Spatial Economics
Spatial Economics
I. The Partial Equilibrium ApproachEdgar M. Hoover
II. The General Equilibrium ApproachLeon N. Moses
I. THE PARTIAL EQUILIBRIUM APPROACH
Spatial economics deals with what is where, and why. The “what” refers to every type of economic entity, i.e., production establishments, other kinds of businesses, households, and public and private institutions. “Where” refers basically to location in relation to other economic activity, i.e., to questions of proximity, concentration, dispersion, and similarity or disparity of spatial patterns. The “where” can be defined in broad terms such as regions or metropolitan areas, or in microgeographic terms such as zones, neighborhoods, or sites. The “why” refers to explanations within the somewhat elastic limits of the economist‘s competence.
Location theory describes this kind of analysis when the emphasis is upon alternative locations for specified kinds of activities, such as industry. Regional analysis is concerned with groupings of interrelated economic activities in proximity, within specified areas or types of areas; and the theory of interregional trade refers to the economic relationships between such areas.
Decision units and their interdependence. The explanations provided by spatial economic theory are ultimately in terms of the economic motivation and behavior of individual decision units and the ways in which their decisions react upon each other. A decision unit in this context can be, say, a business enterprise, a household, a public institution, or a labor union local. Here, as elsewhere in economic theorizing, simplifying assumptions about motivation are used—for example, the assumption that a business firm will prefer locations that provide higher rates of return to the investments of its owners, or the assumption that households will prefer locations with higher and more dependable levels of real income.
Location theory views a decision unit (most often a business establishment or a household) as weighing the desirability of alternative locations. The unit, wherever located, needs to obtain certain “inputs” (e.g., labor services, materials, electric energy, police protection, information) and needs to dispose of certain “outputs” (e.g., goods produced in a factory, labor services of members of a household, services provided by a hospital). The unit functions as a converter of inputs into outputs within technical limits described by its “production function” (for example, a shoe factory as such can convert various alternative combinations of leather, plastics, labor, energy, and so on into various alternative combinations of shoes and by-products). Finally, the unit derives from its activity a residual “return” which is the measure of satisfaction of its objectives.
From the standpoint of a particular decision unit, with a production function that gives it a limited range of alternative ways of combining inputs and producing outputs, some locations are better than others. Thus, the terms on which out-puts can be disposed of will depend on access to established markets for such outputs; labor and other service inputs of the types required will be available on more favorable terms in some places than in others; land for cultivation or building will be available in different qualities and at different prices in various locations.
The process by which the decision unit weighs all these location factors and makes a choice of location and production technology is describable, as first clearly pointed out by Predohl (1928), in terms of marginal substitutions. Such analysis is not peculiar to location economics but is part and parcel of the more general body of economic theory of rational firm and household behavior. The distinctive task of spatial economics is to identify and account for the development of systematic spatial configurations of advantage for economic activities, as they arise out of the interaction of different decision units upon one another in ways strongly conditioned by distance. There is an analogy here to the work of the physicist who identifies systematic spatial patterns of the microstructure of matter (e.g., in molecules, atoms, or crystals) and explains them in terms of the interaction of attractive and repulsive forces between units.
Some of the more important ways in which decision units interact in a systematic spatial way can be cited. For example, sellers of a product compete for markets; users of a material compete for the source of supply; firms in a labor market compete for labor; economic activities in a city compete for space. Such interrelationships appear as forces of mutual repulsion or dispersion between the competing units. At the same time, when one unit supplies a good or service to another, either or both will have an interest in proximity for the sake of reducing transport cost and inconvenience. And many kinds of production and exchange are subject to important economies of scale, calling for some degree of spatial concentration. Suppliers of complementary products and services find themselves attracted to the same markets, and buyers of jointly produced goods or services find themselves attracted to the same sources. Here we have forces of mutual attraction or agglomeration. Both the repulsive and the attractive forces can apply either as between like decision units (e.g., similar households, or firms in the same industry) or as between unlike units that are complementary or competitive (e.g., a seller and a buyer of a product, a household and an employer, or a supermarket chain and a bank both thinking of buying the same parcel of urban land).
A general equilibrium theory of spatial economic relations takes cognizance simultaneously of all the important types of spatial interdependence of firms, households, and other decision units. A partial equilibrium theory focuses on just one or a few selected relationships, which can then be explored with greater attention to realistic detail, while other elements are taken as given. Thus, by making the necessary simplifying assumptions, we can focus on, say, the way in which complexes of metals industries locate in response to given market, raw material, and technological situations; the allocations of land in an urban central business district; the patterns of residence adopted by people employed in an industrial area; the development of reciprocal trade between two regions; or the choice of a good site for a new suburban shopping center.
The remainder of this article describes some of the various lines of partial equilibrium spatial analysis that have been most extensively developed by economic theorists. In each case the point of departure is the simplest case, in which all but a very few variables are ignored. Some indication is given of the ways in which this type of analysis can gradually approach reality by successive relaxations of the initial simplifying assumptions.
Transport orientation . “Transport orientation” refers to one of the classic cases of location determination under highly simplified assumptions. It was first set forth by the engineer Wilhelm Launhardt in 1885, further developed by the economist Alfred Weber (1909), and later elaborated by Tord Palander (1935) and others [see Weber, Alfred]. It is assumed the producer‘s revenues from the sale of output are determined by the cost of transport to one specified market; that the cost of each transported input is similarly determined by cost of transport from one specified source; and that no other considerations of location preference exist. All costs and prices are assumed to be constant, irrespective of the scale of output.
Under these assumptions, the optimum location is simply the location for which the combined costs of procuring and assembling inputs and delivering outputs is least per unit of output. The principal use of the analysis is in evaluating the effects on location of (1) the relative weight and relative transportability of an industry‘s materials and products and (2) the patterns of variation in transport cost—the existence of route networks and of nodes thereon, the cost or service differentials reflecting length or direction of haul, volume or size of shipments, mode of transport, or other factors. All these considerations can be weighed as determinants of the type of transport orientation of a specified kind of production; that is, whether production is likely to be optimally located at the market, at a source of material, or at some intermediate point, such as a junction of routes, of modes of transport, or of rate zones.
If the various inputs are required in fixed pro-portions, as assumed by Weber and others, the optimum production location will also be the point of minimum total transport costs of inputs and outputs; but as Leon Moses (1958) has shown, this need not be the case if the mix of inputs can be varied in response to spatial differences in their relative unit costs.
