Motivation

• Adverse selection is a central concern in health insurance markets
• But very little empirical work in this space (until 2010 or so)

Basic Model

• Unique feature of adverse selection: demand affects firm costs
• Complicates welfare analysis because we need to know demand curve (how demand varies with price) but also how price changes affect costs

Setup

• Two available contracts: high coverage (\(H\)) and low coverage (\(L\)), with relative price of \(H\) given by \(p\)
• Consumer characteristics \(\zeta\) from distribution \(G(\zeta)\)
• Utility denoted by \(\nu^{H}(\zeta_{i}, p)\) and \(\nu^{L}(\zeta_{i})\) for plan \(H\) and \(L\), respectively
• \(\nu^{H}(\zeta_{i}, p)\) decreasing in \(p\), with \(\nu^{H}(\zeta_{i}, p=0)>\nu^{L}(\zeta_{i})\)
• Costs to insurer given by \(c(\zeta_{i})\)

Demand

• Individual \(i\) choose health insurance if \(\nu^{H}(\zeta_{i}, p)>\nu^{L}(\zeta_{i})\)
• Denote highest price at which individual \(i\) chooses \(H\) as \(\pi(\zeta_{i}) \equiv \max \{p: \nu^{H}(\zeta_{i}, p)>\nu^{L}(\zeta_{i})\}\)
• Aggregate demand, \(D(p) = \int 1(\pi(\zeta) \geq p) dG(\zeta) = Pr(\pi(\zeta) \geq p)\)

Supply

• Bertrand competition with many risk-neutral insurers
• Focus on case of perfect competition as benchmark, so that inefficiency derives from selection and not other factors
• Costs to insurer given by \(c(\zeta_{i})\)
• Average costs are then, \(AC(p) = \frac{1}{D(p)} \int c(\zeta) 1(\pi(\zeta) \geq p) dG(\zeta) = E(c(\zeta) | \pi(\zeta) \geq p)\)
• Marginal costs, \(MC(p) = E(c(\zeta) | \pi(\zeta) = p)\)
• Equilibrium, \(p^{*} = \min \{p : p=AC(p) \}\)

Welfare

Measure welfare using certainty equivalent: amount that would make consumer indifferent between obtaining outcome for sure versus obtaining outcome with uncertainty, denoted \(e^{H}(\zeta_{i})\) and \(e^{L}(\zeta_{i})\), so that WTP for health insurance is reflected by \(\e^{H}(\zeta_{i}) - e^{L}(\zeta_{i})\)

• \(CS = \int \left[ (e^{H}(\zeta) - p)1(\pi(\zeta) \geq p) + e^{L}(\zeta) 1(\pi(\zeta) < p) \right] d G(\zeta)\)
• \(PS = \int \left[ (p - c(\eta)) 1(\pi(\zeta) \geq p) \right] d G(\zeta)\)
• \(TS = CS + PS = \int \left[ (e^{H}(\zeta) - c(\eta)) 1(\pi(\zeta) \geq p) + e^{L}(\zeta) 1(\pi(\zeta) < p) \right] d G(\zeta)\)
• Socially efficient to purchase health insurance only if \(\pi(\eta_{i}) \geq c(\eta_{i})\)

Textbook Adverse Selection

• \(D>MC\) because willingness to pay is expected costs plus risk premium
• Relationship between demand and AC reflects problems due to adverse selection

Main points

• Key assumptions:
  • Simplifying assumption that insurers earn 0 profit (at least approximately)
  • Individuals select plans based on health needs (which is private information)
  • Common price to all enrollees of a given plan (community rating)
• Possible outcomes:
  • Full insurance (Demand always above AC)
  • Partial unravelling (Demand intersects with AC somewhere)
  • Full unravelling (Demand always below AC)

Incorporating data

• Goal is to estimate demand curve, \(D(p)\), and average costs, \(AC(p)\)
• \(MC(P)\) derives from product of \(D(p)\) and \(AC(p)\)
• Need data on:
  • prices and quantities
  • expected costs, such as claims or medical spending
  • exogenous shifters in prices

Possible sources of variation

• State regulations introduce variation across people and over time
• Tax policies such as health insurance subisides
• Field experiments and “idiosyncrasies of firm pricing behavior”
• Shifts in administrative costs of handling claims
• Common demand instruments such as plausibly exogenous changes in market conditions

