost Elasticit. Common Wheat Durum Wheat Maize Rice Horticulture Sunflower Olive trees Vineyard Fruits Permanent astures Forage Fallow Forests Set-aside Beef cattle Shee Swine WADC WADA Source: Results of PMP models The results obtained for the 4 scenario show that the rule of eogenous elasticities is suerior to the others. The weighted absolute deviations are smaller on cro activities (7.3%) (WADC), and on animal activities (7.8%) (WADA). For Paris Standard rule the deviations are 8.3% on cro activities and 4.4% on th Congress of the Euroean Association of Agricultural Economists EAAE 8

5 5 animal activities. Standard and average cost rules have weighted absolute deviations resectively of 39.% and 6.% on cro activities, and 88% and 63.4% on animal activities. These results show the oor rescrition caacity of these two methods. When the eogenous elasticities aroach is used only three activities resent an absolute deviation above the 5% indicated by Hazell & Norton [8], as the maimum value for a desirable calibration. These activities are rice (-3.5%), sunflower (76.4%) and vine (-48.6%). The observed values, in terms of area, for and 4, of those activities do not change, only the area of vineyard had a light increase. For Paris standard rule there are si activities resenting absolute deviations above the 5%, four cro activities and two livestock activities. Particularly big are the absolute deviation registered on the area of common wheat (9.9%) and of sunflower (63.7%). Concerning livestock activities, the absolute deviation of.9% on beef cattle determines an increase of this activity bigger than that have actually haened in the beef cattle sector. Regarding standard and average cost secification rules, ten activities resent absolute deviations above the 5%, being articularly big on animal activities. For instance, etensive swine roduction registered an absolute deviation of more than %. Along with these deviations, there are also big absolute deviations on intermediate of asture and forage. The variability of the obtained results with the different secification rules of the cost function can be elained by the imlicit suly elasticities in each one of the cro or animal activity (Table ). In general, the results obtained from the secification rule of eogenous elasticities and of Paris standard resent smaller values, in average, in imlicit suly elasticity, such that these are the rules that ehibit the best rescrition caacity of the results on 4 scenario. The secification rules standard and average cost resenting the biggest, in average, imlicit suly elasticities on activities, and showing results far from observed reality, have the oorest rescrition caacity. Table - Suly elasticity of agricultural activities Cro activities Standard Paris Average Eogen. Standard Cost Elasticit. Common wheat Durum wheat Maize.... Rice Horticulture and Fruit culture Sunflower Olive tree Vine Fruit culture Set-aside Forests.... Beef cattle Shee Swine Source: PMP model results V. CONCLUSION Mathematical rogramming (MP) models have been largely utilized in the area of agricultural economics, because their structure can easily suit to the economic roduction theory. In general, MP models area aimed to evaluate economic, technical and institutional scenarios, imlying changes in rices, technologies and available inuts. Their quality is checked by the sensitivity and ost-otimal analysis to changes in their coefficients. In this contet, this aer evaluates the calibration and rescrition caacities of a PMP model, develoed for the agriculture suly conditions of the Alenteo region. The considered cost function secification rules were standard, Paris standard, average cost and eogenous elasticities. The results showed that the PMP model reroduces eactly the observed activity levels on the base year, whatever the rule used to secify the cost function. This roerty is due to the condition Cm = c + ρ and to the functional form of the cost function. There are an infinite number of arameters satisfying the conditions of a non-linear PMP roblem. Regarding the rescrition caacity of future results, PMP revealed being a feasible methodological otion, mainly if eogenous elasticities or Paris th Congress of the Euroean Association of Agricultural Economists EAAE 8

6 6 standard aroaches were used to secify the cost function. Secification rules of cost function based on standard method or average cost method showed a smaller rescrition caacity of future results. These methods resent, in average, big imlicit suly elasticities on agricultural activities. We can conclude that the roerties of PMP do not only ehaust ust in the eact calibration of the agriculture suly models. Those roerties also resect the rescrition caacity of future results. In this case, the eogenous elasticities aroach showed being suerior to the others, even though Paris Standard method be also a good alternative. REFERENCES. Mccarl B. And Sreen T. (98) Price Endogenous Mathematical Programming as a Tool for Sector Analysis. American Journal of Agricultural Economics, Throsby C.D. (974) New methodologies in agricultural rodution economics: A review. In The Future of Agriculture. Paers and reorts, 5 th international Conferece of Agricultural Economics, S. Paulo, Brazil, 974, Martin L.R. (977) A Survey of Agricultural Economics Literature vol. University of Minnesota Press, Minneaolis. 4. Heckelei T., Britz W. (5) Models Based on Positive Mathematical Programming: State of the Art and Further Etensions, EAAE Proc. 89 th Euroean Seminar of the Euroean Association of Agricultural Economists, Parma, Italy, 5, Howitt R. (995) Positive Mathematical Programming. American Journal of Agricultural Economics, 77 (): House R. (987) USMP Regional Agricultural Model. Washington DC: USDA. National Economics Division Reort, ERS, Kasnakoglu H., Bauer S. (988) Concet and Alication of an Agricultural Sector Model for Policy Analysis in Turkey. In: Agricultural Sector Modelling. S. Bauer and W. Henrichsmeyer (Eds.), Vauk Verlag: Kiel. 8. Bauer S., Kanakoglu H. (99) Non Linear Programming Models for Sector Policy Analysis. Economic Modelling 7: Horner G., Corman J., Howitt R, Carter C., Macgregor R. (99) The Canadian Regional Agriculture Model: Structure, Oeration and Develoment. Agriculture, Canada. Technical Reort /9, Ottawa.. Arfini F., Donati M., Zuiroli M., Paris Q. (5) Eost evaluation of set-aside using Symmetric Positive Equilibrium Problem. EAAE Proc. 89 th Symosium of the Euroean Association of Agricultural Economists, Parma, Italy, 5.. Cyris C. () Positiv Mathematische Programmierung (PMP) im Agrarsektormodells RAUMIS. Dissertation, University of Bonn.. Gohn A., Chantreuil F. (999) La rogrammation mathématique ositive dans les modèles d eloitation agricole. Princies et l imortance du calibrage. Cahiers d Economie et de Sociologie Rurales 5: Barkaqui A., Butault J. (999) Positive Mathematical Programming and Cereals and Oilseeds Suly with EU under Agenda. Proc. 9 th Euroean Congress of Agricultural Economists, Warsaw, Baskaqui A., Butault J., Rousselle J. () Positive Mathematical Programming and Agricultural Suly within EU under Agenda. EAAE Proc. 65 th Euroean Association of Agricultural Economists, Bonn, Vauk Verlag Kiel, :. 5. Graindorge C, Henry B., Howitt R. () Analysing the effects of Agenda using a CES Calibrated Model of Belgian Agriculture. EAAE Proc. 65 th Euroean Association of Agricultural Economists, Bonn, Vauk Verlag Kiel,, : Helming J., Peeters L., Veendendaal P. () Assessing the Consequences of Environmental Policy Scenarios in Flemish Agriculture. EAAE Proc. 65 th Euroean Association of Agricultural Economists, Bonn, Vauk Verlag Kiel,, : Paris Q. (988) PQP, PMP, Parametric Programming and Comarative Static. Chater in Notes for AE 53. Deartment of Agricultural Economics, University of California, Davis. 8. Hazell P., Norton R. (986) Mathematical Programming for Economic Analysis in Agriculture. Mac Millan Publishing Comany, New York. th Congress of the Euroean Association of Agricultural Economists EAAE 8