Spatial competition for markets . One of the most drastic simplifications in Weber‘s basic transport orientation case is that all prices and costs are independent of quantities produced and sold. Relaxing this highly artificial assumption makes it possible to analyze the various ways in which producers compete for markets and the ways in which location patterns are affected by economies of scale and geographic concentration.
If the amount of output that a producer can sell in any one market without lowering his price is limited, he is likely to find it advantageous to sell in more than one market, and perhaps in a whole range of markets constituting his market area. Thus, one kind of situation in which producers in different locations interact through competition for markets is that in which each supplies a market area wherein he can deliver the product at a lower price than his competitor can. One branch of spatial economic analysis considers the way in which the size and shape of contiguous market areas are determined for producers whose locations are taken as fixed. The key factors here are (1) the difference between the f.o.b. (before transport cost is added) prices at the producers‘ locations; (2) the way in which transport costs are related to length of haul; (3) whether or not the same tariff for transportation applies to all producers. If, for example, one producer must ship at a higher tariff because his product is less compactly packed, more perishable, or shipped in smaller lots than that of his competitor, he will be under an added disadvantage at markets at longer distances, and his market area may be entirely surrounded by that of his rival who ships at a lower tariff.
The laws of market areas, first set forth in systematic form by Frank A. Fetter (1924) and subsequently elaborated by others, permit useful insights into some of the ways in which the structure of transport costs influences the location of producers in relation to their markets and the extent to which a reduction of either production costs or transport rates may enlarge the market that can be economically served from a given production location. An enterprise facing a choice among locations can use these principles under some circumstances in estimating the relative advantages, in terms of the market area and net sales revenues, of alternative locations.
In its most simplified form, market-area theory assumes that market areas are discrete because (1) the products from competing centers are highly interchangeable rather than differentiated, (2) transport costs rise continuously with distance, and (3) the output of a producer sells at a uniform f.o.b. price plus freight, rather than under a discriminatory delivered-price system involving freight absorption by the seller. This combination of conditions is rather uncommon in practice and is perhaps most closely approached in the case of the sales territories of the several separated branches of a given firm that seeks to minimize total delivery expense.
But where different firms are competing for markets and are selling somewhat differentiated products, the market areas of different production centers often overlap to a high degree. To some extent this reflects the fact that transport charges do not always rise continuously with added distance but stay constant over substantial ranges of added length of haul. More important is the fact that sellers can and do discriminate among buyers according to the buyer‘s location, most often by partial or complete absorption of the added transport cost of sales to the more distant markets. Full freight absorption means selling at a uniform delivered price to all markets and tends, of course, to produce a very great degree of market-area overlap and cross-hauling of products. This and a great variety of other systems of setting prices in space, such as basing-point systems, have been documented and analyzed in great detail by many writers in terms of (1) theoretical rationale from the standpoint of the seller‘s interest; (2) historical origins and evolution; and (3) conformity with norms of “workable competition” and with the public interest in efficient location and allocation of resources.
Market areas and supply areas. The types of market-area analysis just discussed apply essentially to products that are produced at fewer points than those at which they are consumed or bought. For certain products, however—mainly agricultural ones—the characteristic situation is that of widely dispersed producers selling to a relatively small number of consuming or collecting centers. This is the inverse of the characteristic market-area situation. Accordingly, the various simplified and complex types of market-area analysis have their counterparts in the field of supply-area analysis. The most familiar examples of rather discrete supply areas are urban milksheds. The effects of various transport rate patterns and pricing policies of buyers have been worked out, in fairly close analogy to the effects of transport rate patterns and pricing policies of sellers in the market-area analysis discussed above.
Many situations in the real world are composite, with a single seller or production point serving several markets while at the same time a market is supplied by several sellers at different locations. This is particularly likely to occur where transport costs of the product in question are small relative to either (1) other considerations of production location, such as labor costs, or (2) qualitative differences between the brands of rival producers.
Competition for space. Another classic approach to spatial analysis, pioneered by Thiinen, focuses upon the competition between producers for space on which to operate and upon the role of land rent as the price and allocator of space [see the biography of Thnen]. This approach is the main root of those branches of spatial analysis which, under the broad rubric of “land utilization theory,” address themselves to the question of how to use a specified area rather than where to locate a specified kind of activity.
This line of analysis uniquely points up the dual economic role that space plays—it provides utility as a necessary and generally scarce production input, and it causes disutility by imposing costs of transport or communication to bridge distances.
In the simplest case, the choice of location for the producer is assumed to rest on just two factors: the net price per unit received for his output and the price he has to pay (per acre) for the use of land, i.e., rent. The net prices realized for outputs are assumed to depend only on transport costs to a single specified market. All other location factors (such as cost of transported inputs or labor) are ignored.
The factor of access to market thus acts centripetally on producers, while the balancing centrifugal force is the higher rent resulting from competitive bidding for the space nearer the market. Each industry or kind of land use, depending on its technical production characteristics and the transportability of its output, strikes its own compromise between nearness to market and cheap land; and the equilibrium pattern of land uses is envisaged, in the simplest such case, as a systematic series of concentric ring-shaped zones, each devoted to a particular use.
Given the net price receivable for outputs at a specified location, the individual producer‘s profit possibilities will be greater the lower is the rent charged for the land. In general, his production function will allow considerable substitution between land and other production factors (e.g., more or less intensive cultivation of a crop, or high-rise versus lower buildings in a city). The higher the rent, the more “intensive” is the most efficient way of producing the specified product at that location. There is a maximum rent that the producer can afford to pay to occupy that specified location. The pattern of such maximum, or ceiling, rents tolerated by a specified land use at a series of different locations is described by a “rent gradient” (along one line) or a “rent surface” (over a whole area).
A rent gradient or surface rises to a peak at the point of best access to market (e.g., a town where produce from the surrounding countryside is consumed, or the heart of the central business district in the case of many types of urban commercial land uses). The gradients or surfaces corresponding to different land uses have different heights and slopes. Competition in the real estate market, together with the incentive for owners of land to realize maximum returns, implies that each land use will tend to pre-empt those areas for which its rent surface is the highest one.
This theoretical approach is most applicable to situations in which the main factors affecting the choice of location for a variety of competing uses are (1) rents and (2) some other spatial differential that is common to all the principal alternative uses and is related systematically and continuously to distance—e.g., market access, in the illustrations cited. In practice, this applies to extensive extractive land uses like agriculture and forestry and (on a much more local scale) the main classes of urban land use within a city or metropolitan area.
