## LEARNING OBJECTIVES

- What is an agency?
- How should a principal compensate an agent?

An **agent** is a person who works for, or on behalf of, another. Thus, an employee is an agent of a company. But agency extends beyond employee relationships. Independent contractors are also agents. Advertising firms, lawyers, and accountants are agents of their clients. The CEO of a company is an agent of the board of directors of the company. A grocery store is an agent of the manufacturer of corn chips sold in the store. Thus, the agency relationship extends beyond the employee into many different economic relationships. The entity—person or corporation—on whose behalf an agent works is called a **principal**.

Agency theory is the study of incentives provided to agents. Incentives are an issue because agents need not have the same interests and goals as the principal. Employees spend billions of hours every year browsing the Web, e-mailing friends, and playing computer games while they are supposedly working. Attorneys hired to defend a corporation in a lawsuit have an incentive not to settle, to keep the billing flowing. (Such behavior would violate the attorneys’ ethics requirements.) Automobile repair shops have been known to use substandard or used replacement parts and bill for new, high-quality parts. These are all examples of a conflict in the incentives of the agent and the goals of the principal.

Agency theory focuses on the cost of providing incentives. When you rent a car, an agency relationship is created. Even though a car rental company is called an agency, it is most useful to look at the renter as the agent because it is the renter’s behavior that is an issue. The company would like the agent to treat the car as if it were her own car. The renter, in contrast, knows it isn’t her own car and often drives accordingly.

rented P. J. O'Rourke, Republican Party Reptile (Boston: Atlantic Monthly, 1987), 242.

How can the car rental company ensure that you don’t put its car into reverse while going forward at a high rate of speed? It could monitor your behavior, perhaps by putting a company representative in the car with you. That would be a very expensive and unpleasant solution to the problem of incentives. Instead, the company uses outcomes—if damage is done, the driver has to pay for it. That is also an imperfect solution because some drivers who abuse the cars get off scot-free, and others who don’t abuse the car still have cars that break down and are then mired in paperwork while they try to prove their good behavior. That is, a rule that penalizes drivers based on outcomes imposes risk on the drivers. Modern technology is improving monitoring with GPS tracking.

To model the cost of provision of incentives, we consider an agent like a door-to-door encyclopedia salesperson. The agent will visit houses and sell encyclopedias to some proportion of the households; the more work the agent does, the more sales that are made. We let x represent the average dollar value of sales for a given level of effort; x is a choice the agent makes. However, x will come with risk to the agent, which we model using the variance δ^{2}.

The firm will pay the agent a share s of the money generated by the sales. In addition, the firm will pay the agent a salary y, which is fixed independently of sales. This scheme—a combination of salary and commission—covers many different situations. Real estate agents receive a mix of salary and commission. Authors receive an advance and a royalty, which works like a salary and commission.

The monetary compensation of the agent is \(\begin{equation}s x + y\end{equation}\). In addition, the agent has a cost of effort, which we take to be x 2 2a . Here, a represents the ability of the agent: more able agents, who have a higher value of a, have a lower cost of effort. Finally, there is a cost of risk. The actual risk imposed on the agent is proportional to the degree he shares in the proceeds. If s is small, the agent faces almost no monetary risk, but if s is high, most of the risk is imposed on the agent. We use the linear cost of risk model, developed earlier, to impose a cost of risk, which is s \(\begin{equation}\lambda \delta^{2}\end{equation}\). Here, \(\begin{equation}\delta^{2}\end{equation}\) is the variance of the monetary risk, λ defines the agent’s attitude or cost of risk, and s is the share of the risk imposed on the agent. This results in a payoff to the agent of \(\begin{equation}u=sx+y− x 2 2a −sλ σ 2\end{equation}\).

The part of the equation represented by sx + y is the payments made to the agent. The next term is the cost of generating that level of x. The final term is the cost of risk imposed on the agent by the contract.

The agency game works as follows. First, the principal offers a contract, which involves a commission s and a salary y. The agent can either accept or reject the contract and accepts if he obtains at least u_{0} units of utility, the value of his next best offer. Then the agent decides how much effort to expend; that is, the agent chooses x.

As with all subgame perfect equilibria, we work backward to first figure out what x an agent would choose. Because our assumptions make u quadratic in x, this is a straightforward exercise, and we conclude x = sa. This can be embedded into u, and the agent’s optimized utility u* is

\begin{equation}u^{*}=s 2 a+y-(s a) 22 a-s \lambda \sigma 2=y+1 / 2 s 2 a-s \lambda \sigma 2\end{equation}

The agent won’t accept employment unless u* ≥ u_{0}, the reservation utility. The principal can minimize the cost of employing the agent by setting the salary such that u* = u_{0}, which results in \begin{equation}y=u 0-1 / 2 s 2 a+s \lambda \sigma 2\end{equation} .

Observe that the higher the salary, the greater is the risk \(\begin{equation}\delta^{2}\end{equation}\). That is, the principal has to cover the cost of risk in the salary term.