4.1 Principal-Agent model: the notion of efficiency and incentives for the risk-neutral case

monetary incentives, broadly defined as compensation schemes that award higher pay for better performance results.

Why is performance important?

• The harder the worker works, the more output he creates.

• Effort is costly. Therefore, a higher effort must be rewarded with a higher wage.

• But the observability of effort is limited, thus linking wages to effort is difficult. We need to use observable performance measures as proxies for effort. Hence “performance-based" rather than “effort-based" pay.

• Principal (P) hires Agent (A) to produce output y. A must send effort e to produce output 𝑦 = 𝑝(𝑒), where 𝑝´(𝑒) > 0, 𝑝´´(𝑒) ≤ 0.

• A has effort related costs 𝑐(𝑒), 𝑤ℎ𝑒𝑟𝑒 𝑐´(𝑒) > 0, 𝑐´´(𝑒) > 0

• A receives a fixed wage w, so A´s net gain is: 𝑤 − 𝑐(𝑒)

• P net gain is the difference between A´s output and wage, i.e. 𝑝(𝑒) − 𝑤

• How much effort should A exert? And how much P should pay to A?

• Incentives for P: maximize 𝑝(𝑒) − 𝑤 by choosing w and, indirectly, A´s effort e.

• Incentives for A: given w, maximize 𝑤 − 𝑐(𝑒) by choosing e.

  A agrees to work for P if

  participation constraint (PC) 𝑤 − 𝑐(𝑒) ≥ 𝑤̅ is fulfilled

  (𝑤̅ is A´s reservation wage).

• P moves first choosing w in anticipation of A´s effort. The choice of w by P affects A, and the choice of e by A affects P.

Conflict of interests:

P´s profit increase with e, but A´s net gain decreases with e (reversed for w).

Notes: In general, the choices of w by P, and of e by A, depend on effort observability (output is only an imperfect measure of effort because of uncertainty) and risk preferences (most of us are risk-averse). But let us assume, so far, that all are risk-neutral.

Effort is perfectly observed and there is no uncertainty

• For A to work: 𝑤 − 𝑐(𝑒) ≥ 𝑤̅ → 𝑤 = 𝑤̅ + 𝑐(𝑒) (PC).

• P maximize 𝑝(𝑒) − 𝑤,

  substituting (PC) we get 𝑝(𝑒) − 𝑤̅ − 𝑐(𝑒).

  So P would like A to exert effort e* that gives 𝑝´(𝑒*) − 𝑐´(𝑒*) = 0.

• P will then offer A a wage 𝑤* = 𝑤̅ + 𝑐(𝑒*).

  Will A work for w*?

  Yes, because his participation constraint is (just) satisfied.

—>So we have the contract (w*, e*). This contract can be controlled and enforced because effort is observed. A wage w* will be paid only if an effort e* is exerted. —>Is the contract (w*, e*) efficient?

Efficiency is an important characteristic of economic outcomes. Here we are interested in efficiency of produced output by effort e at the cost w.

Three equivalent definitions of efficiency:

An outcome of economic activity is fully efficient if ...

1. The maximum total gain is achieved that is possible given the available resources.

2. Output minus the total costs of effort is maximized.

3. No resources are wasted or used unproductively.

   
   

• An outcome (w; e) is often inefficient, meaning that there is another outcome (w´, e´)that could result in a higher value of output net of costs of effort, but is not achieved through A and P acting in the best of their individual interests.

• Basic example: while it may be efficient for A to work hard, it is not what A will want or do without incentives.

• Inefficiency means lost profit —>competitive advantage to firms that are more efficient —>strategic choices of compensation schemes depending on the efficiency gains they bring.