With this general type of analysis it is possible to derive useful insights, for planning or prognosis, into the shifts in land-use patterns likely to result from changes in demand for products, technological and transport changes, and land-use controls. Some of the more obvious variations on the simplest case have been developed by Thiinen and succeeding generations of theorists; they include, for example, the existence of cheaper transport along certain routes, variations in the desirability of land other than those due to access to markets, labor cost differentials, economies of scale, multi-crop farming systems and other land-use combinations, trade barriers, and imperfections in the real estate market.
Competition for labor . Many important and interesting questions of spatial economics relate to the spatial interaction between people and jobs. Labor is an essential input of all productive activi-ties, and variations in the cost and availability of manpower influence the choice of location for many activities. At the same time, location theory includes the analysis of the locational behavior of the household as a decision unit with labor services for sale and with certain environmental preferences and other “input requirements.” Commuting and migration are two ways in which the labor supply adapts itself to job location.
One of the oldest components of the theory of labor cost differentials rests on the proposition that living costs are lower in predominantly agricultural areas, so that a lower money wage in such areas is consistent with the equality of real wages, which is a condition for equilibrium under full labor mobility. “Equalizing differences” in money wages between areas are defined by Ohlin (1933) as those which simply reflect differences in the cost of living. In the more advanced countries, however, locally produced foodstuffs account for a smaller part of the consumer budget, and interregional differences in consumer prices within the country are narrower. Moreover, the cost of living as measured by statistical indexes omits some important considerations, such as amenity and style of life, that enter into choice of residence.
A second well-established component of the theory explicitly involves demographic behavior (fertility, mortality, and migration). Since various economic and social impediments to labor mobility exist, labor tends to be abundant and cheap where natural increase outruns the growth of labor demand. Further insight here calls for evaluation of the complex ways in which fertility and mortality are influenced by income level and the local pattern of economic opportunities, and also for more de-tailed analysis of the determinants of spatial mobility. The selectivity of migration plays a vital role here: mobility depends to such an extent on the age, family status, financial resources, education, and other characteristics of the individual that the pattern of manpower characteristics in areas of high unemployment and heavy out-migration contrasts sharply with the pattern in flourishing areas with heavy in-migration.
Certain important locational effects arise from the fact that (since most adults are members of households) labor is often a jointly supplied service. Specifically, labor markets highly specialized in activities employing mainly men are likely to have a surplus of female “complementary” labor which may be attractive to a quite different range of industries.
Still another principal component of the theory of labor cost differentials involves the factors of size, diversity, and productivity in a labor market. This is mentioned in the next section, in the discussion of local external economies. [See Wages, article on Structure.]
Agglomeration. Among the most important questions to which spatial economic theory is ad-dressed is the degree to which a particular economic activity, or a complex of closely related activities, is concentrated in a small number of locations. The term “agglomeration” refers in a broad sense to such concentration.
Perhaps the simplest basis for spatial concentration is economies of scale for the individual production unit, such as a steel works or oil refinery [see Economies of Scale]. If large units are much more efficient than small ones, one large unit can serve a number of market locations more cheaply than a number of smaller decentralized units can, even though the total delivery cost is greater. (Similarly, the economies of concentration in a single large plant can outweigh extra costs of material assembly involved in drawing materials from a larger range of sources of supply.) In the case where scale economies and access to markets are the principal locational factors, production units will be larger when markets are more concentrated and when transport is cheap. Some degree of spatial concentration through scale economies underlies the class of situations discussed earlier in regard to spatial competition for markets, in which individual producers sell to many markets and can adopt discriminatory systems of delivered prices.
The agglomeration of a single activity by virtue of internal economies of scale, as just described, often has important indirect agglomerative effects by providing external economies to related activi-ties [see External Economies and Diseconomies]. For example, fully equipped commercial testing lab-oratories can operate economically only where they can command a sizable volume of business; and firms using such services can save time and money by locating in a center where such service is available. Similarly, the sheer volume of demand for interregional transport (freight and passenger) to and from a large metropolitan area provides the basis for much more efficient, varied, and flexible transport services than a smaller center can sup-port; and this advantage in transport service is an attraction to a wide range of transport-using activities.
Scale economies within individual units, specialization (division of labor among production units), and close contact among units are the elements in an external-economies type of agglomeration. Where there is a large concentration of activities in proximity, more and more particular operations, processes, or services can be undertaken on an efficient scale by separate units that serve other units in the area. In smaller areas, such specialized activities are either absent altogether or have to be provided within the firms that use them—at higher costs, because they are on a relatively smaller-scale basis. To return to the initial example of testing services for industrial products or materials, in a major industrial center various fully equipped and efficient commercial laboratories are available to provide quick service, while in a distant small town the manufacturer needing such testing service has to choose among (1) providing it himself on a small and less efficient scale, (2) having it done at a distance by a commercial facility, with costly delay and inconvenience, or (3) doing without it.
From the standpoint of the user of specialized goods and services, there are three advantages of quick access to a large source of supply—cheapness, variety, and flexibility. First, the specialized producers or providers of services who exist in large agglomerations can provide their goods or services at lower cost because of their own scale economies. Second, more different grades and varieties are available at any one time, which is an important attraction to the buyer who needs to make selective comparisons (the shopper for fashionable clothes, the theatergoer, the employer who needs unusual types of labor, the manufacturer who needs highly specialized technical assistance on production or sales-promotion problems). Third, access to a large source of supply gives greater assurance of ability to meet rapidly changing and unforeseen needs and reduces the penalty arising from instability of requirements. Costly delays and inventory requirements are lessened when a firm‘s sudden need for additional labor, repair services, or materials is so small relative to local supplies that the firm can count on its need being met.
Agglomeration via external economies is manifest both on a large interregional distance scale (urban versus nonurban regions, large metropolitan areas versus smaller urban areas) and on a micro-geographic intraregional scale (downtown versus outlying areas of a city or even smaller specialized areas within a city, such as shopping districts, garment districts, financial districts, or automobile rows). The distance scale depends partly on the urgency of the need for close personal contact and proximity among the units involved, and partly on the extent to which advantages of spatial concentration are offset by various disadvantages of crowding, such as high cost of space, traffic congestion, noise, and pollution.
The role of external economies in agglomeration and urbanization has been particularly well described by Florence (1948) and Lichtenberg (1960). Its importance as a field of analysis is rapidly in-creasing because of the increasingly urban and interdependent character of economic activities and the emergence of distinctively urban economic and social problems of the first magnitude; the increasing importance of activities that are related to others through transmission of information requiring quick, close, and detailed contact; and the increasing awareness of the important role that the industrial structure of a region plays in determining its opportunities for growth and adjustment to changing conditions.
