An essay on health insurance reform

1. Suggested Implementation Details of
Laurence Kotliko¤’s Purple Health Plan
Henry Buist, Ph.D.
April 7, 2017
1 Introduction
Problems with Obamacare have included health insurance companies exiting
the regional insurance exchanges because of claims exceeding premiums. Con-
versely, attempts to set premiums high enough to cover claims have discouraged
participation in the insurance exchanges. By 2017, this basic price dynamic had
resulted in a media meme that Obamacare had become doomed to implode.
The March 2017 attempt by Republicans to pass an insurance reform bill failed
in the House of Representatives.
To overcome the politics of premium increases to extend health insurance
coverage to more people, MIT Professor Jonathan Gruber suggested that a lack
of transparency was needed to get Obamacare to pass. In view of the need
for further reform, a new approach tacks in the opposite direction. To achieve
full transparency, Section 2 speci…es a detailed mathematical model of health
insurance. The market fundamentals include the insurance plan parameters, fu-
ture medical billings as the source of uncertainty, the insured’s total cost of
healthcare as a function of the plan parameters and medical billings, and the
insurer’s net pro…t as a function of plan parameters and medical billings. Dis-
tributional assumptions about uncertain medical billings enable statistical risk
and cost assessments across the diverse population of U.S. households. Section
3 uses the model to contextualize several of the market failure problems in the
health insurance market and to discuss policy goals. The root problems of Oba-
macare and reform measures will continue to be the disincentive for healthy
people to subsidize less healthy people and the inability of many households to
a¤ord the insurance premiums and out-of-pocket expenses. Boston University
Professor Laurence Kotliko¤’s Purple Health Plan and his advocacy of expand-
ing Medicare Part C to all Americans o¤er a viable insurance reform framework.
Section 4 uses the mathematical model to provide suggested implementation de-
tails of Professor Kotliko¤’s reform measures. The proposal’s credibility stems
from its incisive and comprehensive mitigation of the market failure problems
in the health insurance market.
1

2. 2 A Basic Model of Health Insurance
Consider the class of health insurance plans that specify deductibles, coinsurance
rates, and out-of-pocket limits but eschew …xed-dollar copayments per visit.
Also limit the class of plans to HMOs or EPOs which do not cover out-of-
network medical expenses. Collectively, these restrictions signi…cantly simplify
the plan mathematics and enable more clear reform measures. The table below
lists the key variables and parameters related to HMO/EPO insurance plans.
P The annual health insurance premium paid by the insured.
r The insured’s total annual medical billings before insurance.
D Plan deductible; the insured pay all annual expenses (billings) up to D.
L Plan out-of-pocket limit; the insured pay at most annual expenses up to L.
The coinsurance rate (0%, 10%, 20%, 30% most typically).
h The insured’s total annual cost of health care including premiums.
The insurer’s annual net pro…t for a given household.
The known plan parameters are fP; D; L; g whereas h and are functions of
the random variable r representing the only source of uncertainty. These annual
billings r re‡ect the negotiated in-network discounted prices o¤ered by medical
providers. The out-of-pocket limit L includes the deductible D. For simplicity,
the insurer’s annual net pro…t will exclude operating expenses. Many plans
allow pharmacy costs to be treated like any other medical expense and thus
they count toward satisfying the deductible and out-of-pocket limit.
So-called Bronze insurance plans tend to have lower premiums but higher
deductibles and out-of-pocket limits. Gold plans tend to have higher premiums
but lower deductibles and out-of-pocket limits. Silver plans fall in the middle
ranges. Picking a plan often manifests itself as picking a metal. The table
below represents real-world health insurance plan information derived from the
Obamacare website healthcare.gov. Using the pricing tool at this web site in
2016, the author inputted his own family-of-four’s demographics to generate
plan o¤erings. Aetna and United Healthcare o¤ered six plans consistent with
the program class assumptions above. The most credible examples were:
HMO Plan P D L
Bronze $17,076 10% $10,400 $13,000
Silver $20,172 0% $7,200 $7,200
Gold $21,876 10% $4,800 $6,850
Based on the universal mathematics of plan rules given the various class
restrictions above, the insured’s total annual cost of health care h including
premiums P follows the formula:
2

