Modeling Agglomeration and Dispersion in City and Country: Gunnar Myrdal, Francois Perroux, and the New Economic Geography - Critical Essay

American Journal of Economics and Sociology, The, Jan, 2001 by Stephen J. Meardon

The most notable change has been the shift from the somewhat discredited mathematics taken over from Lagrange's classical mechanics to topological mathematics, following G. Debreu and K. J. Arrow. The former describes the movement in space of indeformable objects and their halting (point of equilibrium when two equal and contrary forces are applied to them). The latter admits of spaces lending themselves to contraction, expansion and deformation and representing the operations of economic agents; it describes the successive equilibria while setting limiting conditions; it amplifies all the models of monopolistic competition, while retaining all they have to teach us (1983, p. 186).

Trying to impart the feel of the sort of model he had in mind, a model of interdependence among 'active units,' Perroux (1962 [1960], ch. 2; 1967 [1965], ch. 1) took a basic static general equilibrium model as point of departure. Considering a profit function for a firm employing n inputs, [pi] = [P.sub.v]Q - [p.sub.1][q.sub.1] - [p.sub.2][q.sub.2],.., -- [p.sub.n][q.sub.n]

he posited several such firms (or other units) which, rather than taking market prices as given and choosing optimally their output, instead find themselves subject to an external power that imposes the prices they face or the output they produce:

a power (C) imposes, for example, the quantity...that must be sold by unit U1 [indicated by a " " above the variable]:

[[pi].sub.1] = [P.sub.v1][Q.sub.1] -- [p.sub.2][q.sub.2],...,-[p.sub.n][q.sub.n]

or, for example, the price of inputs purchased by [U.sub.2]:

[[pi].sub.2] = [P.sub.v2][Q.sub.2] -- [p.sub.1][q.sub.1] -- [p.sub.2][q.sub.2],...,-- [p.sub.n][q.sub.n]

or, for example, the price of the product sold by [U.sub.3]:

[[pi].sub.3] = [P.sub.v3][Q.sub.3] -- [p.sub.1][q.sub.1] -- [p.sub.2][q.sub.2], ..., -- [p.sub.n][q.sub.n]

It is possible to conceive of diverse combinations [of C's influence on the subordinate units] and translate them into this notation (1967 [1965], p. 18. Author's translation.).

Diagrammatically, Perroux depicted the relationships between C, [U.sub.1], [U.sub.2], and [U.sub.3] as shown in Figure 1.

Explaining further, he wrote,

The preceding implies an action that is asymmetric and irreversible during a time period. The action extends from C towards [U.sub.1], [U.sub.2], ..., [U.sub.n], and not in the opposite direction....This asymmetric action admits gradations: from total domination of C over a unit ([U.sub.1], [U.sub.2] [U.sub.n]), to a very limited influence exercised by C over one of the units (ibid., pp. 18-19).

Following Perroux's rejection of the Walrasian model and his call for a new model of interdependence to replace it, the exposition above can be frustrating to read. It is not, after all, a replacement. There is no way to measure, for example, the response of [U.sub.2] to changes in its input prices--much less the response of [U.sub.3] to [U.sub.2]'s response. Exercises in comparative statics or dynamics are impossible because the model is not specified mathematically; rather it is described textually with the aid of some algebraic notation. That by no means invalidates Perroux's ideas, but he leaves the reader expecting something more revolutionary: a mathematically specified general equilibrium model that fulfills the same functions as the Walrasian model, but in addition captures the exercise of market and extra-market power that the Walrasian model entirely misses. The reader never finds it.