Pay for Performance with Motivated Employees

Benchmark

Proof of Proposition 1. The principal maximizes her profits subject to eq. (3). Profits can be rewritten as:

Taking the first-order condition with respect to eA and eB yields the following:

Under complete information, the first-best effort levels satisfy the following conditions.

1. If both employees are motivated, i.e. θA=θB=θ¯, eH=∂f(eH,eH)∂eH(1+2θ¯);

2. If both employees are selfish, i.e. θA=θB=θ_, eL=∂f(eL,eL)∂eL;

3. If only one employee is motivated, i.e. θi=θ¯ and θj=θ_, eH=∂f(eH,eL)∂eH(1+θ¯) and eL=∂f(eL,eH)∂eL(1+θ¯).

Irrespective of the function we consider, the principal requires the same level of effort from both agents, even when the team is heterogeneous.

1. If both employees are motivated,

   eH={1+2θ¯2iff(eA,eB)=eAeB;1+2θ¯iff(eA,eB)=eA+eB

2. If both employees are selfish,

   eL={12iff(eA,eB)=eAeB;1iff(eA,eB)=eA+eB

3. If only one employee is motivated,

   eH=eL={1+θ¯2iff(eA,eB)=eAeB;1+θ¯iff(eA,eB)=eA+eB

Screening Problem

The participation and incentive constraints are:

with i,j∈{A,B} and i≠j.

First, if equations (ICH) and (PCL) are satisfied, then

Equation (4) reflects the fact that a motivated employee receives a higher surplus from the project than a selfish one. (PCiH) is satisfied and cannot be binding because

must be satisfied as well. In contrast, (PCiL) must be binding. Next, (ICiH) must be binding, otherwise the principal could increase ωiH slightly and keep all the constraints satisfied. Finally, the incentive constraint for the selfish type cannot be binding given that

must be satisfied.

Proof of Proposition 2. The participation constraint for the self-interested employees (PCL) and the incentive constraint for the motivated employees (ICH) are binding. Using (PCL) and (ICH), the optimal wages satisfy the following equation:

where θ¯[μf(eiL,ejH)+(1−μ)f(eiL,ejL)] is the information rent paid to motivated employees.

The principal maximizes profits subject to eq. (5). Profits can be rewritten as follows:

Taking the first-order conditions with respect to eiL and eiH yields the following:

Analogous expressions could be derived for agent j. By comparing the first-order conditions with those obtained under the first best, it can be noted that if teams consist of homogeneous agents with certainty, i.e. μ=1, both motivated and selfish employees exert a lower effort than in the first best. In contrast, if teams consist of heterogeneous agents with certainty, i.e. μ=0, the effort is distorted only at the bottom, as in the standard adverse selection model. Therefore, an increase in μ has a negative impact on the effort of motivated employees under individual incentives: it magnifies the distortion at the top.

For the Cobb–Douglas and linear production functions:

1. If the employee is selfish,

   eL{∈(12(1−θ),12)iff(eA,eB)=eAeB;=1−θ¯iff(eA,eB)=eA+eB.

2. If the employee is motivated,

   eH{∈(12,12(1+θ))iff(eA,eB)=eAeB;=1+θ¯iff(eA,eB)=eA+eB.

When f(eA,eB)=eA+eB, the effort of the selfish employee is lower than the first-best effort level of a selfish employee, whether his teammate is selfish or motivated. In contrast, the effort of the motivated employee equals the first-best effort of a motivated employee with a selfish teammate, but is lower than the first-best effort of a motivated employee with a motivated teammate. The latter implies that, besides the distortion at the bottom, there is also a distortion at the top.

When f(eA,eB)=eAeB, the effort of the selfish (motivated) employee is lower than the first-best effort level of a selfish (motivated) employee, irrespective of his teammate’s type. Thus, there is a distortion at the bottom and at the top.

Proof of Proposition 3. Also under team incentives (PCL) and (ICH) must be binding.^[6] The incentive constraint for a motivated employee requires that the expected utility that he receives is higher than the expected utility that he obtains by pretending to be selfish. In expectation, a motivated employee knows that with probability μ he will be in a team with another motivated teammate and with probability (1−μ) in a team with a selfish one. Participation constraints guarantee that in expectation both types accept the contract.

From (PCL) the optimal wage for selfish employees satisfies the following equation:

and from (ICH) the optimal wage for motivated employees satisfies the following equation:

where θ¯[μf(eiLH,ejHL)+(1−μ)f(eiLL,ejLL)] is the information rent paid to motivated employees. The principal chooses effort levels to maximize her profit function subject to eqs (7) and (8). After some simple computations, profits can be rewritten as follows:

Taking the first-order conditions, the effort level must satisfy the following:

1. If both employees are motivated, eiHH=∂f(eiHH,ejHH)∂eiHH(1+2θ¯);

2. If both employees are selfish, eiLL=∂f(eiLL,ejLL)∂eiLL[1−2θ¯(1−μμ)];

3. In heterogeneous teams, eiHL=∂f(eiHL,ejLH)∂eiHL[1+θ¯(1−2μ1−μ)]andeiLH=∂f(eiLH,ejHL)∂eiLH[1+θ¯(1−2μ1−μ)].

Analogous expressions could be derived for agent j. By symmetry, in what follows we can rewrite each effort level as eiHH=eHH, eiLL=eLL, eiLH=eLH and eiHL=eHL. For the Cobb–Douglas and linear production functions:

1. If both employees are motivated,

   eHL={1+2θ¯2iff(eA,eB)=eAeB;1+2θ¯iff(eA,eB)=eA+eB.

2. If both employees are selfish,

   eLL={12[1−2θ¯(1−μμ)]iff(eA,eB)=eAeB;1−2θ¯(1−μμ)iff(eA,eB)=eA+eB.

3. In heterogeneous teams,

   eHL=eLH={12[1+θ¯(1−2μ1−μ)]iff(eA,eB)=eAeB;1+θ¯(1−2μ1−μ)iff(eA,eB)=eA+eB.

Similarly to the first best, the principal requires the same level of effort from both agents, even when the team is heterogeneous. For either production function, under team incentives the distortion at the top when both employees are motivated disappears. However, there is a distortion for the motivated employee with a selfish teammate. This distortion allows the principal to pay a lower information rent to motivated employees, as stated by Corollary 1. Overall, the distortion at the top is mitigated under team incentives.

It is worth noting that an increase in μ has a negative impact on the effort exerted by employees in heterogeneous teams, but positively affects the effort exerted by selfish employees working together:

To guarantee that effort levels are not negative, it is assumed that μ belongs to the following interval: μ∈[2θ¯1+2θ¯,1+θ¯1+2θ¯].

Proof of Corollary 1. Suppose that f(eA,eB)=eA+eB. The information rent paid to the motivated employee under individual incentives is given by

The information rent paid to the motivated employee under team incentives is given by

We find that the information rent paid to the motivated employee is always lower under team incentives than under individual incentives for any θ¯ and any μ∈(0,1).

Substituting the respective effort levels into eqs (6) and (9), we obtain the principal’s profits under individual and team incentives:

Comparing the profits, we find that ΠTI>ΠII if θ¯2[2−6μ+5μ2](1−μ)μ>0, that is always the case for any θ¯ and any μ∈(0,1).

This result holds also when f(eA,eB)=eAeB. The proof is similar and available upon request.