Health Insurance Optimal Ins (Theory) Supply of Medical Services

Medical Insurance, Technological Change, and Welfare

Jason Shafrin

Can technological change make people worse off? Most economists think technical improvements are always good. Producing more of the output with fewer input is considered a more efficient use of resources. But is this the case in the medical field? John Goddeeris shows that this may not always be the case in his 1984 paper.

The Model

Let us assume that individuals maximize utility of the for developed by Arrow (1976):

  • V=Σi pi ui(xi, hi(mi))

Here, i indexes the state of illness, where the probability that each stat of illness occurs is pi. Individuals can spend their income on consumer goods, xi, or medical care, mi, where medical care is translated into health by the function hi(mi). A technological advancement is defined as hia(mi)≥hib(mi), for all mi, and strict inequality for some mi.

We can now introduce insurance into the model. Individuals who buy insurance pay a premium equal to π and a coinsurance rate z. The price of the medical premium must be equal to the expected value that the insurance company expects to pay out in medical benefits (less the copayments).

One would think that V*a>V*b, but this may not always be the case. For instance, let us assume that a person can either be healthy or sick (i.e., i=2). Further, assume the following utility functions:

  • V=(1-p)u1(x1) + p u2(x2,h2)
  • u1(x1) = -exp[-x1]
  • u2(x2,h2) = -exp[-(x2+h2)]

If individuals are endowed with income x0, then:

  • x1 = x0 – π,
  • x2 = x0 – π – zm2,
  • π = p(1-z)m2.

Assume p=.1, x0=10 and the original technology is:

  • h2(m2) = -10 if m2 < 5
  • h2(m2) = -4 if m2 > 5

This means that if medical spending is above 5, health will be partially restored. Goddeeris finds that the optimal coinsurance rate to maximize utility is no coinsurance (i.e., z=0). With no coinsurance, sick individuals choose m2=5. The utility level under the original technology (i.e., V*b) equals -.000476. What happens when there is a positive technological changes as follows:

  • h2(m2) = -10 if m2 < 5
  • h2(m2) = -4 if 5 ≤ m2 <15
  • h2(m2) = -3 if m2 > 15

Again, the author finds that no coinsurance (i.e., z=0) is optimal. With no coinsurance, individuals of course choose m2 = 15. However , tutility level under the new technology (i.e., V*a) equals -.000592. How can this technological improvement have decreased utility?

In this example, the true cost of the innovation is so large relative to its benefits are so large, people only choose to use it since coinsurance is 0. A higher coinsurance rate would have induced individuals to choose m2 = 5. According to Goddeeris, “the larger added expenditures in the ill state leads to an even greater reduction in expected utility. A ero co-insurance rate remains optimal after the innovation. Thus V*a < V*b, and the innovation –which clearly expands productive capabiities and is in fact adopted–is welfare reducing by our standard.”

The reason this occurs, is that individuals act ex post as if their expenditure decisions have no impact on insurance premiums. While no individual person’s actions will affect insurance rates, since all sick individuals act similarly, health insurance premiums increase much more after the technological innovation than before it.

Despite the finding that technology is welfare reducing in this particular case, technological improvement are of course welfare improving in other cases. One question that remains is how to operationally decide when a technology is welfare enhancing and when is it welfare reducing. In which category do MRI machines fall? What about CT scanners?