### Moral hazard, for lack of a better word, is good.

6/12/2014 06:02:00 PM
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There's two semi-recent blog posts that have been distracting me lately. The first is Noah Smith's blog post about Steve Levitt's efforts to convince UK Prime Minister David Cameron that healthcare is just like the car industry, and that therefore Cameron should abolish the UK's system of free socialized medicine (Cameron, understandably, stormed out of the meeting). Then there's this post from Steve Randy Waldman arguing the rather iconoclastic point that market-clearing isn't always the optimal condition of a market. The point on market clearing was interesting, I guess, but I think his post could be simplified to this: ex-ante willingness-to-pay is not a meaningful metric of welfare.

Which brings me back to Steve Levitt's spiel to David Cameron. Levitt argued that buying healthcare was just like buying a car--if the government pays a large share of the cost of everyone's cars, then everyone is going to buy too many cars that are too expensive. This is a fairly common argument about "moral hazard:" insurance makes markets inefficient, because it will cause people to "spend too much" on goods and services in those markets. I will take it as uncontroversial that subsiding a product--whether cars or healthcare--with insurance increases the amount people buy. However, implicit in this argument that they'll buy "too much" is that willingness-to-pay is a meaningful measure of welfare--according to Levitt, if an individual wouldn't be willing to pay the full price for a healthcare treatment, then it is not worth providing that treatment to him at all. This is wrong, wrong, wrong.

I don't mean that Levitt is wrong about healthcare, though he is. I mean he's just wrong about how economics works. Let's ignore healthcare for a moment and examine why "think like a freak" isn't thinking like an economist. Here's an example of the basic model underlying how Levitt thinks about the world:

There are two goods $x$ and $y$, over which people have convex preferences given by a utility function of the form $U=\alpha ln\left(x\right)+\beta ln \left(y\right).$ An individual with wealth $w$ must decide how much of each good to buy at the market prices $p$, giving rise to the budget constraint $x+py\leq w$ (note:I've standardized everything here in terms of the nominal price of $x$, so $w$ has the interpretation of the number of units of $x$ the individual's wealth is worth, and $p$ is the ratio of the nominal price of $y$ to the nominal price of $x$. I can do this because, in general, aggregate price level doesn't affect real economic variables.) This is all to say that the individual's problem is to maximize utility subject to the budget constraint: $$\max_{x,y}U=\alpha ln\left(x\right)+\beta ln \left(y\right)~s.t.~w-x-py\geq 0.$$ When you do the calculus and solve for $y$, it turns out that $$y^*= \frac{\beta}{\alpha+\beta}\frac{w}{p}.$$ If you're an economics student this should look familiar to you, it's the oldest result in the book. Now $p$ can be interpreted as the maximum amount the individual is willing to pay per unit for $y^*$ units, which is the "willingness-to-pay" (WTP): $$WTP\leq\frac{\beta}{\alpha+\beta}\frac{w}{y}.$$ Ok, now let's illustrate the point with some numerical examples. Suppose that $\alpha=\beta=0.5$, that the individual has wealth $w=100$ and that the actual price of $y$ is $p=30$ units. The individual is willing to pay at most 50 units for one unit of $y$, but only 25 per unit for 2 units of $y$. So, the individual would decide that it is not worth buying two units of $y$ at the price $p=30$.

But does this mean that it would be socially wasteful to give this individual more units of $y$? NO! Not necessarily. Consider the same case as above, but endow the individual with an extra unit of $y$ at the outset, so that his budget set is now $x+py\leq w+1p$. Solving as before yields the solution that $$WTP\leq \frac{\frac{\beta}{\alpha+\beta}w}{y-\frac{\beta}{\alpha+\beta}}$$ which in our numerical illustration simplifies to $WTP\leq \frac{100}{2y-1}$. First, note that we do have moral hazard here: without subsidizing this individual with a unit of $y$, he would have bought less than 2 units at the market price, while with the subsidy he still wants to buy at least one more, leaving him with at least units, more than he would have bought without the subsidy. But, note that his willingness-to-pay for 2 units of $y$ is now 33.33--more than the price per unit of 30! That is, the value to this individual of 2 units of $y$ exceed the social costs of producing them, even though this individual would not have been willing to purchase them without the subsidy.

