Alternatives for Modeling Results from the RAND Health Insurance Experiment

Alternatives for Modeling Results from the RAND Health Insurance Experiment

Order Code RL33296
CRS Report for Congress
Received through the CRS Web
Alternatives for Modeling Results from the RAND
Health Insurance Experiment
March 6, 2006
Chris L. Peterson
Specialist in Social Legislation
Domestic Social Policy Division
Congressional Research Service ˜ The Library of Congress

Alternatives for Modeling Results from the RAND
Health Insurance Experiment
Summary
The RAND Health Insurance Experiment (HIE) was ongoing from the mid-
1970s to the early 1980s. Two thousand nonelderly families from six urban and rural
areas were randomly assigned health insurance plans with different levels of cost-
sharing (that is, with various levels of deductibles, coinsurance, and out-of-pocket
maximums). The results from this unprecedented health insurance experiment
showed that people facing higher cost-sharing (that is, they had to pay a higher
proportion of total health care costs out of their own pockets) had lower health care
spending than those in plans with lower cost-sharing. No similar experiment has
been performed since the HIE, so it remains the epochal analysis for understanding
the link between health insurance cost-sharing and total health care spending. This
report examines the methods used to apply the HIE results in health policy analyses.
The key variable used to try to explain health care spending in the HIE was the
plans’ coinsurance — that is, the percentage of total health care costs that the
individual must pay. Understanding these results from the HIE was complicated by
the fact that for each coinsurance rate, there were multiple plans, each with a different
out-of-pocket maximum (although the maximum never exceeded $1,000). For
example, a person may have been enrolled in the “25% coinsurance plan,” but after
that person had spent $1,000 (or less) out of pocket, the plan effectively became a 0%
coinsurance plan. Thus, the nominal coinsurance could not be used as the sole cost-
sharing variable for explaining the impact of cost-sharing in the HIE plans.
The HIE results have been particularly useful for policy analysts estimating what
effect changes in cost-sharing might have on health care spending — in public health
insurance programs, for example. Microsimulation modeling is one tool used by
health policy analysts to estimate the impact of cost-sharing changes. “Micro” refers
to the fact that the modeling takes place on an individual level rather than an
aggregate level, based on a database of individuals representative of a certain
population (the U.S. population or a smaller subset, such as individuals enrolled in
Medicaid). If one wanted to estimate the impact of an increase in coinsurance, for
example, a microsimulation model would apply that increase to every person in the
data along with a concomitant drop in total health care spending.
In most health insurance modeling, the HIE results remain the basis for
adjusting total health care spending in response to cost-sharing changes. However,
applying those results in a model is not always straightforward. One of two methods
is typically used — elasticities, generally preferred by health economists, and
induction, preferred by actuaries. Each has benefits and shortfalls, but little
comparative analysis has been done. This report begins by generally describing and
comparing elasticities and induction factors. The report then summarizes key
findings from the HIE and discusses how elasticities and induction factors can be
used to replicate those results. Because of the limitations of these methods in
modeling, this report offers a third alternative that appears to better replicate the HIE
results. This method, called the cubic formula, is simply a formula that produces
HIE-reported spending levels from the experiment’s four coinsurance levels.

Contents
Theoretical Explanation of Cost-Sharing Methods . . . . . . . . . . . . . . . . . . . . . . . . 2
Elasticity of Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Health Insurance Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Induction Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Health insurance applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Additional Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
RAND Health Insurance Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Selected Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Calculating and Applying Cost-Sharing Methods Based on HIE Results . . . . . . 17
Arc Elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Predicting Quantity with Point-elasticity and Arc-elasticity
Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Effect of Arc Elasticity’s Lack of Path Neutrality . . . . . . . . . . . . . . . . 20
Induction Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Predicting Quantity with Induction Formulas . . . . . . . . . . . . . . . . . . . 22
Cubic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Estimating Free-Plan Spending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Predicting Spending From Estimated Free-Plan Spending . . . . . . . . . . . . . 30
Pure Coinsurance Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Typical Plan Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Predicting Spending by Type of Care . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Constant Induction Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Inpatient and Outpatient Care . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Prescription Drugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
List of Figures
Figure 1. Predicted Expenses Using Various Cost-Sharing Methods, from
Example Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Figure 2. Predicted Expenses Using Various Cost-Sharing Methods and
Values, from Example Case Results at 15% Coinsurance . . . . . . . . . . . . . . 11
Figure 3. Effect of Nominal Coinsurance on Annual Per-Person Medical
Expenses, in Dollars, from RAND Health Insurance Experiment . . . . . . . . 13
Figure 4. Effect of Nominal Coinsurance on Annual Per-Person Medical
Expenses, as a Percentage of Free Plan Expenses . . . . . . . . . . . . . . . . . . . . 14
Figure 5. Effect of Average Coinsurance on Annual Per-Person Medical
Expenses, as a Percentage of Free Plan Expenses . . . . . . . . . . . . . . . . . . . . 15
Figure 6. Analysis of Arc Elasticity’s Lack of Path Neutrality in HIE
Results: Predicting Quantity Varying By Beginning HIE Data Point . . . . . 19
Figure 7. Illustration of Arc Elasticity’s Lack of Path Neutrality:
Predicting Quantity Varying Whether Beginning Data Point Was an
Original HIE Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Figure 8. Predicting Quantity Using Induction Factors and Original Data
Points from HIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 9. Predicting Quantity of Inpatient Care, By Induction Factor and
Beginning HIE Data Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Figure 10. Comparison of Results of Constant Induction Factor from
Free-Plan Spending with Induction Factors and Original Data Points
from HIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Figure 11. Predicting Quantity Using Cubic Formula, Compared to Ideal
Arc Elasticities and Induction Factors, From Original HIE Data Points . . . 27
Figure 12. Effect of Average Coinsurance on Spending, by Type of Service . . . 35
Figure 13. Effect of Coinsurance on Annual Per-Person Total Medical
and Prescription Drug Expenses, as a Percentage of Free Plan Expenses . . 42
List of Tables
Table 1. Estimated Pure Price Effects of Coinsurance on Medical Expenses,
as a Percentage of Free Plan Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Table 2. Arc Elasticities Between Average Coinsurance Amounts, by
Type of Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Table 3. Predicted Free-Plan Spending Using Arc Elasticities, by Elasticity
Formula for Predicted Spending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Table 4. Induction Factors Between Average Coinsurance Amounts, by
Type of Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Table 5. Predicted Spending of Example Person in Plan With $1,000
Deductible and 25% Coinsurance, Based on Predicted Free-Plan
Spending, By Cost-Sharing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Table 6. Example Person’s Predicted Spending at 95% Coinsurance, by
Type of Service and Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Table 7. Average Predicted Spending, by Plan and Factor, Based
on 2002 MEPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Alternatives for Modeling Results from the
RAND Health Insurance Experiment
Changes in health insurance plans’ cost-sharing (for example, the deductible and
coinsurance) affect the quantity of health services used, according to results from the
seminal RAND Health Insurance Experiment (HIE) of the 1970s and 1980s.
Generally speaking, if a person’s cost-sharing increases, less health care will be used;
if cost-sharing decreases, more health care will be used. Economists account for such
changes with a measure called a demand elasticity; actuaries have a related measure,
called induction factors.
These two methods are used in health insurance models, usually with the
purpose of replicating the HIE results. Neither of these methods is perfect or even
perhaps inherently preferable. Moreover, converting the HIE results into appropriate
elasticities or induction factors is not always straightforward. The use of each has
benefits and shortfalls, but little comparative analysis has been done.
The Congressional Research Service (CRS) has partnered for more than a
decade with actuaries from the Hay Group to formulate microsimulation models that
provide estimates of the actuarial value of health insurance plans. To account for
changes in cost-sharing, these models use induction factors. CRS and Hay are in the
process of a significant overhaul of these models. As part of that process, the
application of the models’ induction factors was assessed and compared to elasticities
and another alternative presented in this report, the cubic formula. This report is the
documentation of that assessment.
This report begins with a basic explanation of elasticities and their health
insurance applications. Induction factors are similarly described then contrasted with
elasticities from a theoretical standpoint. Such an elementary explanation is intended
to ensure that the key distinctions and limitations of elasticities and induction factors
are not missed, particularly for their applications in modeling. The next section of
the report reviews some of the key findings from the RAND Health Insurance
Experiment. Finally, the report discusses how best to replicate HIE results using
elasticities, induction factors, and the cubic formula.
In short, this report assesses the ability of the three methods to consistently
replicate certain RAND Health Insurance Experiment results. Because health care
and people’s responsiveness to its costs may have changed in the decades since the
HIE’s implementation, the method that best replicates the HIE results may not in fact
best represent current responsiveness to health care cost-sharing. Whether people’s
responsiveness has changed is difficult to know without another experimental study
like the HIE. However, on the specific question of which method is best for
replicating the HIE results, this report points to the cubic formula.

CRS-2
Theoretical Explanation of Cost-Sharing Methods
Elasticity of Demand
The elasticity of demand is a number that approximates the effect that a change
in price has on the quantity purchased of a good or service. In other words,
elasticities are used to answer this question: If the original quantity (Q ) of a good or
0
service is purchased at the original price (p ), how many units would be purchased
0
(Q ) at a different price (p )?
1
1
For two given prices (p and p ) and the two associated quantities (Q and Q ),
0
1
0
1
the elasticity is defined as the percentage change in quantity resulting from a one
percent change in price. Starting from a particular point (p , Q ), this is represented
0
0
algebraically as follows:
Q −Q
1
0
Q
(1) E
=

0
point
p −p
1
0
p0
For example, a car dealership knows that if its price on a particular model is
$30,000, it will sell 500 of those cars in the year; however, if it drops its price to
$27,000, it will sell 600. The point elasticity would then be calculated as follows:
600 −500
500
(2)
E
=
20%
27,000 −
=
30,000
10
−
= −
% 2
point
30,000
Because the absolute value of the elasticity is greater than one, it denotes that
people are very responsive to price changes for this model. Specifically, a 10% drop
in price would yield a 20% increase in quantity demanded.
Those dealing with elasticities aspire to apply a particular value, say -2, to all
different prices and quantities for that good. However, point elasticities do not yield
consistent results; as a cost-sharing factor, they lack certain desirable properties. One
such property is reversibility, which means that the calculation of a cost-sharing
factor (based on two points) yields the same result regardless of which point is
considered the starting point (p , Q ). Point elasticities are not reversible, as
0
0
demonstrated below by using the same two points used in Eq. (2) but switching the
starting point. The elasticity below, -1.5, does not match the previous one, -2:
500 600
−
600
(3) E
= 30,000 27,000 1.5
point
−
= −
27,000
To obtain the same elasticity from a given pair of points, arc elasticities are used
instead of point elasticities. In other words, arc elasticities are reversible. Arc

CRS-3
elasticities are calculated as (change in quantity divided by average quantity) /
(change in price divided by average price), or:
Q − Q
1
0
(Q
+ Q )/2
(4)
1
0
E
=
arc
p − p
1
0
(p
+ p )/2
1
0
Using the prior example, the arc elasticity is the following, and does not vary
regardless of which is chosen as (p , Q ):
0
0
600 500
−
(600 500)
+
/2
(5) E
= −
27,000 30,
−
000
1.73
arc
(27,000 30,
+
000)/2
As in this example, the arc elasticity (-1.73) is often close to the average of the
point elasticities (-1.75).
In addition to reversibility, another advantage of the arc elasticity is that it is
defined even if any of the parameters equals zero. In the point-elasticity formula, if
either p or Q is zero, the elasticity cannot be calculated. For these reasons, arc
0
0
elasticities are generally favored by health economists over point elasticities.
Although the arc elasticity may be nearly as easy to calculate, the point elasticity
is often easier to apply when predicting quantity (Q ). Eq. (6) shows the formula that
1
results from solving the point-elasticity formula in Eq. (1) for Q .
1
(6)
Q
= Q (1 + E(p -p )/p )
1 point
0
1
0
0
Eq. (7) shows the formula that results from solving the arc-elasticity formula in
Eq. (4) for Q .
1
(7)
Q
= Q (Ep - Ep - p - p ) / (Ep - Ep - p - p )
1 arc
0
0
1
1
0
1
0
1
0
The temptation is to take an arc elasticity and predict Q based on the point-
1
elasticity formula in Eq. (6). However, this does not yield proper results. To
illustrate, apply the previously calculated arc elasticity (-1.73) to predict Q when p
1
1
is $27,000 and the starting point is ($30,000, 500). Although the actual Q is 600,
1
the point-elasticity version of Q yields 586. For the point-elasticity version of Q to
1
1
yield 600, the original point elasticity of -2 would have to be used. Although it is
more unwieldy, the arc-elasticity version of Q in eq. (7) yields the proper result (in
1
this case, 600) when applying the arc elasticity.

