Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design (1991) By Bengt Holmstrom and Paul Milgrom Presented by Group von Neumann Morgenstern Anita Chen, Salama Freed, Jian Zhai, and Liming Zheng Notation C(t)=cost B(t)=benefit of principal V(t)= net asset gain of agent t=effort contributed by agent w(x)=wage a=income/price of an agent x=information signals β=intercept of the linear wage function r=measurement of agent s risk aversion =arbitrary function of effort CE=certainty equivalence =covariance matrix K= pool of potential personal activities A=subset of allowable tasks out of personal tasks K I= task assignment variable Section 1: Introduction This paper discusses a principal-agent model contextualized to the work environment. Unlike previous papers, which were limited to agents who only performed a single job task, Holmstrom and Milgrom introduce agents whom are responsible for a variety of tasks. Keeping in line with previous models, Holmstrom and Milgrom also assume a risk-averse employee (or agent), which results in job incentives that sometimes force the agent to bear unwanted personal risk and costs. This assumption then brings up the issue of how an employer (or principal) can motivate an agent to perform a particular given some degree of risk-aversion. In theory, job design should play an important part in controlling and motivating incentives for agents. However, one-dimensional models previously had issues explaining why job contracts sometimes omit explicit incentive clauses, a counter-intuitive result given the assumption of risk aversion. What distinguishes Holmstrom and Milgrom s model is that either the agent has multiple dimensions to his or her job OR, that the principal assigns multiple tasks to a particular agent(s).
This important feature comes from the linear nature of Holmstrom and Milgrom s principalagent model (an idea that was originally proposed in their 1987 paper). The linearity property ensures that changes in compensation in one task will have effects on how an agent will allocate attention to other tasks. Other central ideas discussed are: 1. How asset ownership affects incentives under different employment schemes (e.g. firm employment vs. independent contracting). 2. When firm employment is preferable over independent contracting (and vice versa), 3. How to determine the optimal limits of mixing business with pleasure. As we will see, this limit is again dependent on whether an agent s job tasks are difficult to measure by the principal. 4. And a sketch of how to optimally divide up tasks amongst two agents. Section 2: The Linear Principal-Agent Problem Assumptions: 1. Costs are a strictly convex function 2. Benefits are strictly concave and accrue to principal 3. Principal is risk-neutral 4. Agent is risk-averse; risk aversion is measured by r 5. Agent s utility function is exponential 6. Agent s wages are a linear function of job performance 7. Agent makes a one-shot choice of how he allocates effort during employment (e.g. agent can see his performance before acting). First introduce a principal who hires an agent for a job. This agent has a vector of efforts or actions denoted as The agent s utility is gained from making wages. In order to earn money, the agent must exert effort at a personal cost, C(t). The agent also creates a vector of signals x, which is defined as following: so that is some concave function and each ε is normally distributed. Here, signaling will indicate how much effort the agent is putting into his/her work. Hence we have that the agent s wages are a function of x. We can substitute the previous equation into the agent s expected utility.
u(ce)=e{u[w( where utility 1 is an exponential function u(w)=-e -rw. Recall that the certainty equivalence (CE) is the level of compensation that the agent would require to be indifferent between a risky asset (the expected income). If we have that wages are linear, then Because w(x) is a function of t, we can rewrite the principal s expected profits in terms of t: so or equivalently, under the linear compensation scheme. If we aggregate the principal and agent s certainty equivalences, we can get the following sum: where the third term is the risk premium on the agent s wages. Now let s denote the optimal incentive scheme as: (1) Subject to the condition that t maximizes: (2) Equation (2) says that given the incentive scheme, the agent will pick an effort t that maximizes his utility. This type of constraint is generally called the incentive compatibility constraint (ICC). The solution to (1) will be some optimized pair (a,t)*,which comes from the feasible set of employment contracts characterized by (t,a,β). Interactions among Tasks Suppose we have that so that. Then the ICC becomes 1 Note that utility is an exponential function and is written as u(w)=-e -rw. This relationship implies that the agent has constant absolute risk aversion (CARA). Recall the definition of absolute risk aversion: A(w)= so in this case, the agent has A(w)=r. The implication of CARA then is that the agent s risk aversion is independent of wealth effects. While this assumption is not the most realistic, it is often included in principalagent problems for mathematical convenience.