The example above is quite trivial, and some version of this always appears in any kind of policy proposal. What you just witnessed is an example of an "endowment effect"--giving a person a good increases his willingness-to-pay for that good, because receiving that good represents an increase in his net worth. You may recognize the term from behavioral economics where it is more famous, but make no mistake: the endowment effect is not only real, and often huge, but also totally rational. Because of the endowment effect, it's actually never correct to assume that willingness-to-pay has welfare significance. No, in order to make judgements about the welfare effects of a policy of subsidization, we must also examine the sources of those funds. Had I stipulated that the individual above must pay a premium equal to the expected cost of his subsidy, he would have been unambiguously worse off because he would have preferred to spend those premiums on $x$ instead of $y$. But that's not what I assumed--the consumer above is unambiguously better off than without the subsidy, because the subsidy represents an increase in wealth(ie, an endowment). To determine whether this policy increases aggregate welfare, we need to compare the increase in utility to this individual to the decrease in utility of the individual that pays for it. Often, such as when the payer is far wealthier than the beneficiary, subsidies actually net out a gain in aggregate welfare, because the endowment is worth more to the latter than the former.

Of course, that a subsidy is welfare improving doesn't make it optimal--even in cases where a policy is welfare improving solely because it is correlated with a large disparity in wealth, it is usually better to opt for income redistribution instead, on the basis that income is even more highly correlated with the disparity in marginal utilities. But I will show you a case where this is not true, and in the process, show why, exactly, Levitt and others are largely incorrect about the implications of moral hazard in healthcare. Lest you think I've concocted a funky model just to disprove Levitt, I will play by the rules and use a data-driven model published in the top health economics journal (and one of my all time favorite papers): Blomqvist (1997) in the Journal of Health Economics. Here's the model calibrated to the data from the very famous RAND health insurance experiment: $$U_i=\frac{1}{-c_i}+\frac{0.001098}{-\left(h_i-\theta_i \right)}$$ where $h_i$ is spending on healthcare, $c_i$ is spending on all other goods and services, and $\theta_i$ represents individual $i$'s state of health. We'll consider two types of individuals (there are more types in the paper): 91.4 percent of the population are $L$-types with $\theta_L=4.86$, while the rest are $H$-types with $\theta_H=61.07$. Let's give them a universal insurance policy that pays 70 percent of all health spending $h$, so that the budget constraint is given by $c\leq y-m-0.3h$ where $y$ is income and $m$ is the insurance premium. We'll go further and say that everyone's income is 150 (Measured in hundreds of dollars. This comes from the RAND data. All quantities are measured in hundreds of dollars.) so that the healthiness parameter $\theta_i$ represents the only source of heterogeneity in our example.

I ran the computations and it turns out that the equilibrium with insurance is for $L$-types to consume $h_L^*=12.95$ hundred dollars worth of healthcare, while $H$-types consume $h_H=68.16$, and the premium works out to $m=12.35$, which fully finances all of the insurance payouts. Total social welfare, defined as the summed utilities across all individuals $W=0.914U_L+(1-0.914)U_H$ turns out to be $W=-0.007703$.

Now let's consider this healthcare market under Steve Levitt's proposal that it operate like a car market--everyone pays out of pocket for their own care, and there is no insurance. So now we have the exact same scenario as above, except $m=0$ and the budget constraint is $c\leq 150-h$. We now have $h_L^*=9.52$, $h_H=63.92$ yielding social welfare of $W=-0.007750$ which is actually less than before.

Take a moment to reflect on the magnitude of what I just showed there. Insurance increased aggregate welfare even though 1) there was moral hazard, 2) there was no uncertainty or risk involved, and 3) everyone was equally wealthy to begin with! In healthcare, there's a huge disparity in marginal utilities caused not by wealth inequality, but health inequality--some people are quite healthy, while others are extremely sick. What this says is that not only can we increase aggregate well-being by redistributing through health insurance, it is actually optimal to do so, because ordinary income redistribution won't deal with this kind of heterogeneity. Health inequality is a kind of inequality that you can't find in Piketty's new book, and it's not a kind of inequality you can see in IRS tax data. But it's real, and it's a huge freaking deal.

(Side note: I typed this out on my kindle while hiking in the Smokey mountains, so I appologise for any errors.)