Hybrid linear programming to estimate CAP impacts of flatter rates and environmental top-ups

Transcription

1 AUA Working Paper Series No ?? September 2010 Hybrid linear programming to estimate CAP impats o latter rates and environmental top-ups The ase o arable ropping in Kopais plain, Greee Stelios Rozakis Department o Agriultural Eonomis & Rural Development Agriultural University o Athens s.rozakis@aua.gr Agriultural University o Athens Department o Agriultural Eonomis & Rural Development

2 Hybrid linear programming to estimate CAP impats o latter rates and environmental top-ups The ase o arable ropping in Kopais plain, Greee Stelios Rozakis Agriultural Eonomis and Rural Development Department Agriultural University o Athens Iera Odos 75, Athens GR Tel. : , Fax : ABSTRACT This paper examines evolutions o the Common Agriultural Poliy (CAP) deoupling regime and their impats on Greek arable agriulture. Poliy analysis is perormed by using mathematial programming tools. Taking into aount inreasing unertainty, we assume that armers pereive gross margin in intervals rather than as expeted risp values. A bottom-up hybrid model aommodates both proit maximizing and risk prudent attitudes in order to aurately assess armers response. Marginal hanges to rop plans are expeted so that latter single payment rates ause signiiant hanges in inomes and subsidies. Nitrogen redution inentives result in moderate hanges putting their eetiveness in question. Key-words: Interval Linear Programming, Min-Max Regret, Common Agriultural Poliy, Arable ropping, Greee JEL lassiiation : C61, D81, Q12, Q18 Aknowledgments The author would like to thank the Laboratoire d'analyse et Modélisation de Systèmes pour l'aide à la Déision - LAMSADE in University o Paris IX Dauphine or unding his sabbatial stay (April 2010) during whih this researh was undertaken. Valuable remarks by George Vlahos, Stathis Klonaris and an anonymous reeree as well as prooreading by John Stedman are appreiated. 1

3 Introdution Ater several periods o implementation o CAP 2003 reorm it is questionable i important objetives suh as re-alloation o subsidies to the beneit o low-inome armers, enhanement o viable and diverse ativities in rural areas, ood seurity and environmental preservation are attained. Disussion on the CAP uture beyond 2013 has started, mainly driven by budgetary restraint priorities. The ost o the CAP is subjet to severe ritiism, imposing strit aountability on soial and environmental ost eetiveness. Nevertheless, various events suh as signiiant derease o arm inomes due to prie dereases, the eonomi risis and ood shortages, arguments on the soial role o agriultural ativities and assoiated externalities have attenuated the risk o adoption o propositions or drasti derease o resoures earmarked to CAP. However, even i the total amount o subsidies remain onstant, a re-alloation among member ountries and/or ativities seems inevitable. As a matter o at, there are signiiant deviations among EU members i payments reported on an area basis. Greee appears to reeive rom pillar I an average o 544 /ha when the mean payment in EU15 amounts at 295/ha with 185/ha or the 12 late member states. For these reasons various studies have been undertaken to evaluate impats o dierent poliy measures to replae the urrent single arm payment regime. A omprehensive analysis in the ontext o the Health Chek (EC, 2007) on behal o the European Commission alulates impats on alloation o the Net Value Added at the arm level in the EU25 or main produts using Farm Aounting Data Network (FADN) data. Despite its broad sope and valuable results this study onstitutes an aounting assessment without taking into onsideration armers response to restruturing o the ropping plan or minimizing the negative impats o poliy measures on their welare. In order to obtain reliable estimates useul or poliy analysis, appropriate setor and regional models are required. Classi analytial tools, suh as rop supply and proit untions used or deriving onditional arm inome estimates and ator demand untions, require onsiderable amounts o data to estimate all ross-prie supply elastiities. Moreover eonometri estimates are valid only or the observed range o variation o relative pries and other variables. Mathematial models may ill this gap and derive response untions or output, inomes, 2

4 employment and other variables impliitly by means o parametri optimization (Kuther and Norton, 1982). Espeially in ase o substantial poliy hanges suh as the Common Agriultural Poliy latest reorm, with deoupling subsidies rom prodution, mathematial programming models have been widely suggested to agriultural eonomists (Salvatii et al., 2000). In Europe, suh setor and regional models have been used to estimate impats o CAP through subsequent poliy hanges in the last ten years (i.e. Akril et al., 2001, Sourie et al., 2001, Wilson et al., 2003, Guindé et al., 2005). In Greee, examples inlude analyses ousing on the tobao and otton, staple rops that absorbed major alterations, ollowing onventional linear programming (Mattas et al., 2006), multi-riteria methods (Manos et al. 2009) and also positive models inorporating downward sloping demand (Rozakis et al., 2008) or inreasing ost untions (Positive Mathematial Programming (PMP), Petsakos and Rozakis, 2010) in the objetive untion. Multi-riteria methods with non-interative eliitation o the utility untion and PMP have dominated the reent literature onerning CAP analysis. These methods, broadening eonomi rationality, manage to transorm the objetive untion so that optimal solutions inlude not only rop plans on the verties o the easible polyhedron but also points on hyper-plans enabling the model to approah observed levels o ativities, thus outperorming its LP ounterparts. Alternatively, risk inorporation into the model may also yield optimal plans beside easible polygon verties. A review o methods introduing risk in mathematial programming an be ound in Hardaker et al (2004). One ould mention the E-V model as well as its linear versions suh as MOTAD and target-motad and also models based on game theory reasoning suh as maximin, minmax, saety-irst and other models that seek eiient diversiiation among ativities as a means o hedging against risk (or early appliations in the Greek ontext see Manos and Kitsopanidis, 2006, 2008). For all these models in order to introdue non-linear risk-related terms in the objetive untion, availability o ovariane matries that require gross margins o individual rops related to dierent states o nature or years- is undamental. Consequently, it is extremely diiult to apply these methods to setor or regional models ontaining numerous arms, thus relevant publiations while theoretially appealing are applied to only a limited number o representative arms (Petsakos et al., 2008) or to limited ativities or produts (Katranidis & Kotakou, 2008). 3

