# Vicarious liability

Now suppose that shareholders of the company are held vicariously liable for the full amount of harm to the victim, h, which occurs as a result of the manager's lack of care. Under this legal rule, the objective of the shareholders is again to design a contract to minimise their total expected payment, which now incorporates payments to the manager and to the victim of the accident if it occurs.

*Figure 10.3.3* An increase in the manager's assets reduces the wage paid to him if an accident does not occur

We again assume that shareholders can observe the level of care taken by the manager. Then the wage that they pay the manager can be conditioned on the level of care. The shareholders' problem is to solve:

subject to:

It is important to remember throughout the analysis that although shareholders can set *x _{t} indirectly* by developing a compensation scheme for the manager, they cannot choose

*x*the manager's level of care is chosen by him and him alone. The critical point is that the shareholders may be able to induce manager to choose the level of care that they desire, by creating appropriate incentives in the compensation scheme that the manager agrees to.

_{t}directly:Since the shareholders would not want pay the manager more than they need to, they will again arrange payments so that in expectation, the manager (in expectation) receives his reservation wage:

or:

The manager's expected wage payment on the right-hand side of (10.17) is equal to his reservation wage, plus the level of care that he takes, less his assets. Note that the expected wage is increasing in the level of care chosen by the manager.

Shareholders are only interested in the level of care that makes them as well off as they can possibly be. Substituting (10.17) into (10.15) gives us the shareholder's new problem:

Since this is equivalent to choosing *x _{t}* to minimise

*x*, the solution to this problem is x=x*, the efficient level of care. Thus, if the level of care can be observed by the shareholders and they are made vicariously liable for the harm caused by the manager's lack of care, then the shareholders end up fully internalising the external costs of the manager's lack of care indirectly, via the compensation scheme that is paid to the manager. Intuitively, under vicarious liability, the joint welfare of the shareholder and the manager depends on the costs of care and the expected damages inflicted on the victim by the manager, and so it is in mutual interests of both the shareholders and the manager to write contracts which make the sum of these two things as small as possible.

_{i}+p(x_{{})hOnce the shareholders have decided that they want the manager to choose a level of care that is equal to x*, they must design a compensation scheme to incentivise the manager to actually choose this level of care. This can be done as follows. First, the let the wage in the event of an accident be zero. Next, let:

be the total expected payment that the shareholders must make if the efficient level of care is taken by the manager. Finally, let the wage in the event of an accident be:

Then, for any *x,* the expected wage received by the manager is:

The manager's total expected wage, net of the expenditure that he takes on care, is:

Under the compensation scheme in (10.20) and (10.22), the manager's expected remuneration is maximised when he chooses the efficient level of care. Therefore, the compensation scheme induces the manager to behave efficiently, even though the manager's objective is to simply maximise his expected wage, less his expenditure on care, and even though under a rule of vicarious liability he faces *no* liability for the harm to victims that his lack of care may cause.

When the efficient level of care is chosen, the manager receives an expected wage that is equal to *W - a + x** which, if we subtract his expenditure on care, leaves him with *W - a* and makes him just indifferent between accepting the compensation scheme and not accepting. Given this level of care, the shareholders' expected wage payment to the manager is then:

and their total expected payment (including expected payments to victims as a result of the manager's lack of care) is:

Thus, the compensation scheme in which the manager receives nothing if an accident occurs, and the payment in equation (10.20) if an accident does not occur, achieves the lowest possible expected payout for the shareholders, and induces the manager to choose the efficient level of care. Note that the shareholders' expected payout is a function of the reservation utility of the manager, his level of assets, the costs of care, and the expected harm to victims.

Recall from (10.20) that this managerial compensation scheme pays a wage of:

if an accident does not occur. We have:

where the last inequality follows from the fact that *V* =* [1 - p( x* )]W_{n} + p(*x*)h < h.* We also have:

Thus, in contrast to the situation under strict liability, the wage if there is no accident is an *increasing* and concave function of the level of care taken by the manager. This is shown in Figure 10.3.4.

*Figure 10.3.4* The wage paid to the manager if an accident does not occur under vicarious liability

It is worth considering the expected wage of this optimal compensation scheme in a little more detail. In equilibrium, the expected wage is equal to *W **+ **x*** - **a*. Suppose that the level of harm rises. Then we know from the analysis in Chapter 4 that must x* rise. Hence, *the manager's expected wage at the optimum level of care rises if the level of harm rises**.*

On the other hand, what can we say about the *actual* wage, W_{n}? Choose any level of care, *x** _{i}*. For this level of care, we have:

Now

The first term in the numerator of (10.23) is positive (negative) if *x*_{i}

exceeds (is less than) x*. The last term is positive. Therefore ^{dW}”^{(x}'^{)} > 0

*dh*

for all *x*_{t} > *x** and for some *x*_{t} < *x**** as well, as long as x_{i} is sufficiently close to x*. The situation is shown in Figure 10.3.5. If the level of harm

*Figure 10.3.5* If the level of harm increases under vicarious liability, the wage paid to the manager when an accident does not occur must rise

rises, the shareholders - who are vicariously liable - would like the manager to take more care. The manager requires higher compensation to still find the contract acceptable.

Summarising our results:

- • Under strict liability, the manager is induced to choose an inefficiently low level of care and receives his reservation utility in expected terms. Shareholders pay an expected amount of
*V°,*which is given in equation (10.11). - • Under vicarious liability, the manager is induced to choose the efficient level of care and receives his reservation utility in expected terms. Shareholders pay an expected amount of V*, which is given in equation (10.19).

Since the shareholders are assumed to possess all of the bargaining power in this situation, the manager receives his reservation utility under either legal rule and is indifferent between the two schemes. However, note that comparing (10.11) and (10.19), we can see that a move from strict liability to shareholder liability will result in the shareholders paying out more in total (in expected terms), since:

which follows from the fact that *x° + p(x° )a < x* + p(x*)h* for any *a < h.*

Therefore, although vicarious liability results in an efficient allocation of resources (and improves the welfare of victims), shareholders are made worse off as a result of a move from strict liability to vicarious liability. The expected loss to shareholders is less than the expected gain to victims, and so from an overall efficiency point of view, vicarious liability is preferable to strict liability when managers have assets that are less than the harm that they cause.