Censored Quantile Instrumental Variable Estimates

of the Price Elasticity of Expenditure on Medical Care

Amanda Kowalski, Yale University

October 17, 2014

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 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: censoring, control function, endogeneity, medical care.

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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 increas-

ingly important public policy issue. Existing efforts to control costs depend critically on how

insured individuals respond to the prices that they face for medical care. The Medicare Mod-

ernization 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.

Econometricians 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 ex-

penditures 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 litera-

ture, 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 paramet-

ric assumptions that these methods require. (See Buntin and Zaslavsky (2004), Duan et al.

(1983), Mullahy (1998), and Newhouse et al. (1980).) Perhaps even more important than cen-

soring 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. In my estimation sample, drawn from a large recent

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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 address heterogeneity across the conditional expenditure distribution, censoring,

and endogeneity. I use a censored quantile instrumental variable (CQIV) estimator, devel-

oped specifically for this application by Chernozhukov et al. (2014). 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 ex-

penditure 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 dis-

tribution, 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 distribu-

tional 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 estima-

tors, the CQIV estimator is the same regardless of the model because it assumes that the zeros

arise from the same decision process as the uncensored observations, so I refer to expenditures

at zero as “censored” without loss of generality. In contrast, traditional censored mean esti-

mators 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

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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 a control function ap-

proach to an instrumental variable identification strategy. In traditional health insurance

policies, the price of an additional dollar of care is a function of expenditure. Thus, the esti-

mated relationship between price and expenditure will be biased away from zero unless this

endogeneity is taken into account. The intuition behind my instrumental variable identifica-

tion 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). Moving

beyond Eichner (1997, 1998), I 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. I also use new data that allows me to observe individuals who have zero

medical expenditures, and I develop and implement several tests of robustness.

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 elasticity estimates to vary, the estimates are relatively stable

across the conditional deciles.

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

mean elasticity of -0.22 (Newhouse and the Insurance Experiment Group (1996)). 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

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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 speci-

fication, and 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.

I discuss the CQIV model and estimation in the next section. In Section 3, I describe

my data, I motivate the instrumental variable identification strategy that arises from my

empirical context, and I present graphical evidence of the variation that drives my results.

In Section 4, I present results based on CQIV and other estimators. In Section 5, I describe

the results from several robustness tests, I discuss potential mechanism behind my results,

and I provide intuition for why my results differ from those obtained from the Rand Health

Insurance Experiment. I conclude and discuss directions for future research in Section 6.

2 CQIV Model and Estimation

Here, I describe the CQIV model and estimation, and I provide intuition to the applied

researcher. For further detail, consult Chernozhukov et al. (forthcoming).

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2.1 CQIV Model

The CQIV model is based on the following triangular system of equations:

Y = max(Y ∗, C) = T (Y ∗) (1a)

Y ∗ = QY ∗(U |P,W, V ) (1b)

P = QP (V |W,Z) (1c)

where Y is the observed dependent variable and T (x) ≡ max(x,C) is the transformation func-

tion that censors the unobserved uncensored dependent variable Y ∗ from below at censoring

point C. In my empirical application, I examine robustness to specifying Y in two ways: Y can

represent observed medical spending censored at C = 0, or the logarithm of observed medical

spending censored at C = −0.7. (The logarithm of zero is negative infinity, so I censor it at

C = −0.7, a value that is smaller than the logarithm of the smallest observed expenditure in

the data: 50 cents. Results are robust to the choice of censoring point for values that do not

eliminate information.) P is the year-end marginal price of medical care, which is potentially

endogenous; W is a vector of covariates; the function u 7→ QY ∗(u | P,W, V ) is the conditional

quantile function of Y ∗ given (P,W, V ); v 7→ QP (v | W,Z) is the conditional quantile func-

tion of P given (W,Z); Z is an indicator for family injury, an instrumental variable that is

excluded from Equation (1b); U is a Skorohod disturbance that satisfies the full independence

assumption

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

and V is a Skorohod disturbance that satisfies

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

The full independence assumption states that the disturbance is distributed uniformly on

the interval (0,1), conditional on P,W,Z, V, and C. This assumption requires that the entire

6

distribution of the disturbance in the equation determining Y ∗ is independent of P,W,Z, V

and C. 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 dis-

cussion in Section 3.2. The main other assumption required for identification is the relevance

condition of Z in Equation (1c). Given these assumptions, we can recover the conditional

quantile function of Y ∗ using V as a “control variable.”

The CQIV model allows for nonparametric functional forms for Equations (1b) and (1c).

These functional forms could include terms for nonparametric estimation such as power series

and splines, and the quantile functions need not be additive in U and V. For simplicity of

computation, 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 δ(U) = (α(U), β(U), γ(U)) is the coefficient function of interest. In particular, α(U)

yields the marginal effect of price on latent expenditure.

Since the censoring in my application arises from a corner solution decision, I can perform

a “corner calculation” to obtain the marginal effect of price on observed expenditure at each

U as follows:

α(U)1{Y ∗ > C},

where 1{Y ∗ > C} is an indicator for latent expenditure greater than the censoring point at

the given U. This calculation is consistent with the recommendation of Wooldridge (2010)

and the updated version of Machado and Silva (2008). Applying the corner calculation is not

equivalent to estimating an uncensored quantile model.

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

role. Bitler et al. (2006), who analyze welfare reform experiments, can compute simple quan-

tile 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. (See Gilleskie and Mroz (2004)

for an approach to estimating the effects of covariates on health expenditures.) Also, covari-

ates are important in the CQIV estimation as discussed below - they are used to predict the

observations for which the conditional quantile function is above the censoring point.

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

quantiles of expenditure 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. Given this linearity, we expect results that specify the dependent

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

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

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. As I discuss below, I observe two price changes in my application, first from 1

to 0.2 and then from 0.2 to 0. It is unclear if the response to both price changes should

be proportional to those price changes, as required by the previous specification. Therefore,

I also investigate robustness to the following functional form, which allows me to test the

appropriateness of the previous specification, which it nests:

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

X ′ = (1{P = 1}, 1{P = 0},W ′, V ) (3b)

P = ρZ +W ′θ + V (3c)

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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 et al. (1999). Because the price P only takes on three values, (3a) is nonparametric

in P . However, I continue to use the parametric specification of (3c) to estimate the control

term. In traditional instrumental variable models of the conditional mean, a rank condition

generally implies that a model with two endogenous variables requires two instruments. As

in Newey et al. (1999), two instruments are not necessary here because there is only a single

endogenous variable, specified nonparametrically in the structural equation.

As in traditional models, we can interact the marginal price variable with a covariate

to examine heterogeneity 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. If unobserved heterogeneity is one-dimensional

and if the quantiles of unobserved heterogeneity are the same as the quantiles of the expen-

diture distribution conditional on covariates, then the estimated coefficients at the highest

quantiles will yield price responsiveness for individuals who spend the most. Alternatively, U

could represent the quantiles of unobserved health or hypochondria.

2.2 Estimation

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

advantages of the CQIV estimator relative to other models in my empirical context, 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

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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 conditional quantile, if

some observations on Y ∗ are censored, the conditional quantile regression lines can be biased

toward zero at all conditional quantiles. The Powell (1986) estimator overcomes this difficulty

by incorporating censoring directly into the estimator as follows:

β(u) minimizesn∑

i=1

ρu(Yi − 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 values of Y equal

to zero arise from a censoring process or through a separate decision process, without the

transformation function, T (x) ≡ max(x,C), the conditional quantiles of X ′iβ(U) could be

infeasible (beyond the censoring point C) at some quantiles. The transformation function is a

nonparametric way to assure that infeasible predictions do not introduce bias into the objective

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 re-

gression (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 cen-

sored. The algorithm is a three-step procedure that selects the observations for which the

conditional quantile function is above the censoring point. The first step involves a paramet-

ric 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 regres-

sion in the second step. After the second step, a larger set of observations is retained based

10

on the estimated conditional quantiles values of the dependent variable. This sample gets

asymptotically close to the ideal sample of “quantile-uncensored,” and consistent estimates

are obtained through a third step of quantile regression on this sample. The CQIV compu-

tational 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 tra-

dition of Hausman (1978). 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 not

technically satisfied when the endogenous variable is discrete. The handling of discrete en-

dogenous variables in control function models is an area that requires more study. Recent

work by Frandsen (2014) examines a censored outcome and a binary endogenous regressor in

a quantile model. To address the discreteness of my endogenous variable, I estimate a version

of the CQIV estimator that uses a Chernozhukov and Hansen (2008) moment condition ap-

proach to endogeneity in lieu of the control term approach. This approach does not require

continuity in the endogenous variable. The results, reported in Kowalski (2009), are almost

identical to those presented here.

As discussed in Chernozhukov et al. (forthcoming), the CQIV estimator does not require

an additive error in the first or second stages, which sets it apart from the alternative cen-

sored quantile instrumental variable estimator proposed by Blundell and Powell (2007). The

Blundell and Powell (2007) estimator requires additive errors in the first and second stages,

while allowing for a local nonparametric endogenity correction in the second stage. In the

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

from the first stage equation. The results that follow are robust to the inclusion of higher

order functions of the control term in the structural equation. Results that use an alternative

11

quantile specification of the first stage as implemented in Chernozhukov et al. (forthcoming)

are similar.

I obtain 95% confidence intervals on the coefficients via a nonparametric bootstrap. Online

Appendix 1 provides more information on the bootstrap procedure and shows that the results

are robust to a weighted bootstrap procedure shown to be consistent by Chernozhukov et al.

(forthcoming). 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. To facilitate comparison across estimators, I follow the

same practice for all estimators in the paper unless otherwise noted.

3 Data

3.1 Data Description

To estimate my model, my data must include medical expenditure and marginal price, and

they must allow me to observe family structure so that I can construct my instrument. Data

compiled by MEDSTAT Group Inc. (2004) meet these criteria. 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.

