26  The Principal-Agent Problem

For more information on these topics, see Allen, Doherty, Weigelt, and Mansfield Chapter 15: Principal-Agent Issues and Managerial Compensation.

The principal-agent problem arises whenever one party (the agent) is hired to act on behalf of another (the principal), but their interests don’t perfectly align. The classic example is the relationship between company shareholders and managers. While shareholders primarily want to maximize company value, managers often have competing personal interests that can lead them to make different decisions than shareholders would prefer.

When interests naturally align, there’s no principal-agent conflict. For instance, on a ship caught in a storm, the captain and crew’s interests are perfectly aligned - everyone wants to survive. The crew can trust the captain to make good decisions because his life is also at stake. Problems emerge when interests diverge, like in warfare where a general safely removed from combat might prioritize winning battles while soldiers on the ground want to stay alive. Similarly, elected officials may serve special interests rather than constituents.

The challenge is magnified by information asymmetry - agents typically know more about their own actions than principals can observe. If principals could perfectly monitor agents’ behavior, they could simply reward desired actions and punish undesired ones. But in reality, principals often can’t directly observe critical factors like effort level. A business owner wants managers to work hard but can’t always tell how much effort they’re actually putting in.

Managers may pursue various personal goals that conflict with maximizing firm value:

  1. Minimizing their own effort while maintaining their position
  2. Protecting their job security by avoiding necessary but risky decisions
  3. Preventing any failures that could damage their reputation
  4. Building their personal brand and future career prospects
  5. Maximizing perks and privileges of their position

Consider how the goal of job security affects decision-making: A manager choosing between a risky venture with potential for high profits and a safer option with modest returns might select the safer path to protect their position. However, shareholders with diversified portfolios would likely prefer the riskier venture with higher potential returns. This illustrates how different incentives can lead to suboptimal decisions from the principal’s perspective.

Example: The Basic Trade-off

Consider a simple case where:

  • Revenue \(R(e) = 100e^{0.5}\) (\(e\) is effort)
  • Manager’s disutility of effort \(u(e) = 10e\)
  • Fixed salary \(S = 50\)

The manager’s net benefit is their salary minus their disutility of effort: \(B(e) = S - u(e) = 50 - 10e\).

Of course, \(B\) is maximized when the manager chooses \(e = 0\), because any effort reduces their net benefit. But this leads to \(R(0) = 0\), which is not optimal for the principal.

To address this misalignment, principals typically use incentive-compatible compensation schemes that link the agent’s pay to observable outcomes.

Example: Profit Sharing

Let’s modify the previous example by adding profit sharing:

  • Revenue \(R(e) = 100e^{0.5}\)
  • Costs \(C = 20\)
  • Profit \(\pi(e) = R(e) - C = 100e^{0.5} - 20\)
  • Manager gets \(\alpha\) share of profits
  • Manager’s total compensation: \(S(e) = 50 + \alpha \pi(e)\)
  • Manager’s disutility of effort \(u(e) = 10e\)
  • Manager’s net benefit: \(B(e) = 50 + \alpha \pi(e) - 10e = 50 + 100 \alpha e^{0.5} - 20 \alpha - 10e\)

The manager will choose \(e\) to maximize \(B(e)\):

\[\begin{align} \frac{\partial B(e)}{\partial e} &= 50 \alpha e^{-0.5} - 10 = 0\\ \frac{50 \alpha}{e^{0.5}} &= 10\\ 50 \alpha &= 10 e^{0.5}\\ e^{0.5} &= 5 \alpha\\ e &= 25 \alpha^2 \end{align}\]

So if the manager gets 40% of profits, they will choose effort \(e = 25 (0.4^2) = 4\), generating a profit of \(100 4^{0.5} = 200\).

26.1 Classwork 26

1) Profit Sharing and Effort

Suppose:

  • Revenue \(R(e) = 50e^{1/3}\)
  • Costs \(C = 30\)
  • Profit \(\pi(e) = R(e) - C\)
  • Manager’s disutility of effort \(u(e) = 50e\)

  1. If the manager gets paid a fixed salary of $500, what effort will the manager choose, and what profit will the firm earn?

  2. Suppose instead that the manager gets paid 30% of the firm profit as a bonus. She is also compensated for her disutility of effort. Show that the effort that maximizes profit is 0.192. Calculate the profit at that level of effort.

    • Revenue \(R(e) = 50e^{1/3}\)
    • Costs \(C = 0\)
    • Manager’s disutility of effort \(u(e) = 50e\)
    • Profit \(\pi(e) = R(e) - C - u(e)\)

  3. Continuing from part b, if the manager wants to maximize her bonus, what level of effort should she choose?

  4. Interpret: when profit is certain, firms compensate managers for their disutility of effort, and managers earn a bonus that is a share of firm profit, incentives (are/are not) aligned. The manager chooses an effort level that is (too low/too high/the same) compared to what the owner would want. The principal/agent problem (is/is not) resolved.

2) Uncertainty

Another issue in managerial compensation is that if:

  • The firm’s profits are uncertain,
  • The manager is risk-averse,
  • And the stockholders have a diversified portfolio and are therefore risk-neutral,

the stockholders are the party that can bear the risk cheaply, not the manager. Let’s see this in an example:

  • Suppose the firm profits are $10M with probability 0.5 and $20M with probability 0.5.
  • The manager’s utility is \(U = \sqrt{\text{Income}}\) (they are risk averse)
  • The firm must offer the manager a compensation package with an expected utility of 1000, or the manager will work at another company
  • The shareholders are risk-neutral with a utility function of \(U = \text{Income}\).

  1. Suppose the manager gets a flat salary of $1M. Calculate their expected utility, and calculate the utility of the shareholders, who now get $9M w.p. 0.5 and $19M w.p. 0.5.

  2. Now suppose the manager gets a salary of some proportion of the firm profits \(\alpha \pi\). Find the \(\alpha\) that would give the manager an expected utility of 1000, and find the shareholder’s utility for the amount left over.

  3. The manager gets an expected utility of 1000 for the compensation plan in part A and for the compensation plan in part B, so the manager is indifferent between both these plans. But the shareholders are not: which compensation plan would they prefer? Who bears risk more cheaply: the manager or the shareholders? How does this result conflict with the result from the previous question, where we found that the manager should be given a proportion of profits to induce them to expend effort?


26.2 Practice Questions

Question 1: What is the fundamental cause of the principal-agent problem?






Question 2: In the ship captain example from the text, why is there no principal-agent problem?






Question 3: Which of the following best illustrates a personal goal that might conflict with maximizing firm value?






Question 4: When profit sharing is introduced in the text’s example, what is the primary purpose?






Question 5: According to the text, why does information asymmetry make the principal-agent problem more challenging?