Harsh occupations, health status and social security

Abstract

We study the optimal design of a social security system when individuals differ in health status and occupation. Health status is private information but is imperfectly correlated with occupation: individuals in harsh occupations are more likely to be in poor health. We explore the desirability of letting the social security policy differ by occupation and compare the results with those obtained when disability tests are used instead. We show that tagging by occupation is preferable to testing when the audit technology is relatively expensive and/or the proportion of disabled workers differs markedly across occupations. Expected utility differences between occupations could induce workers to switch occupations if they were able to. We explore the implications of imposing equality of expected utility across occupations.

Appendix: Joint testing and tagging second-best problem

In this appendix we present the formal derivation of the second-best problem when the planner simultaneously employs differentiated policies by occupation and disability tests. The planner chooses then a combination of consumption and retirement age $\left( c_{ij},z_{ij}\right)$, one for each type ij, and now an audit probability $\pi _{j}$, one for each occupation j. We assume however that the penalty for individuals caught lying is the same in both occupations, which we have represented by a minimum utility $\underline{u}$. There is a self-selection contraint for each occupation that ensures that healthy workers do not have incentives to mimic disabled ones. These self-selection constraints take into account that those claiming to be disabled are audited with a probability $\pi _{j}$ that depends on the occupation and are fined with a minimum utility $\underline{u}$ if caught lying. The Lagrangian for this problem is:

$$\begin{aligned} \pounds= & {} \underset{j=1,2}{\sum n_{j}}\left[ \left( 1-p_{j}\right) \left( u(c_{Hj})-v(z_{Hj};h_{H})\right) +p_{j}\left( u(c_{Dj})-v(z_{Dj};h_{D})\right) \right] \\&+\mu \underset{j=1,2}{\sum n_{j}}\left[ w\left( \left( 1-p_{j}\right) z_{Hj}+p_{j}z_{Di}\right) -\left( \left( 1-p_{j}\right) c_{Hj}+p_{j}c_{Dj}\right) \right] \\&-k\left( n_{1}p_{1}\pi _{1}\right) -k\left( n_{2}p_{2}\pi _{2}\right) \\&+\lambda _{1}\left[ u(c_{H1})-v(z_{H1};h_{H})-\left[ \left( 1-\pi _{1}\right) \left[ u(c_{D1})-v(z_{D1};h_{H})\right] +\pi _{1}\underline{u} \right] \right] \\&+\lambda _{2}\left[ u(c_{H2})-v(z_{H2};h_{H})-\left[ \left( 1-\pi _{2}\right) \left[ u(c_{D2})-v(z_{D2};h_{H})\right] +\pi _{2}\underline{u} \right] \right] \end{aligned}$$

Rearraging the FOCs we obtain

$$\begin{aligned} u^{\prime }\left( c_{Hj}\right) =\frac{\mu n_{j}\left( 1-p_{j}\right) }{ \lambda _{j}+n_{j}\left( 1-p_{j}\right) }, \text { and }, \frac{v^{\prime }(z_{Hj};h_{H})}{u^{\prime }\left( c_{Hj}\right) }=w \end{aligned}$$

for the healthy workers in occupation j, and

$$\begin{aligned}&u^{\prime }\left( c_{Dj}\right) =\frac{\mu n_{j}p_{j}}{\lambda _{j}\left( 1-\pi _{j}\right) +n_{j}p_{j}} \text { and } \\&\frac{v^{\prime }(z_{Dj};h_{D})}{ u^{\prime }\left( c_{Dj}\right) }=w-\frac{\lambda _{j}\left( 1-\pi _{j}\right) }{n_{j}p_{j}}\left( w-\frac{v^{\prime }(z_{Dj};h_{H})}{u^{\prime }\left( c_{Dj}\right) }\right) <w \end{aligned}$$

for the disabled workers in occupation j.