2 Details of the Paper

2.1 Preliminary Analysis

Descriptive analysis to see the existence of inertia and adverse selection.

Existence of Inertia 1: New Employees

Compare the choices over time of the cohorts of new enrollees from years $t_{−1}$, $t_{0}$, and $t_{1}$.
Results: while it is evident that the $t_{0}$ and $t_{−1}$ cohorts make very similar choices with the default option at $t_{1}$, the new enrollees making active choices in that year have a very different choice profile that reflects the price changes for $t_{1}$.

Existence of Inertia 2: Dominated Plan Choice

Leverages a specific situation caused by the combination of plan characteristics and plan price changes in our setting.
As a result of the large price changes for year $t_{1}$, $PPO_{250}$ became strictly dominated for certain combinations of family size and income, which determine employee premium contributions.
Without inertia, employees must switch their options to monetary advantageous one.
Results show that only a fraction of employees change their insurance option.

Adverse Selection

The more the employees are riskier, the more likely they enroll in more comprehensive plan.

2.2 Model

Choice Model: consumer’s choice conditional on predicted family-level ex ante medical cost risk
Cost Model: generates the expenditure distributions

2.2.1 Choice Model

$k \in \mathcal{K}$: a family unit
$j \in \mathcal{J}$: one of the three PPO insurance plans available after the $t_{0}$ menu change
$t \in \mathcal{T}$: one of the three years from $t_{0}$ to $t_{2}$
$F_{kjt}(\cdot)$: family-plan-time specific distributions of out-of-pocket health expenditures (output by the cost model)
Each family has the following latent, v-NM expected utility: \[ U_{kjt} = \int^{\infty}_{0} f_{kjt}(OOP)u_{k}(W_{k}, OOP, P_{kjt}, 1_{kj, t-1})dOOP, \]
$OOP$: a realization of medical expenses from $F_{kjt}(\cdot)$
$W_{k}$: family-specific wealth
$P_{kjt}$: family-time-specific premium contribution for plan $j$
$1_{kj, t-1}$: an indicator of whether the family was enrolled in plan $j$ in the previous time period
$u_{k}(\cdot)$: v-NM utility index and assumed to be CARA preferences:

\[ u_{k}(x) = - \frac{1}{\gamma_{k}(\mathbf{X}^{A}_{k})}e^{\gamma_{k}(\mathbf{X}^{A}_{k})x}. \] - $\gamma_{k}$ is a family-specific preference parameter that is known to the family but unobserved to the econometrician - A family’s overall level of consumption $x$ conditional on $OOP$ from $F_{kjt}(\cdot)$: \[ x = W_{k} - P_{kjt} - OOP + \eta(\mathbf{X}^{B}_{kt}, Y_{k})1_{kj, t-1} + \delta_{k}(Y_{k})1_{1200} + \alpha H_{k, t-1}1_{250} + \epsilon_{kjt}(Y_{k}) \]

2.2.2 Cost Model

The cost model is developed and estimated the following procedure:

Leverage the Johns Hopkins ACG Case-Mix software package
For each individual and open enrollment period, we use the past year of diagnoses, drugs and expenses, along with age, gender, to predict mean total expenditures for the upcoming year
Grouping individuals into cells based on mean predicted future utilization, estimate a spending distribution for the upcoming year
Combine individual total expense projections into the family out-of-pocket expense projections used in the choice model $F_{kjt}$

The cost model assumes that there is no private information and no moral hazard (total expenditures do not vary with $j$). While both of these phenomena have the potential to be important in health care markets, these assumptions are believed to have less impact the results, relative to inertia.

2.2.3 Identification

The model separately identify inertia from persistent unobserved preference heterogeneity by leveraging three features of the data and environment.

The plan menu change and forced re-enrollment at year $t_{0}$ ensures that we observe each family in our final sample making both an “active” and a “passive” choice from the same set of health plans over time, in the context of meaningful relative price changes
The three $PPO$ plan options we study have the exact same network of medical providers and cover the same medical services, implying that differentiation occurs only through preferences for plan financial characteristics (here, risk preferences)
Since insurance choice here is effectively a choice between different financial lotteries, our detailed medical data allow us to precisely quantify health risk and the ex ante value consumers should have for health plans, con- ditional on risk preferences and the assumption that beliefs conform to $F_{kjt}$

We separately identify the two sources of persistent preference heterogeneity $\gamma$ (risk preferences) and $\delta$ ($PPO_{1200}$ differentiation) by leveraging the structure of the three available choices.