CRS-4
In other words, when predicting quantity using elasticities, it is critical to use the
Q formula that corresponds with the elasticity used, whether arc or point.1 The
1
additional complication of applying the point elasticity is that, because it is not
reversible, one must determine which point elasticity to use; when predicting Q from
1
a particular (p , Q ), it is best to choose the point-elasticity value based on the starting
0
0
point closest to the one it is being applied to.
Health Insurance Applications. Calculating elasticities when individuals
are covered by health insurance is complicated by the fact that the price paid by
consumers for health care (that is, their cost-sharing) is usually not the full price of
that care. For example, the average price of a hospital stay may be $5,000, but the
insured’s effective “price” would be only the cost-sharing — a $750 deductible, for
example. This effective price, the person’s out-of-pocket liability, is what influences
their behavior rather than the total price. Thus, when looking at elasticities for health
insurance purposes, the prices (p and p ) are generally the prices paid by the
0
1
individual out of pocket rather than the actual total price for the good or service.
In addition, it is often difficult to measure the quantity of health care purchased.
When deriving an elasticity for health care, what should be used for “quantity”?
Fortunately, there is a way around this dilemma, using the fact that the total amount
spent on health care (i.e., the person’s out-of-pocket payments plus payments by
insurance) is the actual price of the good or service (not just the out-of-pocket
amount) multiplied by the quantity used. Therefore, the percentage change in total
spending would be as follows, with P and P representing the actual price of the
0
1
good or service, not just the out-of-pocket amount:
P Q − P Q
(8) % ) in total spending = 1 1
0
0
P Q
0
0
However, in calculating elasticities based on cost-sharing changes, we assume
that the actual price of the health good service does not change — that is, that P and
0
P are equal. As such, price falls out of the equation so that the percentage change
1
in total spending is equivalent to the percentage change in quantity demanded:
PQ
− PQ
Q − Q
(9) % ) in total spending =
1
0
1
0
=
= % ) in quantity demanded
PQ
Q
0
0
Thus, the easiest way empirically to calculate an elasticity for health care is to
use the percentage change in total spending as the percentage change in quantity
demanded. This applies to all of the uses of “quantity” for the remainder of this
report.
For example, an insurance company contracts with physicians so that a typical
office visit is $100, or P. On average, its enrollees make three visits per year, for
1 It is possible to use an arc elasticity in the point-elasticity formula for Q and obtain the
1
appropriate Q by using a multiplicative factor. However, calculating that factor is more
1
cumbersome than simply using the appropriate formula in Eq. (7).

CRS-5
total spending of $300 per enrollee for the office visits. The enrollees must pay a
copayment per visit of $5 (a coinsurance of 5%). The following year, the plan keeps
its contract arrangements the same with physicians but increases its copayments to
$25 for office visits (a coinsurance of 25%). The result is that the average number
of visits drops to 2.25, for total average spending of $225. From these figures, the
arc elasticity is calculated as follows:
225 30
− 0
(225 30
+ 0)/2
(10) E
=
= −
25 5
0.21429
arc
−
(25 5
+ )/2
Because the absolute value of this elasticity (0.21429) is less than one (and
closer to zero), one would say the demand for this type of care is relatively inelastic.
That is, a substantial change in the effective price of care led to a relatively small
change in the amount of care demanded.2
Nevertheless, the example is a much-simplified version of the cost-sharing
structure for the range of services in most health insurance plans, which have
deductibles and out-of-pocket maximums in addition to copayments and coinsurance
that may vary by type of service. As a result, precisely determining p for actual
1
health insurance plans is not straightforward; p becomes a function not only of the
1
new cost-sharing structure but of Q , the variable we seek to derive from p . Because
1
1
of this simultaneous (some might say circular) relationship, elasticities appear to be
limited outside of applications to plans with only coinsurance cost-sharing.3
Induction Factors
Induction factors are not discussed as commonly as elasticities in health policy
circles. A thorough literature review on the topic came up with only a handful of
references, all several years old.4 Like elasticities, induction factors are used to
2 With an explicit measure of quantity (number of visits), because the underlying total price
($100) remained the same, the arc elasticity would yield the same result whether using that
quantity or total spending, affirming what was shown in Eq. (9). From the two data points
in this example, the point elasticities would be -0.0625 and -0.4167, depending on which
point was chosen as the start. This is quite a large range for describing the impact of cost-
sharing changes.
3 A plan’s average coinsurance is often estimated in an effort to make elasticities applicable,
which is discussed later in this report.
4 Daniel Zabinski et al., “Medical Savings Accounts: Microsimulation Results from a Model
with Adverse Selection,” Journal of Health Economics, volume 18 (1999), pp. 195-218.
(Hereafter cited as Zabinski, et al., Medical Savings Accounts.) Edwin Hustead et al.,
“Medical Savings Accounts: Cost Implications and Design Issues,” American Academy of
A c t u a r i e s ’ P u b l i c P o l i c y M o n o g r a p h N o . 1 , M a y 1 9 9 5 , a t
[http://www.actuary.org/pdf/health/msa_cost.pdf]. Hustead is also the principal actuary at
the Hay Group for the CRS contract on the valuation models. Documentation on those
models regarding induction factors is very similar to the write-up in the MSA monograph.
(continued...)

CRS-6
predict the impact of cost-sharing changes on total health care spending. The key
advantage of induction often touted over elasticities is that its factors are relatively
easy to apply on all plan types, even those with complicated cost-sharing structures.
“Moving beyond the simplified case of pure coinsurance, the actuarial method [of
induction] offers a tractable, albeit imperfect, approximation to the actual change in
medical care.”5 It avoids elasticities’ circular conundrum by applying the new cost-
sharing structure to the original spending. It does this by calculating the dollar
amount of out-of-pocket payments (OOP) under the old and the new cost-sharing
structures, holding total spending (Q ) constant. Where I is the induction factor and
0
OOP denotes the dollar amount paid out of pocket based on the new cost-sharing
1*
structure but the old quantity demanded, Q is predicted as follows:
1
(11)
Q = Q + I (OOP - OOP )
1
0
0
1*
Solving this equation for the induction factor yields the following:
(12)
I = (Q - Q ) / (OOP - OOP )
1
0
0
1*
Eqs. (11) and (12) illustrate that an induction factor is a very different measure
from an elasticity, even though both may try to replicate similar impacts of cost-
sharing changes on total health care spending. The value of an elasticity represents
the percentage change in quantity resulting from a one percent change in price. The
value of an induction factor is the percentage of the difference in two plans’ out-of-
pocket payments that directly affects total health care spending.
For example, a person has total health care spending of $5,000, of which $4,000
is paid by a health insurance plan and $1,000 out of pocket. Another plan in which
the person had total spending of $5,000 may require $1,500 out of pocket, a $500
increase. An induction factor of 70%, or 0.7, means that total health care spending
would be reduced by 70% of the out-of-pocket difference between the plans (70% of
$500, or $350). Thus, under the new plan with higher cost-sharing, total health care
spending for the person would be predicted to drop to $4,650 (that is, $5,000 - $350).
Although induction factors and elasticities are very different measures, there are
cases in which they can be shown to be closely related. For example, in plans where
the cost-sharing can be represented as a pure coinsurance, the induction formula’s
OOP can be written as Q p , and OOP as Q p . In that case, Eq. (11) can be written
1*
0 1
0
0 0
as shown in Eq. (13), which can be solved for the induction factor, as shown in Eq.
(14):
4 (...continued)
“Methodological Description of Health Care Reform Premium and Discount Estimates,”
addendum to The White House Domestic Policy Council, “Health Security Act: The
P r e s i d e n t ’ s R e p o r t t o t h e A m e r i c a n P e o p l e , ” O c t . 1 9 9 3 , a t
[http://www.ibiblio.org/pub/academic/medicine/Health-Security-Act/supporting
-documents/method2.txt].
5 Zabinski et al., Medical Savings Accounts, p. 200.

CRS-7
(13) Q = Q + Q I(p - p )
1
0
0
0
1
Q − Q
1
0
Q0
(14) I
= p − p
0
1
The following emerges by dividing both the numerator and denominator of the
right-hand side of Eq. (14) by -p :
0
Q − Q
1
0
Q
− 1
(15)
0
I
=
x
= - E
/ p
point
0
p − p
1
0
p
p
0
0
Through similar algebraic manipulations, the induction factor can also be
written as follows:
(16)
I = - E (Q + Q ) / [Q (p + p )]
arc
1
0
0
1
0
These last two equations may not have widespread practical applications.
However, they both illustrate that induction factors can be expressed as a function of
elasticities, both point and arc, in their pure coinsurance forms.6 They also highlight
an issue that is indicated by the Q in the denominator of Eq. (16) and in the upper
0
denominator of Eq. (14) — that induction factors are not reversible. That is,
induction factors calculated from two points will yield different values depending on
which of the two points was chosen as the starting point. This is the same
shortcoming that point elasticities have, and Eq. (15) does not fix this. Arc
elasticities do not have this shortcoming, thus the induction-factor equation as a
function of the arc elasticity in Eq. (16) necessarily places Q in the denominator to
0
introduce it. Because two values for the induction factor result from any two price-
quantity data points, care must be taken to choose the correct one, depending on the
Q used to create the factor and the Q used to predict cost-sharing changes in a new
0
0
plan.
Health insurance applications. Using the office-visit example presented
earlier, Eq. (14) yields the following induction factor:7
225 − 300
300
(17) I =
1.25
=
5% 25%
−
6 Zabinski et al., Medical Savings Accounts, shows another version of the induction factor
as a function of the arc elasticity.
7 Eqs. (12), (15) and (16) also yields an induction factor of 1.25.

CRS-8
Induction factors’ lack of reversibility is demonstrated once more by reversing
the parameters of the office-visit example and calculating that induction factor:
300 − 225
225
(18) I
=

= 1.67
25% − 5%
In sum, although induction factors appear to enable a better analysis of a greater
variety of health insurance plans, compared to arc elasticities, they are not
unambiguously superior — one reason being that they are not reversible.
Specifically, the use of induction factors could lead to flawed results if a single value
is being used across a domain of coinsurance levels.
Additional Comparisons
In elementary algebra, the slope of a line is defined as rise over run —
specifically, the change in the dependent variable (y) divided by the change in the
independent variable (x). Thus, the equation for the linear slope between two price-
quantity data points is (Q - Q )/(p - p ).8 Substituting this in the cost-sharing
1
0
1
0
formulas previously discussed yields the following:
(19) E
= slope * (p / Q )
point
0
0
(20) E = slope * (average p) / (average Q)
arc
(21) I = slope / -Q0
Interestingly, if the predicted quantity (say, Q ) based on some price (p ) is
2
2
calculated from a starting point (p , Q ) based on one of the original points used to
0
0
calculate the factors above, the equation is reduced to the following for both point
elasticities and induction factors:
(22) Q = Q + slope * (p - p )
2
0
2
0
This results in a straight line with the original slope and passing through the
original data points.9 From the office-visit example, moving from 5% to 25%
coinsurance yields an induction factor of 1.25 and a point elasticity of -0.0625.
Applying these factors in the domain of 5% to 25% coinsurance to predict quantity
yields the straight line in Figure 1, as Eq. (22) would predict.
8 Economists tend to flip the axes when dealing with price and quantity to create a demand
curve with a slope that is the reciprocal of this one. For this report, traditional graphs and
slopes are used, with apologies to economists who would prefer straightforward demand
curves.
9 The derivative of Q with respect to p yields the original slope.
2
2

CRS-9
Figure 1. Predicted Expenses Using Various Cost-Sharing Methods,
from Example Case
300
Straight line resulting from point-elasticity
and induction-factor equations
280
s
nse 260
Expe
d
Arc resulting from
e
arc-elasticity equation
ict
d 240
Pre
220
200
5%
10%
15%
20%
25%
Coinsurance
Source: Congressional Research Service (CRS) example and calculations.
This characteristic has important implications for applying the RAND HIE
results, discussed later.10 However, similarly substituting Eq. (20) in the predicted-
quantity formula for arc elasticities, Eq. (7), does not yield a simplified equation of
any sort.11 Using the arc-elasticity formulas, predicted quantity yields a non-linear
curve — an arc. That arc will pass through the original pair of data points if the
starting point (p , Q ) used to predict quantity is one of the original data points. The
0
0
10 The same line results when switching the starting point of the two original data points and
when using the appropriate, other point elasticity or induction factor.
11 In lieu of using the substitution of the slope, calculus can be used to find some simplified
form for the predicted quantity based on the arc-elasticity formula. Let Eq. (7) predict a Q2
based on some p , given a constant elasticity calculated from two original data points. The
2
derivative of Q with respect to p yields a complicated equation. The gist of the resulting
2
2
equation is that p appears only in the denominator, as a quadratic formula. This is
2
consistent with the arc that results from applying Eq. (7), as in Figure 1.

