Censored Quantile Instrumental Variable Estimates

of the Price Elasticity of Expenditure on Medical Care∗

Amanda Kowalski, Yale and NBER†

November 27, 2013

Abstract

Efforts to control medical care costs depend critically on how individuals respond to prices. I estimate

the price elasticity of expenditure on medical care using a new censored quantile instrumental variable

(CQIV) estimator. CQIV allows estimates to vary across the conditional expenditure distribution, relaxes

traditional censored model assumptions, and addresses endogeneity with an instrumental variable. My

instrumental variable strategy uses a family member’s injury to induce variation in an individual’s own

price. Across the conditional deciles of the expenditure distribution, I find elasticities that vary from -0.76

to -1.49, which are an order of magnitude larger than previous estimates.

Keywords: censored, quantile, instrumental variable, medical care.

∗I am grateful to Amy Finkelstein, Jonathan Gruber, and Jerry Hausman for their guidance. I would like to thank the fol-lowing individuals for thoughtful comments: Joseph Altonji, Michael Anderson, Joshua Angrist, David Autor, John Beshears,Soren Blomquist, Tonja Bowen-Bishop, David Card, Amitabh Chandra, Victor Chernozhukov, David Cutler, Ian Dew-Becker,Deepa Dhume, Peter Diamond, Joseph Doyle, Esther Duflo, Jesse Edgerton, Matthew Eichner, Ivan Fernandez-Val, BrighamFrandsen, John Friedman, Michael Greenstone, Raymond Guiteras, Matthew Harding, Naomi Hausman, Panle Jia, Lisa Kahn,Jonathan Kolstad, Fabian Lange, Blaise Melly, Derek Neal, Whitney Newey, Joseph Newhouse, Douglas Norton, Edward Norton,Christopher Nosko, Matthew Notowidigdo, James Poterba, David Powell, Andrew Samwick, David Seif, Hui Shan, Mark Showal-ter, Erin Strumpf, Heidi Williams, and several anonymous referees who gave very constructive suggestions. Seminar participantsat Chicago Booth, Brookings, Cornell, Duke, Kellogg, Maryland, Michigan, MIT, NBER Summer Institute, Notre Dame, Rand,Stanford SIEPR, Temple Fox, Texas A&M, Wharton, Wisconsin, Yale, Yale SOM, and the Yale School of Public Health providedhelpful feedback. Dr. Kavita Patel provided a helpful clinical perspective. I thank Toby Chaiken for excellent research assis-tance. Mohan Ramanujan and Jean Roth provided invaluable help with the data. I gratefully acknowledge National Instituteon Aging, Grant Number T32-AG00186. Stata software to implement the quantile estimators used in the paper is available athttp://www.econ.yale.edu/∼ak669/research.html.†Department of Economics, Yale University, Box 208264. New Haven, CT 06520-8264. Office: 202-670-7631.

amanda.kowalski@yale.edu.

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1 Introduction

Following the recent passage of national health reform legislation in 2010, which focuses on expanding health

insurance coverage, controlling costs incurred by the insured is an increasingly important public policy is-

sue. Existing efforts to control costs depend critically on how insured individuals respond to the prices that

they face for medical care. The Medicare Modernization Act of 2003 included provisions to encourage price

responsiveness by establishing tax-advantaged health savings accounts as an incentive for individuals who

enroll in high deductible health insurance plans. The national reform also encourages price responsiveness by

establishing plans with substantial patient cost sharing to be offered in health insurance exchanges. Relative

to traditional plans, plans with high deductibles and other forms of cost sharing require consumers to face

a higher marginal price for each dollar of care that they receive. However, the effects of consumer prices on

medical care utilization are not well understood in the current environment.

Researchers began studying the price elasticity of expenditure on medical care decades ago, but three

limitations persist: a lack of estimates that allow the price elasticity to vary across the conditional distribution

of expenditure, a difficulty in handling censoring of expenditures at zero, and a need for identification strategies

to overcome the insurance-induced endogenous relationship between expenditure and price. Estimates based

on the Rand Health Insurance Experiment of the 1970’s, still widely considered to be the standard in the

literature, address the identification issue by randomizing consumers into health insurance plans with varying

generosities. Although the Rand estimates address censoring using traditional methods, there is a large and

enduring controversy over the appropriateness of the parametric assumptions that these methods require. (See

Newhouse et al. (1980), Duan et al. (1983), Mullahy (1998), and Buntin and Zaslavsky (2004).) Perhaps

even more important than censoring are the issues that arise because medical spending is so skewed. I extend

the literature by allowing for heterogeneity in the price elasticity of expenditure across the distribution of

expenditure conditional on covariates.1 In my estimation sample, drawn from a large recent data set of

employer-sponsored health insurance claims, just 25% of individuals account for 93% of expenditures, and

much variation remains after controlling for observable individual characteristics. It seems reasonable, then,

that individuals with drastically different levels of expenditure conditional on observable characteristics could

respond differently to price changes.

In this paper, I produce new estimates of the price elasticity of expenditure on medical care that ad-

dress heterogeneity across the conditional expenditure distribution, censoring, and identification. I use a new

censored quantile instrumental variable (CQIV) estimator, developed specifically for this application by Cher-

nozhukov, Fernandez-Val, and Kowalski (2009). The CQIV estimator is particularly well-suited to address the

limitations of the literature. First, the CQIV estimator allows me to obtain estimates of the price elasticity of

expenditure on medical care that vary across the conditional expenditure distribution. Unlike estimators of

the conditional mean, which can be very sensitive to values in the tail of the distribution, conditional quantile

estimators are inherently more robust to extreme values, which is particularly advantageous given the skewness

in the distribution of medical expenditures.

Second, the CQIV estimator allows me to handle censoring at zero without any distributional assumptions.

My data are superior to traditional claims data used in previous studies because they allow me to observe

enrolled individuals even if they consume zero care. In my estimation sample, approximately 40% of individuals

consume zero medical care each year, making censoring an important econometric issue. In the literature,

there is controversy over whether to model zero medical expenditures as censored at zero or as “true zeros”

that arise through a separate decision process. Although these models generate different mean estimators, the

1Gilleskie and Mroz (2004) offer a different approach for estimating the effects of covariates on health expenditures.

2

CQIV estimator is the same regardless of the model, so I refer to expenditures at zero as “censored” without

loss of generality. In contrast, traditional censored mean estimators such as the two-part model, which models

the decision to consume any care in one part and the decision of how much care to consume conditional

on consuming any care in another part, imply a parametric relationship between the two parts that is not

straightforward. In contrast, the CQIV estimator handles censoring nonparametrically in the tradition of

Powell (1986).

Third, the CQIV estimator allows me to address endogeneity with an instrumental variable identification

strategy. In traditional health insurance policies, the price of an additional dollar of care is a function of

expenditure. Thus, observed relationships between price and expenditure will be biased unless this endogeneity

is taken into account. The intuition behind my instrumental variable identification strategy is that because of

the cost-sharing provisions that govern family health insurance policies, some individuals face lower prices for

their own medical care when a family member gets injured. As formalized below, the maintained assumption

is that one family member’s injury can only affect another family member’s expenditure through its effect on

his marginal price. Although this assumption cannot be tested directly, I take several steps to increase its

plausibility. In particular, I model a nuanced interaction of cost sharing provisions among family members,

extending the identification strategy used by Eichner (1997, 1998).2 I also focus on injury categories that

appear to be exogenous because employees that have them in their families do not spend more on their own

medical care before the injuries occur.

My main results show that the price elasticity of expenditure on medical care varies from -0.76 to -1.49

across the conditional deciles of the expenditure distribution. Although the CQIV estimator allows the elas-

ticity estimates to vary across the predicted expenditure distribution, the estimates are relatively stable across

the conditional deciles, indicating that price responsiveness changes very little with predicted spending.

My estimates are an order of magnitude larger than the Rand estimate of the conditional mean elasticity

of -0.22 (Newhouse et al. (1983)). Conditional quantile estimates are not directly comparable to conditional

mean estimates without several assumptions, but I consider three types of evidence that suggest that the price

elasticity of expenditure on medical care is larger than previous estimates would suggest. First, I examine

robustness to the choice of estimator. The underlying variation that I use for identification is so pronounced

that I can illustrate it in simple figures. Furthermore, estimates based on traditional estimators in my data

are also much larger than those in the literature. Next, I perform tests of my identification strategy, and they

support its validity. They suggest that much of the price elasticity that I estimate occurs on the outpatient

visit margin. Within-year responses to family injuries drive my results. I also test the robustness of my

empirical specification, but in each setting, the variation in the estimates is small relative to the magnitude

of the main estimates. Third, I compare the assumptions required here to the assumptions required by the

Rand study, and I find differences that could be empirically large.

In the next section, I provide background information on the cost sharing provisions of traditional health

insurance plans, formalize my identification strategy, and provide a simple model. In Section 3, I describe the

data. In Sections 4 and 5, I present the empirical model and results based on CQIV and other estimators. In

Section 6, I examine potential mechanisms for my results, and I perform several tests of robustness. I compare

my results to the results from the Rand Health Insurance Experiment in Section 7. I conclude and discuss

directions for future research in Section 8.

2Relative to Eichner (1997, 1998), in addition to modeling a more nuanced interaction of family cost sharing provisions, I usenew data that allows me to observe individuals who have zero medical expenditures and a new CQIV estimator. I also developand implement several tests of robustness.

3

2 Background

2.1 Marginal Pricing for Medical Care

Traditional employer-sponsored health insurance plans have three major cost sharing parameters: a deductible,

a coinsurance rate, and a stoploss. The “deductible” is the amount that the consumer must pay before the

insurer makes any payments. Before reaching the deductible, the consumer pays one dollar for one dollar of

care, so the marginal price is one. After meeting the deductible, the insurer pays a fractional amount for

each dollar of care, and the consumer pays the rest. The marginal price that the consumer pays is known as

the “coinsurance rate.” After the consumer has paid the deductible and a fixed amount in coinsurance, the

consumer reaches the “stoploss,” and the insurer pays all expenses. For consumers that have met the stoploss,

the marginal price is zero.

For an individual that is not insured as a member of a family, the top subfigure of Figure 1 depicts how

the deductible, coinsurance rate, and stoploss induces a nonlinear budget set for the individual, who faces a

tradeoff between total beneficiary plus insurer spending on medical care, Y , and spending on all other goods,

A. The individual faces three distinct marginal prices, the slopes of each segment.

If a consumer is insured as a member of a family, the general cost sharing structure is the same, but an

additional family-level deductible and stoploss enable one family member’s spending to affect another family

member’s marginal price. For example, in my empirical application, one plan has an individual deductible of

$500, and it also has a family deductible that is $1,500, three times the individual deductible. Each family

member must meet the individual deductible unless total family spending toward individual deductibles exceeds

the family deductible. Since the family deductible is three times the individual deductible, if a family has fewer

than four members, all family members must meet the individual deductible. In that case, only subfigure I

of Figure 1 applies, and it applies separately for each family member. In a family with exactly four members,

when the first, second, and third family members go to the doctor, they each face the individual deductible

of $500, as if they were insured as individuals. However, when the fourth family member goes to the doctor,

if the family deductible of $1,500 has been met through the fulfillment of three individual $500 deductibles,

he makes his first payment at the coinsurance rate, as shown in subfigure II of Figure 1. In families with

more than four members, the family deductible is fixed at $1,500, and it can be met by any combination of

payments toward individual $500 deductibles.3 A similar interaction occurs at the level of the stoploss such

that an individual whose family has met the family stoploss faces the budget set shown by subfigure III of

Figure 1. Given the family-level cost sharing parameters, some individuals will face lower marginal prices than

their own medical spending would dictate.4

The marginal price variation induced by the family cost sharing parameters suggests a simple way to study

price responsiveness: compare expenditures of individuals whose families have and have not met the family

deductible. The flaw with this simple identification strategy is that individuals in families that have met

the family deductible may be more likely to consume medical care for reasons unrelated to its price, such

as contagious illnesses or hereditary diseases. For this reason, instead of comparing individuals according

to whether their family members have met the family deductible, I compare individuals according to an

instrumental variable.

3In the case of more than four family members, intermediate cases between subfigures I and II are possible, in which thebudget sets are shifted to the left because the individual has some spending at the given rate before the family deductible orstoploss is met. For simplicity of exposition, I do not show all of those cases here, and I do not discuss them when I present themodel in Section 2.3. However, the empirical implementation is completely general.

4Eichner (1997,1998) models a simpler family deductible structure in which people in family plans simply have a single familydeductible, but that structure is not appropriate in the plans that I study.

4

Figure 1: Potential Budget Sets for Individuals

price = Coinsurance

price = 0

TotalBeneficiary + InsurerPayments (Y)

price = 1

Deductible [(Stoploss - Deductible)/Coinsurance] + Deductible

Spending on All Other Goods (A)

Income -Premium

Income -Premium -Deductible

Income -Premium -

Stoploss

price = Coinsurance

price = 0

TotalBeneficiary + InsurerPayments (Y)[(Stoploss - Deductible)/Coinsurance]

II. Family Deductible METFamily Stoploss UNMETIncome -

Premium

Income -Premium -Stoploss + Deductible

price = 0

TotalBeneficiary + InsurerPayments (Y)

Spending on All Other Goods (A)

Income -Premium

Spending on All Other Goods (A)

I. Family Deductible UNMETFamily Stoploss UNMET

III. Family Deductible METFamily Stoploss MET

Y1 Y2 Y3 Y4

2.2 Identification Strategy

The fundamental question that I attempt to answer in this paper is: How does an individual’s year-end medical

expenditure respond to his year-end marginal price for care? As current health savings account policy aims to

reduce expenditure in a given year, year-end expenditure is a policy relevant outcome. To identify the effect

of marginal price on an individual’s year-end medical expenditure, I use an instrumental variable – whether a

family member has an injury. The first stage effect of a family member’s injury on the individual’s marginal

price is possible in families of four or more because of the family deductible and family stoploss structure

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described above. When one family member receives expenditure-inducing treatment for an injury, the family

is more likely to meet the family deductible than it otherwise would have been, and any individual in the

family is more likely to face a lower marginal price than his own spending would dictate. Empirically, I find

that one family member’s injury does indeed affect another family member’s marginal price.

Given the first stage, the key to the identification strategy is an exclusion restriction: one family member’s

injury cannot affect another family member’s medical spending outside of its effect on his marginal price. I

aim to rule out mechanical violations of the exclusion restriction in the way that I specify my outcome and

estimation sample. I specify the outcome that I study as the medical spending of an individual in a family, and

not the medical spending of the entire family so that expenditure for the treatment of one family member’s

injury is not included in the outcome variable.5 Furthermore, since one family member’s injury does have a

direct effect on his own medical expenditure, and the injury itself likely influences his decision to consume

follow-up medical care and care for secondary illnesses, I use injured family members only to construct the

instrument, and I do not include them in the estimation sample. If two or more family members are injured,

all injured family members are excluded from the estimation sample. Family injuries have limited persistence

across years, so sample selection issues due to the exclusion of injured parties should not be a cause for concern.

Beyond mechanical violations, other potential violations of the exclusion restriction involve indirect effects

of one family member’s injury on another family member’s medical spending that occur through a mechanism

other than the marginal price. It is possible to think of several mechanisms through which the exclusion

restriction could be violated, but as in all instrumental variable applications, the exclusion restriction is a

maintained assumption that is fundamentally untestable. However, to increase the plausibility of the exclusion

restriction, beyond choosing diagnoses that seem exogenous to a doctor, I use a data-driven method to select

categories of injuries for which the exclusion restriction appears to be supported. My approach is motivated by

that of Card, Dobkin, and Maestas (2009), who select a subset of diagnoses that appear to have similar rates

of appearance in emergency rooms on weekdays and weekends. Using only this subset of diagnosis categories,

they measure whether the start of Medicare eligibility at age 65 results in lower mortality.

In my approach, within the class of all diagnosis categories identified as “injuries” through the International

Classification of Diseases (ICD-9), I select a set of injury categories for which employees in families with injuries

do not appear to spend more than employees in similar families without injuries in the part of the year before

the injury occurred. I estimate my main specification using only those categories of injuries in the specification

of the instrument. The complete set of injury categories that I include are: fractures; injuries of the thorax,

abdomen, and pelvis; injuries to blood vessels; late effect of injuries, poisonings, toxic effects, and other

external injuries; foreign body injuries; burns; injuries to the nerves and spinal cord; poisoning by drugs,

medicinal and biological substances; and complications of surgical and medical care, not elsewhere classified.

As I discuss below, approximately 11% of employees in my sample have a family member with an injury of

one of these types in a given year. Ideally, these injury categories should be severe and unexpected enough

that treatment for an injury in these categories should not be related to an underlying family-level propensity

to seek treatment, which could lead to a violation of the exclusion restriction.

To further avoid violations of the exclusion restriction, and also to avoid measurement error, in my primary

specification I determine the instrument only on the basis of whether an individual was treated for an injury,

and not on the basis of the spending associated with the treatment. If the instrument included a measure

of injury spending, the instrument could be related to another family member’s medical spending through a

family-level propensity to go to expensive doctors, thus violating the exclusion restriction. The disadvantage

5Family medical spending would also be an interesting outcome variable, but it would entail a mechanical violation of theexclusion restriction.

6

of specifying the instrument as a dummy variable is that it reduces the variation in the instrument. In Online

Appendix 4, I incorporate additional variation into the specification of the instrument: variation in the type

of injury, variation in the timing of the injury, and variation in the cost of the injury. Results based on all

instrument specifications are similar.

Beyond the internal validity of the identification strategy, the external validity of my results depends

on whether the price elasticity of expenditure on medical care is homogenous in the population. In any

instrumental variable setting, if the treatment effect of interest is not homogenous in the population, the

estimated effect is a “local average treatment effect,” which is the average effect on “compliers” who would

not have received the treatment absent the intervention of the instrument. In this setting, compliers are

people who have a family injury which causes them to face a lower price than they would have absent the

injury. Although it is not possible to identify compliers because doing so would involve the observation of a

counterfactual state in which a family member did not get injured, Angrist, Imbens, and Rubin (1996) propose

a formal methodology to examine the average characteristics of compliers in a setting with a binary treatment

and a binary instrument. The multivalued treatment in my application precludes the use of the Angrist et

al. (1996) methodology, but I can still informally describe the compliers as the population for which the first

stage is likely to be the strongest.6 For example, the first stage will likely be strongest among people who go

to more expensive doctors, because the higher the expense, the higher the likelihood of meeting the family

deductible. The first stage is also likely to be strongest among large families because large families have more

people to contribute to the fixed family deductible. Furthermore, the first stage is likely to be strongest among

individuals that have a family injury that occurs early in the year.

Some agents will also be more likely to respond to price changes given their expected spending. First,

consider a person whose own expected spending is below the deductible, such as Y 1 in Figure 1. Depending

on the spending of his other family members, a family injury could move him from subfigure I to subfigure II

or III. Next, someone whose predicted spending is in the lower part of the coinsurance region, such as Y 2,

will only see his price change if his family stoploss is met. Third, someone in the higher end of the coinsurance

range, for example at Y 3, will experience a price change when the family deductible is met because he will

then move into the range where the price is zero. Lastly, someone whose own expected spending is above the

individual stoploss at Y 4 will not have his price affected by a family member’s injury because he already faces

a price of zero. This exercise shows that the instrument will have the most power for people who are just

below the deductible who are pushed into the coinsurance region, and people who are just below the stoploss

who are pushed to a price of zero. In the calculation of arc elasticities, and in some specifications, I will take

these two discrete price changes into account.

2.3 Simple Model

In a given year, an individual in a family solves the following problem:

maxY

∑b

πbU(Y,Ab|W )

s.t. pbY ≤ Kb ∀b6Frolich and Melly (2010) articulate the assumptions needed for a local average treatment effect interpretation in a quantile

model with endogeneity - notably a monotonicity assumption. Their assumptions do not apply directly to the CQIV modelbecause they consider a binary instrument and a binary treatment.

7

In this simple model, agents derive utility from total agent plus insurer spending on medical care, which is

given by Y (reflecting an underlying demand for health), and total spending on all other goods, Ab. W denotes

a vector of individual characteristics. The individual faces three potential budget constraints, indexed by b,

which include budget constraints I, II, and III from Figure 1. The probability that the individual places

on facing each budget constraint is given by πb. Following the nonlinear budget set methodology reviewed in

Hausman (1985), pb is the price on the linear segment of budget set b that is relevant for a given Y , and Kb is

the relevant “virtual income” (the y-intercept when each segment is extended to the vertical axis, representing

the marginal income that the individual trades off against medical care on that segment). In this model,

individuals are forward-looking maximizers of expected utility. At the start of the year, each individual forms

an expectation over what his and his family members’ year-end expenditures will be based on expenditures in

the previous year and expected behavior and medical utilization in the coming year.

We can represent the solution to the individual’s problem as a demand curve:

Y = φ(pI , pII , pIII ,KI ,KII ,KIII , πI , πII , πIII ,W ).

As shown, the demand curve will be a function, φ, of the marginal price and virtual income associated with the

utility maximizing segment on each potential budget set, as well as the probability of each potential budget

set and covariates. To make the problem more tractable, I approximate the individual’s demand curve with

the following demand curve:

Y = ψ(P |K,W ),

which is a general function ψ of P , the realized year-end price, income, K, and covariates. This is the

demand curve that forms the basis for estimation in Section 4.1. It assumes forward-looking behavior such

that the realized year-end price is the utility-maximizing price. Models of forward-looking behavior have a

long tradition in the literature on the demand for health care under insurance, including an influential paper

by Ellis (1986), which develops a dynamic model in which agents respond to expected year-end price. Here, I

am assuming that the expected year-end-price is the realized year-end price.7

We can use the model to understand the impact of a family injury on demand, the impact of the within-year

timing of the injury on demand, and the impact of a transitory vs. permanent family injury on demand. We

begin with a simple example, in which there are no unexpected family injuries and the individual forms an

accurate prediction of each πb. Suppose that the individual expects to spend a small amount, given by Y 1 in

Figure 1. Depending on whether his family members meet the family deductible or stoploss before he begins

spending, he could face budget set I, II, or II, as depicted. Based on his predictions of his family members’

spending, he assigns a probability to each of the three subfigures at the start of the year, and he maximizes

his expected utility over all three budget sets.8

Consider the impact of an unexpected injury on demand. Suppose that the agent would have expected

only two of his family members to meet their individual deductibles, but then, on January 1, a third family

member has an injury that is large enough that it will push him over the individual deductible, thus pushing

the family over the family deductible. Now, the agent adjusts his medical consumption relative to the case

7This assumption is true at the end of the year, after all uncertainty is resolved.8If a single agent maximizes utility for the entire family, the optimization process will be similar to the process for each separate

individual, because there is an incentive to spread spending across several family members so that the likelihood that that familydeductible is met increases. (If only one family member responds, even if he responds a great deal, the family deductible might notbe met because the cost sharing rules require spending toward at least three individual deductibles before the family deductiblecan be met.)

8

in which his family member was not injured such that he places a higher probability on budget sets II and

III relative to I. He adjusts the number of visits that he makes to the doctor or his spending per scheduled

visit accordingly (both have the same impact in terms of the model, but we test how much response comes

through the visit margin in Section 6.2), taking into the account the expected price of the visit and the likely

effect that the visit itself will have on his year-end marginal price.

The model that we have discussed so far, and the one that informs the main specifications, is a static

model, but we can extend it to understand the effect of a family injury that happens in January relative to

a family injury that happens in December on the agent’s demand. Suppose that there are multiple periods

indexed by t (days or months) within the year. As time progresses, uncertainty is resolved, and the agent

changes the probabilities πbt that he assigns to the budget sets in each subfigure. The model does not place any

restrictions on the timing of spending - it need only occur before the plan year resets on December 31. However,

we assume that all spending cannot occur exactly on December 31, for example, because appointments might

not be available, because some types of spending require several appointments, or because some types of

spending are incurred after a specific acute event in the middle of the year. If the injury happens on January

1, the agent, his injured family member, and his other family members have the entire year to respond to

the injury. Thus, the agent’s expected year-end price will be lower and his expected year-end spending will

be higher if the injury occurs toward the beginning of the year. Suppose instead that the injury occurs on

December 1. If the injury itself requires several follow-up visits, and it takes time for the agent’s family

members to schedule appointments to react to the injury, it could be that even though the injury would have

been large enough to change the individual’s year-end price if it had occurred on January 1, it will not be

large enough to change the year-end price given its occurrence late in the year. Thus, the first stage effect

of the injury on the marginal price will be smaller, and the reduced form effect of the injury on the agent’s

expenditure will be smaller than they would have been if the injury occurred sooner. If the attenuation of

the first stage effect is the same as the attenuation of the reduced form effect, then the instrumental variable

estimate - the ratio of the reduced form effect to the first stage effect - will be unchanged. I present evidence

consistent with the model that exploits the within-year timing of injuries in Section 6.1.

