Counting the missing poor in pre-industrial societies

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.

Appendix

1.1 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.

1.2 Proof of Proposition 2

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

$$\begin{aligned} \mathbf {Mw=}\lambda {\mathbf {w}} \end{aligned}$$

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:

$$\begin{aligned} \mathbf {Mw-}\lambda \mathbf {w}= & {} \mathbf {0} \\ \left( \mathbf {M-}\lambda {\mathbf {I}}\right) {\mathbf {w}}= & {} {\mathbf {0}} \end{aligned}$$

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:

$$\begin{aligned} \det \left( \begin{array}{cc} f_{p}s_{p}{\bar{m}}_{p}-\lambda &{} f_{n}s_{n}m_{n} \\ f_{p}s_{p}m_{p} &{} f_{n}s_{n}{\bar{m}}_{n}-\lambda \end{array} \right) =0 \end{aligned}$$

Therefore we have:

$$\begin{aligned} \left( f_{p}s_{p}{\bar{m}}_{p}-\lambda \right) \left( f_{n}s_{n}{\bar{m}} _{n}-\lambda \right) -f_{n}s_{n}m_{n}f_{p}s_{p}m_{p}=0 \end{aligned}$$

Hence, after some simplifications:

$$\begin{aligned} \lambda ^{2}-\lambda \left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}} _{n}\right) +f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) =0 \end{aligned}$$

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

$$\begin{aligned} \Delta =\left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}\right) ^{2}-4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) \end{aligned}$$

Note that $\Delta$ can be rewritten as:

$$\begin{aligned} \Delta= & {} \left( f_{p}s_{p}(1-m_{p})\right) ^{2}+\left( f_{n}s_{n}(1-m_{n})\right) ^{2}+2f_{p}s_{p}(1-m_{p})f_{n}s_{n}(1-m_{n}) \\&-4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) \\= & {} \left( f_{p}s_{p}(1-m_{p})\right) ^{2}+\left( f_{n}s_{n}(1-m_{n})\right) ^{2}\\&-2f_{p}f_{n}s_{n}s_{p}(1-m_{p}-m_{n})+2f_{p}f_{n}s_{n}s_{p}m_{p}m_{n} \\= & {} \left( f_{p}s_{p}(1-m_{p})\right) ^{2}+\left( f_{n}s_{n}(1-m_{n})\right) ^{2}\\&-2f_{p}f_{n}s_{n}s_{p}(1-m_{p}-m_{n}-m_{p}m_{n}) \end{aligned}$$

Hence the two eigenvalues are:

$$\begin{aligned} \lambda _{1}= & {} \frac{\left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}} _{n}\right) +\root 2 \of {\left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}} _{n}\right) ^{2}-4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) }}{2} \\ \lambda _{2}= & {} \frac{\left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}} _{n}\right) -\root 2 \of {\left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}} _{n}\right) ^{2}-4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) }}{2} \end{aligned}$$

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:

$$\begin{aligned}&\left( \begin{array}{cc} f_{p}s_{p}{\bar{m}}_{p} &{} f_{n}s_{n}m_{n} \\ f_{p}s_{p}m_{p} &{} f_{n}s_{n}{\bar{m}}_{n} \end{array} \right) \left( \begin{array}{c} N_{p} \\ N_{n} \end{array} \right) \\&\quad =\left( \frac{f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}+\root 2 \of { \begin{array}{l} \left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}\right) ^{2} \\ -4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) \end{array} }}{2}\right) \left( \begin{array}{c} N_{p} \\ N_{n} \end{array} \right) \end{aligned}$$

Hence we have

$$\begin{aligned}&f_{p}s_{p}{\bar{m}}_{p}N_{p}+f_{n}s_{n}m_{n}N_{n} \\&\quad =\frac{f_{p}s_{p}{\bar{m}} _{p}+f_{n}s_{n}{\bar{m}}_{n}+\root 2 \of { \begin{array}{l} \left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}\right) ^{2} \\ -4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) \end{array} }}{2}N_{p} \\&\quad f_{p}s_{p}m_{p}N_{p}+f_{n}s_{n}{\bar{m}}_{n}N_{n} \\&\quad =\frac{f_{p}s_{p}{\bar{m}} _{p}+f_{n}s_{n}{\bar{m}}_{n}+\root 2 \of { \begin{array}{l} \left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}\right) ^{2} \\ -4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) \end{array} }}{2}N_{n} \end{aligned}$$

