Are Invisible Hands Good Hands in Health Care Markets? Extension

1 Introduction

In the health economics literature, it is often suggested that particular distortions in health care markets imply that competition may not be socially desirable (Crew 1969). Health insurance often leads to out-of-pocket usage prices that are less than the marginal costs of medical services. Health insurance thus generates deadweight loss by encouraging too much consumption of the services. Therefore, reducing competition would raise usage prices that are “too low,” and thus may be desirable. Gaynor, Hass-Wilson, and Vogt (GHV) argue that this view cannot be correct if insurance markets are competitive or monopolistic. They find that “provided that price exceeds marginal cost in the medical market, the benefit to consumers of a price decrease outweighs the loss in profits suffered by the medical industry” (GHV 2000, 1001).

GHV’s (2000) finding has important policy implications in health care regulation. However, the analysis is based on a specific exponential Bernoulli utility function, ^[1]v(x)=−exp(−rx) (GHV 2000, 999), which means that consumers have a constant Arrow-Pratt coefficient of absolute risk aversionr>0. It is unclear whether this assumption can easily be satisfied in the real world. The present note shows that the model can be extended to one with a general concave Bernoulli utility function, which means that the finding remains valid as long as consumers are risk averse.

In a related paper, Wigger and Anlauf (2007) study the role of market power in health care markets. They consider a market with a monopolistic drug provider and competitive insurers. Consumers are either healthy or sick. In the latter case, they have demands for drugs. Consumers choose an insurance contract (characterized by a premium and a co-insurance rate) before the monopolist chooses a drug price. Wigger and Anlauf show that when the marginal cost of the drug is strictly positive, a marginal increase in the co-insurance rate, starting from the market equilibrium level, lowers social welfare. The present note shows that a marginal increase in the out-of-pocket usage price from the equilibrium level always reduces social welfare. The findings are in contrast to the view of Crew (1969).

2 Model

Consider a standard insurance model. There is a continuum of consumers who are homogeneous ex ante. They face independent health shocks. Each consumer has the following utility function:

In expression [1], v(⋅) is the Bernoulli utility function, defined on the ex post monetary payoff of the consumer, Y is the consumer’s income, τ is the out-of-pocket usage price faced by an insured consumer, m is the insurance premium, x is the quantity of health care consumed, ε is a random shock to health, and g(x,ε) is the consumer’s ex post benefit from health care. GHV (2000, 999) assume that the Bernoulli utility function is v(b)=−exp(−rb), in order to “guarantee that there are no income effects ex ante.” In this note, function v(.) can be any differentiable function on [0,+∞) that satisfies v′(⋅)>0 and v″(⋅)<0. The health care market is imperfectly competitive and the insurance market is perfectly competitive. The market price and marginal cost of health care are denoted by p and c, respectively.

The game played in the market has the following timing. First, health care providers interact and determine the market price of health care. Second, insurance firms offer insurance policies. Third, consumers decide to accept or refuse the policies. Fourth, health states are realized and consumers choose their quantity demanded of health care. As in GHV (2000), the interaction among health care providers is not explicitly modeled here. A lower market price represents more intensive competition in the health care market.

3 Invisible hands are good hands

The model is solved through backward induction. Conditional on insurance policy (τ,m) being accepted by a consumer at the third stage, the consumer in state ε maximizes her ex post utility v(Y−τx−m+g(x,ε)) at the fourth stage. Because v′(⋅)>0, she only needs to maximize her ex post payoff (Y−τx−m+g(x,ε)). Hence the consumer solves the following problem at the fourth stage.

The demand for health care x∗ satisfies the first-order condition

Hence, the consumer’s ex post demand for health care x∗=x∗(τ,ε) depends on the out-of-pocket price τ, but not on the insurance premium m. We have x∗=0 if and only if τ≥g1(0,ε). When there is a non-degenerated solution, from eq. [4] we have

and

The Envelop Theorem implies that

At the second stage, the insurance firms offer the most attractive policy that is not money-losing. Given health care price p, the insurers solve the following problem:

Constraint [9] must be binding because of competition. The problem can be rewritten as an unconstrained problem

where x∗=x∗(τ,ε) is given by eq. [4]. For any p≥c, it is assumed that the objective function is continuous and differentiable.

If problem [8] or [10] has multiple (or a compact set of) solutions, we shall assumed that the insurers always choose the solution that has the lowest usage price (and the highest premium). Indeed, while the consumers and insurers are indifferent to any of the solutions, the health care providers strictly prefer the solution with the lowest usage price, which induces the highest consumption of health care. Hence, there is a unique Pareto optimal policy. Such a policy is written as (τ(p),m(p)). Given health care price p, function L(τ;p) is strictly increasing in a small-enough left-neighborhood of τ(p), and non-increasing in a small-enough right-neighborhood of τ(p).

