4.2 Incentive schemes for agents with non-neutral risk-preferences

Incentives and risk

• Incentive pay is based on some measure of performance, which is almost always noisy. That is, it depends not only on A's effort but also factors outside their control (general market conditions, weather, luck, etc.)

• The noise in performance measure exposes A to financial risk. No problem if A is risk-neutral (as in the earlier lectures): what matters is just the expected gain.

• However, most of us are risk-averse. Managing this risk in the right way is fundamental to designing effective performance pay schemes.

Risk preferences

Some notation:

x some random variable (e.g. payoff);

U(x) utility function 𝐸(𝑥) = 𝑥̅expected value of x;

Var (x) variance of x

Certainty equivalent: Case of a risk-averse agent

• Certainty equivalent x* is the fixed amount the agent would agree to receive in exchange for the lottery ((𝑥𝑇, 𝑥𝐻)).

• Das Sicherheitsäquivalent x* ist der feste Betrag, den der Vermittler im Gegenzug für die Lotterie erhalten würde ((𝑥𝑇, 𝑥𝐻)).

Efficient risk sharing: Implications:

1. Risk sharing is always more efficient than going it alone for risk-averse agents. Substituting the expressions for 𝛼*, 𝛽* in the total expected utility, we obtain 2

2. The agent with higher risk tolerance gets a higher share of the total income in the partnership.

   Corollary: Suppose A is the worker, and B is the boss, and 𝑟𝐴 > 𝑟𝐵 = 0 (entrepreneurs are mostly riskneutral, otherwise they would work somewhere for a salary instead).

Linear model of performance-related incentive pay

• The Principal (P) and the Agent (A) negotiate over an employment contract. A works, but A's effort (e) is not observed by P, so cannot be contracted on.

• A's effort e brings profit P(e) for P (before wage costs).

• P uses a performance measure 𝑧 = 𝑒 + 𝜂 to reward A, where 𝜂~𝑁(0, 𝜎 2 ). For example, z could be sales or physical output.

• A's wage is 𝑤 = 𝛼 + 𝛽𝑧 where 𝛼 is fixed wage and 𝛽 is the “strength of incentives".

  —>P's net profit is 𝜋 = 𝑃(𝑒) − 𝛼 − 𝛽z.

  P is risk-neutral and wants to maximize E(𝜋) by affecting A's effort through the choice of 𝛼 and 𝛽.

“Incentive effect”

• If there is no incentive pay (𝛽 = 0), no-one will work (or will put in only the minimal detectable effort), because c´(e*) = 0 implies e*= 0.

• After the introduction of incentive pay (𝛽 > 0), all the existing workers will work harder than before, with more hard-working (lower c´´(e)) and less risk-averse (lower r ) workers working even harder.

  
  

  —>The sum of individual outputs will increase.

“Sorting effect”

• P's profit from a linear incentive contract (𝛼*, 𝛽*) goes down with A's costs of effort (c´´(e)) and risk aversion (r).

• As long as there are free workers on the labour market (because of unemployment, moving jobs, etc.), P can increase his profits by attracting more hard-working (lower c´´(e)) and less risk-averse (lower r ) workers than the averages in his existing workforce, offering them a high 𝛽 and low 𝛼.

• At the same time, a given incentive contract is more attractive for the relative more hard-working and less risk-averse workers, because given 𝛽 and 𝛼, the worker's CE decreases with c´´(e) and r.

  
  

  —>So, after the introduction of incentive pay, low-c´´(e), low-r workers will come, and high-c´´(e), high-r workers will leave the firm.

  —>The sum of individual outputs will increase

Is the linear incentive pay scheme for a risk-averse Agent efficient?

• No. The efficient risk sharing implies that all risk should be borne by the risk-neutral Principal (recall earlier this lecture).

• Under incentive pay, the risk-averse A bears some risk through the share β* of output that his pay includes.

• Bearing risk is more expensive for A than for the risk-neutral P. The risk premium paid to the risk-averse A is a source of inefficiency.

• Under first-best, P would pay A based on his effort. Unobservability of effort makes first-best unfeasible. Hence, the incentive pay as a second-best.

  
  

  —>The way to increase the efficiency of an incentive scheme is to reduce the noise 𝝈^𝟐 in the performance measure z. This can be done by better performance measurement (later in the course).