Optimal Ins (Theory)

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=Σ_i p_i uⁱ(x_i, h_i(m_i))

Here, i indexes the state of illness, where the probability that each stat of illness occurs is p_i. Individuals can spend their income on consumer goods, x_i, or medical care, m_i, where medical care is translated into health by the function h_i(m_i). A technological advancement is defined as h_i^a(m_i)≥h_i^b(m_i), for all m_i, and strict inequality for some m_i.

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?

• Goddeeris, John H. (1984) “Medical Insurance, Technological Change and Welfare” Economic Inquiry, 22(1): 56-67.
• Goddeeris, John H. (1984) “Insurance and Incentives for Innovation in Medical Care” Southern Economic Journal, Vol. 51, No. 2, pp. 530-539.
• Arrow (1976) “Welfare Analysis of Changes in Health Coinsurance Rates” NBER.

Typically, economists when economists look at the health insurance market, they focus on the insurance side of it. By this I mean to define insurance as the purchase of a product which will reimburse the buyer in the case of an adverse event. However, one must also look at the concept of protection. Protection is defined as expending a costly effort to reduce the probability of an adverse event. This costly effort, however, will not effect the amount of the loss, only the probability that it occurs.

A seminal paper by Ehrlich and Becker (JPE 1972) finds the optimal levels of self-protection and how optimal self-protection change when insurance markets are introduced. Let us assume that the probability of a loss is p(e) where e is the effort expended and p’<0. An expected utility maximizer optimizes the following function:

• max_e [1-p(e)]*U(I -e) + p(e)*U(I - L - e)

The first order condition is:

• -p’*[U(I -e)-U(I - L - e)]=(1-p)*U’(I -e) + p*U’(I - L - e)

Ehrlich and Becker note that “[t]he term on the left is the marginal gain from the reduction in p; that on the right, the decline in utility due to the decline in both incomes, is the marginal cost.”

When we introduce an insurance market, the expected utility maximizer faces a new objective function.

• max_e,s ,s [1-p(e)]*U(I-e-s*π(e)) + p(e)*U(I - L - e + s)

Here s is the insurance benefit and π(e) is price of the insurance; s*π(e) is the insurance premium. Let U(0)=U(I-L-e+s) and U(1)=U(I-e-s*π(e)). The first order conditions now become:

• -(1-p)U’(1) + p*U’(0)=0
• -p’*[U(1)-U(0)] - (1-p)*U’(1)*[1+s*π'] - p*U’(0)=1

How does self protection change when insurance markets are introduced? According to Ehrlich and Becker “On the one hand, self protection is discouraged because its marginal gain is reduced by the reduction of the difference between the incomes and thus the utilities in different states, on the other hand, it is encouraged if the price of market insurance is negatively related to the amount spent on protection through the effect of these expenditures on the probabilities.”

If insurance companies are actually able to measure self-protection and can price insurance accordingly, then individuals will have some incentive to increase prevention in order to lower their premiums. If insurance is priced in an actuarially fair manner (i.e., π=p(e)/[1-p(e)]) we can show that premiums will drop when self-protection increases:

• ∂π/∂e=p’/(1-p)²<0

However, if insurance companies are not able to observe self-protection efforts, than it is likely that moral hazard will occur–self protection will decrease. In the words of the authors, “Self-protection would then usually be discouraged by market insurance–moral hazard would exist–because the main effect of introducing market insurance would be to narrow the differences between incomes in different states.”

• Ehrlich I, Becker G. 1972. “Market Insurance, Self-Insurance, and Self-Protection.” The Journal of Political Economy, 80(4), pp. 623-648.

Despite much public rhetoric, why is preventative and chronic care so poor in the U.S.? The easy answer is that patients switch plans so frequently that insurance companies who invest in preventative care will incur the cost, but not reap the benefits. The harder question is why patients are switching health plans.

According to a working paper by Cebul, Herschman, Rebitzer, Taylor and Votruba featured on Slate, the answer may be “search frictions.” In the paper, turnover is generated from two sources: 1) from employees leaving the company for new jobs and 2) by having the employer switch to a new health plan. Data from the Community Track Study in 1996/7, 1998/9, 2001 and 2003 show that average annual insurance cancellations are about 21%. More than one third of the turnover is caused by employers switching health plans. Small employers were more likely to switch insurance plans than larger employers. Why don’t they just stay with one plan?

The search friction model is developed from a labor economics paper by Burdett and Mortensen (1998). The authors argue persuasively that extending the model to the case of health insurance makes perfect sense.

