Vrible Growh Impcs on Opiml Mrke Timing in ll-ou Producion Sysems Jy R. Prsons, Dn L. Hog,. Mrshll Frsier, nd Sephen R. Koonz 1 bsrc This pper ddresses he economic impcs of growh vribiliy on mrke iming decisions in n ll-in, ll-ou producion sysem. Mrkeing decisions bsed on he pen verge re deermined o be differen hn hose bsed on he enire disribuion of oupu levels. cse sudy d se of 350 swine provides verificion of our heoreicl consruc. Pper presened he esern griculurl Economics nnul Meeings, Long Bech, C, July, 00. Copyrigh 00 by Jy Prsons, Dn Hog,. Mrshll Frsier, nd Sephen Koonz. ll righs reserved. Reders my mke verbim copies of his documen for non-commercil purposes by ny mens, provided his copyrigh noice ppers on ll such copies. 1 Jy Prsons (jy.prsons@opimlg.com) is grdue reserch ssisn, Dn Hog (dhog@ceres.gsci.colose.edu) is professor, Mrshll Frsier (mfrsier@ceres.gsci.colose.edu) nd Sephen Koonz (skoonz@ceres.gsci.colose.edu) re ssocie professors Colordo Se Universiy.
Inroducion The noion h he oupu from producion process cn vry is no new one. This is especilly rue s i pplies o griculurl producion where numerous fcors such s weher nd geneics joinly deermine he finl oucome. Yield vriions re especilly perinen in he livesock indusry where we ypiclly see enire pens of nimls mrkeed one ime bsed on he verge size in he pen. Idelly, o be enirely confiden bou hese mrkeing decisions, he enire rnge of he d should be undersood (Pringle, 000). verges msk his informion. Informion h migh reurn more hn i coss o collec. Previous reserch on he opiml slugher weigh of livesock hs focused on feeding sregies, geneics, nd pricing sysems. For insnce, i hs been shown h here re higher profis per hog for lener gils relive o he fer brrows nd h he gils py more mrkeed hrough componen pricing sysem while he brrows py more in live weigh pricing sysem (Bolnd, Preckel, nd Schinckel, 1993). Oher sudies hve shown h feed prices nd niml replcemen coss re imporn in deermining he opiml mrke weigh (Chvs, Kliebensein, nd Crenshw, 1985), hve exmined how producers migh modify heir feeding decisions o bes respond o chnges in inpu nd oupu prices (Crbree, 1977), nd used gin isoquns o esblish decision rules for opiml rions hrough vrious growing phses (Hedy, Sonk, nd Dhm, 1976). In generl, ps reserch hs focused on esblishing decision rules bsed on represenive niml from he group. This my be pproprie in indusries like poulry where vribiliy hs been reduced o miniml levels in recen yers. However, hese sme decision rules my be sub-opiml for heerogeneous nimls such s cle, where here re frequen clls o improve quliy nd consisency (Smih e l. 1995 nd NCB). Grid mrkeing nd 1
compliced soring sysems (i.e. Brehour, 1989) h use ulrsound o idenify individul niml ris show h he beef indusry undersnds h economic losses cn occur when pens re sold bsed on verge niml ris. The objecive of he presen pper is o presen model h ccouns for he disribuion of he nimls in he mrke iming decision. For exmple, when pen is mrkeed, vribiliy in niml growh resuls in some nimls being over-finished, while ohers hve no ye reched heir full economic poenil. The impc of his disribuion on he opimliy condiions is explored hrough horough nlysis of he mrginl curves resuling from he producion process. Swine producion provides he pplicion focus of he presen pper bu he mehods exend o oher species. By choosing swine s our pplicion focus, we re ble o uilize exensive d ses vilble from universiy reserchers o es our model. However, s i urns ou, he mrke iming for swine wihin heir producion cycle plces limis on he economic vlue ssocied wih full ccoun of he oupu disribuion. The vlue of our model is no so much in is pplicion o he swine indusry s presened in he presen pper s i is in he heoreicl consruc