The Generalisation of Mandeville and Optimisation

Figure 10.1 is of the most use in illustrating the effect of changes in the two behavioural parameters, y and q. When the supply of labour is independent of the degree of ‘luxury’, y is zero and so therefore is y, and a is simply R/(1 -pu). For given values of u, w, and a, the outputs P and Q are as large as possible. The effect of an increase in ‘luxury’ in these circumstances is unambiguous: S increases and Q(P) rotates clockwise about the origin, determining a higher equilibrium of P and Q. This case is pure Mandeville (1732 [1988]: Book I, 34, 35). The greater the degree of ‘luxury’, the greater total production, income, and population. But should Jove convert the bees from ‘vice’ and turn them to a virtuous frugality, their hive collapses:

As Pride and Luxury decrease...

All Arts and Crafts neglected lie;

Content, the Bane of Industry.

So few in the vast Hive remain,

The hundredth Part they can’t maintain.

For as q (and S) decline, Q(P) rotates anti-clockwise until it lies along the vertical axis. National product falls to the bare minimum of a subsistence, agricultural economy: P = a = R/(l - pu), and Q = 0 .

Paley was deeply aware of the importance of ‘luxury’ in stimulating employment and industry, and thereby farm production and population: ‘The watchmaker, while he polishes the case, or files the wheels of his machine, is contributing to the production of corn as effectually, though not so directly, as if he handled the spade or held the plough’ (Paley ibid.: 610). He discussed the stimulating effect upon agriculture of trade with large urban centres in a passage which resembles the treatment by Smith (ibid.: 610-612; cf. Smith 1776 [1976]: Book III, Chapter i) and may owe something to Hume (1752 [1994]: 98-99). But as I have noted above, Paley was also aware—as Mandeville most certainly was not—that the effect of ‘luxury’ upon production and population cuts two ways. On the one hand, indeed it stimulates demand, production, and population, but on the other, by tending to reduce the supply of labour, it has the opposite effect. This is apparent from Figure 10.1. An increase in q causes Q(P) to rotate clockwise which by itself would increase output. However, it also causes P(Q) to shift downwards and Q(P) to shift to the left, which by themselves would reduce output:

It appears, then, that luxury, considered with a view to population, acts by two opposite effects and it seems probable that there exists a point in the scale to which luxury may ascend...beyond which the prejudicial consequences begin to preponderate. The determination of this point, though it assume the form of an arithmetical problem, depends upon circumstances too numerous, intricate and undefined, to admit of a precise solution (Paley ibid.: 597—598; italics in original).

Though no ‘arithmetical’ determination of optimum q = q* is available, a merely algebraic one may be had. For as Paley’s intellectual grandson Robert Malthus noted 20 years later, ‘many of the questions, both in morals and in politics, seem to be of the nature of the problems de maximis and minimis in fluxions; in which there is always a point where a certain effect is the greatest, while on either side of this point it gradually diminishes’ (Malthus 1814 [1986]: 102). By means of elementary ‘fluxions’, we may differentiate P with respect to q, and so solve for q* from the first-order conditions for a maximum of P (Appendix B).