We can also extend the model to incorporate behavior over multiple years to understand the effect of

transitory vs. permanent shocks to family members. Suppose that the arrival rate of some health events is

uniform across years, but treatment can be delayed until subsequent years with minimal health cost. If an

agent expects that his family members might have higher spending in the subsequent year, resulting in a lower

price for his own care, he has an incentive to delay treatment.9 Suppose a member of the agent’s family has a

transitory shock - he suffers a burn, but the issues related to the burn will be resolved within the calendar year,

so his expected spending next year is the same as it was this year. Such a transitory shock might encourage

family members to shift delayed expenditures from the previous calendar year into the current calendar year

because they expect their marginal prices to be higher next year. If that is the case, my estimate of how

expenditure responds to changes in year-end marginal price will be biased away from zero, relative to what

the true response would be to a permanent decrease in year-end price. In contrast, suppose a member of the

agent’s family has a permanent shock - he develops a new condition this year that is likely to result in higher

medical expenditures over the long term. The agent could also shift planned expenditures forward from the

subsequent year, but he has less incentive to do so than he does if the shock is transitory because next year’s

budget set will likely be very similar to this year’s.

9Cabral (2011) shows evidence of such delaying of dental care treatment. Although dental care is not included in the majormedical expenditure that I study, treatment delay could be possible.

9

3 Data

3.1 Data Description

I use recent proprietary data from a US firm in the retail trade industry with over 500,000 insured employees.

The data for my analysis are merged together from several databases compiled and distributed by Medstat

(2004). I use 2004 data in my main analysis, and I also examine 2003 and 2005 in other analyses. In my

merged data set, in addition to observing inpatient and outpatient claims, I also observe characteristics of

the offered plans and associated enrollment characteristics. The Medstat data are particularly well-suited to

my analysis because the medical claims data identify the beneficiary and insurer contributions on each claim.

Because providers often submit claims automatically on behalf of beneficiaries, and because the firms that pay

the claims collect the data, incentives are aligned to ensure the accuracy and completeness of the data.

A major advantage of the Medstat data over stand alone claims data is that if beneficiaries do not file any

claims or discontinue enrollment, I can still verify their coverage and observe their demographic characteristics

in the enrollment database. These data represent an advantage over Eichner (1997, 1998). Although I

predominantly use cross-sectional variation in the data, I can track individuals and their covered family

members over time as long as the subscriber remains at the same firm. One limitation of the Medstat data is

that I do not observe employees or family members who are not covered, and I do not observe health insurance

options available outside the firm. However, according to the 2006 Kaiser Annual Survey of Employer Health

Benefits, 82% of eligible workers enroll in plans offered by their employers, so I should observe a large majority

of workers at the firm that I study.

I focus on data from one firm to isolate marginal price variation from other factors that could vary by

firm and plan. The main advantage of the firm that I study is that the four plans that it offered in 2003 and

2004 varied only in the deductible and stoploss. Furthermore, one of the offered plans has a $1,000 deductible,

which is coincidentally the initial qualifying amount for a plan to be considered “high deductible” by 2003

legislation. Out of concern that plan selection could be correlated with price sensitivity, I rely on within-plan

price variation for identification by including plan fixed effects. Plan-related local average treatment effects

are possible, and I investigate them in Online Appendix 2.

Table 1 presents a comparison of the cost sharing parameters across plans. The individual deductibles

vary from $350 to $1,000, and the family deductible is always three times the individual deductible, as in

the example described above. The simple cost sharing parameters introduced above provide a very accurate

description of the marginal prices that consumers face at this firm. Almost all covered medical spending counts

toward the deductible and stoploss, except for spending on prescription drugs, which I do not include in my

analysis because it is covered separately. Unlike in many medical plans, there is no fixed per-visit payment.

The only complication in the cost sharing structure at the firm that I study is that the plans offer incentives

for beneficiaries to go to providers that are part of a network. All four plans are preferred provider organization

(PPO) plans. According to the Kaiser 2006 Annual Survey of Employer Health Benefits, 60% of workers with

employer-sponsored health insurance are covered by PPO plans. PPO plans do not require a primary care

physician or a referral for services, and there are no capitated physician reimbursements. However, there is an

incentive to visit providers in the network because there is a higher coinsurance rate for expenses outside of the

network. In the firm that I study, the general coinsurance rate is 20%, and the out-of-network coinsurance rate

is 40%. The network itself does not vary across plans. In the data, there are no identifiers for out-of-network

expenses, but, as demonstrated by Figure 2, which plots beneficiary expenses on total expenses for a sample

10

Table 1: Cost Sharing Comparison

Cost-sharing ComparisonPlan A Plan B Plan C Plan D

Deductible

Individual $350 $500 $750 $1,000

Family* $1,050 $1,500 $2,250 $3,000

Stoploss(Includes

Deductible)

Individual $2,100 $3,000 $4,500 $6,000

Family* $4,550 $6,500 $9,750$10,000

$13,000 in 2004

Coinsurance(Beneficiary)

In-Network 20% 20% 20% 20%

Out-of-Network 40% 40% 40% 40%

*Each family member must meet the individual deductible unless total family spending toward individual deductibles is equal to the family deductible. For example, since the family deductible is three times the individual deductible, if a family has fewer than four members, all family members must meet the individual deductible, and the family deductible does not apply. A similar relationship holds for the stoploss.

of individuals,10 beneficiary expenses follow the in-network schedule with a high degree of accuracy, indicating

that out-of-network expenses are very rare. Accordingly, in my analysis, I assume that everyone who has met

the deductible faces the in-network marginal price for care. My main results do not change when I exclude

the small number of beneficiaries whose out-of-pocket payments deviate from the in-network schedule.

3.2 Sample Selection

Although selection into the firm that I study could affect external validity, the firm has employees in every

region of the United States, and it is large enough that idiosyncratic medical usage should not be a problem.

With over 800,000 people covered by the plans offered by this firm, this firm is large, even among other large

firms in the Medstat data. Furthermore, all of the component Medstat databases are available for this firm

for 2003 and 2004, so I can check for internal consistency by comparing results across both cross-sections.

Beginning in the 2003 data, the Medstat data include fields that make the determination of marginal price

10The sample in this figure only includes 2004 employees in families of two in the $500 deductible plan so that we can focus onvariation around a simple cost sharing relationship. As discussed above, the empirical cost sharing relationship would be morecomplicated if we included families or people in multiple plans on the same graph.

11

Figure 2: Empirical Cost Sharing for Individuals

050

030

00T

otal

Ben

efic

iary

Pay

men

ts

500 5000 10000 13000 20000Total Beneficiary and Insurer Payments (Y)

Sample includes 2004 employees in families of two in $500 deductible plan.Graph depicts 98.5% of observations. Observations with total beneficiary payments greater than $3,100 omitted.Observations with total beneficiary and insurer payments greater than $21,000 omitted.

Empirical Cost Sharing for Individuals

and continuous enrollment very accurate. Since these data are so recent, they should provide an accurate

description of current health insurance offerings and usage.

Within the firm, the main selection criterion that I apply is a continuous enrollment restriction. In my main

results, which use the 2004 and 2003 data as separate cross-sections, I require that the family is enrolled from

January 1 to December 31 of the given year. Selection due to the continuous enrollment restriction eliminates

over 30% of the original sample in each year. Analysis of other firms in the Medstat data suggests that the

rate of turnover at this firm is comparable to the rate of turnover at other large firms. Since my outcome

of interest is year-end expenditure, and family members play a role in the determination of the instrument,

I only include individuals in my main sample if their entire families, with the exception of newborns, are

enrolled for the entire plan year. I discuss summary statistics on the sample before and after the continuous

enrollment restriction in Section 3.3, and I report results that relax the continuous enrollment restriction in

Online Appendix 3. I retain families with newborns on the grounds that child birth is an important medical

expense.

Through selection based on the detailed fields in the Medstat data, I can be confident that my selected

sample consists of accurate records. Since families are important to my analysis, I perform all selection steps

at the family level. To avoid measurement error, I eliminate families that switch plans, families that have

changes in observable covariates over the course of the year, and families that have demographic information

that is inconsistent between enrollment and claims information. I also eliminate families that have unresolved

payment adjustments. Statistics on each step of the sample selection are available in a supplemental data

12

appendix.11 Taken together, these steps eliminate less than seven percent of individuals from the continuously

enrolled sample.

In this clean sample, approximately 25% of employees with other insured family members are insured in

families of four or more. The 2004 main estimation sample includes 29,161 employees insured in families of four

or more. Although the stoploss induces some intra-family interactions in marginal price in families of three, I

restrict the estimation sample to families of four or more so that deductible interactions are also possible.

To better control for unobservables, I limit my estimation sample to the employee in each family, and I use

other family members only in the determination of the instrument. In some specifications, I test robustness to

including other family members in the estimation sample. Restricting the sample to employees or employees

and spouses sacrifices power because it does not take the price responsiveness of all family members into

account, but, arguably, it provides the best control for unobservables on the grounds that employees at the

same firm have some common characteristics that they do not necessarily share with the spouses and children

of their co-workers. Moreover, restricting the sample to employees eliminates the need to address possible

correlations in price responsiveness among family members.

3.3 Summary Statistics

In the 2004 sample, mean year-end medical expenditure by the beneficiary and the insurer is $1,414 in the

sample of employees only. As mentioned above, in my full sample, almost 40% of people consume zero care

in the entire year, and people in the top 25% of the expenditure distribution are responsible for 93% of

expenditures.

The first panel of Table 2 summarizes the expenditure distribution across bins that follow a logarithmic

scale. The first column shows summary statistics for the main analysis sample of employees. As shown,

excluding individuals with zero expenditure, the distribution of positive expenditure among employees is very

skewed, with 31.3% of individuals in the expenditure range between $100 and $1,000, and smaller percentages

of individuals in the bins above and below this range.12 The next three columns show summary statistics

for spouses, children and other dependents, and the full sample. The statistics show that children generally

have lower expenditures than employees and their spouses. Columns 8 and 9 show characteristics of the

sample before the continuous enrollment restriction. As expected, year-end expenditures are smaller in this

sample because these individuals have less than the full year to incur expenditures. In columns 10 and 11,

I compare my sample to the nationally-representative 2004 Medical Expenditure Panel Survey (MEPS), first

to all individuals without injuries under age 65, then to a MEPS sample intended to be comparable to my

main estimation sample: a single respondent without injuries in each family of four or more, aged 18-64. Even

though my sample is not intended to be nationally representative, the number of people with zero expenditures

in my sample is very similar to the number of people with zero expenditures in the MEPS. However, average

expenditures in the MEPS are lower, perhaps because the MEPS includes people in much less generous plans

than the employer-sponsored plans that I study, as well as the uninsured.

Panel B in Table 2 depicts the distribution of the endogenous variable, the marginal price for the next

dollar of care at the end of the year. I calculate the marginal price to reflect the spending of the individual and

his family members. If the individual has not consumed any care and the family deductible has not been met,

the marginal price takes on a value of one because the individual still needs to meet the deductible. In the

employee sample, 57.2% of beneficiaries face a marginal price of one, 39.0% of employees face the coinsurance

11http://www.econ.yale.edu/~ak669/Data_Appendix_07_08_13.pdf12The first rows of Tables A4 and A5 report the unconditional deciles of expenditure.

13

Table 2: 2004 Summary Statistics

Families of Two

Employees Spouses Child/Other Everyone Employees Everyone Employees Everyone Ref.All All All All NO Fam.

InjuryFam. Injury

All All All All All

Variable (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

A. Year-end Expenditure ($)0 35.5 31.0 43.8 39.5 36.1 31.2 27.1 47.8 41.0 35.9 38.7

.01 to 100.00 11.0 10.6 13.1 12.2 10.9 11.5 8.5 11.1 10.3 14.6 12.5100.01 to 1,000 31.3 31.9 32.3 32.0 31.3 31.7 33.8 27.4 28.6 36.5 34.8

1,000.01 to 10,000 19.1 22.0 10.0 14.4 18.8 21.6 25.5 12.0 17.3 12.2 13.210,000.01 to 100,000 3.0 4.5 0.8 2.0 2.9 3.9 5.0 1.7 2.8 0.8 0.7

100,000.01 and up 0.0 0.1 0.0 0.0 0.0 0.1 0.1 0.0 0.0 0.0 0.0Mean 1,414 1,982 610 1,052 1,275 1,689 2,615 893 1,314 631 655

B. Year-end Price0 3.8 4.7 2.2 3.0 3.4 7.4 5.2 2.6 3.5 . .

0.2 39.0 45.4 26.8 33.1 38.1 46.4 45.4 27.4 34.1 . .1 57.2 49.9 71.1 63.9 58.5 46.2 49.4 70.1 62.4 . .

Mean 0.65 0.59 0.76 0.71 0.69 0.57 0.57 0.76 0.69 . .C. Family Injury

0 (NO Family Injury) 89.1 89.6 90.1 89.8 100.0 0.0 96.0 91.3 91.0 88.7 88.21 (Family Injury) 10.9 10.4 9.9 10.2 0.0 100.0 4.0 8.7 9.0 11.3 11.8

D. Family Size2 0.0 0.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0 22.0 0.03 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 19.3 0.04 66.9 64.9 56.0 60.1 67.8 59.0 0.0 57.7 64.7 24.2 58.25 24.4 25.7 29.3 27.5 24.0 28.3 0.0 28.0 25.2 13.8 27.66 6.5 7.1 10.3 8.8 6.2 9.0 0.0 9.7 7.3 5.4 9.37 1.6 1.7 3.2 2.5 1.5 2.9 0.0 3.1 2.0 2.0 2.9

8 to 11 0.5 0.6 1.3 1.0 0.5 0.7 0.0 1.5 0.8 1.7 2.1E. Relation to Employee

Employee 100.0 0.0 0.0 22.7 100.0 100.0 100.0 22.5 100.0 39.6 100.0Spouse 0.0 100.0 0.0 18.9 0.0 0.0 0.0 18.2 0.0 18.9 0.0

Child/Other 0.0 0.0 100.0 58.4 0.0 0.0 0.0 59.3 0.0 41.5 0.0F. Male

0 (Female) 42.6 61.0 49.2 50.0 42.7 42.4 61.5 49.9 44.9 50.3 51.01 (Male) 57.4 39.0 50.8 50.0 57.3 57.6 38.5 50.1 55.1 49.7 49.0

G. Year of Birth1934 to 1943 0.1 0.3 0.0 0.1 0.1 0.2 7.6 0.1 0.2 4.0 1.21944 to 1953 3.9 4.5 0.0 1.8 4.0 3.3 33.2 2.0 4.6 13.6 10.61954 to 1963 30.9 30.7 0.0 12.8 30.8 31.7 28.7 13.1 32.1 17.3 36.31964 to 1973 51.8 47.3 0.0 20.7 51.7 53.2 17.4 18.9 48.6 15.7 35.91974 to 1983 13.2 17.1 1.3 7.0 13.4 11.7 12.9 8.0 14.5 15.5 15.81984 to 1993 0.0 0.1 47.8 27.9 0.0 0.0 0.2 29.4 0.0 16.8 0.21994 to 1998 0.0 0.0 27.5 16.0 0.0 0.0 0.0 15.2 0.0 8.0 0.01999 to 2004 0.0 0.0 23.5 13.7 0.0 0.0 0.0 13.2 0.0 9.3 0.0

H. Employee ClassSalary Non-union 29.8 33.1 29.3 30.2 29.7 30.7 10.3 25.4 25.2 . .Hourly Non-union 70.2 66.9 70.7 69.8 70.3 69.3 89.7 74.6 74.8 . .

I. US Census RegionNew England 1.4 1.5 1.4 1.4 1.5 1.2 1.6 1.5 1.5

Middle Atlantic 1.6 1.7 1.5 1.6 1.6 1.6 1.7 1.8 1.8East North Central 15.7 15.9 15.6 15.7 15.6 16.6 14.2 15.3 15.3

West North Central 11.8 12.1 12.0 12.0 11.9 11.5 10.7 11.5 11.4South Atlantic 19.0 18.3 19.0 18.9 19.2 17.1 23.7 20.1 20.2

East South Central 11.6 11.9 11.2 11.4 11.5 12.1 13.7 11.3 11.4West South Central 28.3 28.4 28.3 28.3 28.3 27.9 25.0 26.8 26.7

Mountain 7.5 7.2 7.8 7.6 7.4 8.2 6.4 8.4 8.2Pacific 3.2 3.0 3.2 3.2 3.1 3.8 2.9 3.4 3.4

J. Plan by Individual Deductible350 60.0 59.2 60.3 60.0 59.4 64.5 66.2 58.9 58.8 . .500 17.0 17.7 16.6 16.9 17.0 16.9 15.0 16.4 16.6 . .750 6.3 6.3 6.2 6.2 6.5 5.1 5.4 6.4 6.4 . .

1000 16.7 16.8 16.9 16.9 17.1 13.4 13.4 18.3 18.2 . .K. Outpatient Visits

0 35.5 31.0 43.9 39.6 36.0 31.1 27.0 47.8 40.9 30.2 32.71 16.5 15.2 18.1 17.2 16.6 16.1 12.5 16.0 15.8 17.8 16.52 10.8 10.5 11.3 11.0 10.6 11.9 9.5 9.7 9.9 12.1 11.4

3 to 4 13.5 14.0 12.2 12.8 13.4 13.8 14.2 10.8 12.1 14.2 13.85 to 10 15.7 17.9 10.6 13.2 15.5 16.7 21.5 10.7 14.1 15.0 15.7

11+ 8.0 11.2 3.8 6.2 7.8 10.4 15.3 5.0 7.2 10.7 10.1L. Inpatient Visits

0 95.1 91.7 96.9 95.5 95.1 94.5 95.0 96.2 95.3 94.6 92.71 4.5 7.3 2.8 4.1 4.5 5.1 4.3 3.4 4.3 4.5 6.4

2+ 0.4 0.9 0.3 0.4 0.4 0.5 0.7 0.4 0.4 0.9 0.9Sample Size 29,161 24,261 74,868 128,290 25,994 3,167 54,889 209,555 47,179 29,972 3,258

Cells report column % by variable unless otherwise noted.All statistics from 2004 MEPS are weighted, and the "employee" refers to the reference person.In 2004 MEPS, Everyone includes the total population below age 65, and Ref. includes the reference person aged 18-64 from families of 4+.In 2004 MEPS, Year-end Expenditure is calculated as the sum of total expenditure on inpatient care, outpatient hospital care, and office based visits.In 2004 MEPS, Outpatient Visits is calculated as the sum of hospital outpatient visits and office based outpatient visits.

Families of Four or More

Before Continuous Enrollment Selection, Families of Four or

More MEPS 2004

Employees

18.48 18.9

22.19 23.2

35.99 34.0

23.33 23.9

14

rate of 0.2, and 3.8% of employees have met the stoploss and face a marginal price of zero. This price variation

should be large enough to be meaningful.

The distribution of the instrument, “family injury,” shows that 10.9% of employees have at least one family

member who is injured in the course of the year. Since injured employees are excluded from the sample, all of

the injuries included in the determination of the instrument in the employee sample are to spouses and other

dependents. In the full sample, injuries to employees are included in the determination of the instrument, and

the same injury can be reflected as a “family injury” for more than one person. Overall, 10.2% of individuals

in the full sample have an injury in the family. I use the same criteria to determine injuries in the MEPS

samples in columns 10 and 11, and I find a similar family injury rate.

People with the injuries included in my instrument are excluded from all estimation samples, but I report

statistics on injured people in Table 3. This table includes all categories of injuries with corresponding ICD-9

Table 3: Individuals with Injuries

ICD-9

Family Injury Instru-ment?

Total Count of Injured

Mean Year-End Expend.

Mean Expend. on Injury

Coeff-icient

Lower bound

Upper bound

2004 Injured Individuals (1) (2) (3) (4) (5) (6) (7) (8)

Fractures 800-829 1 2,601 $5,173 $1,914 28 -85 141

Dislocation 830-839 0 663 $6,890 $1,293 387 -17 790

Sprains and Strains of Joints and Adjacent Muscles 840-849 0 4,347 $3,546 $520 242 -10 493

Intracranial Injuries, Excluding Skull Fractures 850-859 0 331 $9,873 $1,934 1108 -1019 3235Internal Injury of Thorax, Abdomen, and Pelvis 860-869 1 80 $31,353 $7,080 -383 -586 -180Open Wounds 870-899 0 3,274 $2,696 $603 123 5 240Injury to Blood Vessels 900-904 1 20 $5,197 $794 -41 -933 851Late Effects of Injuries, Poisonings, Toxic Effects, and Other External 905-909 1 27 $30,403 $205 93 -999 1186Superficial Injuries 910-919 0 1,276 $2,448 $200 225 14 436Contusion with Intact Skin Surface 920-924 0 2,626 $3,553 $304 236 82 391Crushing Injuries 925-929 0 59 $2,296 $555 1558 53 3063Foreign Body Injuries 930-939 1 536 $2,591 $400 83 -106 272Burns 940-949 1 238 $3,146 $977 -86 -366 194Injuries to Nerves and Spinal Cord 950-957 1 65 $14,322 $601 -227 -581 128Complications of Trauma 958-959 0 3,194 $4,526 $251 244 101 387Poisoning by Drugs, Medicinal and Biological Substances 960-979 1 172 $8,540 $1,759 -384 -580 -189Toxic Effects of Substances Chiefly Nonmedicinal and Other External 980-995 0 1,013 $4,466 $401 267 41 493Complications of Surgical and Medical Care, Not Elsewhere Classified 996-999 1 531 $27,675 $3,594 94 -160 348All Injuries 15,548 $3,759 $1,317No Injury 116,820 $909 $0Everyone 132,368 $1,243 $155

Individiuals with injuries included in the "family injury instrument" excluded from estimation samples.Categories of injuries shown need not be mutually exclusive.

codes in the first column. The second column identifies the injuries included in the instrument through the

selection process discussed in Section 6.1. If a person has any claim for an injury with an ICD-9 code in one of

the listed categories, he is included in the count in the third column first column. Sprains and strains of joints

and adjacent muscles are the most prominent. In column 4, I report the mean year-end total expenditures

for the injured people to demonstrate that their spending should be large enough to have a meaningful effect

on the price that their family members face. Total expenditures could capture injury-related spending that is

induced by the injury but is categorized under different diagnosis codes. In column 5 I only report spending

for which the primary diagnosis code is in a given injury category. Internal injuries of thorax, abdomen, and

pelvis appear to be the most expensive. I exploit variation in spending across injuries in Online Appendix 4.

Panels D through J of Table 2 summarize the distribution of covariates. Family size varies from four to

eleven, with 66.9% of employees in families of four. The full sample is gender balanced, but 57.4% of employees

are male. All employees are between the ages of 20 and 65 in 2004. The distribution of “year of birth” is

15

bimodal because the sample includes parents and their children. Panel H shows that 29.8% of the employees

are salaried, and the remaining employees are hourly. One of the limitations of the Medstat data is that it

does not include any income measures, but the salaried versus hourly classification serves as a crude proxy.

I also investigate potential income effects in other ways, discussed below. The distribution of the sample by

Census region demonstrates that the firm has a very national reach. The largest concentration of employees

is in the West South Central Census region, where 28.3% of the sample resides.

Panel J depicts the distribution of employees and families across the four plans. Each plan has a unique

individual deductible, which I use as the plan identifier. The average deductible is $510, which very close to the

average deductible of $473 among employer-sponsored PPO plans reported in the Kaiser 2006 Annual Survey

of Employer Health Benefits. The most generous plan, which has a $350 deductible, is the most popular,

enrolling 60% of the employee sample. Since this plan is the most popular, and since the low deductible makes

the people in this plan the most likely to experience a price change for a fixed amount of spending, it is likely

that the behavior of the people in this plan has a substantial influence on my results.