(3) where the i s indicate the partial derivatives with respect to each t. Now we take the crosspartials, first of income w.r.t. to effort. Using the inverse function theorem, we can also find the cross partial of effort w.r.t. income: (4) We can substitute (3) and (4) into (2) and we come up with: (5) where B is the vector if the first derivatives of B 1. B n and I is the n x n identity matrix. This equation is another rearrangement of the ICC. Note that we don t usually consider the crosspartials of B since there may be complementarities in the agent s costs which feed directly into the optimal incentive payment scheme. Example: consider a teacher who teaches both basic and advanced thinking skills. Suppose that advanced thinking skills are unobservable to the principal. So we have 2 tasks, one which is observable and one which is not unobservable. Let s call them t 1 and t 2, respectively. Then the signal vector becomes: (6) If we take that the errors of t 1 and t 2 are independent and that t 2 has infinite variance, then we can use equation (5) and find that the ICC becomes: (7) ] This tells us that if t 1 and t 2 are complements, then the teacher has incentive to allocate more effort into teaching basic skills (the sign on the cross-partial C 12 will be negative). In the general case for substitutes, given some t i, the principal would have to reduce incentives for other activities (if t i is unobservable, as in the example) or by providing more reward for t i.
Section 3: Allocation Incentives for Effort and Attention Assumptions 1. Again, we have risk neutral principals and risk averse agents. 2. Working may be pleasurable under some amount of time. Missing Incentive Clauses in Contracts Description of the model: let denote the time spent on the unmeasurable performance and be the time spent on the measurable performance, so. However, the unmeasurable performance is very important, e.g. B if. Proposition 3.1: In the home construction model, the efficient linear compensation rule pays a fixed wage and contains no incentive component ( ), even if the contractor is risk neutral. Proof: The proof in the paper is not completely right, for that, if, agents will set, the situation will be like the case of a fixed payment,. However, if these two sorts of efforts are not completely substitutes, the conclusion remains. Implications of this proposition: if some effort is both important and hard to measure, to stimulate agents by incentives on measurable effort is to encourage them ignoring the others. "Low-Powered Incentives" in Firms Description of the model: let denote the time spent on the behaviors generating receipts and be the time spent on the effort on increasing the net asset value. Thus, let the expected gross profit from the enterprise be the sum of two parts,, where B represents the expected net receipts and V the expected change in the net asset value. The enterprise has two types of organizations: the one where the principal owns the capital is called employment regime. The other is the independent contract regime in which the agents own the assets. In the general case, we can assume that B and V are increasing and concave functions, and the capital gain has a white noise part. Furthermore, denote:
. Proposition 3.2: Assume that, Then, the optimal employment contract always entails paying a fixed wage ( ). Whenever the independent contracting relation is optimal, it involves "high-powered incentives" ( ). Furthermore, there exist values of the parameters r,, and for which employment contracts are optimal and others for which independent contracting is optimal. If employment contracting is optimal for some fixed parameters (r,, ), then it is also optimal for higher values of these parameters. Similarly, if independent contracting is optimal, then it is also optimal for lower values of these parameters. Remarks: The inequality condition means that spending time on both activities is better than only focusing on single activity. This proposition not only shows the optimal choice of payments under different organizations, but also provides a qualitative description of the conditions under which different regimes are applied. Intuitive Proof of Proposition 3.2: It is obvious that principal should not stimulate employee to focus on single activity, for that the inequality condition holds. In case of the independent contract regime, if no incentives, agents will spend all the time on pursuing capital gains and leave principal nothing. Hence, the principal would always choose. If the risk is so large or the agents is extremely risk averse that the profit improvement cannot offset the loss of utility caused by the uncertainty, the risk neutral principal would rather choose certain payments and keep asset in his own hands. On the other hand, if the risk is not large or the agents are willing to take some level of risk, the other type of regime is optimal. However, this is not a strict proof, and neither is the paper s proof. I will leave this in the presentation.
