Academic Articles Health Insurance Optimal Ins (Theory)

The doctor as a double agent

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.


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.