Edgar M. Hoover
[See alsoCentral Place; Geography, article on Economic Geography; Regional Science; RENT; Transportation, article on Economic Aspects.]
BIBLIOGRAPHY
Alonso, William 1964 Location and Land Use: Toward a General Theory of Land Rent. Cambridge, Mass.: Harvard Univ. Press.
Berry, Brian J. L.; and Pred, Allan 1961 Central Place Studies: A Bibliography of Theory and Applications. Bibliography Series, No. 1. Philadelphia: Regional Science Research Institute. → Reprinted in 1965, with additions through 1964.
Chisholm, Michael 1962 Rural Settlement and Land Use: An Essay in Location. London: Hutchinson University Library.
Fetter, Frank A. 1924 The Economic Law of Market Areas. Quarterly Journal of Economics 38:520-529.
Florence, P. Sargant 1948 Investment, Location, and Size of Plant: A Realistic Inquiry Into the Structure of British and American Industries. Cambridge Univ. Press.
Greenhut, Melvin L. 1963 Microeconomics and the Space Economy: The Effectiveness of an Oligopolistic Market Economy. Chicago: Foresman.
Hoover, Edgar M. 1948 The Location of Economic Activity. New York: McGraw-Hill.
Isard, Walter 1956 Location and Space-economy: A General Theory Relating to Industrial Location., Market Areas, Trade and Urban Structure. Cambridge, Mass.: Technology Press of M.I.T.; New York: Wiley.
Isard, Walter et al. 1960 Methods of Regional Analysis: An Introduction to Regional Science. New York: Wiley; Cambridge, Mass.: M.I.T. Press.
Lichtenberg, Robert M. 1960 One-tenth of a Nation: National Forces in the Economic Growth of the New York Region. Cambridge, Mass.: Harvard Univ. Press.
LÖsch, August (1940) 1954 The Economics of Location. New Haven: Yale Univ. Press. → First published as Die räumliche Ordnung der Wirtschaft.
Meyer, J. R. 1963 Regional Economics: A Survey. American Economic Review 53:19-54.
Moses, Leon N. 1958 Location and the Theory of Production. Quarterly Journal of Economics 72:259-272.
North, Douglass C. 1955 Location Theory and Regional Economic Growth. Journal of Political Economy 63:243-258.
Nourse, Hugh O. 1967 Regional Economics. Unpublished manuscript.
Ohlin, Bertil (1933) 1957 Interregional and International Trade. Harvard Economic Studies, Vol. 39. Cambridge, Mass.: Harvard Univ. Press.
Palander, Tord 1935 Beitrdge zur Standortstheorie. Uppsala (Sweden): Almqvist & Wiksell.
Ponsard, Claude 1958 Histoire des théories économiques spatiales. Rennes (France): Colin.
PredÖhl, Andreas 1928 The Theory of Location in Its Relation to General Economics. Journal of Political Economy 36:371-390.
Weber, Alfred (1909) 1957 Theory of the Location of Industries. Univ. of Chicago Press; Cambridge Univ. Press. → First published as Über den Standort der Industrien. Teil 1: Reine Theorie des Standorts.
II. THE GENERAL EQUILIBRIUM APPROACH
Central to most general equilibrium models of location is a system of equations that emphasizes the interdependence of regions due to linkages between economic activities. Thus, in such models an exogenous change in demand for a commodity in one region may affect interregional trade and the spatial distribution of consumption, production, etc., of all commodities in all regions. These models represent a distinct departure from tradition in location theory. Problems in this field were first investigated formally by Weber (1909) and other European scholars who were contemporaries of Walras. Their analyses drew, however, most heavily on the partial equilibrium logic of Marshall. Their theoretical formulations did not lend themselves to consideration of mutual-interdependence issues and tended to obscure the relationship of location theory to branches of economic theory, particularly interregional trade theory, that dealt with general equilibrium problems.
Mutual-interdependence models of location and regional interaction are of two varieties: (1) highly abstract formulations that extend Walras’ reasoning to spatial phenomena; (2) models that are more restrictive in their assumptions and less general in their intent but which lend themselves to empirical application. The latter will be dealt with at greater length here. Not all of the more operational techniques, however, will be reviewed. The approach of some of them is so highly aggregative that they cannot deal adequately with factors affecting the geographic distribution of production, trade, and optimum location of investment in new capacity for individual industries. For this reason, multisector multiplier and regional growth models, while important tools of regional and interregional analysis, will not be discussed (but see Chipman 1951; Isard et al. 1960; Sorts 1960). Moreover, the results of some of the aggregative formulations—economic base analysis is an example—can be derived from those that are reviewed below (Andrews 1958).
This article will concentrate on two types of models: those that employ input-output techniques and those that employ linear programming [see Input-Output Analysis; Programming]. The most important difference between the two is that the former rule out optimization by fixing geographic patterns of production and/or trade. Efforts by scholars to develop techniques that determine relative, as well as absolute, patterns of production and trade and that are concerned with the optimum use of resources led to the application of linear programming to spatial analysis.
We shall begin by considering a group of input-output models. First, the derivation of trade balances for regions is dealt with. Second, the linkages between regions or nations involved in the usual input-output system are described. Third, several full-scale regional input-output techniques that are theoretically capable of analyzing the effects of trade linkages between many regions are considered. A group of linear programming models is then reviewed.
Input-output techniques
Derivation of trade balances. Data on trade between regions of a country are not usually available, and estimates of such trade are thought to be less reliable than estimates of regional output and final demand. Given information on output and final demand and a matrix of technical coefficients, individual industry balances of trade for a region with respect to all other regions can be derived. Let gxj be output of industry j in region g and yCj be total consumption of the output of industry j in region g; then gej the net trade balance of industry j in region g with respect to all other areas, is
gej = gxj - gCj.
A positive e indicates a net export balance, a negative e a net import balance. An over-all trade balance for the region is obtained by summing the e’s for all industries in the region.
Total consumption, c above, is the sum of final demands, y, and intermediate or interindustry demands. The latter are derived by multiplying the known regional outputs by technical coefficients of production, that is,
The technical coefficient gaij is the amount of the output of industry i required to produce a unit of the output of industry j in region g.
