Structural Population Changes

Changes in a populations health-income distribution may occur over time due to immigration, advances in technology, better education or rise and fall of national economies. In this section we study how structural changes in the population’s health and income affect the health insurance market. To this end, we analyze two different changes in the underlying distribution of health and income: a systematic improvement of health and income and an increased correlation between health and income.

Systematic Improvement of Health and Income

First, we investigate the effect of a systematic improvement of the population’s income and health on the contribution rate and PR’s profit. Technically, we consider a stochastic improvement of f(h, e) to a distribution with density function ˜f(h, e) in the sense of (multivariate) first-order

13These are customers who are underinsured, whose health type causes health costs in equal or larger extent to the maximum benefit level written down in the contractc(h)≥L.

stochastic dominance.¹⁴ Intuitively, as the population’s income and health improve, the popu-lation should spend a lower percentage of its income on health insurance given that the benefit level stays constant. Indeed, if the entire population was insured with the budget-balancing PU, the contribution rate could be adjusted downward. To account for the precise organizational structure of the insurance market a more thorough analysis is needed.

We start by analyzing how customer sets change as the distribution changes. It is instructive to divide the population into four subgroups and study the effect of a systematic improvement on each of these subgroups separately.

Profitability and unprofitability are defined relative to the original distribution and the corre-sponding contribution rateα^∗. LetP U⁺(α^∗) be the set of profitable PU customers andP U⁻(α^∗) the set of unprofitable PU customers. Analogously, let P R⁺(α^∗) be the set of profitable PR customers and P R⁻(α^∗) the set of unprofitable PR customers.

Firstly, consider the effect on the subgroup of unprofitable PU customers, P U⁻(α^∗). As health and income improve, PU’s profits on this subgroup unambiguously increase: customers who remain in the group even after the improvement are less unprofitable than before; additionally, some unprofitable customers leave P U⁻(α^∗) to join one of the other subgroups.

Second, consider how PR’s profit is affected on the set of its unprofitable customers, P R⁻(α^∗).

Customers remaining in the group even after the improvement are less unprofitable than before, and some unprofitable customers join P R⁺(α^∗). This effect increases PR’s profit. On the other hand, there is an inflow of new unprofitable customers fromP U⁻(α^∗). These customers are un-profitable before and after the change of distribution but had income lower than the contribution cap before the change and income exceeding the contribution cap after the change. This effect decreases PR’s profit. Which of the two effects dominates depends on the precise change in health and income.

Third, we analyze the effect on PR’s profit generated from P R⁺(α^∗): Customers remaining in the group are more profitable than before. Additionally, there is an inflow of new profitable customers from all other subgroups. Thus, PR’s profit from this subgroup increases.

Finally, consider P U⁺(α^∗). Again, customers remaining in this group are more profitable than before. Also, there is an inflow of new profitable customers from P U⁻(α^∗). These two effects suggest that PU’s profit should increase. There is a countervailing effect though. Profitable PU customers whose income exceeds the opt-out threshold after the change are attracted by PR, decreasing PU’s profit onP U⁺(α^∗). Therefore, the overall change in profit depends on the exact changes in health and income.

In general, we cannot determine the sign of the change in profits onP R⁻(α^∗) andP U⁺(α^∗). We can however derive an upper bound on a potential loss. In fact, a negative change in profits on

14See the Appendix for a definition and technical details.

P R⁻(α^∗) never outweighs the increase in profits onP U⁻(α^∗). To see this, observe that the loss in profit onP R⁻(α^∗) is caused by unprofitable customers switching fromP U⁻(α^∗) toP R⁻(α^∗).

Thus, the loss on P R⁻(α^∗) corresponds to a one-to-one gain in profit on P U⁻(α^∗). All other effects increase profit. Put differently, profit on P R⁻(α^∗)∪P U⁻(α^∗) increases. An analogous argument applies to the change in profit on P U⁺(α^∗).

Our analysis reveals that the systematic improvement of health and income may affect the overall profit of PU and PR positively or negatively, depending on the precise inflow and outflow in P U⁺(α^∗) and P R⁻(α^∗). As noted above, the overall effect is however positive. Thus, it cannot be that both PU’sand PR’s profits decrease. The change in PU’s profit determines whether PU adjusts the contribution rate upward or downward in response to the systematic improvement.

As PR’s profit is increasing in the contribution rate, this effect may reinforce or counteract the initial change in PR’s profit. The following proposition summarizes our findings.

Proposition 3.3.5. Consider a systematic improvement of the population’s health and income.

