1. Introduction. Key Components of Model. Multidimensional Tasks

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1 Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design By Bengt Holmstrom and Paul Milgrom (1991) Modigliani group: Belen Chavez, Yan Huang, Tanya Mallavarapu, Quanhe Wang April 12, Introduction In the standard principal-agent model incentive systems are utilized to allocate risks and reward productive work. However the credibility of this model reduces when the agent is risk averse and would prefer a fixed wage system. This theory has also not been able to explain why employment contracts usually specify fixed wages with very little importance given to incentives. At the same time the model has not been effective in addressing issues such as asset ownership, job design and allocation of authority. What distinguishes this model from most others is that the principal can allocate different tasks to one or many agents, or the agent s single task can consist of several dimensions. The model discussed in this paper overcomes some of the shortcomings in the basic principal agent model. Holmstrom and Milgrom s multitask principal-agent model: Accounts for paying fixed wages even when outputs can be easily measured and agents are highly responsive to incentive pay Examines ownership patterns even when contracts can take account of all observable variables Explains why employment is sometimes chosen over contracting even when there are no productive advantages Determines how tasks get allocated to different jobs One of the prime examples to illustrate the issue the model is trying to address is whether teachers should be paid through incentives based on their students test scores. Supporters of this system say that the incentives will motivate teachers to take a greater interest in their students success. However, opponents argue that the teachers would neglect the importance of deeper critical thinking, creativity, and activities in arts and focus all their time on basic skills that are tested in standardized exams. The opponents suggest that teachers get paid based on a fixed wage. Key Components of Model Multidimensional Tasks Most tasks tend to be multidimensional. For instance, production workers are responsible for producing high volume and high quality goods and may also be required to clean the machines they utilize. In this case, if the agents are paid based on the volume of output since it is easy to measure, they are likely to sacrifice on the quality of the output and focus only on the quantity. Or, if quality can also be measured, the incentive rate system might cause agents to neglect taking care of the machinery they use. Therefore, when an agent is responsible for multiple tasks, 1

2 incentive pay can allocate risks, motivate hard work and direct attention to their various duties accordingly. Going back to the example of teaching to illustrate the point of job design. If the task of teaching basic skills can be separated from teaching higher-level thinking or arts, these tasks can be assigned to different teachers at different periods. Looking at this is the context of production workers, separating the maintenance of a productive asset and the use of the asset for production the use of a piece rate system would be more efficient. Job design is an important aspect for an efficient use of incentive based pay. This is similar to the concept of specialization. The model proposes that an increase in an agent s incentives for any one task will cause him to reallocate some of his attention away from other tasks. The efficiency of providing incentive based pay for an activity decreases with the difficulty of measuring performance in any other activities that are competitive for the agents time and attention. This is because the principal would not be aware of the performance of agent in the other activities that are equally important. This point could help explain why the one-dimensional principal agent model has not been able to explain why incentive based pay is not as common as expected. Asset Ownership The model also examines the case where the unmeasurable aspect of performance is how the value of a productive asset changes over time. The difficulty of valuing assets is recognized. In the case where the principal receives the returns from an asset the optimal compensation system would be to provide a less incentive based on output contract to avoid any abuse of the asset or any deviation of effort away from asset maintenance by the agent. However when the agent owns the asset returns the optimal compensation system would involve an intensive incentive contract to engage in production to avoid the situation where the agent uses the asset too cautiously or pays too much attention to its improvement. The conditions where the agents owns the assets would be successful if i) the agent is not too risk averse ii) the variance of asset returns are low, and iii) the variance of measurement error in other aspects of the agent s performance is low. In recent times, firms have been giving their employees stock options in the company along with their fixed wage. This is an effective way to ensure that the employees are performing well in their allocated tasks but at the same time carrying out activities that improve the asset of the company. This helps explain why franchisees have steep performance incentives, while managers of similar company-owned outlets receive no incentive pay. This also explains why a free-lance writer gets paid for articles by the word, but a reporter for the same publication gets paid a fixed wage. Personal Activities Holmstrom and Milgrom continue to extend the one principal-agent model by incorporating personal tasks into the model. How does a firm optimally set policies limiting personal activities during working hours? They suggest that outside activities should be more severely restricted when the performance of the task for the firm is difficult to measure. Therefore, a salesperson 2