iself. Specificlly, he noion h decresing mrginl reurns resul in siuion where he verge oupu level is no he bsis from which o compue he verge mrginl vlue produc for group of nimls. model ccouning for he enire disribuion of oupu levels provides more ccure ssessmen of he mrginl vlue ssocied wih coninuing o feed pen of nimls. In doing so, his my led o siuions where mrke iming decisions bsed on he verge oupu level re significnly less hn opiml. This pper exends previous reserch in wo wys. Firs, wheres previous reserch hs focused on decision rules s hey perin o represenive niml for given group, we re
considering he enire disribuion of nimls. Therefore, he decision rules developed in his pper re beer represenion for he full economic poenil of ll-in, ll-ou pen mrkeing prcices. Second, by developing his model, we presen frmework o explore he impc of producion vribiliy on ny producion siuion chrcerized by simulneous erminion of he producion process cross muliple producing unis. In doing so, we mke i possible o beer ssess he impc of prcices such s ighening he geneic line or employing sophisiced soring regime on he poenil profis of n ll-in, ll-ou producion sysem. Theoreicl Model The firs sep in developing he heoreicl model is he deerminion of n pproprie producion funcion. The use of Gomperz sigmoidl curve o describe poenil growh in swine hs proved useful (hiemore, p. 56). The curve o give weigh ime is given by k be = e where is he upper sympoic weigh, k is growh consn, nd b is ime scle prmeer. However, Prks (p. 35) poins ou h his form mkes he deerminions of nd k bised. Therefore, we follow he suggesion of Prks nd use he following modificion of he Gomperz funcion s model for poenil growh. e k o = (1) where o is defined o be he iniil weigh nd is defined o be he ime h hs elpsed since he iniil weigh ws observed. Then, s, nd = 0, = o. The prmeer k > 0 serves s shpe prmeer h influences he slope, curvure, nd poin of inflecion of he sigmoidl curve. 3
4 Given oupu s funcion of ime s described in equion (1), we cn hen derive he mrginl physicl produc wih respec o ime s funcion of ime () k e o o e k k = = ln MPP () or s funcion of weigh ( ) = = k ln MPP. (3) Noe h for ll <, we hve < < 0. Therefore, he MPP is lwys posiive. lso, he second derivive ( ) + = = k ln 1 ln MPP is negive precisely when 0 ln 1 > + or 1 > e. (4) Therefore, he producion funcion (1) is chrcerized by posiive bu diminishing mrginl reurns wih respec o ime whenever relionship (4) holds. Furhermore, o nlyze he concviy of he MPP curve, we clcule ( ) + + = = 3 3 3 ln ln 3 1 ln MPP k which is negive precisely when
1+ 3 ln + ln < 0 or 3 5 3+ 0.079 e < < e 0.685. (5) Therefore, under he ssumpion of consn oupu price P w, he mrginl vlue produc curve given by MVP ( ) = P ( ) = k P w MPP w ln (6) is concve over he weigh regions indiced by relionship (5) nd convex oherwise. Jensen s Inequliy To mximize profis from he producion of single niml, we simply feed he niml unil he mrginl vlue produc equls he mrginl cos. Le he uni of ime be dys nd sr wih simplified ssumpion h he mrginl cos is represened by consn δ h cpures he dily cos of feeding he niml. Since Oswld (1883), physiciss nd chemiss hve been sudying differenil equions of he ype in equion (3) (Prks). Th is, he re of chnge in oupu wih respec o he independen vrible is uniquely reled o he vlue of h. Nelder (196) ws mong hose o rgue h his is more likely o led o nurl lws of nure hn differenil equions of he form expressed in equion (). The rgumen is h more fundmenl informion cn be gined by compring mrginl producs he sme vlue of oupu hn he sme vlue of inpu. e dop his concep in using equion (3) ogeher wih consn oupu price P w o produce figure 1 where he mrginl vlue produc is expressed s funcion of weigh. 5 5