Panels K and L give the distribution of outpatient and inpatient visits.13 In the employee estimation

sample, the average number of outpatient visits is 3.4, but 35.5% of the sample has zero visits, and 8.0% of the

sample has eleven or more visits. Inpatient visits are more rare - the average number of inpatient visits is 0.05,

4.5% of the sample has a single visit, and only 0.4% of the sample has two or more visits. Across all visits in

the data, the average outpatient visit costs $244, and the average inpatient visit costs $6,704. Patients with

no inpatient visits spend $603 on average, and patients with inpatient visits spend $10,641 on average. Given

the small number of inpatient visits, we expect that patients have more discretion over outpatient visits, but

given the high amount of inpatient spending, we expect that changes in inpatient visits have a large effect on

expenditure. I examine results with visits as the dependent variable in Section 6.2.

3.4 Graphical Evidence

The raw variation in the data that drives my instrumental variable approach is so pronounced that it can

be seen graphically, without the assistance of complicated estimators. In instrumental variable parlance, the

effect of family injury on expenditure is the “reduced form,” and the effect of family injury on the year-end

price is the “first stage.” The simple instrumental variable estimate is the ratio of the reduced form to the first

stage. To show the variation that drives the instrumental variable strategy, I present graphical depictions of

the reduced form and the first stage in the 2004 sample of employees.

To demonstrate the reduced form, in the top panel of Figure 3, I present the cumulative distribution (cdf)

of expenditure conditional on family injury.

The cdf of expenditure for employees with no family injury is represented by a solid line, and the cdf of

expenditure for employees with a family injury is represented by a dashed line. In this depiction, each quantile

on the y-axis is associated with a value of the logarithm of expenditure on the x-axis. Median expenditure

is $125 among employees with no family injuries and $170 among employees with family injuries. Since the

lines never cross, it is clear from the figure that employees with family injuries have higher expenditures at all

quantiles. Similarity in the curvature of the two cdfs provides reassurance that not all individuals with family

injuries have extremely high expenditures, thus driving the results. The y-intercepts of each line indicate that

family injuries affect the extensive margin decision of whether to consume any care; only 31% of people with

family injuries consume zero care, as opposed to 36% of people with no family injuries. To examine whether

13In the Medstat data, inpatient claims are organized into visits, but outpatient claims are not. To calculate outpatient visits,I calculate the number of unique days for which a patient has outpatient claims.

16

Figure 3: Reduced Form and First Stage

.2

.31

.36

.5

.75

1

0 2.5 5 7.5 10 12.5Ln(Expenditure)

NO Family Injury Family Injury

Reduced Form

0.07

.41

.54

1

0 .2 1Year−end Price

NO Family Injury Family Injury

First Stage

Sample includes 29,161 employees.

the difference between the lines at all quantiles is driven by effects on the extensive margin, I create a similar

figure, not shown here, that depicts cumulative distributions conditional on positive expenditure. The lines of

the new figure do not cross, indicating that even among employees with positive expenditure, employees with

family injuries have higher expenditure at each quantile. Columns 5 and 6 of Table 2 present the underlying

conditional probability density functions in tabular form.

To demonstrate the first stage effect of family injury on the year-end price, in the bottom panel of Figure

3, I present the cumulative distribution of year-end price conditional on family injury. Since the year-end price

takes on only three values, the cdf is a step function. The figure shows that employees with family injuries are

more likely to face lower prices than their counterparts without family injuries. Labels on the y-axis show that

54% of employees with family injuries spend more than the deductible, while only 41% of employees without

family injuries spend more than the deductible. Similarly, 7.4% of employees with family injuries spend more

than the stoploss, while only 3.5% of employees without family injuries spend more than the stoploss.

The depiction in the bottom panel also allows us to assess which price change, the change from 1 to 0.2

or the change from 0.2 to 0, yields the most identification. Following Angrist and Imbens (1995), the vertical

difference between the cdfs at the new price is proportional to the weight in an instrumental variable estimate

formed from a weighted combination of separate Wald estimates for each price change. Since the difference in

the cdfs is largest between 1 and 0.2, the figure indicates that most identification comes from the price change

between 1 and 0.2, and some identification comes from the price change between 0.2 and 0. I investigate

17

results based on both price changes in Section 4.

As a more formal alternative to the bottom panel of Figure 3, a simple ordinary least squares (OLS)

regression of year-end price on family injury and the set of covariates discussed below indicates that having an

injury in the family decreases the year-end price by 10 percentage points, with a standard error of 0.7 percentage

points. The R-squared of this first stage regression with the covariates partialled out is 0.0060, implying a

concentration parameter (defined as NR2/(1−R2)) of 176. Based on this evidence, “weak instruments bias”

is unlikely to be a problem in this application.

The inclusion of covariates should not have a substantial impact on the estimate, but should merely make

it more precise. One way to assess the importance of covariates to the instrumental variable strategy is to

examine the distribution of each variable conditional on the values of the instrument. Ideally, in this setting,

individuals who have an injured family member would be similar in all observable ways to those who do not

have an injured family member.

Panels D through J of columns 5 and 6 of Table 2 give the distribution of covariates conditional on family

injury. The distribution of family size shows that individuals with family injuries are slightly more likely to

be from larger families, as is to be expected if the incidence of injures is distributed evenly across individuals.

Unreported regression results confirm that measures of family size, as well as some of the other covariates,

predict the probability of family injury. Given this discrepancy, I include flexible controls for family structure

in my formal estimates. It is more likely that the exclusion restriction is satisfied conditional on the covariates.

In all of my regressions, I include a dummy for the presence of a spouse on the policy, the year of birth of the

oldest and youngest dependent, and the count of family members born in each of the year ranges in the table,

with the 1999-2004 range saturated by year. In the remaining panels of Table 2, the distribution of the other

control variables appears much less sensitive to the instrument. I control for them in my formal estimates

because complex interactions between these variables might not be visible in the table.

4 CQIV Model and Estimation

I formalize the above graphical evidence using a censored quantile instrumental variable estimator. Here, I

describe the CQIV model and estimator, and I provide intuition to the applied researcher. Technical readers

should consult Chernozhukov et al. (2011).

4.1 CQIV Model

The CQIV model is as follows:

Y = max(Y ∗, C) = T (Y ∗) (1a)

Y ∗ = QY ∗(U |P,W, V ) (1b)

P = φ(V,W,Z) (1c)

where Y is the observed dependent variable and T (x) ≡ max(x,C) is the transformation function that censors

the unobserved uncensored dependent variable Y ∗ at censoring point C. In my empirical application, I examine

robustness to specifying Y in two alternative ways: Y can represent observed medical spending censored at

C = 0, or the logarithm of observed medical spending censored at C = −0.7.14 P is the year-end marginal

14The logarithm of zero is negative infinity, so I censor it at C = −0.7, a value that is smaller than the logarithm of the smallestobserved expenditure in the data: 50 cents. In results not reported, I demonstrate that the CQIV results are robust to the choiceof censoring point for values that do not eliminate information by censoring values above zero.

18

price of medical care, W are covariates described above, Z is an indicator for family injury (the instrumental

variable), φ is a general function, V is a latent unobserved regressor called the “control function,” and U is a

Skorohod disturbance that satisfies the full independence assumption

U v U(0, 1)|P,W,C, V.

This independence assumption states that the disturbance is distributed uniformly on the interval (0,1),

conditional on P,W,C, and V . The quantiles of any distribution always follow a uniform distribution, so the

uniform distribution is completely general and is not a parametric assumption of the model. The content of

the assumption is in requiring that the entire distribution of the disturbance in the equation determining Y ∗ is

independent of P,W,C, and V . This assumption is stronger than the assumption required by related models

of the conditional mean, which only require that the mean of the disturbance is independent of the regressors.

In my application, the full independence assumption is a formalization of the exclusion restriction that one

family member’s injury cannot affect another family member’s expenditure outside of its effect on marginal

price, and it should be plausible given the discussion in Section 2.2.

Even though the CQIV model allows for nonparametric functional forms for Equations (1b) and (1c), some

functional forms for these equations must be specified for estimation. These functional forms could include

terms for nonparametric estimation such as power series and splines. I specify the following main functional

forms:

Y ∗ = α(U)P +W ′β(U) + γ(U)V = X ′δ(U), X = (P, V,W ) (2a)

P = ρZ +W ′θ + V (2b)

where α(U) are the coefficients of interest, yielding the marginal effect of price on latent expenditure. Since the

censoring in my application arises from a corner solution decision, the marginal effect on observed expenditures

could also be of interest. The marginal effect of price on observed expenditure can be calculated as follows:

α(U)1{Y ∗ > C},

where 1{Y ∗ > C} is an indicator for latent expenditure greater than the censoring point. In practice, the

corner marginal effect can be calculated for a given observation by predicting expenditure and setting the

marginal effect to zero if predicted expenditure is below the censoring point. The average corner marginal

effect can be calculated by taking the average of the corner marginal effect across all observations.15 This

“corner calculation” is consistent with the recommendation of Wooldridge (2010).16 Applying the corner

calculation is not equivalent to estimating an uncensored quantile model.

The functional form of (2a) allows the coefficients to vary with the quantiles of U (the quantiles of ex-

penditure distribution, conditional on the covariates and the control function). The linearity of (2a) in the

parameters is an important feature of this functional form, which is common in quantile models. It requires

that the conditional quantiles of the dependent variable are linear in the price. Given this linearity, we expect

results that specify the dependent variable as the level of medical expenditure could be very different from re-

sults that specify the dependent variable as the logarithm of medical expenditure. The choice of a logarithmic

15To calculate the average corner marginal effect for a discrete, rather than infinitesimal, change in the marginal price, predictexpenditure for each observation at each of the two price values, setting predicted expenditure to the censoring point if it is belowthe censoring point. The average of the difference between predicted expenditure at both price values gives the corner marginaleffect. I use this approach in the results that I discuss below.

16An updated version of Machado and Santos Silva (2008) provides further discussion of “quantiles with corners.”

19

versus a level specification is not well informed by theory, so I investigate robustness to both specifications.

The choice of how to specify price as an independent variable is also not well informed by theory: it is

unclear if the response to the price change from 0 to 0.2 should be proportional to the price change from 0.2

to 1, as required by the previous specification. Relaxing this assumption, I also investigate robustness to the

following functional form:

Y ∗ = λ1(U)1{P = 1}+ λ2(U)1{P = 0}+W ′β(U) + γ(U)V = X ′δ(U), X = (P, V,W ) (3a)

P = ρZ +W ′θ + V (3b)

where 1{P = 1}, is an indicator for a price of one, 1{P = 0} is an indicator for a price of zero, and P = 0.2

is the omitted value. In this specification, I follow the control function approach of Newey, Powell, and Vella

(1999). Because the price P only takes on three values, (3a) is nonparametric in P . However, I continue to

use the parametric specification of (3b) to estimate the control term.17

As in traditional models, we can interact the marginal price variable with a covariate to examine hetero-

geneity in price responsiveness along an observed dimension. The CQIV model also allows us to examine

heterogeneity in price responsiveness along the unobserved dimension U . In what follows, I maintain the

agnostic interpretation that the coefficients are a function of the quantiles of unobserved heterogeneity. For

stronger interpretations, we can make assumptions about the heterogeneity represented by U . For example,

in this application, income is not observed, and if we assume that income is the only dimension of unobserved

heterogeneity, then the estimated coefficients will allow us to examine price responsiveness at varying quantiles

of the income distribution. Alternatively, U could represent the quantiles of unobserved health or hypochon-

dria. If unobserved heterogeneity is one-dimensional and if the quantiles of unobserved heterogeneity are the

same as the quantiles of the expenditure distribution conditional on covariates, then the estimated coefficients

at the highest quantiles will yield price responsiveness for individuals who spend the most.

4.2 Estimation

I estimate this model using the CQIV estimator. Here, I provide more intuition for the advantages of the

CQIV estimator relative to other models, and I provide practical implementation details. I have already shown

that the CQIV model allows the coefficients to vary with the quantiles of interest. In addition, CQIV handles

censoring nonparametrically, and it allows for endogeneity.

Censoring induces attenuation bias in quantile regression much in the same way it induces bias in mean

regression: when C is observed in the place of a value that should be much smaller, a line that fits the observed

values will be biased toward zero. Since quantile regression uses information from the entire sample to generate

the estimate at each quantile, if some observations on Y ∗ are censored, the quantile regression lines can be

biased toward zero at all quantiles. The Powell (1984) estimator overcomes this difficulty by incorporating

censoring directly into the estimator as follows:

β(u) minimizes

n∑i=1

ρu(lnYi − T (X ′iβ(u))),

where i indexes individuals, and ρu(x) = {(1 − u)1(x < 0) + u1(x > 0)}|x| for a realization u of U . From

this objective function, it is clear that regardless of whether zeros arise from a censoring process or through

17In traditional instrumental variable models of the conditional mean, a rank condition generally implies that a model withtwo endogenous variables requires two instruments. As in Newey et al. (1999), two instruments are not necessary here becausethere is only a single endogenous variable, specified nonparametrically in the structural equation.

20

a separate decision process, without the transformation function, T (x), the predicted values of X ′iβ(U) could

be infeasible (beyond the censoring point), introducing bias. The transformation function is a nonparametric

way to assure that infeasible predictions do not introduce bias into the objective function. This method works

for quantile estimators because a function of a quantile is equal to the quantile of the function. However, since

the mean does not share this property, mean estimators such as the two-part model require more complicated

techniques to relate the function of a mean to the mean of the function. Despite its theoretical appeal, direct

estimation of this model is rare because the function T (x) induces nonconvexities in the objective function

that present computational difficulties.

Chernozhukov and Hong (2002) devised a tractable computational censored quantile regression (CQR)

algorithm for Powell’s estimator based on the idea that Powell’s censored regression model estimates the

coefficients using observations that are not likely to be censored. The algorithm is a three-step procedure

that predicts which observations are least likely to be censored and estimates the coefficients based on those

observations. The first step involves a parametric prediction of the probability of censoring based on a probit

or logit model. A set fraction of observations that are unlikely to be censored are retained for estimation via

quantile regression in the second step. After the second step, a larger set of observations is retained based on

the predicted values of the dependent variable. This sample gets asymptotically close to the ideal sample of

non-censored observations, and consistent estimates are obtained through a third step of quantile regression

on this sample. The CQIV computational algorithm uses an analog of the Chernozhukov and Hong (2002)

algorithm to handle censoring, with an additional pre-step to handle endogeneity.

The CQIV estimator uses a control function approach to handle endogeneity in the tradition of Hausman

(1978). The control function approach is based on the observation that endogeneity between the price variable

P and the expenditure variable Y ∗ results in a lack of orthogonality between P and the structural disturbance

U . Given this lack of orthogonality, estimates based only on Equation (1b) would be inconsistent. However,

if Z is orthogonal to U conditional on covariates W , then the structural disturbance U is a function of the

first stage disturbance V as follows: U = φV + η. By construction, η is orthogonal to V . Therefore, when V

is included along with P in Equation (1b), the new structural error term is strictly independent of the price

variable P . The estimated first stage error term V is referred to as the estimated “control function,” because

it “controls” for endogeneity in the structural equation.

One advantage of the control function approach to endogeneity, in contrast to the moment condition

approach to endogeneity used by Chernozhukov and Hansen (2008) in their quantile instrumental variable

estimator, is that the control function approach does not require a rank invariance condition on the structural

equation. However, a disadvantage is that the assumptions necessary for the control function approach are

less likely to be satisfied when the endogenous variable is discrete, as it is in this application. The handling of

discrete endogenous variables in control function models is an area that requires more study and is one that

might benefit from new methods developed to handle set identification. To address this issue, I estimate a

version of the CQIV estimator that uses a Chernozhukov and Hansen (2008) moment condition approach to

endogeneity in lieu of the control term approach. This approach does not require continuity in the endogenous

variable. The results, reported in Kowalski (2008), are almost identical to those presented here.

In the reported estimates, I obtain an estimate of the control term by predicting the OLS residuals from

the first stage equation. As discussed in Chernozhukov et al. (2009), CQIV, unlike related estimators, does

not require a linear first stage. The results that follow are robust to the inclusion of higher order functions

of the control term in the structural equation, and results that use an alternative quantile specification of the

first stage as implemented in Chernozhukov et al. (2009) are similar. I obtain 95% confidence intervals on the

21

coefficients via a nonparametric bootstrap.18 In practice, I report the mean of the confidence interval as the

point estimate because the discreteness of the covariates can hinder convergence of the quantile estimator at

specific combinations of covariates.19

The CQIV estimator is a conditional quantile estimator, so covariates play an important role. Bitler,

Gelbach, and Hoynes (2006), who analyze welfare reform experiments, can compute simple quantile treatment

effects by subtracting the quantiles of the control group outcomes from the corresponding quantile of the

treatment group because assignment to the treatment group is random. In my application, I only claim that

my instrument is as good as random conditional on covariates, so I need to include them in all specifications.

Also, covariates are necessary for the CQIV estimation algorithm to work - they predict which observations

are least likely to be censored.

Beyond these considerations, the covariates hold characteristics of individuals constant so that I can es-

timate a policy-relevant parameter: the effect of a change in price on expenditure, holding characteristics

fixed. Recently, methods have been developed that use conditional quantile estimates and distributions of

covariates to predict unconditional distributions of outcomes and calculate unconditional quantile treatment

effects. See Machado and Mata (2005) for a seminal example. In my context, I make use of analogous methods

by predicting the quantiles of expenditure conditional on price to transform my estimated coefficients into arc

elasticities.

5 Results

Table 4 reports the main CQIV elasticities, derived from estimating equations that vary the specification of

expenditure and price. Specifications A and C model the logarithm of expenditure, and specifications B and

D model the level of expenditure. In specifications A and B, price is a single variable that can take on three

values: 0, 0.2, and 1. Specifications C and D specify the three values of price nonparametrically as two dummy

variables.

I do not specify year-end price in logarithmic form because it can take on a value of zero. Thus, I must

transform all of the estimated coefficients into elasticities to obtain the price elasticity of expenditure on

medical care. In Appendix A, I discuss several approaches to transform the estimated coefficients into arc

elasticities, which are preferable to point elasticities for large price changes. I demonstrate that the choice

of arc elasticity transformation can have a large impact on the results, so researchers aiming to compare

elasticities across studies should take the elasticity calculation seriously. I show that the midpoint elasticity

that I report facilitates comparison to the Rand Health Insurance Experiment, but it has an undesirable

property that makes large elasticities approach -1.5. Since most of my identification comes from the larger

price change from before to after the deductible - from 1 to 0.2, I report arc elasticities from this range.

18I draw 200 samples the size of the estimation sample, with replacement. I estimate the model on each sample, retaining theestimated coefficients. I report the 0.025 and the 0.075 quantiles of each estimate as the lower and upper bounds of the 95%confidence interval. In practice, estimates on several of the samples do not converge because one of the quantile estimation stepsthat relies on the Stata quantile regression algorithm does not converge, particularly at the lowest quantiles. I report the numberof bootstrap replications that converge by quantile for the main specifications in the bottom rows of Tables A1 and A2. I excludethe estimates from the replication samples that do not converge from the calculation of confidence intervals. Although I wouldprefer larger bootstrap replication samples, I am limited by computational constraints. When I compute the confidence intervalsin the main specification using 1000 bootstrap reps, the results are very similar.

19Specifically, the first quantile regression step does not converge at the .50 or .90 quantiles in my application, so I cannotobtain point estimates. However, I can obtain estimates from the vast majority of bootstrap replication samples, so I report themean of the estimates from the replication samples as the point estimate. Tables A1 and A2 report the point estimates as wellas the mean of the confidence intervals and demonstrate that both are very similar when they are both available. These tablesalso show the elasticities that are derived from the point estimates and the mean elasticities derived from the replication samples,which are also similar. To facilitate comparison across estimators, I follow the same practice for all estimators in the paper unlessotherwise noted.

22

Table 4: Main CQIV Elasticities with Alternative Specifications of Expenditure and Price

2004 Employee Sample 10 20 30 40 50 60 70 80 90 Tobit IV

A. Dependent variable: Ln(Expenditure).

N= 29,010 Elasticity -1.40 -0.76 -1.16 -1.46 -1.49 -1.41 -1.38 -1.41 -1.40 -1.42

lower bound -1.49 -1.49 -1.46 -1.49 -1.50 -1.49 -1.45 -1.46 -1.44 -1.49

upper bound -1.32 -0.02 -0.86 -1.43 -1.48 -1.33 -1.30 -1.35 -1.35 -1.36

Stoploss Elasticity -0.47 -0.33 -0.56 -0.74 -0.68 -0.48 -0.40 -0.43 -0.41 -0.47

lower bound -0.59 -0.65 -0.78 -0.80 -0.77 -0.63 -0.48 -0.50 -0.46 -0.59

upper bound -0.34 0.00 -0.33 -0.68 -0.58 -0.34 -0.32 -0.35 -0.35 -0.36

B. Dependent variable: Expenditure.

Elasticity -0.26 0.14 -0.64 -0.67 -0.50 -1.21 -1.35 -1.33 -1.39 -1.50

lower bound -1.50 -0.66 -0.68 -0.73 -1.47 -1.46 -1.44 -1.42 -1.44 -1.50

upper bound 0.98 0.94 -0.60 -0.62 0.47 -0.97 -1.27 -1.25 -1.34 -1.49

Stoploss Elasticity 0.17 0.03 -0.35 -0.38 -0.13 -0.10 -0.11 -0.11 -0.11 -0.23

lower bound -0.12 -0.35 -0.38 -0.41 -0.40 -0.12 -0.12 -0.12 -0.11 -0.26

upper bound 0.47 0.41 -0.32 -0.35 0.14 -0.09 -0.10 -0.10 -0.11 -0.21

C. Dependent variable: Ln(Expenditure). Price specified as two dummy variables.

Elasticity -0.43 -1.49 -1.50 -1.50 -1.50 -1.31 -1.31 -1.33 -1.35

lower bound -1.49 -1.49 -1.50 -1.50 -1.50 -1.46 -1.42 -1.42 -1.42

upper bound 0.62 -1.49 -1.50 -1.50 -1.49 -1.15 -1.19 -1.25 -1.28

Stoploss Elasticity 0.22 -0.73 -0.89 -0.88 -0.82 -0.77 -0.74 -0.71 -0.69

lower bound -0.43 -0.84 -0.91 -0.88 -0.85 -0.82 -0.78 -0.74 -0.74

upper bound 0.88 -0.62 -0.87 -0.87 -0.80 -0.72 -0.69 -0.67 -0.64

D. Dependent variable: Expenditure. Price specified as two dummy variables

Elasticity -0.16 -0.17 -0.26 -0.29 -0.36 -0.84 -1.27 -1.28 -1.27

lower bound -1.50 -1.50 -1.50 -1.50 -1.50 -1.47 -1.42 -1.38 -1.34

upper bound 1.19 1.17 0.99 0.92 0.78 -0.20 -1.13 -1.19 -1.20

Stoploss Elasticity 0.03 -0.57 -0.91 -0.87 -0.88 -0.85 -0.77 -0.69 -0.69

lower bound -0.34 -0.91 -0.96 -0.96 -0.93 -0.91 -0.79 -0.72 -0.71

upper bound 0.40 -0.24 -0.87 -0.79 -0.83 -0.80 -0.75 -0.66 -0.66

Unless otherwise noted, price specified as a single variable.

Lower and upper bounds of 95% confidence interval from 200 bootstrap replications.

Censored Quantile IV

Controls include: employee dummy (when applicable), spouse dummy (when applicable), male dummy, plan

(saturated), census region (saturated), salary dummy (vs. hourly), spouse on policy dummy, YOB of oldest

dependent, YOB of youngest dependent, family size (saturated with 8-11 as one group), count family born 1944 to

1953, count family born 1954 to 1963, count family born 1974 to 1983, count family born 1984 to 1993, count

family born 1994 to 1998, count family born 1999, count family born 2000, count family born 2001, count family

born 2002, count family born 2003, count family born 2004 (when applicable).

