5  The principal-agent problem

The allocation of decision rights creates a problem as the interests of the owners and employees are not always aligned. This is known as the principal-agent problem.

Principal-agent problems are typically modelled as a game between two players, the principal and the agent, who face a conflict of interest. In the case of an employer and employee, the employer is the principal and the employee the agent. The misalignment of interests creates a challenge in designing incentives to align those interests.

Below I examine two classes of principal-agent problems.

The first is where the action of the agent is observable by the principal. In this case, the employer can observe the effort of the employee. I will show that incentive conflicts do not cause problems when effort is contractible. The employer can simply specify the appropriate level of effort for which a bonus is payable. In choosing the optimal action the costs to the employee from higher effort and the benefits to the firm in terms of higher profits need to be balanced.

However, outside of idealised problems, an agent’s effort is not usually costlessly observable by the firm. They can’t simply contract to deliver a set amount of effort. Further, output or other proxies for effort may not reflect effort or be difficult to measure. It’s likely that there are other effects on output besides the agent’s effort.

Therefore, a second class of principal-agent problems involves hidden actions, where the conflict between the principal and the agent is over an action that may be taken by the agent, and this action cannot be detailed in a complete contract. The agent knows what action is taken, but the principal doesn’t. For our employer-employee problem, the hidden action is the employee’s effort. There is a problem because the employer would like the employee to work hard, whereas the employee would prefer to take it easy.

To illustrate the trade-offs between risk sharing and incentives, below are illustrations of this challenge with observable and unobservable effort . In each case, I provide one narrative and one mathematical description of the problem. The following are based on examples in Brickley et al. (2020).

5.1 Observable effort

5.1.1 A narrative exploration

The owner of Firm A wants their employees to put in higher effort and work diligently. However, their employee Agent A, while liking to be paid, doesn’t like to put in effort. As Agent A’s income goes up, Agent A is happier, but Agent A is also happier the more they can lounge around in the office and cruise Facebook.

Agent A won’t get out of bed and turn up to work for less than $1,000 a week. That is what we call his reservation wage. Agent A then needs to be paid for Agent A’s efforts in the office. Agent A has a mentally taxing job, so each hour of effort is harder than the last. As a result, Agent A will demand a salary of at least $1000 plus a compensating amount for whatever level of effort he provides.

The firm benefits from Agent A’s effort, receiving around $100 for every hour that he applies himself.

If they could tell whether Agent A was working or not, they could just agree a level of effort, pay him an agreed sum for it, and then monitor to make sure he delivers. What is that sum?

The firm will want to pay Agent A the sum that maximises its profit. Agent A wants to set his level of effort such that he balances his desire for income and that increasing cost of effort. As the firm receives $100 for every hour he works, they will pay Agent A up to the point where he demands more than $100 for each additional hour that he works.

Putting this into classical economic terms, the marginal benefit to the firm of each hour that Agent A works is $100. Their profit will be maximised when that marginal benefit is equal to the marginal cost of inducing an additional hour of effort out of Agent A. The firm could pay him more to get even more effort and output, but the cost of inducing this effort would exceed the benefit they receive.

5.1.2 A mathematical exploration

Agent A’s utility (U) is a function of the agent’s wage income (W) and effort (e, the hours actually spent working). Utility is increasing in income (+ve) and decreasing (-ve) in effort. Let:


Agent A’s reservation wage is $1,000.

The firm gets $100 of output for every hour Agent A works:


Assume that effort is observable at zero cost and it is verifiable. It is possible to contract over effort. In this case, the firm will offer Agent A a contract requiring Agent A puts in a specified level of effort \hat e.

The contract will be acceptable to Agent A if they receive at least their reservation level of utility. That is, the contract will be acceptable as long as it pays:

W=\$(1000+\hat e^2)

If Agent A accepts the contract they get:

U(.)=1000+\hat e^2-\hat e^2=1000

The firm’s challenge is to maximise profit (\pi). That is:

\begin{align*} max\:\pi_e&=Q-W \\ &=\$100\hat e-[\$1000+\hat e^2] \end{align*}

First Order Condition (the derivative of the profit function equals zero at the maximum): \:100-2\hat e=0

So the profit maximising level of effort is:

\begin{align*} e^*&=50 \\ W &= 1000+\hat e^2 \\ &=1000+2500 \\ &=3500 \\ \pi&=50(1000)-3500 \\ &=1500 \end{align*}

At effort level \hat e=50, the marginal benefit of higher effort is equal to Agent A’s marginal cost of inducing higher effort. This can be seen in the diagram.