5 In this exerise we opted or a suiiently detailed tehno-eonomi representation at the arm level ontaining a priori inormation on tehnology, ixed prodution ators, resoure and agronomi onstraints, prodution quotas and set aside as well as environmental regulations, along with expliit expression o physial linkages between ativities. A bottomup approah is adopted to relet the diversity o arable agriulture, artiulating numerous arm sub-models in a blok angular orm (Williams, 1999), that have neither the same produtivity nor the same eonomi eiieny so that the prodution osts are variable. Thus, ex-post aggregation helps to relax the proportionality hypothesis o LP (onerning the Leontie tehnology) and to avoid problems suh as disontinuous response and overspeialization arising in single representative arm models. Moreover, we attempted to relax the ertainty assumption inorporating risk onsiderations o the deision makers, in this ase armers, or two important reasons. Firstly, under deoupling reorm muh more than beore, prie and yield variations inluene gross margins, as no rop speii subsidies exist anymore. Seondly, and more importantly, the radial inrease o ereal pries o 2007 ollowed by their ollapse in 2008 boosted prie volatility. This situation obliges modelers to pay speial attention to unertainty o pries, whih ombined with the vagaries o nature and the new institutional environment, make armers very autious. As our intention is to use large samples o arms, we seleted a novel method that is not data greedy, namely interval LP. The unertainty element in the objetive untion is brought about via the introdution o intervals in the gross margin oeiients in the objetive untion. To speiy intervals the sole requirement is an idea o gross margin variation range. It is proved that interval linear programming (ILP) models are equivalent to a speii lass o multi-objetive (MO) models with objetives generated by the extreme interval values. Consequently, there is a need to selet an appropriate riterion to resolve the MO problem and obtain a ompromise solution. By means o experiments, an attempt was suessively made to all elementary arm models to hek whether it is reasonable to represent armers' behavior using the min-max regret riterion. This riterion suggests that the deision-maker regrets ater all about the osts o missing opportunities resulted by inal deision ompared with alternative ations that ould be hosen. For arm sub-models whose observed behavior is explained better when unertainty is taken into aount in the orm o ILP then minimizing maximum regret (optimal plan approahes loser to the base year rop mix that the optimal plan resulted by its LP ounterpart), we adopt hereater the ILP speiiation. When the gross 4

6 margin maximization rule reprodues satisatorily reality, it is retained as a deision rule and the orresponding arm models remain LP speiied. Thus, a hybrid blok angular arable setor model is ormed with an improved preditive ability than the initial LP. The main drawbak is the exponential inrease o omputing time lapse to solve the ILP as or n interval oeiients the min-max optimization o the ILP requires the solution o 2(n-1) LP and 0-1 models. In this study speiied or the Kopais region however, arm models ontain one-digit objetive untion terms keeping the model size manageable. Results indiate hanges in rop mix or senarios examined inluding ounter-intuitive indings in the ase o environmental top-ups that lead to less area ultivated applying redued nitrogen less than expeted mainly due to the speii onsideration o unertainty in the model and the signiiant number o armers that adopt min-max regret behavior in the hybrid model. The paper is organized as ollows: A onise presentation o the mathematial struture o the LP model is given in the next setion. Formal aspets o the "Interval Linear Programming (ILP)" approah are presented in setion 3. The use o the min-max regret riterion within the ILP ramework is explained in setion 4. The ase study and the results thereo are the ous points o setion 5. Finally, onlusions and remarks or urther researh omplete the artile. Modeling the Farmers' Behavior: The mathematial ormulation General arhiteture o the model A otton growing arm () is supposed to hoose a ropping plan (x ) and input use among tehnially easible ativity plans optimisation problem or the armer appears as: A x b so as to maximise gross margin gm. The max x gm x,, g, x p s y sub v x s. t. A x, x, A 0 x b m n n _( I ) ( II ) (1) The setor model ontains arm problems suh as the one speiied above. The basi arm problem is linear with respet to x, the primal n 1-vetor o the n ropping ativities. The m n-matrix A and the m 1-vetor b represent respetively the tehnial oeiients and the apaities o the m onstraints on prodution. The vetor o parameters θ haraterizes the th 5