CRS-10
arc in Figure 1 illustrates this for the coinsurance domain of 5% to 25%, using (5%,
300) as the starting point and -0.21429 as the arc elasticity.
The concavity of the arc also indicates that responses to cost-sharing will not be
constant, as is the case using point elasticities and induction factors. Specifically, the
concavity results in greater responsiveness at lower prices than at higher prices,
within the applicable domain. Whether this is preferable to a constant responsiveness
probably depends on the good or service being analyzed. With respect to the demand
for health care, this concavity is consistent with the HIE results discussed later —
that people are more responsive to cost-sharing changes when their cost-sharing is
relatively small, compared to the response when their cost-sharing is higher.
Another desirable property of cost-sharing factors in addition to those
mentioned earlier is path neutrality — that is, when using a factor’s particular value
(based on two data points), the predicted quantity for a given price should be
identical regardless of which point is chosen as the start (p , Q ).12 For example, the
0
0
straight line in Figure 1 (from induction factors and point elasticities) results whether
using (5%, 300) or (25%, 225) as the starting point. However, in spite of the
reversibility of the underlying factor, arc elasticities are not path neutral. The arc in
Figure 1 would be slightly different had (25%, 225) been used as the starting point.
Beginning at the higher price, the resulting arc would not bulge as much from the
straight line. At 15% coinsurance, for example, the quantity at the straight line is
262.5; on the arc in Figure 1, it is 241.9; if the starting point had been (25%, 225),
the quantity on the arc would have been at 250.5. At 15% coinsurance, the difference
in the estimated quantity between the arc-elasticity results is approximately 3.5%.
In the previous examples, the starting point (p , Q ) used to predict quantity has
0
0
always been one of the two original points used in calculating the cost-sharing factor.
However, when applying a given cost-sharing factor, the data may require beginning
from a known point that is not one of the original points (if those original points were
even known). In the office-based example — given the induction factor of 1.25, the
point elasticity of -0.0625, and the arc elasticity of -0.21429 — assume a plan has
15% coinsurance (p ) and that an analyst wants to predict a range of quantities based
0
on changes to that coinsurance. If a cost-sharing factor is path neutral, then the
predicted quantities in this example should be the same as illustrated in Figure 1.
Assuming this would be the case, the Q associated with the 15% coinsurance was
0
taken from the points on the lines in Figure 1 — 262.5 for the point-elasticity and
induction formulas, and 241.9 for the arc-elasticity formula.
Figure 2 shows the predicted quantities generated from a beginning coinsurance
of 15% and applying the various cost-sharing methods. The dark lines are the
original ones from Figure 1 with which the results would coincide if they were path
neutral. However, none of the three cost-sharing methods — point elasticities,
induction factors or arc elasticities — is path neutral. The dashed lines show the
predicted quantities based on the elasticities, with the straight line based on the point
elasticity and the arc based on the arc elasticity; the light, solid line shows the
12 Tom Selden, an economist at the Agency for Healthcare Research and Quality (AHRQ),
came up with this name for the concept.

CRS-11
predicted quantities based on the induction factor. All of the methods reproduce the
Q on which the predicted quantities in the figure were based, at 15% coinsurance.
0
Only the arc elasticity matches the original data point (5%, 300). Note that none of
the methods obtains the original point of (25%, 225).13 This lack of path neutrality
is a serious limitation of all these cost-sharing methods, the implications of which are
discussed in detail in the section on the methods’ ability to replicate HIE results.
Figure 2. Predicted Expenses Using Various Cost-Sharing Methods and
Values, from Example Case Results at 15% Coinsurance
300
280
Predicted based on
s
point elasticity
260
Expense
d
Predicted based
icte
on induction
d 240
e
Pr
220
Predicted based on
arc elasticity
200
5%
10%
15%
20%
25%
Coinsurance
Source: Congressional Research Service (CRS) example and calculations.
Note: The two dark lines are the original results shown in Figure 1. Ideally, the factors would be
path neutral; because they are not, each of the three factors yields predicted quantities that do not
coincide with the original lines. See text for description of results.
13 As demonstrated previously, arc elasticities are not path neutral even when using one of
the factor’s original points as the starting point for predicting quantity. The other predicted
quantity associated with 15% coinsurance (Q =250.5) would have yielded yet another
0
distinct arc in Figure 2, this one passing through the original point of (25%, 225) but not
(0%, 300).

CRS-12
RAND Health Insurance Experiment
The RAND Health Insurance Experiment (HIE) was ongoing from the mid-
1970s to the early 1980s. Two thousand nonelderly families from six urban and rural
areas were randomly assigned health insurance plans with different levels of cost-
sharing. The results from this unprecedented health insurance experiment showed
that people facing higher cost-sharing (that is, they had to pay a higher proportion of
total health care costs out of their own pockets) had lower health care spending that
those in plans with lower cost-sharing. No similar experiment has been performed
since the HIE, so it remains the epochal analysis for understanding the link between
health insurance cost-sharing and total health care spending.
The key variable used to explain health care spending in the HIE was the plans’
coinsurance — that is, the percentage of total health care costs that the individual
must pay. Four coinsurance rates were used: 0% (called the free plan, in terms of
there being no cost-sharing), 25%, 50%, and 95%. However, for each coinsurance
rate, there were three plans, each with a different out-of-pocket maximum: 5%, 10%,
or 15% of family income, up to a maximum of $1,000.14 A person may have been
enrolled in the “25% coinsurance plan,” but after that person reached the out-of-
pocket maximum, the plan effectively became a 0% coinsurance plan. Thus, the HIE
results need to be understood in the context of other variables besides the nominal
coinsurance, particularly the out-of-pocket maximum.
Selected Results
From the HIE data, the RAND authors calculated annual per-person medical
spending, controlling for factors such as location and factors that affect likelihood of
having a medical expense.15 These results are shown in Figure 3 for the four
coinsurance levels (incorporating all the out-of-pocket maximums), with the four
points connected by line segments.
As shown in the figure, average free-plan spending was $1,019 (in 1991
dollars). Plans with a nominal coinsurance of 25% averaged $826 in spending,
significantly less than in the free plan (p<0.001). At the 50% nominal coinsurance
level, spending averaged $764, significantly less than in the 25% plans (p=0.05,
t=1.97). Plans with a nominal coinsurance of 95% averaged $700 in spending, less
than in the 50% plans (p=0.06, t=1.93). Note that for the remainder of this report,
many results are shown rounded, even if their usage later on is based on the
unrounded amounts. This may cause others’ results to differ slightly.
14 More information on the design of the experiment is available from many sources,
including Joseph P. Newhouse et al., Free for All? Lessons from the RAND Health
Insurance Experiment (Cambridge, Massachusetts: Harvard University Press, 1993).
(Hereafter cited as Newhouse, Free for All?). Authors of the HIE often refer to the out-of-
pocket maximum as the Maximum Dollar Expenditure (MDE). There were a couple other
plans with slightly different cost-sharing designs, but they received less attention in the HIE
results.
15 All of the results in this section are based on the average annual per-person medical
spending based on the four-equation model, described in Chapter 3 of Newhouse et al.

CRS-13
Figure 3. Effect of Nominal Coinsurance on Annual Per-Person
Medical Expenses, in Dollars, from RAND Health Insurance
Experiment
(0%, $1,019)
1,000
(25%, $826)
$)
91
800
9
1
(
(50%, $764)
ses
n
(95%, $700)
e
600
Exp
an
) Me
ed
400
ict
red
P
(
al
u
n
200
An
0
0%
20%
40%
60%
80%
100%
Nominal Coinsurance (before out-of-pocket maximum)
Source: Table 3.3, Joseph P. Newhouse et al., Free for All? Lessons from the RAND Health
Insurance Experiment (Cambridge, Massachusetts: Harvard University Press, 1993).
Note: Expenses exclude dental and outpatient psychotherapy. For each coinsurance rate, there are
multiple plans, each with its own out-of-pocket maximum (none exceeding $1,000).
Because the dollar amounts are for 1991, it is useful to standardize the results,
with the free-plan spending ($1,019) as the base for comparing health care spending.
Spending in the 25% coinsurance plan would be represented as having 81% of the
free-plan spending, and so on. This is shown by the points connected by the heavy
solid line segments in Figure 4.
As previously mentioned, the effect of the plan’s coinsurance is diminished by
its particular out-of-pocket maximum. The RAND authors noted, for example,
“individuals who exceeded the [out-of-pocket maximum] tended to increase their
spending on all [health care] episode types.”16 The authors then took steps to
estimate the “pure price effect” of the coinsurance, estimating how much spending
16 Newhouse et al., Free for All?, p. 105.

CRS-14
would occur in each coinsurance in the absence of any deductibles or out-of-pocket
maximums. This is shown in Figure 4 by the points connected by the lighter solid
line segments.
As expected, average health care spending at a given coinsurance rate is lower
if there is no out-of-pocket maximum, compared to levels with an out-of-pocket
maximum. These estimates are from Table 4.17 of Newhouse et al., in which
Chapter 4 provides a detailed description of how these estimates were obtained.
Figure 4. Effect of Nominal Coinsurance on Annual Per-Person
Medical Expenses, as a Percentage of Free Plan Expenses
100%
(0%, 100%)
With out-of-pocket maximum
(25%, 81%)
80%
(50%, 75%)
ng
(95%, 69%)
endi
(25%, 71%)
p
S
n
60%
la
(50%, 63%)
ee P
(95%, 55%)
Without out-of-pocket maximum
of Fr
40%
entage
rc
e
P
20%
0%
0%
20%
40%
60%
80%
100%
Nominal Coinsurance
Source: Tables 3.3 and 4.17, Joseph P. Newhouse et al., Free for All? Lessons from the RAND
Health Insurance Experiment (Cambridge, Massachusetts: Harvard University Press, 1993).
Note: Expenses exclude dental and outpatient psychotherapy. For each coinsurance rate, there are
multiple plans, each with its own out-of-pocket maximum (none exceeding $1,000).
Using only the nominal coinsurance from plans with an out-of-pocket maximum
to estimate total spending raises concerns, since it overlooks the plan’s out-of-pocket
maximum, which can have a large impact on total spending, as illustrated in Figure
4. An alternative is to calculate the average coinsurance individuals faced in the
plans. For example, a person with $10,000 in spending in the typical HIE 25% plan
would not have had $2,500 in out-of-pocket spending; because the HIE plans limited

CRS-15
out-of-pocket expenditures, the most the person would have spent out of pocket was
$1,000. Had the person’s out-of-pocket expenditure been $1,000, the average
coinsurance would have been 10%, not the nominal 25%. Because it reflects all cost-
sharing (nominal coinsurance as well as deductibles and out-of-pocket maximums),
the average coinsurance, rather than the nominal coinsurance, is arguably a preferable
single value for representing a plan’s overall cost-sharing.
Figure 5. Effect of Average Coinsurance on Annual Per-Person
Medical Expenses, as a Percentage of Free Plan Expenses
100%
(0%, 100%)
Dark line is with out-of-
(16%, 81%)
pocket maximum
80%
g
(24%, 75%)
in
(31%, 69%)
end
(25%, 71%)
Sp
60%
(50%, 63%)
Plan
(95%, 55%)
ree
Without out-of-pocket maximum
f F
o
40%
ge
ta
Percen
20%
0%
0%
20%
40%
60%
80%
100%
Average Coinsurance
Source: From the preceding figure, except the average coinsurance for plans with an out-of-pocket
maximum is from Table 9, Willard G. Manning et al., “Health Insurance and the Demand for Medical
Care: Evidence from a Randomized Experiment,” The American Economic Review, vol. 77, no. 3
(June 1987), pp. 251-277.
Notes: Compare the dark line in this figure to the dark line in the preceding figure. The only
difference is that the x-values in Figure 4 are based on the nominal coinsurance while in this figure
they are based on the average coinsurance. Over its applicable domain, the average coinsurance for
plans with an out-of-pocket maximum yields spending in line with the pure price effects from pure-
coinsurance plans. For pure-coinsurance plans, the nominal coinsurance is equal to the average
coinsurance. Expenses exclude dental and outpatient psychotherapy.

CRS-16
Using the HIE data, the 25% plans’ average coinsurance is 16%, the 50% plans’
is 24%, and the 95% plans’ is 31%. This is shown by the points connected by the
heavy solid line segments in Figure 5, along with the previous points from the pure
price effects (that is, for pure-coinsurance plans) from Figure 4. Over its applicable
domain, the average coinsurance for plans with an out-of-pocket maximum yields
spending in line with the pure price effects from pure-coinsurance plans.17
The concordance of these results is what one might have expected. It also
makes applying these results convenient. First, microdata on health care
expenditures rarely provide cost-sharing information like a plan’s nominal
coinsurance. The results in Figure 5 suggest that such information may not be
necessary — that the average coinsurance paid by a person, even in a plan with a
complicated cost-sharing structure, should be as reliable in deriving and applying
cost-sharing factors as that of a pure-coinsurance plan.
In addition, the average coinsurance of the typical HIE plans did not exceed
31%. However, in dealing with individuals’ expenditure data, some people may have
faced high cost-sharing. For example, many people may have had health care
expenses that never reached their plan’s deductible, thus facing an average
coinsurance of 100%. Cost-sharing factors calculated from the HIE at an average
coinsurance of 31% may not be appropriate when applied at 100% coinsurance.
Because the pure price effects are estimated for coinsurance up to 95% and are
consistent with the typical HIE plans through their average coinsurance domain of
31%, one can justify using cost-sharing factors from the pure price effects for
application to all average coinsurance levels, regardless of the complexity of a plan’s
underlying cost-sharing structure.
Because of the potentially broader application of the HIE’s estimated pure price
effects, Table 1 is provided, which shows these effects for outpatient and inpatient
care as well as the total. It is worth noting that for up to 25% coinsurance, spending
does not differ by outpatient versus inpatient care, relative to free-plan spending.
Table 1. Estimated Pure Price Effects of Coinsurance on
Medical Expenses, as a Percentage of Free Plan Expenses
Pure coinsurance
Outpatient
Inpatient
Total medical
Dental
0%
100%
100%
100%
100%
25%
71%
71%
71%
79%
50%
58%
68%
63%
68%
95%
49%
60%
55%
50%
Source: Table 4.17, Joseph P. Newhouse et al., Free for All? Lessons from the RAND Health
Insurance Experiment (Cambridge, Massachusetts: Harvard University Press, 1993).
Note: Computed assuming estimates weighted by shares of spending occurring when enrollees are
far from reaching the plans’ out-of-pocket maximum. For total medical, the shares are 46% outpatient
and 54% inpatient.
17 For pure-coinsurance plans, the nominal coinsurance is equal to the average coinsurance.