For single niml wih he mrginl curves depiced in figure 1, he profi mximizing weigh o ermine producion is represened by * where MVP=MC. Now, ssume here re wo nimls in pen h re o be mrkeed ogeher. Le heir weighs be represened by 1 nd wih ( 1 + ) * =. By Jensen s Inequliy (Milehmmer, p. 10), we know h he verge of he mrginl vlue producs ( MVP ) will be less hn he mrginl vlue produc of he verge weigh ( ( *) = δ ) MVP over ny concve region of he mrginl vlue produc curve. In he cse of mximizing profis for he pen mrkeed ogeher, i is he verge of he mrginl vlue producs h we wish o eque o he consn δ represening he verge of he mrginl coss. Therefore, s figure 1 indices, wih wo nimls in he pen, profis re mximized by shifing he mrke weigh o he lef. The resul is lower mrke weigh for ech 1 niml ( nd, respecively) nd lower verge weigh **. Figure 1 $ MVP 1 δ MVP MC MVP MVP 1 ' 1 ** * ' eigh 6
The mgniude of he shif nd is subsequen effec on mrke iming decisions will be influenced by wo hings, he curvure of he mrginl vlue produc curve nd he disribuion of he niml weighs. In (5), we deermined h, under he ssumpion of consn oupu price, he MVP curve (6) would be concve when he weigh is beween 0.079 nd 0.685. hiemore (p. 6) poins ou h, prime me is found from pigs slughered beween 30% nd 60% of mure size. One cn conclude h, in he cse of swine, i is likely h he MVP curve will be in he ler sges of concviy round he profi mximizing mrke weigh. In erms of he effec of he disribuion, we cn expec ll symmeric disribuions lying wihin he concve region o behve similr o he wo nimls depiced in figure 1. Obviously, he lrger he sndrd deviion of he disribuion, he lrger he difference beween δ nd MVP. Thus, he degree of dispersion will ffec he mgniude of he shif from * o **. If he disribuion is symmeric, we cn expec * o lie closer o eiher 1 or where 1 nd represen he minimum nd mximum weighs, respecively, in he disribuion. Thus, n symmeric disribuion will likely decrese he difference beween δ nd MVP. If he disribuion is no conined wihin he concve region of he MVP curve, hen we cn expec o see furher decrese in he difference beween δ nd δ could be less hn MVP. Incresing Mrginl Coss MVP wih he possibiliy exising h ih regrds o mrginl cos, we hve limied ourselves o esiming he dily cos of feed. In figure 1, we nively ssumed his consn δ. This served is purpose s simplifying ssumpion in he bove exposiion bu, in reliy, he dily cos of feeding n niml grows wih he size of he niml. One of he lws of niml science is he long held belief h o 7
minin body weigh, nimls should be fed in proporion o heir mebolic body size (Prks; Kleiber). Therefore, funcion of he form dily feed inke, is he weigh of he niml, nd is some consn. 0.75 0.75 F = seems pproprie where F is hiemore (p. 589) poins ou h mos empiricl esimes of feed inkes of pigs of vrious weighs involve pigs growing posiively. He suggess vlue for beween 0.09 nd 0.11 when he weigh unis re mesured in kilogrms nd he pigs re being fed under commercil condiions. doping he lower bound nd convering o English unis leves us wih nive bu prcicl formul 0.75 F = 0.0 (7) o represen pounds of dily feed inke, F, s funcion of weigh,. If we ssume consn posiive feed price P f per pound of feed, hen he mrginl cos wih respec o ime, represening he cos of feeding he niml noher dy, cn be wrien s funcion of weigh MC 0. 75 ( ) P F = 0.0 P =. (8) f f Exmining he chrcerisics of he mrginl cos curve, we firs noe he obvious h (8) is posiive for ll posiive vlues of. Second, we noe h he mrginl cos wih respec o ime is monooniclly incresing since [ ( )] MC 0. = 0.15 P k 75 f ln > 0 for ll 0 < <. Finlly, we nlyze he concviy of he mrginl cos curve by clculing [ ( )] MC 0. 75 = 0.15 Pf k ln 1 + 0.75ln which is posiive when 8