7 representative arm (y yields or rop, v variable osts, p pries dependent on quality). κ stands or the vetor o general eonomi parameters (p pries not dependent on arm, sub area subsidies and s on pries speii to rops). The onstraints an be distinguished as resoure, agronomi, demand and poliy ators. The model enables a omparative stati analysis, but it does not allow or arm expansion, as it takes as given land resoure endowments and land rent o the base year. Dierent sets o parameters are applied to denote the CAP 2000 and the urrent CAP (reorm 2003). Speiially or the year 2008, a onstant term denotes the deoupled subsidies enjoyed by the arm ater the reorm (this amount is ixed based on historial data on subsidies reeived by the arm during the ultivation period) subjet to additional onstraints that modiy easible prodution plans: I. Cross ompliane i obligation in order to reeive the single payment (rop rotation with legumes in 20% o the eligible land). II. Atual arm land must be greater than or equal to eligible land. Unertainty and Interval Programming In mathematial programming models, the oeiient values are oten onsidered known and ixed in a deterministi way. However, in pratial situations, these values are requently unknown or diiult to establish preisely. Interval Programming (IP) has been proposed as a means o avoiding the resulting modelling diiulties, by proeeding only with simple inormation on the variation range o the oeiients. Sine deisions based on models that ignore variability in objetive untion oeiients an have devastating onsequenes, models that an deliver plans that will perorm well regardless o uture outomes are appealing. More preisely, an ILP model onsists o using parameters whose values an vary within some interval, instead o parameters with ixed values, as is the ase in onventional mathematial programming. Many tehniques have been proposed to solve the resulting problem. Shaoheng studied the ase where all the model parameters are represented by intervals and the deision variables are non negative. Later, Chinnek and Ramadan generalized their approah to the ase where variables are without sign restrition. The ase whih is o greater interest or our purpose is the one where only the objetive untion oeiients are represented by intervals. This partiular problem is the most requently onsidered in ILP literature (Bitran, Inuiguhi 6

8 and Sakawa, Ishibuhi and Tanaka, Mausser and Laguna (1998, 1999a, 1999b), Rommelanger, Steuer). We now introdue some deinitions and notations and briely present the ormal problem. Interval Linear Programming Problem Let us onsider a Linear Programming (LP) model with n (real and positive) variables and m onstraints. The objetive untion is to be maximized. Formally: max {x : Γ, x S} (ILP) where n : i l, u i i, i 1.. n n m n S x : Ax b, x 0, A, b m Let x S : x arg max y : y S, be the set o potentially optimal solutions. Let Υ be the set o all the extreme objetive untions: Y : i li, ui, i 1.. n.to give insight into what the problem beomes when intervals are introdued, we reall the ollowing theorem (Inuiguishi and Sakawa, Steuer): Theorem Let us onsider the ollowing multiobjetive linear programming problem: υ max{x : x S; Υ} (MOLP) where the υ-max notation stands or the vetor maximization. Then, a solution is a potentially optimal solution to (ILP) problem i, and only i, it is weakly eiient to the (MOLP) problem. Theoretially, this result enables us to mobilize all the tools and onepts o multi-objetive linear programming literature, espeially to hoose/propose suitable solution onepts or (ILP) problem. In the literature, two distint attitudes an be observed. The irst attitude onsists o inding all potentially optimal solutions that the model an return in order to examine the possible evolutions o the system that the model is representing. The methods proposed by Steuer as well as Bitran ollow this type o logi. The seond attitude onsists o adopting a speii riterion (suh as the Hurwiz's riterion, the maxmin gain o Falk, the minmax regret o Savage, et.) to selet a solution among the potentially optimal solutions. Rommelanger, Ishibuhi and Tanaka, Inuiguhi and Sakawa and also Mausser and Laguna 7

9 proposed dierent methods with this seond perspetive. Following this perspetive, the next setion introdues the approah that we have seleted, namely the minimization o the maximum regret approah, and the proedure we adopted or its implementation. Minimizing the Maximum Regret Minimizing the maximum regret onsists o inding a solution whih will give the deision maker a satisation level as lose as possible to the optimal situation (whih an only be known as a posteriori), whatever situation ours in the uture. The armers are aed with a highly unstable eonomi situation and know that their deisions will result in unertain gains. It seems reasonable to suppose that they will deide on their surae alloations prudently in order to go through this time o eonomi instability with minimum loss, while trying to obtain a satisying proit level. This is preisely the logi underlying the minmax regret riterion; i.e. seletion o a robust solution that will give a high satisation level whatever happens in the uture and that will not ause regret (Loomes and Sugden, 1982). Thereore, we make the hypothesis that the armers o the onsidered region adopt the minmax regret riterion to make their surae alloation deisions. The mathematial translation o this hypothesis or the arable setor supply model was to implement the minmax regret solution proedure proposed in the literature (Inuiguhi and Sakawa, Mausser and Laguna, 1998, 1999a, 1999b). The presentation o the ormal problem and the algorithm o minmax regret are presented in the ollowing paragraphs. The MinMax Regret (MMR) Problem Suppose that a solution x S is seleted or a given. The regret is then: R, x max y S y x The maximum regret is: max R, x The minmax regret solution xˆ is then suh that orresponding problem to be solved is: R xˆ max Rmax x or all x S. The 8