CRS-17
Calculating and Applying Cost-Sharing Methods
Based on HIE Results
In this section of the report, cost-sharing factors (arc elasticities and induction
factors) are calculated from the HIE results in Table 1.18 In this section, once the arc
elasticities and the induction factors are calculated, variations in their application and
the results are discussed.
One goal of this analysis is to demonstrate how the factors might be applied in
a microsimulation model, using the structure of the CRS/Hay models as an example.
The CRS/Hay models use expenditure data from individuals who are enrolled in
health insurance plans with innumerable (and unknown) cost-sharing structures.
Using the expenditure data by source of payment (that is, out-of-pocket versus
insurance-paid expenses), the first step is to standardize the data, applying cost-
sharing factors to produce expenditure levels as if everyone were in a free plan. The
estimated free plan data then becomes the baseline against which cost-sharing
arrangements are applied.
Table 2. Arc Elasticities Between Average Coinsurance
Amounts, By Type of Service
Average
coinsurance range
Outpatient
Inpatient
Total medical
Dental
0% - 25%
-0.17
-0.17
-0.17
-0.12
0% - 50%
-0.27
-0.19
-0.23
-0.19
0% - 95%
-0.34
-0.25
-0.29
-0.33
0% - 25%
-0.17
-0.17
-0.17
-0.12
25% - 50%
-0.30
-0.06
-0.18
-0.22
50% - 95%
-0.27
-0.20
-0.22
-0.49
25% - 95%
-0.31
-0.14
-0.22
-0.39
Source: Congressional Research Service (CRS) calculations on data in Table 1, which is from Table
4.17, Joseph P. Newhouse et al., Free for All? Lessons from the RAND Health Insurance Experiment
(Cambridge, Massachusetts: Harvard University Press, 1993).
Arc Elasticities
Table 2 displays the arc elasticities calculated according to Eq. (4) and based
on the coinsurance (p and p ) and spending levels (Q and Q ) in Table 1. The first
0
1
0
1
three rows show the arc elasticities when compared to the free plan. The next three
display the arc elasticities between consecutive coinsurance rates. The last row
shows the arc elasticity between 25% and 95% coinsurance.
18 Point elasticities are no longer in the discussion for a couple reasons. First, one of the
four coinsurance levels used in the HIE is 0%, a value which makes the point-elasticity
formula undefined. When the point elasticity performs best, its results coincide with those
using induction factors. Otherwise, as illustrated in Figure 2, the application of the point
elasticity is far from optimal.

CRS-18
The key column in Table 2 is the shaded one showing elasticities for “total
medical.” These values vary, depending on the average coinsurance range used.19
Even if an analyst were to carefully select an elasticity according to these results,
choosing a value is not always straightforward. Consider a person in a plan with
15% pure coinsurance on $5,000 total spending ($750 out of pocket). Which
elasticity should be used if estimating the impact of moving to a 40% coinsurance
plan? An elasticity of -0.22 would predict total spending at $4,097, using Eq. (7).
If an elasticity of -0.17 were used, total spending in the new plan would be estimated
at $4,284, nearly 5% higher. Thus, even the most fastidious analysts can reasonably
use different elasticities and come up with different results on that basis alone.
Predicting Quantity with Point-elasticity and Arc-elasticity
Formulas. Earlier in this report, it was noted that “when predicting quantity using
elasticities, it is critical to use the Q formula that corresponds with the elasticity
1
used, whether arc or point.” This point merits repeating in the context of the HIE
results because arc-elasticity factors are so often applied in the point-elasticity
formula for predicting quantity. That is, arc-elasticity factors like those in Table 2
are often applied in Eq. (6) instead of the more appropriate Eq. (7).
Based on the total medical elasticities in Table 2, Table 3 shows the results if
one were to use those elasticities to estimate free-plan spending from the other three
coinsurance rates. Specifically, for each row in Table 3, the appropriate elasticity is
taken from the first three rows of Table 2. Since the purpose of the calculation is to
estimate free-plan spending, the result should be 100%. Because the arc-elasticity
formula is rarely written in terms of Q , arc elasticities are often applied using the
1
point-elasticity formula in Eq. (6). Those results are shown in Column A of Table
3 and do not yield the target amount of 100% shown in Column C. Column B shows
the results of using the arc elasticities with the formula for Q from Eq. (7), which
1
yields the targeted results. Eq. (23) illustrates how these results were calculated,
applying the 0%-95% elasticity to the point-elasticity formula; Eq. (24) does the same
but applies the elasticity in the arc-elasticity formula. The results in Table 3 again
illustrate why arc elasticities should not predict quantity using the point-elasticity
formula. Although the arc-elasticity formula for predicting quantity is more
complicated than the point-elasticity one, it is the correct one.
(23)
Q = Q (1 - E), where Eq. (6) is reduced because p =0
1
0
1
= 55% (1 + 0.29) = 71%
(24)
Q = - Q (E - 1) / (E + 1), where Eq. (7) is reduced because p =0
1
0
1
= - 55% (-0.29 - 1) / (-0.29 + 1) = -55% (-1.29)/(0.71)
= 100%
19 The variation in this column tends to be smaller than that in the service-specific columns.
The total medical elasticities reflect the combination of outpatient and inpatient spending
and therefore have the result of tempering the differences in those elasticities.

CRS-19
Table 3. Predicted Free-Plan Spending Using Arc Elasticities,
by Elasticity Formula for Predicted Spending
A
B
C
Applying point-
Applying arc-
Beginning
elasticity formula
elasticity formula
Target amount for
Coinsurance (p )
for total spending
for total spending
total spending
0
25%
83%
100%
100%
50%
77%
100%
100%
95%
71%
100%
100%
Source: Congressional Research Service (CRS) calculations, using arc elasticities from Table 2 and
quantities from Table 1.
Figure 6. Analysis of Arc Elasticity’s Lack of Path Neutrality in HIE
Results: Predicting Quantity Varying By Beginning HIE Data Point
100%
(0%, 100%)
Predicted based on
higher p0
80%
g
(25%, 71%)
(50%, 63%)
60%
an Spendin
Pl
Predicted based
e
on lower p0
(95%, 55%)
re
F
40%
rcentage of
Pe
20%
0%
0%
20%
40%
60%
80%
100%
Average Coinsurance
Source: Congressional Research Service (CRS) analysis.

CRS-20
Effect of Arc Elasticity’s Lack of Path Neutrality. As previously
mentioned, the arc-elasticity formula for predicting quantity is not path neutral —
that is, when using a factor’s particular value (based on two data points), the
predicted quantity for a given price will vary depending on which point is chosen as
the start (p , Q ). Using the HIE data points as (p , Q ) and the appropriate elasticities
0
0
0
0
between consecutive points yields the two curves in Figure 6.
The dashed line in the figure is predicted quantity based on the higher of two
consecutive HIE data points. For example, if predicting quantity at a coinsurance
price of 70%, this is between the HIE data points of (50%, 63%) and (95%, 55%),
where the arc elasticity for total medical care is -0.22.20 Using Eq. (7) and (50%,
63%) as (p , Q ) predicts a quantity of 58.6%. Using the same equation but the
0
0
beginning point of (95%, 55%) predicts a quantity of 58.8%. This is a relatively
small difference. In fact, over the domain of 25% to 95%, the difference in the
predicted quantities, varying which pair of consecutive points is chosen as (p , Q ),
0
0
averages about two-tenths of one percent.21 This is shown graphically in Figure 6
by the nearly imperceptible difference between the lines over the 25% to 95%
domain.
Figure 6 also illustrates a serious limitation of predicting quantity using an arc
elasticity when the price is zero (that is, free), as was the case in the RAND Health
Insurance Experiment. Applying the free-plan data from Table 2 to predict quantity
between a price of 0% and 25% coinsurance causes Eq. (7) to be reduced as follows:
(25)
Q = Q (Ep - Ep - p - p ) / (Ep - Ep - p - p ), where p =0 and Q =1
1
0
0
1
1
0
1
0
1
0
0
0
= (- Ep - p ) / (Ep - p )
1
1
1
1
1 + E
=
, where E = -0.17
1 − E
= 71%
The constant predicted quantity of 71% is shown by the horizontal solid line
over the 0% to 25% domain in Figure 6 for the predicted quantity based on the lower
p . Using a constant arc elasticity to predict quantity where price (either p or p ) is
0
0
1
zero yields a constant quantity across the domain of prices. This is unacceptable for
the present structure of the CRS/Hay models, which derive their results from a
baseline of free-plan values.22 Even analysts not predicting free-plan values must
cope with this issue where a person’s average coinsurance is 0%.
20 In predicting quantity, the unrounded elasticity was used.
21 This is determined by calculating the percentage difference in the integrals of Eq. (7)
alternating the consecutive pairs of HIE points for the beginning point:
I Q (Ep - Ep - p - p ) / (Ep - Ep - p - p ) dp
0
0
1
1
0
1
0
1
0
1
where E, Q and p are constants based on the HIE data, over the domain between the two
0
0
HIE data points.
22 It is not possible to apply the point-elasticity formula for total spending because p =0,
0
making Eq. (6) undefined. It is also not advisable to use, for example, (1%, 99%) instead
of (0%, 100%) because the shape of the resulting curve is not intuitive.

CRS-21
It was illustrated in Figure 2 that using a (p , Q ) that was not one of the original
0
0
data points created a separate arc that did not line up with either of those based on the
original data points. This is another potential area of concern for applying the arc
elasticities. However, this appears to be less of a practical concern when using arc
elasticities derived from the HIE results. The dashed line in Figure 7 is the same one
in Figure 6. The difference between the figures is that in Figure 7 the solid line is
based on the (p , Q ) points shown in the figure, which are on the dashed line.
0
0
Figure 7. Illustration of Arc Elasticity’s Lack of Path Neutrality:
Predicting Quantity Varying Whether Beginning Data Point Was an
Original HIE Point
100%
Predicted based on
points as shown
80%
(13%, 79%)
Predicted
nding
based on
(38%, 66%)
Spe
original HIE
(73%, 58%)
n
60%
points
la
P
e
40%
ge of Fre
rcenta
e
P
20%
0%
0%
20%
40%
60%
80%
100%
Average Coinsurance
Source: Congressional Research Service (CRS) analysis.
Note: Dashed line from Figure 6, based on moving from higher original HIE data points to
consecutively lower ones. Solid line predicted using arc-elasticity formulas and based on (p , Q )
0
0
points on the dashed line.
The biggest differences between the curves occur when price p is less than the
1
13% used for p . The smaller the coinsurance, the larger the difference. From a
0
beginning point of (13%, 79%), the free-plan value is predicted at 111% of the actual

