## Abstract

Under income-differentiated mortality, poverty measures suffer from a selection bias: they do not count the missing poor (i.e., persons who would have been counted as poor provided they did not die prematurely). The Pre-Industrial period being characterized by an evolutionary advantage (i.e., a higher number of surviving children per household) of the non-poor over the poor, one may expect that the missing poor bias is substantial during that period. This paper quantifies the missing poor bias in Pre-Industrial societies, by computing the hypothetical headcount poverty rates that would have prevailed provided the non-poor did not benefit from an evolutionary advantage over the poor. Using data on Pre-Industrial England and France, we show that the sign and size of the missing poor bias are sensitive to the degree of downward social mobility.

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## Notes

Besides the discussion of King’s table (for year 1688), Lindert and Williamson (1982) provide also a critical presentation of other social tables, such as the ones of Joseph Massie (for year 1759) and Patrick Colquhoun (for years 1801–1803). Lindert and Williamson (1983) revisits the social table of Dudley Baxter (for year 1868), and provide a picture of England’s development during two centuries.

Several mechanisms may explain why the rich had, at that time, a higher number of surviving children than the poor. These mechanisms include the various channels through which material living conditions affect mortality and fertility outcomes. Throughout this paper, we will take that evolutionary advantage

*as given*and examine its impact on poverty measurement, without trying to identify the particular mechanisms at work behind this advantage.See Malthus (1798), chapter 1, pp. 71–72 and chapter 5, p. 93.

For instance, while poverty above age 60 is higher in Portugal than in Estonia on the basis of standard headcount poverty rates, the ranking of poverty is reversed once the missing poor are taken into account.

Given that this paper assumes that skills are redistributed randomly in each new cohort, it underestimates the persistence of cumulative selection effects, unlike the present paper.

In a recent work, de la Croix et al. (2019) show that the reproductive success of the middle class exceeded the one of the poor and the very rich classes in Pre-Industrial England.

Campbell (2013) showed that per capita income increased in Italy during its twelfth and thirteenth century commercial revolution, in Holland during its fifteenth and sixteenth century golden age, and in England during the seventeenth and eighteenth century runup to its industrial revolution. During these episodes, expanding trade sustained output and employment growth in the manufacturing and service sectors, developments that were not reflected in real wage data.

The effects of that pattern on family welfare are studied by Horrell et al. (2021).

We consider here a closed economy and we abstract from migrations. Abstracting from migrations is a simplification. However, this assumption makes sense for the Pre-Industrial epoch, during which most movements of populations took place

*within*a country rather than*between*countries (see Wrigley 1969).We have \({\bar{m}}_{p}=1-m_{p}\).

We have \({\bar{m}}_{n}=1-m_{n}\).

See Caswell (2001), pp. 83–84.

See Caswell (2001), pp. 84–85.

Various mechanisms may explain why \(\mu >1\). On the mortality side, bad material living conditions can lead to higher mortality of the poor because of low quality/quantity of food or lack of health care. On the fertility side, the effects of poverty on fertility may take distinct forms. For instance, poverty may cause physiological stress that reduces fertility. However, it is also possible that humans, like other living beings, react to a poor environment by adopting a particular life history strategy involving a larger number of descendants (r-strategy), which may counterbalance the previous effect. We will not try here to quantify all these effects. Rather, we will take \(\mu\)

*as given*and consider its effect on the measurement of poverty.On the evolutionary advantage of the rich in Pre-Industrial times, see (Clark 2007).

The headcount poverty rate HC is fully determined by parameters \(\left\{ m_{n},m_{p},\mu \right\}\). Hence the derivative \(\frac{\partial {\rm HC}}{ \partial \mu }\) measures the full effect of a marginal variation of the strength of the evolutionary advantage of the non-poor on the prevalence of poverty in the long-run.

The intuition behind that result goes as follows. If we were considering the

*size*of the population in the long-run, the absolute levels of group-specific fertility rates and survival rates would matter. However, from the perspective of the*structure*of the population in the long-run, only the*relative*levels of fertility and survival rates across groups matter. The long-run structure of the population depends on whether one group has a larger number of surviving offspring than another.For instance, under a low downward mobility (\(m_{n}=0.100\)), a probability of leaving poverty of 0.186 can replicate a headcount poverty rate of 24.2%, but to reproduce \({\rm HC}=48.2\)%, one needs, under \(m_{n}=0.100\), a lower upward mobility for the poor, \(m_{p}\) being extremely low (0.001).

