Download Measuring the Cost of Poor Choices among Risky

MEASURING THE COST OF POOR CHOICES AMONG RISKY PROSPECTS
Daniel McFadden 1
Schaeffer Center for Health Policy and Economics, USC
Department of Economics, UC Berkeley
December 10, 2013
INTRODUCTION
The economic arguments that competitive markets are Pareto-efficient, and that self-organized
markets are the standard to beat in applied mechanism design, assume that consumers are rational
in their expectations and preference maximization. Tests of revealed preference, and laboratory
and field experiments from behavioral economics, indicate that in some circumstances many
consumers are not perfectly rational, at least in a narrow neoclassical sense. A natural question to
ask is how much damage is done to individual welfare and to the efficiency of resource allocation
by imperfectly rational consumers.
Some economic evidence supports the idea that most harm from faulty decision-making is selfinflicted, and that consumers learn through market discipline to be self-protective. Studies in
financial markets suggest that “noise” traders may harm themselves, but do not degrade resource
allocation when rational counter-parties can through leverage offset and correct their positions.
Evidence from laboratory experiments is that “irrationality” often errs on the side of caution, where
refusals to trade provide satisficing status quo protection against losses. However, there appears
to have been little systematic study of the consequences of faulty consumer decision-making. This
paper takes a first step, asking how to quantify deviations from perfect rationality of consumer
behavior in choice among risky prospects in insurance markets, and how to relate these deviations
to losses in individual welfare. I concentrate on insurance markets for three reasons. First, these
are markets where expectations are important, and faulty perceptions can be critical. Second, the
primary purpose of insurance is to transfer income between states of nature, so that core utility
1
This research is supported by the Behavioral and Social Research program of the National Institute on Aging
(grants P01AG033559 and RC4AG039036), and by the Presidential Fund of USC. I am indebted to Florian Heiss,
Kevin Murphy, Kenneth Train, and Joachim Winter for useful comments, and to Bo Zhou for computational
assistance.
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should be dominated by expected net benefits, and idiosyncratic tastes for contract terms and
providers should be relatively unimportant. Third, consumer choice among uncertain prospects
seems to bring out glaring departures from rationality, distortions in expectations, hyperbolic
discounting, ambiguity and loss aversion, superstition, and procrastination; see Kahneman and
Tversky (1979) and Kahneman (2011).
FIGURE 1. The Elements of Decision-Making
How can choices among risky prospects break bad? Start from Figure 1, a schematic of
consumer decision-making. The “Chicago” model of rationality insists that perceptions be
consistent with objective statistical evidence and public facts, without influence from private
experience and memory, that the decision-maker have a well-defined, stable utility for outcomes
that is not influenced by affect and attitudes, context, or “animal spirits”, and that the process of
decision-making be utility-maximization. This model does not question tastes, “de gustibus non e
disputantum”, and does not recognize limits of reasoning capacity as a problem for decisionmaking.
As one moves toward behavioral views of decision-making, perceptions become
subjective, and may be statistically inconsistent (e.g., prospect theory) or excessively sensitive to
personal experience and memory (e.g., representativeness, prominence, and primacy/recency
effects). Utility may be influenced by affect, context, and “animal spirits”, making it stochastic,
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or even chaotic, and the idea that risky prospects are evaluated in terms of their expected utility
may break down. Finally, heuristics, analogies, satisficing, and other shortcuts may replace utility
maximization in the decision-making process.
There are three possible reactions to behavioral evidence against the “Chicago” model of
consumer behavior. The first is to discard the concept of utility as a factor in choice and a measure
of well-being, as in Danny Kahneman’s summation: “Economists have preferences, psychologists
have attitudes”. The second is to simply expand the definition of “rationality” to include and
excuse any observed choice behavior, the classical view summarized by Taussig (1912):
“An article can have no value unless it has utility. No one will give anything for an article
unless it yield him satisfaction. Doubtless people are sometimes foolish, and buy things, as
children do, to please a moment’s fancy; but at least they think at the moment that there is a
wish to be gratified. Doubtless, too, people often buy things which, though yielding pleasure
for the moment, or postponing pain, are in the end harmful. But here ... we must accept the
consumer as the final judge. The fact that he is willing to give up something in order to procure
an article proves once for all that for him it has utility – it fills a want.”
