RESOURCE ALLOCATION AND A FITTED PRODUCTION FUNCTION

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Gwendolyn Blankenship
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1 RESOURCE ALLOCATION AND A FITTED PRODUCTION FUNCTION J. H. DULOY* I. Itroductio Over a cosiderable period of time, ecoometricias have bee attemptig to estimate the margial productivities of resources by the use of regressio aalysis. The first use of the most commoly used fuctioal form, the Cobb-Douglas or logarithmic fuctio, was i a attempt to aswer some questios cocerig the distributio of the product of the ecoomy betwee capital ad labour, usig time-series data.l Curretly, regressio methods of estimatig margial productivities have bee largely cofied to cross-sectioal studies of agricultural or pastoral regios2 There exists a cosiderable body of literature dealig with the statistical problems ivolved i fittig regressio fuctios for this purpose. There has also bee discussio of the validity of the results from a fuctio fitted to a sample of farms whe applied co a idividual farm. This preset article has othig to say o these problems, ad will, i fact, assume that the regressio coefficiets i the fuctios fitted provide ubiased estimates of the populatio coefficiets.q However, there exists a quite differet problem, largely utouched i the literature i relatio to estimated margial value products. This problem may be defied i a questio: Give that the productio fuctio has bee estimated, how is a optimum allocatio of resources defied, ad how should resources be moved to obtai it? I the ext sectio the most commo method of iterpretatio of the * Uiversit of Sydey. The author s thaks are due to Dr. H. S. Koij, who read the first drai ad made some useful suggestios. C. W. Cobb ad P. H. Douglas, A Theory of Productio. America Ecoomic Review, Vol. XVIII, Supplemet (March, 1928), pp *For a recet Australia example where a fuctio is fitted to time-series, see G. 0. Gutma, Ivestmet ad Productio i Australia Agriculture, Review of Marketig arid Agricrtltural Ecoomics, Vol. 23, No. 4 (December, 1955), pp See, for e:ample, H. S. Koij, Estimatio of a Average Productio Fuctio from Surveys, The Ecoomic Record, Vol. F V, No. 70 (April. 1959). pp The literature has bee reviewed by R. M. Parish ad J. L. Dillo, Recet Applicatios of the Productio Fuctio i Farm Maagemet Research, Review of Marketig ud Agricrtltural Ecoomics, Vol. 23, No. 4 (December, 1955), pp This assumptio, of course, implies the further assumptio that such populatio coefficiets i fact do exist; that is, that all farms i the sample are operatig o the same productio fuctio. 75

76 estimated fuctio will be described, ad i subsequet sectios a attempt will be made to provide a better approach to the iterpretatio, based o aswerig the questio posed above. 11.

2 76 estimated fuctio will be described, ad i subsequet sectios a attempt will be made to provide a better approach to the iterpretatio, based o aswerig the questio posed above. 11. Margial Productivities ad Market Prices At some stage i all studies usig Cobb-Douglas fuctios which the author has see, the margial value products of the resources eterig the fuctio have bee computed at the geometric mea level of icome of the sample, ad at the geometric mea level of the particular resource beig cosidered. That is, the M.V.P. of a particular resource is computed at its geometric mea, all other resources beig held costat at their geometric meas. Computatioally, this is very simple. If we have a fuctio ai Y = c TT xi... (1) where Y is gross farm icome i moey terms ad the Xi are the idividual resources either i value or physical uits, the the M.V.P. of the ith resource is simply MWxl = ai - MY Mxi where My ad Mxi are the geometric meas of icome ad of the ith resource, as defied above. The usual practice is the to compare MVPxi withpxi the market price of the ith resource. Where MVPxi > pxi, it is suggested that the use of the ifh resource should be expaded, ad, coversely, cotracted where MVPxi < pxi. This is doe successively for all resources XI,..., XI, eterig the fuctio. It is apparet that this procedure ivolves several special assumptios i additio to the assumptios regardig the fuctio itself made i this paper. The first of these special assumptios is that all resources are cosidered to be variable. This assumptio uderlies the examiatio of the relatioship betwee the M.V.P. ad the market price of each of the resources take successively. Geerally, however, authors go further tha this, ad suggest that, whereas the use of oe resource, for example, should be expaded, the use of aother should be cotracted. A ecessary coditio for the validity of this type of recommedatio is that the resource adjustmets beig cosidered are purely margial. Geerally, to brig resource use to a optimum, fairly large, ad certaily o-margial, adjustmets are ecessary. Summarisig, we ca state that if all resources are cosidered variable, ad the chages i resource allocatio required or implied are omargial, the, from a give fuctio, recommedatios ca be made for oe resource oly, usig the covetioal approach to these recommedatios. Where the above circumstaces hold, ad more tha oe resource is cosidered, the a error is itroduced, as the margial (2)