“The market for health insurance is a natural place to expect search frictions. Health insurance is a complex, multi-attribute product and this complexity makes it difficult for clients to meaningfully compare more than a handful of proposals. Informal discussions with insurers suggest that they offer customers hundreds if not thousands of different policies. This complexity also makes the marketing of insurance costly so that companies can make only a limited number of appeals to employer groups in a period.”

The authors explain how the price friction mechanism works. The price of the insurance policy is p, the marginal cost of the policy is c, and the firm’s reservation price for buying insurance is p^R.

Suppose all firms made the same price offers p=c and so earned zero profit. Then one maverick firm could clearly increase profits by charging some discretely higher price (less than or equal to the reservation price p^R). This high offer would be rejected more frequently than the going price because any potential client who fielded more than one offer in a period would obviously reject the high offer. But on occasion the contacted client would have no other offers, and a policy would be sold. This would produce positive profit for the firm. Similarly, in a candidate equilibrium in which all firms were charging the same price (a price such that c<p<p^R ), a maverick firm could always increase profit by undercutting slightly the price charged by competitors, thereby increasing the number of clients while reducing by profit per client by only a trivial amount. In short, an equilibrium must entail a distribution of price offers.

Once market friction reach a sufficient level, in equilibrium we will observe a churning of employers going through different insurance policies each year. Introducing the issues that come along with adverse selection is likely to only increase market frictions because insurance companies now will want to screen employees.

Possible Solutions

The authors offer arguments made that may be solutions to the problem.

• Patient-financed health investments. Health care investments (i.e.: preventative care) should be financed by the client. This way, the person reaping the rewards from preventative care will also incur the costs. If the patient switches insurance plans, this will not be a problem since they will be eligible for lower premiums because of their preventative care history. On the other hand, determining what type of care is an expense and what is an investment may be very difficult practically. Further, shifting more risk onto the patient is the antithesis of what insurance is supposed to do. Finally, when switching insurers, it will be very difficult to verify the actual amount of preventative care received without a nation-wide standard for electronic medical records.
• Long term contracts. Another simple solution is just for employers to purchase long term contracts from insurance companies. This way, insurers will be able to reap more of their rewards from earlier years’ health investments. On the other hand, “given constantly-evolving medical technology and treatment protocols, as well as hard to predict changes in governmental regulation and mandates, it is difficult to see how long-term contracts might be implemented.”
• Price Caps. Government set price caps are probably the worse option. As Slate states, “But where should the government set the ceiling? If it’s too low, the government could end up destroying insurance companies’ incentives to stay in business at all.”
• Legislate a basic insurance package. The authors conclude the paper with the following: “It follows from this that much of the distortions resulting from frictions could be mitigated if there were a simple, easily understood and reasonably priced alternative insurance policy that would be available to all market participants. In the context of our search models, we believe we can prove that by making this alternative insurance available on a voluntary basis to all purchasers the inefficiencies resulting from search frictions could be greatly reduced.” Another option would be to offer everyone the choice of a nationalized health plan (a la Medicaid). People who did not want Medicaid, could choose to have vouchers (see Healthcare Vouchers) used to pay towards a private insurance plan of their choice. Many of the basic private insurance plans will likely mimic the nationalized Medicaid, but some plans will offer alternatives which will be more flexible and easier to adapt to new technology advances.

Should hospitals with long waiting times have higher or lower budget transfers? Offering hospitals who have low wait times more money will increase a hospital’s incentives to decrease wait times. On the other hand, thus policy may hurt the busier hospitals and may not alleviate the wait times of those who are waiting the longest. In the case of public school transfers, if the best schools are rewarded, this encourages achievement, but may punish the worst off kids (i.e.: those at poorer schools). Transfers to low-performing school may mute incentives to increase achievement.

The issue of hospital payment structure is analyzed by Luigi Siciliani in his article on optimal contracts B.E. Journal of Economic Analysis & Policy. As with any thesis which claims to give an optimum solution, this optimum is based on some assumptions. This paper uses four major assumptions.

1. Demand for treatment can be controlled by dumping some patients. Doctors can tell patients who wish to have medical treatment that they either a) don’t really need it or b) that they will not provide it
2. The purchaser (i.e.: NHS, an insurance company, Medicare, etc.) can not observe the number of people dumped.
3. Dumping is costly for the specialist. By dumping patients, the specialist receives more complaints about their service level. Thus, either the physician’s reputation is tarnished (a cost) or the physician must spent more time (another cost) convincing the patient that they do not need treatment.
4. Hospitals differ in potential demand for treatment, either due to the catchment area of the hospital or from having a better or worse reputation.