The third is to take the “benevolently paternalistic” view of Thaler and Sunstein (2008) that an
intelligent observer can ascribe utilities to individuals, detect behavior that is inconsistent with
reasonable self-interest (e.g., errant perceptions, erratic decision processes, and even expected
utility maximizing behavior with inappropriate utility functions), and determine when “nudges”
are justified to “buck up” decision-making and reduce the likelihood that people harm themselves
or degrade market efficiency.
There are difficulties with all three of these reactions. The first two effectively preclude pursuit
of a consumer-side parallel to the theory of industrial organization that would study whether
consumer conduct in markets is consistent with efficient allocation, and what market rules promote
healthy conduct. The third response is disturbing to people who fear Orwellian erosion of
consumer sovereignty. And it is true that while I think I could improve your choices, I am skeptical
that you could improve mine. However, the third response can be viewed more benignly as an
extension of social judgments that already establish limits on behavior, such as laws that mandate
primary education and forbid recreational drug use, or consumer protection legislation that
provides transparency and limits deception. The approach of this paper is in the spirit of ThalerSunstein, starting with an assumption that people have, or can be imputed to have, core utilities
that are sufficiently stable and context-free to provide a basis for welfare judgments, but also make
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decisions cluttered by errant perceptions and careless optimization that do not necessarily
maximize or reveal core preferences. Key questions are then what can be learned about core utility
from observed behavior, and what market interventions can improve core consumer welfare.
How can the quality of choices among risky prospects be evaluated? The man who buys a
lottery ticket feels he made a good choice if it wins, and a bad choice if it loses. However, this ex
post evaluation is neither a good predictor of future payoffs nor a reliable metric for judging
whether it was ex ante rational to buy the ticket. More useful is an answer to the question: “Using
statistically realistic expectations, how did expected utility from actual choices compare with
expected utility that would have been obtained from a benchmark decision-making rule that used
only information available at the time of choice?” If an answer can be phrased in terms of
consumer surplus foregone, or excess cost of obtaining the benchmark outcome, then performance
relative to the benchmark can be used as a measure of welfare loss, and used to guide the design
of managed markets. The most rigorous neoclassical benchmark decision rule is “expected utility
maximization with rational expectations”, but of more practical interest are heuristics and decision
aids promoted by market managers that “nudge” consumers toward “boundedly rational” decisionmaking.
SOME CONSUMER THEORY
Suppose a consumer faces a binary choice between self-insuring against a specific casualty
loss and buying an insurance contract that provides complete coverage in the event of the loss.
What are the implications of neoclassical consumer theory if individuals have core utilities, but
faulty perceptions? A first issue is entanglement of perceptions and tastes for risky prospects: Do
consumer’s choices among lottery tickets reflect risk preferences, or misperceptions of the
probabilities of payoffs? Prospect Theory, the Kahnemann and Tversky (1979) summary of
empirical regularities in choice experiments, notes that people are inconsistent in the editing
process that defines their “status quo”, the perception of small probabilities, and the manipulation
of the probabilities of events. Another element of their summary is representation of choice as a
maximand of “expected” value function using consumers’ errant probabilities and a valuation
function that has a “fourfold pattern of risk attitudes”, as depicted in Figure 2, with loss aversion,
risk aversion for modest gains, risk affinity for modest losses, and reversed risk attitudes for large
gains or large losses.
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Figure 2. The Valuation Function from Prospect Theory
Losses
Gains
However, a characterization that makes these effects entirely a result of errant perceptions has
consumers maximizing stable core utilities with mild risk aversion that are insensitive to the status
quo, combined with superstitious perceptions that conservatively underestimate the moderate
probabilities of small gains or losses from the status quo, with more underestimation for gains, but
overestimate or ignore very small probabilities of large gains or losses. This interpretation is a
best case for economic analysts of errant consumers, as the core utility foundations for the welfare
calculus remain intact, and the analyst’s task is reduced to clearing away the effects of faulty
perceptions and optimization errors to recover core preferences. However, a deeper question is
whether ambiguity in distinguishing perceptions and preferences makes it necessary to resort to
attribution of core preferences by a social planner, or recovery of core utility through experiments
that directly reveal perceptions in various contexts.
Proceeding with the assumption that core utility is meaningful, consider a consumer facing a
voluntary decision between insuring completely at a premium C against a casualty loss, or selfinsuring and facing an uncertain loss L. Let G denote the objective CDF for L. Suppose the
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consumer has a core CARA utility function of disposable income with a risk attitude parameter .
