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variable cost analysis

  An approach to industrial location (or the location of facilities in general) concerned with spatial variations in production costs. This is one of two major alternative theoretical approaches to industrial location in the classical tradition, the other being variable revenue analysis.

The variable cost model in its simplest form may be expressed as follows:

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where TCi is total cost of producing a given volume of output at location i, Qj is the input coefficient or required quantity of input j and Uij is the unit cost of the input in question at location i. Total cost is simply the summation over n inputs of the product of required quantity and unit cost.

As the expression above shows, total cost (for a given output) depends on two major considerations: the input coefficients and spatial variations in input cost. The input coefficients arise from the technique adopted in manufacturing the goods in question: they may vary from place to place, within the technical constraints that may require certain minimum quantities, as the quantity of a particular input used increases where its cost is low relative to that of other inputs. The capacity to substitute among inputs, along with the fact that input combinations may also vary with scale of output, greatly increases the complexity of variable cost analysis in both theory and practice.

The unit cost of the required input will vary in geographical space in obvious ways. For many materials these will be a reflection of transport costs (see also pricing policies), but for other inputs all manner of complications can arise to influence the spatial cost pattern. For example, the cost of labour per unit of output can vary with actual wage rates, fringe benefits paid, training costs and welfare facilities provided by the firm, as well as with the productivity of labour. The intricacy of modern systems of input-output linkages, as manufacturing processes become more sophisticated technically, is a major complication in calculating input costs. Added to this is the difficulty of incorporating more general external economies and advantages arising from agglomeration.

Variable cost analysis proceeds under one or other of two assumptions concerning the incidence of alternative locations: that they are discrete points or that a continuous surface exists. The assumption of relatively few discrete points is the more realistic, in the sense that actual location practice generally involves the evaluation of a small number of alternatives: this is the usual framework for comparative cost analysis. However, industrial location theory often proceeds on the implicit assumption that those locations actually considered and costed are selected from an infinite number of possible locations. Total cost is thus conceived of as a continuous spatial variable.

The concept of a cost surface is central to variable cost analysis. A cost surface depicts spatial variations in the cost of production as a three-dimensional surface, with distance along the two horizontal axes and cost in pecuniary units on the vertical axis (see figure). Cost surfaces are typically portrayed by the pecuniary equivalent of contour lines. The surface may represent spatial variations in the cost of single inputs, such as labour, land or individual materials. It may also represent total operating costs at a given scale, though this is more difficult to identify empirically .

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variable cost analysis

The cost surface reveals, in effect, the topography of cost of production, any section through which can be depicted as a space cost curve, or plot of spatial variations in production cost along one distance dimension. This is the spatial analogy of the cost curve of conventional production theory in economics which depicts the relationship between cost of production and volume of output. The space cost curve can portray single-input costs or total costs at a given scale. The form of the space cost curve — whether it is steep or shallow — can give some indication as to the degree of restriction likely to be imposed on locational choice if plant viability is to be achieved. With appropriate assumptions as to the form of the revenue surface (see variable revenue analysis), an optimal or profit-maximizing location can be identified, along with the spatial margins to profitability constraining freedom of choice of location.

The spatial margin is the locus of points where the total cost of producing a given volume of output is equal to the total revenue obtainable from selling that output. This defines the area within which profitable operation is possible. The concept of the spatial margin to profitability was introduced by E.M. Rawstron (1958), as one of the few really original spatial-economic concepts devised by a geographer. Its derivation from the cost and revenue surfaces is shown in the figure. The significance of the spatial margin is that it permits the incorporation of sub-optimal decisions into a theoretical framework previously directed towards searching for the single optimal profit-maximizing location (O in the figure). Anywhere within the margin offers some profit, so a firm would be able to exercise freedom of choice within these limits, trading-off profits for personal or other considerations. Within the margin a firm can locate in total ignorance yet still survive. The shape and extent of the spatial margin will vary with the prevailing cost and revenue surfaces. Some industries operate within wide margins, while others will be confined by tight and localized limits to viability. The margins can vary with the nature of the industrial organization — they may be wider for highly skilful entrepreneurs than for the less able, for example.

If revenue is assumed to be a spatial constant, then the optimal location for the profit-maximizing firm will be where total cost is minimized, i.e. at the lowest point on the total cost surface. How this point arises and how it may be identified are the problems around which Alfred Weber (1929) built his classical approach to industrial location theory. Much of the work of later exponents of the variable cost approach, such as T. Palander (1935) and E.M. Hoover (1948), was greatly influenced by Weber\'s theory. For almost half a century the variable cost model constituted the core of industrial location theory, but its lack of realism resulted in a broadening perspective with more attention given to revenue and decision-making considerations. Nevertheless, variable cost analysis is still highly relevant to actual location practice, from the small unit of production to the multinational firm. Cost of production is still very important to locational viability. And in the field of industrial development planning, variable cost analysis still provides a useful framework for the design of spatial strategy. (DMS)

References Hoover, E.M. 1948: The location of economic activity. New York: McGraw-Hill. Palander, T. 1935: Beitrage zur Standortstheorie. Uppsala: Almqvist and Wiksell. Rawstron, E.M. 1958: Three principles of industrial location. Transactions, Institute of British Geographers 25: 132-4 2. Weber, A. 1929: Alfred Weber\'s theory of the location of industries, trans. C.J. Friedrich. Chicago: University of Chicago Press. (Reprinted 1971, New York: Russell and Russell; first German edition, 1909.)

Suggested Reading Smith, D.M. 1981: Industrial location: an economic geographical analysis, 2nd edn. New York: John Wiley. Smith, D.M. 1987: Neoclassical location theory. In W. Lever, ed., Industrial change in the United Kingdom. London: Longman, 23-37.



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