In order to explain the paradox in the long run, when generations of people succeed each other, the change in the relative sizes of three groups must be accounted for. The thresholds (s,S) clearly play a key role in determining the size of the groups, but a link between generations must be established. First, recall that r, and hence the degree of disappointment, depends on the proportion of D^M/D^m in the set of the n close relationships of each youth.

Secondly, the proportion D^M/D^m in the population depends, in turn, on r and on S. Therefore, the sample distribution of D^M/D^m depends on the proportion D^M/D^m in the population, so that an endogenous dynamic across generations arises. Is this dynamic stable? Let us also observe that income growth depends not only on exogenous productivity growth, but also on labour time, which is governed by σa, and hence on the group composition of population. In these regards, let us state the following propositions.

Proposition 1. A level of S exists, i.e. S*, such that the size of group I remains unchanged through periods, having kept n and N constant and sufficiently great. If S_t=1>S*, where t=1 is the starting period, then the size of group I tends to zero in the subsequent periods. Income per head grows at a greater rate than productivity growth, although this difference disappears in the asymptotic steady state growth. If S_t=1<S*, then the size of group I in the subsequent periods tends to the size of the adult population (0.5N). Income per head grows at a lesser rate than productivity growth, converging to an inferior asymptote of positive growth rate.See the Appendix for proof.

Proposition 2.The threshold s fixes only the proportion of group III relative to group II, having kept n and N constant and sufficiently great. See the Appendix for proof.

Propositions 1 and 2 thus define the possibility of a constant composition of the population in the three groups over a long period of time, with S performing a key role. However, this “equilibrium composition of the population” is dynamically unstable. If S in the starting period is larger than the equilibrium level S*, then group I tends to vanish, more economic goods are produced because of the greater propensity of groups II and III for these goods, economic growth rises because σa>1 of group III.

Consequently, aggregate utility declines and working time increases across generations of individuals at rates that depend on s, on the level of σa of group III, and on the rate of technical progress. If S in the starting period is smaller than the equilibrium level S*, then groups II and III tend to vanish, economic growth slightly decelerates, but aggregate utility increases and working time decreases. In this case technical progress contributes to aggregate utility growth through complementarity between the economic good and the relational good.

Group III, which appears to be more successful than group II in yielding greater utility, although at the cost of longer working hours, is not successful in the long run. Therefore, the model is able to explain different possible patterns in the dynamics of income, well-being, and labour input depending on definite parameters. In particular S* and, secondarily, s* both make youths’ vulnerability effective so that they fall into groups II and III, and determine the diffusion of this vulnerability in the population.

It can thus be argued that S* and s* are able to capture the influence of those social forces that shape individuals’ preferences and expectations, like materialistic social models, or the market system itself as the provision of extrinsic incentives which superimpose themselves on intrinsic motivations (see Section 3.1) (Bowles 1998)

By Prof. Maurizio Pugno

Next: Conclusions

Summary: Index