essays in public finance and industrial organization a dissertation ...

CHAPTER 2. HEALTH PLAN CHOICE 69
risk, denoted θ. Each consumer chooses between a high-cost plan A and a low cost
plan B. We can think of the plans, for now, as vertically differentiated. The plans’
expected costs of covering a type-θ consumer are cA(θ) and cB(θ). Let ∆c (θ) =
cA(θ) − cB(θ) denote the cost differential. We assume ∆c is strictly positive and
increasing in θ.
Let vA(θ) and vB(θ) denote a type θ’s expected (dollar) value from being covered
by each of the plans. For the moment, the benefits of coverage are determined only
by forecastable health risk. We assume that contributions vary across plans but
not across consumers. A consumer who makes a contribution pj to enroll in plan
j ∈{A, B} gets a net benefit vj(θ) − pj. 3 Define ∆v(θ) =vA(θ) − vB(θ) to be the
additional amount a type-θ consumer would pay for the high-cost plan.
The efficient assignment places a type-θ consumer in plan A if and only if
∆v(θ) − ∆c(θ) ≥ 0.
At the same time, a type-θ consumer will select plan A if and only if
∆v(θ) − ∆p ≥ 0,
where ∆p = pA − pB is the incremental contribution for plan A.
Are there prices that lead to an efficient outcome? Assume that ∆v(θ) is increasing
in θ, which seems appropriate if plan A simply offers more generous coverage or easier
access to care. Then for any incremental contribution ∆p, atype-θ consumer will
choose A if and only if θ ≥ θ(∆p), where θ (∆p) is a threshold that can be varied
arbitrarily with ∆p. 4 Therefore it is possible to achieve efficient sorting if and only if
the efficient assignment also involves a threshold rule, i.e. if ∆v(θ)−∆c(θ) is negative
up to some θ ∗ and positive above it. Intuitively, the requirement for efficiency is that
willingness to pay increases more quickly with risk than the cost differential between
3 Here we make the simplifying assumption, which we maintain in our econometric model, that
plan preferences are additively separable in the plan premium. See ? for an extensive discussion of
this assumption.
4 An empirical prediction of this model is that plan A will experience unfavorable selection, and
its risk composition will be worse the larger is ∆p.