Disproportionate Joint Cost Allocation. at Individual-Farm Level Using Maximum Entropy

Transcription

1 Dsproportonate Jont Cost Allocaton at Indvdual-Farm Level Usng Maxmum Entropy Marus Lps Insttute for Sustanablty Scences, Agroscope, 8356 Ettenhausen, Swtzerland Poster paper prepared for presentaton at the EAAE 204 Congress Agr-Food and Rural Innovatons for Healther Socetes August 26 to 29, 204 Ljubljana, Slovena Copyrght 204 by Marus Lps. All rghts reserved. Readers may mae verbatm copes of ths document for non-commercal purposes by any means, provded that ths copyrght notce appears on all such copes.

2 Abstract Ths paper addresses the allocaton of jont cost among enterprses also called producton branches or actvtes and presents an approach based on maxmum entropy and standard costs from farm-management lterature as allocaton factors. The approach allows us to dscard the wdely appled assumpton of a proportonal jont-cost allocaton. Snce t provdes a dsproportonate jont cost allocaton, the dstnctve feature of the approach s that t favours the adjustment of large standard costs rather than of small ones. ey words: jont cost allocaton, maxmum entropy Introducton When allocatng jont costs among enterprses (also referred to n the lterature as producton branches or actvtes ), whch sgnfes an mportant challenge n the analyss of full cost, we normally struggle wth a data gap or an under-determned (cost) model, snce a scarcty of avalable resources such as tme and money means that the true allocaton s not avalable. As a method for overcome data gaps and allowng nformaton recovery, maxmum entropy represents a promsng tool for addressng the jont-cost allocaton problem. Ths paper ams to present an approach that maes the vrtue of maxmum entropy avalable for an emprcal applcaton of jont-cost allocaton at ndvdual-farm level. To our nowledge, there s no farm- specfc jont-cost allocaton approach based on maxmum entropy, though there are several maxmum-entropy-based analyses that address jont-cost allocaton on a regonal or country level (e.g. Lence and Mller, 998; Léon et al., 999; Garvey and Brtz, 2002; Peeters and Surry, 2005; Fragoso and da Slva Carvalho, 202). There are two major dfferences between a cost allocaton at regonal level and at ndvdual-farm level. Frstly, a common producton technology s assumed n all regonal analyses when performng the cost allocaton. As regards the dversty of Swss farms n general and crop farms n partcular, a common producton technology cannot be assumed. Secondly, the man objectve dffers. Whle a regonal analyss focuses on nput coeffcents representng a farm type, a farm-level cost analyss ams to allocate the jont costs entrely among the enterprses of a partcular farm. 2 Method 2. Proportonal allocaton For smplcty s sae, the followng cost allocaton s presented for one jont-cost tem and one farm only,.e., machnery costs. The farm produces arable crops (=,2,..,I). An arable crop s consdered an enterprse (e.g. potatoes). Each crop s grown on an area x measured n hectares. From the accountancy fgures, we now the total jont costs y,.e. the total machnery costs at farm level n Swss Francs (CHF). As a result of the jont-cost allocaton, we are loong for β, the (machnery) cost n CHF per hectare of crop where μ serves as an allocaton factor. For the latter standard costs per hectare, also referred to as budgeted costs or forecast costs, are used.

