2 Details of the Paper (Theoretical Part)

2.1 Model

2.1.1 Setup and Notation

  • Consumers choose from two available insurance contracts:
    • Contract \(H\), full insurance with price \(p\)
    • Contract \(L\), no insurance available for free
  • Characteristics of the insurance contracts as given, although we allow the price of insurance to be determined endogenously
  • \(G(\zeta)\): distribution of the population, where \(\zeta\) is a vector of consumer characteristics
  • \(v^{H}(\zeta_{i}, p)\), \(v^{L}(\zeta_{i})\): consumer \(i\)’s (with characteristics) utility from buying contracts \(H\) and \(L\), respectively
  • Assume that \(v^{H}(\zeta_{i}, p)\) is strictly decreasing in \(p\) and that \(v^{H}(\zeta_{i}, p = 0) > v^{L}(\zeta_{i})\)
  • \(c(\zeta_{i})\): the expected monetary cost associated with the insurable risk for individual \(i\)

2.1.2 Demand for Insurance

  • Each individual makes a discrete choice of whether to buy insurance or not
  • Assume that firms cannot offer different prices to different individuals
  • Then, individual \(i\) chooses to buy insurance if and only if \(v^{H}(\zeta_{i}, p) \geq v^{L}(\zeta_{i})\)
  • Define \(\pi(\zeta_{i}) \equiv \max \{ p: v^{H}(\zeta_{i}, p) \geq v^{L}(\zeta_{i}) \}\)
  • Aggregate demand for insurance is therefore given by:

\[ D(p) = \int 1(\pi(\zeta) \geq p)dG(\zeta) = \Pr(\pi(\zeta_{i}) \geq p) \]

  • Assume that \(D(p)\) is strictly decreasing, continuous, and differentiable

2.1.3 Supply and Equilibrium

  • Consider \(N \geq 2\) identical risk-neutral insurance providers, who set prices in a Nash equilibrium
  • Assume that when multiple firms set the same price, individuals who decide to purchase insurance at this price choose a firm randomly
  • Assume that the only costs of providing contract \(H\) to individual \(i\) are the insurable cost \(c(\zeta_{i})\)
  • The above assumptions imply that at the average (expected) cost curve in the market:

\[ AC(p) = \frac{1}{D(p)} \int c(\zeta)1(\pi(\zeta) \geq p)dG(\zeta) = E(c(\zeta) \mid \pi(\zeta) \geq p) \]

  • The marginal (expected) cost curve in the market

\[ MC(p) = E(c(\zeta) \mid \pi(\zeta) = p) \]

Add two further simplifying assumptions to straightforwardly characterize equilibrium.

  1. Assume that there exists a price \(\overline{p}\) such that \(D(\overline{p}) > 0\) and \(MC(p) < p\) for every \(p > \overline{p}\). In words, we assume that it is profitable (and efficient, as we will see soon) to provide insurance to those with the highest willingness to pay for it1
  2. Assume that if there exists \(\underline{p}\) such that \(MC(\underline{p}) > \underline{p}\), then \(MC(p) > p\) for all \(p < \underline{p}\). That is, we assume that \(MC(p)\) crosses the demand curve at most once2
  • These assumptions guarantee the easy to verify that these assumptions guarantee the existence and uniqueness of equilibrium
  • In particular, the equilibrium is characterized by the lowest break-even price, that is, 3

\[ p^{*} = \min \{ p: p = AC(p) \} \]

2.2 Measurig Welfare

  • Measure consumer surplus by the certainty equivalent.
  • The certainty equivalent of an uncertain outcome is the amount that would make an individual indifferent between obtaining this amount for sure and obtaining the uncertain outcome.
  • Denote by \(e^{H}(\zeta_{i})\) and \(e^{L}(\zeta_{i})\) the certainty equivalents for consumer \(i\) of an allocation of contract \(H\) and \(L\), respectively.
  • Under the assumption that all individuals are risk-averse, the willingness to pay for insurance in given by \(\pi(\zeta_{i}) = e^{H}(\zeta_{i}) - e^{L}(\zeta_{i})\)
  • Consumer welfare is given by:

\[ CS = \int [(e^{H}(\zeta) - p)1(\pi(\zeta_{i}) \geq p) + (e^{L}(\zeta) - p)1(\pi(\zeta_{i}) < p)]dG(\zeta) \]

  • Producer welfare is given by:

\[ PS = \int (p - c(\zeta))1(\pi(\zeta_{i}) \geq p)dG(\zeta) \]