3. h(r) = P +
8
>>>><
>>>>:
r if r D
D + (r D) if D < r < (L D)= + D
L if (L D)= + D r
(1)
and the insurer’s annual net pro…t for a given household follows the formula:
(r) = P
8
>>>><
>>>>:
0 if r D
(1 ) (r D) if D < r < (L D)= + D
(r L) if (L D)= + D r
: (2)
Uncertainty within this framework is resolved by the realization of annual
billings r. In 2016, there was an estimated 324 million people in the U.S. form-
ing an estimated 126 million households. Regardless of insurance, 126 million
examples of r are thus generated every year. Who pays for r varies greatly, de-
pending on being covered by one of an employer’s group plans, by Medicare, by
the Veterans Administration, or by Medicaid. The size of r itself is controlled
in part by who pays and having insurance naturally increases r. Coughlin and
colleagues [1] document who bears the expense of uncompensated care. The an-
swer includes the above federal agencies, state agencies, and the private sector
providers.
The key information needed to quantify the health insurance market is the
probability density of medical billings r. We know that r is a non-negative num-
ber. The parameters of the Weibull density can be ‡exibly chosen to represent
variations in r caused by household variations in demographics and pre-existing
medical conditions. For some non-negative random variable x, the Weibull den-
sity function Y is
Y (x; ; ; ) =
x
1
exp
x
!
given parameters , , and : The mean value follows
E(x) = (1 +
1
) + v
where is the gamma function. For example, if = = 2 and v = 0, the chart
of the density is:
3

4. 0 1 2 3 4 5 6 7 8 9 10
0.0
0.1
0.2
0.3
0.4
x
y
Weibull density: = = 2, v = 0
and has a mean value of 1:7725. To map x to an intuitive level of medical
expenses r use the formula r = 10; 000 x. For consistency, all factors will
now be denominated as ten thousand dollar units. For example, E(x) = 1:7725
simple means that the household has an expected level of medical billings over
the year of $17,725. Similarly, if D = 0:61 the deductible is $6100.
Choosing > 0 shifts the chart to the right by that amount after setting
y = 0 for x < . However, variations in parameter are su¢ cient to capture
medical expense distribution di¤erences across households. For example, the
case = 2; = 0:25; and v = 0 can capture the expense distribution of very
low risk households having E(x) = 0:2216 (r = $2216) with a low maximum
(about $8000):
4

5. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
1
2
3
4
x
y
Weibull density: = 0:25, = 2, = 0
Now consider a …ve-member household with two members having chronic ill-
nesses. Setting = 4 with = 2 and v = 0 can generate a plausible distribution
of annual billings:
0 1 2 3 4 5 6 7 8 9 10 11 12
0.00
0.05
0.10
0.15
0.20
0.25
x
y
Weibull density: = 4, = 2, v = 0
where the mean translates to r = $35,449 and there is probability mass greater
than zero above $100,000 in billings.
5

6. For practical purposes we can assume the expense risk parameter is bounded
above by 10 where r = $88,623. In e¤ect, for all 126 million U.S. house-
holds, their fundamental medical risk is characterized by the Weibull parameter
2 (0; 10] given the …xed assumptions = 2 and = 0. The parameter itself
could have an assumed or calibrated distribution such as Y ( ; 2; 2; 0) where the
average U.S. household thus would have = 1: 772 5 so that r = $15,708.
Under Bertrand price competition, insurance pro…ts would be driven to zero
for each . That is, (r) = 0. Using the Weibull density function for = 2
and = 0 and setting Equation 2 to zero allows expressing the zero-pro…t
equilibrium insurance premium for any as
P( ) =
Z (L D)= +D
D
(1 ) (r D)
2
2
r exp
r 2
dr + (3)
Z 1
(L D)= +D
(r L)
2
2
r exp
r 2
dr
For example, if = 0:2, D = 0:5 and L = 1 and if = 1:7725 as before then
P( ) = 0:87227 or a $8722.70 annual premium.
Using Equation 1, Equation 3 and the Weibull density, the household’s
expected total cost of health care under insurance and including the premiums
can be expressed as
E(h(r)) = P( ) +
Z D
0
r
2
2
r exp
r 2
dr + (4)
Z (L D)= +D
D
(D + (r D))
2
2
r exp
z 2
dr +
Z 1
(L D)= +D
L
2
2
r exp
r 2
dr:
We can calculate that, when P( ) is priced so that (r) = 0, E(h(r)) =
r. In words, under competitive health insurance markets, the average annual
medical billings r that the household generates given their risk pro…le equals the
household’s expected total cost of health care under insurance, E(h(r)), which
is the sum of the ex ante insurance premium and the average out-of-pocket
expense computed using the three integrals in Equation 4. If the condition was
not true, the insurance company would realize a gain or a loss on average.
3 Problems and Policy Considerations in the
Health Insurance Market
Healthcare insurance reform should start as an exercise in data science. Instead
of assuming that expenses follow a certain density with assumed parameters, a
6