There are difficulties in deriving commodity trade balances by the above technique. Coefficients of production vary between regions as a result of differences in methods of production, differences in prices, which affect the coefficients because they are in dollar terms, differences in the product mix of industries, etc. Yet, since regional input-output coefficients are not usually known, national coefficients are often employed, with resulting errors in the estimates of intermediate demands. It is also difficult to obtain information on consumption by the final demand sectors, particularly investment and government.
Import and export balances for individual industries have sometimes been derived as a first step in studies whose object is to determine the industries that might be encouraged to locate or expand in a region. Although the size of the import balance, or regional excess demand for the output of an industry, is only one element in such an analysis—costs of producing and transporting the inputs and outputs of an industry are also considered—it is viewed as particularly important. This outlook reflects a belief that narrowing regional differentials in factor prices and production costs are leading to an increased market orientation of nonprimary industry. Some studies, unfortunately, appear to settle the issue of a region’s comparative advantage by a ranking of industries in terms of the size of regional excess demand. Virtue is found in those patterns of investment that contribute most to self-sufficiency, an approach that partakes of regional mercantilism.
Locational analyses based on balances of trade employ general equilibrium techniques to only a very limited degree. After the initial input—output calculations are made, an industry-by-industry approach is adopted. In general such analyses do not take into account in a systematic way the manner in which the introduction of new capacity for any one industry will affect the costs of production of all other industries in a region. Changes in outputs that are induced by cost changes are, therefore, not taken into account. Also neglected are the constraints that may exist on expansion. While factors of production are more mobile for regions of a country than they are internationally, regional supply functions for some factors may be inelastic unless there is substantial unemployment. If factor supplies and other things impose constraints on industry expansion, a pattern of investment should be selected that minimizes or maximizes some meaningful economic variable. Such an objective requires a general equilibrium system of an optimizing character. Several systems of this type are discussed below.
Trade linkages in national models . As was indicated above, balance of trade studies begin with known outputs. The main objective of most interindustry studies is, however, to determine outputs, and the objective of regional input-output systems is to determine what effects a change, say, in final demand, in one or more regions will have on all others. The simplest approach to the latter issue is that adopted in national input-output analyses. These are in effect two-region systems, one region being the nation under study and the other the rest of the world. In such models we are given the technical coefficients of production of the region or nation, gaij, for i, j = 1, 2, ..., n industries; the final demands, gyj of all exogenous sectors other than foreign trade; and a set of imports, gmj and exports, gej. In matrix notation these are represented by gA , gY , gM , and gE . Imports are arranged in two categories, competing and noncompeting goods, the latter being those for which there are no counterparts in the region under study. Imports of these goods are entered in an exogenous import row and have no effect on regional output. Output of each of the industries in the nation or region, g, being studied is found by solving the following set of simultaneous equations:
gX = [I - gA ]-1 [gY + (gE - gM )].
Here gX is the n × 1 vector of outputs in region g; [I — gA]-1 is the inverse of [I — gA], the n × n identity matrix minus the n × n matrix of technical coefficients; gY is the n × 1 vector of final demands; (gE — gM) is the n × 1 vector of the differences between imports and exports for each of the competing-goods industries. Negative values for these differences are precluded. Thus, if the imports of any good exceed the sum of exports and all other final demands, a zero value is assigned to the final demand for the good. Effectively, it is treated as a noncompeting good.
In the above approach imports and exports are not explained within the model. There are, for example, no functions relating region gr’s imports of all goods to its income. Since regional linkages are external to the system, it is not possible to determine the effects on region g of such things as an exogenous increase in demand for a non-traded good in the rest of the world. We shall now consider a variety of regional input-output techniques that can determine the effects of such changes.
The Leontief intranational model . In its most general form Leontief’s intranational model (see Research Project . . . 1953) involves a complex hierarchy of goods and regions. The place of any particular good within the hierarchy depends upon the degree of spatial aggregation required to obtain a balance, or equality, between regional production and consumption. An empirical application of the model employs the simpler hierarchy traditional to international trade theory, that of traded and nontraded goods, although they are referred to as national and local goods. National goods are those for which a balance between production and consumption can be struck only at the national level. Local goods are those for which there is a balance between production and consumption for each region into which the nation is divided for purposes of the study. The outline of the model’s structure presented below is in terms of this simpler hierarchy.
Assume there are n industries, the first h being national and the remaining n — h local. There are r regions. A national bill of final demand, Y, for all goods, a national matrix of technical coefficients, A, and a set of locational constants, gβd, for the national goods are given. The locational constant gβd indicates the proportion of total output of a national good, d, produced by region g. These constants may be determined from past data on the geographic distribution of production. By definition, the locational constants for each national good sum to unity.
Total outputs of both national and local goods are determined by solving
X = [I - A]-1y .
A regional allocation of total output of each national good is then obtained by applying the locational constants as follows:
gxd = gβd Xd, d = 1,2, ..., h, national industries;
g = 1,2, ... ,r, regions.
In this expression gxd represents the amount of national good d that is produced in region g and xd represents the total output of national good d. Since the locational constants for each national good sum to unity, we are assured that the above procedure precisely allocates total output of each national good to the regions.
Derivation of regional outputs of nontraded goods is somewhat more involved. For this purpose it is convenient to think of the national matrix of technical coefficients as having been arranged so that all national-goods industries appear first. The coefficients are then in four meaningful blocks:
d, f = 1,2, ...,h, national industries;
l, q = h + 1, h + 2, ..., n, local industries.
Technical coefficients in the {adf} or A1 block pertain to the requirements of national-goods industries for the outputs of other national-goods industries per unit of their output; coefficients in the {adl} or A2 block describe the requirements of local-goods industries for the outputs of national-goods industries per unit of the former’s outputs, etc.
Regional outputs of local goods are determined as follows:
g = 1,2, ..., r
gxX = [I - A4]-1 [A3gXd +gYl], d = 1,2, ..., h
l = h+1, ..., n.
Here [I - A4]-1 is the inverse of the block of coefficients that pertain to the requirements of local-goods industries for the outputs of local-goods industries; A3, gXd gives the intermediate requirements of local-goods industries in region g for production of the already determined regional outputs of national goods; gYl is the final demand for local goods in region g.
The intranational model being considered determines all outputs. It also determines individual commodity and, therefore, aggregate trade balances for each region with respect to all others. Balances of trade between individual pairs of regions remain unknown. This is one of the senses in which the model is intranational rather than interregional. It is part of the logic of the system that the effect on a particular region of a change in final demand for national goods is the same regardless of the region in which the change takes place. Similarly, the effect on any given region of a change in final demand for local goods is the same regardless of the region (if other than itself) in which the change takes place.