3 µ β = I y () x µ = Together, the numerator and denomnator on the rght-hand sde of Equaton represent the share of one hectare of enterprse out of the farm-wde jont costs. Taen as a whole, all shares form the so-called apportonment formula or allocaton ey. To perform a Proportonal cost allocaton, we reformulate Equaton to: β = αµ (2) The factor alpha (α) s defned as follows: y α (3) µ = I x = Fgure graphcally llustrates jont-cost allocaton (.e. for machnery costs), and underscores the adjustment of allocaton factors, whch s the core element of the allocaton procedure. Specfcally, the costs of two crops grassland (G) and potatoes (P) wth potatoes representng a crop wth maredly hgher standard costs are adjusted. If we assume that the farm as a whole has lower costs than suggested by the farm management lterature (α < ), the proportonal lne (Prop) whose slope s equal to α les below the angle bsector (45 ). Applyng the standard costs μ Grassland and μ Potatoes, the proportonal jont-cost allocaton leads to β G_prop and β P_prop, respectvely. Costs per hectare (β ) 45 Prop (α) β P_prop β P_dsprop Dsprop β G_dsprop β G_prop μ Grassland μ Potatoes Fgure. Proportonal and dsproportonate jont-cost allocaton G = Grassland; P = Potatoes; Prop = proportonal; Dsprop = dsproportonate Standard costs per hectare (μ ) 2

4 2.2 Core maxmum entropy model The followng outlne of a jont-cost allocaton model usng maxmum entropy s based on Golan et al. (996: Chapter 3) and ams to derve the jont cost (.e. machnery) of enterprse per hectare (β ). Snce we use standard costs per hectare (μ ) n the model, we must focus on the sngle-hectare level. For the model specfcaton, each ndvdual hectare of crop s treated as an ndependent actvty. Accordngly, the number of hectares s cropspecfc and denoted as N(), whle j refers to the ndvdual hectare s number [j =,2,,N()]. Assumng that N() s an nteger, the ndvdual hectares of crop are denoted as x,j : N ( ) x = (4) x, j j= β, the cost per hectare as already mentoned above, s provsonally defned for each hectare ndvdually as β,j. The allocaton of jont cost y can be formulated as follows: y = I N ( ) = j= β (5), j We assume that β,j les n a range charactersed by support ponts (z, ). Wth regard to the dversty wthn crop farmng, we assume that the true value for β,j les wthn a range of μ ± μ. Followng Howtt and Reynaud (2003), three support ponts are defned ( = 3). The three support ponts are 0, μ and 2μ, respectvely. Snce support ponts (z, ) refer to crops and do not dffer among the dfferent hectares of crops, the hectare-wse dstncton need not be consdered for support ponts. Each β,j s defned as a weghted sum of ts support ponts:, j = p, j, z, = β (6) p,j, represents the probablty of support pont of hectare j of enterprse beng appled. For each hectare j of all enterprses, the probabltes must add up to : = p, j (7), j, = Maxmsng the Shannon Entropy measure H allows us to determne the probabltes p,j, : I N ( ) max H = p, j, ln p, j, (8) = j= = Gven that support ponts wthn the dfferent hectares of a specfc crop are dentcal, the resultant probabltes must also be dentcal. Snce there are no dfferences among the several hectares of crop, t holds that p, j, = p, j Consequently, Equaton 7 can be reformulated, whch facltates the subsequent model formulaton: = p (9), = 3

