13.2 Limited Liability and Risk Aversion

In the previous section, we saw that the unobservability of the manager’s effort is a necessary condition for the contract between the manager and the firm to be inefficient: if effort is observable (and can be the subject of legally binding agreements), then there exists a contract that maximizes total surplus. We also saw that the unobservability of effort is not in itself sufficient to generate inefficiency: if the agent is risk-neutral, a second-best contract can replicate what happens in the first-best situation. This latter result relies on two assumptions that are often unrealistic. The first is that the firm can transfer all risk to the manager by paying them very little (or even nothing) in case of failure. The second is that the manager is risk-neutral. In this section, we will see that the result does not hold when even one of these two assumptions fails.

Limited Liability

The analysis in the last part of the previous section assumed that the firm could transfer the risk associated with uncertain outcomes entirely to the manager. The contract could specify very low wages in the event of failure, as long as these remained compatible with incentives and participation. In many real-world situations, however, this is not feasible: the firm may be unable to pay less than a certain minimum wage (e.g., due to legal constraints).

We formalize this situation with a limited liability constraint: the wage the manager receives, even if the project fails, cannot fall below a certain minimum level. In the figure below, we assume this minimum is $100$ euros.

The limited liability constraint has no impact on total surplus if the firm’s reservation profit is less than 1800 euros. In this case, the firm undertakes the project and creates the same surplus as in the first-best scenario. In fact, the constraint only affects the distribution: the firm’s profit is 75 euros lower than in the first-best case, but this reduction is not wasted — the worker’s surplus increases by exactly the same amount (expected wage is 700 euros, compared to the 625 euros in a first-best contract).

Where, then, does the inefficiency arise? It appears when the firm’s reservation profit is between 1800 and 1875 euros. In this case, the firm’s optimal choice is not to offer a contract (or to offer one that the manager finds unacceptable). As a result, the firm does not undertake the project, generating zero surplus, even though it would have done so in the absence of the constraint.

Risk Aversion

When the manager is risk-averse rather than risk-neutral, it becomes more costly for the firm to incentivize effort through performance-based pay. In the following example, we assume the manager’s utility is the square root of their wage and the disutility of effort is $10$. To ensure a sufficient level of expected utility that compensates for both risk and the disutility of effort, the firm must raise wages compared to the risk-neutral case. This increase results in a lower expected profit for the firm.

In the graph, we can see that in order to induce effort, the contract must include a relatively large bonus in the event of success. This implies greater variability in the manager’s income, and because the manager is risk-averse, they require a higher expected wage to accept such variability. The firm is therefore forced to offer higher wages compared to the first-best case.

However, this additional cost for the firm does not translate into greater benefit for the manager: the increase in expected wage merely compensates for the negative effect of risk on the worker’s welfare — it does not transfer surplus from the firm to the manager. As a result, even when the firm proceeds with hiring, the resulting allocation is less efficient than in the case of a risk-neutral worker.

Unlike the limited liability case — where inefficiency occurs only when the firm decides not to hire — here the loss of efficiency arises even when the firm does hire the manager and the project is carried out. The informational asymmetry forces the parties to bear higher costs to achieve the same outcome, thereby reducing the total surplus generated by the exchange.