Empirical application

• Health insurance for Alcoa employees in 2004
• “High” versus “low” coverage PPO plans
• Exogenous variation:
  • 7 different pricing menus for same coverage
  • Focus on relative difference, \(p= p_{H} - p_{L}\)
  • Price menus determined by president of business unit
  • 40 units in company

• Denote medical expenditures by \(m_{i}\)
• Denote incremental costs by \(c_{i}=c(m_{i}; H) - c(m_{i}; L)\)
• \(c(m_{i}; H)\) is observed
• \(c(m_{i}; L)\) computed based on terms of contract \(L\)

• Baseline equations: \[\begin{align*} D_{i} & = \alpha + \beta p_{i} + \epsilon_{i} \\ c_{i} & = \gamma + \delta p_{i} + u_{i} \end{align*}\]

• From those: \[MC = \frac{1}{\beta} \frac{ \partial D_{i} c_{i}}{\partial p} = \frac{1}{\beta} (\alpha \delta + \gamma \beta + 2\beta \delta p)\]

• Find intersection points and compare to equilibrium points (\(AC(p) = D(p)\)) and efficient points (\(MC(p) = D(p)\))

Motivation

• Prices often not reflective of costs of coverage
• Insurers have incentive to attract specific types of enrollees
• Enrollees have incentive to select specific types of plans

Setup

• “Standard” setup with 2 plans
• Assumes WTP $u $ perfectly correlated with health risk \(\theta\)
• Assumes WTP increasing in \(\theta\) faster than costs (i.e., slope of demand curve steeper than slope of AC curve)

Relaxing assumptions yields potential inefficient plan allocation

Data

• “Private firm that helps small and mid-sized employers manage health benefits”
• 11 employers in one metropolitan area in “western United States” from 2004-2005
• Data on 6,603 enrollees (3,683 employees), claims, plan selection, etc.

Model: Demand

Household utility: \[u_{hj} = \alpha_{1} x_{j} + \alpha_{2} x_{h} + \psi (r_{h} + \mu_{h} ; \alpha_{3}) - p_{j} + \sigma_{\varepsilon} \varepsilon_{hj}\]

• \(x_{j}\): plan characteristics
• \(x_{h}\): household characteristics
• \(r_{h}\): health risk score (based on observable demographics)
• \(\mu_{h}\): unobserved health status

• Household \(h\) selects plan \(j\), \(q_{hj} =1\), if \(u_{hj} \geq u_{hk} \forall k \in J_{h}\)
• Standard logit model for plan choice: \[Pr(q_{hj}=1 | x_{h}, \mu_{j}) = \frac{ \exp(v_{hj}) }{ \sum_{k} \exp(v_{hk}) },\]

for \(v_{hj} = u_{hj} - \sigma_{\varepsilon} \varepsilon_{hj}\)

Model: Costs

Costs assumed: \[c_{ij} = a_{j} + b_{j}(r_{i} + \mu_{i} -1) + \eta_{ij}\]

• \(a_{j}\): baseline cost for standard enrollee
• \(b_{j}\): marginal cost of insuring a higher/lower risk enrollee
• Costs for firm-year \(f\) and insurer \(k\) are sum of all relevant \(c_{ij}\)

Model: Other things

• Plan bids: markup over expected costs
• Employer contribution: employers pass on share of lowest-cost plan plus fraction of incremental cost for high-cost plans

Identification and estimation

• Need variation in prices separate from unobserved household tastes or private health risks \(\mu\)
• Instrument for actual plan contribution using predicted contribution
• Estimate using method of simulated moments (simulation comes from draws of \(\mu\) from normal distribution with mean 0 and variance \(\sigma^{2}\):
  • Consumer choice: \(E[q_{hj} - Pr(q_{hj}=1 | x_{h}, \mu_{j})|z_{h}, \mu_{h}] = 0\), with instruments \(z_{h}\)
  • Plan costs: \(E[C_{kf} - \hat{C}_{kf}|x_{kf}, \mu_{kf}] = 0\)
  • Plan bids: \(E[B_{if} - \hat{B}_{if}|x_{f}] = 0\)

Results

Takeaways

• Welfare loss of 2-11% of total cost of coverage under current pricing and contributions
• Uniform contribution policy (or community rating) offers modest improvements (1-3%)
• Risk-adjusted premium policy can capture entirety of welfare gains