1 + 0.75 ln < 0 or < e 4 3 0.636. (9) Therefore, he mrginl cos curve is convex whenever relionship (9) holds nd concve oherwise. pplying hiemore s observion from bove, i is hen likely h he mrginl cos curve will be concve over he weigh regions in which mrkeing of swine occurs. Jensen s Inequliy hen presens us wih siuion where we cn expec he verge of he mrginl coss o be less hn he mrginl cos of he verge. Figure depics our siuion wih wo nimls weighing 1 nd, respecively. ih boh he MC nd MVP curves being concve over he pplicble region, we cn expec o hve siuion where MVP < MVP( *) nd MC( *) nd MC < where ( ) 0.5 ( ) MVP = 0.5MVP 1 + MVP ( ) 0.5 ( ) MC = 0.5MC 1 + MC. The ne effec his hs on he mrginl profi will be deermined by he relive curvure of he wo curves over he pplicble region. 9
Figure $ MVP(*) MVP MC MVP MC MC(*) 1 * eigh Inuiively, we migh expec he siuion s i is depiced in figure where he curvure of he MVP curve is more pronounced hn h for he MC curve. Then, he verge mrginl profi *, would be less hn he mrginl profi of he verge, π = MVP MC (10) ( *) = MVP( *) MC( *) π. (11) This would led us o he conclusion h he verge mrginl profi would rech zero prior he weigh which he mrginl profi of he verge is zero. s in he cse of consn mrginl coss explored erlier, we cn expec profis for his pen of wo nimls wih incresing mrginl coss o be mximized n verge mrke weigh somewhere o he lef of. However, he couner blncing effec of concve mrginl cos curve will mke h shif less pronounced hn he shif from * o ** indiced in figure 1. 10
Empiricl pplicion pnel d se consising of welve weigh observions individully idenified for 350 hogs every 1-3 weeks from 14 dys of ge o 171 dys of ge ws obined from Purdue Universiy. The swine in he d se re ll gils king pr in Purdue Universiy sudy on nibioic remens. Two differen genoypes re represened in he d nd he pigs re divided ino 3 pens of pproximely 10-1 pigs per pen. ny poin in ime, ech pen is receiving he sme rion fed d libium. Excly hlf of he nimls re given n nibioic remen. However, he selecion of he remen nimls is done by rndom drw he beginning of he ril nd gin he beginning of he finishing phse. Therefore, he nimls fll ino one of four cegories concerning nibioic remens: (1) remen in boh he nursery nd finishing phse, () remen in he nursery nd no remen in finishing, (3) no remen in he nursery nd remen in finishing, or (4) no remen in eiher he nursery or finishing. The d se is firs nlyzed s if one growh ph exised for he enire se of 350 hogs. Our d se ws plgued by common problem in niml growh modeling. The fses growing pigs were mrkeed prior o he welfh weigh observion resuling in significn moun of missing d. Including ll welve observions o esime our model prmeers would downwrdly bis he pek of he sigmoidl growh curve (Crig nd Schinkel). Therefore, he group verge from observion welve ws no used in he growh curve esimion. Using he men vlues for he enire group ech of he firs eleven observions, we fied Gomperz growh curve (1) o he d. This resuled in he model 11
0.0148 1.55 e 370 ( = R = 0.9999) (1)' 370 s represenion of he growh ph of he pen verge. The fied curve from equion (1)' is grphed long wih he cul d of men weighs in figure 3. Figure 3 Fied Growh Curve cul Men. 400 350 300 50 eigh (lbs.) 00 150 100 50 0 0 50 100 150 00 50 300 (dys) The growh curve prmeers = 370 nd k = 0.0148 resuling from he esimion of (1)', combine o yield he mrginl vlue produc nd mrginl cos equions MVP( ) = 0.00651 ln (6)' 370 MC 0. 75 ( ) 0.01 = (8)' where he oupu price is ssumed consn P w = $0.44 per pound nd he feed cos is ssumed consn P f = $0.06 per pound. These re grphed in figure 4. e cn solve numericlly for 1