The elasticity at the 0.6 quantile in specification A is -1.41. To interpret this conditional quantile elasticity,

suppose that there are two groups of people. Each group has the same characteristics as controlled for by the

regressors, but they differ in the prices that they face. The coefficient at the 0.6 quantile of -1.41 indicates that

if we increase price by one percent from the first group to the second group, the 0.6 quantile of expenditure

in the second group will be 1.41 percent higher than the 0.6 quantile in the first group. The discussion of

Table A4 in Appendix A gives us another way of thinking about this coefficient in the context of the predicted

spending estimates from which this arc elasticity is derived: if we have two groups of people as controlled for by

the regressors, but they face different prices, the group with a price of 1 will have a 0.6 quantile expenditure

of $23, but the group with a price of 0.2 will have a 0.6 quantile of expenditure of approximately $1,728.

The coefficients at other quantiles can be interpreted analogously. The coefficient of -1.38 at the 0.7 quantile

23

implies that in two groups of people with the same characteristics, a group with a one percent higher price

would have a 1.38 percent higher 0.7 quantile of expenditures; the 0.7 quantile of expenditures would be $83

in a group that faces a price of 1 and $2,566 in a group that faces a price of 0.2.

It is important to hold group characteristics fixed in the interpretation of the elasticities, because they

are derived from a conditional quantile model.20 The 0.7 quantile of expenditure is $531, which is different

from the 0.7 conditional quantile of expenditure of $2,825. This difference arises because the people at the 0.7

quantile of expenditure spend more than 70% of all people, but the people at the 0.7 conditional quantile of

expenditure spend more than 70% of people only after controlling for their characteristics. People with higher

expenditures conditional on their covariates might have higher income, higher propensity for hypochondria,

or more severe illnesses. In practice, controlling for characteristics makes a difference. As shown in Table 5,

coefficients on some covariates have plausible signs but vary dramatically, sometimes to the extent that they

are statistically significantly different across conditional quantiles, indicating that there is merit in permitting

coefficients to vary with the conditional quantiles in this application.

By comparing the four specifications in Table 4, we see that heterogeneity in price responsiveness across

the conditional quantiles appears sensitive to whether expenditure is specified in logarithms or levels. Because

we have transformed the coefficients into elasticities, we can compare the elasticities across specifications. We

see in specifications A and C that when we specify the dependent variable as the logarithm of expenditure, we

do not see much heterogeneity in price responsiveness across the conditional quantiles. In specification A, the

smallest elasticity is -0.76, and the largest is -1.49. In specification C, the -0.43 elasticity at the 0.10 conditional

quantile is much smaller than the others, which vary from -1.31 to -1.50. All of the elasticities are generally

statistically different from zero but not from each other. There is no strong pattern across the quantiles, but

the elasticity at the conditional median is the largest, and the elasticities generally decreases away from the

median in either direction. However, in specifications B and D, when we specify the dependent variable as

the level of expenditure, we see much more variation in the elasticity across the conditional quantiles. In

specification B, aside from one imprecise positive elasticity at the 0.20 quantile, the estimates vary from -0.26

to -1.39, and the elasticities generally get more negative at higher conditional quantiles. The coefficients in

specification D follow a similar pattern. As discussed in Appendix A, the elasticities in the levels models are

smaller at smaller quantiles because the corner calculation has a larger impact at the lower quantiles when

predicted expenditure is more likely to be zero. This pattern arises because changes in observed expenditure

are less likely when observed expenditure is zero, and observed expenditure of zero is more likely in the

levels model, not because people at the lowest conditional quantiles have lower latent price responsiveness

or because the two models give fundamentally different results. Since theory does not inform the choice of

specification, I focus on the logarithmic specification of the dependent variable, and I present results from the

level specification in appendix tables.

Although price responsiveness appears sensitive to the specification of expenditure, it is not as sensitive

to the specification of price. Instead of specifying price as a single variable in the structural equation, spec-

ifications C and D specify price as two separate variables following (3a) to allow for heterogeneity in price

responsiveness for each of the two price changes. The elasticities, which are calculated for a price change from

1 to 0.2, are similar to their counterparts in which price is specified as a single variable.

In Table 4, I also report “Stoploss Elasticities” that are based on the price change from before to after

the stoploss is met - from 0.2 to 0.21 The magnitudes of the stoploss elasticities are somewhat larger in the

20Tables A4 and A5 report the actual quantiles of expenditure in the data, as well as expenditure at each conditional quantilefrom models that specify the logarithm and the level of expenditure, respectively. Comparing the first two rows of each tabledemonstrates that the actual quantiles of expenditure are different from the conditional quantiles of expenditure.

21How do we expect the stoploss elasticity to compare to the elasticity from 1 to 0.2? If the true curve is isoelastic, the

24

Table 5: CQIV Coefficients on Selected Covariates

Dependent variable: Ln(Expenditure).

2004 Employee Sample 10 20 30 40 50 60 70 80 90 Tobit IVYear-end price -5.17 -3.94 -6.99 -9.72 -8.42 -5.47 -4.25 -4.59 -4.31 -5.26 lower bound -6.76 -7.82 -10.51 -11.13 -10.18 -7.39 -5.22 -5.48 -4.94 -6.76 upper bound -3.58 -0.07 -3.48 -8.32 -6.66 -3.55 -3.28 -3.70 -3.68 -3.76Control Term -1.55 -4.97 -5.65 -3.73 -1.81 -0.64 -0.16 0.34 0.19 NA lower bound -3.09 -9.93 -7.66 -4.99 -3.52 -2.23 -1.11 -0.55 -0.46 NA upper bound 0.00 0.00 -3.65 -2.47 -0.10 0.96 0.79 1.22 0.84 NAMale -0.28 -0.77 -0.79 -0.56 -0.30 -0.28 -0.23 -0.11 -0.06 -0.98 lower bound -0.56 -1.54 -1.19 -0.81 -0.58 -0.58 -0.45 -0.26 -0.19 -1.25 upper bound 0.00 0.00 -0.38 -0.30 -0.02 0.02 -0.02 0.04 0.07 -0.72$500 Deduct -0.16 -0.54 -0.38 -0.16 0.00 0.33 0.17 0.23 0.25 0.05 lower bound -0.32 -1.07 -0.64 -0.32 -0.17 0.09 0.05 0.12 0.15 -0.16 upper bound 0.00 0.00 -0.12 0.01 0.18 0.58 0.30 0.34 0.34 0.27Pacific -0.07 -0.41 -0.39 -0.21 -0.02 0.26 0.25 0.25 0.13 0.15 lower bound -0.14 -0.88 -0.77 -0.48 -0.25 -0.14 -0.09 -0.17 -0.15 -0.37 upper bound 0.00 0.05 -0.01 0.07 0.21 0.66 0.59 0.66 0.41 0.67Salaried Subscriber 0.03 0.11 0.06 0.01 0.05 0.22 0.14 0.11 0.11 0.35 lower bound 0.00 -0.01 -0.07 -0.08 -0.03 0.11 0.07 0.04 0.06 0.24 upper bound 0.05 0.24 0.19 0.09 0.12 0.33 0.21 0.17 0.17 0.45Spouse on Policy 0.00 -0.27 -0.27 -0.23 -0.16 0.10 -0.04 -0.07 -0.09 0.05 lower bound -0.05 -0.68 -0.57 -0.47 -0.35 -0.26 -0.26 -0.28 -0.26 -0.29 upper bound 0.05 0.14 0.02 0.01 0.04 0.46 0.19 0.13 0.08 0.39YOB of Oldest Dependent 0.00 -0.02 -0.02 -0.02 -0.01 -0.02 0.00 0.00 0.00 0.00 lower bound -0.01 -0.05 -0.04 -0.04 -0.02 -0.04 -0.02 -0.01 -0.01 -0.03 upper bound 0.00 0.00 0.01 0.00 0.00 0.01 0.01 0.01 0.00 0.02YOB of Youngest Dependen 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 lower bound 0.00 -0.02 -0.02 -0.02 -0.01 -0.02 -0.01 -0.01 -0.01 -0.02 upper bound 0.00 0.03 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.03Family Size of 8 to 11 -0.78 3.99 0.20 2.55 2.09 0.87 0.95 0.97 1.66 3.00 lower bound -6.50 -0.48 -6.20 0.23 0.39 -1.15 -0.45 -1.01 0.21 0.62 upper bound 4.94 8.45 6.60 4.88 3.79 2.89 2.34 2.94 3.11 5.38Count family born 2004 -0.36 0.64 0.52 0.60 0.41 0.68 0.39 0.10 -0.21 -0.50 lower bound -0.88 -0.65 -0.21 0.02 -0.14 0.17 0.05 -0.24 -0.51 -1.16 upper bound 0.16 1.92 1.25 1.19 0.96 1.20 0.73 0.45 0.10 0.17

Censored Quantile IV

Omitted categories in estimation: $350 Deduct, Family Size of 4, Northeast, count family born 1934 to 1943.Omitted catgories from table: $750 Deduct, $1000 Deduct, Family Size of 5, Family Size of 6, Family Size of 7, Middle Atlantic, East North Central, West North Central, South Atlantic, East South Central, West South Central, Mountain, count family born 1944 to 1953, count family born 1954 to 1963, count family born 1974 to 1983, count family born 1984 to 1993, count family born 1994 to 1998, count family born 1999, count family born 2000, count family born 2001, count family born 2002, count family born 2003.

specifications that include price nonparametrically - the largest elasticity is -0.91 in specifications C and D,

in contrast to -0.74 in specifications A and B. Given that the elasticities are sensitive to the specification of

expenditure, but they are not very sensitive to the specification of price, I consider the robustness of my results

elasticities over both ranges should be approximately equal (as discussed in Appendix A, the logarithmic elasticities would beexactly equal). However, we might not expect the true curve to be isoelastic because the people who face the two price changesare different people. The people who face the second price change are higher spenders, for example the people at Y 2 and Y 3 inFigure 1 as opposed to the people at Y 1. We might expect high spenders to be more or less price responsive relative to lowerspenders. On the one hand, high spenders might have less control over their spending because care decisions for high spendersmight be mostly driven by doctor decisions. On the other hand, high spenders might exercise more control over their spendingbecause they have more money at stake. In addition to considering the amount of spending, we might also want to consider themagnitude of the price change. Chetty (2011) argues that labor supply elasticities in response to smaller wage changes are smallerbecause of “optimization frictions” - people change their behavior most in response to the largest incentives. In this context, itis unclear which price change should be the most meaningful. In absolute terms, the price change around the stoploss is muchsmaller than the price change around the deductible, but it is larger in percentage terms. Empirically, the estimates show thatthe stoploss elasticity is smaller than the main elasticity at almost all quantiles in all four specifications.

25

with respect to specifications in logarithms and levels in the specification with a single price. The specification

with a single price allows me to estimate other models such as Tobit IV that cannot allow for two endogenous

price variables with a single instrument. I report Tobit IV estimates for comparison throughout the paper,

and they are generally in the same range as the CQIV estimates.22

Although the magnitudes of the estimated elasticities vary with the specification and the price change used

for identification, all of my estimated elasticities are much larger than the elasticity estimate of -0.22 obtained

by the Rand Health Insurance Experiment. In Appendix A, I show that this finding is not merely an artifact

of the way that I am calculating the elasticity, and in Online Appendix 1, I show that this finding is robust

to the choice of estimator.

6 Additional Analysis and Robustness

In this section, I consider variations on the main specification to examine the mechanisms behind my results.

In Section 6.1, I show key evidence in support of my identification strategy: spending does not respond before

an family injury occurs, but it does respond after the injury occurs. In Section 6.2, I explore the mechanisms

behind my estimated elasticity, and I show that agents mainly respond to price on the outpatient visit margin.

I also include additional robustness analysis in the Online Appendix. In Online Appendix 1, I show

that my results are robust to alternative estimators. Regardless of the estimator, the elasticities that I find

are large relative to the literature. In Online Appendix 2, I examine heterogeneity in treatment effects by

observable characteristics because although one advantage of the CQIV estimator is that allows estimates

to vary with unobserved heterogeneity, price responsiveness along observed dimensions is also of interest. I

find some evidence that females, salaried employees, and people in the least generous plans are more price

responsive than their counterparts along the same dimension, but sample size limitations limit the statistical

significance of these results. In Online Appendix 3 and Online Appendix 4, I show that my results are robust

to several alternative estimation samples and specifications of the instrument. I show that within-year injury

timing has a limited impact on my results in Online Appendix 5. Finally, in Online Appendix 6, I conduct an

analysis using data on smaller families on the grounds that because of the structure of plans at the firm that

I study, cost sharing interactions cannot occur in families of two. The results do not provide support for the

identification assumption, but, as I discuss, the analysis has several limitations.

6.1 Mechanisms: Timing of Injury Response

A key feature of my identification strategy is that the response to injuries should occur after they happen but

not before. Accordingly, I restrict my instrument to categories of injuries for which employees that have them

in their families do not spend more on their own medical care before the injuries occur. I first describe the

process of selecting the injury categories and show that spending does not respond during the portion of the

year before the injuries occur. Next, using longitudinal data, I show that spending also does not appear to

respond in the textityear before the injuries occur.

My approach is motivated by that of Card, Dobkin, Maestas (2009), who select a subset of diagnoses that

appear to have similar rates of appearance in emergency rooms on weekdays and weekends. Using only this

subset of diagnosis categories, they measure whether the start of Medicare eligibility at age 65 results in lower

mortality. In my approach, I select a set of injury categories for which employees in families with injuries do

22To obtain the Tobit IV estimates, I use the Stata command ivtobit.

26

not appear to spend more than employees in similar families without injuries in the part of the year before

the injury has occurred.

To select the categories of injuries listed in column 2 of Table 3, I use data on spending patterns within

the year. I create a dataset such that each employee with a family injury in a given category has a single

observation for spending before the month of the family injury. For example, if the injury is in March, the

observation reports cumulative spending through February. To control for seasonality in medical spending, I

organize the data so that each employee without any family injury can serve as a control for any employee

with a family injury, regardless of injury month. Therefore, each employee without any family injuries has

an observation for cumulative spending before each month. For example, a given employee without any

family injuries has one observation for January spending, another observation for January through February

spending, another observation for January through March spending, etc. For each injury category, I then

run a regression in which I predict spending with a dummy variable for the eventual injury, clustering by

employee since employees without any family injuries have multiple observations. I repeat the regression for

all categories of family injuries shown in Table 3. The sample size varies slightly across specifications because

I eliminate the employees that themselves will have the injury as I do in the baseline elasticity estimates, but

mean spending is always approximately $700 before the injury.

I report the coefficient on the dummy variable for the eventual injury by category in column 6 of Table 3,

and I report the associated upper and lower bounds of the 95% confidence intervals in columns 7-8. The injuries

included in the instrument are those for which the value of the coefficient is less than 100, indicating that

employees with upcoming family injuries in the given category spend no more than $100 more than employees

with no upcoming family injuries in that category in the period before the injury. Note that spending appears

to be lower, sometimes much lower, in families with upcoming injuries in some categories. However, as the

primary concern with my identification strategy is that employees in families with injuries could be higher

spenders than employees in families without injuries, leading to elasticity estimates that are too large, I include

all categories with negative coefficients in the instrument.23

Having selected the injury categories for the instrument, I test that once I have included all of these

categories, the result still holds that spending does not respond to an injury before it occurs. Further, I also

examine spending during the part of the year after the injury occurs to examine the mechanism behind my

main results. Spending after the injury occurs should be higher for families with injuries.

To implement the test, I create a new dataset. For each employee with a family injury, there is one

observation for the period before the first injury and one observation for the period after the injury, excluding

the injury month itself. For example, an employee with a family injury in March has an observation for

spending from January through February and another observation for spending from April through December.

Each employee without a family injury has an observation for the part of the year before as well as after each

month. For example, an employee without a family injury has one observation for January spending and one

observation for March through December spending; that same employee also has one observation for January

through February spending and one observation for April through December spending, etc.

I run a regression in which I compare families with injuries to families without injuries, after the injury

relative to before the injury, controlling for all of the covariates in the main specification, as well as a full

set of injury month fixed effects. I report the results using the new selected injuries as a single category

in the top panel of Table 6. In column 1, the coefficient on “Family Injury,” defined as in the main CQIV

23I do not select the new categories of injuries based on the confidence intervals because the size of the confidence intervalshas a strong relationship to the total count of injured individuals; if I used the confidence intervals in my selection criteria, Iwould be much more likely to reject the null hypothesis of no difference for the categories with higher counts, retaining only thosecategories that offer little power.

27

specifications, shows that individuals with family injuries spend about 23 cents more in the period before

the injury relative to people without injuries. We can be 95% confident that they spend between 80 dollars

less and 80 dollars more than similar families without injuries in the period before the injury, relative to an

average of approximately 700 dollars. The coefficient on “After,” a dummy variable for the period after the

injury, indicates that there are statistically significant seasonal patterns in medical spending. The coefficient

on “Family Injury x After” indicates that after controlling for seasonality by comparing employees with family

injuries to employees without family injuries, employees with family injuries spend approximately 400 dollars

more after the injury occurs. Taken together, these results suggest that people with family injuries as defined

by my instrument do not spend more before the injuries occur, but they do respond to incentives by spending

more after the injury occurs.

Table 6: Within-Year Test

Dependent Variable: ExpenditureExpenditure

>0Outpatient

Visit >0Inpatient Visit >0

Family Injury x After 363.08 0.048 0.047 0.007 lower bound 166.62 0.025 0.024 -0.001 upper bound 559.54 0.071 0.071 0.015Family Injury 0.23 0.011 0.009 0.000 lower bound -79.33 -0.007 -0.008 -0.005 upper bound 79.78 0.028 0.026 0.006After 37.27 0.028 0.008 -0.001 lower bound 11.62 0.024 0.004 -0.002 upper bound 62.92 0.033 0.012 0.000

The next two columns show that a similar pattern holds for dependent variables that indicate any expen-

diture or any outpatient visit. Although people with injuries in their families do not tend to have statistically

more inpatient visits before the injury occurs, they also do not tend to have statistically more inpatient visits

after the injury occurs, as shown in the fourth column. This finding is consistent with the evidence in Section

6.2 that shows that employees generally respond to injuries on the outpatient visit margin. Overall, the results

of this test lend support to the main identification assumption.

I perform another exercise using the longitudinal data, which also lends some support to the main iden-

tification assumption. To perform the test, I create a dataset that includes all employees whose full families

are enrolled in 2003 and 2004, and I drop individuals with family injuries in 2003 and with injuries themselves

in either year, resulting in a dataset with 17,092 observations (requiring longitudinal data severely limits the

sample, just as requiring continuous enrollment limits the main sample). I run a regression to predict whether

having a family injury in 2004 predicts spending in the previous year of 2003. For this test, I regress spending

in 2003 on a dummy variable for having a family injury in 2004, controlling for all of the 2003 and 2004

values of the covariates. The coefficient on “New Family Injury, 2004 Only,” a dummy variable that indicates

having a family injury in 2004 but not 2003, suggests that individuals who will have a family injury in 2004

spend around $109 less in 2003 than similar individuals who will not have a family injury in 2004, with a 95%

confidence interval of -$505 to $286. This result is not statistically significant, likely because restricting the

sample to employees for whom longitudinal data are available severely restricts the sample size from 29,886

employees in the main 2003 estimation sample to 17,092. However, the coefficient is small relative to mean

2003 spending in the sample of approximately $1,000. Overall, the results of this test are consistent with the

28

main identification assumption.

6.2 Mechanisms: Visits and Outpatient Spending

By estimating models that specify the dependent variable in terms of visits instead of expenditure, we can

examine one potential mechanism through which agents respond to prices: by deciding whether to go to

the doctor. Table 7 reports results that specify the dependent variable in Equation 2a as the logarithm or

number of total visits, outpatient visits, and inpatient visits. The first row in each specification shows the

coefficients. The CQIV corner coefficients across all quantiles suggest that as the marginal price goes from

one to zero, patients reduce total visits by up to 2.83 percent, or up to 18.78 visits. If we transform these

coefficients into elasticities, we can compare the price elasticity of visits to the price elasticity of expenditure

from previous specifications. Holding covariates constant, the median elasticity in specification A1 suggests

that if we have two groups of individuals and one group faces prices that are 1% higher, the median number

of visits in that group will be 1.02% lower. Tobit IV arc elasticities have a similar order of magnitude, as

do arc elasticities calculated from a Poisson regression.24 As compared to the expenditure elasticity, the visit

elasticity is of roughly the same order of magnitude but slightly smaller. Also, the visit elasticity is larger

at higher conditional quantiles, just as the expenditure elasticity in the levels specification is larger at higher

conditional quantiles. The comparison between the visit elasticity and the expenditure elasticity suggests that

a large part of the expenditure elasticity occurs on the visit margin, and a smaller part occurs on the intensive

margin of spending within a given visit.

Specifications B and C in Table 7 show that most of the visit elasticity comes from outpatient visits, and

there is a very small elasticity of inpatient visits. The inpatient visit results are much less reliable because few

bootstrap replications converge at all quantiles except the 0.9 quantile. Unreported elasticity estimates from

a CQIV specification with outpatient expenditure as the dependent variable are very similar to the elasticity

estimates in the main specification.

Quantile estimators are less sensitive to extreme values than mean estimators. However, to demonstrate

that individuals with the highest expenditures are not driving the results, I estimate the main specification

at the very highest conditional quantiles. Even at very highest conditional quantiles, where we expect more

inpatient expenditures conditional on observed characteristics, the unreported estimated elasticities remain

fairly stable.

7 Comparison to Rand

My estimates are an order of magnitude larger than those commonly cited from the Rand experiment. There

could be a multitude of reasons for this discrepancy, including a possible change in the underlying expenditure

elasticity over the decades between the Rand study and my study or a difference in behavior between people

in experimental plans and people in actual plans. Here, I examine differences in methodology between my

estimates and the Rand estimates.

To induce subjects to participate in the Rand experiment, researchers had to guarantee that participants

would be subject to very low out-of-pocket costs, so all plans in the experiment had a yearly stoploss of

$1,000 or less in 1974-1982 dollars. Furthermore, each year, all families were given lump sum payments that

equaled or exceeded their out-of-pocket payments. The experimenters randomized families into plans with

24Poisson regression, also known as quasi-MLE Poisson regression if the researcher only believes that he has specified theconditional mean correctly, is a popular regression for count data models. The raw coefficients are not simple to interpret, butthe arc elasticities should be directly comparable to the other arc elasticities.

