Allocating Liability among Several Physicians: A Theoretical Model

Appendix A: Punitive Damages

Contrary to the current law in most countries, let us assume that the judge can impose punitive damages on tortfeasors. The existence of punitive damages makes the question of the apportionment of medical damage between physicians of little use. That is why we consider the simplest case, in which the fine is independent of the precaution levels. However, an analysis with the same conclusions could be investigated for a fine which varies with the precaution levels.

Under a strict liability regime, each physician must pay a fine F if medical damage occurs. The reaction of physician i to physician j’s precaution level can be rewritten as

which leads to the following first-order condition

The first-best precaution level can be implemented by making the private marginal benefit (left-hand side of the last condition) equal to the social marginal benefit given in (3). Therefore, the optimal fine is F^o = H/π.

Under a fault-based liability regime, a fine is payable by the physician only if he was negligent. We assume that the policy-maker set the due care standard at the first-best precaution level, defined by (3). Each physician chooses between two strategies: to comply with the due care standard, or not to comply. If physician j complies, the payoffs of physician i associated with each strategy are respectively s _i − c(x^o) and s i − c ( x i ) − π F p x i , x o . We can demonstrate that punitive damages are required to implement the first-best precaution level only when detection of damage is low (i.e., below a threshold probability of detection). Let us set F = H. If the probability of detection is sufficiently high, s _i − c(x^o) is greater than s i − c ( x i ) − π H p x i , x o with x ⁱ < x^o, and physician i chooses to comply with the due care standard. If the probability of detection is low, s _i − c(x^o) is less than s i − c ( x i ) − π H p x i , x o with x ⁱ < x^o, and physician i prefers to be careless. In this case, the fine should be set to its optimal amount F^o, in order to encourage the physician to choose the first-best precaution level. The fine amount depends on a threshold probability of detection π ̄ , defined by

or, after rearranging,

If π ≥ π ̄ , damages equal to H are sufficient, such that each physician complies with the due care standard. If π < π ̄ , punitive damages equal to F^o are required. The proof is similar to that of Proposition 2, provided in Appendix B.

Therefore, under both liability regimes, the existence of punitive damages allows the policy-maker to provide physicians with the optimal incentives for care-taking. Strict liability and the negligence rule are efficient for any value of the probability of detection. The only difference between both liability regimes is the amount of punitive damages, depending on the probability of detection.

Appendix B: Proof of Proposition 2

1. The efficient weight of proportional allocation α* = ϵ(x*) follows the same rule as with the perfect detection of damage.

We denote by ϵ(x) the scale elasticity of the probability of damage with respect to both physicians’ carelessness, given that both physicians are equally careful. Formally,

At the second-best precaution level x*, the Eq. (7) leads to define an efficient weight of proportional allocation, as α* = ϵ(x*). The rule is close to the efficient one in the perfect detection case (Leshem 2017). As in Leshem, we can demonstrate that this equilibrium is unique. For that, it is sufficient that the marginal benefit of precautions is linear. As the marginal cost is strictly increasing, then both functions cross only once.□

1. The efficient weight of proportional allocation α* = ϵ(x*) increases with the degree of substitutability between the precaution levels of the physicians γ.

We first assume equally careful physicians; that is, x i c = x c . We replace the terms and we rearrange to obtain

The substitution of 1 − x to x ^c leads to

where one argument of ϵ is not indicated, for sake of clearer presentation. Taking the first derivative of ϵ(x*) with respect to γ yields

By the implicit-function theorem, we can express ∂x*/∂γ. Let us denote by W(x₁, x₂) the expression in (4). We can then derive

We see that ∂x*/∂γ = πH(1/2 − x*)/[γπH + c″(x*)] < (1 − x*)/γ if γ > 0. As a consequence, we can write

for x* < 1.

∂[ϵ(x*(γ), γ)]/∂γ > 0 if γ ≤ 0 for x* < 1.□

1. The efficient weight of proportional allocation α* = ϵ(x*) increases with the probability of detection π for γ ∈ [−1, 0) and decreases with π for γ ∈ (0, 1).

The first derivative of ϵ(x*) with respect to π is written as

It can be derived from (5) that the second term ∂x*/∂π is positive. Then, the expression is positive for γ ∈ [−1, 0), negative for γ ∈ (0, 1) and equal to 0 for γ ∈ {0, 1}.□

Appendix C: Proof of Proposition 3

Let us demonstrate that there exists a probability of detection π ̃ such that the expression in (9) is binding, and that this probability is unique. π ̃ is defined by

From the Eq. (5), we know that x* increases with π; that is, ∂x*/∂π > 0. Moreover, the cost function is strictly increasing; that is, c′(x) > 0. Then, it follows that the additional cost c(x^o) − c(x*) is strictly decreasing with π.

The derivative of the expected compensation in the event of negligence (left-hand side) is (1/2)Hp(x*, x*) + (1/2)πH[(∂p(x*, x*)/∂x*) (∂x*/∂π)]. The first term is positive. As ∂p(x*, x*)/∂x* < 0, the second term is negative. For low values of π, the marginal increase in compensation induced by a higher detection (first term) dominates the marginal decrease in compensation induced by a lower occurrence of damage (second term). For high values of π, the opposite occurs. In other words, the expected compensation is increasing for low values of π and decreasing for high values of π.

At π = 0, damage is never compensated. It follows that x* = 0. Then, it is obvious that the additional cost–which is the cost of socially optimal precautions – is greater than the expected compensation (which is 0). As π tends to 1, the expected compensation is positive, whereas the additional cost tends to 0. Then, the expected compensation is greater than the additional cost. Therefore, there exists a unique probability of detection π ̃ for which the expected compensation and the additional cost cross only once. For π < π ̃ , the expected compensation is less than the additional cost. For π ≥ π ̃ , the expected compensation always exceeds the additional cost.