Section 4: Limits on Outside Activities Assumptions 1. Constant returns to time for improving performance measurement and benefit to principals. Assume the agent has an infinite pool K={1,,N} of potential activities that only benefit the agent and not the principal. The principal controls these whether the agent can perform these activities during business hours by allowing them (where the task is or excluding them (. The agent can perform the set of allowable tasks A as much as desired. The benefits to the agent can be described as: (8) ) ) Assuming constant returns to time for profits and improving performance, we can describe them as: (9) We are interested in determining the commission rate α and what tasks in K should be allowed in subset A. This will be done by first fixing the commission rate α and setting A(α) based on that, then finding the best α. Stage 1: Fix the commission rate α and find A(α) (10) (11) with first order conditions of and Our first order conditions reveal that the time the agent spends on all tasks and the time the agent spends on personal tasks is not related to A, the tasks allowed. Therefore, if the number of tasks increases, with a fixed α, the agent will spend more time on personal tasks and less time on the principal s tasks. As such, the agent stands to gain while the principal stands to lose )
The optimal set of allowable tasks, illustrated in figure 1, is: (12) (Note the strict inequality here. Even if the gain to the agent and the cost to the principal are equal, the principal reserves the right to restrict that task.) Figure 1 Proposition 3 tells us that, assuming α allows for t(α)>0: The agent should be allowed those tasks A(α) such that, or those in which the agent s average product is strictly greater than the principal s marginal product. Increasing an agent s commission rate means that the list of personal tasks allowed to that agent increases as well. o Responsibility and authority go hand in hand. When the employee is financially more responsible for his performance, it is easier to give more freedoms.
o In the extreme case, when the error term in the performance metric is zero, the agent will be able to engage in any personal tasks desired, because there will be perfect measurement of performance. o Note also that the marginal cost of the employee of spending time on work is reduced by excluding private tasks. If we exclude one task for not meeting the previously noted criteria, then we should exclude all tasks where the average product does not exceed the marginal product. o This shows the social value of personal activity and the likelihood that it will be excluded are not necessarily closely related. o Note on the graph that even though there is higher value in task 2, it is excluded, and it invites more excess attention. Stage 2: Determining the optimal choice of α Proposition 4, assuming t(α)>0), says: The best commission rate is given by ] where o The employee s responsiveness to increased incentives increases as the set of allowable tasks increases (because dt/da is a function of a summation of A times the inverse marginal productivity of personal tasks. o So it is best to raise commission in response to the increase in allowable tasks make the stakes higher if you allow more personal freedom. o An increase in one leads to an increase in the other. If the error in performance measurement decreases due to a decrease of the variance of the agent becomes more risk averse, the principal can relax the set of allowable tasks and increase the commission rate. o There will be more restrictions on an employee s personal tasks as long as performance is not able to be accurately and precisely measured. Any task that will be excluded from first best arrangement (where personal benefit does not outweigh benefit to the company) should be excluded from other arrangements as well (second best)
Section 5: Allocating Tasks between Two Agents Assumptions 1. There are two homogenous agents 2. Personal costs of agents are convex 3. The principal can group jobs in any way and all tasks are considered small Holmstrom and Milgrom describe the model in this section as a first pass attempt at studying the optimal grouping of tasks. So while the model is incomplete, it still provides valuable insight into this problem. Here, two propositions are proved under ideal conditions: 1) Proposition 5: It is never optimal for the two agents to be jointly responsible for any task k. 2) Proposition 6: There is a NEGATIVE trade-off between incentives (here, we consider attention) and risks (here, we consider hard to measure tasks). We want to minimize the objective function of the two agents: ( t 1 ) t ) ( 2 Subject to: ( t i ), i=1, 2 And incentives constraints: We have the Lagrangian equations: And ( ( t i ) if ( t i ) if
(21) Caveats to the Proposed Model As previously mentioned, the principal can group jobs in any way and we consider all tasks as small. However, when tasks require a certain amount of time to be completed and vary in size, the need to minimize costs might lead to a reverse conclusion. Some other problems with the model include: 1) The error term of measurements in tasks is assumed to be independent. While this is mathematically tractable, Holmstrom (1982) shows a positive relation between errors will reduce risk premium. Thus in the incentive domain, this model is incomplete. 2) Requirement of equal substitutes in the agent s cost function and excludes the possibility of complementary relation in activities. 3) Agent focus one job of all the time, but in reality, issue of job rotation is an important aspect of job design. Another Example on this Topic Prendergast (2002) 2 mentions that in a more risky environment, given the assumption of reliable reports, there is a POSITIVE relation between risk and output based payment. 2 See The Tenuous Trade Off of Risk and Incentives (JPE 2002) by Canice Prendergast.