29

Table 7: Visits

2004 Employee Sample 10 20 30 40 50 60 70 80 90 Tobit IV Poisson

A1. Dependent Variable: Ln(All Visits)N= 29,161 Year-end price -0.01 -0.51 -1.11 -1.26 -2.03 -2.25 -2.52 -2.74 -2.83 -1.94

lower bound -1.24 -1.27 -1.60 -1.72 -3.23 -2.99 -2.60 -3.55 -3.50 -2.29 upper bound 1.22 0.24 -0.61 -0.80 -0.82 -1.52 -2.43 -1.93 -2.16 -1.59Elasticity -0.10 -0.20 -0.58 -0.76 -1.02 -1.03 -1.15 -1.15 -1.19 -1.01 lower bound -0.69 -0.54 -0.83 -1.07 -1.29 -1.25 -1.17 -1.33 -1.33 -1.18 upper bound 0.49 0.14 -0.33 -0.44 -0.75 -0.81 -1.12 -0.97 -1.05 -0.85

B1. Dependent Variable: Ln(Outpatient Visits)N= 29,161 Year-end price -0.21 -0.46 -0.90 -1.11 -2.05 -2.24 -2.46 -2.68 -2.71 -1.94

lower bound -1.48 -1.15 -1.43 -1.61 -3.19 -2.86 -2.48 -3.47 -3.39 -2.28 upper bound 1.05 0.24 -0.38 -0.61 -0.90 -1.63 -2.43 -1.90 -2.02 -1.59Elasticity -0.11 -0.15 -0.53 -0.69 -1.05 -1.04 -1.13 -1.14 -1.16 -1.01 lower bound -0.71 -0.45 -0.83 -1.05 -1.29 -1.22 -1.14 -1.32 -1.31 -1.17 upper bound 0.48 0.14 -0.22 -0.33 -0.81 -0.86 -1.12 -0.96 -1.00 -0.85

C1. Dependent Variable: Ln(Inpatient Visits)N= 29,161 Year-end price 0.00 . . 0.00 0.00 0.00 0.00 -0.20 -1.34 -0.14

lower bound 0.00 . . 0.00 0.00 0.00 0.00 -0.39 -2.31 -0.25 upper bound 0.00 . . 0.00 0.00 0.00 0.00 0.00 -0.37 -0.04Elasticity 0.00 . . 0.00 0.00 0.00 0.00 -0.03 -0.10 0.00 lower bound 0.00 . . 0.00 0.00 0.00 0.00 -0.05 -0.18 0.00 upper bound 0.00 . . 0.00 0.00 0.00 0.00 0.00 -0.03 0.00

A2. Dependent Variable: All VisitsN= 29,161 Year-end price 0.05 -3.50 -6.63 -9.23 -7.20 -3.85 -8.34 -11.31 -18.78 -4.99 -1.23

lower bound -1.40 -5.39 -8.34 -11.39 -14.30 -6.65 -8.75 -13.35 -22.27 -6.39 -1.33 upper bound 1.51 -1.61 -4.92 -7.07 -0.11 -1.06 -7.93 -9.27 -15.29 -3.58 -1.13Elasticity -0.56 -0.65 -0.75 -1.02 -0.79 -1.00 -1.16 -1.15 -1.19 -1.49 -1.11 lower bound -1.50 -0.68 -0.88 -1.39 -1.49 -1.36 -1.20 -1.31 -1.32 -1.50 -1.26 upper bound 0.37 -0.62 -0.63 -0.64 -0.08 -0.64 -1.12 -0.99 -1.06 -1.49 -0.95

B2. Dependent Variable: Outpatient VisitsN= 29,161 Year-end price -0.35 -2.90 -5.68 -8.36 -6.30 -7.04 -8.09 -11.55 -18.59 -4.90 -1.22

lower bound -2.11 -4.66 -7.44 -10.53 -12.75 -12.28 -8.75 -13.35 -22.46 -6.28 -1.32 upper bound 1.40 -1.14 -3.91 -6.19 0.15 -1.80 -7.43 -9.74 -14.72 -3.52 -1.12Elasticity -0.56 -0.64 -0.85 -1.04 -0.70 -1.12 -1.15 -1.18 -1.19 -1.49 -1.10 lower bound -1.50 -0.74 -1.07 -1.44 -1.49 -1.50 -1.20 -1.31 -1.32 -1.50 -1.26 upper bound 0.38 -0.55 -0.63 -0.64 0.08 -0.74 -1.10 -1.05 -1.06 -1.49 -0.94

C2. Dependent Variable: Inpatient VisitsN= 29,161 Year-end price . 0.00 . 0.00 0.00 0.00 0.00 -0.19 -0.49 -0.14 -0.07

lower bound . 0.00 . 0.00 0.00 0.00 0.00 -0.38 -0.60 -0.24 -0.12 upper bound . 0.00 . 0.00 0.00 0.00 0.00 0.00 -0.37 -0.04 -0.03Elasticity . 0.00 . 0.00 -0.26 -0.50 0.00 -0.75 -0.77 -0.01 -1.50 lower bound . 0.00 . 0.00 -0.52 -0.50 0.00 -1.50 -1.50 -0.01 -1.50 upper bound . 0.00 . 0.00 0.00 -0.50 0.00 0.00 -0.03 0.00 -1.49

Censored Quantile IV

initial marginal prices of 0%, 25%, 50%, and 95%, but after family spending reached the stoploss, marginal

price was zero for the rest of the year, regardless of plan. In practice, the stoploss was binding for a large

fraction (roughly 20%) of participants. Approximately 35% of individuals in the least generous plan exceeded

the stoploss, as did approximately 70% of individuals who had any inpatient care. To put these rates in a

broader context, less than 4% of individuals met the stoploss in my non-experimental data.

Rand researchers recognized that the stoploss affected their ability to calculate the price elasticity of

expenditure on medical care based on the experimentally randomized prices:

“In order to compare our results with those in the literature, however, we must extrapolate

to another part of the response surface, namely, the response to coinsurance variation when there

30

is no maximum dollar expenditure. Although any such extrapolation is hazardous (and of little

practical relevance given the considerable departure from optimality of such an insurance policy),

we have undertaken such an extrapolation rather than forego entirely any comparison with the

literature.” (Manning et al. (1987), page 267)

Manning et al. (1987) cited three sources of estimates of the price elasticity of expenditure on medical care

in the Rand data, the most prominent of which was based on a simulation by Keeler and Rolph (1988) and

not on the Manning et al. (1987) four-part model. Keeler and Rolph (1988) recognized that a comparison

of year-end expenditures based on the experimentally induced coinsurance rates across plans could be mis-

leading because behavior was influenced by stoplosses. They therefore used the experimental data to simulate

year-end-expenditures in hypothetical plans without stoplosses, and they based their elasticity estimates on

this simulated behavior. To conduct the simulation, they assumed myopic responses to marginal price and

examined the frequency of visits for all participants in the period for which their families still had over $400

remaining before meeting the stoploss. Notably, they included people in families that far exceeded the stoploss

in the simulation. Based on calibrated parametric assumptions on the frequency of visits by type and the cost

per visit by type, they forecasted year-end expenditures, and they compared forecasted expenditures across

coinsurance plans relative to the free plan to attain their elasticity estimates using the following midpoint arc

elasticity formula:

η =(Ya − Yb)/(Ya + Yb)

(a− b)/(a+ b),

where p denotes the coinsurance rate and Y denotes simulated expenditures. The often-cited Rand elasticity

estimate of -0.22 comes from a comparison of predicted expenditures across plans with 95% and 25% coin-

surance rates with a = 25, b = 95 Ya = 71 and Ya = 55. Similar calculations based on the predictions from

the four-part model and the experimental means yield estimates of -0.14 and -0.17, respectively. The 95%

to 25% price change that forms the basis for this arc elasticity should be roughly comparable to the price

change on which I base my arc elasticities - from 100% before the deductible to the 20% coinsurance rate. One

key methodological difference, however, is that I use within-plan price variation instead of across-plan price

variation.

Another key difference between the Rand methodology and my methodology comes from the underlying

treatment of myopia vs. foresight. While I assume forward looking behavior, the Keeler and Rolph (1988)

methodology assumes complete myopia. Recent research by Aron-Dine, Einav, Finkelstein, and Cullen (2011)

examines myopic vs. forward-looking responses to price and rejects the null of myopic behavior. If consumers

are forward-looking, it is problematic to assume that the initial statutory marginal price ever governs behavior

of participants who expect to meet the stoploss, even in the period before the stoploss is met. Including

these participants in the elasticity calculation should bias estimates of price responsiveness downward because

variation across plans will be less pronounced among participants who expect to meet the stoploss and thus

do not respond at all to the statutory marginal price. Furthermore, participants with the highest coinsurance

rates are more likely than participants with the lowest coinsurance rates to meet the stoploss, and thus they

are more likely to behave as if medical care is free, which further attenuates elasticity estimates toward zero.

The relative treatment of myopia and foresight is one potential explanation for the difference between my

estimates and the Rand estimates.

31

8 Conclusion

This paper makes several contributions. Using recent, detailed data and a rigorous identification strategy,

I estimate the price elasticity of expenditure on medical care using a new censored quantile instrumental

variable (CQIV) estimator. With the CQIV estimator, I go beyond standards in the literature by allowing

the elasticity estimate to vary with the conditional quantiles of the expenditure distribution, relaxing the

distributional assumptions traditionally used to deal with censoring, and addressing endogeneity.

My main results show that the price elasticity of expenditure on medical care varies from -0.76 to -1.49

across the conditional deciles of the expenditure distribution. Although the CQIV estimator allows the elas-

ticity estimates to vary across the predicted expenditure distribution, the estimates are relatively stable across

the conditional deciles, indicating that price responsiveness changes very little with predicted spending. My

estimated elasticities are an order of magnitude larger than those in the literature. I take several steps to

compare my estimates to those in the literature, and I consider several sources of heterogeneity in the esti-

mates. I conclude that the price elasticity of expenditure on medical care is much larger than the literature

would suggest.

32

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[32] “Medical Expenditure Panel Survey.” Agency for Healthcare Research and Quality. 2004.

[33] “MarketScan Database,” Ann Arbor,MI: The MEDSTAT Group Inc., 2004.

[34] Mullahy, John. “Much ado about two: reconsidering retransformation and the two-part model in healtheconometrics.” Journal of Health Economics, 1998, 17, pp. 247-281.

[35] Newey, Whitney K. “Efficient Estimation of Limited Dependent Variable Models with Endogenous Ex-planatory Variables.” Journal of Econometrics, 1987, 36, pp. 231-250.

[36] Newey, Whitney K., James L. Powell, and Francis Vella. “Nonparametric Estimation of TriangularSimultaneous Equations Models.” Econometrica. 1999. 67(3). pp. 565-603.

[37] Newhouse, Joseph P., Charles E.Phelps, and M. Susan Marquis. “On Having Your Cake and Eating ittoo: Econometric Problems in Estimating the Demand for Health Services. ” Journal of Econometrics,1980, 13, pp. 365-390.

[38] Newhouse, Joseph P and the Insurance Experiment Group. Free for All? Lessons from the Rand HealthInsurance Experiment. Harvard University Press. Cambridge: 1993.

[39] Powell, James L. “Censored Regression Quantiles.” Journal of Econometrics, 1986, 23, pp.143-155.

[40] Tobin, J. “Estimation of Relationships for Limited Dependent Variables.” Econometrica, 1958, pp. 24-36.

[41] Wooldridge, Jeffrey M. Econometric Analysis of Cross Section and Panel Data. MIT Press. Cambridge,MA. Second Edition. 2010.

34

A Transforming Coefficients into Arc Elasticities

The CQIV coefficients are not elasticities because year-end price is specified in levels - price cannot be specified

in logarithmic form because it can take on a value of zero. Tables A1 and A2 report the CQIV coefficients at

the conditional deciles for the logarithmic and levels specifications of expenditure, respectively.

Table A1: Coefficients and Elasticities

Dependent variable: Ln(Expenditure). Price specified as a single variable.

2004 Sample 10 20 30 40 50 60 70 80 90 Tobit IVCoefficients

Year-end Price -4.54 -3.86 -3.05 -4.17 -5.22 -5.47 -4.25 -4.59 -4.31 -5.25point estimate -2.81 -0.78 -3.55 -4.25 . -5.39 -4.28 -4.61 . -5.50

lower bound -6.50 -7.69 -4.60 -4.77 -7.11 -7.39 -5.22 -5.48 -4.94 -6.74upper bound -2.57 -0.03 -1.49 -3.56 -3.33 -3.55 -3.28 -3.70 -3.68 -3.76

Year-end Price (without corner) -5.17 -3.94 -6.99 -9.72 -8.42 -5.47 -4.25 -4.59 -4.31 -5.26point estimate -6.52 -1.62 -8.30 -9.92 . -5.39 -4.28 -4.61 . -5.51

lower bound -6.76 -7.82 -10.51 -11.13 -10.18 -7.39 -5.22 -5.48 -4.94 -6.76upper bound -3.58 -0.07 -3.48 -8.32 -6.66 -3.55 -3.28 -3.70 -3.68 -3.76

Elasticities (Midpoint)Elasticity -1.40 -0.76 -1.16 -1.46 -1.49 -1.41 -1.38 -1.41 -1.40 -1.42

point estimate -1.48 -0.81 -1.29 -1.48 . -1.46 -1.41 -1.43 . -1.46lower bound -1.49 -1.49 -1.46 -1.49 -1.50 -1.49 -1.45 -1.46 -1.44 -1.49upper bound -1.32 -0.02 -0.86 -1.43 -1.48 -1.33 -1.30 -1.35 -1.35 -1.36

Elasticity (without corner) -1.41 -0.77 -1.41 -1.50 -1.49 -1.41 -1.38 -1.41 -1.40 -1.42point estimate -1.48 -0.86 -1.50 -1.50 . -1.46 -1.41 -1.43 . -1.46

lower bound -1.49 -1.49 -1.50 -1.50 -1.50 -1.49 -1.45 -1.46 -1.44 -1.49upper bound -1.34 -0.04 -1.33 -1.50 -1.49 -1.33 -1.30 -1.35 -1.35 -1.36

Stoploss Elasticity -0.47 -0.33 -0.56 -0.74 -0.68 -0.48 -0.40 -0.43 -0.41 -0.47point estimate -0.57 -0.16 -0.68 -0.76 . -0.49 -0.40 -0.43 . -0.50

lower bound -0.59 -0.65 -0.78 -0.80 -0.77 -0.63 -0.48 -0.50 -0.46 -0.59upper bound -0.34 0.00 -0.33 -0.68 -0.58 -0.34 -0.32 -0.35 -0.35 -0.36

Stoploss Elast. (without corner) 0.70 0.50 0.84 1.11 1.01 0.73 0.60 0.64 0.61 0.71point estimate -0.57 -0.16 -0.68 -0.76 . -0.49 -0.40 -0.43 . -0.50

lower bound 0.52 0.01 0.50 1.02 0.87 0.51 0.48 0.53 0.53 0.54upper bound 0.88 0.98 1.17 1.21 1.15 0.94 0.72 0.75 0.69 0.88

Rand Range Elasticity -1.57 -0.87 -1.58 -1.71 -1.70 -1.57 -1.51 -1.56 -1.54 -1.58point estimate -1.68 -0.88 -1.70 -1.71 . -1.64 -1.55 -1.58 . -1.64

lower bound -1.68 -1.70 -1.71 -1.71 -1.71 -1.69 -1.63 -1.64 -1.61 -1.68upper bound -1.46 -0.04 -1.44 -1.70 -1.68 -1.45 -1.40 -1.47 -1.47 -1.48

Rand range Elasticity (without corner) -1.56 -0.86 -1.25 -1.65 -1.70 -1.57 -1.51 -1.56 -1.54 -1.58point estimate -1.68 -0.84 -1.37 -1.67 . -1.64 -1.55 -1.58 . -1.64

lower bound -1.68 -1.70 -1.65 -1.69 -1.71 -1.69 -1.63 -1.64 -1.61 -1.68upper bound -1.44 -0.02 -0.85 -1.60 -1.68 -1.45 -1.40 -1.47 -1.47 -1.48

Logarithmic ElasticitiesElasticity -2.53 -1.95 -2.25 -3.35 -3.96 -2.72 -2.11 -2.28 -2.14 -2.62

point estimate -3.22 -0.76 -2.45 -3.46 . -2.68 -2.13 -2.29 . -2.74lower bound -3.31 -3.88 -3.23 -3.81 -4.70 -3.67 -2.60 -2.72 -2.46 -3.36upper bound -1.76 -0.02 -1.26 -2.89 -3.22 -1.77 -1.63 -1.84 -1.83 -1.87

Elasticity (without corner) -2.57 -1.96 -3.48 -4.83 -4.18 -2.72 -2.11 -2.28 -2.14 -2.62point estimate -3.24 -0.81 -4.12 -4.93 . -2.68 -2.13 -2.29 . -2.74

lower bound -3.36 -3.89 -5.22 -5.53 -5.06 -3.67 -2.60 -2.72 -2.46 -3.36upper bound -1.78 -0.03 -1.73 -4.14 -3.31 -1.77 -1.63 -1.84 -1.83 -1.87

Rand Range Elasticity -2.69 -2.05 -2.41 -3.61 -4.22 -2.87 -2.23 -2.40 -2.26 -2.76point estimate -3.41 -0.81 -2.54 -3.78 . -2.83 -2.25 -2.42 . -2.89

lower bound -3.51 -4.09 -3.49 -4.18 -4.96 -3.87 -2.74 -2.87 -2.59 -3.55upper bound -1.86 -0.02 -1.34 -3.04 -3.47 -1.86 -1.72 -1.94 -1.93 -1.97

Rand range Elasticity (without corner) -2.71 -2.07 -3.67 -5.10 -4.41 -2.87 -2.23 -2.40 -2.26 -2.76point estimate -3.42 -0.85 -4.35 -5.20 . -2.83 -2.25 -2.42 . -2.89

lower bound -3.55 -4.10 -5.51 -5.83 -5.34 -3.87 -2.74 -2.87 -2.59 -3.55upper bound -1.88 -0.04 -1.83 -4.36 -3.49 -1.86 -1.72 -1.94 -1.93 -1.97

Boostrap reps converged 94 153 154 142 156 157 145 143 163 200

Censored Quantile IV

35

Table A2: Coefficients and Elasticities

Dependent variable: Expenditure. Price specified as a single variable.

2004 Sample 10 20 30 40 50 60 70 80 90 Tobit IVCoefficients

Year-end Price 163 -2254 -13856 -18299 -10151 -1373 -3210 -5501 -9607 -3372point estimate 206 -2471 -15593 -18815 -176 -1427 -3248 . . -3369

lower bound -143 -5338 -17222 -19396 -21373 -2127 -3790 -6474 -10266 -4612upper bound 468 829 -10489 -17202 1071 -619 -2630 -4527 -8948 -2133

Year-end Price (without corner) 187 -5829 -32647 -43278 -24144 -1664 -3409 -5848 -9715 -7873point estimate 348 -5902 -36821 -44015 -390 -1621 -3383 . . -7869

lower bound -288 -12785 -40779 -45578 -50056 -2441 -3996 -6693 -10290 -10771upper bound 662 1127 -24516 -40978 1769 -886 -2822 -5003 -9139 -4975

Elasticities (Midpoint)Elasticity -0.26 0.14 -0.64 -0.67 -0.50 -1.21 -1.35 -1.33 -1.39 -1.50

point estimate 0.57 -0.63 -0.64 -0.64 -0.21 -1.20 -1.38 . . -1.50lower bound -1.50 -0.66 -0.68 -0.73 -1.47 -1.46 -1.44 -1.42 -1.44 -1.50upper bound 0.98 0.94 -0.60 -0.62 0.47 -0.97 -1.27 -1.25 -1.34 -1.49

Elasticity (without corner) -2.89 0.01 -1.46 -1.47 -13.71 -2.60 -3.20 -3.90 -1.87 -226.33point estimate 1.42 -1.34 -1.45 -1.47 -1.02 -1.42 -1.54 . . -59.03

lower bound -7.38 -1.46 -1.50 -1.51 -28.97 -3.94 -5.10 -6.49 -2.39 -418.32upper bound 1.59 1.49 -1.41 -1.43 1.56 -1.26 -1.30 -1.31 -1.35 -34.35

Stoploss Elasticity 0.17 0.03 -0.35 -0.38 -0.13 -0.10 -0.11 -0.11 -0.11 -0.23point estimate 0.17 -0.80 -0.95 -0.94 -0.05 -0.10 -0.11 . . -0.22

lower bound -0.12 -0.35 -0.38 -0.41 -0.40 -0.12 -0.12 -0.12 -0.11 -0.26upper bound 0.47 0.41 -0.32 -0.35 0.14 -0.09 -0.10 -0.10 -0.11 -0.21

Stoploss Elast. (without corner) -2.89 55.15 2.67 2.03 3.74 0.15 0.17 0.17 0.16 0.37point estimate 0.40 -1.12 -1.29 -1.15 -0.16 -0.10 -0.11 . . -0.22

lower bound -6.31 -1.80 1.59 1.50 -0.34 0.13 0.15 0.15 0.16 0.31upper bound 0.53 112.09 3.75 2.57 7.82 0.17 0.18 0.19 0.17 0.44

Rand Range Elasticity -2.89 0.01 -1.46 -1.47 -13.71 -2.60 -3.20 -3.90 -1.87 -226.33point estimate 1.42 -1.34 -1.45 -1.47 -1.02 -1.42 -1.54 . . -59.03

lower bound -7.38 -1.46 -1.50 -1.51 -28.97 -3.94 -5.10 -6.49 -2.39 -418.32upper bound 1.59 1.49 -1.41 -1.43 1.56 -1.26 -1.30 -1.31 -1.35 -34.35

Rand range Elasticity (without corner) -0.33 0.12 -0.06 0.00 -0.56 -1.28 -1.39 -1.37 -1.40 -1.70point estimate 0.51 -0.36 0.00 0.00 -0.21 -1.21 -1.38 . . -1.71

lower bound -1.52 -0.61 -0.12 0.00 -1.59 -1.54 -1.50 -1.48 -1.46 -1.71upper bound 0.86 0.85 0.00 0.00 0.46 -1.02 -1.27 -1.25 -1.34 -1.69

Logarithmic ElasticitiesElasticity -9.26 0.19 0.00 0.00 -0.20 -2.74 -1.54 -1.42 -1.80 0.00

point estimate 0.30 0.00 0.00 0.00 -0.12 -1.40 -2.06 . . 0.00lower bound -18.95 0.00 0.00 0.00 -0.65 -4.86 -2.11 -2.16 -2.23 -0.01upper bound 0.44 0.38 0.00 0.00 0.25 -0.61 -0.97 -0.68 -1.37 0.00

Elasticity (without corner) -9.56 0.76 2.01 2.33 0.44 -3.01 -2.04 -1.88 -2.14 0.75point estimate -0.28 1.10 1.96 2.15 0.04 -1.60 -2.27 . . 1.43

lower bound -20.91 -0.16 1.68 1.78 -1.80 -5.16 -2.41 -2.36 -2.40 -1.04upper bound 1.79 1.68 2.34 2.87 2.69 -0.87 -1.68 -1.41 -1.89 2.54

Rand Range Elasticity -2.81 -3.30 -3.59 -2.92 -3.93 -5.17 -5.80 -6.20 -6.61 -5.72point estimate . -3.26 . . -4.13 -5.26 -5.81 . . -5.85

lower bound -3.46 -3.76 -5.89 -4.79 -5.17 -5.55 -5.92 -6.30 -6.65 -6.12upper bound -2.16 -2.85 -1.30 -1.06 -2.70 -4.79 -5.68 -6.11 -6.56 -5.32

Rand range Elasticity (without corner) -3.26 -6.33 -7.50 -7.73 -5.29 -5.20 -5.81 -6.22 -6.61 -6.40point estimate . -6.24 -7.61 -7.74 -4.20 -5.27 -5.82 . . -6.45

lower bound -4.34 -6.98 -7.68 -7.77 -7.84 -5.58 -5.94 -6.33 -6.65 -6.69upper bound -2.18 -5.68 -7.31 -7.69 -2.73 -4.82 -5.68 -6.11 -6.56 -6.11

Boostrap reps converged 132 172 180 174 128 54 89 88 95 200

Censored Quantile IV

To interpret the coefficients, suppose that we have one group of people with a price of 1 (full insurance) and

another group with a price of 0 (no insurance), holding the characteristics controlled for by the regressors fixed.

The CQIV coefficient at the median in Table A1 implies that median expenditure will be 522% higher for the

group that faces a price of 0 than it is for the group that faces a price of 1. The coefficient at the 0.90 quantile

implies that the 0.90 conditional quantile of expenditure will be 431% higher. The second set of coefficients

36

in the table is calculated without the corner calculation. As expected, predicted expenditures are more likely

to be below zero at the lowest conditional quantiles of expenditure, so the corner calculation attenuates the

coefficients at the lowest conditional quantiles the most and has almost no impact on the highest conditional

quantiles. If we were to rely on the coefficients without the corner calculation, we would predict larger price

responsiveness at all quantiles, but such responsiveness would not be feasible because negative expenditure is

not possible.

The above claims about behavior in response to a 100 percentage point change in marginal price are

misleading because such a change is not well defined at marginal prices other than 0 and 1, and this measure

is not unit free. To convert this coefficient into an elasticity, I use two arc elasticity formulas. Arc elasticities

are generally preferred to point elasticities for large changes. My preferred arc elasticity formula is the following

midpoint formula:

η =(Ya − Yb)/|Ya + Yb|

(a− b)/(a+ b),

where Yx is predicted expenditure at price x. Allen and Lerner (1934) discussed a version of this formula. I

have extended it to include the absolute value so that it will not yield wrong-signed elasticities for negative

predicted expenditures (prices are always positive in my application). The midpoint formula is a means of

calculating a percentage change that depends on both the starting and ending values instead of just one or

the other. For example, in my application, the main price change is from before to after the deductible: 1 to

0.2. From 1 to 0.2 is an 80% change, but from 0.2 to 1 is a 400% change. The midpoint arc elasticity formula

computes the change relative to the middle of the price range (0.6) and arrives at a 133% change in price.

(There is a 66% change in price from 0.2 to 0.6 and another 66% change in price from 0.6 to 1, resulting in a

133% change in price over the entire range.)

I also examine arc elasticities following the alternative logarithmic elasticity formula introduced by Gallego-

Diaz (1944):

ηlogarithmic =ln(Ya

Yb)

ln(ab ).