3 77 productivity of ay oe resource is a fuctio, ot oly of the level of that resource, but of the level also of all other resources i the fuctio. It seems tedious to spell out this commo kowledge i such detail, but, as will be see i Sectio V, o-observace of these cosideratios ca lead to wrog coclusios. The secod special assumptio is that capital is ot limited, i.e. that the coclusios usually draw idicate movemet towards maximisig icome, without capital restraits, by obtaiig a equatio of M.V.P. ad market price of resources. The use of this assumptio ca be questioed o two grouds. Firstly, the maximum icome positio ca be defied oly whe the sum of the elasticities i the fuctio is less tha oe, or, whe the sum is greater tha or equal to oe, oe or more of the resources are held costat, such that the sum of the elasticities of the variable resources is less tha oe. These cosideratios limitig the geerality of the criterio of equality betwee M.V.P. ad market price have by o meas always bee observed. Secodly, capital-ratioig, self-imposed to some extet, is geerally recogised i agriculture. The extet of re-allocatio of resources ope to most farmers thus usually ivolves the re-allocatio of existig resources, or at best, the use of a limited quatity of resources previously ot used o the farm.6 If this be the case, the the appropriate stadard agaist which to compare observed margial value products is the opportuity cost, rather tha the market price.o Some defiitios eed be clarified before proceedig to the more systematic treatmet i the ext Sectio. All variables will be cogsidered to be i value terms, that is, each idepedet variable Xi (which i practice will be a aggregate of iput categories) is the sum of the physical quatities of resources multiplied by the appropriate market prices ad icludig ay complemetary costs associated with the use of resource aggregate Xi. With this defiitio of XI, the test geerally used as described above of comparig the margial value products of resources with their market prices ow becomes a compariso of the margial value product of the resources with uity. I the ext sectio, the movemet towards the fial optimum of all margial value products equal to oe will be cosidered i two steps. For some empirical evidece o this poit, see E. 0. Heady ad E. R. Swaso, Resource Productivities i Iowa Farmig with Special Referece to Ucertaity ad Capital Use i Souther Zowa, Research Bulleti 388, Agricultural Experimet Statio, Iowa State College (Jue, 1952). 61 this paper, the opportuity cost of resource use will be computed from the fuctio, assumig, iitially, that all resources are readily trasferable. The assumptio that all resources ca be expressed i terms of capital ad thus allocated amog various aveues of expediture accordig to some profit maximisatio criteria raises the questio of the stickiess ad lumpiess of may resources. These questios are discussed i G. L. Johso, Classificatio ad Accoutig Problems i Fittig Productio Fuctio to Farm Record ad Survey Data, Chapter 9 i E. 0. Heady et al. (eds.), Resource Productivity, Returs to Scale, ad Farm Size, The Iowa State College Press, Ames Iowa, I the mai, these problems wiil be igored i the preset paper (see. however, Sectio IIIB below). This defiitio is adopted to simplify otatio i succeedig sectios. A slightly more complex otatio would eable a examiatio of the effects of chages i the prices of resources or of capital.