10 min max max y x (MMR) x S y S The main diiulty in solving MMR lies into the ininity o objetive untions to be onsidered. Shimizu and Aiyoshi proposed a relaxation proedure to handle this problem. Instead o onsidering all possible objetive untions, they onsider only a limited number among them and solve a relaxed problem (hereater alled MMR ) to obtain a andidate regret solution. A seond problem (alled hereater CMR) is then solved to test the global optimality o the generated solution. I the solution is globally optimal, the algorithm terminates. Otherwise, CMR generates a onstraint whih is then integrated into the onstraint system o MMR to solve it again or a new andidate solution. This proess ontinues in this manner until a globally optimal solution is obtained. The relaxed MMR problem is: min max max y x (MMR ) x S C y S where min r 1 C,,..., 2 p. This problem is equivalent to: (MMR ) s.t. k k r x x k, k = 1,, p r 0, x S, k C where x is the optimal solution o k k k max y. A onstraint o type r x x k is alled k y S a regret ut. Let us denote x the optimal solution o MMR and r the orresponding regret. Sine all possible objetive untions are not onsidered in MMR we annot be sure that there is no belonging to Γ \ C whih an ause a greater regret by its realization in the uture. Hene, we use the ollowing CMR problem to test the global optimality o x : max max y x (CMR) y S Observe that the objetive untion value o CMR represents the maximum regret or x over, denoted by x R max. I the optimal solution p 1 x p 1 S, o CMR gives x r, it R max means that p 1 an ause a greater regret than r by its realization in the uture and that it has p 1 p 1 to be onsidered also in C while solving MMR. So, the regret ut r x x p 1 is added to the previous onstraint set o the MMR to solve it again and obtain a new andidate. The proess is iterated until the generated andidate regret solution is ound to be optimal by CMR. This solution proedure idea is summarized by the ollowing algorithm: 9

11 The MinMax Regret Algorithm Step 0: r 0, k 0, hoose an initial andidate x. For the initial regret andidates to start the algorithm, the LP optimal solutions may be used. Step 1: k k 1, Solve CMR to ind k and R max x : I R max x r then END. x minimizes the maximum regret. Step 2: Add the regret ut k k r x x k to the onstraint set o MMR' Step 3: Solve (MMR') to obtain a new andidate x and r. r r. Go to Step 1. The diiulty in this resolution proess lies in the quadrati nature o the CMR problem. Inuiguhi and Sakawa investigated the properties o the minmax regret solution to ind a more suitable way to solve CRM. Mausser and Laguna (1998) used their results to ormulate a mixed integer linear program equivalent to CMR whih is less omplex to solve. As Mausser and Laguna (1999a) notied that the omplexity o that mixed integer program severely limits the size o problems to be addressed, thereore they suggested to use heuristis. In the problem studied here, unertain objetive untion oeiients are in no arm deision making unit more than ive. Thus, in our experiments we used this equivalent problem mixedinteger ormulation ii. Let us onsider the ollowing ILP model solved in the two dimensional variable spae to illustrate how the algorithm works and its underlying logi. max 1 x x 2 subjet to x 1 + x 2 60 land availability 70 x x own labour availability 12 x x working apital and x 1, x 2 0 where 1 [7.2, 10.4] and 2 [3, 5.5]. 10

12 X1 + 3 X2 + R > = (11.1, 48.9) (13.9, 45.3) X X2 + R > = (12.85, 44.01) 20 (20, 24) 10 0 (25, 0) Figure 1. Variable Spae in the Example and Regret Cuts. This problem has a easible region delimited by the ive verties (Fig. 1). The set o all the extreme objetive untions is Y={(7.2, 3) ;(7.2, 5.5) ;(10.4, 3) ;(10.4, 5.5)}. The orresponding MOLP problem, by denoting S the easible region deined by the onstraints, is υ max{ 7.2 x x 2, 7.2 x x 2, 10.4 x x 2, 10.4 x x 2 : (x 1,x 2 ) S } When onsidered, separately, to eah o these objetive untions orresponds a dierent optimal solution (respetively to Y, (11.1, 48.9) ; (11.1, 48.9); (20, 24) ; (11.1, 48.9)). Along with the verties (25, 0) ; (60, 0), those solutions onstitute basi eiient solutions or the MOLP. The set o potentially optimal solutions or the ILP (the eiient solutions or the MOLP) is given by onvex linear ombinations o every adjaent ouple o these our solutions. Let us apply the algorithm to this problem and disuss the results. Initialisation Step 0 : r 0, k 0 x., Let us hoose (11.1, 48.9) as the initial andidate Iteration 1 Step 1 : k 1, Solving CMR leads to R max x = and, R max x r. Step 2 : The regret ut 10.4 x x 2 + r 1 is (10.4, 3) (10.4*20+3*24) = 280 is then added to the onstraint set o the MMR. In this way, the program will return a new andidate whih will try to minimize the potential regret ( x 1-3 x 2 ) 11