CRS-22
HIE free-plan value. Over the domain from 13% to 95%, the largest difference
between the curves is two-tenths of one percentage point. Thus, even when the
beginning data point is not one of the original points used to calculate the elasticity,
applying the arc elasticity to predict HIE results yields relatively consistent results,
except when predicting based on small coinsurance rates. In other words, even
though arc elasticities are generally not path neutral, their application for HIE results
produces curves that appear relatively path neutral, except at smaller coinsurance
levels. Except when dealing with free-plan levels, arc elasticities appear useful in
reasonably replicating HIE results.
Induction Factors
Table 4 displays the induction factors calculated according to Eq. (14) and
based on the coinsurance (p and p ) and spending levels (Q and Q ) in Table 1. As
0
1
0
1
previously mentioned, the value of an induction factor is the percentage of the
difference in two plans’ out-of-pocket payments that directly affects total health care
spending. Table 4 expresses these percentages as decimals (for example, 1.16 rather
than 116%). As in Table 2, which shows the arc elasticities, the first three rows of
Table 4 show the factors when compared to the free plan. The next three display the
factors between consecutive coinsurance rates. The last row shows the factors
between 25% and 95% coinsurance. As previously discussed, induction factors are
not reversible — their value depends on which of the two data points in the
calculation is chosen as the starting point (p , Q ). As a result, for each pair of HIE
0
0
coinsurance rates, there are two induction factors — one using the lower coinsurance
as p and one using the higher coinsurance as p .
0
0
Table 4. Induction Factors Between Average Coinsurance
Amounts, By Type of Service
Average
Lower coinsurance as starting point
Higher coinsurance as starting point
coinsurance
Total
Total
range
Outpatient Inpatient medical Dental Outpatient Inpatient medical
Dental
0% - 25%
1.16
1.16
1.16
0.84
1.63
1.63
1.63
1.06
0% - 50%
0.84
0.64
0.74
0.64
1.45
0.94
1.17
0.94
0% - 95%
0.54
0.42
0.47
0.53
1.10
0.70
0.86
1.05
0% - 25%
1.16
1.16
1.16
0.84
1.63
1.63
1.63
1.06
25% - 50%
0.73
0.17
0.45
0.56
0.90
0.18
0.51
0.65
50% - 95%
0.34
0.26
0.28
0.59
0.41
0.30
0.32
0.80
25% - 95%
0.44
0.22
0.32
0.52
0.64
0.26
0.42
0.83
Source: Congressional Research Service (CRS) calculations on data in Table 1, which is from Table
4.17, Joseph P. Newhouse et al., Free for All? Lessons from the RAND Health Insurance Experiment
(Cambridge, Massachusetts: Harvard University Press, 1993).
Predicting Quantity with Induction Formulas. Induction factors are path
neutral (that is, they consistently predict quantity, using Eq. (13)) when the following
conditions hold:
! the beginning point (p , Q ) is one of the original data points that
0
0
was used in calculating the induction factor;

CRS-23
! the induction factor was calculated based on that (p , Q ) as the
0
0
starting point; and
! the price in the other original data point on which the induction
factor was based (p , Q ) has the same relationship to starting point
1
1
(p , Q ) as does the price being used to predict quantity (that is, the
0
0
p used to calculate the induction factor and the price for which
1
quantity is being predicted are both greater than or are both less than
p ).
0
Figure 8 shows the results when abiding by these conditions for total medical
(solid line) and inpatient care (dashed line). They are simply line segments drawn
between the original data points in Table 1.
Figure 8. Predicting Quantity Using Induction Factors and Original
Data Points from HIE
100%
(0%, 100%)
80%
g
Inpatient care
din
(50%, 68%)
(25%, 71%)
(95%, 60%)
Spen
n
60%
la
(50%, 63%)
(95%, 55%)
ree P
Total medical
f F
care
40%
ge o
a
Percent
20%
0%
0%
20%
40%
60%
80%
100%
Average Coinsurance
Source: Congressional Research Service (CRS) analysis.

CRS-24
Figure 9. Predicting Quantity of Inpatient Care, By Induction Factor
and Beginning HIE Data Point
100%
Inpatient care using HIE
induction factors
80%
Inpatient care with 0.3 induction
ing
d
factor, from lower p0
n
Spe
60%
Plan
e
Inpatient care with 0.3
induction factor, from
f Fre
higher p0
o
40%
ge
ta
n
rce
Pe
20%
0%
0%
20%
40%
60%
80%
100%
Average Coinsurance
Source: Congressional Research Service (CRS) analysis.
Note: The lines are predicted by applying the induction factors and the HIE price-quantity data points
to the appropriate formula (Eq. (13)), where p is the average coinsurance along the x-axis and the
1
percentage of free-plan spending is predicted by the formula and graphed against the y-axis.
The induction factors in the CRS/Hay models, however, do not vary by
coinsurance. For each type of care, the induction factors are held constant across the
domain of coinsurance levels. The CRS/Hay models use the induction factors most
commonly cited: 0.3 for inpatient hospitalization, 1.0 for prescription drugs, and 0.7
for all other medical care.23 Using the original HIE data points as the starting points
(p , Q ), Figure 9 shows the predicted quantity of inpatient care using a constant
0
0
induction factor of 0.3. The thin line is predicted quantity based on the lower HIE
23 See, for example, Table II-2A in Edwin Hustead, et al., “Medical Savings Accounts: Cost
Implications and Design Issues,” American Academy of Actuaries’ Public Policy
Monograph No. 1, May 1995, at [http://www.actuary.org/pdf/health/msa_cost.pdf].
(Hereafter cited as Hustead, et al., Medical Savings Accounts: Cost Implications.)

CRS-25
coinsurance between two points; the heavier solid line is based on the higher HIE
coinsurance. The dashed line is the same as in Figure 8.
When using the 0.3 induction factor, the lines that result in Figure 9 have
relatively constant slopes (not counting the graph’s vertical lines that link the line
segments). Solving Eq. (21) for the slope and holding the induction factor I constant
illustrates why the slope is relatively flat and constant. Based on the four original
HIE data points (for Q ) and a constant induction factor of 0.3, the slope of the
0
resulting two lines is between -0.18 and -0.30. This is consistent with the slope of
the line segments between the HIE data points where the coinsurance is not zero.
However, between the HIE data points of (0%, 100%) and (25%, 71%), the slope is
-1.16. As a result, the two lines based on the 0.3 induction factor vary substantially
from the line based on the HIE data points, although each of the two lines based on
the 0.3 induction factor passes through one original HIE point — the one that served
as (p , Q ) in the formula for predicting quantity.
0
0
Figure 10. Comparison of Results of Constant Induction Factor from Free-
Plan Spending with Induction Factors and Original Data Points from HIE
100%
Inpatient care predicted from
0.3 induction factor and
(0%, 100%)
80%
ng
ndi
e
p
S
Inpatient care predicted
n
60%
from HIE induction
Pla
e
factors and HIE data
points
of Fre
40%
tage
n
e
c
Per
20%
0%
0%
20%
40%
60%
80%
100%
Average Coinsurance
Source: Congressional Research Service (CRS) analysis.

CRS-26
There are two lines predicted from the original HIE points and the 0.3 induction
factor because the use of a constant induction factor effectively assumes the factor
is reversible. Of course, it is not reversible, as illustrated by the two lines that emerge
between two data points when holding the induction factor constant.
The dissimilarity between the slopes of the lines based on the HIE factors versus
the 0.3 factor causes particular concern for the CRS/Hay models, which produce
results based on cost-sharing additions to a free plan. Along with the same dashed
line in the previous two graphs based on the HIE data, Figure 10 shows predicted
inpatient care from a constant 0.3 induction factor and from the free-plan spending
data point of (0%, 100%). Where average coinsurance is 25%, the difference
between the two lines is greatest: The predicted quantity of inpatient care is 30%
higher than the HIE results. These results suggest that, if induction factors are to be
used, they should not be constant against the entire domain of cost-sharing.
Cubic Formula
Both arc elasticities and induction factors have limitations when trying to apply
them to replicate HIE results. These limitations are severe when doing analyses
based on free-plan levels, as illustrated in Figure 6, or when using a constant factor
for all levels of cost-sharing, as in Figure 10. This has huge implications for the
CRS/Hay models because they are built on a base dataset that represents free-plan
spending and uses constant factors regardless of cost-sharing. These findings suggest
the need for another method to estimate the impact of cost-sharing on demand for
health care.
Some other method that yields better model results may lack otherwise desirable
characteristics not critical in modeling. For example, an arc elasticity has
applications for analyzing all kinds of goods and services. It produces a standardized
value for comparing people’s price sensitivity for all kinds of goods and services.
Standardization is obtained by taking the slope of the line (or of the demand curve)
and applying the original or average price and quantity combinations so that the units
are no longer a part of the factor, as demonstrated in Eq. (19) and Eq. (20). For
modeling purposes, however, such standardization is not only unnecessary but is also
problematic in that it leads to the problems observed in Figure 6. A better method
for replicating the HIE results may not ultimately produce factors that are
standardized for comparing other goods and services. It is more important, however,
that the method produce values that are reversible, path neutral and consistent with
the HIE results.
The HIE results of interest for our modeling purposes are the four price-quantity
data points, in terms of average coinsurance and percentage of free-plan spending:
(0%, 100%), (25%, 71%), (50%, 63%), and (95%, 55%). With only four points, a
cubic formula can be calculated using least squares fit, which will create a curve
passing through all four HIE points. For the remainder of this report, this formula is
referred to as “the cubic formula,” where Q is the percentage of free-plan spending
predicted from a particular average coinsurance, p:
(26) Q = -1.5546p3 + 2.8459p2 - 1.7743p + 1

CRS-27
Figure 11 shows the results of this formula as a heavy line, along with the ideal
arc-elasticity and induction results presented earlier. The dashed line is based on the
arc elasticities, and the lighter solid line is based on the induction factors. All three
are for total medical care.
Figure 11. Predicting Quantity Using Cubic Formula, Compared to Ideal
Arc Elasticities and Induction Factors, From Original HIE Data Points
100%
(0%, 100%)
80%
g
(25%, 71%)
din
(50%, 63%)
Spen
60%
(95%, 55%)
Free Plan
of
e
40%
tag
Arc elasticities
Induction factors
rcen
e
Cubic formula:
P
20%
Q = -1.5546p3 + 2.8459p2 - 1.7743p + 1
R2 = 1
0%
0%
20%
40%
60%
80%
100%
Average Coinsurance
Source: Congressional Research Service (CRS) analysis.
Note: The arc-elasticity curve is the same as the dashed line in Figure 6, which predicts quantity
based on the higher of consecutive original HIE points as (p , Q ) and using the unique arc elasticity
0
0
between each pair of points, as shown in Table 2. The induction-factor curve is the same as the solid
line in Figure 8, which predicts quantity based on the higher of consecutive original HIE points as (p ,
0
Q ) and using the unique induction factor between each pair of points and for moving from higher to
0
lower cost-sharing, as shown in Table 4. While these are the ideal scenarios for the arc-elasticity and
induction-factor curves, the cubic formula always yields the same results, without needing a (p , Q )
0
0
and regardless of whether one is moving from higher or lower cost-sharing.
The cubic formula has several improvements over the other methods. For
example, because (p , Q ) is not part of the formula (since the original HIE points are
0
0
inherent in the cubic formula), path neutrality and reversibility are not concerns. The

CRS-28
quantity predicted based on a particular price will always be the same using the cubic
formula. Moreover, calculating the slope (that is, change in quantity / change in
price) at any point along the curve is quite simple, using the derivative of Q with
respect to p:24
(27) slope of Q at a given coinsurance (p) = Q’(p) = -4.6638p2 + 5.6918p - 1.7743
A potential concern one might note in Figure 11 is the difference between the
cubic formula and the other two curves between 50% and 95% coinsurance.
Although the arc-elasticity and induction curves are close to one another in this
domain, the cubic-formula curve appears much higher. The difference between the
cubic-formula curve and the induction curve is 3.1%.25 Between 0% and 50%
coinsurance, the cubic-formula curve is lower than the induction-factor line segments
so that, overall, the area under both curves is nearly identical — 64.6% and 64.7%
respectively.26 The area under the arc-elasticity curves is 63.3%, approximately 2%
less than the area under the other two curves.27 Because the HIE data provided only
four data points, it is not known which of the three curves best reflects the impact of
cost-sharing between those points. Considering this as well as the relatively small
overall differences between the curves, the shape of the curves should not be an
overriding criterion in determining which method to use.
Estimating Free-Plan Spending
As presented here, each of the methods take a particular average coinsurance to
predict the percentage of free-plan spending associated with that coinsurance. This
section discusses how these results would be applied to create free-plan spending
levels, such as those used in the CRS/Hay models, and how their results vary.
The example case is a person who was enrolled in a high-deductible health plan.
The deductible was $1,000, after which a 25% coinsurance applied. During the year,
the person had total medical expenses of $1,500.28 Of this total, the person paid
$1,125, or 75%. If this person had been in a free plan, her spending would likely
have been more than $1,500.
24 The slope of the demand curve would be the reciprocal of Eq. (27).
25 This is determined by calculating the percentage difference in the integral of Eq. (26) and
the area under the straight line created by the ideal induction-factor results:
1 - I(-1.5546p3 + 2.8459p2 - 1.7743p + 1 )dp / [(95%-45%)(55% + ½ (63% - 55%))]
where p ranges from 50% to 95%.
26 These numbers were calculated as in Footnote 24, using the integral of Eq. (26) for the
cubic formula and the area under the straight lines for the induction factors.
27 The area under the arc-elasticity curves was obtained by integrating Eq. (7) between
consecutive pairs of HIE points, using the higher coinsurance of each pair as p :
0
I Q (Ep - Ep - p - p ) / (Ep - Ep - p - p ) dp
0
0
1
1
0
1
0
1
0
1
where E, Q and p are constants based on the HIE data, over the domain between the two
0
0
HIE data points.
28 The assumption for all of these examples is that all expenses are covered medical
expenses.