The hierarchy of goods and regions, whether of a simple or complex variety, assumed by the intranational model is difficult to establish in reality. The empirical application of the system finally settled upon a definition of goods and regions that in some sense minimized departures from strict production-consumption balance for the entire group of industries designated as local (see Isard 1953). The empirical difficulty raises a question as to the kind of theoretical framework that would produce a strict hierarchy of goods and regions. The answer is provided by Lósch’s general equilibrium model of location (1940).
Lösch assumed a uniform transport surface, every point being on a straight-line connection with every other; freight rates given and the same in all directions; population evenly distributed and with identical tastes, so that each point in space has an identical demand function for each commodity; factor supplies everywhere the same; identical production functions and U-shaped cost curves for each commodity everywhere; and intermediate inputs, if required, present everywhere, so that only final goods are transported. Economies of scale in production ensure that the economy is not composed of self-sufficient households. Transport costs, on the other hand, impose limits on market areas and the extent of specialization. Losch concluded that under competitive equilibrium conditions, with all consumers served and profits of all firms zero, the surface is covered by a network of hexagonal market areas of different sizes or orders, one for each good. The hexagons pertaining to any given good are everywhere the same. Each hexagon of a given size contains an equal number of hexagonal market areas of all lower-order goods and is itself completely contained in the next-higher-order market. This is the strict hierarchy involved in the Leontief intranational model.
Interregional models . To describe an interregional input-output system, we assume a closed economy divided into b, g = 1,2, ..., r regions and producing i,j = 1,2, ... ,n goods. Technical coefficients, gaij, are given for each region and may differ between them. For each region there is also given a set of trade or supply coefficients, bgti, for every good, showing the relative regional composition of that region’s purchases of every commodity. Thus, the coefficient 34t2 = .40 indicates that for every dollar spent on the output of industry 2 by the sectors of region 4, 40 cents’ worth is purchased in region 3. Each region’s trade coefficients for each commodity add to unity: , i = 1,2, ..., n. Regional final demands, gyi, are given.
From the trade and technical coefficients a new set of coefficients is derived, which describes the interregional and interindustry structure of every region:
Ā ≡ {bgaij} = {bgtigaij},
i,j = 1, 2, ... ,n;
b,g = 1, 2, ... ,r.;
Hereafter Ā will be described as the interregional input-output matrix. Consider one element of this nr × nr matrix, say, 34a,12 = .20. The coefficient indicates that for each dollar’s worth of commodity 2 produced in region 4, 20 cents’ worth of commodity 1 is purchased from region 3. Since the trade coefficients for each good in a region add to unity, the sum of all a’s in any column of the interregional input-output matrix yields a technical coefficient for the relevant region and industry:
In interregional systems, outputs of an area are determined by the amounts that the endogenous sectors of an area ship on final demand account rather than the amounts that its exogenous sectors consume. Shipments on final demand account, by*i, are determined from the usual final demands and the trade coefficients:
j = 1,2, ..., n.
Regional outputs of all goods are then obtained in the usual way, namely,
X = [I - Ā]-1Y*
Here X and Y* are the vectors, respectively, of regional outputs and of shipments on final demand account, and [I - Ā]-1 is the inverse of the interregional input-output matrix. The system yields balances of trade for individual commodities and for the aggregate of all commodities between pairs of regions or for a single region with respect to all others.
The intranational model described above determines all outputs by fixing the relative regional outputs of national goods and the trading patterns of a class of goods, i.e., those designated as local. Interregional input-output systems allow relative, as well as absolute, levels of regional output of all goods to vary but fix relative trading patterns of all goods in one way or another. The model developed by Isard (1951) has a set of trade coefficients for every industry in every region. The models of Chenery and Clark (1959) and of Moses (1955), on the other hand, have a single set of trade coefficients for all sectors within a region—the assumption employed in the above description of an interregional input-output system. There are, however, differences between the latter two models.
Chenery and Clark assume that any region is more efficient than any other region in supplying its requirements of all goods that it actually produces. Each area, then, fully utilizes existing capacity before resorting to imports. Unless a capacity constraint is encountered, trade coefficients must therefore be zero or one. If such a constraint is encountered, trade coefficients will take on intermediate values, but they then cannot be stable. Moreover, this approach can be used only for a two-region system, since it has no mechanism for assigning the imports of a region to a number of different areas.
The theory of spatial competition of traditional location analysis provides the basis for the determination and stability of the coefficients employed by Moses and, with some adaptation, Isard. Thus, assume that each industry produces a single homogeneous product, so that the output of a given industry from one region is a perfect substitute in production or consumption for the output of the same industry from any other region. Each industry in each region is assumed to have excess capacity and to produce at constant cost. Regional supply functions of the factors of production are perfectly elastic at given factor prices, up to some limit representing regional endowments. Factors of production are perfectly mobile within each region, industry is perfectly competitive, and transport costs per unit of output increase continuously with distance but do not vary with the quantity shipped. In these circumstances there are perfectly defined market and supply areas for every site where a good is produced and consumed. Trade coefficients then reflect the aggregation of market and supply areas into regions. The world of reality departs from perfect competition, transport rate structures have many peculiarities, etc. There may, as a result, be significant differences in market and supply boundaries of different industries in a region and, therefore, intraregional differences in trade coefficients, as suggested by the Isard model.
The most questionable aspect of interregional input-output systems is their assumption of stable trading patterns. A theoretical framework can be developed, as was done above, under which trade coefficients would be stable, but it is extremely unlikely that the conditions of this framework will be met in reality. The evidence for stability that has been presented is not convincing, because all economic activity was grouped into a few gross industries and regions, with changes over time in individual market and supply areas probably balancing out (Moses 1955).
The main virtue of regional input-output models is their operational character. This is achieved at a heavy cost in terms of theoretical interest. They have no mechanism for explaining trade patterns. Aside from reductions in final demand, they have no way of explaining the disappearance of an industry from a region. Nor are they so constructed that their internal logic points to situations in which new industries will emerge in a region. The above criticism is not removed by the introduction of dynamic elements. In dynamic formulations investment in new capacity is typically related to rates of change in output of industries found in each region in the initial period. In this respect the best that has so far been done with the various input-output models is to incorporate and investigate the conclusions of separate locational and interregional trade analyses. Thus, in one study (Isard & Kuenne 1953) the ideal location of a new integrated steel plant was determined from a separate location study. The impact on the area of the mill’s output was then investigated by a regional input-output analysis that involved a limited application of the trade-coefficient approach. In this approach the indirect effects that the region would have on itself because of its effects on other regions were ignored. This limited impact approach has been adopted in a number of studies (see, for example, Hirsch 1964; Moore & Peterson 1955).