Within the Medstat data, I focus on a US firm in the retail trade industry with over

500,000 insured employees. I use 2004 data in my main analysis, and I also examine 2003 and

2005 in other analyses. I provide details on sample selection in Online Appendix 2. In my

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

but almost 40% of people consume zero care. 57.2% of beneficiaries face a marginal price of

one, 39.0% of employees face the coinsurance rate of 0.2, and 3.8% of employees have met

the stoploss and face a marginal price of zero. 10.9% of employees have at least one family

12

member who is injured. I provide more extensive summary statistics in Online Appendix 3.

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 ad-

vantage 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 Annual Survey

of Employer Health Benefits conducted by Kaiser Family Foundation and Health Research &

Educational Trust (henceforth, “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. 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 pay-

ments. 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.

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

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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 by estimating separate specifications by plan.

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.

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. Note that each family member must meet the individual deductible

before the family deductible can be met. Online Appendix 4 depicts the insurance-induced

budget sets for individuals and families, and Online Appendix 5 provides a simple model of

how agents maximize utility subject to those budget sets, resulting in a demand curve.

14

The cost sharing parameters 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. In contrast, Duarte (2012) considers Chilean plans that have much more

compilicated cost sharing structures.

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 2006 Kaiser

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

in Online Appendix 6, 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 Identification

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? A simple

identification strategy would 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

15

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 – 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 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. Family medical spending would also be an interesting

outcome variable, but it would entail a mechanical violation of the exclusion restriction. 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

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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 et al. (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 in Online Appendix 3, 10.9% 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

17

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 of specifying

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

alternative specifications detailed in the Online Appendix 12, 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.

3.3 Graphical Evidence

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

that it can be seen graphically, before implementing any 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.

To demonstrate the reduced form, in the left panel of Figure 1, I present the cumulative

distribution (cdf) of the logarithm of expenditure conditional on family injury. The cdf for

employees with no family injury is represented by a solid line, and the cdf 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 reas-

surance 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 exten-

18

Figure 1: 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.

sive 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

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.

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

panel of Figure 1, 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

19

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 right 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.

As a more formal alternative to the right panel of Figure 1, 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 make the estimate 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.

In Online Appendix 3, I report 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,

20

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. 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 Results

Table 2 reports the main CQIV elasticities, derived from estimating equations that vary the

specification of expenditure and price 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. Specifications A and C model the logarithm

of expenditure, and specifications B and D model the level of expenditure.

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 Online Appendix 7, I discuss several approaches

to transform the estimated coefficients into arc elasticities, which are preferable to point elas-

ticities 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 identifica-

tion comes from the larger price change from before to after the deductible - from 1 to 0.2, I

report corner arc elasticities from this range.

21

Table 2: 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 in Online

22

Appendix 7 related to Table OA7 gives us another way of thinking about this coefficient in

the context of the estimated conditional quantiles 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.

It is important to note that the quantiles of expenditure are very different from the con-

ditional quantiles of expenditure. Tables OA7 and OA8 in Online Appendix 7 report the

quantiles of expenditure, as well as expenditure at each conditional quantile from models that

specify the logarithm and the level of expenditure, respectively. The 0.7 quantile of expendi-

ture is $531, which is much smaller than 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. As discussed in Section

2, people with higher expenditures conditional on their covariates might have higher income,

higher propensity for hypochondria, or more severe illnesses. Hence, covariates are an impor-

tant part of the model. As shown in Table 3, 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 2, we see that heterogeneity in price respon-

siveness 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 het-

erogeneity 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

23

Table 3: 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.

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.

In specifications B and D, when we specify the dependent variable as the level of ex-

penditure, 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

24

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 Online

Appendix 7, the elasticities in the levels models are smaller at smaller quantiles because the

corner calculation has a larger impact when the conditional quantiles of 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 in Table 3, and I present results from the level specification in Online

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, specifications 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, suggesting that the main

specification is appropriate.

In Table 2, 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. Online Appendix 8 discusses how the

stoploss elasticities might compare to the main elasticities, demonstrating that the comparison

could go in either direction. Empirically, the estimates show that the stoploss elasticity

is smaller than the main elasticity at almost all quantiles in all four specifications. The

magnitudes of the stoploss elasticities are somewhat larger in the 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

25

not very sensitive to the specification of price, I investigate the robustness of my results using

specifications with expenditure in logarithms and levels but 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 obtained

through the Stata command ivtobit for comparison, and they are generally in the same range

as the CQIV estimates.

5 Additional Analysis and Robustness

I peform extensive additional analysis, which I include in the Online Appendix. Although

the magnitudes of my estimated elasticities vary across specifications, all of my estimated

elasticities are much larger than the elasticity estimate of -0.22 obtained by the Rand Health

Insurance Experiment. In Online Appendix 9, I examine results derived from alternative

estimators, and I show that, regardless of the estimator, the elasticities that I find are large

relative to the literature.

In Online Appendix 10, I examine heterogeneity in treatment effects by observable charac-

teristics. 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, but estimates even among their counterparts are large relative to

those in the literature.

In Online Appendix 11 and Online Appendix 12, 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 13. In Online

Appendix 14, 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.

26

I also consider variations on the main specification to examine the mechanisms behind my

results. In Online Appendix 15, 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 Online Appendix 16, I explore the mechanisms behind my estimated elasticity, and

I show that agents mainly respond to price on the outpatient visit margin.

All analyses considered, my main results are an order of magnitude larger than those from

the Rand Health Insurance Experiment. In Online Appendix 17, I discuss methodological

differences between my approach and the Rand approach. The relative treatment of myopia

and foresight is one potential explanation for why the Rand results are so much smaller than

mine.

6 Conclusion

This paper makes several contributions. Using detailed data and a rigorous identification

strategy, I estimate the price elasticity of expenditure on medical care using a censored quantile

instrumental variable 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 elasticity estimates to vary across the conditional quantiles of

expenditure, the estimates are relatively stable across the conditional deciles. 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 estimates. I conclude that the price elasticity of expenditure on medical care is much

larger than the literature would suggest.

27

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Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (seconded.). Cambridge: MIT Press.

Acknowledgments

I am grateful to Amy Finkelstein, Jonathan Gruber, and Jerry Hausman for their guidance.I would like to thank the following individuals for thoughtful comments: Joseph Altonji,Michael Anderson, Joshua Angrist, David Autor, John Beshears, Soren Blomquist, TonjaBowen-Bishop, David Card, Amitabh Chandra, Victor Chernozhukov, David Cutler, Ian Dew-Becker, Deepa Dhume, Peter Diamond, Joseph Doyle, Esther Duflo, Jesse Edgerton, MatthewEichner, Ivan Fernandez-Val, Brigham Frandsen, John Friedman, Michael Greenstone, Ray-mond 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, Gerardo Sanchez, David Seif, Hui Shan, Mark Showalter, Erin Strumpf,Heidi Williams, and several anonymous referees who gave very constructive suggestions. Sem-inar participants at Chicago Booth, Brookings, Cornell, Duke, Kellogg, Maryland, Michigan,MIT, NBER Summer Institute, Notre Dame, Rand, Stanford SIEPR, Temple Fox, TexasA&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

29

Chaiken for excellent research assistance. Mohan Ramanujan and Jean Roth provided invalu-able help with the data. I gratefully acknowledge National Institute on Aging, Grant NumberT32-AG00186.

Amanda Kowalski, Yale and NBERDepartment of Economics, Yale University, Box 208264.New Haven, CT 06520-8264. Office: 202-670-7631. amanda.kowalski@yale.edu. Stata soft-ware to implement the quantile estimators used in the paper is available at https://ideas.repec.org/c/boc/bocode/s457478.html.

30

Online Appendix 1 Confidence Intervals

I draw 200 samples the size of the estimation sample, with replacement. I estimate the modelon each sample, retaining the estimated coefficients. I report the 0.025 and the 0.075 quantilesof each estimate as the lower and upper bounds of the 95% confidence interval. Although Iwould prefer larger bootstrap replication samples, I am limited by computational constraints.When I compute the confidence intervals in the main specification using 1000 bootstrap reps,the results are very similar.

In practice, estimates on several of the samples do not converge because one of the quantileestimation steps that relies on the Stata quantile regression algorithm does not converge, par-ticularly at the lowest quantiles. I report the number of bootstrap replications that convergeby quantile for the main specifications in the bottom rows of Tables OA4 and OA5. I excludethe estimates from the replication samples that do not converge from the calculation of confi-dence intervals. Specifically, the first quantile regression step does not converge at the 0.50 or0.90 quantiles in my application, so I cannot obtain point estimates. However, I can obtainestimates from the vast majority of bootstrap replication samples, so I report the mean of the95% confidence intervals obtained from the replication samples as the point estimate. TablesOA4 and OA5 report the point estimates as well as the mean of the confidence intervals anddemonstrate that both are very similar when they are both available. These tables also showthe elasticities that are derived from the point estimates and the mean elasticities derivedfrom the replication samples, which are also similar.

Table OA1: Nonparametric Bootstrap vs. Weighted Bootstrap

2004 Sample 10 20 30 40 50 60 70 80 90Dependent 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)CQIV, weighted bootstrap TobitN= 29,161 Elasticity -1.40 -0.73 -1.20 -1.47 -1.49 -1.41 -1.39 -1.41 -1.40 -1.43

lower bound -1.49 -1.49 -1.46 -1.49 -1.50 -1.49 -1.46 -1.45 -1.44 -1.49 upper bound -1.31 0.04 -0.94 -1.45 -1.48 -1.33 -1.31 -1.36 -1.35 -1.37

Quantile

Chernozhukov et al. (2014) provide a proof of the consistency of a weighted bootstrapprocedure, as opposed to a nonparamteric bootstrap procedure, because “it has practicaladvantages over nonparametric bootstrap to deal with discrete regressors with small cell sizesand the proof of its consistence is no overly complex.” I estimate results using a weightedbootstrap procedure for comparison. In each bootstrap replication sample, I draw a new setof weights (e1, ...en) from a standard exponential random variable, and I recompute the CQIVestimator in the weighted sample. I report the results in Table OA1. As shown in specificationA2, results computed using the weighted bootstrap are very similar results computed thenonparametric bootstrap. The reported point estimates differ because they reflect they reflectthe means of the 95% confidence intervals, as discussed above.