For the sake of making that point, let us assume that the measures of poverty in Sects. 7 and 8 are comparable. If, for instance, one adopts the intermediate benchmark for poverty in Pre-Industrial England, and compare it with the measure of poverty in Pre-Industrial France (lower bound), one can see that HC is

*higher*in France than in England. However, this may not be the case when considering hypothetical poverty rates \({\rm HC}^{H}\) where all selection biases are neutralized. Under a low downward social mobility, \({\rm HC}^{H}\) is*lower*in France than in England. Note that such an inversion of the ranking does not occur when a high downward social mobility is assumed (since in that case \({\rm HC}^{H}\) is close to HC in each country).One limitation of this matrix population model is that it cannot account for potential second-order effects due to interactions between fertility, mortality and intergenerational mobility, because these are modelled as parameters, and not as variables. We leave the exploration of more general models for future research.

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The authors would like to thank Arnaud Deseau, Claude Diebolt, Sylvain Funck, Oded Galor, Gilles Grandjean and two anonymous reviewers, as well as participants of seminar at Université Saint Louis (Brussels), for their comments on this paper. This work was supported by the French National Research Agency Grant ANR-17-EURE-0020, and by the Excellence Initiative of Aix-Marseille University—A*MIDEX.

## Appendix

### Appendix

### Proof of Proposition 1

Irreducibility prevails when the life cycle graph associated to the matrix admits at least one path from each node and toward each node. This is the case for matrix \({\mathbf {M}}\). As shown in Fig. 2, the life cycle graph associated to our model includes two distinct populations, i.e.,, poor and non-poor, which contribute to each other through the social mobility process, which can go upwards or not for the poor, and downwards or not for the non-poor. Thus each population makes contributions to the other population, implying irreducibility of \({\mathbf {M}}\).

Primitivity arises when there exists a power *k* such that raising the matrix to that power makes it positive. This is clearly the case for matrix \({\mathbf {M}}\), which is a positive matrix.

### Proof of Proposition 2

Let us characterize eigenvalues of the matrix \({\mathbf {M}}\). We look for solutions for the equation:

where \(\lambda\) is the eigenvalue (a scalar) while \({\mathbf {w}}\) is the associated eigen vector, a vector that makes matrix multiplication and scalar multiplication equivalents. From the definition of the eigen vectors, it follows that:

Nonzero solutions require \(\left( \mathbf {M-}\lambda {\mathbf {I}}\right)\) to be a singular matrix, that is, that it has a zero determinant.

Hence eigenvalues are solutions to:

Therefore we have:

Hence, after some simplifications:

Eigenvalues can be found as the roots of this polynomial. We have:

Note that \(\Delta\) can be rewritten as:

Hence the two eigenvalues are:

We have \(\lambda _{1}>\lambda _{2}\), so that the dominant eigenvalue is \(\lambda _{1}\).

We can then derive the long-run population structure by calculating the eigenvector \({\mathbf {w}}_{1}\) associated to the dominant eigenvalue \(\lambda _{1}\). The associated eigenvector is such that:

Hence we have

Two equations and two unknowns. Normalizing to \(N_{p}+N_{n}=1\), the second equation can be rewritten as:

From which it follows that

Hence the eigen vector associated to \(\lambda _{1}\) is

From the Strong Ergodic Theorem, we have that

that is, the asymptotic population structure is given by the eigen vector \({\mathbf {w}}_{1}\), while the precise size of the different population groups can always be scaled as desired, since eigenvectors are always defined up to a multiplicative constant.

Hence the long-run headcount is given by:

### Proof of Proposition 3

Let us substitute for \(f_{n}s_{n}=\mu f_{p}s_{p}\) in the long-run poverty rate of Proposition 2. We obtain:

Let us define \(\phi \equiv \root 2 \of {\left( {\bar{m}}_{p}\right) ^{2}+\left( \mu {\bar{m}}_{n}\right) ^{2}+2{\bar{m}}_{p}\mu {\bar{m}}_{n}-4\mu {\bar{m}} _{n}-4\mu {\bar{m}}_{p}+4\mu }\). Hence the headcount ratio is:

The derivative of the headcount with respect to \(\mu\) is:

Whose sign depends on the sign of \(-{\bar{m}}_{n}+\phi ^{\prime }.\)

Hence we have:

Since \({\rm HC}=\frac{{\bar{m}}_{p}-\mu {\bar{m}}_{n}+\phi }{2-{\bar{m}}_{p}-\mu {\bar{m}} _{n}+\phi }\rightarrow \frac{{\rm HC}\left( 2-{\bar{m}}_{p}-\mu {\bar{m}}_{n}\right) - {\bar{m}}_{p}+\mu {\bar{m}}_{n}}{(1-{\rm HC})}=\phi .\)

Hence the condition can be written as:

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Lefebvre, M., Pestieau, P. & Ponthiere, G. Counting the missing poor in pre-industrial societies.
*Cliometrica* (2022). https://doi.org/10.1007/s11698-022-00243-y

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DOI: https://doi.org/10.1007/s11698-022-00243-y

### Keywords

- Poverty
- Measurement
- Selection effects
- Missing poor

### JEL Classification

- I32
- N33