Conjunctive Effects of Economies of Scale and Rate Structures in Establishing the Geographical Milk Supply Area of the Plant

Transcription

1 JOURNAL OF ECONOMIC TIlEORY 3, (1971) Conjnctive Effects of Economies of Scale and Rate Strctres in Establishing the Geographical Milk Spply Area of the Plant R. KENNETH DEHAVEN Department of Agricltral Economics, Clemson University, Clemson, Soth Carolina Received Jne 9, 1910 The precrsor of economies of scale was technological innovation. The innovators have not retired nor has the importance of their contribtion diminished. However, the near miracles of modern technology provide no sanctary, for technological feasibility is still sbservient to economic feasibility. Economies of scale in prodction have come to be viewed by some as the panacea of efficiency problems. The failre to recognize increased assembly costs for farm prodcts as the size of the processing plant increases is tantamont to an overly optimistic expectation of cost redction de to economies of scale. The dairy indstry is a case in point. The feasibility of cost redcing innovations in dairy processing plants has been largely dependent on the volme of milk available for processing. A redction in the nmber of processing plants and/or an increase in the qantity of milk spplied were prereqisite to the realization of economies of scale. Pressres for the assembly of ever-greater qantities of milk in a single location (plant) gave rise to a system of assessing farmto-market transportation charges called the "flat rate." This rate system holds the dbios distinction of perpetal criticism from its inception. Yet, the flat rate system predominant in the 1920's persists, the criticisms of prodcers, dairy indstry leaders, and economists notwithstanding. Under this ~ate strctre all prodcers within a given geographical spply area pay an identical charge per 100 ponds of milk shipped (i.e., a "flat rate"). However, the rate system is only one, easily observed, variable in an interdependent system of three sch variables. Few, if any, attempts have been made to explain this interdependence in theoretical terms. Sch an attempt reqires a delineation of the relationships between the geographical expansion of the milk spply area, the optimm size of processing plant, and the haling rate. This is the task assmed in.this paper. 199

2 200 DEHAVEN The analysis is divided into three theoretical cases. An f.o.b. farm price for milk is assmed in Case I (i.e., the processor pays farm-to-market transportation cost). A plant delivered price, where each prodcer pays the actal farm-to-market cost of transportation, is assmed in Case II. A plant delivered price where each prodcer pays the average per nit farm-to-market cost is assmed in Case III. THEORETICAL PERSPECTIVE Von Thiinen developed a theory of agricltral prodction patterns based on competition among alternative land ses. His theory, applied to the dairy indstry, dictates that milk prodced far from poplation centers will be manfactred into btter, cheese, etc., while only the milk prodced in close proximity to the city will be sold in flid form. When all of the milk prodced on von Thiinen's prodction plain may be processed in a single plant, the theory is inadeqate. Ba bb [l, p. 57] reports that the optimm scale in flid mil k processing plants seems to be between 40,000 and 60,000 qarts per day and diseconomies of size in distribtion do not begin to offset economies of scale in processing ntil. mch larger plant sizes are reached. Ths, the distribtion cost parameter is not assmed to be a constraint. Only the assembly and processing sides of the assembly-processing-distribtion triangle are considered. ANALYSIS "Spplies of raw prodcts reqired to increase the otpt of agricltral processing plants normally mst be secred over widening spply areas or by raising prices to nearby farmers to obtain additional prodction. In either case, procrement costs increase with plant volme':.. " [3, p. 767]. The impets for expanded spply areas, as opposed to increased prices ) to nearby prodcers, was fathered by the introdction of the motor trck as an efficient farm-to-market transportation vehicle and mothered by economies of scale in processing plants. The physical restrictions concomitant to von Thiinen's prodction zones have long since vanished. It is technologically feasible to locate a processing plant in Florida to process Wisconsin milk. However, this is no act of magic; there is an associated cost. The relevant variable (constraint) in fixing the size of