3 who is paid on commission will optimally be permitted to engage in personal activities during business hours than a bureaucrat who is paid a fixed wage. This is because the commissions direct the salesperson to inside activities, which cannot be done for a bureaucrat. However, this seems to be changing for many companies today. For instance, Google, which pays most of its employees through fixed wages, gives them a lot of freedom with how they decide to engage in personal activities. Google even goes to the extent of encouraging its employees to engage in outside activities by providing them with a gym, game room and TV s around their campus. According to the analysis on outside activities, incentives for tasks can be provided in two ways: the main task is rewarded or the marginal opportunity cost for the task can be lowered by reducing the incentives on other competing tasks. Constraints are used instead of incentives when it is difficult to measure the performance of an agent. This helps understand large-scale organizations. The larger a firm gets the more difficult it becomes to constantly monitor their employees and therefore imposing constraints on personal activities is the most efficient system. Job Design In this case, Holmstrom and Milgrom extend the model to where the employer can divide the responsibility for many tasks between two agents that further allows the employer to decide how performance for each task is compensated. Each task must be allocated to just one agent. The tasks should then be grouped into jobs in a way that tasks that are easily measured are assigned to one worker and the other tasks are assigned to the second agent. This relates back to the argument of specialization, where the differences between measurability of quality and quantity in production make the incentive problems difficult. But here by grouping all the tasks for quality as one job and the tasks for quantity as one job the effectiveness of incentive pay increases. Even if the agents are identical before they start their tasks, they still should be separated to have measurement characteristics that are as different as possible in their jobs. The principal should then provide the agent whose performance is easily measured with a more intensive incentive pay and require more work from them. 2. The linear Principal-Agent Model Notation: t=effort contributed by agent C(t)=cost of agent B(t)=benefit of principal x=information signals w(x)=wage of agent =parameter of incentive wage β=minimum wage r=measurement of agent s risk aversion =representative of effort CE=certainty equivalence =covariance matrix V(t)=asset value 3

4 Purpose of the model: Choosing the (t, ) to maximize the total benefits, meanwhile, maximizing the agent s profit. Assumptions: (1) The wage payment is a linear function of measured performance. (2) Agent is required to make one-shot choice of how to allocate his efforts. (3) The principal is risk neutral, which means, he only pays attention on the expected return. In other words, the risk will only influence the agent s choice rather than the principal s. (4) C is strictly convex, while B is strictly concave. Description of the Model In this model, the agent is asked to decide how to allocate his efforts between k tasks 1 : t=(t 1,t 2,, t k ). The agent s efforts will generate both cost C to himself and benefit B to principal. Meanwhile, this effort will also produce a kind of signal x= which will help principal to decide how much wage this agent deserves to receive in an incentive strategy: w(x)=, where denotes the base pay the agent will receive even in an incentive strategy. Two important things to note are, first is that only serves to allocate the total profits between agents and principals--since it is a constant, it will disappear when taking the derviative That s why we don t observe in the latter equations. Second, denotes the risk an agent has to bear in an incentive wage. This parameter reflects the uncertainties, such as fortune, bias, etc. which will influence the evaluation of an agent s effort. Since there exists uncertainty, the authors here adopt expected utility theory to get the agent s certainty equivalent wage (CE), where (we know from the Arrow-Pratt equation that this utility function has CARA: constant absolute risk aversion with constant r), r measure the agent s risk aversion and denotes the variance of the agents income that the agent needs to bear. From now, we can evaluate the expected profit of both agent and principal, (1), (2) 1 Note: The paper has a typo, the n in this model donates the amount of agents 4