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

$$\begin{aligned}&f_{p}s_{p}m_{p}N_{p}+f_{n}s_{n}{\bar{m}}_{n}\left( 1-N_{p}\right) \\&\quad =\frac{ f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}+\root 2 \of { \begin{array}{l} \left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}\right) ^{2} \\ -4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) \end{array} }}{2}(1-N_{p}) \end{aligned}$$

From which it follows that

$$\begin{aligned} N_{p}=\frac{f_{p}s_{p}{\bar{m}}_{p}-f_{n}s_{n}{\bar{m}}_{n}+\root 2 \of {\left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}\right) ^{2}-4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) }}{ 2f_{p}s_{p}-(f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n})+\root 2 \of {\left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}\right) ^{2}-4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) }} \end{aligned}$$

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

$$\begin{aligned} {\mathbf {w}}_{1}= & {} \left( \frac{N_{p}}{N_{n}}\right) \\= & {} \left( \frac{N_{p}}{1-N_{p} }\right) =\left( \begin{array}{c} \frac{f_{p}s_{p}{\bar{m}}_{p}-f_{n}s_{n}{\bar{m}}_{n}+\root 2 \of {\left( f_{p}s_{p} {\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}\right) ^{2}-4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) }}{2f_{p}s_{p}-(f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}} _{n})+\root 2 \of {\left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}\right) ^{2}-4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) }} \\ \frac{2f_{p}s_{p}\left( 1-{\bar{m}}_{p}\right) }{2f_{p}s_{p}-(f_{p}s_{p}{\bar{m}} _{p}+f_{n}s_{n}{\bar{m}}_{n})+\root 2 \of {\left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n} {\bar{m}}_{n}\right) ^{2}-4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) }} \end{array} \right) \end{aligned}$$

From the Strong Ergodic Theorem, we have that

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{{\mathbf {N}}_{t}}{\lambda _{1}^{t}}=c_{1} {\mathbf {w}}_{1} \end{aligned}$$

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:

$$\begin{aligned} {\rm HC}=\frac{N_{p}}{N_{p}+N_{n}}=\frac{f_{p}s_{p}{\bar{m}}_{p}-f_{n}s_{n}{\bar{m}} _{n}+\root 2 \of { \begin{array}{l} \left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}\right) ^{2} \\ -4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) \end{array} }}{2f_{p}s_{p}-f_{p}s_{p}{\bar{m}}_{p}-f_{n}s_{n}{\bar{m}}_{n}+\root 2 \of { \begin{array}{l} \left( f_{p}s_{p}{\bar{m}}_{p}+f_{n}s_{n}{\bar{m}}_{n}\right) ^{2} \\ -4f_{p}f_{n}s_{n}s_{p}\left( 1-m_{n}-m_{p}\right) \end{array} }} \end{aligned}$$

1.3 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:

$$\begin{aligned} {\rm HC}=\frac{{\bar{m}}_{p}-\mu {\bar{m}}_{n}+\root 2 \of {\left( {\bar{m}}_{p}+\mu {\bar{m}} _{n}\right) ^{2}-4\mu \left( 1-m_{n}-m_{p}\right) }}{2-{\bar{m}}_{p}-\mu \bar{m }_{n}+\root 2 \of {\left( {\bar{m}}_{p}+\mu {\bar{m}}_{n}\right) ^{2}-4\mu \left( 1-m_{n}-m_{p}\right) }} \end{aligned}$$

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:

$$\begin{aligned} {\rm HC}=\frac{{\bar{m}}_{p}-\mu {\bar{m}}_{n}+\phi }{2-{\bar{m}}_{p}-\mu {\bar{m}} _{n}+\phi } \end{aligned}$$