For a fully rational consumer, the utility of buying the insurance contract is [1 – e-(y-C)]/, where
y denotes income, and the expected utility of self-insuring is [1 – ∫ e−(y−L) 𝐺𝐺(𝑑𝑑𝑑𝑑)]/. Let m(t)
= log ∫ etL 𝐺𝐺(𝑑𝑑𝑑𝑑) denote the cumulant generating function of the CDF G, with an expansion m(t)
= t+ 2t2/2 + O(t3), where  is the mean and 2 is the variance of L. The fully rational consumer
buys the insurance contract if [1 – e-(y-C)]/ > [1 – e-(y-m()/)]/, or in other words if the premium
C is less than the certainty-equivalent actuarial value R  m()/ =  + 2/2 + O(2) of the
income loss when self-insured, C < R. Now suppose consumers have errant perceptions that selfinsuring results in a certainty-equivalent expected income loss Q, and buy insurance when C < Q.
The values (R,Q) are heterogeneous in the population; let H(R,Q) denote their CDF. The table
below gives the possible outcomes and consequences:
Condition
Choice
Optimal
Choice?
Penalty
Probability
1. Q ≤ C & R ≤ C
Self-insure
Yes
0
H(C,C)
2. Q > C & R ≤ C
Buy
No
R-C
H(C,) – H(C,C)
3. Q ≤ C & R > C
Self-insure
No
C-R
H(,C) – H(C,C)
4. Q > C & R > C
Buy
Yes
0
1 - H(C,) - H(,C) + H(C,C)
Rows 2 and 4 are conditions where the consumer buys insurance; the population share buying
insurance is 1 – H(,C). Rows 3 and 4 are conditions under which buying insurance is optimal;
the share optimally insured is 1 – H(C,). When consumers self-insure, rows 1 and 3, their ex ante
perception was a net certainty-equivalent income gain C – Q > 0. These consumers experience ex
post net income shocks C – L relative to buying insurance, but these measures of regret are
confounded by luck and provide unreliable evidence to correct perceptions. The mean of C – L
among self-insurers is C – E(R|Q>C), while they expect C – E(Q|Q>C). Then, E(Q-R|Q>C) is a
mean surprise to the self-insured, but one they will not generally learn since Q’s are not reliably
reported public information.
A benchmark for the value of insurance is the mean expected certainty-equivalent income loss
if no one insures,
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(1)


− ∫0 𝑅𝑅𝐻𝐻(𝑑𝑑𝑑𝑑, ) = − ∫0 �1 − 𝐻𝐻(𝑅𝑅, )�𝑑𝑑𝑑𝑑.
Rows 2 and 3 are conditions where faulty perceptions lead to sub-optimal choices. Consumers in
row 2 incur a mean certainty-equivalent income penalty

(2)
𝐶𝐶
∫𝑄𝑄=𝐶𝐶 ∫𝑅𝑅=0(𝑅𝑅−𝐶𝐶)𝐻𝐻(𝑑𝑑𝑑𝑑,𝑑𝑑𝑑𝑑)
H(C,) – H(C,C)
=
𝐶𝐶
∫𝑅𝑅=0(𝑅𝑅−𝐶𝐶)𝐻𝐻(𝑑𝑑𝑅𝑅,)−𝐻𝐻(𝑑𝑑𝑅𝑅,𝐶𝐶))
H(C,) – H(C,C)
=−
𝐶𝐶
∫𝑅𝑅=0(𝐻𝐻(𝑅𝑅,)−𝐻𝐻(𝑅𝑅,𝐶𝐶))𝑑𝑑𝑑𝑑
H(C,) – H(C,C)
.
Consumers in row 3 incur similarly a mean certainty-equivalent income penalty
(3)
𝐶𝐶

∫𝑄𝑄=0 ∫𝑅𝑅=𝐶𝐶(𝐶𝐶−𝑅𝑅)𝐻𝐻(𝑑𝑑𝑅𝑅,𝑑𝑑𝑑𝑑)
H(,C) – H(C,C)

=
∫𝑅𝑅=𝐶𝐶(𝐶𝐶−𝑅𝑅)𝐻𝐻(𝑑𝑑𝑑𝑑,𝐶𝐶)
H(,C) – H(C,C)

= −
∫𝑅𝑅=𝐶𝐶(𝐻𝐻(,𝐶𝐶)−𝐻𝐻(𝑅𝑅,𝐶𝐶))𝑑𝑑𝑅𝑅
H(,C) – H(C,C)
.
