4.3 Incentives schemes with a performance target

Introduction

Performance targets are frequently used in incentive contracts. Here are some examples.

• McDonald's, 2010. Target: operating income ≥ 7.24 bn USD. CEO bonus: 2,160,000 USD if above, 0 if below.

• Bank of America, 2011. Target: return on assets ≥ 50 bps. CEO bonus: an extra 33% of the total bonus. Source: Hu et al. (2019) Optimal performance targets. WP

• Facebook, 2019. An average worker earns 150,000 USD a year. Additionally there is an individual bonus target (15%), there is an individual performance multiple (varies widely) and there is a company performance multiplier. Adding the percentages together, multiply them by your base salary is the individual quarterly bonus. In the case of an average worker it might be 22,500 USD. Source: https://www.theverge.com/2019/2/7/18214769/facebook-employee-bonus-incentives-social-progress

Incentive schemes types with a performance target

Incentive schemes with a performance target are often practiced. The scheme can be in the form of a bonus or piece rate paid per unit of output starting with a certain pre-determined output level.

• In the reward scheme employee earns a base salary for low levels of output, only earning a bonus if performance rises above some target level T.

• In the penalty scheme employee earns a base salary for high level of output, but the reward is reduced if performance falls below T.

• In the lump-sum reward scheme, there is a discrete jump in reward, if performance is above the target.

• In the floor and cap incentive scheme the employee earn based on a continuous scheme with a cap. There is a base salary (floor) too, regardless of the performance.

  —>Why complicate things beyond simple piece rate by introducing targets? Because there are efficiency gains (to be seen today).

Piece rate incentive scheme with a target

• Consider an incentive scheme under which a wage 𝑠 ≥ 0 is paid in any case and, additionally, a piece rate r is paid for every unit of output starting from the target 𝑦̅. That is

• All other conditions are the same as above: A's costs of effort are 𝑒^2/2𝑐 , participation constraint is

Agent's optimal effort choice under given incentives:

• As before, higher rate (r ) means higher optimal effort and greater costs of effort (lower c) means lower optimal effort.

• Substitute e* to calculate A's expected wage:

• A´s expected wage net of costs of effort:

Principal's optimal choice of incentives given A's optimal choice of effort:

• Recall A's optimal effort choice:

• P anticipates this in deciding on the parameters of the incentive scheme with a target: s, r and 𝑦̅

• Subject to A´s participation constraint:

• Substituting which gives

• FOC (try at home: take first derivative wrt. 𝑦̅):

—>This means that the optimal piece rate and the target are interconnected. Choosing r*, one will get the target level 𝑦̅* implied by it.

• For example, r* = 1 (i.e the efficient piece rate) corresponds to the optimal target 𝑦̅* = 𝑐 − 𝑎 (i.e. a bit below the efficient output level, to take uncertainty into account).

Comparison of the piece rate incentive contracts with and without a target.

• Without a target: efficiency requires a negative fixed wage s*, which may be hard to implement in practice (liability constraint).

• With a target: Efficiency does not require a negative fixed wage. For example, if a is small (i.e. not too much noise) and c is large (A is hard working, his costs of effort are low), both A and P enjoy positive profit and fixed wage (profit: 𝑐/2 − 𝑤̅, fixed wage: 𝑠* = 𝑤̅ − 𝑎 + 𝑐 2 ).

  
  

  —>Implication: Performance targets is a popular management tool because a contract with a target can restore efficiency and can accommodate the limited liability constraint.

Incentive scheme with a target bonus

Linear incentive scheme with a target for a risk-averse agent

In this case, the target does not restore full efficiency but does bring the solution closer to efficient. Why?

1. Full efficiency is only achievable when A bears no risk, but this contradicts the incentive constraint.

2. Thanks to the performance target, optimally chosen by the Principal, the Agent will end up bearing less income risk. Principal can introduce steeper incentives, which result in more effort of the agent.

—>Intuitively, think of the target level as an extra contract parameter principal can choose. More parameters to optimize means (usually) a better solution.

Solution process:

• Is the same as in the case of the linear piece rate scheme without a target

• The only difference is that we need to use the formulas for the mean and variance of truncated normal distribution. Why truncated normal?

  —->Because A gets uncertain income 𝛽 ⋅ (𝑦 − 𝑦̅) only when 𝑦 − 𝑦̅, so the noise term z in the wage equation is truncated to zero below 𝑧 = 𝑦̅ − 𝑒. Hence its distribution is truncated normal.

• Performance targets improve efficiency of incentive contracts.

  —>For the risk-neutral A: target can help satisfy the limited liability constraint, leading to implementable first-best solution. —>For the risk-averse A: no first-best, but still an efficiency improvement compared to the contract without a target.

• The target is an additional parameter that P can use to optimize the contract, in particular, to reduce the risk born by A, and thus to sharpen incentives for A to stimulate A's effort.