• Yes, because the same solution is obtained by maximizing the total of A+P´s gain:

  𝑝(𝑒) − 𝑤 + 𝑤 − 𝑐(𝑒) = 𝑝(𝑒) − 𝑐(𝑒) ⇒ maxe 𝑝(𝑒) − 𝑐(𝑒)

  ⇔ 𝑝´(𝑒) = 𝑐´(𝑒)

• The efficient solution is sometimes called the first-best, i.e. a solution that would have been implemented by an all-knowing body who has all the relevant information and does not need to rely on P or A to reveal this information.

• We live in the world of second-best solutions, where the highest possible outcome for all is not achieved.

—>But economists can, sometimes, design mechanisms and especially contracts that bring the outcomes close to first best, by exploiting the incentives of the agents in the right way.

• The efficient solution is sometimes called the first-best, i.e. a solution that would have been implemented by an all-knowing body who has all the relevant information and does not need to rely on P or A to reveal this information.

• We live in the world of second-best solutions, where the highest possible outcome for all is not achieved.

Incentive contract if effort is not observed

• In our basic setup output ≈ effort, thus there is no uncertainty. But, effort often cannot be observed. So let us allow some uncertainty in the relationship between effort and output.

• Suppose A can choose between high and low effort, 𝑒ℎ > 𝑒𝑙 , 𝑐(𝑒ℎ) > 𝑐(𝑒𝑙).

  P cannot observe A´s effort choice but can observe the outcome, 𝜋ℎ 𝑜𝑟 𝜋𝑙 , where 𝜋ℎ > 𝜋𝑙 .

—>Uncertainty: If A exerts 𝜋ℎ, 𝜋𝑙 will be achieved with a probability p, while if A exerts 𝑒𝑙 , 𝜋ℎ can still be achieved, but with a lower probability q, i.e. 𝑝 > 𝑞. Thus relationship between A´s effort and outcome is as follows:

• Suppose, as before, A gets a fixed wage w from P. Which effort level will A prefer to exert? Given A´s effort choice, how much money will P offer to A?

• A´s net gain under 𝑒ℎ will be lower than under 𝑒𝑙 because 𝑐(𝑒ℎ) > 𝑐(𝑒𝑙), so A will choose 𝑒𝑙 .

• Anticipating this, P will offer 𝑤 = 𝑤̅ + 𝑐(𝑒).

• Is this outcome efficient? It depends. If the expected marginal benefit of effort is higher than the marginal cost.

  That is, when 𝑝 ⋅ 𝜋ℎ + (1 − 𝑝)𝜋𝑙 − (𝑞 ⋅ 𝜋ℎ + (1 − 𝑞)𝜋𝑙) > 𝑐(𝑒ℎ) − 𝑐(𝑒𝑙).

then the low wage-low effort contract is inefficient, because it prevents the realization of the highest possible net gain to A+P.

Incentives based on outcomes

Suppose the high-effort outcome is efficient and P offers A the wage 𝑏ℎ if the outcome is 𝜋ℎ and 𝑏𝑙 if the outcome is 𝜋𝑙 , 𝑏ℎ > 𝑏𝑙 as an incentive to exert 𝒆𝒉 rather than 𝒆𝒍 . Will this work?

—>Yes, under following conditions:

1. Incentive constraint (IC):

   A must prefer 𝑒ℎ to 𝑒𝑙 , i.e.

1. Participation constraint (PC):

   A must prefer to work for P rather than get the reservation wage outside, i.e.

1. Limited liability constraints:

   wages must be nonnegative, i.e.

• P will choose the values of 𝑏𝑙 , 𝑏ℎ that maximize P´s profit and satisfy all the constraints, i.e. choose the lowest possible 𝑏ℎ, 𝑏𝑙 that still satisfy A´s constraints.

  
  

Case 1: Both the incentive and participation constraints are binding (hold with=rather than >), and the limited liability constraints hold.

Then solving for 𝑏ℎ, 𝑏𝑙 from the constraints:

This case corresponds to the fully efficient solution because A gets precisely how much it costs to exert 𝑒ℎ (no waste of money).