Both arc elasticity formulas are unit free, and symmetrical with respect to prices a and b, so that a price

increase and a symmetric price decrease generate the same elasticity. It can be shown that logarithmic arc

elasticity gives the elasticity of the isoelastic curve, lnY = ηlogarithmic ln p, that connects points (a,Ya) and

(b,Yb). By an appeal to the mean value theorem, it can be shown that there is at least one point on the

curve between these two points at which the point elasticity is exactly the log arc elasticity. See Gallego-Diaz

(1944-1945). Both arc elasticity formulas yield a value of 0 when expenditure does not change regardless of

the price, and both arc elasticity formulas yield a value of 1 when outlays are equal: aYa = bYb.

However, when outlays are not equal, particularly when the elasticity is greater than one, both formulas

yield very different values. Table A3 shows how the arc elasticities vary with expenditures over three different

price changes: the change from before to after the deductible in my application, the change from before to after

the stoploss in my application, and the “Rand range” from 0.25 to 0.95 (the range used for the calculation of an

elasticity from the Rand Health Insurance Experiment). The table shows that as the difference in expenditure

at each price grows large, the midpoint arc elasticity approaches a limiting value, but the logarithmic arc

elasticity continues to grow.25 Given this property, the logarithmic elasticity is probably more informative

25In the price range that I study from 1 to 0.2, the midpoint arc elasticity approaches -1.5. Over the Rand range, the midpointarc elasticity approaches approximately -1.71. Given this property, the upper bound of the confidence interval is not informativeif its value is -1.5.

37

Table A3: Coefficients and Elasticities

Elasticity - Before to After DeductibleExpenditure | P=1 1 1 1 1 1 1 1 1 1 1 1 1 0Expenditure | P=.2 0.2 1 1.29 1.35 3.05 5 10 30 199 1000 10000 125 1

Midpoint 1.00 0.00 -0.19 -0.22 -0.76 -1.00 -1.23 -1.40 -1.49 -1.50 -1.50 -1.48 -1.50Logarithmic 1.00 0.00 -0.16 -0.19 -0.69 -1.00 -1.43 -2.11 -3.29 -4.29 -5.72 -3.00 .

Stoploss Elasticity - Before to After StoplossExpenditure | P=.2 1 1 1 1 1 1 1 1 1 1 1 . 0Expenditure | P=0 0 1 1.29 1.35 3.05 5 10 30 199 1000 10000 . 1

Midpoint 1.00 0.00 -0.13 -0.15 -0.51 -0.67 -0.82 -0.94 -0.99 -1.00 -1.00 . -1.00Logarithmic . . . . . . . . . .

Rand Range Elasticity - From .25 to .95Expenditure | P=.95 1 1 1 1 1 1 1 1 1 1 1 1.1664 0Expenditure | P=.25 25/95 1 1.29 1.35 3.05 5 10 30 199 1000 10000 64 1

Midpoint 1.00 0.00 -0.22 -0.26 -0.87 -1.14 -1.40 -1.60 -1.70 -1.71 -1.71 -1.65 -1.71Logarithmic 1.00 0.00 -0.19 -0.22 -0.84 -1.21 -1.72 -2.55 -3.97 -5.17 -6.90 -3.00 .

in my application because the elasticities that I find are almost always greater than one. However, to the

extent that readers would like to compare my results to the elasticity derived from the Rand Health Insurance

Experiment, the midpoint elasticity formula is more appropriate.

Elasticities are a unit free measure, but if the underlying relationship is not isoelastic, the range over which

they are taken matters. If the underlying relationship is isoelastic (it has the form lnY = ηlogarithmic ln p),

where ηlogarithmic is a constant, then the range over which the logarithmic arc elasticity is taken will not

matter for the estimate. However, with the midpoint formula, even if the underlying relationship is isoelastic,

the range over which the midpoint arc elasticity is taken will matter. For example, in the penultimate column

of Table A3, the underlying relationship is isoelastic with ηlogarithmic = −3, however, η = −1.5 for the range

from 1 to 0.2, and η = −1.65 for the Rand price range.26 If the underlying relationship is linear (Y = γp),

where γ is any constant, the range over which the arc elasticities is taken does not matter; as shown in the

first column of Table A3, both arc elasticities yield a value of one.

I present results from both elasticity measures with and without the corner calculation in Tables A1 and

A2. When I specify the dependent variable as the level of expenditure, the corner calculation involves setting

predicted expenditure for some observations to zero for one or both values of the price. As shown in the

last column of Table A3, when one value of expenditure is zero and the other value is nonzero, the midpoint

arc elasticity takes on its limiting value, and the logarithmic arc elasticity cannot be computed because the

logarithm of zero is undefined. This is relevant in my application because predicted values of expenditure are

censored at zero in the calculation of corner arc elasticities. In the corner arc elasticities, if both predicted

values are censored at zero, I set the elasticity equal to zero.

The lower panels of Table A1 show the midpoint and logarithmic elasticities, with and without the corner

calculation, over the same three price changes shown in Table A3. The elasticities vary dramatically based

on the arc elasticity formula and the price change used. The midpoint corner arc elasticities, the first set of

elasticities in the table, vary from -0.76 to -1.49 across the conditional quantiles of expenditure. These elasticity

estimates convey that if price increases by one percent, expenditure will decrease between .76 percent and 1.49

percent.

Table A4 provides more information on how each of the elasticities in Table A1 can be interpreted in terms

of predicted medical expenditures in my context. The first row shows the actual quantiles of expenditure in

the data. The second row shows average predicted expenditure at each quantile, conditional on all covariates,

including the observed year-end price.27 Comparing the two rows makes clear that the conditional quantiles of

26The example cannot be extended to the stoploss price change because one of the prices is zero.27To predict expenditure for the estimates derived from the logarithmic model, I exponentiate the predicted value for each

38

Table A4: Actual Expenditure, Predicted Expenditure By Price, And Elasticities

Dependent variable: Ln(Expenditure).

2004 Employee Sample Price Corner 10 20 30 40 50 60 70 80 90 Tobit IV

Average actual expenditure, mean shown in last column0 0 0 60 130 261 531 1215 3939 1414

Average predicted expenditure using point estimate from model with dependent variable: Ln(Expenditure)Actual 1 94.5 300.1 902.0 1,277.6 . 1,931.0 2,825.2 4,344.8 . 95Actual 0 95.1 300.6 902.0 1,277.6 . 1,931.0 2,825.2 4,344.8 . 95

1 1 0.5 37.5 0.6 0.5 . 23.1 83.3 131.5 . 11 0 0.5 37.5 0.3 0.1 . 23.1 83.3 131.5 . 0

0.95 1 0.7 40.6 0.7 0.5 . 30.3 103.2 165.6 . 10.95 0 0.7 40.6 0.5 0.2 . 30.3 103.2 165.6 . 10.25 1 65.0 126.3 150.8 207.4 . 1,319.8 2,070.8 4,164.3 . 650.25 0 65.0 126.3 150.8 207.4 . 1,319.8 2,070.8 4,164.3 . 65

0.2 1 90.0 137.0 228.3 340.5 . 1,728.2 2,565.6 5,242.9 . 900.2 0 90.0 137.0 228.3 340.5 . 1,728.2 2,565.6 5,242.9 . 90

0 1 331.5 189.4 1,199.4 2,473.4 . 5,080.8 6,044.6 13,173.2 . 331Elasticities 0 0 331.5 189.4 1,199.4 2,473.4 . 5,080.8 6,044.6 13,173.2 . 331 Elasticity (Midpoint) 1 -1.48 -0.86 -1.49 -1.50 . -1.46 -1.41 -1.43 . -1.48

Elasticity (Midpoint) 0 -1.48 -0.86 -1.50 -1.50 . -1.46 -1.41 -1.43 . -1.48Stoploss Elasticity (Midpoint) 1 -0.57 -0.16 -0.68 -0.76 . -0.49 -0.40 -0.43 . -0.57Stoploss Elasticity (Midpoint) 0 -0.57 -0.16 -0.68 -0.76 . -0.49 -0.40 -0.43 . -0.57

Rand Range Elasticity (Midpoint) 1 -1.68 -0.88 -1.70 -1.71 . -1.64 -1.55 -1.58 . -1.68Rand Range Elasticity (Midpoint) 0 -1.68 -0.88 -1.70 -1.71 . -1.64 -1.55 -1.58 . -1.68

Elasticity (Logarithmic) 1 -3.22 -0.81 -3.69 -4.06 . -2.68 -2.13 -2.29 . -3.22Elasticity (Logarithmic) 0 -3.24 -0.81 -4.12 -4.93 . -2.68 -2.13 -2.29 . -3.24

Rand Range Elasticity (Logarithmic) 1 -3.41 -0.85 -3.99 -4.49 . -2.83 -2.25 -2.42 . -3.41Rand Range Elasticity (Logarithmic) 0 -3.42 -0.85 -4.35 -5.20 . -2.83 -2.25 -2.42 . -3.42

Quantile

expenditure do not correspond to the actual quantiles of expenditure. The next rows show average predicted

expenditure at prices of 1, 0.95, 0.25, 0.20, and 1, all of the actual prices in my empirical setting as well as

the actual prices used in the calculation of the Rand elasticity.

The corner calculation has a much more pronounced impact on the elasticities from the model of the level of

expenditure than it does in the model of the logarithm of expenditure. The rows with a value of 1 for “corner”

censor predicted expenditure from below at zero and then average over all observations. In the logarithmic

specification, the corner calculation has a negligible impact on average predicted expenditure, indicating that

most predicted values are positive. However, in the corresponding Table A5, based on estimates from the level

specification, the corner calculation has a much larger impact. The differences between the corner elasticities in

the logarithmic and level specifications reflect differences in the models more than they reflect real phenomena.

In the levels model, if predicted expenditure is less than zero, the corner calculation requires that it is set to

zero. In contrast, in the logarithmic model, if the predicted logarithm of expenditure is less than -0.7, the

corner calculation requires that it is set to zero; many predicted negative values will not have to be censored.

In practice, predictions less than the censoring point occur much less frequently in the logarithmic model

than they do in the levels model, perhaps because the logarithmic transformation reduces the dispersion in

the dependent variable. Thus, the elasticities in the levels model are smaller at smaller quantiles because the

corner calculation has a larger impact at the lower quantiles when predicted expenditure is more likely to be

zero. This pattern arises because changes in observed expenditure are less likely when observed expenditure

is zero, not because people at the lowest conditional quantiles have lower latent price responsiveness.

observation, using the point estimates at each quantile. Because the estimator did not converge at the 0.5 or 0.9 quantiles, thereare no predictions at those values. After exponentiating, all expenditures are positive, so I do not need to apply the cornercalculation to values of predicted expenditure below zero. However, before I exponentiate, some values are below the censoringpoint. For the corner predictions, I first set predicted log expenditure equal to the censoring point for people with values belowthe censoring point, and then I exponentiate.

In the health economics literature, it is common to apply a Duan (1983) “smearing” transformation to predicted medicalexpenditures derived from a logarithmic model of the conditional mean because the logarithm of the expectation is not equalto the expectation of the logarithm, especially under heteroskedasticity. However, the smearing correction is not merited in mycontext because the logarithm of the quantile is equal to the quantile of the logarithm.

39

Table A5: Actual Expenditure, Predicted Expenditure By Price, And Elasticities

Dependent variable: Expenditure.

2004 Employee Sample Price Corner 10 20 30 40 50 60 70 80 90 Tobit IV

Average actual expenditure, mean shown in last column0 0 0 60 130 261 531 1215 3939 1414

Predicted expenditure using point estimate from model with dependent variable: ExpenditureActual 1 80 173 486 644 562 986 1376 . . 1320Actual 0 57 -3120 -17135 -20155 57 977 1363 . . -634

1 1 202 0 0 0 425 423 204 . . 01 0 178 -5187 -30031 -35572 -80 410 178 . . -3390

0.95 1 192 0 0 0 434 500 368 . . 00.95 0 161 -4892 -28190 -33371 -60 491 347 . . -29970.25 1 80 26 0 0 565 1625 2715 . . 25120.25 0 -83 -761 -2416 -2560 212 1625 2715 . . 25110.2 1 73 128 207 310 574 1706 2884 . . 29050.2 0 -100 -466 -575 -360 232 1706 2884 . . 2905

0 1 44 720 6789 8443 614 2030 3560 . . 4478Elasticities 0 0 -170 715 6789 8443 310 2030 3560 . . 4478

Elasticity (Midpoint) 1 0.70 -1.50 -1.50 -1.50 -0.22 -0.90 -1.30 . . -1.50Elasticity (Midpoint) 0 5.33 -1.25 -1.44 -1.47 -3.08 -0.92 -1.33 . . -19.44

Stoploss Elasticity (Midpoint) 1 0.25 -0.70 -0.94 -0.93 -0.03 -0.09 -0.10 . . -0.21Stoploss Elasticity (Midpoint) 0 0.26 -4.74 -1.19 -1.09 -0.14 -0.09 -0.10 . . -0.21

Rand Range Elasticity (Midpoint) 1 0.70 -1.71 . . -0.22 -0.91 -1.31 . . -1.71Rand Range Elasticity (Midpoint) 0 5.33 -1.25 -1.44 -1.47 -3.08 -0.92 -1.33 . . -19.44

Elasticity (Logarithmic) 1 0.63 . . . -0.19 -0.87 -1.65 . . .Elasticity (Logarithmic) 0 . 1.50 2.46 2.85 . -0.89 -1.73 . . .

Rand Range Elasticity (Logarithmic) 1 0.65 . . . -0.20 -0.88 -1.50 . . -9.54Rand Range Elasticity (Logarithmic) 0 . 1.39 1.84 1.92 . -0.90 -1.54 . . .

Quantile

The final rows of Table A4 show all of the elasticities reported in Table A1, calculated directly from

the average predicted expenditures in the previous rows.28 The elasticity of -1.48 at the 0.10 quantile of

expenditure conveys that, holding covariates fixed, the predicted 0.10 quantile of expenditure is 50 cents

among a group of people with a price of 1, but it increases to $331.50 at a price of 0. The elasticity of -1.43 at

the 0.80 conditional quantile conveys that, holding covariates fixed, the predicted 0.80 quantile of expenditure

is $131.5 for a group of people with a price of 1, but it increases to $13,173.2 at a price of 0. Even though

the difference in predicted expenditures between a price of 1 and a price of 0 is vastly different between the

0.10 and 0.80 quantiles, the elasticities only differ by 0.05 (-1.48 vs. -1.43). In contrast, the first logarithmic

elasticity reported, which is calculated using the exact same expenditures and prices, is 0.93 larger at the 0.10

quantile than it is at the 0.80 quantile (-3.22 vs. -2.29). These calculations demonstrate that the choice of arc

elasticity formula can have a large impact on the estimated elasticity. The other elasticities in the table show

that the price range over which the arc elasticities are calculated can have a dramatic impact on the reported

elasticity in this application. I cannot calculate the sloploss elasticities with the logarithmic formula because

one of the prices is zero. The midpoint stoploss elasticities calculated over the stoploss range from 0.2 to 1

are much smaller - from -0.16 to -0.76 across the quantiles, and the elasticities calculated over the Rand range

from 0.25 to 0.95 are slightly larger - from -0.88 to -1.71 across the quantiles.

The issues inherent in the arc elasticity measures complicate the comparison between arc elasticity estimates

across the quantiles and the comparison between elasticity estimates in my application and the elasticity

estimate obtained from the Rand Health Insurance Experiment. However, estimates obtained from both

elasticity calculations, the logarithmic and the level specifications of the dependent variable, and my price

range and the Rand price range are generally an order of magnitude larger than -0.22, the Rand elasticity

estimate. As Table A3 shows, the Rand elasticity estimate of -0.22 implies that spending is 1.29 times higher

28The elasticities in this table are obtained by applying the various arc elasticity formulas to average predicted expenditure forease of exposition. In all other tables, I predict expenditure and calculate the elasticity for each individual and then take theaverage. Comparison of the elasticities in the “point estimate” row of Table A1 to the elasticities reported here show the averageelasticity is very similar to the elasticity at the average predicted expenditure.

40

if agents face a price of 0.25 instead of a price of 0.95. My CQIV estimates obtained from the logarithmic

specification of -.76 to -1.49 imply that spending is 3.4 to 199 times higher at a price of 0.2 than it is at a price

of 1. The first lesson learned from this comparison are that studies that attempt to compare estimates to the

Rand elasticity should take the particular arc elasticity calculation used in Rand in their empirical exercise

seriously. The second is that no matter the means of comparison, my elasticity estimates are much larger than

the Rand elasticity estimate.

The choice of the arc elasticity calculation is not well-informed by theory. However, to the extent that my

elasticity estimates will be compared to the Rand elasticity estimate, in subsequent specifications, I report

elasticities obtained using the arc midpoint elasticity formula with the corner calculation. As identification in

my setting comes mainly from the price change from 1 to 0.2, I focus on midpoint corner elasticities calculated

over this range, and I also discuss midpoint corner elasticities calculated from the price change from 0.2 to

0. To inform comparison with Rand, it could be argued that I need the same price change as well as the

same formula. However, the Rand price range would involve out-of-sample predictions in my data, so I prefer

the price change from 0.2 to 0, which does occur in my data. As the above tables have shown, if I calculate

elasticities in my setting over the counterfactual range of the Rand elasticity, my estimates would be even

larger.

41

Online Appendix 1 Robustness to Alternative Estimators

For comparison with previous literature, I compare my conditional quantile CQIV estimates to conditional

mean estimates. However, conditional quantile estimators and conditional mean estimators are not likely to

yield the same point estimates because they do not estimate the same quantities. Quantile estimates and

mean estimates are only similar to the extent that the underlying treatment effect is linear and the error

distribution is symmetric and homoskedastic. In this application, CQIV estimates are particularly likely to be

different from estimates obtained with mean estimators because medical expenditures are skewed and censored.

Compared to conditional mean estimates, CQIV estimates are less sensitive to extreme values, and they are

not based on parametric assumptions.

One of the most popular censored estimators, the Tobit estimator, developed by Tobin (1958), is based

on the parametric assumption that the error term is homoskedastic and normally distributed. The Tobit

IV estimator, developed by Newey (1987) provides a good comparison to the CQIV estimator because it

incorporates endogeneity. Eichner (1997, 1998) used a version of the Tobit IV estimator. Relative to the Tobit

estimator, the Tobit IV estimator requires additional parametric assumptions: a homoskedasticity assumption

on the first stage error term and a joint normality distributional assumption on the structural and first

stage error terms. In this application, it is unlikely that the Tobit IV assumption of homoskedasticity in the

structural equation holds given the discreteness of the endogenous variable, year-end price.

A Hausman (1978) joint test of the Tobit IV normality and homoskedasticity assumptions can be conducted

through comparison of the Tobit IV and CQIV estimates at each quantile. Under the null hypothesis, Tobit

IV is consistent and efficient, and CQIV is consistent. Although it would be intuitive to compare the Tobit

IV estimate, which is an estimate of the mean conditional elasticity, to a CQIV conditional median elasticity

estimate, a conditional median estimate is not necessary for the comparison. Since Tobit IV imposes a constant

treatment effect across all quantiles, the single Tobit IV coefficient can be compared directly to the CQIV

coefficients at each quantile. The first row of Table 5 in the paper shows the CQIV and Tobit IV coefficients

before the elasticity transformation. The Tobit IV coefficient is within the range of the CQIV coefficients,

and the 95% confidence intervals overlap, indicating that the null hypothesis that the assumptions required

by Tobit IV hold is not rejected. As shown in specification A1 of Table OA1, the Tobit IV coefficient implies

a larger elasticity than the CQIV coefficients at most quantiles, indicating that the use of the CQIV estimator

alone does not explain the large size of my estimates relative to other estimates in the literature.

For comparative purposes, I also estimate instrumental variable adaptations of two other common censored

mean estimators with endogeneity: a truncated model and a two-part model. The truncated arc elasticity

estimate is -0.93 with a bootstrapped 95% confidence interval of (-1.16,-0.71).29 The two-part model arc

elasticity estimate is -0.86 with a bootstrapped 95% confidence interval of (-1.19,-0.53).30 As with the Tobit

IV estimate, these estimates are generally much larger than those in the literature. However, Eichner (1998)

reports a Tobit IV elasticity of -0.8 (the Eichner (1997) elasticity estimates vary from -0.22 to -0.32).31

Given the insurance-induced mechanical relationship between price and expenditure, we expect elasticity

29To obtain this estimate, I estimate the control function using OLS, and then I include it as an independent variable usingthe Stata command truncreg. The dependent variable is the logarithm of expenditure, truncated for the observations with zeroexpenditure. To predict expenditure for use in the arc elasticity calculation, I use the estimated coefficients to predict expenditure.

30It is unclear how to extend the two part model for endogeneity in both parts, and it is also unclear how to use it to predictexpenditure for use in arc elasticity calculations. To obtain a two part model estimate with endogeneity, I estimate the controlfunction using OLS, and then I include it as an independent variable in the first part probit regression as well as in the secondpart OLS log expenditure regression estimated only on the observations with positive expenditure. To predict expenditure foruse in the arc elasticity calculation, I use the coefficients from the second part to predict expenditure for all observations. I thenmultiply this prediction by the predicted probability of being nonzero from the first part.

31The Eichner (1997,1998) elasticities are point elasticities, so caution must be applied in comparing them to the arc elasticitieselsewhere reported in this paper.

1

Table OA1: CQIV vs. CQR, QIV, and QR

2004 Sample 10 20 30 40 50 60 70 80 90

Dependent Variable: Ln(Expenditure)A1. CQIV Tobit IVN= 29,161 Elasticity -1.40 -0.76 -1.16 -1.46 -1.49 -1.41 -1.38 -1.41 -1.40 -1.42

lower bound -1.49 -1.49 -1.46 -1.49 -1.50 -1.49 -1.45 -1.46 -1.44 -1.49 upper bound -1.32 -0.02 -0.86 -1.43 -1.48 -1.33 -1.30 -1.35 -1.35 -1.36

A2. CQR (no endogeneity) TobitN= 29,161 Elasticity -1.47 -1.48 -1.50 -1.50 -1.50 -1.47 -1.42 -1.41 -1.39 -1.50

lower bound -1.49 -1.49 -1.50 -1.50 -1.50 -1.50 -1.42 -1.41 -1.40 -1.50 upper bound -1.46 -1.48 -1.49 -1.50 -1.50 -1.45 -1.41 -1.40 -1.39 -1.50

Dependent Variable: ExpenditureB1. CQIV Tobit IVN= 29,161 Elasticity -0.26 0.14 -0.64 -0.67 -0.50 -1.21 -1.35 -1.33 -1.39 -1.50

lower bound -1.50 -0.66 -0.68 -0.73 -1.47 -1.46 -1.44 -1.42 -1.44 -1.50 upper bound 0.98 0.94 -0.60 -0.62 0.47 -0.97 -1.27 -1.25 -1.34 -1.49

B2. CQR (no endogeneity) TobitN= 29,161 Elasticity -1.15 -1.28 -1.43 -1.48 -1.43 -1.40 -1.36 -1.30 -1.25 -1.50

lower bound -1.50 -1.49 -1.50 -1.50 -1.50 -1.45 -1.40 -1.39 -1.41 -1.50 upper bound -0.80 -1.06 -1.37 -1.47 -1.36 -1.34 -1.31 -1.22 -1.09 -1.50

B3. QIV (no censoring) IVN= 29,161 Elasticity -1.20

lower bound -1.49 upper bound -0.90

B4. Quantile regression (no censoring or endogeneity) OLSN= 29,161 Elasticity -1.25 -1.18 -0.99 -0.99 -1.34 -0.42 -0.83 -0.56 -0.90 -1.42

lower bound -1.50 -1.50 -1.50 -1.50 -1.50 -1.44 -1.38 -1.38 -1.41 -1.43 upper bound -1.00 -0.85 -0.48 -0.49 -1.17 0.60 -0.29 0.27 -0.39 -1.41

Quantile

still running

estimates that do not account for endogeneity to be biased upward. Cutting in the other direction, we

expect elasticity estimates that do not account for censoring to be biased downward. Table OA1 reports

a variety of estimators that account and do not account for endogeneity and censoring. The first panel

specifies the logarithm of expenditure as the dependent variable, and the second panel specifies the level of

expenditure as the dependent variable. Because of endogeneity, we expect that the CQR estimates should be

larger than the CQIV estimates and that the Tobit estimates should be larger than the Tobit IV estimates.

