The spatial impossibility theorem
I begin the discussion by considering the assignment problems introduced by Koopmans and Beckmann (1957). Assume that n firms are to be assigned to n locations. Each firm is indivisible, and the amount of land available at each location is such that a single firm can be set up there.
Hence, every firm must be assigned to a single location, and every location can accommodate only one firm. Each firm produces a fixed amount of goods and uses one unit of land. Suppose further that the technology used by each firm is not affected by the chosen location. Koopmans and Beckmann first considered the linear assignment problem in which firms receive revenues from the rest of the world, which are location specific. They showed that this problem can be expressed as a linear program, the solution of which is given by integer numbers. Since the shadow prices generated by the dual of this program are location specific, these prices have the nature of land rents. Thus, a competitive equilibrium exists since the optimal solution may be decentralized through a competitive land market, very much as in Thunen.Koopmans and Beckmann then turned to the quadratic assignment problem in which each firm uses the goods produced by the others and bears the corresponding transportation costs. Because of the exchange of goods, this problem cannot be expressed as a linear program anymore. When locations generate similar revenues, Koopmans and Beckmann showed that no feasible location pattern of firms can be sustained as a competitive equilibrium, thus implying that there exists no competitive equilibrium. Revisiting the quadratic assignment problem, Heffley (1972) showed that decentralization is possible when sites have very different comparative advantages. Hence, as Hamilton (1980: 38) put it: “Stability is lent to the system by having plants differ from one another in their preferences for the sites qua sites, and instability arises from a large volume of trade among plants.”
In the long debate concerning the comprehensiveness of general equilibrium theory for the spatial economy, Starrett (1978) has made the fundamental contribution.
The question is whether the competitive price mechanism is able to explain the endogenous formation of economic agglomerations and the existence of large trade flows. Because they are not perfectly divisible, agents are not ubiquitous and, therefore, must choose an “address”. Space is then said to be homogeneous if (1) the utility function of each household is identical no matter what its location and (2) the production function of each firm is independent of its location. In other words, the location choice made by a consumer or a producer does not affect her preferences or the technologies that are available. The spatial impossibility theorem may then be stated as follows:Theorem 1 Consider an economy with a finite number of locations. If space is homogeneous, transport is costly, and preferences are locally non-satiated, then there exists no competitive equilibrium involving the transport of goods between locations.
Consequently, the perfectly competitive price mechanism alone is unable to deal simultaneously with cities and trade. This has a fundamental implication for economic geography: if the purpose is to build a theory explaining the formation of economic agglomerations, then such a theory must depart from general competitive analysis. What is the meaning of this result? Whenever economic activities are perfectly divisible, the spatial impossibility theorem implies that the mobility of production factors is a perfect substitute for trade. Such a result is hardly surprising because every activity can be carried out on an arbitrarily small scale in every possible place, without any loss of efficiency. Firms and households are then induced to suppress all distance-related costs by producing exactly what they need where they are. In contrast, as pointed out by Starrett (1978: 27), “so long as there are some indivisibilities in the system (so that individual operations must take up space) then a sufficiently complicated set of interrelated activities will generate transport costs”.
In this case, the spatial impossibility theorem tells us something really new and important: whenever agents have to choose an address, there is no competitive equilibrium (hence the term “impossibility” in the name of the theorem) such that places trade goods. In other words, factor mobility and interregional trade are incompatible in the standard neoclassical world. This result is especially meaningful in so far as it is internal to the theory itself.Intuitively, the reason for this is that the only location factor that matters to an agent is its position with respect to the others. In this case, the price system must play two different roles: (1) it must allow trade between locations while guaranteeing that all local markets clear and (2) it must give firms and households the incentives not to change location. Once the economy is competitive and space homogeneous, the spatial impossibility theorem tells us that it is impossible to kill two birds with one stone: prices that sustain commodity flows between places send incorrect signals from the point of view of the stability of locations, and vice versa. The fundamental reason for the spatial impossibility theorem is the non-convexity of the set of feasible allocations caused by positive transport costs and the fact that agents have an address in space, even though the individual land consumption is endogenous. Hence, in the absence of external factors that drive firms’ and households’ locations, such as the existence of a market town or of spatial externalities, a sound spatial economic theory cannot be built within the competitive general equilibrium framework by differentiating goods through their locations and adding land as a new commodity.