$\gamma$ is identified by the choice between $PPO_{250}$ and $PPO_{500}$, which are not horizontally differentiated
$\delta$ is identified by examining the choice between the nest of those two plans and $PPO_{1200}$

2.3 Estimation

Statistical Assumptions

Assume that the random coefficient $\gamma_{k}$ is normally distributed with a mean that is linearly related to observable characteristics $\mathbf{X}^{A}_{k}$: \[ \gamma_{k}(\mathbf{X}^{A}_{k}) \rightarrow N(\mu_{\gamma}(\mathbf{X}^{A}_{k}), \sigma^{2}_{\gamma}) \\ \mu_{\gamma}(\mathbf{X}^{A}_{k}) = \mu + \mathbf{\beta}(\mathbf{X}^{A}_{k}) \]
Denote the mean and variance of $\delta_{k}$, the random intercept for $PPO_{1200}$, as $\mu_{\delta}(Y_{k})$ and $\sigma^{2}_{\delta}(Y_{k})$
Inertia, $\eta(\mathbf{X}^{B}_{kt}, Y_{k})$, is related linearly to $Y_{k}$ and linked choices and demographics $\mathbf{X}^{B}_{kt}$

\[ \eta(\mathbf{X}^{B}_{kt}, Y_{k}) = \eta_{0} + \mathbf{\eta}_{1}\mathbf{X}^{B}_{kt} + \eta_{2}Y_{k} \]

Assume that the family-plan-time specific error terms $\epsilon_{kjt}$ are i.i.d. normal for each j with zero mean and variances $\sigma^{2}_{\epsilon_{j}}(Y_{k})$

Estimation Method

Estimate the choice model using a random coefficients simulated maximum likelihood approach similar to that summarized in Train (2009)
The likelihood function at the family level is computed for a sequence of choices from $t_{0}$ to $t_{2}$, since inertia implies that the likelihood of a choice made in the current period depends on the previous choice
See online appendix B in more detail

2.4 Estimation Results

See Table 5 in this paper
Inertia is substantial, whose mean is $2,032

2.5 Counterfactual Simulations

Conduct counterfactual analysis to study the welfare consequences of reduced inertia in the following settings
- a “naive” setting where the price of insurance does not change as a consequence of incremental selection
- a “sophisticated” setting where plan prices change to reflect the new risk profile of employees enrolled in the different options

2.5.1 Model of Reduced Inertia and Plan Pricing

Assume the policy being implemented reduces inertia to a fraction $Z$ of the family-specific estimate $\eta_{k}$
The expected utility of family $k$ for plan $j$ at time $t$ is

\[ U_{kjt}(P_{kjt}, Z\eta_{k}, 1_{kj, t-1}) = \int^{\infty}_{0} f_{kjt}(OOP)u_{k}(OOP, \widehat{P_{kjt}}, Z\eta_{k}, 1_{kj, t-1})dOOP, \]

Next, we model insurance plan pricing, following the pricing rule used by the firm during the time period studied
The total premium paid by employer and employee, $TP^{y}_{jt}$, for each plan and year was set as the average plan cost for that plan’s previous year of enrollees, plus an administrative markup, conditional on the dependent coverage tier denoted $y$:

\[ TP^{y}_{jt} = AC_{\mathcal{K}^{y}_{j, t-1}} + L = \frac{1}{\|\mathcal{K}^{y}_{j, t-1}\|}\sum_{k \in \mathcal{K}^{y}_{j, t-1}}PP_{kj, t-1} + L \]

$\mathcal{K}^{y}_{j, t-1}$ refers to the population of families in plan $j$ at time $t−1$ in coverage tier $y$
$PP_{kj, t−1}$ is the total plan paid in medical expenditures conditional on $y$ and $j$ at $t − 1$
$TP^{y}_{jt}$ is the amount an employee in dependent category $y$ enrolling in plan $j$ would have to pay each year if they received no health insurance subsidy from the firm
The firm subsidizes insurance for each employee as a percentage of the total $PPO_{1200}$ promium conditional of the family’s income tier, $I_{k}$
Denote the subsidy $S(I_{k})$
The family-plan-time specific out-of-pocket premium $\hat{P_{kjt}}$ from the choice model is

\[ \widehat{P_{kjt}} = TP^{y}_{jt} - S(I_{k})TP^{y}_{PPO_{1200}t} \]