heir poin of inersecion = 30.58 which represens he profi mximizing mrke weigh for single verge niml. This corresponds o = 13.96 or pproximely 147 dys of ge. Figure 4 1 0.05 0.9 MC 0.05 0.8 0.7 = 30.58 lb 0.04 0.04 0.6 0.03 $ 0.5 MVP 0.03 Probbiliy 0.4 0.3 0. µ = 9.56 MVP = 0.7070 MC = 0.7070 π = 0.0000 = 13.33 µ = 30.58 MVP = 0.7035 MC = 0.7094 π = -0.0059 = 13.96 0.0 0.0 0.01 0.1 0.01 0 75 85 95 105 115 15 135 145 155 165 175 185 195 05 15 5 35 45 55 65 75 85 95 305 eigh 0.00 Our d se conins n observion of he cul weighs 146 dys of ge. Chisqure nlysis provides srong evidence h we cnno rejec he null hypohesis h hese weighs re normlly disribued (figure 5). nlysis of weigh d 13 dys of ge nd 153 dys of ge provided similr evidence of normlly disribued weighs (p-vlues of 0.903 nd 0.174, respecively). Therefore, we will opimize under he ssumpion h he niml weighs re normlly disribued wih men weigh of sndrd deviion of 1.4. deermined by model equion (1)' nd 13
Figure 5 80 70 men = 30.116 s. dev. = 1.399 medin = 30 skewness = 0.167 Chi-squre sisic = 8.083 p-vlue = 0.706 60 Frequency Norml 50 40 30 0 10 0 <=180 180-190 190-00 00-10 10-0 0-30 30-40 40-50 50-60 60-70 70-80 80-90 90-300 >300 min = 179 cul eigh Cegory Frequencies vs. Norml Disribuion 146 dys of ge mx = 99 Profi mximizion occurs when he verge mrginl profi, π = MVP MC, is equl o zero. In oher words, he opimizion problem is o deermine he men weigh such h where N(,σ ) = ( ) N (, σ ) d = 0 π = π (10) = o is norml probbiliy disribuion of wih men of nd sndrd deviion of σ = 1. 4. Dillon nd nderson (p. 14) poin ou h only if he probbiliy disribuion is of simple form, such s discree or ringulr, is n lgebric expression such s (10) convenienly pprised. e were ble o pprise i using he symbolic compuionl pckge clled Mple nd numericlly find he h mde equion (10) hold. However, we found i esier o nlysis nd conduc sensiiviy nlysis on our resuls by convering he norml disribuion in (10) ino discree disribuion in n Excel spredshee. The resuls were idenicl, o hree deciml plces, o hose obined in Mple. The deils of his conversion re 14
conined in ppendix nd he reder should ssume h ll resuls repored here re rrived using he Premium Solver for Excel. Resuls Our resuls indice h he opimum men weigh is indeed less hn he 30.58 lbs. which he mrginl vlue produc curve inersecs he mrginl cos curve. Figure 4 shows he implied shif o he lef from men weigh of 30.58 o men weigh of 9.56 lbs. h is necessry o opimize profis for his group of 350 swine sold s one uni. e clcule MVP(30.58) = MC(30.58) = 0.7101, MVP = 0. 7035, nd MC = 0. 7094 dollrs per dy he poin of inersecion. The relionship, ( 30.58) MC( 30.58) MVP MC < MVP = (1) < indices he concviy of boh mrginl curves wih he mrginl vlue produc curve slighly more concve hn he mrginl cos curve. In fc, he closeness of MC o he vlue of MC(30.58) indices h he mrginl cos curve is nerly liner. Mos impornly, however, relionship (1) indices he nonopimliy of feeding o men weigh of 30.58 lbs. The fc h, men weigh of 30.58 lbs., we hve MVP < MC indices h nimls hve been fed ps he poin of profi mximizion. How fr ps is deermined by solving equion (10) for he opimum men weigh. hen we solve equion (10), we deermine n opimum men weigh of 9.56 lbs. This produces he clculed vlues MC = 0. 7070 ( 9.56) 0. 7136 MVP, ( 9.56) 0. 7077 = MC =, nd MVP =. gin, his indices he relive concviy of he wo curves. However, i lso displys he difference beween he cul mrginl profi, π = 0, nd he 15
