### When is moral hazard adverse selection?

7/10/2014 10:00:00 AM
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[Update 7/15/2014: this post has been edited to fix an error. See explanation below.]

Much of the debate surrounding Obamacare has been about how to minimize moral hazard and adverse selection, while decreasing the inequalities in the system that, for example, lead to higher costs for women than for men, and higher costs for those with higher health risk than those with lower risks. Moral hazard is the effect by which providing insurance increases total health expenditure. While this is often viewed as a type of inefficiency, I explained previously why this is actually often a good thing (which it turns out is closely related to John Nyman's research). Adverse selection is the effect by which individuals with lower health risks get priced out of the insurance market because their low risk reduces their willingness to pay for insurance.

Einav and Finkelstein (2011) laid out the textbook methods for detecting and quantifying these effects, and note that the "positive correlation test" which tests whether those who opt to buy insurance have higher average costs than those who do not, cannot discern between moral hazard and adverse selection, as both lead to higher expenditures for those who have insurance than those who don't. They then summarize how one might distinguish between the two: you need a dataset with exogenous variation in premiums, so you can see whether lowering premiums--inducing more people to buy insurance--causes average costs to decline. If it does, the theory goes, you have adverse selection in the market. This, in turn, comes with a couple possible policy recommendations: either risk-rate the insurance premiums, if possible, or enforce a mandate requiring that everyone buy insurance plans that provide some mandatory (relatively high) minimum amount of coverage, or even prohibiting heterogeneity of plan types. Kifmann (2002) and others (myself included) have also argued that a tax-and-subsidy scheme could achieve our policy objectives more effectively than command-and-control approaches to plan coverage requirements.

In my own simulations of the health insurance market, I keep coming across a problem in the above analysis. Even using the revised methodology from Einav and Finkelstein--using identifying variations in premiums to differentiate between moral hazard and adverse selection--doesn't always work. Despite having drastically different policy implications, moral hazard and adverse selection aren't always distinguishable from each other in reduced-form estimators. For illustration let's return to the healthcare model discussed in my moral hazard post, originally taken from Blomqvist (1997) and calibrated to the RAND health insurance experiment data: $$U_i=E_i\left[\frac{1}{-c_i}+\frac{0.001098}{-\left(h_i-\theta \right)}\right]$$ where $h_i$ is spending on healthcare, $c_i$ is spending on all other goods and services, and $\theta$ is a two-state random that takes the value $\theta=4.86$ when the individual is healthy, and $\theta_H=61.07$ when he is sick. Sickness occurs with probability $p_i$, which varies across individuals. There is insurance available with no deductibles and a 15 percent coinsurance rate, but insurers can perfectly observe risk types, so that the insurance plans are perfectly risk-rated so that for each individual $i$, $$m_i=0.85E_i\left[h_i\right]$$ where $m_i$ is the insurance premium. As in Blomqvist, we'll assume individuals have income of 150 (measured in hundreds of dollars), so that the budget constraints are given by $$c_{i,\theta}+m_i+0.15h_{i,\theta}\leq 150$$ if they buy insurance, or $$c_{i,\theta}+h_{i,\theta}\leq 150$$ if they do not buy insurance.

Now, it is clearly efficient for everyone to buy insurance. Insurance itself has no loading costs and are risk rated, while all individuals are strictly risk-averse and have health risk--that is, we can costlessly make everyone of these individuals better off by reducing their risk. Yet, when you do the math, it turns out that all individuals with $p_i<0.1$ (including the benchmark case in my previous post with p=0.086!) will opt out of this insurance despite the fact that plans are perfectly risk rated. These individuals are selecting out of the insurance market even though the number of higher-risk individuals in the market has absolutely no impact on their own premiums!

The reduced-form methodology described in Einav and Finklestein would indicate the presence of adverse selection in our fully-risk rated insurance market, because exogenously lowering premiums would induce lower risk types to buy insurance, thereby reducing the average costs of those who have insurance. Here's a graph of the relationship between risk type $p_i$ and the minimum coinsurance rate at which individuals are willing to buy insurance: But what's actually happening in our market is not adverse selection as we typically think of it, but rather a problem of moral hazard: the 15 percent coinsurance rate would induce individuals to buy a lot more healthcare than they would if they had to pay for a larger share of it, which in turn drives up the premium insurers have to charge. In this way, the excessively generous benefits of insurance actually drives up spending so much as to drive the individual out of the market.

So when does moral hazard cause adverse selection? This is an important question because, especially post-Obamacare, many individuals face essentially near-zero coinsurance rates for large medical bills above the out-of-pocket maximums, which could be pricing some of them out of the market if their preferences resemble the utility function above. Intuitively, in the utility function above, we are guaranteed to have some minimum coinsurance threshold because the additional utility from more health spending is always strictly positive, meaning that the choice of how much to consume explodes as coinsurance drops to zero, which drives up premiums and therefore drives down total utility, eventually so much that it is less than without insurance at all. This is probably not a reasonable assumption, as I'd be willing to bet that people wouldn't consume an infinite amount of healthcare if it was totally free--while moral hazard certainly exists to some extent, people don't generally want more healthcare than they feel that they need. Obviously, when the coinsurance rate is 100 percent (the insurer pays zero percent) expected utility without insurance and expected utility with insurance are equal. If expected utility with insurance remains higher than without insurance for all coinsurance rates less than that (when insurers pay more than zero percent), then we can say that moral hazard never causes selection effects; otherwise, there exists some coinsurance rate low enough so that some individuals will select out of the market.

It turns out that mathematicians already have a name for exactly this condition: it's called "single-crossing," which you may recall from the Topkis Theorem, or the optimal tax literature, or as that thing your microeconomics professor mentioned while you weren't paying attention. Our case is a rather trivial application: if we define $z$ as the percentage of health costs that the insurer pays, $F\left(z\right)$ as the expected utility with insurance, and $G\left(z\right)$ as expected utility without insurance (which is a constant function over $z$), then single-crossing as applied here says simply that there exists $x \in \left[0,1\right]$ such that for all $z\in \left[0,1\right]$, $z\geq x$ implies $F\left(z\right) \geq G\left(z\right)$ and for all $z$, $z\leq x$ implies $F\left(z\right) \leq G\left(z\right)$. We all ready have established that the two functions intersect at $z=0$, which implies that $x=0$. Graphically, we can easily see that this is not true of the utility functions specified above:So when might the single crossing condition be satisfied? In our case, the answer is basically only if utility from health spending is bounded, and if that bound is sufficiently low as to avoid too much moral hazard.

The point of all this is to say that moral hazard and adverse selection are not as distinct from each other as we might think, and that it can be empirically difficult to disentangle. In fact, under the right conditions--low cost-sharing with non-single crossing utility functions--the two concepts actually merge into one, a condition which will not show up in reduced-form econometric modeling. And because the policy implications differ, this points to a need to supplement our reduced form estimates with fully estimated structural models before making policy recommendations.

Feel free to check my math or play with the model here.

[Update explanation: the original code I used for this post miscalculated the insurance premiums. It had $m_i=E_i\left[h^*_i\right]$ instead of $m_i=zE_i\left[h^*_i\right]$. The result of this error was an overestimation of the optimal insurance rates and, consequently significantly underestimating the thresholds at which individuals would drop out of the market. For the benchmark case with p=0.086, the actual threshold below which he will drop out of the market is 17 percent out-of-pocket costs, which is a lot higher than the 2.5 percent mentioned in the original version of this post.]