Demand heterogeneity in insurance markets: Implications for equity and efficiency

In many markets insurers are barred from price discrimination based on consumer characteristics like age, gender, and medical history. In this paper, I build on a recent literature to show why such policies are inefficient if consumers differ in their willingness-to-pay for insurance conditional on the insured losses they generate. Using administrative claims data, I then show that this type of demand heterogeneity is empirically relevant in a consumer health plan setting. Younger and older consumers and men and women reveal strikingly different demand for health insurance, conditional on their objective medical spending risk. This implies that these groups must face different prices so as to sort themselves efficiently across insurance contracts. The theoretical and empirical analysis highlights a fundamental trade-off between equity and efficiency that is unique to selection markets.

For group a consumers with costs c, only prices p ≤ c sort efficiently.For group b consumers with costs c, only prices p > c sort efficiently.Therefore, no uniform price can sort all consumers in both groups efficiently.Now consider setting separate prices for groups a and b.By the single-crossing and montonicity assumptions, within groups a or b all consumers can be sorted efficiently with the correct group-specific price.This within-group case is a standard case considered in the literature (see, e.g., Einav and Finkelstein (2011)).Therefore, optimal group-specific prices that sort all consumers efficiently are welfare-improving relative to any uniform price because no uniform price can sort consumers with costs c efficiently.
Note that the case above is constructed so that the types z break up the market into submarkets with monotone marginal cost curves.In this case, the types with higher valuations optimally face lower prices.While the simplicity of this case is instructive, a more conceptually difficult case occurs when one allows for z to be only an imperfect signal of demand conditional on costs, so that submarkets defined by z do not contain strictly decreasing, monotone marginal cost curves.In such cases, optimal pricing could be higher, lower, or-in a knife-edge case-equal for z types with higher valuations, depending on the exact pattern of the residual demand heterogeneity.See Figure 4 for an example of the complex case in which optimal prices are higher for the higher-valuation types.

A.2. Worker and firm characteristics
The firm is a manufacturer of consumer nondurables, with plant locations throughout the United States.Files are purged of information that could be used to identify the employer or employees, though the "anonymized" employees are traceable across plan years.Workers are predominately male.Most are hourly.Workers who decline coverage are not in the data.Although no information is known regarding employees who waive coverage, data from the Kaiser Family Foundation (2007) Survey of Employer Health Benefits indicates that 15% of workers in firms of similar size and in the same industry (consumer nondurables) around the study period waived coverage.

A.3. SMLE estimation details for Section 6
Simulated maximum likelihood estimation (SMLE) is used to estimate Equation ( 7).Given the choice model outlined above, there is no closed-form expression for the choice probabilities.Therefore, simulation is needed to calculate likelihood values.Here, I assume a normal error that enters the model nonlinearly, but even if the error structure admitted a closed-form solution to the choice probabilities (as would be the case, for example, with an additive extreme value error, which yields logit probabilities), simulation would still be necessary so as to numerically integrate over the risk distribution F ij (OOP).
Estimation begins by fixing the parameter vector θ and taking Q draws from the distribution of out-of-pocket expenses, F ij (OOP), to calculate an expectation over health risk.Conditional on a set of values from the parameter vector (including a single draw of the error term ij ), the expected utility for person i in plan j is The superscript r denotes a draw from the normal error term.
A basic accept-reject (AR) simulator would assign a value of 0 or 1 as the choice probability of plan j depending on whether j generated the highest utility given the simulated value of the error, r ij .Here, I use a smoothed AR simulator.For each draw r, a smoothing function is applied to ensure that the probability falls strictly between 0 and 1, avoiding problems otherwise arising from flat portions of the log-likelihood function.I employ a logit smoothing kernel that follows Train (2009): This smoother approaches an unsmoothed AR simulator as λ approaches 0.1 The output of the smoother is the probability of selecting a plan, conditional on values from the parameter vector, including a single draw from the normal error term.So as to calculate the simulated choice probability, many draws are taken from the error (still for a fixed parameter value of σ ).Averaging the smoother values across these draws yields the simulated probability of person i choosing plan j: Finally, the simulated log-likelihood function is defined by where d ij is an indicator function for the observed choice of each individual.Standard numerical maximization routines are used to search for the maximizing parameter vector.In practice, I set Q = 50, R = 100, and λ > 5.In this figure, rather than using the observed plan choice, the dependent variable is the simulated probability of choosing plan H. Simulated choices are predicted by the expected utility model parameters reported in Table 6.See the notes to Figure 6 for additional details.

Figure A1 .
Figure A1.Model implied by the common wisdom that uniform prices can sort efficiently.The figure illustrates the model implicit behind the common wisdom that uniform prices can achieve an optimal allocation.Here, a single price sorts efficiently, even though types differ in their costs on average.The common wisdom embodied in the figure implicitly relies on the assumption that consumer types differ only because they differ in the costs they generate, but not demand conditional on costs.See the notes to Figure1for additional documentation.

Figure A2 .
Figure A2.Data: Expected costs ( C i ) and realized costs (C i ).The figure plots scatters of realized healthcare spending versus expected spending, including the insurer and consumer shares.Each panel contains a separate age group.Expected costs are predicted by the Johns Hopkins Adjusted Clinical Grouper (ACG) on the basis of prior year diagnoses and procedures.See the text for full details.The dashed line indicates the intercept and slope from a log-log regression of actual spending on expected spending.

Figure A3 .
Figure A3.Robustness/mechanisms: Revealed demand heterogeneity, alternative cost measures.The figure plots local polynomial regressions of plan choice against expected healthcare consumption.The dependent variable is an indicator for choosing the more generous plan, H.Each panel of the figure exactly replicates panel A of Figure6, except that along the horizontal axes, alternative cost measures are used.In panel A, the current year's actual realized costs are the conditioning cost variable, rather than an unbiased measure of expected costs for the coming year.This would represent perfect foresight on the part of consumers, who choose plans prior to this realization.In panel B, last year's actual costs are used as the conditioning variable.In the left panels, zeros are transformed to allow taking logs.In the right panels, zeroes are dropped from estimation.Consult the text and Figure6for additional details.

Figure A4 .
Figure A4.Model fit: Simulated plan choices.The figure demonstrates the fit of the expected utility model.The plot replicates panel of A of Figure6in plotting local polynomial regressions of plan choice on expected healthcare consumption.In this figure, rather than using the observed plan choice, the dependent variable is the simulated probability of choosing plan H. Simulated choices are predicted by the expected utility model parameters reported in Table6.See the notes to Figure6for additional details.