perceived mrginl profi, π ( 9.56) = 0.7136 0.7077 = 0. 0059 niml., indiced by he verge The finl sk is o deermine wh difference his pproximely one pound difference in men weigh mkes in he mrke iming decision. Plugging men weigh of = 9. 56 ino equion (1) nd solving for yields he opiml mrke iming of = 13. 33 dys. This represens n pproximely seven-enhs of dy difference in he mrke iming obined men weigh of 30.58 lbs. In oher words, = 13. 33 dys, he mrginl profi o be gined by feeding he pen of nimls one more dy is zero. Obviously, we would no expec he sevenenhs of dy difference beween opiml mrke iming for he group nd opiml mrke iming for he verge niml o significnly impc profis. However, one cn envision where relxion of some of he resricions of his model s i perins o swine could led o siuions where his gp is more significn. Sensiiviy nlysis Our bseline exmple for hogs urns ou o show h mrke iming bsed on he verge size is probbly sufficien decision rule. However, how would he mrke iming chnge for pen h is more heerogeneous such s we commonly see wih cle or wih smller operions? One wy o represen more heerogeneiy is by expnding he vrince in our model. In our hog exmple, we ssumed he sndrd deviion ws consn 1.4. Tble 1 summrizes he resuls if we ssume he sndrd deviion is held consn 15, 0, 5 or 30 lbs. Tble 1: Sndrd Deviion Opiml Men eigh MVP =Mc 15 30.076 0.7086 13.65 0 9.68 0.7075 13.41 5 9.9 0.7056 13.13 30 8.903 0.7018 131.93 16
Noe h even wih sndrd deviion of 30 lbs., he difference in he opiml mrke iming of = 131. 93 nd mrke iming deermined by he verge niml of =13. 96 is only bou dy. lso, noe h he chnge in opiml mrke iming s he sndrd deviion moves from 0 o 5 lbs. is greer hn he chnge in opiml mrke iming s he sndrd deviion moves from 5 o 30 lbs. This indices he influence of he convex porion of he mrginl vlue produc curve s more of he weigh disribuion moves beyond 0.68 which is pproximely 5 lbs. s he disribuion widens, weighs disribued in he convex porion of he curve will couner blnce he influence of he weighs disribued in he concve porion of he curve. This offseing effec will limi he size of he downwrd shif mde possible by n expnding sndrd deviion. Summry nd Conclusions This reserch provides useful insigh ino he opiml mrke iming for pens of livesock. In he presence of decresing mrginl reurns, he mrginl vlue ssocied wih he verge oupu level is no represenive of he verge mrginl vlue produc for he pen. The degree of his seprion is dependen upon he degree of concviy in he mrginl vlue produc curve nd he degree of dispersion ssocied wih he disribuion of oupu levels exising in he pen. This seprion my be prilly offse by n nlogous concviy in he mrginl cos curve ssocied wih he decresing mrginl increse in he cos of feeding growing niml. The ne effec cn be expeced o be such h he opiml mrke ime for he pen ken s whole rrives prior o he opiml mrke ime for he verge sized niml in he pen. Our empiricl pplicion o swine verified our heoreicl consruc bu provided n insignificn difference in he opiml mrke iming. Therefore, in he cse of he swine 17
indusry, one cn conclude h mrkeing groups of hogs bsed on he group verge ppers o be n economiclly sound echnique. The insignificnce of he differenil in mrke iming for our bseline cse sudy d is no olly unexpeced. The swine indusry hs homogenized he geneics o he poin h few disinguishble breeds exis in he feeding secor. Therefore, one would expec he verge pig o be very represenive of he group. Furhermore, he iming of he opimum mrke weigh wihin he growh cycle of pig is such h he concviy of he mrginl curves is miniml. However, he heoreicl consruc of our model ppers o be sound. Fuure reserch pplying he principls of our model o more diverse producion populions my likely yield significn insighs ino mrke iming decisions. ppendix for The norml disribuion in equion (10) ws convered ino discree form by clculing Nw = + 0.5 (, σ ) = N(, σ ) = 0.5 = o o. Then he verge of he mrginl vlue producs nd mrginl coss cn be clculed s wih he mrginl profi MVP MC = = o d ( ) = MVP ( ) N (, σ ) = = o ( ) = MC ( ) N (, σ ) ( ) = MVP ( ) MC ( ) π. w w 18
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