13 that might our i (20, 24) is not seleted as a solution. Notie that this is logial onsidering sine we have seleted (11.1, 48.9) as the initial andidate solution. The algorithm detets that the objetive untion or whih the other end o the eiient rontier, the point (20, 24), is optimal, may ause an important regret i this turns out to be the real objetive untion in the uture. Step 3 : (MMR ) returns another andidate x = (20, 24) and r = 0. r r. Obviously, this solution minimizes the potential regret ( x 1-3 x 2 )! It will be tested next. Iteration 2 Step 1 : k 2, Solving CMR leads to R max x = and 2 is (7.2, 5.5), R max x r. Step 2: Following the results o step 1, 7.2 x x 2 + r (7.2* *48.9) =348.9 is added as the new regret ut to onstraint system o the MMR. As beore, the aim is to take into onsideration the last regret possibility that CMR has returned. Now, MMR will try to return a new andidate by onsidering both potential greatest regrets ( x 1-3 x 2 ) and ( x x 2 ). Step 3: Under these onstraints, MMR returns x = (12.85, 44.01) and r = r r. This time the regret is positive and the orresponding solution is not a vertex (see in igure 1). Itération 3 Step 1 : k 3, Testing the andidate by CMR leads to R max (x*) = = r. END. Thus, x *= (12.85, 44.01) minimizes the maximum possible regret by r = Graphially this regret equals to the minimum distane between the intersetion o regret ut lines (igure 1) and the easible rontier. The ILP solution orresponds to the projetion o the intersetion point to the rontier diretion towards point (0,0) in the variable spae. It an also be noted that the min-max regret solution is a well balaned solution, an eiient solution o the MOLP, whih has been obtained by taking into aount extreme ases that might prove atal or a deision maker. 12

14 Case study Surveyed arms are loated in Kopais plain (in Sterea Hellas, about 100 km north o Athens) o a total surae o 25 thousand ha. These arms are representative o arable agriulture (OTEX otton and OTEX other arable rops ). Farm data onerning prodution plans or years 2005 and 2006 were olleted by personal interviews in the ontext o a dotoral dissertation (Lyhnaras, 2008) aiming at evaluating perennial energy rop penetration in the area. For this reason questionnaires have eliited detailed inormation about the value and quantity o agriultural inputs (i.e. water, ertilizers and pestiides), yields and subsidies per rop, land ownership, entitlements or the single payment regime, arm mahinery and buildings, as well as speii inormation about human and mahinery labor used per hetare or eah rop and ield operation. A ollow-up survey has been onduted in 2008 limiting the sample to 41 arms (out o 52 initially surveyed in 2006) with updated inormation on atual rop mix o the period It was the third ultivation period ater the implementation o the CAP reorm presumably revealing the armers responses to the adopted arrangements aeting relative rop proitability and imposing ross ompliane rules (onstraints materializing these rules in the region o study are detailed in the next sub-setion). Table 1. Cropping patterns and harateristis in the sample arms rops % o arms Area (ha) % o area % o arms Area (ha) ( ) ( ) ( ) ( ) ( ) Set aside % Cotton % Cotton dry D. Wheat % d.wheat irrig % Maize % Maize odder % Tomato % Alala % % o area ( ) 0.5% 30.7% 0.0% 1.6% 3.2% 7.2% 10.3% 3.0% 40.4% Other arable rops* % Olive trees % *inludes water melons, onions, oats, potatoes, dry tomatoes and witloo 13

15 Farming in Kopais involves mainly arable rops suh as otton, durum wheat, oats, alala, tomatoes, maize or seed and odder (Table 1). About 90% o arms ultivated otton in 2005, whih was the main rop in terms o land overage (36% o total area), along with alala (hal o the arms with 39% o total area), 40% o durum wheat (irrigated and dry inluded), about 30% produing tomatoes and 24 and 20% o the arms ultivated maize or seed and odder respetively and one arm had set aside land. As an be seen in Table 1, the CAP 2003 reorm has not aused signiiant hanges o ativity levels in the sample. The only serious hange one observe is a 15% redution o ultivated area by otton. About 10% o this area is replaed by alternative ultivations suh as melons, onions, oats, potato, non-irrigated tomato and witloo as well as land set aside. Only a ew hetares (0.5% o total) an be onsidered subjet to permanent land use hange (olive trees) whereas durum wheat, alala, maize and set aside land inrease ompensate or the rest o otton land derease. Variables and onstraints All rops ultivated in the region have been treated as alternative ativities or every arm in the sample. In the 2006 senario, observed rop yields and pries or the same year were used or eah arm. For rops not present in a arm prodution plan, the orresponding sample averages rom the observed data o the same year were used. For the base year senario a set o arm speii poliy onstraints was inluded or the analysis. More speiially, otton areas or the previous CAP regime are onstrained to the ultivated areas observed in the base year releting national poliy to attenuate oresponsibility harges or exeeding a maximum guaranteed national quantity. These onstraints were dropped or the simulation o the new CAP regime. It should be noted that although total deoupling was hosen or most agriultural rops in Greee, otton remains partially oupled as a land subsidy o /ha still applies iii. The total area utilized by eah arm in 2005 was divided into eligible and non-eligible land. The single payment that orresponds to an eligible hetare or eah arm was also added to the objetive untion as a onstant term, so that the model alulates the total single payment per arm, bounded at a level ditated by the total eligible surae For tomatoes (ontrats with anning industry) and maize or odder, a market onstraint was imposed: These rops are onsidered only or the arms that ultivated these rops in the base year but with possibility o a 10% inrease in ultivated area. This restrition, whih is atually veriied by the 2008 observations, is due to the estimate or weakness o disposal o 14