3 ) CONJUNCTIVE EFFECTS OF ECONOMIES 201 the spply area for a given plant (or grop of plants in a given location) is the farm-to-market trarisportation cost. Case I Assme a processing firm bying flid milk f.o.b. the farm and paying the same price to all prodcers. The cost of transportation is not a constraint in the prodcer's decision model; ths, the maximm distance that milk will be transported is the processor's perogative. Let the transportation cost fnction be a simple linear fnction of distance, say y = t(x).l Qantity delivered to the plant at any price eqal to or greater than per nit farm prodction cost is a fnction of distance, say q = fey) or q = J[t(x)]. Thenf(y) represents the transportation cost of varios qantities of milk, and this cost is a fnction of ' distance. With a single departre from conventional analysis, this convenient reslt allows a rather simple graphical analysis. The horizontal axis in Fig. 1 is labeled "qantity processed" (qantity LRAC' LRAC ~ e o f (yl o I.o=::=------,!-!: ,;o"""a:::n..-t,:-;:t"'"y-'P;:;:ro:::c=essed FIG. 1. Theoretical effects of transportation costs on cost minimization assming an F.O.B. farm price. of factor). By convention, this axis represents "qantity prodced" (qantity of prodct). In this analysis, only cost is considered; therefore, congrency between the processing cost and transportation cost schedles is soght. 2 (The alternative procedre wold be to convert fey) by the appropriate processing ratios.) In Fig. 1, "qantity processed" is assmed to be a fnction of fey) and the processing cost (LRAC). The sal assmptions, inclding constant factor prices, apply to the LRAC crve;f(y) is added to LRAC.. I Size of individal shipment and spatial density of prodction are probably as important as distance from plant in the farm-to-market transportation cost fnction. For simplicity, distance from plant is inclded here (see [2]).. "Cost per cwt.," as measred on the vertical axis in Fig. 1, may refer to transportation cost, processing cost, or the sm of the two as in LRAC'.

4 202 DEHAVEN yielding LRAC'. This allows a graphical view of how transportation costs, paid by the firm, affect the cost crve [4, p. 1]. The interval of increasing retrns to scale is redced from oql to oqo, and the range of possible cost redctions de to economies of scale is narrowed (i.e., in the range of economies to scale, LRAC' is everywhere above, and has less slope than, LRAC). The reslt is an optimm constrained at a smaller scale and a higher cost than wold occr if transportation costs are ignored. The least-cost soltion on LRAC' occrs at qantity qo. Recall that qo = j[t(x)]; the soltion of this eqation in terms of x, where x = distance from plant, fixes the oter bondary of the spply area: Under the assmed conditions, factor spply as a fnction of distance can be shown as the, crves in Fig. 2. Let Pi, where i = 1, 2, 3, 4, or 5, 2,. 9 " O~ ~------D-i-.t-=-c-e-f-r-om--p-la--nt FIG. 2. Theoretical relationship of qantity spplied and distance assming an F.O.B. farm price. represent an fo.b. farm price (i.e., assme Pi constant and distance variable). The se of linear crves is an analytical convenience; that the crves are monotonically increasing with distance is factal. Case II The more relevant assmption, however, is a plant delivered price for milk where transportation cost is a variable in the farmer's decision model. Let L; represent a constant plant delivered price and assme that the prodcer pays farm-to-market costs on the fnction t(x). Given a plant delivered price, the farm price received becomes an inverse fnction of distance from plant.'

5 CONJUNCTIVE EFFECTS OF ECONOMIES 203 In Fig. 3, L 1, L 2,..., and L5 represent the plant delivered prices associated with the specific spply fnctions. Now, however, transportation costs mst be, sbtracted from the plant delivered price to obtain a farm price. Or, price received by prodcers decreases with their distance from plant. This gives rise to spply fnctions that increase with distance ntil the plant delivered price mins per nit transportation costs eqals per nit farm prodction cost (i.e., L/ - t(x) = C). These points are denoted by the nmbers 1,2,3,4, and 5 in Fig. 3.!, 'tj -~! l, LS 0. ~ '" L4 ;:-.~ ~ 6 L2 L3 Ll Distance from Plant FIG. 3. Theoretical relationship of qantity spplied and distance assming a plant delivered price. From Fig. 3 a more conventional spply crve can be derived. The prices Li are constant on all crves; therefore, the maximm qantity ' spplied at any given price can be observed on the vertical axis. These prices and qantities are plotted in Fig. 4. By carrying forward the LRAC crve of Fig. 1, the analytical rdiments concomitant to' the constrained economies of scale analysis, assming a constant plant delivered price, are complete. L Pactor spply ' Crve l L4 L3.l! " L2 G,~ Ll ' k Do 0 Oantity FlO. 4. Theoretical spply fnction assming a plant delivered price.