2.5.2 Welfare

We analyze welfare using a certainty equivalent approach that equates the expected utility for each potential health plan option, $U_{kjt}$, with a certain monetary payment $Q$
Formally, the certainty equivalent $Q_{kjt}$ is determined for each family, plan, and time period by solving

\[ u(Q_{kjt}) = -\frac{1}{\gamma_{k}(\mathbf{X}^{A}_{k})} e^{\gamma_{k}(\mathbf{X}^{A}_{k})(W - Q_{kjt})} = U_{kjt}(P_{kjt}, Z\eta_{k}, 1_{kj, t-1}) \]

This welfare measure translates the expected utilities, which are subject to cardinal transformations, into values that can be interpreted in monetary terms¹

Since our empirical choice framework does not distinguish between sources of inertia, we study a range of welfare results spanning the case where inertia is not incorporated into the welfare calculation at all to the case where it is fully incorporated as a tangible cost
Formally, we calculate the certainty equivalent loss as a function of the proportion of the cost of inertia that enters the welfare calculation, denoted $\kappa$:

\[ u(Q^{\kappa}_{kjt}) = -\frac{1}{\gamma_{k}(\mathbf{X}^{A}_{k})} e^{\gamma_{k}(\mathbf{X}^{A}_{k})(W - Q^{\kappa}_{kjt})} = U_{kjt}(P_{kjt}, \kappa Z \eta_{k}, 1_{kj, t-1}) \]

We investigate the welfare consequences of policies that reduce inertia to $Z\eta$, for $\kappa$ between zero and one
Conditional on $\kappa$, the welfare impact for consumer $k$ of policies that reduce inertia to $Z\eta_{k}$ is

\[ \Delta CS^{Z}_{kjt} = (W - Q^{\kappa}_{k, jZ, t}) - (W - Q^{\kappa}_{kjt}) = Q^{\kappa}_{kjt} - Q^{\kappa}_{k, jZ, t} \]

Since total premiums relate directly to average costs, the total welfare change differs from the consumer welfare only if the sum of employee contributions $P_{kjt}$ differs between policy $Z$ and the benchmark model
Given this, the mean per-family welfare change with inertia reduced to $Z\eta$ is

\[ \Delta TS^{Z}_{t} = \frac{1}{\|\mathcal{K}\|}\sum_{k \in \mathcal{K}}\Delta CS^{Z}_{kjt} + \frac{1}{\|\mathcal{K}\|}\sum_{k \in \mathcal{K}}(\widehat{P^{Z}_{kj, t}} - P_{kjt}) \]

2.5.3 Results: No Plan Re-Pricing (The Naive Case)

See Figure A-3 and Table A-6 in online Appendix
Since price is fixed, reduced inertia only leads to increased welfare
At $t_{1}$,
- $Z = 1$ (the benchmark model): 639 consumers choose $PPO_{500}$
- $Z = \frac{1}{2}$: 780 consumers choose $PPO_{500}$ (21 percent increase)
- $Z = \frac{1}{4}$: 913 consumers choose $PPO_{500}$ (44 percent increase)
- $Z = 0$: 1,052 consumers choose $PPO_{500}$ (65 percent increase)
Similar results At $t_{2}$

2.5.4 Results: Endogenous Plan Re-Pricing (The Sophisticated Case)

See Figure 3, Table 6 (and Figure A-4 and Table A-7 in online Appendix)
With endogenous plan re-pricing, premiums change as consumers switch plans due to reduced inertia
Reduced inertia decreases $t_{6}$ enrollment in $PPO_{250}$ from 744 to 385 and increases enrollment in $PPO_{500}$ from 1,134 to 1,501, indicating that the improved choices over time substantially increase incremental adverse selection
The average cost of $PPO_{250}$ increases relative to $PPO_{500}$ with reduced inertia for all years from $t_{1}$ to $t_{6}$, with a maximum relative change of $4,619

確実性等価（Certainty Equivalent）は不確実性なOutcomeに対する期待効用と同等の効用をもたらす（確実な）金額。期待効用をそれに対応する金額（Monetary Value）で評価を行うことができるため、不確実性のあるマーケットの厚生評価に用いられる（Einav, Finkelstein and Cullen (2010)も参照。）。↩︎