Yesterday I put an early draft of a paper on SSRN concerning DGSE.
In a short and almost throwaway section I looked at how the axiomatic assumptions imported into general equilibrium theory from Debrau didnt demonstrated General Equilibrium at all because spatially and temporally stamped goods always existed on a surface of the earth with differential features which would always generate Von Thunen land rent.
The moment I had put the paper online though I thought surely someone had thought of this before as it is pretty obvious, general equilibrium if it exists, does not occur in an asptial and atemporal prior ether but from an existing disposition of resources, firms, property and individuals. In the jargon there will be non-convexity.
Indeed it had been thought of before in a result from 1978 called Starret’s Spatial Impossibility theorem, but outside the very narrow field of spatial economics is little known about.
Starrets considered a surface of islands, the firm on one island and the firm on another. It is like the archipelago of Robinson Crusoe’s only with trade.
the consumer living in A and the ﬁrm locating in B cannot be an equilibrium . It turns out that the economic agents always want to move closer to each other, as opposed to being apart and having to bear the transport cost….
Or the theorem formally put
Suppose an economy with a ﬁnite number of locations and homogeneous space. If transportation consumes scarce resources (and preferences are locally non-satiated), there is no competitive equilibrium with positive transport costs….this result is true for any number of islands and economic agents
The simplifying assumption is homogenious space, which enables the producer and consumer to swap positions and still hold the same result.
Now this result is very powerful and led subsequent spatial theorists to produce theories based on non-convexity, and of course that enabled you to show comparative advantage, gains to trade, increasing returns to scale, economies of urban agglomeration, growth of urban areas etc – and with increasing returns also must go perfect competition. So with the core assumptions of neoclassical economics gone we get realsitic results. Indeed it directly led to the New Economic Geography based on concepts of gains from trade and which of course led to Krugman getting his Nobel Prize.
From the New Palgrave chapter on Spatial Economics
In the Theory of Value, Debreu (1959) answers affirmatively. A commodity is defined by all its characteristics including its location: the same good traded in different locations must be treated as different commodities. This ‘answer’ runs into serious problems, as pointed most clearly by Starrett (1974). Consider the extreme case of homogenous space where firms face the same convex production set, and consumer preferences are the same (and locally not satiated). Transporting commodities between locations is costly. Then the spatial impossibility theorem states that, with a finite number of locations, consumers, and firms, no equilibrium involves transportation. The intuition behind this result is straightforward: since economic activities are perfectly divisible and agents have no objective reason to distinguish between 3 locations, each location operates in autarky to save on transport costs. To avoid this very counterfactual result (no trade), one of the assumptions behind the spatial impossibility theorem needs to be relaxed. If one takes transport costs as an unavoidable fact of life, one must assume either some non-homogeneity of space or some non-convexity of production sets.
Now my instinct would be that if you relax the homogeneity assumption then you are forced to accept non-convexity, which implies dynamic equilibrium or disequilibrium rather than fixed general equilibrium. Without even distribution of costs how can you have topologically convex costs. Without convex costs and with imperfect competition the Arrow-Debrau route to proving the existence of general equilibrium is closed off – and so is DGSE. In focussing on DGSE I missed the wider potential result. Proving this however is another matter entirely.
It implies that the surface of all prices should be seen as a surface of rents and prices a -manifold which can be studied using the maths of non-smooth analysis. Price vectors can then be seen as like gravitational attraction on this manifold, towards points of maximum revenue or minimum cost..
The consequence of the spatial impossibility theorem is that where there are islands of homogeneous costs for a firm they will be unstable as firms will be seeking to move slightly to gain comparative advantage balancing this against the costs of reloacting. Seen like this Hotelling’s famous result of ice ceam sellers on beach can be seen as a special case of this wider phenomenon.
There may be islands of partial equilibrium in some markets – such as at a livestock market – but because receipt’s will be spent in other markets where all buyers don’t gather at a point or are distributed homogeneously, and where because of dislocation costs firms and consumers are not in constant movement, there will never be a general equilibrium in all markets all at once. What we may find are islands is relative stability on this manifold, settlements, cities, towns and villages. Indeed where transport costs relative to production are high then we may find islands of autarky where no competitive markets exist at all. In these islands of relative Lyapunov stability, with constantly shifting pints of attraction. Where you have many small flexible firms these may achieve near or quasi dynamic equilibrium. Again because these islands – settlements, towns and cities, have agglomeration economies of scale general equilibrium, or even island stability for more than short period of time, is never fully reached, cities grow and this shifts the comparative trade advantages of existing firms and whether they can outbid others for the same piece of land. This conception of the geographic landscape as a manifold of stability as it potentially looks beyond the New Economic Geography through being able to better analyse disequilibrium phenomena such as explosive city growth and city decline – using tools of stability analysis on manifolds such as von Neumann/Fourier techniques, for which well established mathematical techniques are known.