To illustrate the calculation, consider enrollment by seniors in 2008 in the Medicare Part D
drug insurance program.
This insurance exchange offers a standard contract at a heavily
subsidized, competitively determined premium. Let T denote a consumer’s total drug bill in 2008,
a random variable at the time of an enrollment decision at the time of open enrollment at the end
of 2007, and let T-1 denote the 2007 total drug bill, which we assume was known to the consumer
at the time an enrollment decision for 2008 was made. For this illustration, we ignore other
information available to consumers when they make their enrollment choices, such as age, gender,
and health conditions; this is empirically reasonable since the correlation of total drug bills across
years is about 0.8. A Part D standard plan has a premium C and a benefit schedule B(T) =
0.25min(2325,max(0,T-275)) + 0.95max(0,T-5726.75). 2 Make the assumption, consistent with
considerable evidence, that the consumer is essentially neutral to risks in drug needs, and assume
the limiting risk parameter  = 0. Then the certainty equivalents (R,Q) that enter the model above
are simply expected losses relative to the partial insurance provided by the Part D standard plan;
R = E(B(T )| T-1), and it is optimal to buy insurance when its actuarial value exceed the premium,
R > C. I estimate H(R,|T-1) and the density f(T-1) of T-1 from a sample of 991,930 seniors whose
enrollment in a Part D plan is “mandatory” and hence not selected by enrollment choice, the result
of inclusion in a retiree or union health plan, or in Medicaid or Medicare Advantage coverage. I
2
This calculation ignores the presence of actuarially equivalent and enhanced alternatives to the standard plan that
are offered at an unsubsidized additional premium, and ignores premium variations by region and low-income
subsidies.
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estimate H(R|Q>C,T-1) and the density f(T-1|Q>C) from a sample of 1,926,035 seniors who
voluntarily enrolled in a stand-alone Part D drug plan in 2008. The population share 1 - H(,C)
of voluntary decision-makers who enroll is a number that can be inferred from surveys or from
Medicare market statistics. Then, 1 – H(,C|T-1) = (1 – H(,C))f(T-1|Q>C)/f(T-1) is the share of
voluntary decision-makers who enroll, given T-1. Cautions about the realism of this example are
that it does not take into account penalties for late enrollment that make enrollment attractive even
if it is first-year actuarially unfair, it uses a calibration step is used to reconcile the somewhat
different distributions of health status and drug needs between the populations of mandatory and
volunteer enrollees, and it calibrates the average premium to match the statistic that 76 percent of
seniors with an voluntary choice buy Part D insurance. The model then establishes that it is optimal
for 95.6 percent of consumers to buy insurance, 1.8 percent optimally self-insure, 2.6 percent buy
insurance when it is a poor choice, 22.2 percent self-insure when this is a poor choice, and 73.4
percent buy insurance and it is optimal. The mean penalty per capita for the small share of people
making a poor choice to buy insurance is $50, while the mean penalty for the relatively large share
of people making a poor choice to self-insure is $172, or about 13.7 percent of their mean total
drug bill. Thus, poor choices in this example lead to an economically significant loss in demand
for insurance, and an economically significant loss in individual welfare.
BENCHMARK ASSESSMENT OF EX ANTE EXCESS COST OF POOR INSURANCE
PLAN CHOICES
In this section, I describe a benchmark method for ex ante assessment of the quality of
insurance contract choices, with an application to plan choices of volunteer enrollees in the
Medicare Part D drug insurance exchange.
The application is drawn from Heiss, Leive,
McFadden, and Winter (2013). Plan choices are made during an open-enrollment period at the
end of each year for the contract that will be in place during the coming year, before actual health
conditions and drug needs in the coming year are realized. Useful evaluations of the quality of
plan choice have to be correspondingly ex ante, based on expected utility in the year ahead, given
information available to the decision-maker at the time of choice.
In overview, we assess whether people could have achieved higher expected utility than they
did if instead of their actual decision rule, they had used a benchmark rule that required only
information they had available at the time of their actual plan choice. A leading benchmark, and
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the focus of our empirical analysis, is the recommendation of the Medicare’s “Plan Finder”, an
internet tool that ranks the cost of plans in the coming year when given a person’s medicine cabinet
for the current year. This decision rule is available to any senior who can access the Medicare
website, personally or through a helper. Other possible benchmark rules are naive heuristics such
as “choose a minimum premium plan”, and the fully rational decision rule that maximizes
statistically realistic expected utility conditioned on information available at the time of choice.