CRS-29
To calculate free-plan spending using the arc-elasticity formulas, the appropriate
elasticity for total medical spending is needed from Table 2. Because the intent is
to predict free-plan spending (that is, p =0), one of the values in the calculation
1
should be based on a coinsurance of zero, as in the first three rows of Table 2. The
other value relevant for choosing the appropriate elasticity is the other coinsurance
of 75%. In the table, the nearest coinsurance rates to 75% are 50% (with an elasticity
of -0.23) and 95% (with an elasticity of -0.29). Based on its distance between 50%
and 95% coinsurance, the 75% coinsurance is estimated through simple imputation
to have an elasticity of -0.26333.29 The (p , Q ) to be applied in Eq. (7) is the (75%,
0
0
$1,500) given in the example, predicting free-plan spending as follows:
(28)
Q = Q (Ep - Ep - p - p ) / (Ep - Ep - p - p ), where p =0
1
0
0
1
1
0
1
0
1
0
1
= Q (Ep - p ) / (-Ep - p )
0
0
0
0
0
1 − E
=
, where E = -0.26333
1 + E
= $1,500 (1+0.26333) / (1-0.26333)
= $2,572
As shown in Eq. (28), predicting free-plan spending — that is, predicting Q1
where p =0 — causes p to cancel out of the formula. Thus, the elasticity is the only
1
0
variable remaining that reflects the original plan’s 75% coinsurance. As a result,
when dealing with free-plan information, constant results can be avoided only by
calculating precise elasticity values. The results are quite sensitive to the value of the
elasticity. An arc elasticity of -0.23 would have yielded free-plan spending of
$2,396. An arc elasticity of -0.29 would have yielded free-plan spending of $2,725.
When dealing with a free plan, the use of a continuous arc elasticities is required
to predict spending that is not constant over a given domain. In other words, the arc
elasticity must be used to create unique elasticities at each given point, like a point
elasticity. Obviously, this is not the role of the arc elasticity. The approach applied
in Eq. (28) to work around the factor’s limitation when dealing with a free plan is
arguably a misuse of its factors. Thus, when applying arc elasticities to free-plan
information, an analyst must choose between possibly misusing the factors or
predicting constant spending regardless of cost-sharing. Neither is desirable.
In applying the induction-factor formulas, not only should the factor be chosen
from the first three rows of Table 4 but should also be taken from the penultimate
column, for “higher coinsurance as starting point.” This matters because induction
factors are not reversible. As with the elasticities, there are two induction factors to
choose between — 1.17 (based on coinsurance from 0% to 50%) or 0.86 (based on
coinsurance from 0% to 95%). These constant factors would predict unique amounts
of spending for every coinsurance in the domain, unlike elasticities. As a result,
29 Calculated as -0.23-[(75%-50%)/(95%-50%)*(0.29-0.23)]. Also, Figure 7 showed little
difference in the results when the starting point for predicting quantity was not one of the
original HIE points. However, those results were calculated between consecutive HIE
points with the corresponding elasticity. In this example, there is greater variation due to
the larger coinsurance domain used.

CRS-30
calculating an induction factor for a given coinsurance is not as crucial, but doing so
produces continuous factors that seem more accurate than the factors at 50% and
95%. Based on its distance between 50% and 95% coinsurance, the 75% coinsurance
is estimated to have an induction factor of 0.9978. The free-plan spending level is
then calculated using both Eq. (11) and Eq. (13) respectively below:
(29)
Q = Q + I (OOP - OOP )
1
0
0
1*
= $1,500 + 0.9978 ($1,125 - $0)
= $2,623
(30)
Q = Q (1+ I(p - p ))
1
0
0
1
= $1,500 (1 + 0.9978 (75%-0%))
= $2,623
This is 2% higher than the comparable arc-elasticity amount. These results are
also sensitive to the value of the factor. An induction factor of 0.86 would have
yielded free-plan spending of $2,468. An induction factor of 1.17 would have
yielded free-plan spending of $2,816.
Applying the cubic formula to the same example is more straightforward in that
there are no factors to decide among. Thus, the free-plan spending level is calculated
by applying the price of 75% coinsurance as follows:
(31)
Q = -1.5546p3 + 2.8459p2 - 1.7743p + 1
= -1.5546(75%)3 + 2.8459(75%)2 - 1.7743(75%) + 1
= 61.4%
In other words, the cubic formula estimates that an average coinsurance of 75%
yields a quantity that is 61.4% of free-plan spending. Thus, the $1,500 in total
spending is 61.4% of free-plan spending, and this free-plan spending results:
(32)
$1,500 / 61.4% = $2,442
Although there is a range of possible results from the other two methods, the
cubic formula produces a single result. That result is 5% lower than the arc-elasticity
result in Eq. (28) and 7% lower than the induction-factor result in Eq. (30).
Predicting Spending From Estimated Free-Plan Spending
Pure Coinsurance Plan. Based on the free-plan spending estimates above,
one can predict spending in a plan with nominal coinsurance of 75% and no
deductible or out-of-pocket maximum. In this case, of course, the nominal
coinsurance also serves as the average coinsurance. A priori, one might expect total
spending to be predicted at $1,500 because the average coinsurance of 75%, as
before. Once more, the arc elasticity of -0.26333 is used in Eq. (7), with p =0%,
0
Q =$2,572 and p =75%:
0
1

CRS-31
(33)
Q = Q (Ep - Ep - p - p ) / (Ep - Ep - p - p ), where p =0
1
0
0
1
1
0
1
0
1
0
0
= Q (- Ep - p ) / (Ep - p ), where p cancels out
0
1
1
1
1
1
= Q (1+E) / (1-E)
0
= $2,572 (1 - 0.26333) / (1 + 0.26333)
= $1,500
As illustrated in Figure 6, using constant arc elasticities to predict spending
from free-plan values (that is, p =0) also yields a constant value. The arc-elasticity
0
formula causes spending to be predicted based solely on the value of the elasticity,
with its impact on the free-plan spending of $2,572. Calculating continuous
elasticities is not only problematic, as was mentioned before, but it is also
complicated, requiring one to add a methodology to a methodology.30
Predicting spending from the estimated free-plan level using the induction
factors requires choosing a different induction factor than was used in estimating the
free-plan value, even though the analysis is of the same coinsurance amounts. A new
factor must be chosen because induction factors are not reversible and, in this case,
the analysis is of moving from no cost-sharing to substantial cost-sharing of 75%
rather than vice versa. The applicable induction factors from Table 4 are 0.47 and
0.74. For this example, the induction factor is calculated as 0.59 and is applied in Eq.
(13) as follows, with p =0%, Q =$2,623, and p =75%:
0
0
1
(34)
Q = Q (1+ I(p - p ))
1
0
0
1
= $2,623 (1 + 0.59(0%-75%))
= $1,462
This result is approximately 2.5% less than the $1,500 total spending that should
have been produced, in spite of the precise calculation of the induction factor. The
difference is due to induction factors’ lack of path neutrality and the issues
surrounding the calculation of the induction factor itself. An induction factor of
0.571 would have predicted quantity of $1,500.
The cubic formula produces the percentage of free-plan spending from the
coinsurance. Eq. (31) already predicted that a 75% coinsurance yields quantity that
is 61.4% of free-plan spending. Since this led to a free-plan value of $2,442 (shown
in Eq. (32)), the predicted spending in a 75% coinsurance plan would be calculated
as $2,442 * 61.4% = $1,500. This is not surprising considering that the $2,442 was
calculated as $1,500 / 61.4%, so $1,500 emerges by necessity when multiplying it by
61.4%. This demonstrates the cubic formula’s path neutrality.
Typical Plan Structure. Pure coinsurance plans are virtually nonexistent.
Today’s health insurance plans have not only coinsurance but copayments,
deductibles and out-of-pocket maximums. This example illustrates how the
30 There is an interesting byproduct of the price levels falling out of the arc-elasticity
formula when dealing with a free plan. The arc elasticities become path neutral, as
demonstrated in this example by the predicted spending of $1,500 emerging by identity.

CRS-32
estimated free-plan spending can be used to predict spending for the same person in
a plan with a deductible of $500 followed by coinsurance of 62.5%, with an out-of-
pocket maximum of $2,000. Although not a pure coinsurance plan, this plan would
also have 75% average coinsurance at $1,500 total spending. Thus, assuming
average coinsurance is an accurate predictor of total spending, one might expect the
methods to predict $1,500 of total spending from the free-plan level for this person.
In this instance,$1,500 may be the desired total spending under the new plan.
However, the only information that would be known in the CRS/Hay models is the
free-plan spending for this person. Because applying the elasticities to predict
spending requires an average coinsurance, the only information for calculating this
is the free-plan spending level of $2,572. In other words, calculating the average
coinsurance of a plan that does not have pure coinsurance requires using the given
spending level as the basis. In this example, the first step would be to calculate out-
of-pocket spending under the new plan given total spending of $2,572. This is the
same calculation that would be done for the induction-factor formula, as follows:
(35)
OOP = 500 + 62.5% (2,572-500) = 1,795
1*
The average coinsurance based on free-plan spending (Q ) would then be as
0
follows, requiring the introduction of a new variable (p ):
1*
(36)
p = OOP / Q = 1,795 / 2,572 = 69.8%
1*
1*
0
The actual average coinsurance will likely be different once quantity is adjusted
downward because of the cost-sharing. Nevertheless, in lieu of any other known
number, one must then decide which elasticity to use based on Table 2. Again,
because calculating the elasticity is critical when dealing with a free plan, a precise
elasticity is used, -0.25639, based on the average coinsurance. This produces total
spending of $1,523, approximately 1.5% higher than $1,500:
(37)
Q = Q (Ep - Ep - p - p ) / (Ep - Ep - p - p ), where p =0
1
0
0
1*
1*
0
1*
0
1*
0
0
= Q (1+E) / (1-E)
0
= $2,572 (1 - 0.25639) / (1 + 0.25639)
= $1,523
Using induction factors to predict spending is relatively straightforward using
Eq. (11), since it uses OOP , although its value differs from the one calculated in
1*
Eq. (35) because the free-plan spending estimated from the induction factors in Eq.
(29) is different than the free-plan spending based on the elasticities, as shown in Eq.
(28). The induction factor’s free-plan spending of $2,623 for this person produces
the following:
(38)
Q = Q + I (OOP - OOP )
1
0
0
1*
= $2,623 + I ($0 - $1,827)

CRS-33
The challenge is deciding the value of the induction factor, I. Again, the average
coinsurance will likely be different once quantity is adjusted downward because of
the cost-sharing, but in lieu of any other known number, one must use p to choose
1*
an induction factor based on Table 4. (The CRS/Hay models, and most other models
using induction factors for modeling purposes, use constant induction factors
regardless of cost-sharing level, already shown to be problematic.) The average
coinsurance (p ) based on the free-plan spending in the new plan would be
1*
approximately 69.6%. Based on where 69.6% falls between 50% and 95%
coinsurance, the induction factor is estimated at 0.6221. Plugging this value into Eq.
(38) then yields $1,486, which is 1% smaller than the $1,500 expected a priori.
Alternatively, predicting quantity by directly applying the 69.6% average
coinsurance in the induction-factor formula (based on Eq. (13)) based on free-plan
spending also yields $1,486:
(39)
Q = Q (1+ I(p - p )), where p = OOP /Q
1
0
0
1*
1*
1*
0
= $2,623 (1+0.6221(0%-69.6%))
= $1,486
This should not be surprising, since setting the first line in Eq. (38) equal to the
first line in Eq. (39) yields an identity:
(40)
Q + I (OOP - OOP ) = Q (1+ I(p - p )), where p = OOP /Q and p = OOP /Q
0
0
1*
0
0
1*
0
0
0
1*
1*
0
Q + I (OOP - OOP )= Q (1+ I (OOP /Q - OOP /Q ))
0
0
1*
0
0
0
1*
0
Q + I (OOP - OOP )= Q + I (OOP - OOP )
0
0
1*
0
0
1*
Thus, when using induction factors, the average coinsurance derived from
applying a plan structure to a given price-quantity combination yields the same
results as using the nominal out-of-pocket dollar amounts. One of the most common
arguments for using induction factors is that it enables one to handle complex plan
designs by virtue of the nominal out-of-pocket amounts. In actuality, its
methodology is no different than using the average coinsurance. As a result, the
limitations of the arc elasticity from calculating the average coinsurance in a typical
plan structure are the same limitations faced by the induction-factor formulas. This
point is typically obscured, however, because the induction factors rely on the
nominal dollar amounts.
As with arc elasticities and induction factors, the cubic formula also relies on
the average coinsurance of the new plan calculated from the spending levels of the
old plan (Q ). Applying the cubic formula’s average coinsurance of approximately
0
70.2%, based on its free-plan spending level, produces the following:
(41)
Q = -1.5546p3 + 2.8459p2 - 1.7743p + 1, where p=70.2%
= 61.9%
Applying this percentage-of-free-plan spending to the free-plan spending of
$2,442 (from Eq. (32)) yields $1,512, not quite 1% higher than $1,500.

