## 19.2 Cost of Providing Incentives

### Learning Objective

1. How much does it cost to motivate agents?

The principal obtains profits, which are the remainder of the value after paying the agent minus the salary:

$π=(1−s)x−y =(1−s)sa−( u 0 −½ s 2 a+sλ σ 2 ) =sa− u 0 −½ s 2 a−sλ σ 2$

Note that the principal gets the entire output x = sa minus all the costs—the reservation utility of the agent u0, the cost of providing effort, and the risk cost on the agent. That is, the principal obtains the full gains from trade—the value of production minus the total cost of production. However, the fact that the principal obtains the full gains from trade doesn’t mean the principal induces the agent to work extremely hard because there is no mechanism for the principal to induce the agent to work hard without imposing more risk on the agent, and this risk is costly to the principal. Agents are induced to work hard by tying their pay to their performance, and such a link necessarily imposes risk on the agent, and risk is costly.There is a technical requirement that the principal’s return π must be positive; otherwise, the principal would rather not contract at all. This amounts to an assumption that u0 is not too large. Moreover, if s comes out less than zero, the model falls apart, and in this case, the actual solution is s = 0.

We take the principal to be risk neutral. This is reasonable when the principal is economically large relative to the agent, so that the risks faced by the agent are small compared to those faced by the principal. For example, the risks associated with any one car are small to a car rental company. The principal who maximizes expected profits chooses s to maximize π, which yields $s=1− λ a σ 2 .$

This formula is interesting for several reasons. First, if the agent is neutral to risk, which means λ = 0, then s is 1. That is, the agent gets 100% of the marginal return to effort, and the principal just collects a lump sum. This is reminiscent of some tenancy contracts used by landlords and peasants; the peasant paid a lump sum for the right to farm the land and then kept all of the crops grown. Because these peasants were unlikely to be risk neutral, while the landlord was relatively neutral to risk, such a contract was unlikely to be optimal. The contract with s = 1 is known as selling the agency because the principal sells the agency to the agent for a lump sum payment. (Here, y will generally be negative—the principal gets a payment rather than paying a salary.) The more common contract, however, had the landowner and the tenant farmer share the proceeds of farming, which gives rise to the name sharecropper.

Second, more risk or more risk aversion on the part of the agent decreases the share of the proceeds accruing to the agent. Thus, when the cost of risk or the amount of risk is high, the best contract imposes less risk on the agent. Total output sa falls as the costs of risk rise.

Third, more able agents (higher a) get higher commissions. That is, the principal imposes more risk on the more able agent because the returns to imposition of risk—in the form of higher output—are greater and thus worth the cost in terms of added risk.

Most real estate agencies operate on a mix of salary and commission, with commissions paid to agents averaging about 50%. The agency RE/MAX, however, pays commissions close to 100%, collecting a fixed monthly fee that covers agency expenses from the agents. RE/MAX claims that their formula is appropriate for better agents. The theory developed suggests that more able agents should obtain higher commissions. But in addition, RE/MAX’s formula also tends to attract more able agents because able agents earn a higher wage under the high commission formula. (There is a potential downside to the RE/MAX formula: it discourages agency-wide cooperation.)

Consider what contracts attract what kinds of agents. For a fixed salary y and commission s, the agent’s utility, optimizing over x, is $u*=y+½ s 2 a−sλ σ 2 .$

The agent’s utility is increasing in a and decreasing in λ. Thus, more able agents get higher utility, and less risk-averse agents get higher utility.

How do the terms of the contract affect the pool of applicants? Let us suppose that two contracts are offered, one with a salary y1 and commission s1, the other with salary y2 and commission s2. We suppose y2 < y1 and s2 > s1. What kind of agent prefers Contract 2, the high-commission, low-salary contract, over Contract 1?

$y 2 +½ s 2 2 a− s 2 λ σ 2 ≥ y 1 +½ s 1 2 a− s 1 λ σ 2$

or the equivalent:

$½a( s 2 2 − s 1 2 )−( s 2 − s 1 )λ σ 2 ≥ y 1 − y 2$

Thus, agents with high ability a or low level of risk aversion λ prefer the high-commission, low-salary contract. A company that puts more of the compensation in the form of commission tends to attract more able agents and agents less averse to risk. The former is a desirable feature of the incentive scheme because more able agents produce more. The latter, the attraction of less risk-averse agents, may or may not be desirable but is probably neutral overall.

One important consideration is that agents who overestimate their ability will react the same as people who have higher ability. Thus, the contract equally attracts those with high ability and those who overestimate their ability.

Agency theory provides a characterization of the cost of providing incentives. The source of the cost is the link between incentives and risk. Incentives link pay and performance; when performance is subject to random fluctuations, linking pay and performance also links pay and the random fluctuations. Thus, the provision of incentives necessarily imposes risk on the agent, and if the agent is risk averse, this is costly.

In addition, the extent to which pay is linked to performance tends to affect the type of agent who is willing to work for the principal. Thus, a principal must not only consider the incentive to work hard created by the commission and salary structure but also the type of agent who would choose to accept such a contract.

### Key Takeaways

• The principal chooses the salary to minimize the cost of the agent; thus, the principal nets the total output, minus the cost of the agent.
• The agent’s cost must be at least as large as what the agent would get in an alternative occupation and thus includes a risk adjustment.
• The optimal commission offered by the principal is decreasing in the risk aversion of the agent and the level of risk and increasing in the agent’s ability.
• If the agent is neutral to risk, the principal gets a lump sum, and “sells the agency.”
• Total output falls as the costs of risk rise.
• A company that puts more of the compensation in the form of commission tends to attract more able agents and agents less averse to risk. A principal must not only consider the incentive to work hard created by the commission and salary structure but also the type of agent who would choose to accept such a contract.

### Exercise

1. Describe how a principal would go about hiring agents who are willing to take risks.