5 So now the problem is to find the (t, ) to maximize (1) and (2) subject to: t maximizes, which is the same as to maximizes Features of the model: a) Application: (1) Different activities can be applied in this model, no matter how difficult it is to measure. (2) We can study cases where performance measures can be influenced by activities, rather than by principal s desire. The agent has the freedom to allocate their effort, and sometimes the effort allocation will not be what the principal desire. (3) We can study cases where number of observables is smaller than that of activities. In other words, the principals cannot fully gain the information of agents behavior. b) The return to the principal is not necessarily to be observed. (1) If B=C, then it is optimal for principal to set =0. (2) If B is different than C, especially when B is difficult to measure, then the incentive wage will not be adopted. Simple Interactions Among Tasks: Assuming (the effort is equal to the signal) then we can take the derivative of equation, gaining following equation Further differentiating (3) and (4), we can get and,, (5) where B =(B 1,,B k ) is the first derivatives of B. Two things worthy of noting: (1) When the error terms are stochastically independent ( ) and the activities are technologically independent ( ), then the function (5) can be further simplified as. The commissions are set independently of each other since the cost of each task is independent, in other words, the incentive of a particular task will not influence the opportunity cost of other tasks. As expected, the incentive is decreasing with people s attitude towards risk and risk itself but increasing with marginal profit gained by principal. Moreover, the incentive is also increasing with the, which means the more responsive the agent is to incentives, the more ambitious the incentive put forth by principal will be. (2) However, most of time, we cannot neglect the cross-partials of C ( ). When, situation of complementary between different tasks will occur. That is, if agent increases his input in one task, he will also increase his input in other task. (When 5

6 ( ), if t 2 increase, then ( ) will decrease, which means, under the same amount of cost C, the input t 1 should increase. In a word, the incentive to task 2 increases the both input in task 1 and task 2.) So at this situation, increase the incentive to a particular task will benefit the whole profit gained by principal. When, situations of substitution between different tasks will occur. At this time, increasing the incentive in a particular task will drive agent to reallocate the time input from less incentive one to more incentive one. So in general when inputs are substitutes, incentives for any given activity t i can be provided either by rewarding that activity or by reducing its opportunity cost. That is, when some tasks are hard to measure, the only way to increase the incentives for these tasks is to reduce the incentive to other easy to measure tasks. Above arguments are based on the assumption that t>>0 (attention to all tasks in the vector are positive) (the necessary condition for the existence of equation (5)). Lastly, the author discusses the situation that t is 0. In this situation, the cost of providing positive incentives for a small amount of effort has a minimum cost, which is. This value will not be zero, since (Marginal private cost to agent will be greater than zero.) This observation plays a significant role on job design. 3. Allocation Incentives for Effort and Attention In this section, the author tries to analyze realistic observations based on his model. One extra assumption in this section is that an agent can take pleasure in working up to some limit (C (t) for t ). This means that there exists a range of effort allocation among which the cost of agent is indifferent to whatever he chooses to allocate his effort. In other words the agent s effort is homogeneous and can be allocated among the tasks in whichever way the agent likes. Case 1: Missing Incentive Clauses in Contracts Problem: In daily life, it is uncommon for the principal to set incentive clauses in actual contracts. The author cites the example of a contract for home remodeling as an example, in which the incentives for timely completion of construction is seldom seen, even though construction delays will harm the profit of homeowner. Explanation: If one task (quality of construction) is important but hard to measure, then adopting the incentive wage in the other easy-to-measure task (speed of construction) will drive the agent (construction workers) to pay all their effort on the easy-to-measured one and neglect the hardomeasure one, which will eventually reduce the profit of the principal (homeowner). The mathematical proof is in the previous section. Proposition 1: For the home contractor model, the efficient linear compensation rule pays a fixed wage and contains no incentive component ( ), even if the contractor is risk neutral. This proposition argues that piece-rate may be infeasible in the job design, which includes tasks with different degree of difficulty in measurement. 6