CRS-34
Using the same approaches to predict spending in the original plan, which had
a $1,000 deductible and 25% coinsurance, based on the free-plan spending levels
shown in Eqs. (28-32) yields the values shown in Table 5. The target amount is
$1,500. Once again, the cubic formula produces results closest to the target amount.
To re-emphasize the point, the induction-formula results were calculated using the
first line of both Eq. (38) and Eq. (39). As expected, the results were identical.
Although induction factors are not reversible, this added complication was
previously justified by the factors’ capacity to predict spending based on a new plan’s
cost-sharing structure as applied to the original plan’s spending. The arc-elasticity
and cubic formulas work best when the new plan is only pure coinsurance. Any other
plan structure forces one to resort to calculating an average coinsurance based on the
original plan’s spending level, which introduces some error. However, it was shown
above that the induction factor’s methodology is no different than using the average
coinsurance calculated from the original plan’s spending level, introducing the same
kind of error. This seems to nullify the case for tolerating the induction factors’ lack
of reversibility and the concomitant complexity. Moreover, the induction-factor
methodology is the only one of the three that does not replicate a plan’s original
spending level when modeling a pure-coinsurance plan, as shown in Eq. (34). This
is because the induction-factor results are based on p even when p is known.
1*
1
Table 5. Predicted Spending of Example Person in Plan With
$1,000 Deductible and 25% Coinsurance, Based on Predicted
Free-Plan Spending, By Cost-Sharing Method
Arc elasticity Induction factor Cubic formula Target amount
Average
54.2%
53.6%
55.7%
coinsurance (p )
1*
Value of factor
-0.23554
0.7184
Not applicable
Predicted
$1,592
$1,613
$1,529
$1,500
spending (Q )
1
Difference from
6.1%
7.5%
1.9%
target amount
Source: Congressional Research Service (CRS) calculations.
Notes: Average coinsurance is based on applying the plan design to the predicted free-plan spending
for each factor, shown in Eqs. (28-32). The “value of factor” for the arc elasticity and induction factor
is calculated from the values in Table 2 and Table 4, based on the location of the average coinsurance
between 50% and 95% coinsurance. The cubic formula requires no factor. “Predicted spending” is
based on the calculated “value of factor.” If the given values from Table 2 and Table 4 were used
instead of the calculated values, the “difference from target amount” would be -5.6% or 7.4% for
elasticities and, for induction factors, 5.5% or 30.8%. All exceed the cubic formula’s difference.
The quantity predicted by the cubic formula was closer to the desired amount
than the quantity predicted by the other methods. This is even after taking extreme
care to calculate seemingly the best possible values for the induction factors and arc
elasticities. If such care is not taken (that is, if constant factors are used across cost-
sharing levels), then the cubic formula would certainly produce results more
consistent with the HIE. Not having to calculate values for a factor makes the cubic
formula preferable from a practical standpoint as well.

CRS-35
Predicting Spending by Type of Care
Most of the preceding discussion has focused on how the methodologies can be
used to predict total medical care spending. One might expect, however, that the
effect of cost-sharing will vary depending on the type of service — whether inpatient
or outpatient, for example. One of the most interesting HIE results is that spending
in the 25% plan averaged 71% of free-plan spending for both outpatient and inpatient
care, as shown in Table 1 and illustrated in Figure 12. Because of this concordance,
the corresponding arc elasticities (Table 2) and induction factors (Table 4) do not
vary by type of care between 0% and 25% coinsurance. Between 25% and 95%
coinsurance, spending does differ by type of care, as do the factors.
Figure 12. Effect of Average Coinsurance on Spending, by Type of
Service
100%
80%
ing
Spend
60%
lan
Total medical
P
Inpatient
ree
Outpatient
f F
Dental
40%
tage o
en
rc
Pe
20%
0%
0%
20%
40%
60%
80%
100%
Average Coinsurance
Source: Table 4.17, Joseph P. Newhouse et al., Free for All? Lessons from the RAND Health
Insurance Experiment (Cambridge, Massachusetts: Harvard University Press, 1993).
Note: “Total medical” excludes dental.

CRS-36
Constant Induction Factors. Although the CRS/Hay models do not vary
the induction factors by cost-sharing levels, the factors do vary by type of service:
0.3 for inpatient hospitalization, 1.0 for prescription drugs, and 0.7 for all other
care.31 As was illustrated in Figure 10, these values predict spending levels that do
not always line up well with the HIE. It is arguable that medical care has changed
so that the constant factors are more in line with current utilization patterns than the
HIE results. However, that is not the argument made. These constant factors “were
based largely on the RAND study.”32 It is difficult to determine how these constant
factors were obtained from the range of HIE-based induction factors in Table 4.
A decade ago, a team of actuaries reexamined these constant factors, since “the
management, delivery, and mix of health care services have changed dramatically
since the study was performed.” Some of the workgroup members thought the
factors should be higher, while others thought they should be lower. Ultimately they
decided to leave the constants unchanged but noted that “(o)ne set of factors is not
appropriate for all uses. The factors used should be carefully considered in the
context of the specific situation.”33 The induction factors currently used in the
CRS/Hay models still have these values.
Inpatient and Outpatient Care. For practical modeling purposes, one must
consider whether the differences in the HIE results by type of care would
substantially affect model results. If not, it is probably not worth varying the factors
by type of care. Again, between 0% and 25% coinsurance, there is no difference
between inpatient and outpatient care whatsoever, in terms of percent of free-plan
spending. The largest difference is at 95% coinsurance. Differences at the 95%
coinsurance are compared in this section.
From the example person above, a 95% pure-coinsurance plan predicted from
free-plan levels would produce total medical spending shown in the gray column of
numbers in Table 6. The table also shows what spending would be predicted if all
spending had been either outpatient or inpatient care, and varying the factors
accordingly. The table also shows the results for dental care.
Table 6. Example Person’s Predicted Spending at 95%
Coinsurance, by Type of Service and Factor
Factor
Outpatient
Inpatient
Total medical
Dental
Elasticity
$1,267
$1,543
$1,416
$1,296
Induction
$1,277
$1,576
$1,452
$1,302
Cubic formula
$1,196
$1,465
$1,343
$1,221
Source: Congressional Research Service (CRS) calculations.
Note: “Total medical” excludes dental.
31 These factors are used to calculate a weighted induction factor for each person in the
models, based on their health care utilization by type of care.
32 Hustead, et al., Medical Savings Accounts: Cost Implications, p. 4.
33 Ibid.

CRS-37
Although this section focuses on variation by type of service, the differences
in the total medical column merit some discussion, even if it is somewhat repetitive.
First, these amounts are based on the predicted free-plan values, which were based
on the total spending of $1,500 in the original plan with an average coinsurance of
75%. The three approaches produced different free-plan values. Applying the
factors to the estimated free-plan spending to a 95% coinsurance should yield total
medical spending that is 55.0% of the free-plan spending. This was so for the
elasticity and cubic formulas; for the induction formula, it was 55.4%. Thus, the bulk
of the difference among the factors was due to the difference in creating the free-plan
values. As mentioned in that section, the results from the arc elasticities and the
induction factors vary widely depending on the values chosen for the factors.
Moreover, as illustrated in Figure 11, the methods’ functions yield dissimilar results
between the HIE coinsurance levels, and which one is superior is not known.
The cubic formula does not require the calculation of factors when predicting
quantity. Because it does not have factors of its own, predicting spending by various
types of care requires estimating new formulas. For each type of care, a different
cubic formula must be estimated to be consistent with the HIE results. This is one
practical limitation compared to elasticities and induction, which use the same
formulas but different factors for different types of care. For inpatient care, the
formula is estimated as Q = -2.2764p3 + 3.7873p2 - 1.9645p + 1. For outpatient care,
it is Q = -0.8662p3 + 1.9296p2 - 1.5883p + 1. For dental care, it is Q = -0.782p3 +
1.3865p2 - 1.1377p + 1. For inpatient care in particular, the resulting curve is not
ideal. Between 41.4% and 69.6%, the slope of the curve is actually positive.34 While
the HIE results are not known between the four original data points, it is
counterintuitive that higher levels of cost-sharing would lead to higher inpatient
spending. In spite of its other advantages, the cubic formula is problematic for
producing spending levels by type of service, at least for inpatient care. The results
in Table 6 are not affected, however, because the coinsurance is 95%, one of the
original HIE data points, which is replicated by the cubic formulas.
If the example person’s spending were all inpatient, each of the three approaches
would predict total spending that is 22%-23% higher than if it were all outpatient.
Though large, this difference does not by itself merit accommodating type-of-service
factors. First, this difference occurs at the 95% coinsurance level, an unusual amount
of cost-sharing for a plan. When calculating average coinsurance on individual
records, many may be at this coinsurance level, or even 100% if they are still in the
plan’s deductible range. Of course, in the deductible range, dollar amounts are
relatively small. This leads to the second point that, in the aggregate, the differences
resulting from using separate factors by type of care may largely offset. For example,
the total medical spending amounts are based on the HIE data in which outpatient
care made up 46% of the total and inpatient care made up 54%. Applying those
percentages to the numbers for outpatient and inpatient in Table 6 yields $1,416 in
total medical for the elasticities, $1,439 for the induction factors, and $1,342 for the
cubic formulas. Only the calculated results using the induction factors vary from the
predicted total medical amount in Table 6 by more than a dollar. Thus, as long as
34 These coinsurance points were calculated by taking the derivative of the equation and
solving it for Q’(p)=0.

CRS-38
one is predicting total spending, rather than by type of care, it is arguable that the
total medical factors produce adequate results.
Example Using Actual Data. The preceding hypothesis was tested
informally by using person-level expenditure data from the 2002 Medical
Expenditure Panel Survey (MEPS) for those under age 65 with any health care
spending. These results are shown in Table 7.
Table 7. Average Predicted Spending, by Plan and Factor,
Based on 2002 MEPS
Plan
Cubic formula
Elasticity
Induction factor
Free plan (total medical values)
$2,186
$2,764
$2,161
95% coinsurance (total medical
$1,202
$1,520
$1,196
values)
95% coinsurance
$1,161
$1,473
$1,145
(inpatient/outpatient values)
95% coinsurance (traditional
Not applicable
Not applicable
$1,031
induction values)
Source: Congressional Research Service (CRS) calculations on 2002 Medical Expenditure Panel
Survey (MEPS).
Note: Among persons under age 65 with health care spending. The “traditional induction values” are
0.7 for outpatient care and 0.3 for inpatient care.
Using the total-medical cubic formula, free-plan values were calculated,
producing an average of $2,186. Based on those estimates, total medical spending
was predicted to average $1,202 in a 95% coinsurance plan, or 55% of the free-plan
average.35 Separately predicting outpatient and inpatient care, using the respective
cubic formulas, yields total plan spending of $1,161, a difference of 3.5% from the
total-medical cubic formula result. This result emerges by outpatient spending being
49% of the free-plan level and inpatient spending being 60% of the free-plan level,
replicating the result in Table 1. The total-medical result of $1,202 differs from
$1,161 to the extent that the ratio of outpatient-to-inpatient spending in MEPS
differed from that in the HIE.
These results will also vary depending on which services are included in each
respective category. For example, if prescription drugs were not modeled separately
but were lumped into outpatient care, which is a reasonable decision, the difference
in predicted total spending would be 5.7%, depending on whether type-of-service
formulas were used instead of the total medical formula.36
35 For this analysis, “total medical” consists of inpatient care (inpatient hospital stays,
including zero-night stays and separately billing doctor expenses) and outpatient care
(office-based, outpatient-hospital and emergency-room visits).
36 It is not clear whether or not the HIE outpatient results included prescription drugs. The
separate effect of cost-sharing on prescription drugs is discussed separately, which would
suggest they were not included in outpatient care. The next section of this report discusses
those results separately as well, so prescription drugs are left out of outpatient care here.

CRS-39
As was previously mentioned, when applying elasticities with a free plan, one
must decide whether to misuse the factors, calculating quasi-point elasticities, or
whether to use the constant elasticity values and thus predict constant spending
across a broad domain of cost-sharing. For this example (and for simplicity’s sake),
the constant elasticities in Table 2 will be used to predict free-plan values based on
the MEPS data.
To create the free-plan values, the three elasticities from the gray column of the
first three rows in Table 2 were used. All records in the MEPS data with an average
coinsurance between 0% and 25% had their total expenditures increased by 41% to
create the free-plan spending level.37 This was the case whether the person’s average
coinsurance was 0%, 25% or anything in between, which is a flaw in the arc elasticity
when dealing with free-plan information. For coinsurance above 25% to 50%, the
increase to the estimated free-plan spending was 60%. For coinsurance above 50%,
the free-plan adjustment was 82%. This led to average free-plan spending estimated
at $2,764, 26% higher than the free-plan level predicted by the cubic formula.
Beginning with the free-plan spending estimated using the elasticity formula,
flawed as it is, predicting total spending in the 95% coinsurance plan is less
problematic. Even though a constant adjustment emerges, this is acceptable since the
elasticity on which that adjustment is based relies on the same coinsurance levels (0%
and 95%) used to create the elasticities from the HIE. As with the cubic formula, the
elasticity in this case will predict 55% of the free-plan value using the total-medical
elasticity. The outpatient and inpatient elasticities will yield 49% and 60% of free-
plan spending for those services, respectively. Based on the free-plan estimates, the
total-medical elasticity for a 95% coinsurance plan predicts average spending of
$1,520. At the same coinsurance level, applying the outpatient and inpatient
elasticities to those types of care separately produces an average value of $1,473.
The arc elasticities’ estimate of the 95% coinsurance vary by 3% depending on
whether the type-of-service elasticities are used versus the total-medical elasticity.
Of course, these results still differ dramatically from the cubic formula’s largely
because of the flawed nature of the arc elasticity’s formula for creating free-plan
values. Predicting total spending from free-plan spending is also problematic except
when the coinsurance is at one of the HIE levels, which is the case here.
As with elasticities, continuous induction factors can be calculated over a range
of coinsurance levels in an attempt to improve precision, though its impact on
accuracy may be questionable. In applying induction factors to the MEPS data, the
factors given in Table 4 are used, turning to the next induction factor once the upper
coinsurance level of the pairs is exceeded. For example, to create free-plan levels
using the total-medical induction factors, 1.63 is used for coinsurance between 0%
and 25%. For coinsurance above 25% to 50%, 1.17 is used. For coinsurance above
50%, 0.86 is used. This led to average free-plan spending estimated at $2,161, about
1% lower than the free-plan level predicted by the cubic formula and substantially
lower than that predicted by the arc elasticity.
37 (1-E)/(1+E) = (1+0.17)/(1-0.17)=1.41.