5 The fact that probabltes of all hectares of a partcular crop are equal leads to a further smplfcaton: β, j = β Thus, Equaton 6 s reformulated as follows: p, z, = j β = (0) Usng Equatons 4 and 0, Equaton 5 can also be formulated dfferently: y = I x = β () Fnally, the Shannon Entropy equaton (8) s reformulated, mang use of Equaton 4: I max H = x p, ln p, (2) = = The varable x serves as a weghtng factor that taes account of the dfferng number of hectares of the farm s enterprses. Together, Equatons 9 to 2 form the CoreModel, even allowng the use of non-nteger values for crop areas. Unle the above descrpton, whch refers to one cost tem and a sngle farm, the model s solved for all jont-cost tems and all farms. Returnng to the jont-cost allocaton ssue, the queston arses as to how the CoreModel dffers from a Proportonal cost allocaton. The Shannon measure of entropy (Equaton 2) reaches ts maxmum value when the dstrbuton of all probabltes p, s unform. Thus, the maxmum s attaned when each of the support ponts of all crops s assgned the probablty of /,.e. f β s equal to the standard costs μ. The approach therefore mnmses the devaton from the standard costs. Bearng n mnd that the adjustment s performed at the one-hectare level, the absolute dfferences between support ponts are of mportance. For nstance, f total costs y are smaller than suggested by the standard costs from farm management lterature, the model must cause a reducton of the allocaton factors. In absolute terms, a % probablty shft has a stronger mpact on a crop wth hgh standard costs (such as potatoes) than on one wth low costs (grassland). Fgure, whch also ncludes the adjustment of allocaton factors va maxmum entropy cost wth Dsprop, produces the results β G_dsprop and β P_dsprop for grassland and potatoes, respectvely. The probablty dstrbuton of the maxmum entropy approach therefore leads to a dsproportonate adjustment of standard costs. It s mportant to note that Dsprop never ntersects the angle bsector (45 ), because all standard costs are adjusted n ether a dmnshng or ncreasng drecton. From a producton technology perspectve, a dsproportonate adjustment better addresses the adjustment of costs of the producton processes than does a proportonal adjustment. Potatoes, for nstance, ncur much hgher machnery costs than grassland. If farm-wde machnery costs dffer greatly from the expected values n the farmmanagement lterature, there are more possbltes n practce for adjustng machnery costs for potatoes, snce more operatonal steps are appled (e.g. for plant protecton). Generally speang, the hgher the standard costs μ, the greater the possbltes for adjustng costs. 4

6 2.3 Inequalty restrctons The dsproportonate adjustment of the CoreModel can lead to a stuaton n whch crops wth hgh standard costs are so strongly reduced that they even undercut crops wth low standard costs. Such a result s only possble f the devaton factor alpha has an extremely low value (e.g. α < 0.5). Potatoes, for nstance, would have lower absolute machnery costs than grassland, whch s useless from a producton technology pont of vew. To ensure a plausble ran order among producton branches we mpose nequalty restrctons as suggested by Campbell and Hll (2006). Although t would be possble to apply an nequalty restrcton for each crop, we defne groups ncludng smlar crops. Accordngly, the Inequalty applcaton allows us to mantan the ran order between groups, whle the ran order wthn groups may change. 3 Data Data from 36 crop farm observatons of the Swss Farm Accountancy Data Networ (FADN) s used to apply the dfferent jont allocatons. 2 dfferent enterprses are consdered whle we focus on the allocaton of two dfferent jont cost tems, labour (measured n normal worng days) and machnery costs (n CHF), respectvely. The allocaton factors (μ ) are taen from farm management lterature. Based on μ s the factor alpha s calculated for all farm observatons. The mean value of labour nput (alpha = 2.5) ndcates that far more labour s used than suggested by farm management lterature. As regards machnery costs, the mean value of alpha s Results Table presents the results of Proportonal jont-cost allocaton and the two maxmum-entropy applcatons CoreModel and Inequalty, respectvely. The results refer to the average of the enterprse cases nvolved, the number of whch s ndcated n the second column. For labour, the results for Proportonal on the one hand and both maxmum-entropy applcatons on the other dffer sgnfcantly. Bearng n mnd that the farms n queston use much more labour than suggested by the farm-management lterature, the dfference due to the dsproportonate allocaton under maxmum entropy becomes obvous. For crops wth low standard labour costs such as forest and fallow land, the results are lower for maxmum-entropy applcatons than for Proportonal. Conversely, potatoes and other actvtes exhbt hgher results under the maxmum-entropy applcatons. The CoreModel exhbts a slghtly stronger devaton from Proportonal than does Inequalty, wth CoreModel devatons fallng wthn a range of -22 % (forest) to +9 % (potatoes), whlst the Inequalty applcaton shows a devaton of between -5 % (forest) and +8 % (other actvtes). Gven a mean value for factor alpha of 0.8, the allocaton factors must n general be reduced for machnery. For crops wth low standard machnery costs (e.g., forest and fallow land), the Proportonal allocaton leads to a stronger reducton and hence lower results than both maxmum-entropy applcatons. Conversely, maxmum-entropy applcatons show a more substantal reducton for crops wth hgh standard costs (e.g., 5