4 78 The first coditio for the optimum, the iteral coditio is satisfied whe the margial value products of all variable resources are equal to some umber, p. Where p # 1 the iteral coditio oly is satisfied. The exteral coditio is satisfied whe p = The Iteral Coditios for Equilibrium A. Cobb-Douglas Fuctio-All Resources Variable Give that there exists a productio fuctio ai Y = c lt x,... (1) we shall defie the coditios for iteral equilibrium as MVPxi = p i = 1,...,... (2) where p is some umber ad MVPxi is the margial value product of the ith resource. Xy is defied as the level of the ith resource whe (2) holds. I this sectio, we are cocered with the situatio where capital is limited, ad we wish to defie the optimum use of the give total quatum of resources existig o a idividual farm. At this stage, we shall assume that all resources are variable ad i moey terms as defied at the ed of Sectio 11. The T, the total quatum of resources, is simply T = C X i = %X:... (3) Note that X; P, are fuctios of T, hece, istead of p, we write pr as * the equilibrium value of all MVPX,, for C XIT = T. The, (2) ca be re-writte as 5 I =pr,...., (4) 8Xi * Whe this equality holds, the maximum gross icome, YT is forthcomig. Thepr = ai - YT X% y1 Xh = ai - pr i = 1,., ,..."_...-_... ~ l_l l... l.... ".. (5)... ".-...". ".""..."..(6)

5 I Substitute (6) ito (3), Y*, z ai T= _. pt = 1 ~,..- ".._._I ~ "". ". "...* _ adm Y% = - C ai. T... (8) Substitute (8) ito (6), ai T. 1 = 1,."..."...,. ".,.."_ ~..-..._... By comparig X$ with Xi, the observed level of use of the ith resource, it is possible to recommed resource movemets towards the optimum positio defied for the existig level of resources. IIIB. I the short ru, ot all resources are variable.s The method of aalysis developed above ca readily be exteded to iclude the situatio where the task is to determie the optimum combiatio of a give quatum of resources amog several resource categories, give that the level of oe or more resources is fixed. Cosider that XI, j = 1,..., m are resources which are fixed, ad i = 1,..., are variable resources, ad that there exists a that XI, fuctio Cobb-Douglas Fuctio-with Some Fixed Resources m aj a1 Y=cTTXXJ TT xi... (1) j=1 Let T be the total level of all resources ad T be the total level of variable resources. The, m T=T- C Xj, j=1-5 Xi -? X:T' As : before 1 =pr' Xff' i-1,. (7) (9)... (2)...l.l.l.l.._...l--ll.l " (3) A example of such a resource is family labour. A rather more iterestig example is lad which ca be cosidered as fixed, except i the extremely log period, by the way i which may samples are defied. Properties with o-cotiguous lad are frequetly excluded from the sample, ad thus, ay icrease of the farm area b the additio of o-cotiguous lad meas that the farm is ow operatig o a dif&ret productio fuctio. Hece, the oly icrease i the lad iput geerally cosidered is of cotiguous lad, which is geerally ot available except i the very log ru.

6 I this case p~' is a umber equal to the margial value products of the variable resources, Xi, at a equilibrium positio the level of which is determied by equatios ( 1) ad (2). pr' ai Y? = -..._... ". "..."...". " (4) XiT' ad Y;, x;, ai.... PT Rewritig (2) by substitutig (5), we have that, T' Yk = -,E ai ~ PT' ". ad p~' = 7 c ai... "-..."."...".".--.-."."--." (7) From (6) ad (7) (5) (6)...-. "..."".."l..."....,."-" (8) Note that g', theproportio of the total expediture o variable T' resources expeded o the ith resource, is ot a fuctio of either the level or the combiatio of fixed resources. I practice oe would ot expect that this property of the Cobb-Douglas fuctio would occur o farms where, for example, the proportios of variable costs represeted by expediture o labour ad fuel would vary with the level of ivestmet i machiery, etc. 111C. The Trascedetal Fuctio The results from IIIA (ad IIIB) are readily exteded to the situatio where the productio fuctio is of the form9 al bixi Y=clT Xi e... (11 As i IIIA, let T be the give quatum of resources available o a particular farm, so that, as before, T = C XI = C Xh... (2) As before, we defie SY mz-1,...,... (3) sxi Xi", "See A. N. Halter, H. 0. Carter, ad J. G. Hockig, "A Note o the Trascedetal Productio Fuctio," Joural of Farm Ecoomics, Vol. XXXIX, No. 4 (November, 1957). pp