16 additional prodution. We note that tomatoes and odder maize are not onsidered as eligible rops in this exerise, sine these rops were not inluded in the reorm agenda at the time o data olletion. In order to satisy ross-ompliane obligation (new CAP), at least 20% o the eligible area is required to be ultivated with a leguminous rop. The only leguminous rops onsidered in this exerise are alala and the ommon veth interropped ativity iv. For all rops, exept durum wheat, this ross-ompliane requirement is ulilled by either or both o the above legumes. Durum wheat oinides with veth in the ield or several months during every ultivating season thus the rotation is modeled only with alala. The idle land ativity was also added so that the model alulates at the optimum the arm land that is not ultivated but is maintained with the minimal required are in order to reeive the deoupled payment v. Both ativities were added as linear variables in the objetive untion, assoiated with a negative parameter that represents their estimated variable ost ( 100 /ha or the idle land and 150 /ha or the veth ativity). The resoure onstraints used in both senarios onern the availability o land and water. The onstraint or total arm land was deined as an equation and not as a weak inequality, allowing the replaement o the onstraint slakness with the idle land ativity, in order to impose the ross ompliane obligation o maintaining idle hetares in good agriultural ondition. Water resoures were modelled in terms o both irrigated area and total water quantity. For the ormer, a onstraint bounding total irrigated land in eah arm at the observed levels or the base year was used. For the latter, personal ommuniation with experts provided inormation about average water requirements per hetare or eah rop, whih allowed the ormulation o a onstraint bounding the total water quantity in every arm to its 2005 estimated level. Most arms in the sample are onsidered large arms by Greek standards, sine the average land used is 32.4 hetares, while the national average is only 4.8 hetares per arm. The land entitlements or the new CAP regime amount at more than 67% o the total land used, while the single payment reeived in 2006 varies between 100 and 1200 per ha (average 370 /ha), signiying the importane o the single payment or the survival o the arms in the sample. Gross margin, hereater net o subsidies, modiies the risk and return onditions within whih arable arms operate. As a matter o at, in the present ontext, with subsequent CAP reorms that downgrade subsidy stability ator in the ormation o gross margin, the natural 15

17 unertainty about yields, ombined with an inreasing unertainty about pries, enlarge the gross margin variation range. Table 2 illustrates variability o gross margins or rops observed in the sample in both poliy ontexts. For only 10% prie variation, ereals suer o a tenold inrease in gross margin relative importane while signiiant inreases are observed or maize and otton. Thus, we assume that unitary gross margins are pereived by armers as impreise numbers rather than risp values o expeted gross margins. Thereore, they will be represented in the model by intervals transorming the original LP to an interval linear programming problem. Intervals o ±25-50% have been used in the model or wheat, otton and maize (produts exposed in exogenous shoks) while or odder maize, alala, oats and tomatoes, expeted gross margins are retained (pries lear in national markets) so that the number s o interval-valued oeiients are up to ive. rops Average Gross margin 2005 Table 2. Gross margins and risk dependent on CAP ( /ha). Coupled subsidies as % o gross margin 2005 ±10% sales impat to gross margin 2005 Average Gross margin 2008 Coupled subsidies as % o gross margin 2008 % impat to gm 2008 Maize-od 39 0% 47% 570, % 47% 570, 200 Maize % 26% 700, % 103% 280, -10 Cotton % 5% 2110, % 23% 500, 310 Oats % 15% 520, % 24% 360, 220 Alala % 20% 1120, % 20% 1120, 750 Wht-irr % 18% 350, % 142% 90, -20 Wht-dry % 16% 310, % 555% 50, -40 tomato % 25% 3390, % 25% 3390, 2050 Model validation The validity o the arable setor model has been heked by omparing optimal ativity level outomes o the LP model with the atual ones in the base year (2005). Then interval linear programming approah using the min-max regret riterion has been implemented to investigate i the model s validity an be improved. The CPLEX solver linear and mixedinteger algorithms have been used or this purpose vi. To evaluate the proximity o the optimal solution opt x k to the observed ativity level obs x k or the rop k, several indiators are suggested in the literature suh as the sum o absolute distanes o individual rops in the plan, the mean 16

18 absolute distane, the Theil index and others. In this exerise, we used the ollowing distane (FK) measure that indiates the similarity o rop plan patterns proposed by Finger and Kreinin (1979): opt obs opt obs S x, x 100 min x, x (2) i i I ultivated area o rop i in the observed and the optimal set are idential (X obs i = X opt i or eah i) the index will take on a value o 100. I rop plan patterns are totally dissimilar (or eah X obs i > 0, X opt i = 0 and vie versa) the index will take on a value zero. As table 3 shows, both models satisatorily predit base year that was rather expeted sine alternative rops are limited to those already ultivated in eah arm in the observed rop plan. When LP and ILP models are updated aording to the new institutional ontext, both speiiations lose about 15 FK index units when prediting year Examining results at the arm level, one observes that in the 2005 period the ILP model has perormed better only in 6 arms, whereas in , the ILP model predits more aurately in 29 arms. Table 3. Model preditive apaity ha alala otton wheat Maize odder tomato maize Irrig wheat oats FK INDEX areas ,6 474,7 23, , ,9 4 LP 474,2 442,2 28,2 99,8 34,1 198,4 41,3 6,9 90,43% ILP ,6 31,3 108,2 34, ,2 19,4 92,50% hybrid 488,6 453,6 28,3 108,2 34,1 163,5 30,3 19,1 92,97% areas ,5 416,4 21,6 139,4 40,5 97,9 43,9 10 LP ,88% ILP % hybrid % i The prinipal eet o the ILP approah with the min-max regret is: when the dierene between the gross margins is relatively small, the min-max regret approah gives more "balaned" solutions, more so when the interval oeiients get larger. In at, as the intervals get larger, the gross margins or dierent rops start to overlap or, i they already have an intersetion, this inreases. It then beomes more diiult or the armer to antiipate whih rop will be more proitable. Hene, the min-max regret approah tends to return more and more balaned solutions as the size o the intervals inrease. A detailed disussion on this point is presented by Kazaki and Vanderpooten (2002). 17

