JOURNAL OF ECONOMIC TIlEORY 3, 199-206 (1971)

Conjunctive Effects of Economies of Scale and Rate Structures in Establishing

the Geographical Milk Supply Area of the Plant

R. KENNETH DEHAVEN

Department of Agricultural Economics, Clemson University, Clemson, South Carolina 29631

Received June 9, 1910

The precursor of economies of scale was technological innovation. The innovators have not retired nor has the importance of their contribution diminished . However, the near miracles of modern technology provide no sanctuary, for technological feasibility is still subservient to economic feasibility.

Economies of scale in production have come to be viewed by some as the panacea of efficiency problems. The failure to recognize increased assembly costs for farm products as the size of the processing plant increases is tantamount to an overly optimistic expectation of cost reduc­tion due to economies of scale. The dairy industry is a case in point.

The feasibility of cost reducing innovations in dairy processing plants has been largely dependent on the volume of milk available for processing. A reduction in the number of processing plants and/or an increase in the quantity of milk supplied were prerequisite to the realization of economies of scale. Pressures for the assembly of ever-greater quantities 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 dubious distinction of perpetual criticism from its inception. Yet, the flat rate system predominant in the 1920's persists, the criticisms of producers, dairy industry leaders, and economists notwithstanding.

Under this ~ate structure all producers within a given geographical supply area pay an identical charge per 100 pounds of milk shipped (i.e., a "flat rate"). However, the rate system is only one, easily observed, variable in an interdependent system of three such variables. Few, if any, attempts have been made to explain this interdependence in theoretical terms. Such an attempt requires a delineation of the relationships between the geographical expansion of the milk supply area, the optimum size of processing plant, and the hauling rate. This is the task assumed in .this paper.

199

200 DEHAVEN

The analysis is divided into three theoretical cases. An f.o.b. farm price for milk is assumed in Case I (i .e., the processor pays farm-to-market transportation cost). A plant delivered price, where each producer pays the actual farm-to-market cost of transportation, is assumed in Case II. A plant delivered price where each producer pays the average per unit · farm-to-market cost is assumed in Case III.

THEORETICAL PERSPECTIVE

Von Thiinen developed a theory of agricultural production patterns based on competition among alternative land uses. His theory, applied to the dairy industry, dictates that milk produced far from population centers will be manufactured into butter, cheese, etc., while only the milk produced in close proximity to the city will be sold in fluid form. When all of the milk produced on von Thiinen's production plain may be processed in a single plant, the theory is inadequate.

Ba bb [l, p. 57] reports that the optimum scale in fluid mil k processing plants seems to be between 40,000 and 60,000 quarts per day and diseconomies of size in distribution do not begin to offset economies of scale in processing until . much larger plant sizes are reached. Thus, the distribution cost parameter is not assumed to be a constraint. Only the assembly and processing sides of the assembly-processing-distribution triangle are considered.

ANALYSIS ·

"Supplies of raw products required to increase the output of agri­cultural processing plants normally must be secured over widening supply areas or by raising prices to nearby farmers to obtain additional production. In either case, procurement costs increase with plant volume': .. " [3, p. 767].

The impetus for expanded supply areas, as opposed to increased prices ) to nearby producers, was fathered by the introduction of the motor truck 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 production 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

) CONJUNCTIVE EFFECTS OF ECONOMIES 201

the supply area for a given plant (or group of plants in a given location) is the farm-to-market trarisportation cost.

Case I

Assume a processing firm buying fluid milk f.o .b. the farm and paying the same price to all producers. The cost of transportation is not · a constraint in the producer's decision model; thus, the maximum distance that milk will be transported is the processor's perogative.

Let the transportation cost function be a simple linear function of distance, say y = t(X).l Quantity delivered to the plant at any price equal to or greater than per unit farm production cost is a function of distance, say q = fey) or q = J[t(x)]. Thenf(y) represents the transporta­tion cost of various quantities of milk, and this cost is a function of ' distance. With a single departure from conventional analysis, this conve­nient result allows a rather simple graphical analysis.

