# MANAGED HEALTH CARE: Profit Maximization 3

To see what (6) implies for various services, we make some substitutions. The change in the probability of joining can be written as the product of two derivatives:

for every i. Note that the assumption that for every shadow price qs the elasticity of demand for service s is the same for all individual s does not imply, of course, that all individuals have the same demand curve for that service. It only implies that demand curves of different individuals, for a certain services, are “horizontal multiplications” of some “basic” demand function for the service. Individuals will differ in their relative demands. One interpretation of this assumption, as in Glazer and McGuire (1998), is that given someone is sick, a common function describes valuation of a service, but people differ in the probability that they become ill.

Substituting for m’is from (8), we can rewrite (6) as:

From (10) we can make some observations about qs in profit maximization. The numerator of (10) reflects the incentive the plan has to save money on its expected enrollees. The greater is the numerator, the larger will be qs. The denominator describes the expected gains a plan sacrifices by losing enrollees. The denominator contains a product mis7ij, weighted by the change in enrollment probability, Otj . Some enrollees will be profitable, with > 0 given the risk adjustment formula in use, and some will be unprofitable, with 7tj

For any service provided in profit maximization, the denominator of (9) must be positive, implying that in profit maximization, provision of all services on average attracts profitable enrollees. This observation echoes a conclusion from the health care payment literature where under prospective payment systems, the enrollment response, or more generally, demand response, induces a provider to supply a noncontractible input (corresponding here to qs). See Rogerson (1994), Ma (1994), or Ma and McGuire (1997). Creating profits on the margin in this way to induce firm “effort” is inconsistent with zero profitability unless marginal costs are less than average costs or the payer uses a two-part tariff of some kind to reimburse the provider.

In a first-best allocation, a payer or regulator would induce the plan to set q s = 1, leading to an equality between the marginal benefit of spending on a service and its marginal cost. Equation (10) shows how a payer could do this for this one service by manipulating the payment rj. For a given level of payment Г;, if qs were too high, for example, the payer could simply increase rj by some factor, paying more for every potential enrollee. That would raise the denominator of (10) and induce more spending. In the one service case, risk adjustment is not necessary, simply paying more for all enrollees will do.

As Glazer and McGuire (1998) point out, if plan quality were one dimensional, the right quality could be induced by simply paying more per person, without regard to risk adjustment. Changing the risk adjustment formula, for a given overall level of spending, will also affect qs by changing the product of mis7Cj. If, for example, people with high levels of spending on service s are also those with high levels of spending on other services, then by altering the risk adjustment formula so as to pay more for high users and less for the others, the incentives to set qs high may be reduced. same day payday loans