The results look slightly larger in the logarithmic specifications, but the endogeneity bias is apparent in the

comparison of the CQR coefficients to the CQIV coefficients in the levels specifications. Because of censoring,

we expect that the QIV estimates and the IV estimates should be biased toward zero relative to the CQIV

and Tobit IV specifications. We can only estimate these specifications with the level of expenditure as the

dependent variable, unless we specify a specific value for the logarithm of expenditure at zero in the logarithmic

specifications. Indeed, we find that the IV elasticity is closer to zero than the Tobit IV elasticity. The bias in

the QR and OLS estimates relative to the CQIV and Tobit IV estimates depends on the relative influence of

endogeneity and censoring. At some quantiles, the QIV estimate is biased toward zero relative to the CQIV

estimate, but it is biased away from zero at others. It appears that endogeneity induces more of a bias in

the QR estimates relative to the CQIV estimates and censoring induces more of a bias in the OLS estimates

relative to the Tobit IV estimates. Regardless of the estimator used, the elasticities that I find are large

relative to those in the literature.

As another method of comparison between the CQIV estimates and estimates in the literature, I use a

2

back-of-the-envelope calculation to transform the CQIV estimates into a single estimate. To do this, I take the

average of the CQIV estimates from the 0.10 quantile to the 0.90 quantile, for an average elasticity estimate

of -1.32. This estimate is still much larger than the Rand elasticity estimate of -0.22.

Online Appendix 2 Heterogeneity in Treatment Effects

Although one advantage of the CQIV estimator is that it allows the estimates to vary with unobserved

heterogeneity, we might also be interested in how price elasticities vary along observed dimensions. In this

section, I estimate models that restrict the sample on the basis of observable characteristics. I present models

by gender, salaried/hourly status, and plan.

In Tables OA2 and OA3, specifications B and C, I present the main logarithmic and levels specifications,

restricted to males and females, respectively. Average spending for females is $1,867, which is much higher

than average spending for males of $1,067. Although the elasticities are generally in the same range in the

logarithmic and levels models, women appear to be slightly more price responsive than men, perhaps because

they spend more.

Specifications D and E report separate results for salaried and hourly employees. Although we do not

observe income, salaried/hourly status could serve as a proxy for income in our main results, where we

expect hourly employees to have lower incomes. Hourly employees spend more on medical care than salaried

employees, with average spending of $1,485, as opposed to $1,245 for hourly employees. The results show that

hourly employees are more price responsive than salaried employees, perhaps because of their lower income

and higher spending. However, the results are much less precisely estimated for salaried employees because

of the smaller sample size, which biases the reported elasticity in the positive direction because the reported

elasticity is the average of the upper and lower bounds of the 95% confidence interval, and the lower bound

cannot get more negative than -1.5.

Specifications F through I show separate specifications for each of the four offered plans, starting with

the most generous. We might expect people in the least generous $1,000 deductible plan to be more price

responsive throughout the distribution because they always face higher prices. Since the most generous

plan has the largest enrollment, the confidence intervals are the tightest. In the logarithmic specification,

the estimates are very similar across plans. In the levels specification, people in the least generous ($1,000

deductible) plan appear more price responsive at low conditional quantiles and less price responsive at high

conditional quantiles.

Online Appendix 3 Robustness to Sample Selection

In this section, I examine the robustness of my results to the main sample restrictions. First I examine

the impact of restricting the estimation sample to the employee in each family. Specifications J and K of

Tables OA2 and OA3 present coefficients estimated on samples that include spouses and all other family

members, respectively. The patterns in the estimates across the quantiles are very similar to those in the

employee sample, but the estimates are slightly more precise given the larger sample sizes, even with reported

bootstrapped confidence intervals that account for intra-family correlations.32

32I account for intra-family correlations using a block bootstrap by family. I draw replication samples that have the samenumber of families as the main sample, drawing all observations in a family as a single block.

3

Table OA2: Additional Specifications

Dependent Variable: Ln(Expenditure) unless noted otherwise.

10 20 30 40 50 60 70 80 90 Tobit IV2004 SampleA. EmployeeN= 29,161 Elasticity -1.40 -0.76 -1.16 -1.46 -1.49 -1.41 -1.38 -1.41 -1.40 -1.42

lower bound -1.49 -1.49 -1.46 -1.49 -1.50 -1.49 -1.45 -1.46 -1.44 -1.49 upper bound -1.32 -0.02 -0.86 -1.43 -1.48 -1.33 -1.30 -1.35 -1.35 -1.36

B. Male EmployeeN= 16,730 Elasticity -0.41 -0.50 -0.90 -1.32 -1.47 -1.41 -1.33 -1.33 -1.38 -1.38

lower bound -1.49 -1.49 -1.42 -1.47 -1.50 -1.50 -1.44 -1.44 -1.45 -1.49 upper bound 0.67 0.48 -0.37 -1.16 -1.44 -1.33 -1.21 -1.22 -1.31 -1.27

C. Female EmployeeN= 12,431 Elasticity -1.48 -1.49 -1.49 -1.49 -1.50 -1.27 -1.37 -1.40 -1.35 -1.45

lower bound -1.49 -1.49 -1.50 -1.50 -1.50 -1.50 -1.50 -1.49 -1.47 -1.50 upper bound -1.47 -1.49 -1.49 -1.48 -1.49 -1.05 -1.25 -1.32 -1.22 -1.39

D. Salaried EmployeeN= 8,703 Elasticity -0.41 -0.46 -0.71 -1.01 -1.49 -0.19 -0.93 -1.17 -1.31 -0.16

lower bound -1.48 -1.49 -1.43 -1.48 -1.50 -1.47 -1.46 -1.47 -1.48 -1.48 upper bound 0.65 0.58 0.02 -0.54 -1.48 1.09 -0.40 -0.87 -1.14 1.17

E. Hourly EmployeeN= 20,458 Elasticity -1.12 -0.69 -1.29 -1.48 -1.49 -1.48 -1.42 -1.41 -1.38 -1.47

lower bound -1.49 -1.49 -1.50 -1.50 -1.50 -1.50 -1.47 -1.46 -1.45 -1.49 upper bound -0.75 0.11 -1.08 -1.46 -1.48 -1.47 -1.36 -1.37 -1.32 -1.44

F. $350 Deductible EmployeeN= 17,483 Elasticity -0.84 -0.71 -1.24 -1.46 -1.49 -1.49 -1.34 -1.38 -1.37 -1.33

lower bound -1.48 -1.49 -1.48 -1.49 -1.50 -1.50 -1.45 -1.45 -1.44 -1.48 upper bound -0.20 0.06 -0.99 -1.42 -1.48 -1.48 -1.22 -1.31 -1.30 -1.18

G. $500 Deductible EmployeeN= 4,952 Elasticity -0.46 -0.46 -0.56 -0.97 -1.50 -0.31 -0.69 -1.15 -0.86 -1.16

lower bound -1.49 -1.49 -1.50 -1.50 -1.50 -1.49 -1.48 -1.48 -1.47 -1.50 upper bound 0.58 0.58 0.37 -0.43 -1.50 0.86 0.10 -0.81 -0.25 -0.82

H. $750 Deductible EmployeeN= 1,844 Elasticity -0.97 -0.52 -0.69 -1.21 -1.45 -0.06 -0.02 -0.01 -0.02 -0.02

lower bound -1.49 -1.50 -1.50 -1.50 -1.50 -1.50 -1.50 -1.50 -1.49 -1.50 upper bound -0.46 0.45 0.12 -0.92 -1.40 1.37 1.46 1.47 1.45 1.46

I. $1,000 Deductible EmployeeN= 4,882 Elasticity -1.45 -0.52 -0.60 -0.58 -1.50 -1.37 -1.29 -1.13 -1.30 -1.49

lower bound -1.49 -1.50 -1.50 -1.50 -1.50 -1.50 -1.50 -1.50 -1.50 -1.50 upper bound -1.42 0.45 0.30 0.34 -1.50 -1.24 -1.07 -0.77 -1.10 -1.49

J. Employee and SpouseN= 53,422 Elasticity -1.44 -1.27 -1.44 -1.48 -1.50 -1.47 -1.42 -1.41 -1.42 -1.48

lower bound -1.49 -1.49 -1.50 -1.49 -1.50 -1.50 -1.46 -1.45 -1.45 -1.49 upper bound -1.40 -1.04 -1.39 -1.47 -1.50 -1.45 -1.38 -1.37 -1.38 -1.46

K. EveryoneN= 128,290 Elasticity -1.48 -1.49 -1.45 -1.48 -1.50 -1.43 -1.34 -1.35 -1.36 -1.49

lower bound -1.48 -1.49 -1.50 -1.49 -1.50 -1.46 -1.39 -1.39 -1.39 -1.49 upper bound -1.47 -1.48 -1.40 -1.48 -1.50 -1.40 -1.28 -1.31 -1.33 -1.48

L. Employee including injuredN= 29,764 Elasticity -1.29 -0.94 -1.30 -1.48 -1.49 -1.39 -1.40 -1.42 -1.41 -1.44

lower bound -1.49 -1.38 -1.47 -1.49 -1.50 -1.50 -1.46 -1.46 -1.45 -1.49 upper bound -1.09 -0.49 -1.12 -1.46 -1.48 -1.28 -1.33 -1.38 -1.36 -1.40

Censored Quantile IV

4

Table OA2: Additional Specifications Continued

M. Employee - Do not require employees or their family members continuously enrolledN= 47,179 Elasticity -1.38 -1.01 -1.41 -1.49 -1.50 -1.50 -1.45 -1.44 -1.42 -1.49

lower bound -1.49 -1.38 -1.48 -1.49 -1.50 -1.50 -1.47 -1.46 -1.44 -1.50 upper bound -1.26 -0.65 -1.34 -1.48 -1.49 -1.50 -1.43 -1.41 -1.39 -1.49

N. Employee - Only require employees, and NOT their family members, continuously enrolledN= 36,754 Elasticity -1.36 -0.74 -1.20 -1.47 -1.49 -1.40 -1.36 -1.38 -1.39 -1.44

lower bound -1.49 -1.49 -1.47 -1.49 -1.50 -1.49 -1.44 -1.44 -1.44 -1.49 upper bound -1.23 0.02 -0.93 -1.44 -1.49 -1.31 -1.29 -1.32 -1.34 -1.40

O. Employee, 2003 SampleN= 30,077 Elasticity -0.56 -1.10 -1.43 -1.48 -1.49 -1.36 -1.35 -1.32 -1.35 -1.42

lower bound -1.48 -1.43 -1.49 -1.49 -1.50 -1.49 -1.46 -1.43 -1.43 -1.49 upper bound 0.36 -0.77 -1.37 -1.46 -1.49 -1.23 -1.23 -1.22 -1.28 -1.36

P. Instrument: All Injury CategoriesN= 26,790 Elasticity -0.74 -1.34 -1.43 -1.48 -1.49 -1.50 -1.39 -1.40 -1.40 -1.49

lower bound -1.48 -1.49 -1.50 -1.50 -1.50 -1.50 -1.45 -1.44 -1.44 -1.50 upper bound 0.00 -1.19 -1.36 -1.46 -1.49 -1.50 -1.34 -1.37 -1.37 -1.49

Q. Instrument: Injuries to Children Only, Employees with Spouse Injuries Excluded from SampleN= 28,547 Elasticity -1.39 -0.75 -1.16 -1.46 -1.49 -1.39 -1.39 -1.41 -1.40 -1.41

lower bound -1.48 -1.49 -1.46 -1.49 -1.50 -1.50 -1.46 -1.46 -1.45 -1.49 upper bound -1.30 -0.01 -0.87 -1.43 -1.49 -1.28 -1.31 -1.37 -1.35 -1.33

R. Instrument: Injuries to Spouses Only, Employees with Child Injuries Excluded from SampleN= 26,690 Elasticity -1.12 -0.57 -1.01 -0.99 -1.42 -0.97 -0.97 -0.99 -1.01 -1.20

lower bound -1.49 -1.49 -1.50 -1.50 -1.50 -1.49 -1.42 -1.43 -1.46 -1.50 upper bound -0.75 0.35 -0.52 -0.49 -1.35 -0.45 -0.52 -0.56 -0.57 -0.90

S. Instrument: Simulated Spending on InjuryN= 29,161 Elasticity -1.03 -0.73 -1.01 -1.38 -1.49 -1.39 -1.33 -1.38 -1.39 -1.36

lower bound -1.48 -1.49 -1.44 -1.49 -1.50 -1.49 -1.44 -1.45 -1.44 -1.48 upper bound -0.58 0.03 -0.58 -1.28 -1.47 -1.30 -1.22 -1.32 -1.34 -1.23

T. Instrument: Dummies for Nine Injury TypesN= 29,161 Elasticity -1.47 -1.45 -1.48 -1.49 -1.50 -1.45 -1.41 -1.43 -1.43 -1.49

lower bound -1.49 -1.49 -1.50 -1.50 -1.50 -1.50 -1.46 -1.46 -1.45 -1.50 upper bound -1.46 -1.40 -1.47 -1.49 -1.50 -1.40 -1.37 -1.40 -1.40 -1.48

U. Instrument: Separate Dummies for Injuries in First and Second Half of YearN= 29,161 Elasticity -1.27 -0.72 -1.13 -1.46 -1.49 -1.42 -1.38 -1.40 -1.40 -1.42

lower bound -1.48 -1.49 -1.45 -1.49 -1.50 -1.49 -1.46 -1.46 -1.44 -1.49 upper bound -1.06 0.05 -0.80 -1.44 -1.48 -1.35 -1.30 -1.35 -1.35 -1.36

V. Ln(Outpatient Expenditure)N= 29,161 Elasticity -1.43 -1.48 -1.46 -1.47 -1.49 -1.37 -1.31 -1.34 -1.37 -1.42

lower bound -1.48 -1.49 -1.50 -1.50 -1.50 -1.48 -1.44 -1.43 -1.44 -1.48 upper bound -1.38 -1.47 -1.43 -1.45 -1.49 -1.26 -1.18 -1.24 -1.30 -1.35

91 92 93 94 95 96 97 98 99W. Baseline (Higher Estimated Quantiles)N= 29,161 Elasticity -1.39 -1.39 -1.38 -1.37 -1.36 -1.37 -1.35 -1.35 -1.36

lower bound -1.44 -1.43 -1.43 -1.42 -1.42 -1.41 -1.42 -1.42 -1.46 upper bound -1.34 -1.35 -1.33 -1.32 -1.30 -1.32 -1.28 -1.28 -1.25

Lower and upper bounds for specifications J and K account for intra-family correlations.

Censored Quantile IV

5

Table OA3: Additional Specifications

Dependent Variable: Expenditure unless noted otherwise.

10 20 30 40 50 60 70 80 90 Tobit IV2004 SampleA. EmployeeN= 29,161 Elasticity -0.26 0.14 -0.64 -0.67 -0.50 -1.21 -1.35 -1.33 -1.39 -1.50

lower bound -1.50 -0.66 -0.68 -0.73 -1.47 -1.46 -1.44 -1.42 -1.44 -1.50 upper bound 0.98 0.94 -0.60 -0.62 0.47 -0.97 -1.27 -1.25 -1.34 -1.49

B. Male EmployeeN= 16,730 Elasticity -0.42 0.04 -0.51 -0.56 -0.80 -1.00 -1.37 -1.38 -1.38 -1.45

lower bound -1.50 -0.43 -0.59 -0.68 -1.42 -1.43 -1.45 -1.44 -1.44 -1.50 upper bound 0.67 0.51 -0.43 -0.44 -0.18 -0.56 -1.29 -1.33 -1.33 -1.40

C. Female EmployeeN= 12,431 Elasticity -0.80 -1.01 -1.15 -1.16 -1.13 -1.16 -1.21 -1.30 -1.33 -1.46

lower bound -0.84 -1.25 -1.50 -1.50 -1.50 -1.41 -1.41 -1.42 -1.42 -1.50 upper bound -0.75 -0.77 -0.81 -0.82 -0.75 -0.91 -1.01 -1.17 -1.24 -1.41

D. Salaried EmployeeN= 8,703 Elasticity -0.37 -0.06 -0.56 -0.67 -0.44 -1.17 -1.27 -1.33 -1.36 -1.07

lower bound -1.50 -0.57 -0.59 -0.79 -1.47 -1.45 -1.44 -1.44 -1.44 -1.50 upper bound 0.76 0.46 -0.53 -0.55 0.58 -0.88 -1.11 -1.22 -1.28 -0.64

E. Hourly EmployeeN= 20,458 Elasticity -0.37 -0.15 -0.66 -0.77 -0.63 -1.29 -1.32 -1.35 -1.37 -1.47

lower bound -1.50 -0.65 -0.69 -0.89 -1.39 -1.42 -1.41 -1.42 -1.44 -1.50 upper bound 0.76 0.36 -0.64 -0.65 0.12 -1.17 -1.22 -1.29 -1.30 -1.43

F. $350 Deductible EmployeeN= 17,483 Elasticity -0.19 0.09 -0.75 -0.76 -0.63 -0.44 -1.25 -1.30 -1.40 -1.46

lower bound -1.33 -0.76 -0.76 -0.79 -1.28 -1.46 -1.40 -1.41 -1.46 -1.50 upper bound 0.95 0.93 -0.73 -0.74 0.03 0.57 -1.09 -1.20 -1.33 -1.41

G. $500 Deductible EmployeeN= 4,952 Elasticity -0.21 -0.47 -0.89 -1.00 -1.11 -1.36 -1.33 -1.33 -1.35 -0.34

lower bound -1.50 -1.26 -1.25 -1.47 -1.48 -1.46 -1.45 -1.42 -1.45 -1.50 upper bound 1.08 0.33 -0.52 -0.54 -0.73 -1.27 -1.21 -1.25 -1.26 0.82

H. $750 Deductible EmployeeN= 1,844 Elasticity -0.31 -0.76 -0.50 -0.45 -0.31 -1.01 -1.35 -1.21 -0.72 -0.29

lower bound -1.48 -1.49 -1.48 -1.50 -1.48 -1.45 -1.44 -1.45 -1.44 -1.50 upper bound 0.86 -0.03 0.48 0.59 0.86 -0.57 -1.26 -0.96 0.00 0.92

I. $1,000 Deductible EmployeeN= 4,882 Elasticity -1.38 -0.50 -0.72 -0.69 -1.48 -0.86 -0.67 -0.60 -0.55 -1.50

lower bound -1.50 -1.50 -1.50 -1.50 -1.50 -1.37 -1.13 -1.07 -0.93 -1.50 upper bound -1.27 0.49 0.06 0.12 -1.46 -0.35 -0.21 -0.12 -0.18 -1.50

J. Employee and SpouseN= 53,422 Elasticity -0.29 -0.65 -0.85 -0.96 -0.49 -1.09 -1.16 -1.10 -1.33 -1.50

lower bound -1.50 -0.72 -1.05 -1.21 -1.46 -1.40 -1.43 -1.45 -1.43 -1.50 upper bound 0.93 -0.58 -0.65 -0.71 0.48 -0.77 -0.89 -0.76 -1.22 -1.50

K. EveryoneN= 128,290 Elasticity -0.33 -0.50 -1.01 -0.95 -0.36 -0.94 -1.17 -1.27 -1.28 -1.45

lower bound -1.50 -1.50 -1.50 -1.37 -1.47 -1.46 -1.42 -1.40 -1.44 -1.50 upper bound 0.84 0.49 -0.52 -0.52 0.76 -0.41 -0.93 -1.15 -1.12 -1.40

L. Employee including injuredN= 29,764 Elasticity -0.27 -0.65 -0.67 -0.87 -0.63 -0.48 -1.31 -1.34 -1.37 -1.50

lower bound -1.50 -0.69 -0.69 -1.12 -1.49 -1.47 -1.42 -1.41 -1.44 -1.50 upper bound 0.97 -0.61 -0.64 -0.63 0.24 0.52 -1.20 -1.26 -1.30 -1.49

Censored Quantile IV

6

Table OA3: Additional Specifications Continued

M. Employee - Do not require employees or their family members continuously enrolledN= 47,179 Elasticity -0.39 -0.18 -0.57 -0.85 -0.90 -1.30 -1.34 -1.35 -1.37 -1.50

lower bound -1.50 -0.60 -0.61 -1.14 -1.21 -1.48 -1.43 -1.44 -1.45 -1.50 upper bound 0.71 0.23 -0.53 -0.56 -0.58 -1.12 -1.24 -1.27 -1.29 -1.50

N. Employee - Only require employees, and NOT their family members, continuously enrolledN= 36,754 Elasticity -0.34 -0.60 -0.62 -0.80 -0.52 -0.40 -1.30 -1.34 -1.30 -1.49

lower bound -1.50 -0.67 -0.64 -0.98 -1.49 -1.44 -1.43 -1.42 -1.45 -1.50 upper bound 0.81 -0.53 -0.60 -0.63 0.45 0.64 -1.18 -1.26 -1.15 -1.49

O. Employee, 2003 SampleN= 30,077 Elasticity -0.29 -0.60 -0.67 -0.72 -0.50 -0.41 -1.28 -1.34 -1.37 -1.50

lower bound -0.72 -0.62 -0.74 -0.84 -1.48 -1.47 -1.40 -1.41 -1.43 -1.50 upper bound 0.14 -0.59 -0.60 -0.61 0.48 0.65 -1.17 -1.27 -1.31 -1.49

P. Instrument: All Injury CategoriesN= 26,790 Elasticity -0.54 0.22 -0.60 -0.69 -0.50 -1.36 -1.35 -1.36 -1.38 -1.50

lower bound -1.50 -0.59 -0.62 -0.79 -1.17 -1.46 -1.41 -1.40 -1.43 -1.50 upper bound 0.42 1.02 -0.58 -0.59 0.17 -1.26 -1.29 -1.32 -1.34 -1.50

Q. Instrument: Injuries to Children Only, Employees with Spouse Injuries Excluded from SampleN= 28,547 Elasticity -1.29 -0.77 -1.24 -1.47 -1.36 -0.73 -0.39 -0.38 -0.34 -1.49

lower bound -1.50 -1.50 -1.50 -1.50 -1.50 -1.12 -0.49 -0.44 -0.39 -1.50 upper bound -1.07 -0.04 -0.98 -1.44 -1.22 -0.34 -0.30 -0.32 -0.28 -1.49

R. Instrument: Injuries to Spouses Only, Employees with Child Injuries Excluded from SampleN= 26,690 Elasticity -0.24 -0.61 -0.64 -0.95 -0.45 -0.25 -1.33 -1.27 -1.38 -0.85

lower bound -1.50 -0.63 -0.67 -1.32 -1.50 -1.42 -1.41 -1.34 -1.44 -1.50 upper bound 1.02 -0.59 -0.61 -0.58 0.60 0.93 -1.25 -1.19 -1.32 -0.19

S. Instrument: Simulated Spending on InjuryN= 29,161 Elasticity -0.25 0.13 -0.88 -0.64 -0.53 -0.55 -1.31 -1.32 -1.37 -1.43

lower bound -1.50 -0.63 -1.17 -0.67 -1.48 -1.46 -1.41 -1.41 -1.44 -1.50 upper bound 1.00 0.89 -0.59 -0.62 0.43 0.35 -1.21 -1.24 -1.30 -1.35

T. Instrument: Dummies for Nine Injury TypesN= 29,161 Elasticity -0.38 -0.15 -0.88 -1.06 -0.90 -1.24 -1.29 -1.38 -1.39 -1.50

lower bound -1.50 -1.05 -1.13 -1.47 -1.48 -1.46 -1.41 -1.41 -1.44 -1.50 upper bound 0.74 0.74 -0.63 -0.64 -0.31 -1.01 -1.17 -1.34 -1.34 -1.50

U. Instrument: Separate Dummies for Injuries in First and Second Half of YearN= 29,161 Elasticity -1.50

lower bound -1.50 upper bound -1.49

Lower and upper bounds for specifications J and K account for intra-family correlations.

still running

Specifications L, M, and N explore alternative sample selection criteria in the employee sample. The main

specification, A excludes injured employees on the grounds that the exclusion restriction is less likely to be

satisfied for them - another family member’s injury could be related to their injury through a mechanism

other than price if both family members were injured at the same time, leading to a larger estimated elasticity.