The horizontal axis in Fig. 1 is labeled "quantity processed" (quantity

~

e o u

LRAC'

LRAC

f (yl

o I.o=::=------,!-!:-------,;o"""ua:::n..-t ,:-;:t"'"y -'P;:;:ro:::c=essed

FIG. 1. Theoretical effects of transportation costs on cost minimization assuming an F.O.B. farm price.

of factor). By convention, this axis represents "quantity produced" (quantity of product). In this analysis, only cost is considered; therefore, congruency between the processing cost and transportation cost schedules is sought.2 (The alternative procedure would be to convert fey) by the appropriate processing ratios.)

In Fig. 1, "quantity processed" is assumed to be a function of fey) and the processing cost (LRAC). The usual assumptions, including constant factor prices, apply to the LRAC curve;f(y) is added to LRAC

. .

I Size of individual shipment and spatial density of production are probably as important as distance from plant in the farm-to-market transportation cost function. For simplicity, distance from plant is included here (see [2]).· .

• "Cost per cwt.," as measured on the vertical axis in Fig. 1, may refer to transporta­tion cost, processing cost, or the sum of the two as in LRAC'.

9 "

202 DEHAVEN

yielding LRAC'. This allows a graphical view of how transportation costs, paid by the firm, affect the cost curve [4, p. 1].

The interval of increasing returns to scale is reduced from oql to oqo ,

and the range of possible cost reductions due 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 result is an optimum constrained at a smaller scale and a higher cost than would occur if transportation costs are ignored. The least-cost solution on LRAC' occurs at quantity qo. Recall that qo = j[t(x)]; the solution of this equation in terms of x, where x = distance from plant, fixes the outer boundary of the supply area:

Under the assumed conditions, factor supply as a function of distance can be shown as the, curves in Fig. 2. Let Pi , where i = 1, 2, 3, 4, or 5,

2, .

O~--------~------D-i-.t-=-c-e-f-r-om--p-la--nt

FIG. 2. Theoretical relationship of quantity supplied and distance assuming an F.O.B. farm price.

represent an fo.b. farm price (i.e., assume Pi constant and distance variable). The use of linear curves is an analytical convenience; that the curves are monotonically increasing with distance is factual.

Case II

The more relevant assumption, 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 assume that the producer pays farm-to-market costs on the function t(x). Given a plant delivered price, the farm price received becomes an inverse function of distance from plant.'

CONJUNCTIVE EFFECTS OF ECONOMIES 203

In Fig. 3, L1 , L2 , ... , and L5 represent the plant delivered prices associated with the specific supply functions. Now, however, transportation costs must be, subtracted from the plant delivered price to obtain a farm price. Or, price received by producers decreases with their distance from plant. This gives rise to supply functions that increase with distance until the plant delivered price minus per unit transportation costs equals per unit farm production cost (i.e., L/ - t(x) = C). These points are denoted by the numbers 1,2,3,4, and 5 in Fig. 3. !,

l ,. • 'tJ -~ ! ....

LS 0. 0. ~

'" ;:- L4 .~ L3 ~ 6 L2

Ll

Distance from Plant

FIG. 3. Theoretical relationship of quantity supplied and distance assuming a plant delivered price.

From Fig. 3 a more conventional supply curve can be derived . The prices Li are constant on all curves; therefore, the maximum quantity ' supplied at any given price can be observed on the vertical axis. These prices and quantities are plotted in Fig. 4. By carrying forward the LRAC curve of Fig. 1, the analytical rudiments concomitant to ' the constrained economies of scale analysis, assuming a constant plant delivered price, are complete.

Pactor supply '

L - -5

Curve

l L4

L3

" .l! L2 G

,~ Ll ' k

Do

0 Ouantity

FlO. 4. Theoretical supply function assuming a plant delivered price.

204 DEHAVEN

Part A of Fig. 5 depicts a firm with a typical cost curve (from Fig. I) facing the factor supply curve (from Fig. 4) shown in Part B. Let factor prices Ll , L2 , ... , and Ls be specific to LRAC1 ,LRAC2 , ... , and LRAC5 ,

respectively. At price L1 , the processor is able to move down LRAC1

to point a. The cost curve continues to fall; however, the processor is constrained by the supply function. In order to increase quantity supplied, factor price must be increased, say, to L2 . This increases quantity supplied

PART A

,..: J: u 0: w a. ,... Ul o u

,.: J: u 0: UJ a. UJ u 0: a.

