Chapter 30: The Unit Simplex
John Whalley The unit simplex diagram (Figure 30.1) became a central part of modern economics with the development of general equilibrium theory in the 1950s, and is particularly associated with the proofs of existence of equilibrium following the pioneering work of Arrow and Debreu (1954). The diagram shows the coordinates of relative prices in a three-good economy in which only relative prices matter and in which the absolute price level is of no consequence: the traditional pure barter economy widely used in economics. It is drawn in three-dimensional space, with three axes of prices of goods. The feature that relative prices are all that matters reflects the common assumption made in general equilibrium analysis of homogeneity of degree zero of continuous excess demand functions in the pure exchange economy case. As Arrow showed, this follows directly from an assumption of utility-maximizing behaviour. This implies that the absolute price level plays no role in economic behaviour. Put differently, if all prices were to double, the incomes of households selling endowments of goods would double and at the same time the acquisition prices of goods would double. In relative terms, prices would be unchanged. This central proposition that only relative prices matter is the longstanding feature of microeconomic analysis going back many hundreds of years, but its representation in this diagram was central to mathematical economics literature and discussion of the existence of the equilibrium which followed Arrow and Debreu. The original Walrasian formulation of general equilibrium asserted that a system of...
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