At the time of plan choice in an open enrollment period at the end of year t-1 consumers have
a vector of information, denoted Xt-1, that includes their age, gender, health conditions, a list of
their current drugs and doses which we term their medicine cabinet (MC), their pharmacy total
drug bill for the year (T), and if they have one, their current Part D plan and its premium (C) and
realized out-of-pocket (OOP) cost.
Each consumer is assumed to have sufficient information,
from the Medicare Plan Finder or otherwise, to determine the formulary and benefit design
mapping for each available Plan k in year t that gives the OOP for any given medicine cabinet MC;
we denote this mapping OOP = FBD(MC,k,t). We as analysts can also construct FBD from plan
attributes and drug prices. Given Xt-1, there is a statistically realistic conditional density fR(MC |
Xt-1) of year t medicine cabinets.
Assume that consumers have CARA utility functions Ukt = (1 – exp(-α(y – Ckt – OOPkt)))/α,
where y is discretionary income, Ckt is the (known) insurance premium Ckt for Plan k in year t, and
OOPkt = FBD(MC,k,t)) is the (uncertain) cost out-of-pocket cost that will be realized in year t if
this plan is chosen. Rational expected utility is then
(4)
EUkt =  fR(MC | k,Xt-1) (1 – exp(-α(y – Ckt – FBD(MC,k,t)))) /α
= (1 – exp(-α(y – Ckt – mR(α | k,Xt-1)))/α,
where mR(z | k,Xt-1) = log ∑ exp(z FBD(MC,k,t)) fR(MC | k,Xt-1) is the conditional cumulant
generating function for plan k. When α is small, mR(α | k,Xt-1)/α ≈ R(k,Xt-1), the conditional mean
OOP cost for plan k in year t. Then expected utility is a decreasing function of Ckt + R(k,Xt-1).
Let kA(Xt-1) denote the consumer’s actual plan choice, and IAt(Xt-1) = 𝐶𝐶𝑘𝑘𝐴𝐴(𝑋𝑋𝑡𝑡−1 ),𝑡𝑡 +
𝑅𝑅 (k A (Xt−1 ) , 𝑋𝑋𝑡𝑡−1 ) denote the associated expected cost. Let kB(Xt-1) denote a benchmark rule
plan choice, and IBt(Xt-1) = 𝐶𝐶𝑘𝑘𝐵𝐵 (𝑋𝑋𝑡𝑡−1 ),𝑡𝑡 + 𝑅𝑅 (k B (Xt−1 ) , 𝑋𝑋𝑡𝑡−1 ) denote its associated expected cost.
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We judge ex ante quality of actual choice compared to a benchmark rule by comparing IAt(Xt-1)
and IBt(Xt-1). As noted earlier, we assume  is near zero and higher moments can be ignored. 3 We
use two criteria, the excess expected cost IAt(Xt-1) - IBt(Xt-1), and an indicator for whether the
consumer’s actual choice has an expected cost greater than that of the benchmark rule, IAt(Xt-1) >
IBt(Xt-1). Making these assessments for an individual requires evaluation of the realistic ex ante
conditional expected OOP cost R(k,Xt-1), a difficult inference problem. However, this individual’s
ex post realized OOP cost in plan k , OOPtk determined by her realized medicine cabinet, by
definition satisfies E(OOPtk | k,Xt-1) = R(k,Xt-1). Then, averaging over the population, we have
(5) Share with overspending = 𝑬𝑬𝑋𝑋 𝑡𝑡−1 1(IAt(Xt-1) > IBt(Xt-1)) = E 1(CAt + OOPAt > CBt + OOPBt),
(6) Average overspending = 𝑬𝑬𝑋𝑋 𝑡𝑡−1 ( IAt(Xt-1) – a IBt(Xt-1)) = E (CAt + OOPAt - CBt - OOPBt).