LRAC5

LRAC4 LRAC3 LRAC2 LRAC,

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 supply and product cost assuming a plant delivered factor price.

to ql ; the relevant cost curve is now LRAC2 and another point on LRAC" is established at point b. That is, the increased factor price yielded an increase in quantity and a "jump" to a higher cost curve. The steps are repeated. to reach points c, d, and e.3

The derived curve, LRAC" in Fig. 5 simply connects the points at which further expansion in processing is constrained byquantity supplied. From this series of equilibria (on curve LRAC"), it is possible to select the least cost solution (e.g., point b in Fig. 5). Supply restrictions force expansion along LRAC". No solution as to the optimum size of plant is possible outside a system of simultaneous equations which includes the factor supply function. The cost curves of the firm were not altered.

I The number of possible curves between LRAC1 and LRAC. is, ·of course, infinite'.

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 assumptions of the purely competitive model allow the problem treated here to be easily circumvented. No single firm is assumed large enough to affect the price of inputs. Thus, at a competitively determined price, say, L1 , in Fig. 5, the firm would have no factor supply restriction. ' The spoiler here is that the spatial iimitations imposed by transportation . costs and the large economies of scale in modern processing plants preclude a competitive system.

Case III

Assume the price is still a plant delivered price with producers paying I

transportation costs. However, they do not pay on the function leX) even though this is still assumed to be the actual cost function. Rather, all producers pay leX'), where x' = k/2 and k is the distance from the plant to the outer boundary of the supply area. The more distant producers suffer a differential cost only to the degree that their greater distance from plant increases the cost to all producers.

Price discrimination is commonly defined as the act of charging two or more different prices for identical products or services. Define one unit of milk collection service as the transport of 100 pounds of milk one mile. A single charge per 100 pounds, "treating everybody the same," results in price discrimination when distances transported are variable. For example, both the producer one mile and the producer 10 miles from plant pay x cents per 100 pounds of milk transported. The near producer bought one unit of milk collection service while the more distant · producer bought 10 units. Thus, the price per unit was x to the near producer and x/IO to the more distant producer. '

With one exception, the analysis of economies of scale as constrained by transportation cost under the fiat rate system is identical with that presented in Case II. The supply curve is more elastic due to the fact that transportation costs increase with distance only as the average value of leX) increases.

If all producers paid the actual rates inqicated on leX), an expansion of the supply area would have little or no effect on the rates of near plant producers. Since, with distance, the average value of leX) increases much more slowly than does the functional value, the point where plant delivered price minus transportation cost equals farm production cost (i.e., L( - leX) = C) is reached at much greater distances than would be possible under the assumptions of Case II. . . In fact, the only alternative rate system which would allow milk from

a wider geographical area to enter a single plant (or market) at .a given;

- \

I I

206 DEHAVEN

plan t delivered price would be a system of zero transportation costs. (This statement is based on the assumption that producers would not accept a system charging near plant producers a higher rate) . That is to say that the flat rate system allows a constrained optimum size of plant second in desirability only to the optimum under a system of free transportation.

An f.o.b. farm price obviously allows no discrimination in hauling rates; nor does a variable rate system where each producer pays his proportionate share.4 The one collection rate system consonant with price discrimination is the flat rate system. Rents which would normally accrue to near plant producers are redistributed to distant producers such that · any farm-to-market cost advantage due to location is completely erased. This subsidization of distant producers allows a maximum size of geographical supply area. If the chosen course of action is to increase factor supply through an expansion of the geographical supply area, maximization of the geographical supply area is synonomous with maxi­mization of supply.

REFERENCES

1. E. M. BABB, Changing marketing patterns and competition for fluid milk, J. Farm £COil . 48 (1966), 53-68.

2. R . K. DEHAVEN, "Systems of Variable Rates for Milk Collection Routes-Deriva­tion and Evaluation," unpublished Ph.D. thesis, University of Missouri, 1969.

3. B. C. FRENCH, Some consideration in estimating assembly cost functions for agricul­tural products, J. Farm £COil. 42 (1960), 767-778.

4. W. R. HENRY AND J. A. SEAGRAVES, Economic aspects of broiler production density • . J. Farm £COli. 42 (1960), 1-13.

• It is recognized that an "imperfect" variable rate system may retain some discri­mination.