19 Figure 2. Min-max regret and the LP optimal vs. observed plan or a seleted arm. Thus some armers maximize gross margin while others demonstrate regret-averse attitude as this one in igure 2. Preerenes revealed by the arm-by-arm srutiny lead us to attempt to model arable agriulture assuming dierent preerenes among produers. For eah individual arm elementary model a simple rule replaes the objetive untion with that, between gross margin maximization and min-max regret, perorming better in terms o proximity o the resulted rop mix to the observed one. This way we end up with a hybrid model, by deinition with a higher preditive apaity than the initial LP (igure 3). As a matter o at the FK index at the aggregate level inreases to 91% or the hybrid model approahing FK indies or the base year. We should mention at this point that a study in an arable region with similar harateristis (Thessaly) implementing PMP approah that by deault alibrates peretly to the base year, has predited 2008 rop mix with FK values 85-90% (Petsakos & Rozakis, 2010) LP 2008 ILP 2008 hybrid Figure 3. FK similarity indies or all arms in the model 18

20 CAP poliy senarios towards 2013 and model results The hybrid model will be used to evaluate dierent poliy senarios harvested ater the Health Chek o the CAP in 2008 and preliminary evaluation o the irst years o the implementation o the reorm. Single payment, alulated on historial subsidies reeived by the arm during a reerene period, may simply be realulated on a regional basis resulting in latter rates o diret payments. Eah member state will have a margin to inane environmental preservation, on top o diret payments (top-up), using the rest o subsidies historially reeived under strit environmental justiiations. Table 4. Redued proit by rop due to nitrogen redution wheat tomato otton maize potato % yield redution 7% 15% 15% 19% 16% % Market prie / t Sales loss / ha Fertilizing ost redution / ha Dierential gross margin (loss) / ha Fousing on nitrate pollution, we estimated impats to yields and redued reeipts as well as gains rom redued quantities o ertilizers using growth model algorithms and nitrogen-yield untions (Rozakis et al., 2001) alibrated or soils in Kopais plain (see Appendix). Overall redued proit or seleted rops appears in table 4. These rops along with all relevant parameters have been inluded in the model as additional alternatives. In pratial terms onerning the arable setor possible measures an be summarized in the ollowing propositions: 1. No oupled subsidies anymore; only SFP remains 2. Flatter diret payment rates (national SFP) : average rate o 550 /ha 3. Flatter rates (hist. EU25): average rate o 305 /ha 4. Environmental top-up20: EU25 average rate o 305 /ha plus 200/ha or applying 25% nitrogen redution (otton, maize and wheat) 5. Environmental top-up30: nitrogen redution supplement at 300/ha (otton, maize and wheat) 19

21 Results o hybrid model optimization alibrated against 2008 data onern senario 1, 3, 4 and 5 keeping the order o proposals mentioned in the previous setion. Proposals 2 and 3 yield idential rop plans beause deoupling payment does not by deinition aet a armer s short term deision, but simply hanges the gross margin in aounting terms. Compared with urrent CAP opt situation, otton is dereasing whereas grain and odder maize signiiantly inrease, with wheat, alala and tomato at previous levels (igure 4) urrent CAP (opt) urrent only SFP lat hist EU25 lateu25top-up20 lateu25top-up Figure 4. Total areas ultivated by rop or examined poliy senarios (hybrid model) In the ase o nitrogen redution measures, important areas o otton, maize and irrigated wheat pass into nitrogen-extensive ultivation and to set aside. We alulated total gross margin (GM), budgetary burden (BG) and quantities o water (WQ) and ertilizers (FQ) applied in order to evaluate senarios against onliting objetives. Table 5. Poliy senarios perormane on soial and environmental riteria Senario Gross margin Μ water (k m3) Budgetary burden (k ) Fertilizers (t) Current (yardstik) senario 1 senario 3 senario 4 senario % dierential rom urrent 658 % di % di % di % 656 0% % % % 656 0% % % % 650-1% % % % 635-3% % % Abolition o oupled subsidies (onerning mainly otton and seondly wheat) result in GM redution o along with derease o the amount o subsidies BG o 44%. I the single payment beomes latter ompared with urrent levels at the mean EU25 level, redutions reah around 20

