Sacrificing content for the sake of mathematical tractability

2 September, 2016 at 17:50 | Posted in Economics | Leave a comment

A commodity is a primitive concept (Debreu 1991), and the model comprises a finite set of classes of commodities. The number of distinguishable commodities (Debreu 1959) or commodity labels (Koopmans and Bausch 1959) is a natural number. The Arrow—Debreu model continues by defining these commodities as goods that are physically determined …

31DEP1RAQJL._SX313_BO1,204,203,200_But after asserting that commodities are physically determined, in a number of striking passages Debreu (1959: 30) states that their quantities can be expressed as real numbers. This of course poses a problem, because it means that irrational numbers can be used to express quantities of physical objects, even indivisible things …

Even the use of rational numbers to denote quantities of physical objects implies assuming that these physical objects are perfectly divisible. This, in turn, means that no matter how far we divide the physical objects, we still obtain objects with the same physical properties. Although this may appeal to our intuition when it comes to milk or flour, it is completely devoid of sense when it comes to physical objects that come in discrete units. Who among us owns exactly 1.379 cars, or gets 2.408 haircuts? The use of real numbers, including irrational numbers, implies that consumers can somehow specify quantities of goods that are not even fractions. This diverges even farther from experience and common sense. After all, nobody goes to the local Wal-Mart to purchase √2 vacuum cleaners, or π PCs.

Alejandro Nadal

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