The Paradox of Thrift in an Inegalitarian Neoclassical Economy

[Schilcht 1975] and [Bourguignon 1981] studied the case of a convex saving function in the [Stiglitz 1969] model. They have shown that if one of the two proportions of the rich or the poor is below a certain threshold, there is a two-class equilibrium. However, they have only proved the existence of this threshold. We give here a system of equations to calculate this threshold which we interpret as the maximum proportion of rich for having a stable two-class configuration. If the proportion of rich exceeds this threshold, the economy enters a phase of decline although the golden-rule capital has not yet been reached. This decline is due to a specific articulation between the rate of decrease in the productivity of capital and the rate of increase in the depreciation of capital. The mechanism of this decline recalls the description given in [Keynes 1936], of the decline which happens when there is too much savings in an inegalitarian context. This is an example of what is known as the "paradox of thrift". It is remarkable that this paradox takes place in a neoclassical setting that does not include key Keynesian elements such as saturation of demand, monetization of savings, short-term effects, expectation problems, involuntary unemployment and rigidities. Numerical simulations are given to illustrate and analyze the mechanisms involved.