Compensation in personal injury cases: mean or median income?

If the injurer blinded the victim’s eye, cut off his hand, broke his leg, we view him as if he were a slave sold in the marketplace, and we evaluate how much he was worth prior to the injury and how much he is worth now.

Mishna Baba Kama 8:1 (Circa 200 CE)

Abstract

Courts typically base compensation for loss of income in personal injury cases on either mean or median work income. Yet, quantitatively, mean and median incomes are typically very different. For example, in the US, median income is 65% of mean income. In this paper we use economic theory to determine the relation between the appropriate make-whole (full) compensation and mean and median work incomes. Given that consumption uncertainty associated with compensation generally exceeds that associated with work income, we show that the appropriate make-whole compensation exceeds mean (and therefore median) work income. Hence, if the compensation must be either the mean or the median work income, then mean work income should generally be selected.

Appendix

The main body of this paper presents a simple one-period framework that abstracts from the fact that the mean value of work income is likely to vary over time, and that both work income and the consumption attainable from the compensation may follow random processes that evolve over time. In particular, the uncertainty of both work income and the consumption attainable from the compensation are likely to increase over time. However, our model can be extended to incorporate these considerations.

Thus, suppose that the victim had $T\ge 1$ remaining periods of working life when the accident happened. A natural generalization of the person’s one-period work-income uncertainty is to assume that the work income in period $t=1,2,\ldots ,T$, denoted by $y_{t}$, is lognormally distributed with $y_{t}\sim {\varvec{\Lambda }}\left( \ln M_{t}-\tfrac{1}{2}\sigma ^{2}t,\sigma ^{2}t\right)$. That is, the mean of $y_{t}$ is $M_{t}$, which may vary over time; the coefficient of variation of $y_{t}$ is $\left( e^{\sigma ^{2} t}-1\right) ^{1/2}$, which increases over time; and the median of $y_{t}$ is $M_{t}e^{-\sigma ^{2}t/2}$, which increases (decreases) from period t to period $t+1$ if $\ln (M_{t+1}/M_{t})>(<)\tfrac{1}{2}\sigma ^{2}$.

From the properties of the lognormal distribution, the individual’s no-injury expected utility in period t is

$$\begin{aligned} \left\{ \begin{array} [c]{llll} {\displaystyle \int _{0}^{\infty }} \dfrac{y_{t}^{1-S}-1}{1-S}\Lambda \left( \ln M_{t}-\tfrac{1}{2}\sigma ^{2}t,\sigma ^{2}t\right) dy_{t} &{} =\dfrac{\left( M_{t}e^{-S\sigma ^{2} t/2}\right) ^{1-S}-1}{1-S} &{} \quad \text {if} \ S\ne 1, &{} \\ {\displaystyle \int _{0}^{\infty }} \ln y_{t}d\Lambda \left( \ln M_{t}-\tfrac{1}{2}\sigma ^{2}t,\sigma ^{2}t\right) dy_{t} &{} =\ln M_{t}-\tfrac{1}{2}\sigma ^{2}t &{} \quad \text {if} \ S=1. &{} \end{array} \right. \end{aligned}$$

(4)

Assume that the total lump-sum compensation paid in the period of the accident (i.e., in period $t=1)$ must ensure that the victim can be made whole in each of the T periods. Let the amount $A_{t}$ be the make-whole compensation received (in the period of the accident) for period t. That is, the total compensation A can be divided into $A_{1},A_{2},\ldots ,A_{T}$ with $A=\sum _{t=1}^{T}A_{t}$.³¹ In keeping with the assumptions made in the main body of the paper, let the real value of the make-whole compensation for period t be $A_{t}e^{r(t-1)}z_{t}$, where r is the interest rate and $z_{t}$ is lognormally distributed according to $z_{t}\sim \Lambda \left( -\tfrac{1}{2}s^{2}t,s^{2}t\right)$. Therefore, the mean of $z_{t}$ is unity for all t while its coefficient of variation, $\left( e^{s^{2}t}-1\right) ^{1/2}$, increases over time. The make-whole compensation $A_{t}$ therefore enables the individual to obtain a compensated injury expected utility³² in period t that is equal to

$$\begin{aligned} \left\{ \begin{array} [c]{llll} {\displaystyle \int _{0}^{\infty }} \dfrac{\left[ A_{t}e^{r(t-1)}z_{t}\right] ^{1-S}-1}{1-S}d\Lambda \left( -\tfrac{1}{2}s^{2}t,s^{2}t\right) dz_{t} &{} =\dfrac{\left\{ A_{t}e^{\left[ r(t-1)-Ss^{2}/2\right] t}\right\} ^{1-S}-1}{1-S} &{} \quad \text {if} \ S \ne 1, &{} \\ {\displaystyle \int _{0}^{\infty }} \ln \left[ A_{t}e^{r(t-1)}z_{t}\right] d\Lambda \left( -\tfrac{1}{2} s^{2}t,s^{2}t\right) dz_{t} &{} =\ln A_{t}+r(t-1)-\tfrac{1}{2}s^{2}t &{} \quad \text {if} \ S=1. &{} \end{array} \right. \end{aligned}$$

(5)

In view of (4), (5), certainty-equivalence between the expected utility from work income and from the make-whole income from the accident compensation for each t requires that $A_{t}=M_{t}e^{-r(t-1)+\delta St}$. That is, the accident compensation for the first period is $A_{1} =M_{1}e^{\delta S}$ (the same as in the one-period case) and then changes at a rate of $\ln (M_{t+1}/M_{t})-r+\delta S$ from period t to period $t+1$. The rate of change in $A_{t}$ reflects the rate of change in the mean value of work income and is reduced by the interest rate r, while the period-to-period effects of $\delta$ and S are the same as their one-period effects.

We therefore have that

$$\begin{aligned} A=\sum _{t=1}^{T}M_{t}e^{-r(t-1)+\delta St}. \end{aligned}$$

It follows immediately that the total make-whole compensation, A, increases with each $M_{t}$ and decreases with r. Additionally, just as in the one-period setting, A decreases with $\sigma ^{2}$ and increases with $s^{2}$, with their impacts increasing with S.

Moreover, within a multi-period setting, for each period’s make-whole compensation, the insights presented within the context of the one-period model concerning the relation between mean and median work income and the make-whole compensation remain valid. Hence, the results derived for the one-period model can be generalized.