Another key assumption is that no co-payment charges can be issued. This assumption is plausible, because it basically represents the British NHS system. Thus, the optimal solution must be seen not as the ideal optimal, but as the optimal with a centralized payer and no co-payments.

The Model

Hospitals have parameter θ which describes the public hospital type. This parameter θ indicates potential demand in the absence of a rationing system.

For each treatment, patients differ in the value they would receive from treatment. For instance, healthy patients would not benefit from heart surgery, but individuals with coronary artery blockages likely would benefit from surgery. Thus the author assumes that individuals’ value from treatment is uniformly distributed between v₀ and v₁.

Patients have three options:

1. They can be treated at a public sector hospital after a wait of time w, [u^p(v,p)]={∫^T₀ v dt} -p=vT-p]
2. They can be treated in a private sector hospital with no wait, but pay a price of p, [u^NHS(v,w)]={∫^T_w v*g(w) dt} =vg(w)(T-w)]
3. or they can receive no treatment[u^none=0]

There are two costs to going to the public hospital. First, the individual has to wait w weeks longer, so they do not get to enjoy the benefit of the treatment for as extended a period of time. Secondly, since 0<g(w)<1, the treatment becomes less effective or less valued the longer the patient waits.

Thus, from the math above, we can see that a person will choose a public hospital if and only if:

• v<V(w)=p/{T-g(w)*(T-w)

The comparative statics show that longer wait times decrease the probability of using a public hospital, higher prices, p, decrease the probability of using a private hospital, and higher valuations, v, increase the probability of using a private hospital.

Demand for public hospital services is written as:

• D(θ,w,x)=θV(w)-x
• x is the number of patients who are dumped (i.e.: not added to the waiting list)

The number of treatment supplied by hospital θ is y(θ) and since supply must equal demand, we have:

• θV(w)-x=y(θ)

The authors claim that providers receive disutility from dumping patients. Also, hospitals receive more disutility when they dump patients who value the treatment more (i.e.: high v, this is more likely to be the sicker patients). Thus, we are lead to our first major conclusion.

• Conclusion 1: The patients who are dumped are the ones with the lower benefits from treatment. This means that hospitals dump the patients who don’t really need the treatment.

After some more math, the Dr. Siciliani states a second conclusion:

• Conclusion 2: A mix of explicit rationing (through dumping) and implicit rationing (through waiting) is therefore optimal. Siciliani explains that: “Rationing by waiting alone induces excessive disutility for patients. Rationing by dumping alone generates excessive disutility for the specialists.”

The author continues to conclude that a separating or pooling equilibrium may occur.

“Under symmetric information, the optimal contract is for the purchaser simply to over a transfer in exchange for the provision of the desired level of activity and waiting time, without leaving any rent to the provider…Under asymmetric information, we found that a separating equilibrium exists when it is optimal for the purchaser of health services to contract more activity and higher waiting times to hospitals with higher demand. In this case providers with low potential demand have an incentive to mimic hospitals with high potential demand. To induce hospitals to self-select, the purchaser needs to pay a rent to hospitals with lower potential demand. [But] if it is not optimal for the purchaser to contract more activity and higher waiting time to hospitals with higher demand, then a separating equilibrium may not exist.”

Problems

One main problem with the paper is that it assumes that patients with a high value, v, cost the same to treat as low value patients. If v is a proxy for sickness, this is likely not to be the case; sicker patients with a high v are more expensive to treat. If this were the case, then conclusion 1 would not hold. Public hospitals would instead treat patients with the lowest benefit and dump patients with intermediate benefits–the high benefit patients would still go to the private sector hospital.

Also, the paper does not take into account any strategic interaction between hospitals. “If hospitals with higher potential demand are contracted higher waiting times, then patients will switch from the hospitals with high potential demand to hospitals with low potential demand, increasing excessively the amount of dumping and consequent disutility for hospitals with low potential demand.”

• Luigi Siciliani (2007) “Optimal Contracts for Health Services in the Presence of Waiting Times and Asymmetric Information,” The B.E. Journal of Economic Analysis & Policy: Vol. 7 : Iss. 1 (Contributions), Article 40.

What is the optimal way to pay physicians? If there were a singular variable ‘health’ that could be easily measured, patients could pay physicians for each unit of health they receive. Of course, this is not how the physician-patient relationship operates in the real world. Physicians are paid either a base rate per person per month or receive a fixed fee for each service provided. In physician contracts with health plans, the physician effort level to gather information (diagnosis) is non-contractible and pay based on the patient’s health condition (physicians’ private information) is also non-contractible. In a setting with so much uncertainty, what is the optimal physician contract?