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

$$\begin{aligned} \frac{\partial {\rm HC}}{\partial \mu }=\left( -{\bar{m}}_{n}+\phi ^{\prime }\right) 2\frac{(1-{\bar{m}}_{p})}{\left( 2-{\bar{m}}_{p}-\mu {\bar{m}}_{n}+\phi \right) ^{2}} \end{aligned}$$

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

Hence we have:

$$\begin{aligned} \frac{\partial {\rm HC}}{\partial \phi }\gtrless & {} 0\iff \phi ^{\prime }\gtrless {\bar{m}}_{n} \\\iff & {} \frac{1}{2}\frac{\left[ 2\left( \mu {\bar{m}}_{n}\right) {\bar{m}}_{n}+2 {\bar{m}}_{p}{\bar{m}}_{n}-4{\bar{m}}_{n}-4{\bar{m}}_{p}+4\right] }{\left[ \left( {\bar{m}}_{p}\right) ^{2}+\left( \mu {\bar{m}}_{n}\right) ^{2}+2{\bar{m}}_{p}\mu {\bar{m}}_{n}-4\mu {\bar{m}}_{n}-4\mu {\bar{m}}_{p}+4\mu \right] ^{\frac{1}{2}}} \gtrless {\bar{m}}_{n} \\\iff & {} \frac{\left[ 2\left( \mu {\bar{m}}_{n}\right) {\bar{m}}_{n}+2{\bar{m}}_{p} {\bar{m}}_{n}-4{\bar{m}}_{n}-4{\bar{m}}_{p}+4\right] }{2\phi }\gtrless {\bar{m}}_{n} \\\iff & {} \frac{{\bar{m}}_{n}\left( \mu {\bar{m}}_{n}+{\bar{m}}_{p}\right) -2{\bar{m}} _{n}-2{\bar{m}}_{p}+2}{\phi }\gtrless {\bar{m}}_{n} \end{aligned}$$

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:

$$\begin{aligned} \frac{{\bar{m}}_{n}\left( \mu {\bar{m}}_{n}+{\bar{m}}_{p}\right) -2{\bar{m}}_{n}-2 {\bar{m}}_{p}+2}{\frac{{\rm HC}\left( 2-{\bar{m}}_{p}-\mu {\bar{m}}_{n}\right) -{\bar{m}} _{p}+\mu {\bar{m}}_{n}}{(1-{\rm HC})}}\gtrless & {} {\bar{m}}_{n} \\ (1-{\rm HC})\left[ {\bar{m}}_{n}\left( \mu {\bar{m}}_{n}+{\bar{m}}_{p}\right) -2{\bar{m}} _{n}-2{\bar{m}}_{p}+2\right]\gtrless & {} {\bar{m}}_{n}\left[ 2{\rm HC}-{\rm HC}\left( {\bar{m}} _{p}\right. \right. \\&\left. \left. +\mu {\bar{m}}_{n}\right) -{\bar{m}}_{p}+\mu {\bar{m}}_{n}\right] \\ \left[ {\bar{m}}_{n}\left( \mu {\bar{m}}_{n}+{\bar{m}}_{p}\right) -2{\bar{m}}_{n}-2 {\bar{m}}_{p}+2\right] -{\rm HC}\left[ -2{\bar{m}}_{p}+2\right]\gtrless & {} {\bar{m}}_{n} \left[ -{\bar{m}}_{p}+\mu {\bar{m}}_{n}\right] \\ {\bar{m}}_{n}{\bar{m}}_{p}-{\bar{m}}_{n}-{\bar{m}}_{p}+1-{\rm HC}\left[ -{\bar{m}}_{p}+1 \right]\gtrless & {} 0 \\ {\bar{m}}_{n}({\bar{m}}_{p}-1)+(1-{\bar{m}}_{p})(1-{\rm HC})\gtrless & {} 0 \\ 1-{\rm HC}\gtrless & {} {\bar{m}}_{n} \end{aligned}$$