5.2 Unobservable effort

5.2.1 A narrative exploration

Agent B is another employee of Acme Corporation. Agent B’s output increases with their effort, but Agent B’s output is also subject to other (random) effects outside of their control.

As for Agent A, if Agent B’s effort could be observed, they could simply develop a contract under which Agent B is paid by effort. But in Agent B’s case, they can’t observe it.

Paying Agent B a fixed salary is problematic. If Agent B’s effort is not observable, Agent B could put in low effort. Plus, Agent B could blame the low output on bad luck (the effects outside of Agent B’s control).

An alternative is for the firm to observe and pay based on output. Suppose they pay Agent B a fixed sum plus a share of the output Agent B produces.

Since Agent B’s effort is costly, like Agent A, Agent B will want to be compensated for this. Agent B will set its effort level such that their expected compensation minus the cost of the effort is greatest. This is at the level of effort where the marginal benefit of increased compensation equals the marginal cost of additional effort.

Counter-intuitively, in setting their effort Agent B does not need to take into account the effects on output that are out of Agent B’s control if compensation goes up with every unit of effort. In that case, the marginal cost and benefit of effort are not affected by the random shocks that otherwise affect Agent B’s payment. A random effect that reduced output does not change the fact that extra effort increases output and, accordingly, Agent B’s compensation. (It does, however, have other effects that we will discuss later.)

Agent B’s optimal level of effort is also not linked to the fixed sum component of Agent B’s compensation. The fixed sum does not affect the marginal benefit of effort. Higher pay does not provide an incentive in this model unless it is linked to performance.

Agent B’s effort does increase, however, with an increase if Agent B is paid a higher proportion of their output. This increases the marginal benefit of Agent B’s effort.

5.2.2 A mathematical exploration

Assume that we have a risk neutral employer and a risk averse employee. Agent B is an employee who has output given by the following:

Q=\alpha e+\mu \\ \mu\sim(0,\sigma^2)

where Q is the value of the output (which is observable); e is effort; \alpha is Agent B’s marginal productivity, and \mu is some random effect.

If effort (e) could be observed, we might expect a contract that specifies a level of effort \hat{e} for a fixed salary W.

That contract would deliver profit to the firm of:

\pi=(\alpha e+\mu)-W

But suppose neither e nor \mu is observable.

In that case, as noted above, Agent B may put in low effort or blame the low output on bad luck (a low \mu).

We could instead provide an incentive to Agent B by basing their compensation on output. Consider Agent B’s effort problem if they are paid according to the following linear payment schedule:

W=W_0+\beta Q

0\leq \beta \leq 1

where W_0 is a fixed wage and \beta is the proportion of output (Q) received.

This type of contract might represent a typical compensation scheme. Let W_0=1000 and \beta=0.2



where C(e) is Agent B’s cost of effort.


\begin{align*} W&=1000+0.2(100e+\mu) \\ &=1000+20e+0.2\mu \end{align*}

Setting optimal effort

Solving this, Agent B sets the level of effort where the compensation minus cost of effort is greatest. Note that an extra unit of effort always increases compensation by $20. The random component or shock (\mu) affects the total level of payment, but not the marginal impact of effort. This means that the employee (Agent B) can effectively ignore \mu. In this case the optimal choice of effort is equal to 10.

Increasing the fixed wage

A change in W_0 doesn’t change incentives around effort. With a higher intercept the optimal choice of effort is unchanged. What is important is the marginal benefit and marginal cost of effort.

Increasing the share of output

A change in \beta changes the optimal effort level. The optimal choice of effort is increased as the marginal benefit of effort is increased.

5.3 Optimal incentive schemes

Given this principle-agent problem, what should a firm do? It will try to maximise profit.

To do this, first it needs to ensure that the reservation level of utility is met, otherwise the individual will not work for the firm. One way to do this is to adjust the base pay, w_0, to ensure this is the case.

Then the firm needs to induce effort. It will want to set \beta at the right level. But this comes at a cost to the firm. With greater reward for effort we would expect that Agent B will work harder - this should lead to higher payments for the firm. But with higher \beta Agent B is exposed to increased risk. For a risk averse worker this will generally mean that Agent B will need to be compensated more so that they are willing to bear the higher risk.

What does all this mean? The following five factors favour high incentive pay:

  1. A strong relationship between the employee’s effort and output, meaning the benefits of motivating effort are high

  2. The employee has low risk aversion, as a person with high risk aversion will require higher compensation to bear the risk

  3. A low level of risk that is beyond the control of the employee (\sigma^2), as output is then primarily a function of greater effort

  4. High sensitivity to increased incentives (e.g the cost of effort is not so high as to prevent a response to the incentive)

  5. The employee’s output is measurable at low cost.