7. centralized data repository of actual annual household medical expenses could be
built aggregating across existing insurance and medical records. The actual data
can then be used to …t empirical expense densities for each type of household
distinguishable by their size, member ages and pre-existing conditions. Before
trying to manage the a¤ordability challenge, the …rst goal is to be as accurate as
possible about the riskiness of future annual medical billings across households.
In other words, start with just the facts.
The assumption (r) = 0 represents the policy goal of making the health
insurance market as close to perfect competition as possible. This policy lowers
the annual costs to households at the expense of insurance company investors.
Under perfect competition with (r) = 0, P( ) under di¤erent fD; L; g will
simply adjust to give the same expected value of E(h(r)) = r. A risk-neutral
household would be indi¤erent to higher values of fD; L; g that a¤ord a lower
premium compared to lower values of fD; L; g that require a higher premium.
Plan o¤erings with di¤erent fP; D; L; g thus appear to serve the purpose of
allowing households to sort themselves by risk preference and ability-to-pay the
worst-case cost scenario P + L when medical billings r exceed (L D)= + D.
(The best-case health cost scenario is paying only P with no medical events or
billings.) However, variations in the plan parameters fP; D; L; g can confuse
plan participants, possibly by intent cynically speaking (tricking people into
making plan selection errors and overpaying). Suppose a plan selector speculates
that r might be unusually large in the forthcoming plan year, say $40,000. In
that case, the total plan expense h under the Silver plan in Section 2 is lower
than under Gold by $1354, but the metal color connotation might induce the
plan selector to choose Gold. Not having ready access to the formula for h(r)
could cause a heuristic mistake as another example of Kahneman’s thinking fast
[2]. Such a condition would transfer wealth from less informed plan selectors to
insurance companies and indirectly to more informed plan selectors.
Suppose the household has private information that two of their members
need expensive procedures such as surgery in the upcoming plan year prior to
plan selection. In this case the insurance deforms into an inevitable subsidy to
the insured. Healthcare policy reform must counter this type of intertemporal
gaming characterized by maxing out on medical needs when being insured for
the …rst time, by conditionally maxing out on medical expenses after reaching
out-of-pocket limits, and by strategically timing enrollments before and after
predictable huge expenses.
More generally than the extreme two-surgeries example, households will have
private information about their health conditions that could lead them to be-
lieve that their market assignment to a given level of was too high or too low.
Suppose the market is too high in the opinion of the household making pre-
miums too high. The household would tend to choose the plan with the lowest
premium and to minimize their healthcare use and bear more of their own cost
risk. Now suppose the market is too low in the opinion of the household mak-
ing premiums too generous. The household would tend to choose the plan with
the highest premium and to maximize their healthcare use. These incentives
represent the classic adverse selection problem in information economics.
7

8. As perhaps the biggest challenge to heath insurance markets, many house-
holds simply cannot a¤ord to pay for typical values of r in the absence of insur-
ance or its equivalent value E(h(r)) with insurance. Undesirably from a social
welfare point-of-view, many households thus forego insurance and curtail r to
be as low as possible. Medicare, Medicaid and Obamacare were born in part
out of this fundamental economic dilemma. There is no escaping the fact that
someone must pay for whatever level of r is realized ranging from zero to some
catastrophic example such as $1 million for an extended stay in intensive care.
Any of us could be the $1 million catastrophic case.1
Under insurance, the household has to be able to a¤ord health expenses up
to the maximum expense P( ) + L given the minimum expense of P( ). For
the = 2 households, Figure 1 on the last page shows P( ) and P( ) + L for
values of L from $0 to $35,000. To complete the plan options, the value of D
follows one of two types of plan rules: D = 0:5 L with the value of set to
the common value of 20% and D = L where is irrelevant. Equating D to
L lowers premiums in exchange for the insured absorbing more expenses up to
L without the bene…t of coinsurance. Figure 1 shows how choosing increased
values for L and D lowers premiums. By de…nition the lowest value of P( )+L
is the y-intercept at $17,775 where L = D = 0, and $17,775 is the same value
where E(h(r)) = r. As a less extreme case than buying no insurance, a similarly
pernicious e¤ect is that budget-constrained households might choose high values
of L and D to make the premium more a¤ordable and forcing them to take their
chances that medical billings will be low. To reiterate, E(h(r)) is the same for
all 15 plans on the chart. As a policy goal, a plan should not be o¤ered to a
household which cannot a¤ord P( ) + L under that plan. Paying the premium
is of no help if the household can’t absorb out-of-pocket expenses up to L.
4 Suggested Purple Plan Implementation
The information problems, adverse strategic behavior and lack of a¤ordabil-
ity might lead inevitably to socialized medicine in the U.S.. Before the U.S.
accedes to purely government-controlled healthcare that many other countries
have adopted, Professor Kotliko¤’s Purple Health Plan [3] and his advocacy
of Medicare Part C for all Americans [5] o¤er an insurance reform plan that
preserves market forces subject to various policy interventions. The machinery
of Section 2 is used to enumerate suggested details for implementing the Purple
Plan. How the proposal mitigates the market failure problems of Section 3
motivates the plan con…guration.
(1) The U.S. Department of Health and Human Services reorganizes and
enhances existing resources to run the new Purple Health Plan (PHP).
(2) After transition exemptions, Medicare, Medicaid and Obamacare all con-
vert to PHP. Employers are no longer allowed to subsidize employee health in-
surance premiums. As stated in Kotliko¤ [4], “The government (federal and
1 Note that the Weibull distribution does not capture such extreme tail events as a million
dollar hospital stay.
8