  • Total welfare will then be given by:

\[ \begin{align} TS &= CS + PS \\ &= \int [(e^{H}(\zeta) - c(\zeta))1(\pi(\zeta_{i}) \geq p) + e^{L}(\zeta)1(\pi(\zeta_{i}) < p)]dG(\zeta) \end{align} \] Efficient Allocation

  • It is socially efficient for individual \(i\) to purchase insurance if and only if \(\pi(\zeta_{i}) \geq c(\zeta_{i})\) (a first-best allocation)
  • However, achieving the first best may not be feasible if there are multiple individuals with different \(c(\zeta_{i})\)’s who all have the same willingness to pay for contract \(H\), since price is the only instrument available to affect the insurance allocation
  • Therefore, it is useful to define a constrained efficient allocation as the one that maximizes social welfare subject to the constraint that price is the only instrument available for screening
  • It is constrained efficient for individual \(i\) to purchase contract \(H\) if and only if

\[ \pi(\zeta_{i}) \geq E(c(\widetilde{\zeta}) \mid \pi(\widetilde{\zeta}) = \pi(\zeta_{i})) \]

2.3 Graphical Representation

2.3.1 Adverse Selection

  • See Figure 1 of the paper
  • The individuals who have the highest willingness to pay for insurance are those who, on average, have the highest expected costs
  • This is represented by downward-sloping \(MC\) curve
  • \(AC\) curve is also downward-sloping, and always above \(MC\) curve4
  • Because average costs are always higher than marginal costs, adverse selection creates underinsurance

2.3.2 Advantageous Selection

  • See Figure 2 of the paper
  • Those with more insurance have lower average costs than those with less or no insurance
  • \(MC\) and \(AC\) curve is upward-sloping, and \(AC\) is below \(MC\)5
  • Because average costs are always higher than marginal costs, adverse selection creates overinsurance

2.4 Sufficient Statistics for Welfare Analysis

  • The demand and cost curves are sufficient statistics for welfare analysis of equilibrium and nonequilibrium pricing of existing contracts
  • Different underlying primitives (i.e., preferences and private information, as summarized by \(\zeta\)) have the same welfare implications if they generate the same demand and cost curves6

2.5 Incorporating Moral Hazard

  1. \(\pi(\zeta_{i}) = p\)なるmarginal consumerを考えたとき、当然\(\pi(\zeta_{i}) \geq p\)を満たすので\(H\)を選択する。 このとき、仮定1から\(p > \overline{p}\)について\(MC(p) < p\)であるため、\(p > \overline{p}\)におけるmarginal consumerは\(H\)を選択し、かつ\(MC(p)\)は\(p\)より小さい。 各\(p\)についてmarginal consumerが複数いることはあり得るが（Footnote 3）、すべてのmarginal consumerについて上記が成立するので各\(p > \overline{p}\)についてmarket全体の需要量は\(MC(p)\)を上回る。 したがって、\(p > \overline{p}\)について\(D(p) > MC(p)\)。↩︎

  2. 仮定1と同様にmarginal consumerを考えているので\(H\)を選択する。そのうえで、仮定2より、\(MC(\underline{p}) > \underline{p}\)なる\(\underline{p}\)が存在する場合、\(p < \underline{p}\)について\(MC(p) > p\)が成立する。改定1より\(p > \overline{p}\)について\(D(p) > MC(p)\)が成立しており、仮定2の\(\underline{p}\)が存在しない場合はすべての\(p\)について\(D(p) > MC(p)\)となるため\(D(p)\)と\(MC(p)\)は1回も交差しない。↩︎

  3. 各企業は一律に価格を設定するため、平均費用をすべての消費者に課す。企業はベルトラン競争を行っており、価格が一律でない場合はナッシュ均衡が存在しない。すべての企業の価格が同一であり、利潤0となる\(p = AC(p)\)がナッシュ均衡となる。↩︎

  4. Adverse Selectionでは、（グラフの左側に位置する）WTPが高い消費者ほど（期待）コストが高いため、\(AC > MC\)となる。↩︎

  5. Advantageous Selectionでは、（グラフの左側に位置する）WTPが高い消費者ほど（期待）コストが低いため、\(AC < MC\)となる。↩︎

  6. 上記の通り、需要関数と費用関数さえ推定できればWelfare Lossを推定できるので、両者の元となるPrimitives（Source of Selectionなど）はWelfare Lossを推定するだけであれば不要。↩︎