This is the question which Izabela Jelovac attempts to answer in her 2004 paper in Health Economics. In her model, patients who visit the doctor can have two types of illnesses, one mild and one severe. The doctor chooses an effort level ε, in order to diagnose the illness. The more effort the doctor puts forth, the more likely they will diagnose the disease correctly. In the next stage of the game, the doctor chooses whether to treat the patient with an expensive treatment which will cure the severe disease and or the inexpensive treatment which will cure the serious disease. Prescribing the expensive treatment when the patient has the mild disease causes a health loss. If the patient recovers, then the game ends, but if the patient does not recover then the patient returns to the physician and receives the alternative treatment.

So what does the physician do? Typically, we would guess that the doctor will prescribe the adequate treatment strategy of providing the expensive treatment for the severe disease and the inexpensive treatment for the mild disease. However, “…if a single visit is less profitable to the physician than two visits, the physician is better off providing the most inadequate treatment than the most adequate one in order to increase the likelihood of a second visit.” This is like a car mechanic who damages people’s cars when they come in for an oil change in order to earn more money during the second visit from fixing the problem which they caused in the first.

The author also found that a strategy of always prescribing the inexpensive (or expensive) treatment during the first visit can be optimal as well. For instance, if diagnosing the serious illness is very costly (numerous expensive tests and many physician labor hours are incurred) and if the serious illness isn’t too severe, then giving all patients the inexpensive treatment during the first visit may be optimal.

The authors go through some dense math, but end up concluding that some supply-side cost sharing by the physician is optimal since it induces the physician to provide the most adequate treatment. With the cost sharing, having the patient return for a second visit is never profit maximizing.

This paper does have a few problems. A key assumption in this model is that “the need for a second treatment is necessarily the physician’s fault.” In reality, this need not be the case. Also, the authors do not take into account issues of the physician’s reputation. If 100% of a physician’s patients had 2 visits, patients may decide to leave this physician and sign up with a doctor who choses the “adequate treatment” strategy.

Hillman (Ann Intern Med 1990) writes that “whereas most physicians will act in the patient’s best interest when the medical decision is clear-cut, the effect of financial incentives may be most important in cases where the correct decision is not obvious.”

• Jelovac, Izabela (2001) “Physicians’ payment contracts, treatment decisions and diagnosis accuracy” Health Economics vol 10, pp. 9-25.

Health savings accounts (HSAs) have been a major point of contention for health care reformers. Supporters claim that HSAs can reduce health care costs by decreasing the moral hazard problem inherent when third parties—such as insurance companies or the government—pay for medical services. Opponents claims that HSAs will attract rich and healthy individuals, leaving only poor or sick individuals in the ‘regular’ insurance pool.

One interesting point made in Cardon and Showalter (JHE 2007) is the following:

“Both opponents and advocates of HSAs tend to argue that HSAs will lead to less reliance on insurance, either through higher coinsurance rates and deductibles, or through fewer purchases of policies. This line of reasoning ignores the fact that accumulated HSA balances are wealth, and health insurance protects this wealth. Even individuals with large HSA balances would typically value insurance to protect those balances for future use. HSAs will tend to reduce levels of insurance coverage, but the effect seems unlikely to be as large as some previous researchers suggest.”

The Cardon and Showalter article also gives a nice description of the five main types of tax-preferred health savings accounts.

1. “Archer medical savings accounts (MSAs): accounts in which an individual and/or an employer can contribute pre-tax dollars to pay for most health care services. The tax advantage is the same as for employer-provided health insurance premiums. Unused monies can accumulate over time. An experiment authorized under the Kassebaum-Kennedy bill (Health Insurance Portability and Accountability Act of 1996) allowed for restricted introduction of MSAs which included the requirement of purchasing a catastrophic, (high-deductible) health insurance policy (MSA/CHP).
2. Flexible spending accounts (FSAs): like HSAs, but with no link to insurance coverage. Funds not used by the end of the year revert to the employer.
3. Rollover FSAs: these would allow limited rollover of FSA monies without the restrictions on insurance choices that the current HSA rules require.
4. Health reimbursement arrangements (HRAs): tax-exempt individual accounts used to pay for medical expenditures. Accounts are funded by employers; employee contributions are not allowed. Ownership of the accounts remains with the employer, unlike HSAs and FSAs.
5. Medical IRAs. This proposal would allow consumers to make penalty-free early withdrawals from their retirement plans to pay for allowable medical expenditures.”

• Cardon and Showalter (2007) “Insurance choice and tax-preferred health savings accounts” J Health Econ, v. 26(2), pp. 373-399.

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.�?

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. 

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.