The Public Health Roulette

Abstract

Changes in life expectancies perturb the balance of justice between the young and the old, prompting reallocation of income and health. The long-term consequences are difficult to predict. I report a case where greater life expectancies create greater health disparity between the generations.

Appendix

The Nash Solution with a Fixed Total Resource to Share

Junior and Senior bargain for the biggest share of a resource. The Nash solution to the bargaining problem is the solution to the following mathematical program:

$$MAX\left( {\frac{{R_{y} }}{{T_{y} }} - c_{y} } \right)\left( {\frac{{R_{o} }}{{T_{o} }} - c_{o} } \right)$$

(1)

with respect to R _y and R _o , subject to a resource constraint $R = R_{y} + R_{o}$, where

R _y  :

resource for Junior (the young generation) if bargaining succeeds

R _o  :

resource for Senior (the old generation) if bargaining succeeds

T _y :

Junior’s life expectancy

T _o  :

 Senior’s life expectancy

c _y  :

Junior’s resource if bargaining fails

c _o  :

Senior’s resource if bargaining fails

In the Nash Solution, Junior’s and Senior’s annual incomes are, respectively:

$$\begin{aligned} \frac{{R_{y} }}{{T_{y} }} = \frac{{R + c_{y} T_{y} - c_{o} T_{o} }}{{2T_{y} }} \hfill \\ \frac{{R_{o} }}{{T_{o} }} = \frac{{R - c_{y} T_{y} + c_{o} T_{o} }}{{2T_{o} }} \hfill \\ \end{aligned}$$

(2)

Case 1

How an increase in Junior’s life expectancy affects income distribution.

$$\begin{aligned} \frac{{\partial \left( {\frac{{R_{y} }}{{T_{y} }}} \right)}}{{\partial T_{y} }} = - \frac{{\left( {\frac{{R_{y} }}{{T_{y} }} - \frac{{c_{y} }}{2}} \right)}}{{T_{y} }} < 0 \hfill \\ \frac{{\partial \left( {\frac{{R_{o} }}{{T_{o} }}} \right)}}{{\partial T_{y} }} = - \frac{{c_{y} }}{{2T_{o} }} < 0 \hfill \\ \end{aligned}$$

(3)

Case 2

How an increase in Senior’s life expectancy affects income distribution.

$$\begin{aligned} \frac{{\partial \left( {\frac{{R_{y} }}{{T_{y} }}} \right)}}{{\partial T_{o} }} = - \frac{{c_{o} }}{{2T_{y} }} < 0 \hfill \\ \frac{{\partial \left( {\frac{{R_{o} }}{{T_{o} }}} \right)}}{{\partial T_{o} }} = - \frac{{\left( {\frac{{R_{o} }}{{T_{o} }} - \frac{{c_{o} }}{2}} \right)}}{{T_{o} }} < 0 \hfill \\ \end{aligned}$$

(4)