7 ~ ~~ The, pr. = Y;. I 1 Y; is defied as i From (4), 81 ai... XiT the Cobb-Douglas case. (4) * PT - YT bi Substitutig (5) ito (2) ad re-arragig terms, C a, * T = YT... From (61 * {G m-yk C bi j -- 1 (6) -i- Cbi... (7) pt = Y1. Substitutig (7) ito (S), ai T xi+, = &ti + T (Cbi- bi) ". (8) Note that whe all the bi are zero, the productio fuctio, (l), * becomes a Cobb-Douglas fuctio ad the expressio for XI^, (8), simplifies to the expressio for the Cobb-Douglas case.l0 IV. The Exteral Coditio for Equilibrium I this sectio it is assumed that the estimated fuctio exhibits decreasig returs to scale, that is, i the Cobb-Douglas case, that the sum of the partial elasticities is less tha oe. Ulimited Capital As before, we have a productio fuctio of the form a1 Y=c Tr xi... (1) We wish to maximise profits, that is, to maximise Y - CX,, with o restraits upo the quatity of capital. It is assumed that the price of If bi is positive, Y = f (Xi) icreases at a icreasig rate if a -ai + vai > 1, or, if 0 < a < I, icreases at a decreasig rite util Y =, ad thece at a bi icreasig rate. See Halter, Carter ad Hockig, op. cit. If these values of a, ad b, are observed for oe or more of the idepedet variables i the fitted fuctio, the the method of aalysis developed here may become idetermiate. This is ot surprisig, as we are postulatig icreasig margial returs for at least oe of the factors. 1 0

8 82 capital is ot a fuctio of the quatity of capital demaded by the idividual firm. To maximise profits, we have 6(Y - EX,) 6x1 ai i.e. -Y = 1 xi From (3), Xi = ai Y = o... (2) By takig the aith power of both sides of (4), takig the products over i = 1,..., ad multiplyig both sides by c, we have ai C~I a1 c ii a i Y = CTrXi c ai Divide through by Y ad let Q = 1 - CaI (3) (4) = Y... (5) ai f ctai = Y... (6) - 1 at c - f Y = c Tat... (7) Substitutig from (4), 1 a1 - - f f Xi = ai c lt ai... (8) Xi here is the optimum quatity of the ith resource to use to maximise profits with capital ot limited. As before, we will call this level of the * ih resource, Xi. Lettig K be the optimum quatity of capital to use, the K = CX: 1 ai _. c I?... - = ( 1-c)~ Tai (9) * a1 K The, Xi = - 1-f a1 K - -- C a,... (10) It is a property of the Cobb-Douglas fuctio that the optimum

83 proportio of total resources devoted to the ith resource, - c X; is a costat for give resource prices ad is ot a fuctio of the total level of resources used.

9 83 proportio of total resources devoted to the ith resource, - c X; is a costat for give resource prices ad is ot a fuctio of the total level of resources used. Limited Capital The results from Sectio I11 above eable the defiitio of the best combiatio of a give budle of resources. These results ca be exteded to idicate whether a greater or a smaller total budle of resources ca profitably be employed. Computatioally, by substitutig the values for XT obtaied from the use of equatio (9) ito equatio (1) (Sectio IIIA), the value of Y: is obtaied. Substitutig this ito equatio (8) (Sectio IIIA), the value of PT is obtaied.ll Where pr > 1 the use of additioal reso.jrces is profitable. A itermediate step i the above calculatios yields quatities which are useful whe makig comparisos of the efficiecy of resource allocatio. For istace, do farmers o large properties use the resources uder their cotrol more efficietly tha small farmers?12 Separate fuctios could be fitted to the differet groups of farmers from the same district ad, for each, compute a efficiecy idex, cosistig of actual icome divided by Y; the expected icome whe the observed level of resources is combied optimally. Similarly, a idex measurig the efficiecy of utilisatio of idividual resources could be computed. x4 V. A Example Recommedatios for the re-allocatio of resources which are based o the usual method of iterpretig results from a fitted fuctio may well suggest shifts i the use of a particular resource i the opposite *l From the expressio, PT = a.-- :T These. quatities are easily computed ad have prove readily adapted to automatic treatmet o a electroic computer. IrA similar type of measure was computed for comparig the efficiecy of resource use of share-farmers ad ower-operators. See W. S. Miller, Comparative Efficiecy of Farm Teure Classes i the Combiatio of Resources, Agricultural Ecoomics Research, Vol. IX, No. 1 (Jauary, 1959). pp I this work, however, the idex of efficiecy was the ratio of actual total costs to the miimum total cost of producig a give level of output. This approach does riot allow the use of all the resources at the disposal of the farmer, requires special assumptios about the opportuity costs, ad does ot idicate iefficiecies arisig from operatig at a iappropriate level of productio.

