h�� �pe<sup>���������</sup> (2) where s�� and z�� represent her expected future education and health; p is the probability that a child born today survives; and NG represents the _<u>N</u>_ <u>ext</u> _<u>G</u>_ <u>eneration of workers.</u><sup>3</sup> Multiplying by p captures the loss in future productivity per child born today due to premature mortality, since children who do not survive do not grow up to become productive adults. The survival probability p is the complement of the under‐5 mortality rate.<sup>4</sup> As discussed below, expected future education and health are measured based on the current outcomes. For example, expected future education will be measured as the number of years of school a child progressing through the education system is likely to obtain given prevailing enrollment rates at different levels. Similarly, expected future health will be measured under the assumption that current health conditions prevail into the future.

Human capital in Equation (1) expresses human capital in units of productivity relative to a worker with s� �z� �0 , in which case h� �1 . To express the HCI in more intuitive units, rescale Equation (1) by dividing by a benchmark level of human capital corresponding to complete education and full health. Let p<sup>∗</sup> , s<sup>∗</sup> and z<sup>∗</sup> represent these benchmark values. For survival, a natural benchmark is p<sup>∗</sup> �1 . For years of school, the benchmark is s<sup>∗</sup> �14 years of school, corresponding the maximum possible number of years of school achieved by age 18 by a child who starts school at age 4. For health the natural benchmark corresponding to full health is z<sup>∗</sup> �1 .

With this notation, the HCI is:

> 3 Formally, let h��� represent human capital at some future date t�k . Expected future human capital is given by E��h�����pE��e<sup>�����</sup> �E��e<sup>�����</sup> ��pe<sup>���������</sup> e<sup>���������</sup> , where p is the probability a child does not survive to become a future worker, in which case her human capital as a future worker does not materialize.  The first equality requires the assumption of independence between education and health outcomes across individuals, and E��e<sup>�����</sup> � and E��e<sup>�����</sup> � should be interpreted as expectations conditional on survival (and assuming that human capital conditional on not surviving is zero).  The second inequality is due to the convexity of the human capital function.  Since only the “likely future values” of  health and education, E��s���� and E��z���� , are observable, and not the entire distribution of possible future outcomes which would be required to calculate E��e<sup>�����</sup> � and E��e<sup>�����</sup> � , the last term serves as a lower bound on expected future human capital.  Naturally, given the convexity of the human capital function, a higher variance of education and health across individuals, and a higher covariance between the two, increases the gap between the lower bound and the expectation.   To keep notation simple, s�� and z�� denote the likely future values E��s���� and E��z���� that represent the expected education and health of the next generation of workers.

4 Data on under‐5 mortality are produced by the UN Child Mortality Estimates. Most of the cross‐country variation in mortality under 5 is due to cross‐country variation in under‐1 mortality rates.

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