Since v′(⋅) is decreasing, from eqs [6] and [7] we have both v′(b) and x∗ strictly increasing with ε. Thus, Cov(v′(b),x∗)>0. ^[2] We also have E(∂x∗∂τ)=E(1g11)<0 and E(v′(b))>0. Hence, eq. [11] implies that τ<p in equilibrium. ^[3]Q.E.D.

The insurance makes health care providers better off by increasing consumer demand for health care. The insurance also makes consumers better off by (partially) insuring them from health shocks. Therefore, insurers bring a Pareto improvement to society.

When the health care market is highly competitive, it is likely that health insurance generates ex post deadweight loss. For example, if p=c, then usage price τ<c, which leads to the so-called “moral hazard” problem in the health economics literature (Crew 1969). As GHV (2000, 996) mention, “ex post efficiency is not a sensible welfare criterion here since it ignores the benefits to obtaining insurance ex ante.” Assume that function τ(⋅) is continuously differentiable. We have the following lemma.

However, with eq. [11],

This conflicts with eq. [12]. Hence, τ(p) must be non-decreasing. Q.E.D.

The per capita health care industry’s expected profit is πm=(p−c)E(x∗). As GHV (2000), I add the per capita profits to the consumer’s income in order to conduct a social welfare analysis. The expected social welfare (per capita) is then represented by

Note that the consumer is only a representative of many consumers. She is unable to noticeably influence the per capita profits by adjusting her health care consumption. Hence she takes the per capita profits as exogenously given. Because of the separability inside v(.), the consumers’ ex post maximization problem still leads to x∗=x∗(τ,ε)) as shown in problem [3]. In other words, receiving πm would not affect the consumers’ demands for health care. With πm(p)=(p−c)E(x∗), m=(p−τ)E(x∗), x∗=x∗(τ,ε)) and τ=τ(p),

Corresponding to Proposition 2 of GHV (2000), the following result implies that social welfare W(p) is maximized at p=c.

The insurance policy is endogenous. From eq. [11], ^[4]

Substituting into eq. [16],

Since τ′(p)≥0, p>c, E(∂x∗∂τ)<0, and E[v′(b)]>0, we have W′(p)≤0. Q.E.D.

From eq. [15], it can be seen that the price of health care p influences (expected) social welfare through and only through the usage price τ(p), which is determined by problem

This problem differs from problem [10] only by the πm inside v(.). Since adding per capita profits πm to the consumer’s income does not affect the consumer’s ex post demand x∗(τ,ε), the solutions to the two problems are the same. When insurers solve problem [19], they do not take into account the external effects on health care providers. Note that a lower usage price leads to higher provider profits. Hence the equilibrium usage price tends to be too high from the perspective of social welfare.

where πm=(p−c)E(x∗(τ,ε)) is decreasing in usage price τ, rather than exogenously given. Noting g1(x∗,ε)=τ, the first-order condition of the problem is

Since v′(.)>0 and ∂πm∂τ<0, E[(v′(b)∂πm∂τ]<0. Hence, at the market equilibrium usage price, which satisfies eq. [17], we have ∂W∂τ=E[(v′(b)∂πm∂τ]<0. Therefore, if insurers offered a slightly lower usage price, social welfare would be improved. Q.E.D.

Proposition 2 resembles a finding in Wigger and Anlauf (2007), but the two models differ in the number of health states, modeling of insurance contracts, and timing of the games. Wigger and Anlauf assume two health states (“healthy” and “sick”), an insurance contract is characterized by a premium and a co-insurance rate, and consumers choose an insurance contract before the monopolistic drug provider chooses a price. GHV (2000) and the present note, instead, assume a continuum of health states. An insurance contract is characterized by a premium and an out-of-pocket usage price, and based on the given health care price. Wigger and Anlauf (2007) find that a marginal increase in the co-insurance rate, starting from the market equilibrium level, leaves social welfare unaffected if the marginal cost of the drug is zero, and lowers social welfare if the marginal cost is positive. The present note finds that a marginal increase in the out-of-pocket usage price from the equilibrium level always lowers social welfare, whether the marginal cost is zero or positive.

4 Conclusion

Gaynor et al. (2000) find that “invisible hands are good hands” in health care industries. As the authors suggest, their model ignores income effects on consumer choices: “the separability inside (the Bernoulli utility function) v guarantees that there are no income effects ex post. The assumption of an exponential form for v guarantees that there are no income effects ex ante” (GHV 2000, 999).

I show that the assumption of “an exponential form for v” is unnecessary, i. e., the findings hold as long as consumers are risk-averse. Whether the assumption of “separability inside v” can also be dropped calls for future study. I also show that if competitive insurance firms offered policies with slightly lower usage prices than the equilibrium level, social welfare would be improved.