10 ~ Xl x2 84 directio to the directio of movemet to a optimum positio. This may coveietly be show by a e~amp1e.l~ We shall assume a fuctio of the form y = 5,424 Xl~2068X,~1847)<,~4339)3,~1436 I this example, Cai = 0.969, i.e. there exists dimiishig returs to scale. The "observed" levels of the resources together with their margial value products are set out i Table 1.14 x3 x* Table I OBSERVED RESOURCES AND PRODUCTIVITIES Resource 1 Observed Level of I Observed Margial Value Resource Product a Iterpretig these results covetioally, we ote that the margial value products of resources XI ad Xz are less tha oe, ad those of Xs ad X4 are greater tha oe. We could the suggest that the use of the last two resources be expaded, ad that the use of the first two be curtailed. Such a re-combiatio of resources may certaily be a improvemet o the observed situatio. However, i the case of resource XI, it would ivolve a shift away from the level of use of XI appropriate to the optimum positio of the firm, whether or ot capital is limited. Firstly, assume that capital is ot limited as is implied by the compariso of margial value products with oe. The observed ad the optimum quatities of the four resources are give i Table 2. Table 2 CAPITAL NOT LIMITED Resource XI x2 x3 x.4 Observed Level Optimum Level 124, , ,400 a54 86,500 "A hypothetical example has bee used because the author did ot succeed i fidig i the literature a example which icluded all the iformatio required. A "The margial value products were computed from MVP = ai k/x, where xi Xi is the observed level of the ith resource ad 0 is the expected gross icome computed at the observed levels of all resources.

1 XI 2500 x2 9450 1187 x3 854 x4 I 85 From Table 2, it appears that, if the cosequeces of the assumptio of ulimited capital are traced out, the results obtaied go far beyod the bouds of reasoable

11 1 XI 2500 x x3 854 x4 I 85 From Table 2, it appears that, if the cosequeces of the assumptio of ulimited capital are traced out, the results obtaied go far beyod the bouds of reasoable extrapolatio. The optimum combiatio of resources whe capital is limited to the total of the observed resource levels from Table 1 is set out i Table 3, together with the observed levels of each resource ad its margial value product computed at the observed levels of all resources. Table 3 CAPITAL LIMITED Resource I Observed Level I Observed Margial I Optimum Level Value Product The results tabulated above (Table 3) suggest that, although the observed margial value product of XI is less tha oe, the resource XI is actually beig used at a level below the optimum level as the optimum has bee defied i this paper. (At the above level of total resource use, p~ = 1.12, ad hece the use of additioal capital would be profitable.) VI. Coclusios The commoly used method of computig margial value productivity of resources at some observed level of all resources, ad comparig these with the market prices of resources ca lead to icorrect coclusios about the directio of resource shifts required to brig about a optimum combiatio of resources. The equilibrium positio which is implied by the above procedure may ot exist (where the sum of the partial elasticities of variable resources is equal to or greater tha oe) or may (because of the costat elasticity of productio of the Cobb-Douglas fuctio) exist at a level of resource use far greater tha aythig existig i practice. A preferable approach is oe which allows the defiitio of a optimum positio which is attaiable, ad which, cosiderig all resources simultaeously, specifies the combiatio of resources satisfyig the required optimum.

Estimating Proportions with Confidence

Estimating Proportions with Confidence Aoucemets: Discussio today is review for midterm, o credit. You may atted more tha oe discussio sectio. Brig sheets of otes ad calculator to midterm. We will provide Scatro form. Homework: (Due Wed Chapter