7 Case 2: Low-Powered Incentives in Firms Problem: Williamson argues that the incentives offered to employees in firms are generally lowpowered compared to the high-powered incentives offered to independent contractors. Explanation: The ownership of asset is the key point to this problem. Here the author classifies two modes: contracting and employment. The former one denotes the situation where the change in asset value accrues to the agent, while the latter denotes the situation where change in asset values accrues to the principal. The basic idea is that if the principal owns the asset, then he will prefer a conservative strategy (fixed wage), which will protect his net asset value. On the contrary, if the agent owns the asset, then the principal will adopt incentive wage to encourage agents to increase the usage of their own asset, which eventually will bring net receipts to principal. Good evidence are firms like McDonald s and Burger King which are franchises that provide strong incentives. When the principal owns the asset or he prefers a fixed wage payment, an extra assumption is necessary, which says it is highly desirable for principal to induce the agent to devote a positive amount of effort to both tasks. Proposition 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 for which employment contracts are optimal and others for which independent contracting is optimal. If employment contracting is optimal for some fixed parameters ( ), 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. This proposition argues two opinions. First, when the principal wants to achieve profit from hard-to-measure tasks, it is better for him to hire employment with a fixed wage. This idea is actually similar to that of proposition 1. Second, both risk and agent s attitude towards to risk will also influence the wage payment. When the risk is too high or agent is too risk-averse, the agent will not be encouraged by an incentive wage. In other words, the incentive wage will be infeasible. 7

8 4. Limits on Outside Activities With so many distractions in today s world like ipads, cellphones and the internet, how does the principal set constraints on the agent to avoid these distractions? If I were a worker in a call center or a secretary with access to a phone, I could easily make personal phone calls. Thus, the principal needs to set constraints so that the agent will not neglect the principal s task. It is easier for an employer to prohibit all outside activities than it is to monitor them and limit their extent. For example, Holmstrom and Milgrom state, a rule against personal telephone calls during business hours is easier to be controlled than the rule limits the percentage of personal calls to 2%. This makes sense since we should also keep in mind that monitoring these rule limits would result in an additional cost for the principal. Assumptions: 1. Constant returns to time for improving performance measurement and benefit to the principal. 2. Assume the agent has an finite pool K={1,,N} of potential activities. These activities that the principal can control by exclusion could only benefit the agent and not the principal. The agent s personal business can be allowed in a subset of tasks (A K) or excluding them (k A). 3. The principal can control the incentives (commission rate) and the set of allowable personal tasks A K The personal benefits for the agent can be described as follows: c(t,t 1,,t n )= C(t+ k t k )- A v k (t k ) (8) Let t denote the attention the agent devotes to the principal's task and t k the time he devotes to personal activity k. The notation k stands for summation over k in K. The agent s personal benefits are the same as the cost of his total efforts excluding the cost of his personal efforts. The return from personal activity k is measured by the function v k (t k ); these functions are assumed strictly concave with v k (0)=0. Assuming constant returns to effort for profits and improving performance can be described as: B(t,t 1,,t n )= pt, x(t,t 1,,t n )=t+ε (9) To simplify this problem, we will study the principal s problem in two stages. First, we fix α and consider the optimal choice of A, denoted A(α), and then we determine the best α. Stage 1: we fix α, set A(α) based on that, and find the optimal α. αt+ A v k (t k )-C(t+ A t k ) We can see the amount of time the agent spends on all tasks and personal tasks depends on α not on A. 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 v k (t k (α)), while the principal stands to lose pt k (α). This makes sense as your wage remains the same (compensation), the agent will spend more time on personal tasks if the number of possible tasks increases. 8