22 38% and 59% respetively or GM and BG values. Water onsumption remains at previous levels whereas ertilizer use is slightly inreased. The above hanges result rom internal rop plan hanges made by the armers, who attempt to attain optimal margins taking unertainty into aount. Under senarios 4 and 5 beside lat rate ee supplementary support armers that apply nitrogen redution by 25% versus observed levels ontribute to small but non negligible gross margin inrease (3-6%) without signiiant derease to the total ertilizer quantity. The risk prudent attitude adopted by the majority o armers does not allow or notable hanges in the rop mix under environmental poliy senarios although there is a lear dierene when nitrogen derease top-up area subsidy inreases rom 20 to 30 per ha (igure 4). The linear setor model, i used in all arm sub-models, would result in total nitrogen redutions by 20% or senario 5 due to the quasi-abandonment o otton to the beneit o nitrogen-extensive maize and wheat (igure 5) urrent CAP (opt) urrent only SFP lat hist EU25 lateu25top-up lateu25top-up Figure 5. Total areas ultivated by rop or examined poliy senarios (LP) These ounter-intuitive results by the hybrid model due to the majority o armers that aim at minimizing maximum regret instead o maximizing gross margin may ontribute to design more eetive environmental measures. Assuming that the hybrid model predits muh better as veriied against 2008 observations, poliy makers should question the eetiveness o lat area supplements to enhane environmental poliies. One ould suggest rop dependent rates, sine redued proits due to nitrogen redution are muh higher or maize and otton 21

23 omparing to wheat. Furthermore, poliy makers ould opt to subsidy investments with presumably signiiant N dereases, or instane to promote the adoption o drip ertilization. Conlusions The aim o this study was to improve the representative apaity o a setor supply model in order to provide reliable estimates on impats o poliy measures on ultivated arable land in Kopais plain, Greee. Unertainty was introdued in the optimization proess and has been modeled by means o interval oeiients at the objetive untion level. The resulting model rom this approah is an "Interval Linear Programming Model". Within this ramework, we onsidered 41 elementary linear programming models orresponding to the arms speializing in ereal prodution. Then it was assumed that armers' behavior ould be represented using the min-max regret riterion. To test this hypothesis, the min-max Regret (MMR) algorithm was implemented or eah o the elementary models. The aim o the algorithm is to ind the solution minimizing the maximum regret or a linear programming model with objetive untion oeiients in the orm o intervals. Analysis o the results and the omparison with the optimal solutions o the LP or the elementary models showed that in many ases the MMR approah gave better balaned and distributed solutions, and this more so when the overlapping o the interval proits or various rops inreased. We also observed that our hypothesis was only partially true. Although some improvements were ahieved, the proximities obtained by the MMR approah were not always satisatory enough to support that the armers deide on their surae alloations aording to the logi o min-max regret. Thus the proit maximizing attitude is retained in about 30% o the arms so we ended up with a hybrid blok angular model with two possible objetive untion speiiations or eah arm (blok). The MMR approah sotened the abrupt nature o the linear programming, or whih any minimal dierene between the unitary margins implies the exlusion o the least proitable rop. These ounter-intuitive results by the hybrid model, aused by the majority o armers aiming at minimizing maximum regret instead o maximizing gross margin, may ontribute to produe more eetive environmental measures. Assuming that the hybrid model predits more aurately as veriied against 2008 observations, poliy makers should question the eetiveness o lat area top-ups to ahieve environmental goals. One ould suggest rop 22

24 dependent rates, sine redued proits due to nitrogen redution are muh higher or maize and otton ompared to wheat. Furthermore, poliy makers ould opt to subsidy investments with presumably signiiant N dereases, or instane to promote the adoption o drip ertilization. Further researh ould be oriented in methodologial improvements suh as testing the robustness o the MMR model or various interval levels, and more important to ombine ILP with multi-riteria utility untions that presumably would improve the preditive ability o arm models. On the other hand empirial appliation ould inlude other regions and agriultural ativities, ideally onstruting a national agriultural setor model. Suh a model would take into aount interations among regions and ativities thus resulting in better estimates o senarios under disussion. Furthermore, aggregate impats inluding eonomi welare but also environmental and soial indiators should be treated by multiriteria algorithms able to pinpoint ompromise solutions assisting in the seletion o the most eiient measures in the horizon o 2013 assuming urther drasti CAP reorms. Reerenes Akrill, R.W., Ramsden, S.J., & Gibbons, J.M, (2001). CAP reorm and the rebalaning o support or ereals and oilseeds: a arm-level analysis. European Review o Agriultural Eonomis, 28(2), Bitran G, (1980). Linear multiple objetive problems with interval oeiients. Management Sienes 26: Chinnek J. W. and K. Ramadan (2000). Linear programming with interval oeiients. Journal o Operations Researh Soiety 51: European Commission, DG or Agri. and Rural Development, Impat o a hange towards latter rates o diret payments, in Impat Assessment o the Health Chek o CAP, Annex F: Miroeonomi analyses, Brussels, Deember. Finger J.M., M.E. Kreinin (1979) A measure o Export similarity and its possible uses. Eonomi Journal 89: Guindé, L., Millet, G., Rozakis, S., Sourie, J-C., and D. Treguer (2005) The CAP mid-term reorm impats to Frenh ereal arms: a regional supply modelling approah. in F. Arini (ed.), Modelling Agriultural Poliies: State o the Art and new Challenges, Parma: MUP 23

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