However, the cost of this sample selection is that it might reduce external validity of the findings, or it might

bias the findings through selection on the dependent variable. Specification L leaves the 603 employees with

own injuries in the estimation sample. Since the instrument is defined as an injury of another family member,

not all of these employees have a value of one for the instrument - only 100 have another family injury.

Estimates are similar or only slightly larger in the specifications that include injured employees, suggesting

that sample selection to exclude injured employees does not have a large impact on the results.

Specifications M and N examine robustness to the major sample restriction - the requirement that everyone

be continuously enrolled. I impose this restriction because the dependent variable measures year-end spending.

7

Year-end spending will be artificially low if agents are not in the sample for the entire year, and we will not

capture all responses to family injuries if agents are not in the sample for the entire year. However, as long

as individuals are not differentially continuously enrolled based on family injury status, the results should be

the same as they are in the main specification. In specification M, I do not require employees or their family

members to be continuously enrolled, and in specification N, I only require employees, and not their family

members, to be continuously enrolled. The results are generally similar.

Finally, I examine robustness to estimating the results using 2004 data instead of 2003 data. The elasticities

presented in specification O show remarkably similar patterns. The similarity of the estimates between 2003

and 2004 provides some evidence of robustness, and it suggests that price responsiveness did not change

between 2003 and 2004.

Online Appendix 4 Robustness to Specification of the Instrument

In this section, I examine the robustness of my results to alternative specifications of the instrumental variable.

As discussed in Section 6.1 of the paper, I restrict the categories of injuries in my instrument to those for

which employee spending is similar before the injuries. If employees in families with some categories of injuries

spend more even before the injuries occur, estimates from specifications that use those categories will result

in elasticity estimates away from zero. In specification P of Tables OA2 and OA3, I specify the instrument to

include textitall injury categories. The results are larger than those in the main specification at most quantiles,

but only slightly, suggesting that some injury categories that are not included in my main specification might

be endogenous, but that any bias that they would generate is small. In all other specifications, I consider only

the injury categories included in the main specification.

Next, I explore alternative specifications of the instrument to investigate a specific channel for violation

of the exclusion restriction. There is potential cause for concern if family injuries affect family income and

family income affects expenditure. I cannot control for income directly because I do not observe it. However,

I can informally investigate the role of income effects by estimating separate specifications based on injuries

to spouses and injuries to children. If there are large income effects due to the injury of a wage earner, we

might expect an employee’s response to a spouse’s injury to be different than an employee’s response to a

child’s injury. In specification Q, I only include child injuries in the instrument and drop employees with other

family injuries; in specification R, I only include spouse injuries in the instrument and drop employees with

other family injuries. The logarithmic specification with just child injuries gives almost the exact same point

estimates as the main specification, which is not surprising given that 4/5 of the injuries in the sample are to

children. The specification with just spouse injuries, which is not as well identified, also yields point estimates

that are the similar in magnitude but slightly smaller, suggesting that variation in the estimates due to child

vs. spouse injuries is not large relative to the main elasticity estimates.

In the next three specifications, I incorporate more variation into the specification of the instrument by

exploiting spending on injuries, injury categories, and the timing of injuries. In the main specification, the

instrument is a dummy variable that indicates whether another family member had an injury. I do not specify

the instrument as the amount of spending on the family injury because such a specification could violate the

exclusion restriction if family members see the same expensive doctors or consume care in the same expensive

geographic area. However, by restricting our instrument to a dummy variable, I lose some meaningful variation.

In specification S, I incorporate the amount of spending on the injury using simulation techniques pioneered by

Currie and Gruber (1996a, 1996b). First, within each of the seven injury categories, I calculate mean spending

over all individuals with that injury. Then, I assign this mean to the family members of the injured, removing

8

only their family member’s spending from the mean. In this way, I am able to use variation in the magnitude

of spending on the injury, excluding variation that could be correlated within families. Although it would

have been possible to create a plan-specific measure of average injury spending, in practice, some of the cell

sizes are very small. As compared to the main specification, the results in specification S are slightly smaller,

but in a similar range. The smaller estimates seem to be an artifact of the wider confidence intervals. I might

have expected the confidence intervals to be tighter because of the additional variation in the instrument, but

it seems that this specification sacrifices power because of the greater demands that it puts on the data.

Specification T incorporates more variation into the instrument by specifying it as a vector of dummy

variables for each of the nine injury categories included in the instrument, listed in Table 3. The results are

very similar to the main results. Finally, in specification U, I exploit additional variation in injury timing in

the specification of the instrument. I generate two instruments by interacting the family injury dummy with

an indicator for the half of the year when the injury occurred. Again, the results are very similar to those

from the main specification. The next section considers the implications of within-year injury timing for the

estimates and demonstrates that the estimates should indeed be similar regardless of injury timing.

Online Appendix 5 Implications of Within-Year Injury Timing

In this section, I examine the implications of injury timing for my instrumental variable estimates. To do so,

I examine separate CQIV specifications in which I only include injuries in the first half of the year or only

include injuries in the second half of the year in the determination of the instrument, excluding individuals

with family injuries in the other half of the year from the sample. In the first panel of Tables OA4 and OA5,

I present estimates with the dependent variable in logarithmic and level form. The estimates are very similar,

regardless of the timing of the family injury used. This finding is consistent with the model discussed in

Section 2.3 of the paper: after an injury occurs, it takes time for the family member’s marginal price, as well

as his spending, to respond. On net, if the family member’s marginal price responds at roughly the same rate

as his spending responds, the IV estimate, which reflects the spending response (the reduced form) divided by

the price response (the first stage), will be roughly the same regardless of the injury timing.

Table OA4: Timing of Family Injury

Dependent Variable: Ln(Expenditure).

2004 Sample 10 20 30 40 50 60 70 80 90

Injury in first half of the year Tobit IVN= 27,104 Elasticity -0.74 -0.74 -1.06 -1.45 -1.49 -1.41 -1.38 -1.43 -1.40 -1.43

lower bound -1.49 -1.49 -1.47 -1.49 -1.50 -1.50 -1.47 -1.47 -1.45 -1.50 upper bound 0.00 0.02 -0.66 -1.40 -1.48 -1.33 -1.30 -1.38 -1.36 -1.37

Injury in second half of the year Tobit IVN= 27,030 Elasticity -1.24 -0.58 -1.05 -1.44 -1.49 -1.34 -1.33 -1.32 -1.37 -1.31

lower bound -1.49 -1.49 -1.48 -1.49 -1.50 -1.50 -1.45 -1.46 -1.46 -1.49 upper bound -1.00 0.33 -0.61 -1.39 -1.48 -1.17 -1.21 -1.18 -1.29 -1.13

first stage

Injury in first half of the year - reduced form Tobit OLSN= 27,104 Family Injury . 0.20 0.32 0.52 0.44 0.40 0.35 0.35 0.29 0.65 -0.10

lower bound . -0.20 -0.10 0.23 0.22 0.23 0.21 0.19 0.13 0.36 -0.12 upper bound . 0.60 0.74 0.82 0.66 0.58 0.49 0.51 0.45 0.94 -0.08

Injury in second half of the year - reduced form Tobit OLSN= 27,030 Family Injury . 0.17 0.43 0.37 0.28 0.25 0.23 0.31 0.28 0.50 -0.10

lower bound . 0.00 0.00 0.11 0.10 0.09 0.05 0.14 0.12 0.23 -0.12 upper bound . 0.34 0.86 0.63 0.45 0.41 0.41 0.48 0.44 0.77 -0.08

In first stage results, dependent variable is year-end price.

Censored Quantile IV

Censored Quantile Regression

9

Table OA5: Timing of Family Injury

Dependent Variable: Expenditure.

2004 Sample 10 20 30 40 50 60 70 80 90

Injury in first half of the year Tobit IVN= 27,104 Elasticity -0.26 -0.58 -0.62 -0.78 -0.66 -1.29 -1.34 -0.80 -1.39 -1.49

lower bound -1.50 -0.63 -0.63 -0.93 -1.47 -1.48 -1.41 -1.43 -1.45 -1.50 upper bound 0.98 -0.53 -0.62 -0.62 0.16 -1.09 -1.26 -0.17 -1.33 -1.49

Injury in second half of the year Tobit IVN= 27,030 Elasticity -0.28 -0.64 -0.65 -0.83 -0.55 -1.23 -1.32 -1.31 -1.39 -1.24

lower bound -1.50 -0.67 -0.69 -1.04 -1.50 -1.44 -1.41 -1.41 -1.45 -1.50 upper bound 0.93 -0.60 -0.62 -0.62 0.39 -1.03 -1.22 -1.21 -1.32 -0.98

first stage

Injury in first half of the year - reduced form Tobit OLS OLSN= 27,104 Family Injury . 26 16 46 71 109 209 472 1,175 907 526 -0.10

lower bound . 0 -7 13 27 48 80 190 325 433 187 -0.12 upper bound . 51 40 78 115 170 339 754 2,025 1,380 865 -0.08

Injury in second half of the year - reduced form Tobit OLS OLSN= 27,030 Family Injury . 28 27 29 48 75 154 394 924 603 347 -0.10

lower bound . 0 2 -7 13 5 31 153 234 268 85 -0.12 upper bound . 57 53 66 83 144 277 635 1,613 939 609 -0.08

In first stage results, dependent variable is year-end price.

Censored Quantile IV

Censored Quantile Regression

In the bottom panel of Tables OA4 and OA5, I examine the spending response to the injury (the reduced

form), and the price response to the injury (the first stage) separately. To examine the reduced form, I estimate

CQR, Tobit, and OLS specifications.33 These results, especially the Tobit results, show that the expenditure

response is larger for injuries that occur in the first half of the year than it is for injuries that occur in the

second half of the year. The first stage OLS results also show that the price response is larger for injuries

that occur in the first half of the year. The combination of these findings explains why the CQIV results are

generally not sensitive to injury timing.

Online Appendix 6 Analysis of Data on Smaller Families

Using data beyond my estimation sample, I examine the spending behavior of employees in families with fewer

than four individuals (the main results are based on employees in families of four or more). People in smaller

families have the potential to serve the basis for an indirect test of the exclusion restriction, but the test has

several potential flaws. The results do not yield support for the exclusion restriction, but I report them here

in the interest of transparency.

The exclusion restriction requires that one family member’s injury can only affect another family member’s

expenditure through its effect on his marginal price. If program rules dictate that cost sharing interactions

cannot occur, one family member’s injury should not be related to another family member’s medical expendi-

ture. At the firm that I study, in policies for families of two, one family member’s spending has no mechanical

effect on another family member’s marginal price as shown in Table 1 in the paper. Therefore, any effects

of one family member’s injury on another family member’s spending presumably operate through another

channel. Although the exclusion restriction is not an econometrically testable restriction in the main sample

of families of four or more, evidence that there is no effect of one family member’s injury on another family

member’s spending in a family of two would support the validity of the exclusion restriction in the main

sample.

To formalize this test, I estimate the following specification with censored quantile regression:

33I only estimate an OLS specification of the reduced form for the levels model because it is unclear how to model individualswith zero expenditure in an OLS model of the logarithm of expenditure.

10

Y = max(Y ∗, C) = T ((Yi)∗)

Y ∗ = W ′θ(U) + ξ(U)Z

where the regressors are defined above. This specification differs from the main specification only in that

it examines the reduced form effect of the family injury on Y directly. A traditional instrumental variable

specification would not be informative here because the first stage should not exist in families of two.34

The main limitation of this test is that if employees in families of two are different from employees in

families of four or more, the effect of one member’s injury on another member’s spending could be nonzero

in families of two even if there is no violation of the exclusion restriction in families of four or more. Indeed

employees in families of two have different observable characteristics than employees in families of four or

more, so it is unclear if effects of one family member’s injury on another family member’s spending should

be comparable across samples. I investigate several sample restrictions to make the samples of employees in

families of both sizes similar in terms of observable characteristics, but unobservable characteristics could still

differ.

Column 7 of Table 2 in the paper presents summary statistics on employees in all families of two. Compar-

ison of year of birth with column 1 shows that this population is much older than the population in families

of four - it consists of many older “empty nesters.” Furthermore, only 27% of employees in families of two

consume zero care, as opposed to 36% in families of four or more. Employees in families of two have much

higher average expenditures on medical care than their counterparts in families of four or more ($2,615 vs.

$1,414). Given that employees in families of two consume more medical care, we might be more likely to

observe spurious effects of other family injuries on spending in the families of two than we are in the main

sample. We should keep this caveat in mind when interpreting the results.

I first report results from the sample of employees in all families of two without any sample restrictions.

If the test is valid and the exclusion restriction holds, we expect to see no effect of another family member’s

injury on the employee spending in families of two, but the CQR results in specification A of Tables OA6

and OA7 show that employees with a family injury spend from 0 to 41% more or from $0 to $736 more than

employees without family injuries. In larger families, we expect to see a larger impact of one family member’s

injury on another family member’s expenditure. As shown in specification G of Tables OA6 and OA7, the

coefficients in the family specification suggest that employees with an injured spouse or child spend 15 to 29%

more or from $20 to $813 more on their own medical care than employees without family injuries. Although

the Tobit coefficient is slightly larger in the sample of families of four or more, suggesting that there is more of

an effect of one family member’s injury on the employee’s expenditure in larger families, the Tobit coefficient

in families of two is not a precisely estimated zero. It is possible that employees in families of two are different

from employees in families of four or more, invalidating the test.

One reason why families of two might be different from larger families is that identification in the smaller

families comes mainly from injuries to spouses, and identification in the larger families comes from injuries to

children as well as spouses. In specification B, I restrict the sample to employee-dependent families of two,

and in specification H, I restrict the main sample so that it does not include employees with spouse injuries;

both samples are identified from injuries to non-spouse dependents. In specifications C and I, I restrict the

samples so that they are identified from injuries to spouses. Again, the results are somewhat larger in the

34If we think of the instrumental variable specification as the reduced form estimate divided by the first stage estimate, theinstrumental variable estimate will not exist when the first stage estimate is zero.

11

specifications in families of four, but the results are nonzero in families of two.

Yet another concern with the comparison by family size is that employees in families of two are much older

than the employees in larger families. In specifications D and J, I restrict both samples to include employees

aged 40 and under. Even though the unrestricted sample of employees in families of two is much larger than

the sample of employees in families of four or more, it only includes 6,951 employees under age 40, but the main

sample includes 18,972 employees under age 40. Similarly, in specifications E and K, I restrict both samples to

include employees over age 40. As shown, in families of four or more, the effect of a family member’s injury on

the employee’s expenditure is larger if the employee is older. The mechanism for this comparison is unclear,

but it suggests that we are more likely to find relationships in the older sample of families of two than we are

in the younger main sample based on age. Unfortunately, the younger sample of families of two is so small

that the results are imprecise.

For completeness, I also present results estimated on families of three. As discussed in the paper, there are

some cost sharing interactions in families of three that occur through the stoploss, so the results in families

of three should show some effect of one family member’s injury on the employee’s expenditure. This is indeed

the case, as shown in specification F.

Although the results would be compelling if the specification restricted to families of two yielded a precisely

estimated zero, there are several reasons why it would not in the absence of a violation of the exclusion

restriction. For example, if employees do not understand the interaction between the individual and family

deductibles, then they might think that their own price will go down after their family member has an injury

regardless of family size. Especially since many members of the sample of families of two appear to be empty-

nesters, they might behave as if another family member’s injury will affect their spending, as it did when

they had children on their policy in the past. In sum, this analysis does not provide support for the exclusion

restriction, but there are many reasons to question its usefulness.

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Table OA6: Family Injuries in Smaller and Larger Families

Dependent variable: Ln(Expenditure)

2004 Sample 10 20 30 40 50 60 70 80 90 Tobit

Families of TwoA. Employees in All Familes of TwoN= 54,889 Family Injury 0.00 0.41 0.31 0.36 0.23 0.24 0.20 0.21 0.16 0.48

lower bound 0.00 0.00 0.12 0.19 0.11 0.10 0.07 0.06 0.01 0.29 upper bound 0.00 0.82 0.51 0.53 0.36 0.39 0.32 0.37 0.30 0.67

B. Employees in Employee-Child Families of TwoN= 17,338 Family Injury 0.00 0.59 -0.11 0.15 0.25 0.13 0.20 0.14 0.08 0.35

lower bound 0.00 -0.22 -0.52 -0.14 -0.03 -0.22 -0.04 -0.14 -0.19 -0.06 upper bound 0.00 1.40 0.29 0.45 0.54 0.47 0.43 0.42 0.35 0.77

C. Employees in Employee-Spouse Families of TwoN= 37,551 Family Injury 0.34 0.40 0.46 0.33 0.27 0.27 0.19 0.17 0.14 0.49

lower bound -0.17 0.00 0.18 0.15 0.11 0.09 0.02 0.01 -0.03 0.28 upper bound 0.85 0.79 0.73 0.50 0.43 0.45 0.36 0.32 0.32 0.70

D. Employees 40 and under in Employee-Spouse Families of TwoN= 6,951 Family Injury . 2.41 0.30 0.29 0.27 0.22 0.21 0.22 0.15 0.50

lower bound . 0.00 -0.05 0.00 -0.08 -0.04 -0.03 -0.09 -0.09 0.10 upper bound . 4.81 0.65 0.58 0.62 0.48 0.46 0.53 0.39 0.90

E. Employees over 40 in Employee-Spouse Families of TwoN= 30,600 Family Injury 0.00 0.58 0.37 0.30 0.25 0.25 0.18 0.16 0.09 0.44

lower bound 0.00 0.06 0.13 0.15 0.11 0.10 0.02 0.02 -0.10 0.24 upper bound 0.00 1.11 0.60 0.45 0.40 0.40 0.34 0.30 0.28 0.64

Families of ThreeF. Employees in All Families of ThreeN= 25,482 Family Injury 0.00 2.15 0.38 0.34 0.31 0.20 0.13 0.07 0.04 0.64

lower bound 0.00 0.00 0.00 0.12 0.14 0.07 0.00 -0.10 -0.13 0.39 upper bound 0.00 4.31 0.77 0.55 0.48 0.33 0.25 0.24 0.21 0.90

Families of Four or MoreG. Employees in Families of Four or More N= 29,161 Family Injury . 0.15 0.37 0.46 0.36 0.31 0.30 0.31 0.29 0.57

lower bound . 0.00 0.00 0.25 0.22 0.17 0.18 0.20 0.16 0.36 upper bound . 0.30 0.73 0.66 0.49 0.45 0.41 0.43 0.42 0.78

H. Employees in Families of Four or More - Excluding Employees with Spouse InjuriesN= 28,547 Family Injury . 0.00 0.42 0.44 0.36 0.35 0.33 0.35 0.32 0.59

lower bound . 0.00 0.00 0.18 0.17 0.20 0.20 0.22 0.18 0.36 upper bound . 0.00 0.84 0.69 0.56 0.50 0.46 0.48 0.46 0.82

I. Employees in Families of Four or More - Excluding Employees with Child InjuriesN= 26,690 Family Injury . 0.62 0.30 0.44 0.28 0.27 0.27 0.23 0.10 0.54

lower bound . -0.01 -0.18 0.06 -0.03 0.04 0.07 0.01 -0.18 0.15 upper bound . 1.25 0.78 0.82 0.60 0.49 0.47 0.45 0.39 0.93

J. Employees 40 and under in Families of Four or MoreN= 18,972 Family Injury . -0.03 0.20 0.30 0.40 0.32 0.23 0.29 0.22 0.54

lower bound . -0.11 -0.05 0.09 0.23 0.19 0.09 0.13 0.06 0.31 upper bound . 0.06 0.46 0.51 0.58 0.46 0.36 0.45 0.38 0.76

K. Employees over 40 in Families of Four or MoreN= 10,189 Family Injury . 0.00 0.37 0.59 0.40 0.38 0.44 0.45 0.36 0.61

lower bound . 0.00 0.00 0.15 0.15 0.11 0.24 0.21 0.15 0.30 upper bound . 0.00 0.74 1.02 0.64 0.64 0.65 0.69 0.58 0.91

Censored Quantile Regression

Family of Two controls: male dummy, plan (saturated), census region (saturated), salary dummy (vs. hourly), count family born 1944 to 1953, count family born 1954 to 1963, count family born 1964 to 1973, count family born 1974 to 1983, count family born 1984 to 1993.Family of Four controls: family of two controls, spouse on policy dummy, YOB of oldest dependent, YOB of youngest dependent, family size (saturated with 8-11 as one group), count family born 1994 to 1998, count family born 1999, count family born 2000, count family born 2001, count family born 2002, count family born 2003, count family born 2004.

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Table OA7: Family Injuries in Smaller and Larger Families

Dependent variable: Expenditure

2004 Sample 10 20 30 40 50 60 70 80 90 Tobit

Families of TwoA. Employees in All Familes of TwoN= 54,889 Family Injury 0 5 19 29 121 70 288 83 736 909

lower bound 0 -39 -51 -75 -46 -235 -247 -1,326 -539 445 upper bound 0 48 88 134 288 375 823 1,491 2,011 1,373

B. Employees in Employee-Child Families of TwoN= 17,338 Family Injury 0 25 1 12 39 78 151 401 490 373

lower bound 0 0 -37 -61 -25 -60 -39 -134 -538 -224 upper bound 0 51 39 86 102 216 341 935 1,519 971

C. Employees in Employee-Spouse Families of TwoN= 37,551 Family Injury . 16 47 67 130 181 272 919 342 1,014

lower bound 0 -41 -13 -25 -2 -303 -253 -155 -1,778 419 upper bound 46 73 106 159 262 666 798 1,993 2,461 1,608

D. Employees 40 and under in Employee-Spouse Families of TwoN= 6,951 Family Injury . 35 39 59 54 88 107 449 511 502

lower bound . 0 -2 0 2 -27 -25 -42 -280 8 upper bound . 70 81 118 105 203 238 940 1,302 997

E. Employees over 40 in Employee-Spouse Families of TwoN= 30,600 Family Injury 0 16 38 61 50 84 739 729 237 935

lower bound 0 -30 -16 -23 -256 -542 33 -574 -1,907 385 upper bound 0 62 93 145 356 709 1,445 2,031 2,382 1,484

Families of ThreeF. Employees in All Families of ThreeN= 25,482 Family Injury -29 30 41 79 78 103 122 173 314 524

lower bound -58 0 12 23 28 36 4 -78 -340 171 upper bound 0 61 70 135 128 170 240 424 969 878

Families of Four or MoreG. Employees in Families of Four or More N= 29,161 Family Injury . 23 20 39 54 94 172 379 813 727

lower bound . 0 2 14 21 35 70 91 -148 422 upper bound . 47 38 63 88 154 274 667 1,775 1,032

H. Employees in Families of Four or More - Excluding Employees with Spouse InjuriesN= 28,547 Family Injury . 10 18 38 56 102 189 468 1,269 839

lower bound . 0 0 11 17 52 83 214 633 483 upper bound . 20 35 64 95 152 296 722 1,905 1,195

I. Employees in Families of Four or More - Excluding Employees with Child InjuriesN= 26,690 Family Injury . 59 25 33 52 101 144 430 582 476

lower bound 0 -7 -18 -13 -9 -29 82 -486 33 upper bound 118 56 83 117 211 317 779 1,650 920

J. Employees 40 and under in Families of Four or MoreN= 18,972 Family Injury . 26 18 92 51 85 138 290 602 488

lower bound 0 -4 5 12 19 48 80 35 261 upper bound 53 40 180 89 152 228 500 1,168 715

K. Employees over 40 in Families of Four or MoreN= 10,189 Family Injury . 24 22 41 83 158 336 708 1,737 1,197

lower bound 0 -8 8 24 61 158 221 478 536 upper bound 48 51 74 143 256 515 1,194 2,995 1,859

Censored Quantile Regression

Family of Two controls: male dummy, plan (saturated), census region (saturated), salary dummy (vs. hourly), count family born 1944 to 1953, count family born 1954 to 1963, count family born 1964 to 1973, count family born 1974 to 1983, count family born 1984 to 1993.Family of Four controls: family of two controls, spouse on policy dummy, YOB of oldest dependent, YOB of youngest dependent, family size (saturated with 8-11 as one group), count family born 1994 to 1998, count family born 1999, count family born 2000, count family born 2001, count family born 2002, count family born 2003, count family born 2004.

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