1   The Benchmark Model

As our benchmark, we choose the [blinder-1973-qje] model. This benchmark model is ideally suited to our present project: it is well-known, extremely simple in structure, and explicitly designed for the exploration of the economic implications of family institutions. Additionally, it famously predicts that both primogeniture and class mating merely sustain existing wealth inequality and do not cause it to rise or fall. For researchers interested in explaining the level of wealth inequality, the benchmark model's predictions discourage attention to family institutions.

The benchmark model incorporates standard features of simple overlapping generations models. There is a constant population and sex distribution. Each person lives two periods, denoted "childhood" and "economic life". In childhood, all economic decisions are made by the parents. At the beginning of the second period, an individual's economic life starts with marriage, inheritance, and the commencement of work. All people marry and all married couples bear two children, a son and a daughter. In addition, mating may be random or assortative by wealth. Upon death, an individual may leave an estate. Division of the estate may involve equal sharing or male-preference bequests.

The benchmark model is intentionally very simple, introducing only enough structure to explore some implications of particular familial institutions (Blinder 1973). The goal of the model is not detailed realism but rather transparency in the answers to specific questions. Note in particular that debates about the relative contributions of entrepreneurship and inheritance to the wealth distributions of developed countries lie outside the scope of this model.

The benchmark model has the virtues of simplicity, transparency, attention to family institutions (mating and inheritance), and---for the purposes of the present paper---ease of simulation. The benchmark model produces well-known and surprising predictions (Blinder 1973). Specifically, neither unequal inheritances in the form of male-preference bequests nor wealth concentration in the form of class marriage produce wealth inequality. These family institutions can sustain existing wealth inequality but not modify it. We refer to these as the core predictions of the benchmark model.

While initially surprising, the core predictions of the benchmark are readily explained. Consider first the case of male-preference bequests. Recall that each family has two children, a boy and a girl. The bequest practices dictate that the son inherit everything, so the new generation must have the same wealth distribution as the old generation. Relative inequality---as measured by the coefficient of variation or the Gini coefficient---is unchanged. Consider next the case of class mating. Again, each family has two children, a son and a daughter. The two wealthiest families intermarry, forming the two richest families of the next generation. Similarly for the next two wealthiest, and so on. The benchmark model thus predicts that male-preference bequests and class mating can sustain but not cause wealth inequality.

new_males = []
new_females = []
for male, female in pairup(males, females):
    hh_networth = male + female
    new_males.append(male_share * hh_networth)
    new_females.append(female_share * hh_networth)
males, females = new_males, new_females

2   Sex Is Random

Researchers consider inheritance to be an important determinant of individual financial wealth, especially for very wealthy families.⁴ However the benchmark model shows us that evidence that inheritance practices sustain wealth inequality need not constitute evidence that these practices determine its level. (This crucial distinction is often ignored.) Recall the core predictions of the benchmark model: male-preference bequests sustain but do not change the distribution of wealth, and class mating sustains but does not change the distribution of wealth. Since the benchmark model is simple enough to be relatively transparent, these predictions are readily understandable as implications of the model. Nevertheless, these predictions are often considered surprising or counterintuitive.

We now show that both predictions rest on the neglect of an oft-overlooked yet fundamental demographic factor. Recall a core demographic assumption of the benchmark model: each family has two children, a boy and a girl. Models of wealth accumulation often do not distinguish individuals by sex, but this is a fairly common assumption when they do. It is an analytically tractable way to ensure a constant population and sex ratio. Blinder (p.610) justified this demographic structure by the hope "that it does not stray too far from the truth for a stationary economy." However, even a steady-state population does not mean one boy and one girl per family. We find that an apparently minor modification of this assumption produces large changes in the model predictions. Before we can explore this, however, we need to clarify the nature of class mating and of male-preference bequests.

Marriage is a core social institution that can have important effects on wealth inequality. If marriage is perfectly assortative by wealth ('class mating'), we might expect this to concentrate wealth. If marriage is determined by factors unrelated to wealth ('random mating'), we might expect this to moderate wealth inequality.

We must distinguish two characterizations of 'class mating': spouses paired based on household wealth, and spouses paired based on their own individual wealth. When each household has one child of each sex, failure to draw this distinction is innocuous in the benchmark model. Otherwise, each concept gains independent interest. Here we adopt the second characterization: after ranking men and women by individual wealth, 'class mating' means that spouses share a common rank.⁵

Our project also calls for more care in the discussion of primogeniture. Economists sometimes use the term 'primogeniture' very loosely to refer to a variety of male-preference bequest practices. When families may have no sons, or multiple sons, this usage becomes untenable. If there are multiple sons, do all inherit? If there are no sons, can daughters inherit? If daughters can inherit and there are multiple daughters, do all inherit? If daughters cannot inherit and there are no sons, what is the line of bequest transmission? Since each alternative changes the model, attempts to explain an actual distribution of wealth must naturally be bound to actual social practices.