7 potatoes and other actvtes), leadng to lower results. A substantal dfference between the CoreModel and Inequalty applcatons can be observed for these two crops. The absolute results of CoreModel are much lower. For CoreModel, the devatons from the Proportonal allocaton range between -29 % (other actvtes) and +9 % (fallow land), whle those of Inequalty range between -6 % (other actvtes) and + % (forest). Table. Mean values of labour and machnery-cost results for all enterprses Enterprse No. of Cases Labour n NWD per ha Core Model Machnery n CHF per ha Proportonal Inequalty Proportonal Core- Model Inequalty Wheat Barley Maze Slage Maze Potatoes Sugar Beet Olseeds Peas Grassland Fallow Land Forest Other Actvtes NWD = normal worng days 5 Conclusons Ths paper presents an approach for the allocaton of jont cost among farm enterprses at ndvdual-farm level, based on maxmum entropy and standard costs from farmmanagement lterature as allocaton factors. Addng Inequalty restrctons ensures that a rough ran order between enterprses s mantaned. Accordngly, ths applcaton s suted to farm management analyses at ndvdual-farm level, and n the end allows the potental of maxmum entropy to be used on behalf of jont-cost allocaton. As a normatve approach, the optmal soluton s provded for each farm separately, bearng n mnd the farm-specfc jont-cost stuaton. Compared to a Proportonal jont-cost allocaton, whch s usually appled n the lterature for full cost analyses, the applcaton of maxmum entropy for jont-cost allocaton leads to a dsproportonate adjustment of standard costs, reflectng a probablty dstrbuton n whch the adjustment of hgh standard costs s more lely than the adjustment of low ones. As a result, the maxmum-entropy applcatons brng jont-cost allocaton more n lne wth realty, as well as allowng the strong assumpton of proportonal allocaton to be dscarded. Based on crop-farm observatons from the Swss Farm Accountancy Data Networ (FADN), the average allocated jont costs of the enterprse are compared for a Proportonal 6

8 cost allocaton and for the two maxmum-entropy applcatons CoreModel and Inequalty, respectvely. The shown dfferences hghlght that the choce of allocaton methods s of emprcal mportance. References Campbell, R. C. and Hll, R. C., (2006). Imposng Parameter Inequalty Restrctons usng the Prncple of Maxmum Entropy. Journal of Statstcal Computaton and Smulaton 76(): Garvey, E. and Brtz, W., (2002). Estmaton of Input Allocaton from EU Farm Accountng Data usng Generalzed Maxmum Entropy, Worng Paper 02-0, Unversty of Ireland and Unverstät Bonn. Golan, A., Judge, G. and Mller, D., (996). Maxmum Entropy Econometrcs: Robust Estmaton wth lmted Data. John Wley & Sons, Chchester. Fragoso, R. and da Slva Carvalho, M. L., (202). Estmaton of jont costs allocaton coeffcents usng the maxmum entropy: A case of Medterranean farms, Journal of Quanttatve Economcs. 0(2): 9-. Howtt, R. E. and Reynaud, A., (2003). Spatal Dsaggregaton of Agrcultural Producton Data by Maxmum Entropy. European Revew of Agrcultural Economcs 30(3): Léon, Y., Peeters, L., Qunqu, M. and Surry, Y. (999). The Use of Maxmum Entropy to Estmate Input-Output Coeffcents from Regonal Farm Accountng Data. Journal of Agrcultural Economcs 50(3): Lence, S. H. and Mller, D. J., (998). Recoverng Output-Specfc Inputs from Aggregate Input Data: A Generalzed Cross-Entropy Approach. Am. J. Agr. Econ. 80 (4): Peeters, L. and Surry, Y., (2005). Estmaton d un modèle à paramètres varables par la méthode d entrope crosée généralsée et applcaton à la répartton des couts de producton en agrculture. Actes des Journées de Méthodologe Statstque