• Potential problem with case 1: optimal 𝑏𝑙 may be negative, violating the limited liability constraints. This can happen when the difference 𝑝 − 𝑞 is small (lots of uncertainty in output despite high effort) and/or when extra costs of effort are high. Thus we need a case 2.

  
  

Case 2: 𝒃𝒍 = 𝟎 to satisfy the limited liability constraint. As a consequence: the participation constraint is not binding. Due 𝑏𝑙 = 0 is higher than what would satisfy the participation constraint (𝑏𝑙 < 0).

—>Can P compensate by reducing 𝑏ℎ? No, because the incentive constraint is already binding. Thus P has to pay a rent to compensate for the difference between the optimal 𝑏𝑙 < 0 and the liability constrained 𝑏𝑙 = 0.

• Total costs to the P under case 2:

• R as rent, here the difference between the optimal 𝑏𝑙 < 0 and 𝑏𝑙 = 0 to satisfy PC

• When A gets the rent, full efficiency is not achieved because the actually paid costs are higher than the level required to induce 𝑒ℎ —>But incentives work. Even in case 2 it is possible to motivate A to work hard, but at the price of lower efficiency.

Price rate incentive scheme

It is one of the simplest incentive schemes. The setup is as follows:

• Output produced by A. 𝑦 = 𝑒 + 𝑧, where e is effort and z is zero-mean random noise.

• Costs of effort: 𝑐(𝑒) = 𝑒 2/2𝑐 where c is constant.

• Wages paid P: a fixed wage s and a fraction r of every unit of output produced by A, that is 𝑤 = 𝑠 + 𝑟 ⋅ 𝑦

• A´s participation constraint: 𝐸(𝑤) − 𝑐(𝑒) ≥ 𝑤̅

• We still assume A is risk-neutral.

  
  

Agent´s optimal action given the incentive scheme parameters:

—>Higher rate means higher optimal effort. Lower c (that is, greater costs of effort) means lower optimal effort.

What is P`s optimal choice of incentives for A given A´s optimal effort choice: e*= 𝑟 ⋅ 𝑐? P anticipates e* in deciding on the parameters of the piece rate incentive scheme, s and r.

subject to A´s participation constraint:

Substituting into maximization function gives

• r*=1 implies that P should give ALL output to A to maximize profit.

• Only if r*=1, A will exert the maximum effort 𝑒* = 𝑟 ⋅ 𝑐 = 𝑐

• A´s net expected income is

This is just enough to keep A on the participation constraint.

• P´s profit at r*=1 is

That is, P will only hire A if P´s profit is nonnegative, implying

• So, under the optimal piece rate incentive contract between P and A, A gets all the output (r* = 1) and pays to P (e* = c). We can also say that A “rents" the job from P.

  —>Examples of such working arrangements exist (e.g. taxi drivers, franchise owners) but are exceedingly rare.

Is the optimal piece rate contract efficient?

• Following the efficiency criterion we search for e that maximize output under given resources, or maximize possible difference between the total output and total costs.

• That is, the efficient piece rate incentive contract is found from

𝐹𝑂𝐶: 𝑒* = 𝑐 just like the optimal effort under the contract between P and A when r *= 1.

—>So, the piece rate contract between P and A presented earlier is an efficient contract.

Why do not all firms give workers the full output and charge them for working?

Reason 1:

Agents (workers) are mostly risk-averse, Principals (firm owners) are less risk-averse or even risk-neutral. The fully efficient piece rate contract exposes the agent to “too much" risk, for bearing which a compensation is required. As a result, most piece rate contracts have 𝑟*<1 and 𝑠 > 0. That is, are not fully efficient. (See more in the next section 4.2)

Reason 2:

While the contract with 𝑟*=1, s*<0 is efficient, s<0 violates the limited liability constraint (A's fixed wage cannot be negative). As a consequence a non-binding participation constraint for A is an inefficient incentive contract. We will discuss this case when studying incentives with performance targets. (See more in section 4.3)