1

Online Appendix 2 Sample Selection

Although selection into the firm that I study could affect external validity, the firm hasemployees in every region of the United States, and it is large enough that idiosyncraticmedical usage should not be a problem. With over 800,000 people covered by the plansoffered 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 and2004, 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 ofmarginal price and continuous enrollment very accurate.

Within the firm, the main selection criterion that I apply is a continuous enrollmentrestriction. In my main results, which use the 2004 and 2003 data as separate cross-sections, Irequire that the family is enrolled from January 1 to December 31 of the given year. Selectiondue to the continuous enrollment restriction eliminates over 30% of the original sample ineach year. Analysis of other firms in the Medstat data suggests that the rate of turnover atthis firm is comparable to the rate of turnover at other large firms. Since my outcome ofinterest is year-end expenditure, and family members play a role in the determination of theinstrument, I only include individuals in my main sample if their entire families, with theexception of newborns, are enrolled for the entire plan year. I discuss summary statistics onthe sample before and after the continuous enrollment restriction in Online Appendix 3, and Ireport results that relax the continuous enrollment restriction in Online Appendix 11. I retainfamilies 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 thatmy 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 eliminatefamilies that switch plans, families that have changes in observable covariates over the courseof the year, and families that have demographic information that is inconsistent betweenenrollment and claims information. I also eliminate families that have unresolved paymentadjustments. Statistics on each step of the sample selection are available in a supplementaldata appendix.1 Taken together, these steps eliminate less than seven percent of individualsfrom the continuously enrolled sample.

In this clean sample, approximately 25% of employees with other insured family membersare insured in families of four or more. The 2004 main estimation sample includes 29,161employees 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 tofamilies 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 eachfamily, and I use other family members only in the determination of the instrument. In somespecifications, 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 doesnot take the price responsiveness of all family members into account, but, arguably, it providesthe best control for unobservables on the grounds that employees at the same firm have somecommon characteristics that they do not necessarily share with the spouses and children oftheir co-workers. Moreover, restricting the sample to employees eliminates the need to address

1http://www.econ.yale.edu/~ak669/Data_Appendix_07_08_13.pdf

2

possible correlations in price responsiveness among family members.

Online Appendix 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 expendituredistribution are responsible for 93% of expenditures.

The first panel of Table OA2 summarizes the expenditure distribution across bins thatfollow a logarithmic scale. The first column shows summary statistics for the main analysissample of employees. As shown, excluding individuals with zero expenditure, the distribu-tion of positive expenditure among employees is very skewed, with 31.3% of individuals inthe expenditure range between $100 and $1,000, and smaller percentages of individuals inthe bins above and below this range.2 The next three columns show summary statistics forspouses, children and other dependents, and the full sample. The statistics show that chil-dren generally have lower expenditures than employees and their spouses. Columns 8 and9 show characteristics of the sample before the continuous enrollment restriction. As ex-pected, year-end expenditures are smaller in this sample because these individuals have lessthan the full year to incur expenditures. In columns 10 and 11, I compare my sample to thenationally-representative 2004 Medical Expenditure Panel Survey (MEPS), first to all indi-viduals without injuries under age 65, then to a MEPS sample intended to be comparable tomy main estimation sample: a single respondent without injuries in each family of four ormore, 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 ofpeople with zero expenditures in the MEPS. However, average expenditures in the MEPSare lower, perhaps because the MEPS includes people in much less generous plans than theemployer-sponsored plans that I study, as well as the uninsured.

Panel B in Table OA2 depicts the distribution of the endogenous variable, the marginalprice for the next dollar of care at the end of the year. I calculate the marginal price to reflectthe spending of the individual and his family members. If the individual has not consumedany care and the family deductible has not been met, the marginal price takes on a value ofone 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 rate of0.2, and 3.8% of employees have met the stoploss and face a marginal price of zero. This pricevariation should be large enough to be meaningful.

The distribution of the instrument, “family injury,” shows that 10.9% of employees have atleast one family member who is injured in the course of the year. Since injured employees areexcluded from the sample, all of the injuries included in the determination of the instrumentin the employee sample are to spouses and other dependents. In the full sample, injuries toemployees are included in the determination of the instrument, and the same injury can bereflected as a “family injury” for more than one person. Overall, 10.2% of individuals in thefull sample have an injury in the family. I use the same criteria to determine injuries in theMEPS samples in columns 10 and 11, and I find a similar family injury rate.

2The first rows of Tables OA7 and OA8 report the unconditional deciles of expenditure.

3

Table OA2: 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

4

People with the injuries included in my instrument are excluded from all estimation sam-ples, but I report statistics on injured people in Table OA3. This table includes all categories

Table OA3: 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.

of injuries with corresponding ICD-9 codes in the first column. The second column identifiesthe injuries included in the instrument through the selection process discussed in Section On-line Appendix 15. If a person has any claim for an injury with an ICD-9 code in one of thelisted categories, he is included in the count in the third column. Sprains and strains of jointsand adjacent muscles are the most prominent. In column 4, I report the mean year-end totalexpenditures for the injured people to demonstrate that their spending should be large enoughto have a meaningful effect on the price that their family members face. Total expenditurescould capture injury-related spending that is induced by the injury but is categorized underdifferent diagnosis codes. In column 5 I only report spending for which the primary diagnosiscode is in a given injury category. Internal injuries of thorax, abdomen, and pelvis appear tobe the most expensive. I exploit variation in spending across injuries in Online Appendix 12.

Panels D through J of Table OA2 summarize the distribution of covariates. Family sizevaries from four to eleven, with 66.9% of employees in families of four. The full sample isgender balanced, but 57.4% of employees are male. All employees are between the ages of 20and 65 in 2004. The distribution of “year of birth” is bimodal because the sample includesparents and their children. Panel H shows that 29.8% of the employees are salaried, andthe remaining employees are hourly. One of the limitations of the Medstat data is that itdoes not include any income measures, but the salaried versus hourly classification serves asa crude proxy. I also investigate potential income effects in other ways, discussed below. Thedistribution of the sample by Census region demonstrates that the firm has a very nationalreach. The largest concentration of employees is in the West South Central Census region,where 28.3% of the sample resides.

5

Panel J depicts the distribution of employees and families across the four plans. Each planhas 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 plansreported in the 2006 Kaiser Annual Survey of Employer Health Benefits. The most generousplan, 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 thisplan the most likely to experience a price change for a fixed amount of spending, it is likelythat 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.3 In the employeeestimation sample, the average number of outpatient visits is 3.4, but 35.5% of the sample haszero 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 only0.4% of the sample has two or more visits. Across all visits in the data, the average outpatientvisit costs $244, and the average inpatient visit costs $6,704. Patients with no inpatient visitsspend $603 on average, and patients with inpatient visits spend $10,641 on average. Given thesmall number of inpatient visits, we expect that patients have more discretion over outpatientvisits, but given the high amount of inpatient spending, we expect that changes in inpatientvisits have a large effect on expenditure. I examine results with visits as the dependent variablein Online Appendix 16.

Online Appendix 4 Budget Sets Under Insurance

For an individual that is not insured as a member of a family, the top subfigure of FigureOA1 depicts how the deductible, coinsurance rate, and stoploss induces a nonlinear budgetset for the individual, who faces a tradeoff between total beneficiary plus insurer spendingon medical care, Y , and spending on all other goods, A. The individual faces three distinctmarginal prices, the slopes of each segment.

If a consumer is insured as a member of a family, the general cost sharing structure isthe same, but an additional family-level deductible and stoploss enable one family member’sspending to affect another family member’s marginal price. For example, in my empiricalapplication, one plan has an individual deductible of $500, and it also has a family deductiblethat is $1,500, three times the individual deductible. Each family member must meet theindividual deductible unless total family spending toward individual deductibles exceeds thefamily deductible. Since the family deductible is three times the individual deductible, if afamily has fewer than four members, all family members must meet the individual deductible.In that case, only subfigure I of Figure OA1 applies, and it applies separately for each familymember. In a family with exactly four members, when the first, second, and third familymembers go to the doctor, they each face the individual deductible of $500, as if they wereinsured as individuals. However, when the fourth family member goes to the doctor, if thefamily deductible of $1,500 has been met through the fulfillment of three individual $500deductibles, he makes his first payment at the coinsurance rate, as shown in subfigure II ofFigure OA1. In families with more than four members, the family deductible is fixed at $1,500,

3In the Medstat data, inpatient claims are organized into visits, but outpatient claims are not. To calculateoutpatient visits, I calculate the number of unique days for which a patient has outpatient claims.

6

Figure OA1: 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

and it can be met by any combination of payments toward individual $500 deductibles.4 Asimilar interaction occurs at the level of the stoploss such that an individual whose family hasmet the family stoploss faces the budget set shown by subfigure III of Figure OA1. Given

4In the case of more than four family members, intermediate cases between subfigures I and II are possible,in which the budget sets are shifted to the left because the individual has some spending at the given ratebefore the family deductible or stoploss is met. For simplicity of exposition, I do not show all of those caseshere. However, the empirical implementation is completely general.