CRS-40
Although the arc-elasticity and cubic formula replicate 55.0% of free-plan
spending at 95% coinsurance, the induction-factor results are slightly different for
total medical, at 55.4%. Thus, spending in the 95% coinsurance plan using the 0.47
induction factor averages $1,196. Calculating spending using the inpatient and
outpatient factors (0.42 and 0.54 respectively) instead yields an average of $1,145,
a difference of more than 4%. Had an inpatient induction factor of 0.3 and an
outpatient induction factor of 0.7 been used, spending would have averaged $1,031
in the 95% coinsurance plan, illustrating once again that the induction factors of 0.7
for outpatient care and 0.3 for inpatient care do not replicate HIE results consistently.
Based on the MEPS data, average spending in a 95% coinsurance plan varies
by 3% to 4%, depending on whether the total-medical or type-of-service formulas
and factors are used. However, these differences are also affected by what is
classified as outpatient versus inpatient care. It is important to note that this is the
coinsurance level where the difference by type of service would be greatest. Since
most plans would likely be at lower levels of coinsurance, where there is little or no
difference by inpatient versus outpatient care according to the HIE results, individual
analysts must decide whether such a type-of-service analysis is merited.
Prescription Drugs. Prescription drugs are a component of health care
spending that has received increasing attention from those who follow health
insurance issues. This is not surprising considering the growing proportion of health
care spending that it comprises. In 1980, around the time of the HIE, prescription
drugs made up 6% of health care spending. By 2003, that percentage had doubled,
to 12%.38 Not much attention was given to prescription drugs in the original HIE
results, but a reexamination of the available information is merited in light of
prescription drugs’ growing prominence as a feature of health care coverage. One
caveat of this analysis is that because of the changes over the past 25 years — that is,
the increasing number, variety, price and utilization of prescription drugs — the HIE
results regarding prescription drugs may be particularly out of date and inapplicable.
Considering the popular notion that demand for prescription drugs is most elastic
among health care goods and services, the section examines whether the HIE results
affirm that notion and, to the extent the difference is measurable, whether it should
be accounted for in microsimulation modeling.
Newhouse, et al., presented their finding on the effect of a plan’s cost-sharing
on prescription drugs in this way: “[O]ther than through its effect on [physician]
visits, plan did little to alter drug use; that is, plan did not much affect either the
physician’s tendency to prescribe for a patient in the office or the patient’s tendency
to fill the prescription. ... [A]lthough we saw evidence of medically inappropriate
overprescribing, the proportion of inappropriate prescribing did not vary much by
plan design. ... [C]ost-sharing reduced the use of both prescription and
38 Figure 5 of CRS Report RL31374, Health Expenditures in 2003, by Paulette C. Morgan
[http://www.congress.gov/erp/rl/pdf/RL31374.pdf]. The HIE plans covered prescription
drugs which at the time had “traditionally been poorly covered by health insurance plans.
... (A)bout 8 percent of total spending [in the HIE] was for drugs” (from Newhouse, et al.,
p. 365). This may have been higher than the average for that time because of the HIE
coverage.

CRS-41
nonprescription drugs; there was no evidence of substitution of over-the-counter
drugs for prescription drugs as cost-sharing increased” (p. 365).
These statements would suggest that the effect of cost-sharing on prescription
drug spending would be similar to that of total medical care, or outpatient care more
specifically. Other HIE results indicate that prescription drugs are quite different,
depending on one’s interpretation. To understand this, it is helpful to recall the
discussion of the effect of cost-sharing on total medical care in the section presenting
selected HIE results. Figure 4 showed the association between cost-sharing and
spending in the original HIE plans, which had out-of-pocket maximums. However,
because of the out-of-pocket maximums, the nominal coinsurance failed to capture
all of cost-sharing in the plan. A separate estimate was derived by the HIE authors
to determine the “pure price effects” associated with the HIE coinsurance levels,
shown as “Without out-of-pocket maximum” in Figure 4. The adequacy of these
numbers was affirmed by how they lined up with the average-coinsurance results,
shown in Figure 5, although the analysis was limited in that the average coinsurance
levels did not exceed 31%.
All three sets of estimates — based on nominal coinsurance, average
coinsurance, and pure coinsurance (or pure price effects) — are not available for
prescription drugs. For purposes of this report, the last would seem most important,
which is the piece not contained in the HIE results. The other two sets of estimates
for prescription drugs are shown in dashed lines in Figure 13 along with the original
total-medical lines from Figure 5. The lighter dashed line in shows the relation
between nominal plan coinsurance and prescription-drug spending. The darker
dashed line uses the same prescription-drug spending levels but is based on the
average plan coinsurance. The points labeled in the figure are only for the
prescription-drug lines. No amounts were given for pure coinsurance, or pure price
effects, for prescription drugs. Lacking this, the other two sets of estimates must be
used to judge whether the total-medical values are adequate estimators for
prescription drugs.
As shown in Figure 13, the nominal coinsurance for prescription drugs
corresponds with the average- and pure-coinsurance levels for total medical
spending. Prescription-drug spending based on the average coinsurance appears
much different than for total medical spending. This would seem to lead one to a
different conclusion than that of the HIE authors. The comparisons in the figure may
be suspect, however. For example, the average coinsurance for the dark, dashed line
in the figure is for all medical spending, not specifically for prescription drugs. This
is problematic because the average coinsurance specifically for prescription drugs
may be different, particularly if a disproportionate share of prescription-drug
spending took place below the out-of-pocket maximum, which is believable. If most
prescription drug spending took place below the out-of-pocket maximum, then the
nominal coinsurance would be closer to the actual average cost-sharing for
prescription drugs than would the plan’s total average coinsurance. In that case, the
lighter dashed line may be preferred. This would be consistent with the HIE authors’
conclusions. It would also make calculating the effects of cost-sharing simpler, since
the total-medical values could be used for all types of care, including prescription
drugs.

CRS-42
It is difficult to determine from the HIE what the pure price effects are on
prescription drug spending. Even if it were possible to determine, it would be
questionable as to its applicability today. In lieu of any more recent experimental
information, it is arguable that using total medical factors is no worse than any of the
other options, especially considering the potential challenges of appropriately
applying additional factors in microsimulation modeling.
Figure 13. Effect of Coinsurance on Annual Per-Person Total Medical and
Prescription Drug Expenses, as a Percentage of Free Plan Expenses
100%
(0%, 100%)
80%
(25%, 76%)
ing
(16%, 76%)
Spend
(95%, 57%)
n
60%
(24%, 60%)
(50%, 60%)
Pla
e
(31%, 57%)
Fre
Total medical, based on average
of
coinsurance
e
40%
g
a
Total medical, based on pure
coinsurance
cent
Per
Prescription drugs, based on
20%
nominal coinsurance
Prescription drugs, based on
average coinsurance
0%
0%
20%
40%
60%
80%
100%
Coinsurance
Source: Figure 5 and Congressional Research Service (CRS) calculations from Newhouse et al., Free
for All? Lessons from the RAND Health Insurance Experiment (Cambridge, Massachusetts: Harvard
University Press, 1993), particularly Table 5.13.

CRS-43
Conclusion
Arc elasticities and induction factors are used in health policy circles to replicate
results from the RAND Health Insurance Experiment (HIE). This report showed how
both methods have serious limitations, particularly for their practical applications in
microsimulation modeling.
Results based on arc elasticities are problematic when dealing with a plan, or
even a record in a dataset, in which there is no cost-sharing. (Results from point
elasticities are not even calculable from a plan with no cost-sharing, which is one
reason why point elasticities were quickly dispensed with in this report.) If such a
free plan is used in applying a particular arc elasticity, total spending will be adjusted
by a constant percentage, regardless of the cost-sharing in the non-free plan. A
workaround was presented, adding another level of complexity to the application of
arc elasticities, but its appropriateness is questionable.
Induction factors can handle such free-plan information. Unfortunately, their
values are not reversible. Between two price-quantity points, two induction factors
emerge rather than one. This complicates the appropriate application of induction
factors.
One purported advantage of induction factors is that they can better handle
complicated plan structures. This is because they are calculated and applied based
on the dollar amounts of cost-sharing that individuals face in a plan. In addition,
induction factors predict spending by applying the new cost-sharing structure to the
old total spending, which also makes its application easier. The analysis in this
report showed, however, that the induction factors essentially rely on the average
coinsurance calculated from applying the original total spending to the new cost-
sharing structure. Arc elasticities do so as well with concomitant limitations
acknowledged. That induction factors effectively do the same thing, with the same
concomitant limitations, is typically obscured because the induction factors rely on
the nominal dollar amounts. Thus, in actuality, induction factors have no inherent,
substantive advantage for handling complicated plan structures. Indeed, because of
their use of dollar cost-sharing based on the original total spending, the induction
factor is arguably inferior when dealing with pure-coinsurance plans, upon which the
HIE results in this report were based. This removes the primary rationale for
tolerating the induction factors’ lack of reversibility.
The CRS/Hay models create and use a baseline of free-plan values for thousands
of records/individuals. Although arc elasticities and induction factors can be used
for this purpose, their sensitivity to the specific factors’ values in such circumstances
is additional cause for concern. Special care must be taken to use the appropriate
values. Because the CRS/Hay models currently use induction factors, which are not
reversible, using the appropriate values is even more important and involved.
Presently, however, the models use constant values for the induction factors
regardless of coinsurance levels.39 Either new factors should be used that vary by
39 An adjustment has been built into the models based on the dollar amounts of cost-sharing,
(continued...)

CRS-44
coinsurance or a new method should be applied. Arc elasticities are not superior for
this purpose, given the method’s limitations and the models’ use of free-plan values.
In an effort to apply some method that would be particularly useful for
microsimulation modeling, the cubic formula was derived. Its legitimacy is based
on the notion that a person’s experience in a plan as captured by their average
coinsurance is appropriate for replicating HIE results. The cubic formula has many
of the desirable qualities that arc elasticities and induction factors sometimes lack.
The cubic formula is path neutral. Because it does not have separate factors that
must be calculated, the cubic formula has no issues regarding reversibility. The cubic
formula faces no diminution of predictive power when dealing with free-plan
information. In addition, from a practical standpoint, it is much simpler to
appropriately implement in microsimulation modeling.
The key flaw of the cubic formula is that if one wants to separately model the
impact of cost-sharing changes by type of service (for example, inpatient versus
outpatient), new cubic formulas must be derived, as was done in this report. At least
one of these cubic formulas produced quite undesirable results. The cubic formula
for inpatient care estimated that as coinsurance rises between 41.4% and 69.6%, there
would be higher inpatient spending. While there are no HIE results between its four
original data points, this result is counterintuitive.
The HIE results also showed that, for most cost-sharing levels, there is little or
no difference in total spending resulting from separate application of cost-sharing
factors for certain types of medical care. Even for prescription drugs, the type of
health care popularly believed to be most affected by cost-sharing changes, the HIE
does not provide evidence that people respond dramatically different to changes in
cost-sharing compared to other types of care. This being the case, it is arguable that
the cubic formula for total medical is adequate for predicting all kinds of cost-sharing
changes. Moreover, because predicted values can vary more from the cost-sharing
method used rather than the type-of-service variation, the choice of cost-sharing
method is arguably more important than dealing with type-of-service variations.
None of these methods — arc elasticities, induction factors, the cubic formula
— is perfect. Each has its advantages. Each has its flaws. Analysts face the
decision, given the advantages and flaws of each, of which is best for their purposes.
For the CRS/Hay models, which are presently built on a baseline of free-plan values
and use constant induction factors regardless of cost-sharing, the cubic formula
appears better able to replicate HIE results than the current approach.
39 (...continued)
but this should have little or no impact on replicating the HIE results (or if there were an
impact, it would not be for this purpose), which vary by coinsurance regardless of the
nominal dollar amounts of cost-sharing.