Consider first a legal rule of strict liability without vicarious liability: this means that the manager must personally pay for the harm that his lack of care may cause. The goal of the shareholders is to design a contract to minimise their expected payment to the manager. On the other hand, the manager will want a sufficiently high wage to compensate him for the cost of care and the possibility that he will lose all of his assets if an accident occurs.
Shareholders are assumed to be able to observe the level of care taken by the manager. They want to design a contract which solves:
subject to or:
Since the shareholders would not want to have Wn larger than it needs to be, they will set Wn so that:
Substituting this into the shareholder's objective function yields:
Since these costs are increasing in Wh, the principal should set Wh = 0 for any level of care taken by the manager. Then the shareholders solve:
This states that the level of care that the shareholders would the manager want to take is exactly the same level of care that the manager himself wants to take. Moreover, since a < h, this level of care will be inefficiently low.
To implement this outcome, let x° minimise the shareholders' expected costs in (10.10), and define:
Then, since Wh = 0, by equation (10.7) this must also be the manager's expected wage. Therefore we need to design the wage so that:
This is straightforward to do. Just rearrange (10.12) and let the compensation scheme be:
If the manager chooses xt = x°, then the shareholders will pay out an expected amount of V,0. So let us examine the manager's incentives under this compensation scheme. The manager's expected utility is:
This attains a maximum of W at the point xt = x°. Thus the manager is incentivised to choose the level of care which the shareholders would like him to, and at this level of care the manager is just indifferent between accepting the compensation scheme and rejecting it.
The compensation scheme in (10.13) is decreasing and convex in the level of the manager's care. Thus the wage as a function of the level of the manager's care takes the shape illustrated in Figure 10.3.1.
Figure 10.3.1 The compensation scheme under strict liability when care can be observed
Figure 10.3.2 The shift in the compensation scheme when the manager's assets rise
If the manager's assets rise, we have:
Thus, for each level of x, the wage if an accident does not occur is decreasing in the level of manager's assets. An increase in the manager's assets therefore shifts the compensation schedule downwards, as shown in Figure 10.3.2. This happens because the manager's expected payoff is increasing in his level of assets. If these assets increase then, for a fixed level of care, the shareholders can afford to pay the manager a lower wage if an accident does not occur, and still leave him indifferent between accepting and not accepting the remuneration package.
How does the compensation scheme change at the optimal wage? We have:
Since 1 + p'(x° )a = 0 and since W + x° > a, we have 1 + p'(x°) (W + x°) < 0, and so:
Intuitively, an increase in the manager's assets make him better off at the existing optimal wage: he will wish to take a higher level of care (which is costly), but this reduces the probability of an accident and if an accident does not occur he receives more wealth. If an accident does occur he receives nothing anyway. So overall, at the existing wage, he is better off. The shareholders can therefore afford to keep the manager at his reservation utility by reducing his wage. The compensation scheme shifts downwards for each level of care, and at the efficient level of care, the manager's wage is lower than it was when his assets were lower. This is shown in Figure 10.3.3.