7

the family-level cost sharing parameters, some individuals will face lower marginal prices thantheir own medical spending would dictate.5

Online Appendix 5 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 ∀b

In this simple model, agents derive utility from total agent plus insurer spending on medicalcare, which is given by Y (reflecting an underlying demand for health), and total spendingon all other goods, Ab. W denotes a vector of individual characteristics. The individualfaces three potential budget constraints, indexed by b, which include budget constraints I,II, and III from Figure OA1. The probability that the individual places on facing eachbudget constraint is given by πb. Following the nonlinear budget set methodology reviewed inHausman (1985), pb is the price on the linear segment of budget set b that is relevant for a givenY , and Kb is the relevant “virtual income” (the y-intercept when each segment is extendedto the vertical axis, representing the marginal income that the individual trades off againstmedical care on that segment). In this model, individuals are forward-looking maximizers ofexpected utility. At the start of the year, each individual forms an expectation over what hisand his family members’ year-end expenditures will be based on expenditures in the previousyear 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 incomeassociated with the utility maximizing segment on each potential budget set, as well as theprobability 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 2.1. It assumesforward-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 forhealth care under insurance, including an influential paper by Ellis (1986), which develops adynamic model in which agents respond to expected year-end price. Here, I am assuming thatthe expected year-end-price is the realized year-end price.6

5Eichner (1997, 1998) models a simpler family deductible structure in which people in family plans simplyhave a single family deductible, but that structure is not appropriate in the plans that I study.

6This assumption is true at the end of the year, after all uncertainty is resolved.

8

We can use the model to understand the impact of a family injury on demand, the impact ofthe within-year timing of the injury on demand, and the impact of a transitory vs. permanentfamily injury on demand. We begin with a simple example, in which there are no unexpectedfamily injuries and the individual forms an accurate prediction of each πb. Suppose that theindividual expects to spend a small amount, given by Y 1 in Figure OA1. Depending onwhether 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 familymembers’ spending, he assigns a probability to each of the three subfigures at the start of theyear, and he maximizes his expected utility over all three budget sets.7

Consider the impact of an unexpected injury on demand. Suppose that the agent wouldhave 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 himover the individual deductible, thus pushing the family over the family deductible. Now, theagent adjusts his medical consumption relative to the case in which his family member wasnot 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 scheduledvisit accordingly (both have the same impact in terms of the model, but we test how muchresponse comes through the visit margin in Online Appendix 16), taking into the account theexpected price of the visit and the likely effect that the visit itself will have on his year-endmarginal 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 thathappens in January relative to a family injury that happens in December on the agent’sdemand. Suppose that there are multiple periods indexed by t (days or months) within theyear. As time progresses, uncertainty is resolved, and the agent changes the probabilities πbtthat he assigns to the budget sets in each subfigure. The model does not place any restrictionson 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 severalappointments, or because some types of spending are incurred after a specific acute event inthe middle of the year. If the injury happens on January 1, the agent, his injured familymember, 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 behigher if the injury occurs toward the beginning of the year. Suppose instead that the injuryoccurs on December 1. If the injury itself requires several follow-up visits, and it takes timefor the agent’s family members to schedule appointments to react to the injury, it could bethat even though the injury would have been large enough to change the individual’s year-endprice if it had occurred on January 1, it will not be large enough to change the year-end pricegiven its occurrence late in the year. Thus, the first stage effect of the injury on the marginalprice will be smaller, and the reduced form effect of the injury on the agent’s expenditure

7If a single agent maximizes utility for the entire family, the optimization process will be similar to theprocess for each separate individual, because there is an incentive to spread spending across several familymembers so that the likelihood that that family deductible is met increases. (If only one family memberresponds, even if he responds a great deal, the family deductible might not be met because the cost sharingrules require spending toward at least three individual deductibles before the family deductible can be met.)

9

will be smaller than they would have been if the injury occurred sooner. If the attenuationof the first stage effect is the same as the attenuation of the reduced form effect, then theinstrumental 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-yeartiming of injuries in Online Appendix 15.

We can also extend the model to incorporate behavior over multiple years to understandthe effect of transitory vs. permanent shocks to family members. Suppose that the arrival rateof some health events is uniform across years, but treatment can be delayed until subsequentyears with minimal health cost. If an agent expects that his family members might havehigher spending in the subsequent year, resulting in a lower price for his own care, he has anincentive to delay treatment.8 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 shockmight encourage family members to shift delayed expenditures from the previous calendar yearinto the current calendar year because they expect their marginal prices to be higher nextyear. If that is the case, my estimate of how expenditure responds to changes in year-endmarginal price will be biased away from zero, relative to what the true response would be toa permanent decrease in year-end price. In contrast, suppose a member of the agent’s familyhas a permanent shock - he develops a new condition this year that is likely to result in highermedical expenditures over the long term. The agent could also shift planned expendituresforward from the subsequent year, but he has less incentive to do so than he does if the shockis transitory because next year’s budget set will likely be very similar to this year’s.

Online Appendix 6 Empirical Cost Sharing

Figure OA2 shows the empirical cost sharing schedule. The sample in this figure only includes2004 employees in families of two in the $500 deductible plan so that we can focus on variationaround a simple cost sharing relationship. The empirical cost sharing relationship would bemore complicated if we included families or people in multiple plans on the same graph. FigureOA2 shows that beneficiary expenses follow the in-network schedule with a high degree ofaccuracy, indicating that out-of-network expenses are very rare.

Online Appendix 7 Transforming Coefficients into Arc

Elasticities

The CQIV coefficients are not elasticities because year-end price is specified in levels - pricecannot be specified in logarithmic form because it can take on a value of zero. Tables OA4and OA5 report the CQIV coefficients at the conditional deciles for the logarithmic and levelsspecifications of expenditure, respectively.

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

8Cabral (2011) shows evidence of such delaying of dental care treatment. Although dental care is notincluded in the major medical expenditure that I study, treatment delay could be possible.

10

Figure OA2: 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 97.5% 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

controlled for by the regressors fixed. The CQIV coefficient at the median in Table OA4implies that median expenditure will be 522% higher for the group that faces a price of 0than it is for the group that faces a price of 1. The coefficient at the 0.90 quantile impliesthat the 0.90 conditional quantile of expenditure will be 431% higher. The second set ofcoefficients in the table is calculated without the corner calculation. As expected, the lowestconditional quantiles of expenditure are more likely to be below zero, so the corner calculationattenuates the coefficients at the lowest conditional quantiles the most and has almost noimpact on the highest conditional quantiles. If we were to rely on the coefficients without thecorner calculation, we would estimate larger price responsiveness at all quantiles, but suchresponsiveness would not be feasible because negative expenditure is not possible.

The above claims about behavior in response to a 100 percentage point change in marginalprice are misleading because such a change is not well defined at marginal prices other than0 and 1, and this measure is not unit free. To convert this coefficient into an elasticity, Iuse two arc elasticity formulas. Arc elasticities are generally preferred to point elasticities forlarge changes. My preferred arc elasticity formula is the following midpoint formula at eachconditional quantile:

η =(Ya − Yb)/|Ya + Yb|

(a− b)/(a+ b),

where Yx is the conditional quantile of expenditure at price x. Allen and Lerner (1934)

11

Table OA4: 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

discussed a version of this formula. I have extended it to include the absolute value sothat it will not yield wrong-signed elasticities for negative estimate conditional quantiles ofexpenditure (prices are always positive in my application). The midpoint formula is a meansof calculating a percentage change that depends on both the starting and ending values instead

12

Table OA5: 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

of just one or the other. For example, in my application, the main price change is from beforeto 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 ofthe price range (0.6) and arrives at a 133% change in price. (There is a 66% change in price

13

from 0.2 to 0.6 and another 66% change in price from 0.6 to 1, resulting in a 133% change inprice over the entire range.)

I also examine arc elasticities following the alternative logarithmic elasticity formula in-troduced 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. Itcan 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 whichthe point elasticity is exactly the log arc elasticity. See Gallego-Diaz (1944). Both arc elasticityformulas yield a value of 0 when expenditure does not change regardless of the price, and botharc elasticity formulas yield a value of 1 when outlays are equal: aYa = bYb.

Table OA6: 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 .

However, when outlays are not equal, particularly when the elasticity is greater than one,both formulas yield very different values. Table OA6 shows how the arc elasticities vary withexpenditures over three different price changes: the change from before to after the deductiblein 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 theRand Health Insurance Experiment). The table shows that as the difference in expenditureat each price grows large, the midpoint arc elasticity approaches a limiting value, but thelogarithmic arc elasticity continues to grow.9 Given this property, the logarithmic elasticityis probably more informative in my application because the elasticities that I find are almostalways greater than one. However, to the extent that readers would like to compare my resultsto the elasticity derived from the Rand Health Insurance Experiment, the midpoint elasticityformula is more appropriate.

9In the price range that I study from 1 to 0.2, the midpoint arc elasticity approaches -1.5. Over the Randrange, the midpoint arc elasticity approaches approximately -1.71. Given this property, the upper bound ofthe confidence interval is not informative if its value is -1.5.

14

Elasticities are a unit free measure, but if the underlying relationship is not isoelastic, therange over which they are taken matters. If the underlying relationship is isoelastic (it hasthe form lnY = ηlogarithmic ln p, where ηlogarithmic is a constant), then the range over whichthe logarithmic arc elasticity is taken will not matter for the estimate. However, with themidpoint formula, even if the underlying relationship is isoelastic, the range over which themidpoint arc elasticity is taken will matter. For example, in the penultimate column of TableOA6, the underlying relationship is isoelastic with ηlogarithmic = −3, however, η = −1.5 for therange from 1 to 0.2, and η = −1.65 for the Rand price range.10 If the underlying relationshipis linear (Y = γp), where γ is any constant, the range over which the arc elasticities is takendoes not matter; as shown in the first column of Table OA6, both arc elasticities yield a valueof one.

