Health Insurance Managed Care Optimal Ins (Theory)

Measuring adverse selection in managed health care


Much of health care today is paid for by managed care plans.  If the managed care plans are profit maximizers–which I assume them to be–then they face a tradeoff.  By offering a lower quality of care, they will make more money; but lowering the quality of care reduces the demand for their insurance product.  In their 2000 Journal of Health Economics article, Frank, Glazer and McGuire create a model which employs “shadow prices” to measure the managed care firm’s incentives to provide care.  The shadow price “character[izes] the incentives a plan has to distort services away from the efficient level.  The shadow price captures how tightly or loosely a profit maximizing plan should ration services in a particular category in its own self interest.”


Let us assume there is a vector of medical services (m_i‘) for each individual ‘i‘, and each medical service is indexed by ‘s‘.  Utility for each person is equal to:

  • u_i(m_i)=v_i(m_i) + μ_i
  • u_i(m_i)=[SUM_s  {v_{is}(m_{is})}] + μ_i

The individual will choose a plan if ‘u_i>u_i‘ where u_i is the valuation the individual places on the next preferred plan.  Thus we have:

  • μ_i> u_i-v_i(m_i)

The managed care plan does not know μ_i but does know the distribution of μ_i.  Given u_i, m_i’, the probability individual i chooses the plan is:

  • n_i(m_i)=1- Φ_i[u_i – v_i(m_i)]

The individual maximizes their utility so that:

  • v’_{is}()=q_s

On the firm side, the managed care organization sets a shadow price (‘q_s‘) for each service in order to maximize the following profit function:   

  • π(q)=SUM_i{n_i(q) * [r_i – SUM_s{m_is(q_s)}]}

The first order condition becomes:

  •  SUM_i{(dn_i/ dq_s) * π_i n_i*m’_is}
  • π_i = r_i – SUM_s{m_is(q_s)}

The authors eventually solve this system of equations for ‘q_s‘ and find:

  • q_s = (Sum_i{n_i * m_is})/(SUM_i {Φ’_i * m_is * π_i})

What does all this math mean?  Frank et al. explain it well as follows:

“The use of a shadow price as a description of rationing in managed care permits a natural interpretation of the division of responsibility between the ‘management’ of a plan, presumably most interested in profits, and the ‘clinicians’ in a plan who face the patients. Cost-conscious management allocates a budget or a physical capacity for a service. Clinicians working in the service area do the best they can for patients given the budget by rationing care so that care goes to the patients that benefit most. In this environment, management is in effect setting a shadow price for a service through its budget allocation. It is evident in data that individuals with the same disease get different quantities of service. The constant shadow price assumption is consistent with managed care rationing but with more care being received by patients who ‘need’ it more.”

Now we can return to the dilemma faced by profit maximizing managed care firms. These firms choose the optimal q but face a tradeoff.  By increasing the shadow price of a certain medical service (‘q_s‘) the firm can make more money (- n_i*m’_is) since their costs have decreased as less services will be provided.  On the other hand, firms face the problem that for given per-person profit level (‘π_i‘), increasing the shadow price will decrease the probability that any individual would like to purchase the managed care plan in the first place (dn_i/ dq_s <0).  This model can explain the appearance of the following phenomenon:

“Under simple capitation payments that now exist, providers and plans face strong disincentives to excel in care for the sickest and most expensive patients.  Plans that develop a strong reputation for excellence in quality of care for the sickest will attract high-cost enrollees.” Miller and Luft (1997 p. 20).

It not uncommon to observe an HMO offering free gym memberships (which are a perfectly predictable cost) in order to attract new healthy members, but to provide poor services to very sick patients.

Frank, Richard; Glazer, Jacob; McGuire, Thomas; (2000)  “Measuring adverse selection in managed health careJournal of Health Economics, Vol 19, pp. 829-854.