6 204 DEHAVEN Part A of Fig. 5 depicts a firm with a typical cost crve (from Fig. I) facing the factor spply crve (from Fig. 4) shown in Part B. Let factor prices Ll, L2,..., and Ls be specific to LRAC 1,LRAC 2,..., and LRAC 5, respectively. At price L 1, the processor is able to move down LRAC 1 to point a. The cost crve contines to fall; however, the processor is constrained by the spply fnction. In order to increase qantity spplied, factor price mst be increased, say, to L2. This increases qantity spplied PART A,..: J: 0: w a.,... Ul o LRAC 5 LRAC 4 LRAC 3 LRAC2 LRAC,,.: J: 0: UJ a. UJ 0: a. o '---~---7,--'r--I-----Q-U-AN~T-'T-Y-P-R-OCESSED. I ' I I ' --:--:-~~-r- I I I. L4 - -:- - ~ - ~- I I, L3 - ~ - -t- I Lz --:--',/ /~,. I I I '., I I I o~~~~'--~~ qo ql qz q3 '14 QUANTITY PROCESSED FIG. 5. Theoretical relationship of factor spply and prodct cost assming a plant delivered factor price. to ql ; the relevant cost crve is now LRAC 2 and another point on LRAC" is established at point b. That is, the increased factor price yielded an increase in qantity and a "jmp" to a higher cost crve. The steps are repeated. to reach points c, d, and e. 3 The derived crve, LRAC" in Fig. 5 simply connects the points at which frther expansion in processing is constrained byqantity spplied. From this series of eqilibria (on crve LRAC"), it is possible to select the least cost soltion (e.g., point b in Fig. 5). Spply restrictions force expansion along LRAC". No soltion as to the optimm size of plant is possible otside a system of simltaneos eqations which incldes the factor spply fnction. The cost crves of the firm were not altered. I The nmber of possible crves between LRAC 1 and LRAC. is, of corse, infinite'.

7 CONJUNCTIVE EFFECTS OF ECONOMIES 205 one iota; the inherent economies of scale are intact. Yet the transportation cost constraint is as binding as that of Case T. The assmptions of the prely competitive model allow the problem treated here to be easily circmvented. No single firm is assmed large enogh to affect the price of inpts. Ths, at a competitively determined price, say, L1, in Fig. 5, the firm wold have no factor spply restriction. ' The spoiler here is that the spatial iimitations imposed by transportation. costs and the large economies of scale in modern processing plants preclde a competitive system. Case III Assme the price is still a plant delivered price with prodcers paying I transportation costs. However, they do not pay on the fnction lex) even thogh this is still assmed to be the actal cost fnction. Rather, all prodcers pay lex'), where x' = k/2 and k is the distance from the plant to the oter bondary of the spply area. The more distant prodcers sffer a differential cost only to the degree that their greater distance from plant increases the cost to all prodcers. Price discrimination is commonly defined as the act of charging two or more different prices for identical prodcts or services. Define one nit of milk collection service as the transport of 100 ponds of milk one mile. A single charge per 100 ponds, "treating everybody the same," reslts in price discrimination when distances transported are variable. For example, both the prodcer one mile and the prodcer 10 miles from plant pay x cents per 100 ponds of milk transported. The near prodcer boght one nit of milk collection service while the more distant prodcer boght 10 nits. Ths, the price per nit was x to the near prodcer and x/io to the more distant prodcer. ' With one exception, the analysis of economies of scale as constrained by transportation cost nder the fiat rate system is identical with that presented in Case II. The spply crve is more elastic de to the fact that transportation costs increase with distance only as the average vale of lex) increases. If all prodcers paid the actal rates inqicated on lex), an expansion of the spply area wold have little or no effect on the rates of near plant prodcers. Since, with distance, the average vale of lex) increases mch more slowly than does the fnctional vale, the point where plant delivered price mins transportation cost eqals farm prodction cost (i.e., L( - lex) = C) is reached at mch greater distances than wold be possible nder the assmptions of Case II... In fact, the only alternative rate system which wold allow milk from a wider geographical area to enter a single plant (or market) at.a given; - \ I I

8 206 DEHAVEN plan t delivered price wold be a system of zero transportation costs. (This statement is based on the assmption that prodcers wold not accept a system charging near plant prodcers a higher rate). That is to say that the flat rate system allows a constrained optimm size of plant second in desirability only to the optimm nder a system of free transportation. An f.o.b. farm price obviosly allows no discrimination in haling rates; nor does a variable rate system where each prodcer pays his proportionate share. 4 The one collection rate system consonant with price discrimination is the flat rate system. Rents which wold normally accre to near plant prodcers are redistribted to distant prodcers sch that any farm-to-market cost advantage de to location is completely erased. This sbsidization of distant prodcers allows a maximm size of geographical spply area. If the chosen corse of action is to increase factor spply throgh an expansion of the geographical spply area, maximization of the geographical spply area is synonomos with maximization of spply. REFERENCES 1. E. M. BABB, Changing marketing patterns and competition for flid milk, J. Farm COil. 48 (1966), R. K. DEHAVEN, "Systems of Variable Rates for Milk Collection Rotes-Derivation and Evalation," npblished Ph.D. thesis, University of Missori, B. C. FRENCH, Some consideration in estimating assembly cost fnctions for agricltral prodcts, J. Farm COil. 42 (1960), W. R. HENRY AND J. A. SEAGRAVES, Economic aspects of broiler prodction density. J. Farm COli. 42 (1960), It is recognized that an "imperfect" variable rate system may retain some discrimination.