I present results from both elasticity measures with and without the corner calculation inTables OA4 and OA5. When I specify the dependent variable as the level of expenditure, thecorner calculation involves setting the conditional quantile of expenditure for some observa-tions to zero for one or both values of the price. As shown in the last column of Table OA6,when one value of expenditure is zero and the other value is nonzero, the midpoint arc elastic-ity takes on its limiting value, and the logarithmic arc elasticity cannot be computed becausethe logarithm of zero is undefined. This is relevant in my application because the conditionalquantiles of expenditure are censored at zero in the calculation of corner arc elasticities. Inthe corner arc elasticities, if both conditional quantiles of expenditure are censored at zero, Iset the elasticity equal to zero.

The lower panels of Table OA4 show the midpoint and logarithmic elasticities, with andwithout the corner calculation, over the same three price changes shown in Table OA6. Theelasticities 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 thatif price increases by one percent, expenditure will decrease between 0.76 percent and 1.49percent.

Table OA7 provides more information on how each of the elasticities in Table OA4 canbe interpreted in terms of conditional quantiles in my context. The first row shows theactual quantiles of expenditure in the data. The second row shows the average estimatedexpenditure at each quantile, conditional on all covariates, including the observed year-endprice.11 Comparing the two rows makes clear that the conditional quantiles of expendituredo not correspond to the actual quantiles of expenditure. The next rows shows the average

10The example cannot be extended to the stoploss price change because one of the prices is zero.11To obtain the estimated conditional quantiles of expenditure for the estimates derived from the logarithmic

model, I exponentiate the predicted value for each observation, using the point estimates at each quantile.Because the estimator did not converge at the 0.5 or 0.9 quantiles, there are no conditional quantile estimatesat those values. After exponentiating, all expenditures are positive, so I do not need to apply the cornercalculation. However, before I exponentiate, some values are below the censoring point. For the cornercalculations, I first set the conditional quantile of log expenditure equal to the censoring point for people withvalues below the censoring point, and then I exponentiate.

In the health economics literature, it is common to apply a Duan (1983) “smearing” transformation toestimated medical expenditures derived from a logarithmic model of the conditional mean because the loga-rithm of the expectation is not equal to the expectation of the logarithm, especially under heteroskedasticity.However, the smearing correction is not merited in my context because the logarithm of the quantile is equalto the quantile of the logarithm.

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Table OA7: Actual Quantiles of Expenditure, Conditional Quantiles of Expenditure, AndElasticities

Dependent variable: Ln(Expenditure).

2004 Employee Sample Price Corner 10 20 30 40 50 60 70 80 90 Tobit IVAverage actual quantiles of expenditure, mean shown in last column

0 0 0 60 130 261 531 1215 3939 1414

Average actual quantiles of expenditure, conditional quantiles of 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 . 650.2 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

estimated conditional quantiles of expenditure at prices of 1, 0.95, 0.25, 0.20, and 1, all of theactual prices in my empirical setting as well as the actual prices used in the calculation of theRand elasticity.

Table OA8: Actual Quantiles of Expenditure, Conditional Quantiles of Expenditure, AndElasticities

Dependent variable: Expenditure.

2004 Employee Sample Price Corner 10 20 30 40 50 60 70 80 90 Tobit IVAverage actual quantiles of expenditure, mean shown in last column

0 0 0 60 130 261 531 1215 3939 1414

Average actual quantiles of expenditure, conditional quantiles of expenditure using point estimate from model with dependent variable: Ln(Expenditure)Actual 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 corner calculation has a much more pronounced impact on the elasticities from themodel of the level of expenditure than it does in the model of the logarithm of expenditure. The

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rows with a value of 1 for “corner” censor the estimated conditional quantile of expenditurefrom below at zero and then average over all observations. In the logarithmic specification, thecorner calculation has a negligible impact, indicating that most of the estimated conditionalquantiles are positive. However, in the corresponding Table OA8, based on estimates fromthe level specification, the corner calculation has a much larger impact. The differencesbetween the corner elasticities in the logarithmic and level specifications reflect differencesin the models more than they reflect real phenomena. In the levels model, if the estimatedconditional quantile is less than zero, the corner calculation requires that it is set to zero.In contrast, in the logarithmic model, if the estimated conditional logarithm of expenditureis less than -0.7, the corner calculation requires that it is set to zero; many negative valueswill not have to be censored. In practice, values less than the censoring point occur muchless frequently in the logarithmic model than they do in the levels model, perhaps becausethe logarithmic transformation reduces the dispersion in the dependent variable. Thus, theelasticities in the levels model are smaller at smaller quantiles because the corner calculationhas a larger impact at the lower conditional quantiles of expenditure, which are more likelyto be zero. This pattern arises because changes in observed expenditure are less likely whenobserved expenditure is zero, not because people at the lowest conditional quantiles have lowerlatent price responsiveness.

The final rows of Table OA7 show all of the elasticities reported in Table OA4, calculateddirectly from the previous rows.12 The elasticity of -1.48 at the 0.10 quantile of expenditureconveys that, holding covariates fixed, the estimated 0.10 conditional quantile of expenditureis 50 cents among a group of people with a price of 1, but it increases to $331.50 at a priceof 0. The elasticity of -1.43 at the 0.80 conditional quantile conveys that, holding covariatesfixed, the estimated 0.80 conditional quantile of expenditure is $131.5 for a group of peoplewith a price of 1, but it increases to $13,173.2 at a price of 0. Even though the difference inthe estimated conditional quantiles between a price of 1 and a price of 0 is vastly differentbetween the 0.10 and 0.80 conditional 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 exactsame 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 formulacan have a large impact on the estimated elasticity. The other elasticities in the table showthat the price range over which the arc elasticities are calculated can have a dramatic impacton the reported elasticity in this application. I cannot calculate the sloploss elasticities withthe logarithmic formula because one of the prices is zero. The midpoint stoploss elasticitiescalculated over the stoploss range from 0.2 to 1 are much smaller - from -0.16 to -0.76 acrossthe quantiles, and the elasticities calculated over the Rand range from 0.25 to 0.95 are slightlylarger - from -0.88 to -1.71 across the quantiles.

The issues inherent in the arc elasticity measures complicate the comparison between arcelasticity estimates across the quantiles and the comparison between elasticity estimates in myapplication and the elasticity estimate obtained from the Rand Health Insurance Experiment.

12The elasticities in this table are obtained by applying the various arc elasticity formulas to average esti-mated expenditure at each conditional for ease of exposition. In all other tables, I predict expenditure at eachconditional quantile and calculate the elasticity for each individual and then take the average. Comparisonof the elasticities in the “point estimate” row of Table OA4 to the elasticities reported here show the averageelasticity is very similar to the elasticity at the average .

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However, estimates obtained from both elasticity calculations, the logarithmic and the levelspecifications of the dependent variable, and my price range and the Rand price range aregenerally an order of magnitude larger than -0.22, the Rand elasticity estimate. As TableOA6 shows, the Rand elasticity estimate of -0.22 implies that spending is 1.29 times higher ifagents face a price of 0.25 instead of a price of 0.95. My CQIV estimates obtained from thelogarithmic specification of -.76 to -1.49 imply that spending is 3.4 to 199 times higher at aprice of 0.2 than it is at a price of 1. The first lesson learned from this comparison are thatstudies that attempt to compare estimates to the Rand elasticity should take the particulararc elasticity calculation used in Rand in their empirical exercise seriously. The second is thatno matter the means of comparison, my elasticity estimates are much larger than the Randelasticity estimate.

The choice of the arc elasticity calculation is not well-informed by theory. However, tothe extent that my elasticity estimates will be compared to the Rand elasticity estimate,in subsequent specifications, I report elasticities obtained using the arc midpoint elasticityformula with the corner calculation. As identification in my setting comes mainly from theprice 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 wellas the same formula. However, the Rand price range would involve out-of-sample predictionsin my data, so I prefer the price change from 0.2 to 0, which does occur in my data. As theabove tables have shown, if I calculate elasticities in my setting over the counterfactual rangeof the Rand elasticity, my estimates would be even larger.

Online Appendix 8 Stoploss Elasticities

How do we expect the stoploss elasticity to compare to the elasticity from 1 to 0.2? If thetrue curve is isoelastic, the elasticities over both ranges should be approximately equal (asdiscussed in Online Appendix 7, the logarithmic elasticities would be exactly equal). However,we might not expect the true curve to be isoelastic because the people who face the two pricechanges are different people. The people who face the second price change are higher spenders,for example the people at Y 2 and Y 3 in Figure OA1 as opposed to the people at Y 1. We mightexpect high spenders to be more or less price responsive relative to lower spenders. On the onehand, high spenders might have less control over their spending because care decisions for highspenders might be mostly driven by doctor decisions. On the other hand, high spenders mightexercise more control over their spending because they have more money at stake. In additionto considering the amount of spending, we might also want to consider the magnitude of theprice change. Chetty (2012) argues that labor supply elasticities in response to smaller wagechanges are smaller because of “optimization frictions” - people change their behavior most inresponse to the largest incentives. In this context, it is unclear which price change should bethe most meaningful. In absolute terms, the price change around the stoploss is much smallerthan the price change around the deductible, but it is larger in percentage terms.

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Online Appendix 9 Alternative Estimators

For comparison with previous literature, I compare my conditional quantile CQIV estimates toconditional mean estimates. However, conditional quantile estimators and conditional meanestimators are not likely to yield the same point estimates because they do not estimatethe same quantities. Quantile estimates and mean estimates are only similar to the extentthat the underlying treatment effect is linear and the error distribution is symmetric andhomoskedastic. In this application, CQIV estimates are particularly likely to be differentfrom estimates obtained with mean estimators because medical expenditures are skewed andcensored. Compared to conditional mean estimates, CQIV estimates are less sensitive toextreme 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 andnormally distributed. The Tobit IV estimator, developed by Newey (1987) provides a goodcomparison 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 IVestimator requires additional parametric assumptions: a homoskedasticity assumption on thefirst stage error term and a joint normality distributional assumption on the structural andfirst stage error terms. In this application, it is unlikely that the Tobit IV assumption ofhomoskedasticity in the structural equation holds given the discreteness of the endogenousvariable, year-end price.

