Optimal Ins (Theory)

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Increase copays and increase medical spending?

February 16, 2007 in Academic Articles, Health Insurance, Optimal Ins (Theory), Pharmaceuticals | Permalink

Most economists believe that increasing the price of an item will decrease demand for the item. Health care is no different from any other good. If you increase the copayment or coinsurance rate, people will consume fewer medical services. The famous RAND Health Insurance Experiment (HIE) demonstrated that higher coinsurance rates discourage medical care consumption. As I said, health care is no different from any other good…or is it?

Dana Goldman and Tomas Philipson argue in their 2007 NBER working paper (“Integrated insurance design in the presence of multiple medical technologies“) that the problem of moral hazard in the health insurance market is different from moral hazard under most other insurance markets. For most other types of insurance, only one good is insured (e.g.: a car, a house, etc.). Health insurance, however, covers a wide variety of different services. Thus the authors claim that increasing prescription drug copay costs can actually increase health care spending and make patients worse off. Let us assume that prescription drugs and medical services are substitutes. If the price of prescription drugs increases, it is likely that the individual will consume the more of the expensive medical services which are fully covered by insurance. They suggest that a zero or negative copay may be optimal for some prescription drugs.

The optimal copay is determined by the patients elasticity of demand and the degree to which other medical services are complements or substitutes to the original item in question. The authors give some empirical evidence from other studies to support their claim:

  • Soumerai, Ross-Degnan, Avorn, McLaughlin and Choodnovsky (NEJM 1991) compare Medicaid patients in New Hampshire “who had a three-drug limit per patient” and Medicaid patients in New Jersey without the limit. The authors found a 35% reduction in drug use, but a doubling in nursing home admission rates.
  • Soumerai, McLaughlin, Ross-Degnan, Casteris, and Bollini (NEJM 1994) look at individuals on psychotropic medications and find that a drug cap led to a 15%-49% reduction in the use of drugs but a 43%-57% increase in mental health visits and emergency mental health services.
  • Horn, Sharkey, Tracey, Horn, James and Goodwin (Am. J Man Care 1996) find that formulary limitations in 6 HMOs were associated with increased ER visits and hospitalizations for otitis media, atraumatic arthritis, ulcers, hypertension, and asthma.
  • Gaynor, Li, and Vogt (NBER 2005) find that higher drug co-payments in a given year lead to increased spending during the following year.
  • On the other hand, studies such as Johnson, Goodman, Hornbrook and Eldredge (Med Care 1997) and Tamblyn, Reid, Mayo, McLeod, and Churchill-Smith (J Clin Epidemio. 2000) found that increased co-pays did not increase outpatient visits, hospitalizations or ER visits.

The authors conclude that “the preponderance of evidence suggests strong negative cross-price elasticities between drugs and other medical spending, at least for patients with chronic disease.”

The doctor as a double agent

October 11, 2006 in Academic Articles, Health Insurance, Optimal Ins (Theory) | Permalink

In modern medicine, doctors are agents for two distinct groups. The physician is an agent for the patient, but also an agent for insurance companies-especially in the managed care settings.  In balancing both relationships, the doctor must juggle the conflicting principal-agent problems of information asymmetry and third party payment. 

Ake Blomqvist develops an interesting theoretical model to explain this phenomenon in his 1991 Journal of Health Economics paper. An individual’s expected utility is based on the level of their consumption (c) and their health status (h), which is a function of a health state variable (‘θ‘) and medical expenditures (‘z‘).  Health insurance premiums are given by ‘m‘.

  • E=[Σ_i {π_i*h(z_i-θ_i)}] + u(y-m)]

If we assume perfect information and that insurers must break even (m – Σ_i [π_i*z_i] = 0), the we have the following first order conditions:

  • z_0=0
  • h′(z_i-θ_i)+)=u′(y-m)

These conditions state that a health person (state i=0) will not spend any money on medical expenses, and that the marginal utility of consumption should be the same in each state.  This involves a contingent contract for each health state (θ_i). 

Blomqvist then imposes asymmetric information and institutes a copayment rate of ‘σ’.  Consumption now changes from ‘y-m‘ to ‘y-m-σz‘ and the break-even constraint is ‘m=(1-σ)’Σ_i {π_i*z_i(m,σ)‘.  Since contracts are now incomplete and patients can choose the level of services they desire, variable ‘z’ is now a function of the copayment rate (‘σ‘) and the premium (‘m‘).  The author derives the conclusion that the optimal ‘σ’ is always located between zero and one.

Managed Care model

In managed care, the insurer has an incentive to minimize services, but this desire is counterbalanced by the threat of a competitor offering more generous services and thus attracting their customers.  The new maximization problem and resulting first order condition are:

  • E=π_0*H+[Σ_{1 to N} {π_i*h(z - θ_i)}] + u(y-m)]
    • s.t.: m-(1-π_0)*z
  • FOC: ∑_{1 to N} {π_i*h’} – (1-π_0)*u’ = 0

The managed care firm selects a level of ‘z’ for all states (except complete health where z_0=0; h=H).  The benefit of this new equilibrium is that the problem of moral hazard has been solved since care is now rationed.  On the other hand, there is still the problem of information asymmetry and care is not state-contingent as in the first best scheme. 

Stochastic performance guarantee

Blomqvist now improves the efficiency of his model even more. Since managed care has an incentive to under-provide care, the author proposes that the government fine HMOs if the resulting health of the individual is below the expected level agreed upon in the contract.  If HMOs provide fewer medical services, they will save money on direct expenses, but also increase the risk that they will be fined.  Let us look at this proposition more formally.

Blomqvist defines a term measuring the distance between predicted health–given health state (θ_i) and medical expenditures (z_i)–and the ex post observed health outcome (ξ).

  • ε=h(z_i - θ_i) – ξ

Let us assume ε~f(ε).  An HMO is deemed to have broken its contract to provide a given level of care when the following occurs:

  • h(z_i-Θ) – ξ > γ,

The variable Θ represents the health state reported by the physician and this variable need not equal the value of true health state, θ.  When the above equation holds true, the HMO pays a fine of ‘F’. 

Thus the total cost of providing medical services is:

  • C=z_i + P*F

The variable ‘P’ represents the probability that an HMO is ‘falsely convicted’ [P(ε>γ)].  Proposition 3 of the paper stats that for any value of the conviction criteria ’γ’ there exist an F which will induce HMO physicians to tell the truth if f(ε) is decreasing in the non-conviction range.  Blomqvist demonstrates that this can be a first best solution.

Problems

The author notes that many issues may confound this theory in reality.  Load factors and costs to oversee the system of fines could make the stochastic performance guarantee sub-optimal.  Also, second opinions may be a more efficient means to find out the accurate value of a patient’s health state from the physician. 

Blomqvist, Ake (1991) “The doctor as a double agent: Information asymmetry, health insruance, and medical care,” Journal of Health Economics, vol 10, pp. 411-432. 

Measuring adverse selection in managed health care

September 29, 2006 in Health Insurance, Managed Care, Optimal Ins (Theory) | Permalink

Introduction 

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.”

Model

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 care” Journal of Health Economics, Vol 19, pp. 829-854.

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