Then exactly one of the following three scenarios arises:

(i) If the loss in PU’s profit on P U⁺(α^∗) outweighs the gain in profit on P U⁻(α^∗), the con-tribution rate increases and PR’s profit increases.

(ii) If the loss in PR’s profit on P R⁻(α^∗) outweighs the gain in profit on P R⁺(α^∗), the con-tribution rate decreases and PR’s profit decreases.

(iii) Else the contribution rate decreases and the private insurance may profit or lose.

Proposition 3.3.5 sorts the wide range of possible systematic improvements of health and income into three categories according to their effect on the contribution rate and PR’s profit. Given that the class of improvements we consider unambiguously increase health and income for the entire population, these categories are surprisingly manifold. In particular, there exist cases in which customers have to pay a higher percentage of their income for health insurance. Intuitively, this is because an improvement might allow PR to absorb profitable PU customers, urging PU to increase the contribution rate in order to run a balanced budget.

This observation has important implications. The current organization of the health insurance market might mitigate policy programs and campaigns targeted to improve the population’s health in order to decrease the contribution rate.

Increase in Correlation Between Health and Income

Motivated by empirical studies (Deaton and Paxson, 1998) which document an increase in cor-relation between health and income, we investigate how changes in corcor-relation affect the health insurance market. For a meaningful comparison of correlations, we consider distributions ranked

by correlation according to the supermodular order which have identical marginal distributions of health and income.¹⁵

Start with a distributionf and consider a distributiongthat is larger thanf in the supermodular order. For the case when the opt-out threshold exceeds the contribution cap, we obtain the following clear-cut result.

Proposition 3.3.6. If the opt-out threshold exceeds the contribution cap, an increase in correla-tion between health and income increases the contribucorrela-tion rate.

Note that in Germany, in fact since 2003 the opt-out threshold exceeds the contribution cap and there is hence ’no cream skimming’. To gain intuition for our result, it is instructive to decompose the transition fromf toginto several sub steps. Consider the two-dimensional space of health and income types. Start with the health income distribution f. Now fit a rectangle into the health income space and consider a transformation that shifts probability mass from the bottom right corner of the rectangle to the bottom left corner and the same probability mass from the upper left corner to the upper right corner.¹⁶ This transformation increases correlation between health and income while keeping the marginal distributions of health and income fixed.

Intuitively, we can construct g fromf by applying several of these transformations to f.

e h

K¹ K²

Figure 3.2: Mass shift not affecting the contribution rate

If the correlation-increasing mass transformation is such that all four corners of the rectangle lie within the set of PR customers (see figure ??), PU is unaffected and the contribution rate remains the same. Similarly, in case the four corners of the rectangle lie within the set of PU customers, PU does not need to adjust the contribution rate because the marginal distributions of health and income conditional on being a PU customer are unchanged.

Lastly, consider the case when the left corners of the rectangle are in the set of PU customers whereas the right corners of the rectangle lie within the set of PR customers (see figure??). As a consequence of this transformation, the income distribution of PU customers is not altered since income is on the x-axis and marginals are held constant by the transformation. But the health distribution of PU customers worsens. Therefore, PU has to increase the contribution rate to run a balanced budget. Taking all three cases together, we see that PU increases the contribution rate

15See the Appendix for a formal definition.

16This probability mass shift corresponds to an ’elementary transformation’ as described and analyzed in Meyer and Strulovici (2015) to characterize the supermodular stochastic order.

e

h

K

K

(e

,h

) (e

,h

) (e

,h

) (e

,h

)

Figure 3.3: Mass shift affecting the contribution rate

if correlation between health and income increases. From a broader perspective, PU customers have comparatively low income whereas PR customer have comparatively high income. If the correlation between health and income increases, PU’s low-income customers have also a worse health type, forcing PU to increase the contribution rate.

If the opt-out threshold lies below the contribution cap, there exists an income range where PR cream skims. Graphically, PU and PR customers are not separated any longer by a single cut-off in the income dimension. Thus, we have to consider additional correlation-increasing transformations. Consider the transformation where only the upper right corner of the rectangle is in the set of PR customers; all other corners lie in the set of PU customers. For this transformation there are two conflicting effects. On the one hand, the income distribution of PU customers worsens. On the other hand, unprofitable customers leave PU. It depends on the distribution f which of the two effect dominates, and, consequently, whether PU adjusts the contribution rate upward or downward. All other transformations entail a decrease in the contribution rate.

Overall, in this case it depends on the specific increase in correlation and on how much weight is put on which transformation whether PU adjusts the contribution rate upward or downward.