Regional input-output studies have sometimes had the objective of projecting all economic activity in an area over a considerable period of time. In only one study, however, that of Berman and her co-workers (1960), has the important topic of factor redistribution, and labor migration in particular, been faced. In this study population projections derived by demographic techniques and employment projections derived by input-output analysis were made to square with one another, and it was concluded that a significant change would take place in historical migration trends.
Interregional linear programming models
The intranational model and all of the fixed-trade interregional models treated above face a serious difficulty if a capacity constraint is encountered anywhere in the system. If the “predicted” output of even a single industry in a single region exceeds the region’s capacity to produce the good, then a strict application of the model’s logic requires that the final demand program be declared infeasible. This is surely undesirable if the good is transportable and other regions have sufficient excess capacity to produce the required output. An obvious solution is to alter trading patterns, at least at the margin. In what way, however, should they be altered? If some particular region can deliver the required amounts of the good at a lower price, either because of a production or transport cost advantage, a well-formulated economic model would assign to it the task of supplying the requirements. The issue then is the following: if choice and optimization can be introduced into multiregion systems, why not eliminate the fixed, exogenously given patterns of trade and/or output entirely and formulate models that determine all outputs and trade by means of an optimizing scheme? Several techniques for doing this, all of them of a programming variety, are presented below. Their connections with the theory of interregional trade are very clear. They encompass much of what is valuable in traditional location theory but, because of their general equilibrium character, also go beyond it.
The first model to be considered involves cost minimization and is related to the well-known transportation problem of linear programming. It builds directly on a study by Henderson (1958). The existence of intermediate inputs for current production is ignored at first, and attention is focused on a number of final products. The following notation is employed:
i = 1,2, ... , n, final consumption products;
b,g =1,2, ... , r, regions;
e = 1,2, ... , m, primary factors of production;
bwe = price of a unit of primary factor e in region b;
baei = quantity of primary factor e required to produce a unit of good i in region b;
bki = capacity, or maximum rate of output, of plant and equipment producing good i in region b;
h = 1,2, ..., z, routes or modes of transportation connecting each pair of regions, the routes between regions being entirely independent of one another.
bghSi = shipment of commodity i from region b to region g by route or mode h;
bghK = capacity or maximum quantity of all goods that can be transported by route h between regions b and g, it being assumed that all goods are identical in their use of transport capacity;
bghte = quantity of primary factor e required to transport a unit of any commodity from region b to region g by route h;
bDe = endowment of primary factor e in region b;
byi = final demand for commodity i in region b.
It should be noted that factors of production are perfectly mobile within regions, regional factor supply functions are perfectly elastic up to the limit imposed by endowment, and demand functions are perfectly inelastic with respect to both income and price. The assumption that goods are identical in their use of transport capacity and primary inputs for transport is employed to keep an involved notational scheme from becoming even more complicated.
The problem is to determine a regional allocation of output and trade in all goods that satisfies the fixed demands at minimum total cost, subject to the constraints on capacities and factor endowments. The system is written as follows: minimize
subject to
Equation (1) states the objective—to minimize the total cost of producing and transporting all commodities. The constraints have the following meanings: (2) total shipments of each commodity into each region must be at least great enough to satisfy final demand; (3) total shipments, this being the same as output, of each commodity by each region to all regions must be less than or equal to the maximum possible rate of output; (4) total shipments of all commodities by any route must be less than or equal to the capacity of that route; (5) total requirements of each primary factor in a region for production and transport must be less than or equal to the endowment. All shipments are nonnegative.
The introduction of intermediate inputs changes the model in only one way. The first set of constraints, equations (2) above, must be rewritten to state that the total pool of a good available in a region minus the region’s intermediate demand for the good must be at least as great as its final demand for the good. A region’s pool of a good is defined as total shipments into it from all other areas plus the amount of the good it produces itself minus its total exports of the good.
The minimum-cost, feasible solution is the same as would be achieved by a perfectly competitive economy. The entire system is a set of linked interregional trade problems, the linkages being due to capacities and factor endowments. If these constraints did not exist or were not encountered in a particular problem, total production and transport of each commodity would be assigned to the least-cost region and route, i.e., would be determined by absolute advantage. The existence of binding constraints on factors and transport provides a solution based on considerations of comparative advantage. Moreover, in a multiregion system with positive transport costs, comparative advantage is defined in terms of markets to be served, as well as commodities produced.
The dual to the above minimizing problem is a maximizing problem that determines the delivered price of every good in every region and the quasi rent of each productive capacity, transport capacity, and factor endowment. Delivered price is determined by the marginal supply source. Thus, if any region’s demand for a commodity is satisfied by more than one region, price in the consuming area will be equal to the unit production and transport cost of the least efficient supplying source. Capacity in every other area that supplies this region receives a quasi rent, or return above its cost. Capacities that are not fully utilized earn no such return.
The quasi rents are key variables for analysis of the location of investment in new capacity and the retirement of existing capacity. For this purpose it is convenient to think in terms of a planned economy that has set aside a given sum for investment in additional capacity. It is also convenient to assume that the cost of providing additional capacity to produce each good in each region and of providing additional capacity on each transport route is given. The task of the planning authority is to allocate investment among goods, regions, and routes in an optimal manner. Since final demands are fixed, the obvious criterion of optimality is to allocate investment so as to achieve the greatest reduction in total cost. The significance of the quasi rents can now be seen. Each is in fact the reduction in total system cost that would be realized if the associated capacity were increased sufficiently to permit an additional unit of output or transport. From the initial set of quasi rents, the changes that take place in the quasi rents as additional capacity is added, and the cost of providing each type of capacity, the given total investment can be assigned in an optimal manner. If, instead of taking the total amount of investment as given, we assume that an interest rate is given, the optimum level, as well as allocation, of investment in commodities, regions, and routes can be determined.
Input-output models were criticized above because they had no internal rules governing the emergence of productive capacity for particular goods in regions which have not previously had such capacity. The programming technique does not have this weakness, since in all cases of zero capacity some small fictitious quantity can be assigned. Unit costs of production and transport are determined for these capacities. The model will then determine in what regions, if any, new industry should be introduced.