We can use any of the above options to illustrate the importance of demographic assumptions in models of wealth inequality. In the context of the present paper, we need a practice that is a close match to the benchmark model in two ways: it should be a plausible extension of the notion of male-preference bequests to the new demographic context, and among remaining the options it should appear least likely to cause additional wealth inequality. We therefore consider the case of split inheritances. Specifically, we take 'male-preference bequests' to mean that a lone son inherits everything (matching the benchmark model), multiple sons inherit everything and split it equally, and in the absence of a male heir, daughters inherit everything and split it equally.

The individuals in the resulting model are heterogeneous: they differ by sex, the sex of their sibling, and the wealth of their family. Such heterogeneity is a substantial complication if we attempt to extend the original deductive analysis of the benchmark model. However this degree of heterogeneity can easily be handled by simulation methods. We therefore simulate both the benchmark model and our modifications of it.

Figure 1 presents the results of five simulations. We measure wealth inequality with a Gini coefficient and chart inequality across 160 generations. Each simulation imposes a Gini coefficient of 0.6 for the initial distribution of wealth. (The value of this model parameter proves unimportant.)

The perfectly horizontal line illustrates the benchmark model with male-preference bequests and/or class mating. (All three cases yield the same result, as expected.) The simulation results naturally reflect the analytically derived core predictions of the benchmark model: wealth inequality is sustained at the initial level and shows no tendency to change. The demographic structure of all other reported simulations differ from the benchmark model in an way that might seem minor but proves to be crucial: children's sexes are distributed randomly across families.

Intergenerational Inequality when Sex is Random

The topmost line illustrates the effect of male-preference bequests under class mating. The results diverge markedly from the benchmark predictions: inequality grows rapidly and stays high. In striking contrast with the benchmark predictions, we discover a role for inheritance in producing wealth inequality. Our more realistic demographics imply that male-preference bequests and class mating combine to cause extreme inequality.

The next two simulations show that there is now a critical synergy between bequest practices and mating practices. Consider the line illustrating male-preference bequests under random mating: inequality quickly declines to a modest Gini coefficient of about 0.4 and subsequently varies around that level. Although we do not see extreme inequality in this case, we still find that male-preference bequests are causative for the level of wealth inequality.

Next consider the outcome for equal bequests and class mating. We find a gentle downward trend in inequality. Again we have a striking contrast with the benchmark model: class mating fails to sustain wealth inequality. The explanation is natural: when multiple inheritance is possible, family fortunes must eventually disperse.

Finally, as a point of reference, Figure 1 also displays the outcomes under equal bequests and random mating. The results for this case are unsurprising and are compatible with the benchmark model predictions. There is a rapid decline of wealth inequality from its initial level toward an equal distribution of wealth.

3   Conclusion

Economists have long been interested in measuring and explaining inequality in the ownership of wealth. Research on the sources of wealth inequality remains active, and inheritance and mating practices have occasionally come under scrutiny. [blinder-1973-qje] derived surprising predictions from the benchmark model: male-preference bequests or class mating can sustain existing wealth inequality, but neither cause it. That is, neither male-preference bequests or class mating modify the level of wealth inequality, even in the long run. This paper uses simulation methods to show that the core predictions of the benchmark model depend on an analytically convenient and seemingly innocuous demographic assumption.

Although the distribution of wealth has received increasing attention from economists, the role of demographic assumptions is rarely explored.⁶ Our simulation results demonstrate that even apparently minor aspects of these demographic assumptions can be crucial, both for wealth dynamics and for steady-state outcomes. We make one change in the benchmark model: we allow randomness in sex determination. This seemingly trivial change radically alters the predictions of the model.

Allowing for randomness in the sex of offspring is an appealing increase in realism. As is so often the case, additional realism has a cost: it reduces the analytical tractability of the benchmark model. We therefore approach the model through simulation methods. Our simulation results show that this seemingly trivial change in demographic structure implies a dramatic change in model predictions. Perhaps most notably, male-preference bequests combine with class mating to cause the emergence of extreme inequality. The synergy is crucial. Neither male-preference bequests in the presence of random mating nor class mating in the presence of equal sharing can cause or even sustain substantial wealth inequality. Furthermore, under these slightly more realistic demographics, equal bequests prove a powerful equalizer even in the face of class mating.

The fragility of the provocative core predictions of the benchmark model is of considerable interest. More generally, the present paper illustrates the crucial role of demographic assumptions in models of wealth inequality. Our simulation results show how an seemingly minor change in demographic assumptions can dramatically alter a model's predictions.

4   References

5   Notes