The final equalities are simply ex post unconditional expectations over the population that are
estimated by their empirical equivalents, year t averages over a representative sample of voluntary
decision-makers. The distinction between ex ante and ex post performance metrics comes from
whether or not the benchmark is feasible, since for a given benchmark the estimate of expected
excess spending in the ex ante calculation is identical to the average of regret in an ex post
calculation. If the benchmark is feasible, using only information the consumer has and requiring
only practical calculation, then it provides a useful ex ante standard against which to judge actual
choice behavior. On the other hand, if the benchmark is infeasible, for example the perfect
foresight rule that requires the consumer to know exactly what drugs needs will be in the coming
year, it is potentially quite misleading, as even a consumer who makes a fully rational expectedutility maximizing choice can have large wistful regret relative to the perfect foresight benchmark
when uncertainty is high. However, differences in average ex post excess spending, relative to the
perfect foresight benchmark, between actual choice and a feasible benchmark choice will coincide
3
Heiss et all (2013) estimate a discrete choice model for plan choice as a function of premium, expected OOP cost,
and expected OOP variance, and find that premium and expected OOP cost are weighted similarly, with somewhat
more weight on expected OOP cost, but that the coefficient of OOP variance is volatile and often inconsistent with
risk aversion. The leading interpretation is that enrollees are unable to assess variance, and show some preference
for higher variance plans because these are associated with desirable plan features such as elimination of initial
deductions. Another possibility is that Part D plans fall in a region where underestimates of probabilities of modest
losses lead to apparent risk affinity, as in prospect theory.
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with ex ante average excess spending. (This result depends on linearity, and does not hold for
calculations of shares with excess spending.)
We carry out the calculations described above on a sample of 1,138,105 voluntary enrollees in
Part D in 2008 whose plan choices are not circumscribed by qualification into low-income support
programs, restricted by employer or union based health plans, or by enrollment in Medicare
Advantage plans; details of the sample are given in Heiss et al (2013). Construction of the FBD
mapping and determination of OOPkt for each enrollee in each available plan was a major
undertaking described in Hess et al (2013). To avoid selection effects, we use our calculated value
for OOP even when it is observed in the plan actually chosen. When FBD incorporates therapeutic
substitution, replacing drugs in a medicine cabinet by the least costly therapeutic equivalent in the
formulary of a plan under examination, we find that 80.3 percent of actual plan choices in 2008
had higher cost than the Plan Finder benchmark rule, and that the average excess cost was $314,
25.8 percent of the mean OOP cost $1214 of voluntary choosers. We conclude that poor plan
choice is economically significant and results in substantially higher excess drug costs relative to
the Plan Finder decision rule. We also considered other benchmarks, the naïve “minimize
premium” rule, and our version of a “rational expectations” rule that first estimated R(k,Xt-1) by a
“method of sieves”, and then chose a plan that minimized Ckt + R(k,Xt-1). We found that 61.6
percent of actual choices are worse than the minimize premium rule, with higher mean excess drug
cost of $121. We also found that our version of a rational expectations rule performed only slightly
better than Plan Finder: 80.6 percent of actual choices did worse than the rational expectations
rule, and the mean excess cost relative to this rule was $349.
To summarize, even simple decision rules such as “choose the lowest premium plan” or
“choose the plan suggested by Plan Finder” would have generated considerably lower ex ante
expected spending than the actual plan choices in the Part D market. More sophisticated rules,
such as our version of a rational decision rule that conditions on a small set of observable end-ofprior-year health and drug use characteristics, would have brought moderately larger savings. It
is hard to reconcile these monetary losses with the implicit decision costs associated with using a
tool such as Plan Finder, or with true idiosyncratic tastes for specific insurers and benefit designs.
In this application, there seems to be a strong case for nudging consumers toward lower cost plans,
both for their own benefit, and to sharpen their choices so that poor plans are pressured to reduce
their premiums or exit the market. Heiss et al (2013) suggest, for example, that Medicare take a
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lesson from Geico’s popular advertisement for its car insurance, and tell consumers that “15
minutes with Plan Finder could save you 15 percent or more on your prescription drugs.”
.
References
Heiss, F.; Leive, A.; McFadden, D.; Winter, J. (2013) “Plan Selection from Part D: Evidence from administrative
data,” Journal of Health Economics, 32, 1325–1344.
Kahneman, Daniel. Thinking, Fast and Slow. New York: Farrar, Strauss and Giroux, 2011.
Kahneman, Daniel, and Amos Tversky. "Prospect Theory: An Analysis of Decision Under Risk". Econometrica.
XLVII (1979): 263-291.
McFadden, D. (1999) “Computing Willingness to Pay in Random Utility Models,” in J. Moore, R. Riezman, and J.
Melvin (eds.), Trade, Theory and Econometrics: Essays in Honour of John S. Chipman, Routledge.
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