The quasi rents for factors of production, so far ignored, have interpretations similar to those for capacity. They indicate how much total cost would be reduced if a region had enough of a particular primary factor to produce and transport an additional unit of output. Obviously the quasi rents will be zero in all cases where the optimal solution involves less than full employment for a factor. The optimal solution, therefore, indicates the areas from which mobile factors should migrate—those in which they are less than fully employed and earn zero quasi rents—and those which should attract them. As stated, there is some asymmetry in the system. It determines optimal patterns of interregional trade in goods but not optimal patterns of interregional migration. To include a set of relationships that would change the situation would be misleading, since there is at present no way of quantifying the social, as well as the economic, costs of migration.
The model that has been described determines optimal patterns of output, trade, and employment of primary factors. It determines geographic patterns of price and, when extended to include investment, optimal expansions in capacity for production and transport. It indicates the regions from which mobile factors should migrate and the regions which should attract them, if such decisions are made on the basis of economic considerations alone. With the two extensions suggested above, the results of the model reflect a network of comparative advantages that are defined in terms of production functions for goods and transport, supply prices of primary factors, propensities to consume, and the initial distribution of capacities. An empirical application of this type of model has been attempted by Moses (1960).
A second interregional linear programming model (see Lefeber 1958; Stevens 1958; Kuenne 1963) is almost the converse of the one presented above. It takes as given the prices of final goods in each region, rather than the minimum prices of primary factors of production. Instead of a perfectly inelastic demand for each good in each region, it assumes that demands are perfectly elastic and that a certain minimum quantity of each good is to be delivered to each region. The primal problem of this model maximizes the value of output of final goods, rather than minimizing the cost of satisfying a fixed set of demands. Given a set of final goods prices, it determines an optimal point on the production frontier of the entire economy.
The treatment given demand in both of these programming models is inadequate: one assumes a perfectly elastic demand for each commodity, the other a perfectly inelastic demand. Samuelson (1952), however, has suggested a programming formulation for a single-commodity multiregion spatial competition problem that involved regular supply, as well as demand, functions. In this formulation, total consumer surplus, as defined by Marshall, was maximized, subject to constraints imposed by demand and supply considerations in each region. Smith (1963) has provided an interpretation to the dual of this problem which shows that it involves the minimization of rents. Recently, Takayama and Judge (1964) have demonstrated that the Samuelson spatial competition problem is in reality a quadratic programming problem. They have suggested a method of solution that can be applied either to a single homogeneous good or to a number of goods produced and consumed in many regions. In the latter case, however, all of the goods must be for final consumption only.
Leon N. Moses
[See alsoRegional Science.]
BIBLIOGRAPHY
Andrews, Richard B. 1958 Comment re Criticisms of the Economic Base Theory. Journal of the American Institute of Planners 24:37-40.
Berman, Barbara R.; Chinitz, Benjamin; and Hoover, Edgar M. 1960 Projection of a Metropolis: Technical Supplement to the New York Metropolitan Region Study. Cambridge, Mass.: Harvard Univ. Press.
Borts, George H. 1960 The Equalization of Returns and Regional Economic Growth. American Economic Review 50:319-347.
Chenery, Hollis B.; and Clark, Paul G. 1959 Interindustry Economics. New York: Wiley. → See especially pages 308-332.
Chipman, John S. 1951 The Theory of Inter-sectoral Money Flows and Income Formation. Baltimore: Johns Hopkins Press.
Henderson, James M. 1958 The Efficiency of the Coal Industry: An Application of Linear Programming. Cambridge, Mass.: Harvard Univ. Press.
Hirsch, Werner Z. 1964 [A General Structure for Regional Economic Analysis.] Introduction to Conference on Regional Accounts, Second, Miami Beach, Fla., 1962, Elements of Regional Accounts: Papers. Baltimore: Johns Hopkins Press.
Isard, Walter 1951 Interregional and Regional Input-Output Analysis: A Model of a Space-economy. Review of Economics and Statistics 33:318-328.
Isard, Walter 1953 Some Empirical Results and Problems of Regional Input-Output Analysis. Pages 116-181 in Research Project on the Structure of the American Economy, Studies in the Structure of the American Economy: Theoretical and Empirical Explorations in Input-Output Analysis. New York: Oxford Univ. Press.
Isard, Walter; and Kuenne, Robert E. 1953 The Impact of Steel Upon the Greater New York-Philadelphia Industrial Region. Review of Economics and Statistics 35:289-301.
Isard, Walter et al. 1960 Methods of Regional Analysis: An Introduction to Regional Science. New York: Wiley; Cambridge, Mass.: The Technology Press of M.I.T. → See especially pages 182-231.
Kuenne, Robert E. 1963 Spatial Economics. Pages 395-454 in Robert E. Kuenne, The Theory of General Economic Equilibrium. Princeton Univ. Press.
Lefeber, Louis 1958 Allocation in Space: Production, Transport and Industrial Location. Amsterdam: North-Holland Publishing.
LÖsch, August (1940) 1964 The Economics of Location. New Haven: Yale Univ. Press. → First published as Die räumliche Ordnung der Wirtschaft.
Moore, Frederick T.; and Peterson, James W. 1955 Regional Analysis: An Interindustry Model of Utah. Review of Economics and Statistics 37:368-383.
Moses, Leon N. 1955 The Stability of Interregional Trading Patterns and Input-Output Analysis. American Economic Review 45:803-832.
Moses, Leon N. 1960 A General Equilibrium Model of Production, Interregional Trade, and Location of Industry. Review of Economics and Statistics 42:373-397.
Research Project on the Structure of the American Economy 1953 Studies in the Structure of the American Economy: Theoretical and Empirical Explorations in Input-Output Analysis, by Wassily Leontief et al. New York: Oxford Univ. Press.
Samuelson, Paul A. 1952 Spatial Price Equilibrium and Linear Programming. American Economic Review 42:283-303.
Smith, Vernon L. 1963 Minimization of Economic Rent in Spatial Price Equilibrium. Review of Economic Studies 30:24-31.
Stevens, Benjamin H. 1958 An Interregional Linear Programming Model. Journal of Regional Science 1: 60-98.
Takayama, T.; and Judge, G. G. 1964 Equilibrium Among Spatially Separated Markets: A Reformulation. Econometrica 32:510-524.
Weber, Alfred (1909) 1957 Theory of the Location of Industries. Univ. of Chicago Press; Cambridge Univ. Press. → First published as Über den Standort der Industrien. Teil 1: Reine Theorie des Standorts.