A Hausman (1978) joint test of the Tobit IV normality and homoskedasticity assumptionscan 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. Al-though it would be intuitive to compare the Tobit IV estimate, which is an estimate of themean conditional elasticity, to a CQIV conditional median elasticity estimate, a conditionalmedian estimate is not necessary for the comparison. Since Tobit IV imposes a constant treat-ment effect across all quantiles, the single Tobit IV coefficient can be compared directly to theCQIV coefficients at each quantile. The first row of Table 3 in the paper shows the CQIV andTobit IV coefficients before the elasticity transformation. The Tobit IV coefficient is withinthe range of the CQIV coefficients, and the 95% confidence intervals overlap, indicating thatthe null hypothesis that the assumptions required by Tobit IV hold is not rejected. As shownin specification A1 of Table OA9, the Tobit IV coefficient implies a larger elasticity than theCQIV coefficients at most quantiles, indicating that the use of the CQIV estimator alone doesnot 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 othercommon 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).13 The two-part model arc elasticity estimate is -0.86 with a bootstrapped 95%confidence interval of (-1.19,-0.53).14 As with the Tobit IV estimate, these estimates are

13To obtain this estimate, I estimate the control function using OLS, and then I include it as an independentvariable using the Stata command truncreg. The dependent variable is the logarithm of expenditure, truncatedfor the observations with zero expenditure. To predict expenditure for use in the arc elasticity calculation, Iuse the estimated coefficients to predict expenditure.

14It is unclear how to extend the two part model for endogeneity in both parts, and it is also unclear howto use it to predict expenditure for use in arc elasticity calculations. To obtain a two part model estimate

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Table OA9: CQIV vs. CQR, QIV, and QR

2004 Sample 10 20 30 40 50 60 70 80 90Dependent 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 -1.31 -0.75 -1.50 -1.39 -1.49 -1.34 -0.50 -1.26 -0.87 -1.20

lower bound -1.50 -1.50 -1.50 -1.50 -1.50 -1.46 -1.40 -1.42 -1.44 -1.49 upper bound -1.11 0.00 -1.50 -1.28 -1.49 -1.22 0.41 -1.10 -0.29 -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

generally much larger than those in the literature. However, Eichner (1998) reports a TobitIV elasticity of -0.8 (the Eichner (1997) elasticity estimates vary from -0.22 to -0.32).15

Given the insurance-induced mechanical relationship between price and expenditure, weexpect elasticity estimates that do not account for endogeneity to be biased upward. Cuttingin the other direction, we expect elasticity estimates that do not account for censoring to bebiased downward. Table OA9 reports a variety of estimators that account and do not accountfor endogeneity and censoring. The first panel specifies the logarithm of expenditure as thedependent variable, and the second panel specifies the level of expenditure as the dependentvariable. Because of endogeneity, we expect that the CQR estimates should be larger than theCQIV 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 isapparent in the comparison of the CQR coefficients to the CQIV coefficients in the levelsspecifications. Because of censoring, we expect that the QIV estimates and the IV estimatesshould be biased toward zero relative to the CQIV and Tobit IV specifications. We can

with endogeneity, I estimate the control function using OLS, and then I include it as an independent variablein the first part probit regression as well as in the second part OLS log expenditure regression estimated onlyon the observations with positive expenditure. To predict expenditure for use in the arc elasticity calculation,I use the coefficients from the second part to predict expenditure for all observations. I then multiply thisprediction by the predicted probability of being nonzero from the first part.

15The Eichner (1997, 1998) elasticities are point elasticities, so caution must be applied in comparing themto the arc elasticities elsewhere reported in this paper.

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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 logarithmicspecifications. Indeed, we find that the IV elasticity is closer to zero than the Tobit IVelasticity. The bias in the QR and OLS estimates relative to the CQIV and Tobit IV estimatesdepends on the relative influence of endogeneity and censoring. At some quantiles, the QIVestimate is biased toward zero relative to the CQIV estimate, but it is biased away from zeroat others. It appears that endogeneity induces more of a bias in the QR estimates relativeto the CQIV estimates and censoring induces more of a bias in the OLS estimates relative tothe Tobit IV estimates. Regardless of the estimator used, the elasticities that I find are largerelative to those in the literature.

As another method of comparison between the CQIV estimates and estimates in the liter-ature, I use a back-of-the-envelope calculation to transform the CQIV estimates into a singleestimate. To do this, I take the average of the CQIV estimates from the 0.10 quantile to the0.90 quantile, for an average elasticity estimate of -1.32. This estimate is still much largerthan the Rand elasticity estimate of -0.22.

Online Appendix 10 Heterogeneity in Treatment Effects

Although one advantage of the CQIV estimator is that it allows the estimates to vary withunobserved heterogeneity, we might also be interested in how price elasticities vary alongobserved dimensions. In this section, I estimate models that restrict the sample on the basisof observable characteristics. I present models by gender, salaried/hourly status, and plan.

In Tables OA10 and OA11, specifications B and C, I present the main logarithmic and levelsspecifications, restricted to males and females, respectively. Average spending for femalesis $1,867, which is much higher than average spending for males of $1,067. Although theelasticities are generally in the same range in the logarithmic and levels models, women appearto be slightly more price responsive than men, perhaps because they spend more.

Specifications D and E report separate results for salaried and hourly employees. Althoughwe do not observe income, salaried/hourly status could serve as a proxy for income in our mainresults, where we expect hourly employees to have lower incomes. Hourly employees spendmore 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 responsivethan 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 smallersample size, which biases the reported elasticity in the positive direction because the reportedelasticity is the average of the upper and lower bounds of the 95% confidence interval, andthe 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 de-ductible plan to be more price responsive throughout the distribution because they alwaysface higher prices. Since the most generous plan has the largest enrollment, the confidenceintervals are the tightest. In the logarithmic specification, the estimates are very similar acrossplans. In the levels specification, people in the least generous ($1,000 deductible) plan appearmore price responsive at low conditional quantiles and less price responsive at high conditionalquantiles.

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Online Appendix 11 Robustness to Sample Selection

In this section, I examine the robustness of my results to the main sample restrictions. FirstI examine the impact of restricting the estimation sample to the employee in each family.Specifications J and K of Tables OA10 and OA11 present coefficients estimated on samplesthat include spouses and all other family members, respectively. The patterns in the estimatesacross the quantiles are very similar to those in the employee sample, but the estimates areslightly more precise given the larger sample sizes, even with reported bootstrapped confidenceintervals that account for intra-family correlations.16

16I account for intra-family correlations using a block bootstrap by family. I draw replication samples thathave the same number of families as the main sample, drawing all observations in a family as a single block.

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Table OA10: 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

23

Table OA10: 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

24

Table OA11: 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

25

Table OA11: 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 -0.28 0.15 -0.62 -0.72 -0.62 -1.20 -1.32 -1.36 -1.36 -1.50

lower bound -1.50 -0.66 -0.64 -0.82 -1.48 -1.45 -1.42 -1.42 -1.43 -1.50 upper bound 0.94 0.97 -0.61 -0.63 0.25 -0.95 -1.21 -1.30 -1.29 -1.49

Lower and upper bounds for specifications J and K account for intra-family correlations.

Specifications L, M, and N explore alternative sample selection criteria in the employeesample. The main specification, A excludes injured employees on the grounds that the exclu-sion restriction is less likely to be satisfied for them - another family member’s injury couldbe related to their injury through a mechanism other than price if both family members wereinjured at the same time, leading to a larger estimated elasticity. However, the cost of thissample selection is that it might reduce external validity of the findings, or it might bias thefindings through selection on the dependent variable. Specification L leaves the 603 employeeswith own injuries in the estimation sample. Since the instrument is defined as an injury ofanother 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 spec-ifications that include injured employees, suggesting that sample selection to exclude injuredemployees does not have a large impact on the results.

Specifications M and N examine robustness to the major sample restriction - the require-ment that everyone be continuously enrolled. I impose this restriction because the dependent

26

variable measures year-end spending. Year-end spending will be artificially low if agents arenot in the sample for the entire year, and we will not capture all responses to family injuriesif agents are not in the sample for the entire year. However, as long as individuals are not dif-ferentially continuously enrolled based on family injury status, the results should be the sameas they are in the main specification. In specification M, I do not require employees or theirfamily 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 2003data. The elasticities presented in specification O show remarkably similar patterns. Thesimilarity of the estimates between 2003 and 2004 provides some evidence of robustness, andit suggests that price responsiveness did not change between 2003 and 2004.

Online Appendix 12 Robustness to Specification of the

Instrument

In this section, I examine the robustness of my results to alternative specifications of theinstrumental variable. As discussed in Online Appendix 15, I restrict the categories of injuriesin my instrument to those for which employee spending is similar before the injuries. Ifemployees in families with some categories of injuries spend more even before the injuriesoccur, estimates from specifications that use those categories will result in elasticity estimatesaway from zero. In specification P of Tables OA10 and OA11, I specify the instrument toinclude all injury categories. The results are larger than those in the main specification atmost quantiles, but only slightly, suggesting that some injury categories that are not includedin my main specification might be endogenous, but that any bias that they would generate issmall. In all other specifications, I consider only the injury categories included in the mainspecification.