9. state) ends the tax exclusion of employer-provided health insurance premiums.”
Thus, households covered under employer plans join households covered under
existing government programs to participate in the PHP. Disassociating one’s
health care from one’s employer would sponsor a more e¢ cient labor market and
tax system. Employers can and should convert their prior insurance premium
subsidies to higher wages to o¤set this policy intervention.
(3) PHP forms a consortium with health insurance companies to share
anonymous medical expense data. A data science consulting team hired by
the PHP uses the data to estimate and publish the empirical densities of r
across di¤erent types of households.2
The data warehouses are updated quar-
terly. Health insurance companies have access to the data warehouse in order
to form their own views about uncertain future r distributions.
(4) Health insurance companies can o¤er a plan to any American household
regardless of location. State regulations are reformed to allow insurance compa-
nies to freely “cross state lines.”The data warehouse of step 3 would facilitate
national-level competition.
(5) Based on the data warehouse and the published empirical densities, the
PHP discloses to the household the equivalent of their risk group. For each
distinguishable , the PHP publishes benchmark insurance plans illustrated in
Figure 1 under these plan parameters:
L = $0, $1000; $2000; : : : ; 2 r where r increases with ;
Two types of deductibles: D = 0:5 L with = 0:20 and D = L; and
P( ) priced at zero pro…t according to the empirical density.
(6) Insurance companies o¤er PHP plans fP; D; L; g limited to the class of
HMO plans following Equation 1 to avoid consumer confusion and the apples-to-
oranges comparison problem. This step represents the Medicare Part C for all
idea. As discussed by Kotliko¤ [3] and [5], restriction to HMOs better manages
healthcare costs by incentivizing insurers to provide quality coverage to retain
enrollees while also not over-providing coverage that drives up costs. Insurer
plans are not limited to the benchmark parameters of step 5. Actual o¤ered
premiums will be of course higher than zero pro…t P( ) to induce insurer par-
ticipation, but the market features Bertrand price competition as desired.
(7) The PHP endows every household with an HSA account to purchase
insurance and to pay out-of-pocket expenses. PHP funds the account by an
amount equal to P( )+L for the standardized case D = L = r: In other words,
the household now can a¤ord both the premium and the out-of-pocket expenses
for a zero-pro…t plan constructed around their risk-adjusted expected expenses.
To incentivize the household to shop for less expensive medical services, the
household can carry forward the balance to the next plan year without impacting
the next endowment. Subject to rules to avert gaming, accumulations in the
HSA will allow years where r < r to fund higher quality plan coverage during
years of more acute medical needs.
2 The empirical densities refer to the same idea as experience-ratings in Kotliko¤ [3].
9

10. (8) Equations similar to Equations 3 and 4 can be used to project PHP
funding requirements given the distribution information about and about r
for each . A mix of payroll and income taxes in addition to the possible
introduction of a national sales tax would have to be calibrated to the projected
funding needs.
The risk pro…le re‡ects human behavior and industry supply costs. As a
familiar idea to reduce waste, tort reform should be pursued to mitigate overly
cautious malpractice risk mitigation (defensive medicine). Reducing for all
households year after year can be the subject of future creative solutions.
References
[1] Coughlin, Teresa A, John Holahan, Kyle Caswell and Megan McGrath, "Un-
compensated Care for the Uninsured in 2013: A Detailed Examination," The
Urban Institute and the Kaiser Family Foundation, May 2014.
[2] Kahneman, Daniel. 2011. Thinking, Fast and Slow. New York: Farrar,
Straus, and Giroux.
[3] Kotliko¤, Laurence J. 2007. “The Healthcare Fix: Universal Insurance for
All Americans,”Cambridge, MA: MIT Press.
[4] Kotliko¤, Laurence J.: “The Purple Health Plan,” located at
http://www.thepurplehealthplan.org/node/2:
[5] Kotliko¤, Laurence J.: “Could Medicare Part C be the answer to our health
insurance mess?”Dallas News (online), April 2017.
10

11. 11
$-
$5,000
$10,000
$15,000
$20,000
$25,000
$30,000
$35,000
$40,000
$- $5,000 $10,000 $15,000 $20,000 $25,000 $30,000 $35,000
Out-of-pocket Limit L
Figure 1: Insurance costs for alpha risk parameter=2 under various (D, L)
choices w/ 20% coinsurance
Premium with D = 0.5 L
Premium with D= L
Premium + L with D = 0.5 L
Premium + L with D= L