Next, I explore alternative specifications of the instrument to investigate a specific channelfor violation of the exclusion restriction. There is potential cause for concern if family injuriesaffect family income and family income affects expenditure. I cannot control for income di-rectly because I do not observe it. However, I can informally investigate the role of incomeeffects by estimating separate specifications based on injuries to spouses and injuries to chil-dren. If there are large income effects due to the injury of a wage earner, we might expectan employee’s response to a spouse’s injury to be different than an employee’s response toa child’s injury. In specification Q, I only include child injuries in the instrument and dropemployees with other family injuries; in specification R, I only include spouse injuries in theinstrument and drop employees with other family injuries. The logarithmic specification withjust child injuries gives almost the exact same point estimates as the main specification, whichis not surprising given that 4/5 of the injuries in the sample are to children. The specificationwith just spouse injuries, which is not as well identified, also yields point estimates that arethe similar in magnitude but slightly smaller, suggesting that variation in the estimates dueto 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 theinstrument 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

27

family member had an injury. I do not specify the instrument as the amount of spending onthe family injury because such a specification could violate the exclusion restriction if familymembers see the same expensive doctors or consume care in the same expensive geographicarea. However, by restricting our instrument to a dummy variable, I lose some meaningfulvariation. In specification S, I incorporate the amount of spending on the injury using simu-lation techniques pioneered by Currie and Gruber (1996a,b). First, within each of the seveninjury categories, I calculate mean spending over all individuals with that injury. Then, Iassign this mean to the family members of the injured, removing only their family member’sspending from the mean. In this way, I am able to use variation in the magnitude of spendingon the injury, excluding variation that could be correlated within families. Although it wouldhave 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 inspecification S are slightly smaller, but in a similar range. The smaller estimates seem to bean artifact of the wider confidence intervals. I might have expected the confidence intervalsto be tighter because of the additional variation in the instrument, but it seems that thisspecification 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 vectorof dummy variables for each of the nine injury categories included in the instrument, listedin Table OA3. The results are very similar to the main results. Finally, in specification U, Iexploit additional variation in injury timing in the specification of the instrument. I generatetwo instruments by interacting the family injury dummy with an indicator for the half of theyear when the injury occurred. Again, the results are very similar to those from the mainspecification. The next section considers the implications of within-year injury timing for theestimates and demonstrates that the estimates should indeed be similar regardless of injurytiming.

Online Appendix 13 Implications of Within-Year Injury

Timing

In this section, I examine the implications of injury timing for my instrumental variableestimates. To do so, I examine separate CQIV specifications in which I only include injuriesin the first half of the year or only include injuries in the second half of the year in thedetermination of the instrument, excluding individuals with family injuries in the other halfof the year from the sample. In the first panel of Tables OA12 and OA13, I present estimateswith 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 modeldiscussed in Online Appendix 5: after an injury occurs, it takes time for the family member’smarginal price, as well as his spending, to respond. On net, if the family member’s marginalprice responds at roughly the same rate as his spending responds, the IV estimate, whichreflects the spending response (the reduced form) divided by the price response (the firststage), will be roughly the same regardless of the injury timing.

In the bottom panel of Tables OA12 and OA13, I examine the spending response to theinjury (the reduced form), and the price response to the injury (the first stage) separately. To

28

Table OA12: 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

Table OA13: 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

examine the reduced form, I estimate CQR, Tobit, and OLS specifications.17 These results,especially the Tobit results, show that the expenditure response is larger for injuries thatoccur in the first half of the year than it is for injuries that occur in the second half of theyear. The first stage OLS results also show that the price response is larger for injuries thatoccur in the first half of the year. The combination of these findings explains why the CQIVresults are generally not sensitive to injury timing.

17I only estimate an OLS specification of the reduced form for the levels model because it is unclear how tomodel individuals with zero expenditure in an OLS model of the logarithm of expenditure.

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Online Appendix 14 Analysis of Data on Smaller Fam-

ilies

Using data beyond my estimation sample, I examine the spending behavior of employees infamilies with fewer than four individuals (the main results are based on employees in familiesof four or more). People in smaller families have the potential to serve the basis for anindirect test of the exclusion restriction, but the test has several potential flaws. The resultsdo not yield support for the exclusion restriction, but I report them here in the interest oftransparency.

The exclusion restriction requires that one family member’s injury can only affect anotherfamily member’s expenditure through its effect on his marginal price. If program rules dictatethat cost sharing interactions cannot occur, one family member’s injury should not be relatedto another family member’s medical expenditure. At the firm that I study, in policies forfamilies of two, one family member’s spending has no mechanical effect on another familymember’s marginal price as shown in Table 1 in the paper. Therefore, any effects of one familymember’s injury on another family member’s spending presumably operate through anotherchannel. Although the exclusion restriction is not an econometrically testable restriction inthe main sample of families of four or more, evidence that there is no effect of one familymember’s injury on another family member’s spending in a family of two would support thevalidity of the exclusion restriction in the main sample.

To formalize this test, I estimate the following specification with censored quantile regres-sion:

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 specificationonly in that it examines the reduced form effect of the family injury on Y directly. A traditionalinstrumental variable specification would not be informative here because the first stage shouldnot exist in families of two.18

The main limitation of this test is that if employees in families of two are different fromemployees in families of four or more, the effect of one member’s injury on another member’sspending could be nonzero in families of two even if there is no violation of the exclusionrestriction in families of four or more. Indeed employees in families of two have differentobservable characteristics than employees in families of four or more, so it is unclear if effects ofone family member’s injury on another family member’s spending should be comparable acrosssamples. I investigate several sample restrictions to make the samples of employees in familiesof both sizes similar in terms of observable characteristics, but unobservable characteristicscould still differ.

Column 7 of Table OA2 in the paper presents summary statistics on employees in allfamilies of two. Comparison of year of birth with column 1 shows that this population ismuch older than the population in families of four - it consists of many older “empty nesters.”

18If we think of the instrumental variable specification as the reduced form estimate divided by the firststage estimate, the instrumental variable estimate will not exist when the first stage estimate is zero.

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Furthermore, only 27% of employees in families of two consume zero care, as opposed to 36% infamilies of four or more. Employees in families of two have much higher average expenditureson medical care than their counterparts in families of four or more ($2,615 vs. $1,414). Giventhat employees in families of two consume more medical care, we might be more likely toobserve spurious effects of other family injuries on spending in the families of two than we arein 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 samplerestrictions. If the test is valid and the exclusion restriction holds, we expect to see no effectof another family member’s injury on the employee spending in families of two, but the CQRresults in specification A of Tables OA14 and OA15 show that employees with a family injuryspend 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 anotherfamily member’s expenditure. As shown in specification G of Tables OA14 and OA15, thecoefficients in the family specification suggest that employees with an injured spouse or childspend 15 to 29% more or from $20 to $813 more on their own medical care than employeeswithout family injuries. Although the Tobit coefficient is slightly larger in the sample offamilies of four or more, suggesting that there is more of an effect of one family member’sinjury on the employee’s expenditure in larger families, the Tobit coefficient in families of twois not a precisely estimated zero. It is possible that employees in families of two are differentfrom employees in families of four or more, invalidating the test.

One reason why families of two might be different from larger families is that identificationin the smaller families comes mainly from injuries to spouses, and identification in the largerfamilies comes from injuries to children as well as spouses. In specification B, I restrict thesample to employee-dependent families of two, and in specification H, I restrict the mainsample so that it does not include employees with spouse injuries; both samples are identifiedfrom injuries to non-spouse dependents. In specifications C and I, I restrict the samples sothat they are identified from injuries to spouses. Again, the results are somewhat larger inthe 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 oftwo are much older than the employees in larger families. In specifications D and J, I restrictboth samples to include employees aged 40 and under. Even though the unrestricted sampleof employees in families of two is much larger than the sample of employees in families of fouror more, it only includes 6,951 employees under age 40, but the main sample includes 18,972employees under age 40. Similarly, in specifications E and K, I restrict both samples to includeemployees over age 40. As shown, in families of four or more, the effect of a family member’sinjury on the employee’s expenditure is larger if the employee is older. The mechanism forthis comparison is unclear, but it suggests that we are more likely to find relationships inthe 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 inthe paper, there are some cost sharing interactions in families of three that occur through thestoploss, so the results in families of three should show some effect of one family member’sinjury 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 twoyielded a precisely estimated zero, there are several reasons why it would not in the absence

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of a violation of the exclusion restriction. For example, if employees do not understand theinteraction between the individual and family deductibles, then they might think that theirown 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 didwhen they had children on their policy in the past. In sum, this analysis does not providesupport for the exclusion restriction, but there are many reasons to question its usefulness.

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Table OA14: 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 OA15: 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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Online Appendix 15 Mechanisms: Timing of Injury Re-

sponse

A key feature of my identification strategy is that the response to injuries should occur afterthey happen but not before. Accordingly, I restrict my instrument to categories of injuriesfor which employees that have them in their families do not spend more on their own medicalcare before the injuries occur. I first describe the process of selecting the injury categoriesand show that spending does not respond during the portion of the year before the injuriesoccur. Next, using longitudinal data, I show that spending also does not appear to respondin the textityear before the injuries occur.

My approach is motivated by that of Card et al. (2009), who select a subset of diag-noses that appear to have similar rates of appearance in emergency rooms on weekdays andweekends. Using only this subset of diagnosis categories, they measure whether the start ofMedicare eligibility at age 65 results in lower mortality. In my approach, I select a set ofinjury categories for which employees in families with injuries do not appear to spend morethan employees in similar families without injuries in the part of the year before the injuryhas occurred.

To select the categories of injuries listed in column 2 of Table OA3, I use data on spendingpatterns within the year. I create a dataset such that each employee with a family injury ina 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 throughFebruary. To control for seasonality in medical spending, I organize the data so that eachemployee without any family injury can serve as a control for any employee with a familyinjury, regardless of injury month. Therefore, each employee without any family injuries hasan observation for cumulative spending before each month. For example, a given employeewithout any family injuries has one observation for January spending, another observation forJanuary through February spending, another observation for January through March spend-ing, etc. For each injury category, I then run a regression in which I predict spending with adummy variable for the eventual injury, clustering by employee since employees without anyfamily injuries have multiple observations. I repeat the regression for all categories of familyinjuries shown in Table OA3. The sample size varies slightly across specifications because Ieliminate the employees that themselves will have the injury as I do in the baseline elasticityestimates, 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 column6 of Table OA3, and I report the associated upper and lower bounds of the 95% confidenceintervals in columns 7-8. The injuries included in the instrument are those for which the valueof the coefficient is less than 100, indicating that employees with upcoming family injuriesin the given category spend no more than $100 more than employees with no upcomingfamily injuries in that category in the period before the injury. Note that spending appearsto 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 familieswith injuries could be higher spenders than employees in families without injuries, leading toelasticity estimates that are too large, I include all categories with negative coefficients in the

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instrument.19

Having selected the injury categories for the instrument, I test that once I have includedall of these categories, the result still holds that spending does not respond to an injury beforeit occurs. Further, I also examine spending during the part of the year after the injury occursto examine the mechanism behind my main results. Spending after the injury occurs shouldbe 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 theperiod after the injury, excluding the injury month itself. For example, an employee with afamily injury in March has an observation for spending from January through February andanother observation for spending from April through December. Each employee without afamily 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 spendingand one observation for March through December spending; that same employee also has oneobservation for January through February spending and one observation for April throughDecember 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 mainspecification, as well as a full set of injury month fixed effects. I report the results usingthe new selected injuries as a single category in the top panel of Table OA16. In column 1,the coefficient on “Family Injury,” defined as in the main CQIV specifications, shows thatindividuals with family injuries spend about 23 cents more in the period before the injuryrelative to people without injuries. We can be 95% confident that they spend between 80dollars less and 80 dollars more than similar families without injuries in the period beforethe injury, relative to an average of approximately 700 dollars. The coefficient on “After,” adummy variable for the period after the injury, indicates that there are statistically significantseasonal patterns in medical spending. The coefficient on “Family Injury x After” indicatesthat after controlling for seasonality by comparing employees with family injuries to employeeswithout family injuries, employees with family injuries spend approximately 400 dollars moreafter the injury occurs. Taken together, these results suggest that people with family injuriesas defined by my instrument do not spend more before the injuries occur, but they do respondto incentives by spending more after the injury occurs.

The next two columns show that a similar pattern holds for dependent variables thatindicate any expenditure or any outpatient visit. Although people with injuries in theirfamilies do not tend to have statistically more inpatient visits before the injury occurs, theyalso do not tend to have statistically more inpatient visits after the injury occurs, as shown inthe fourth column. This finding is consistent with the evidence in Online Appendix 16 thatshows 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 tothe main identification assumption. To perform the test, I create a dataset that includes

19I do not select the new categories of injuries based on the confidence intervals because the size of theconfidence intervals has a strong relationship to the total count of injured individuals; if I used the confidenceintervals in my selection criteria, I would be much more likely to reject the null hypothesis of no difference forthe categories with higher counts, retaining only those categories that offer little power.

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Table OA16: 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

all employees whose full families are enrolled in 2003 and 2004, and I drop individuals withfamily injuries in 2003 and with injuries themselves in either year, resulting in a dataset with17,092 observations (requiring longitudinal data severely limits the sample, just as requiringcontinuous enrollment limits the main sample). I run a regression to predict whether havinga family injury in 2004 predicts spending in the previous year of 2003. For this test, I regressspending in 2003 on a dummy variable for having a family injury in 2004, controlling for allof the 2003 and 2004 values of the covariates. The coefficient on “New Family Injury, 2004Only,” a dummy variable that indicates having a family injury in 2004 but not 2003, suggeststhat individuals who will have a family injury in 2004 spend around $109 less in 2003 thansimilar 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 sampleto employees for whom longitudinal data are available severely restricts the sample size from29,886 employees in the main 2003 estimation sample to 17,092. However, the coefficient issmall relative to mean 2003 spending in the sample of approximately $1,000. Overall, theresults of this test are consistent with the main identification assumption.

Online Appendix 16 Mechanisms: Visits and Outpatient

Spending

By estimating models that specify the dependent variable in terms of visits instead of expen-diture, we can examine one potential mechanism through which agents respond to prices: bydeciding whether to go to the doctor. Table OA17 reports results that specify the dependentvariable in Equation 2a as the logarithm or number of total visits, outpatient visits, and in-patient visits. The first row in each specification shows the coefficients. The CQIV cornercoefficients across all quantiles suggest that as the marginal price goes from one to zero, pa-tients reduce total visits by up to 2.83 percent, or up to 18.78 visits. If we transform thesecoefficients into elasticities, we can compare the price elasticity of visits to the price elasticityof expenditure from previous specifications. Holding covariates constant, the median elastic-ity in specification A1 suggests that if we have two groups of individuals and one group faces

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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 calculatedfrom a Poisson regression.20 As compared to the expenditure elasticity, the visit elasticity isof roughly the same order of magnitude but slightly smaller. Also, the visit elasticity is largerat higher conditional quantiles, just as the expenditure elasticity in the levels specification islarger at higher conditional quantiles. The comparison between the visit elasticity and theexpenditure elasticity suggests that a large part of the expenditure elasticity occurs on thevisit margin, and a smaller part occurs on the intensive margin of spending within a givenvisit.

Specifications B and C in Table OA17 show that most of the visit elasticity comes fromoutpatient visits, and there is a very small elasticity of inpatient visits. The inpatient visitresults are much less reliable because few bootstrap replications converge at all quantiles exceptthe 0.9 quantile. Unreported elasticity estimates from a CQIV specification with outpatientexpenditure as the dependent variable are very similar to the elasticity estimates in the mainspecification.

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, Iestimate the main specification at the very highest conditional quantiles. Even at very highestconditional quantiles, where we expect more inpatient expenditures conditional on observedcharacteristics, the unreported estimated elasticities remain fairly stable.

Online Appendix 17 Comparison to Rand

My estimates are an order of magnitude larger than those commonly cited from the Randexperiment. There could be a multitude of reasons for this discrepancy, including a possiblechange in the underlying expenditure elasticity over the decades between the Rand studyand my study or a difference in behavior between people in experimental plans and people inactual plans. Here, I examine differences in methodology between my estimates and the Randestimates.

To induce subjects to participate in the Rand experiment, researchers had to guarantee thatparticipants would be subject to very low out-of-pocket costs, so all plans in the experimenthad a yearly stoploss of $1,000 or less in 1974-1982 dollars. Furthermore, each year, all familieswere given lump sum payments that equaled or exceeded their out-of-pocket payments. Theexperimenters randomized families into plans with initial marginal prices of 0%, 25%, 50%,and 95%, but after family spending reached the stoploss, marginal price was zero for therest 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 planexceeded 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 mynon-experimental data.

Rand researchers recognized that the stoploss affected their ability to calculate the price

20Poisson regression, also known as quasi-MLE Poisson regression if the researcher only believes that he hasspecified the conditional mean correctly, is a popular regression for count data models. The raw coefficientsare not simple to interpret, but the arc elasticities should be directly comparable to the other arc elasticities.

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Table OA17: 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

elasticity of expenditure on medical care based on the experimentally randomized prices:

“In order to compare our results with those in the literature, however, wemust extrapolate to another part of the response surface, namely, the response tocoinsurance variation when there is no maximum dollar expenditure. Althoughany such extrapolation is hazardous (and of little practical relevance given theconsiderable departure from optimality of such an insurance policy), we have un-dertaken such an extrapolation rather than forego entirely any comparison withthe literature.” (Manning et al., 1987, p. 267)

Manning et al. (1987) cited three sources of estimates of the price elasticity of expenditure

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on medical care in the Rand data, the most prominent of which was based on a simulationby Keeler and Rolph (1988) and not on the Manning et al. (1987) four-part model. Keelerand Rolph (1988) recognized that a comparison of year-end expenditures based on the ex-perimentally induced coinsurance rates across plans could be misleading because behaviorwas influenced by stoplosses. They therefore used the experimental data to simulate year-end-expenditures in hypothetical plans without stoplosses, and they based their elasticityestimates on this simulated behavior. To conduct the simulation, they assumed myopic re-sponses to marginal price and examined the frequency of visits for all participants in the periodfor 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 oncalibrated parametric assumptions on the frequency of visits by type and the cost per visitby type, they forecasted year-end expenditures, and they compared forecasted expendituresacross coinsurance plans relative to the free plan to attain their elasticity estimates using thefollowing 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-citedRand elasticity estimate of -0.22 comes from a comparison of predicted expenditures acrossplans with 95% and 25% coinsurance rates with a = 25, b = 95 Ya = 71 and Ya = 55. Similarcalculations based on the predictions from the four-part model and the experimental meansyield estimates of -0.14 and -0.17, respectively. The 95% to 25% price change that formsthe basis for this arc elasticity should be roughly comparable to the price change on which Ibase my arc elasticities - from 100% before the deductible to the 20% coinsurance rate. Onekey methodological difference, however, is that I use within-plan price variation instead ofacross-plan price variation.

A key difference between the Rand methodology and my methodology comes from theunderlying treatment of myopia vs. foresight. While I assume forward looking behavior, theKeeler and Rolph (1988) methodology assumes complete myopia. Recent research by Aron-Dine et al. (2012) examines myopic vs. forward-looking responses to price and rejects thenull of myopic behavior. If consumers are forward-looking, it is problematic to assume thatthe initial statutory marginal price ever governs behavior of participants who expect to meetthe stoploss, even in the period before the stoploss is met. Including these participants in theelasticity calculation should bias estimates of price responsiveness downward because variationacross plans will be less pronounced among participants who expect to meet the stoploss andthus do not respond at all to the statutory marginal price. Furthermore, participants withthe highest coinsurance rates are more likely than participants with the lowest coinsurancerates 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 myopiaand foresight is one potential explanation for the difference between my estimates and theRand estimates.

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