9 The optimal set of allowable tasks, illustrated in figure 1, is: A(α)={k K v k (t k (α))> pt k (α)}, (12) Explanation of figure1: This graph shows the relationship between the agent s return and his efforts, v 1 and v 2 two curves the returns from two private tasks. It is optimal to allow task 1 but to exclude task 2, because if the agent is given t1(α) amount of time, he will yield more from doing his personal tasks than doing the principal s task. Since v1(t1(α))>pt1(α), but v2(t2(α))<pt2(α). Intuitively, A(α) expands as α increases, because tk(α) is decreasing as vk is strictly concave. As α is raised, the agent will spend less time on private business and more time on principal s tasks. Furthermore, we see that the critical value of α at which private task k will be excluded when vk = pt. Since this follows tk(α) tkhat iff vk (tk hat) α. Proposition 3: Assuming α allows for t(α)>0: (i.) If the average product vk(tk(α))/tk(α) exceeds the marginal product p in the principal s task, it is optimal to let the agent pursue exactly their private activities that belong to A(α) defined in equation (12). (ii.) The higher the agent's marginal reward for performance for the principal s task, the greater is his freedom to pursue his personal activity. Formally, if α α, then A(α) A(α ). (iii.) 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. 9

10 Important notes about the commission rate: 1. The incentives make the agent reallocate his efforts in a way he benefits the agent and principal because the agent receive a better incentive by doing his personal tasks. 2. Responsibility and authority should go hand in hand. More specifically, if the agent is more responsible for his performance, he will get more freedom to do his own business. But this is not the same everywhere. Normally, in North America, chairman owns both responsibility and authority, while CEO have more authority than responsibility. In China, the state control a lot, even though you are a leader in this company, you don t have a lot of authority. In Scandinavian corporations, most people may have shared responsibility and authority. 3. Increasing an agent s commission rate will lead to more personal tasks allowed for the agent. Proposition 4: Assuming t(α)>0: i. From equation (5), we got the best commission rate is given by α=p/[1+rσ2/(dt/da)] where dt/da=1/c + A(α)(1/vk ) ii. If the error in performance measurement (σ2) decreases or the agent becomes less risk averse, the principal can relax the set of allowable tasks and increase the commission rate. Otherwise, if the employee s performance cannot be precisely measured, there will be more restrictions on his activities. iii. Any tasks that are excluded from first best arrangement will also be excluded from the second best. Ending remarks 1. α and A(α) complementary instruments: increasing either leads to an increase in the other. 2. Since performance of bureaucracy is hard to measure, the incentive strategy is not easily applied in this situation, so the most effective way to constrain activities to reduce their freedom. 3. If exclusivity is easier to enforce within firms across firms, then poor sales measurement and employment are positively related. 10

11 5. Allocating Tasks between Two Agents Notation: { } is the vector of total attention to be devoted to the various tasks is the total attention devoted by agent i is the commission paid to agent i for task k. C is the agent s private cost From what we have seen so far, we know that the commission rate,, serves to do the following important items: allocate risk, motivate work, and direct the agent s efforts across tasks. When any of these three objectives are in conflict with each we have a trade-off. Holmstrom and Milgrom point out that things such as job restructuring and relative performance evaluation can help mitigate these problems. Optimal groupings In their model there are two identical agents, indexed: i=1,2, who allocate their attention across a continuum of tasks indexed by. Attention given to task k by agent i is: Assume agents can share a task and that their labor inputs are perfect substitutes Profit is a function of total time vector: { } where In other words, total attention is a sum of the agents efforts devoted to task k. Task k has a performance signal which is given by and only depends on total attention devoted to it. Error variance of task k is and the errors are assumed to be independent. Agent i s total labor input is given by the following function: The agent s private cost is C which is assumed to be differentiable and strictly convex. Holmstrom and Milgrom stress how we are interested in the optimal solution which is nonsymmetric so this means we have to be careful to deal correctly with inherent non-convexities of the problem. The problem is set up in the following manner Subject to: (12) The optimal set of allowable personal tasks, (13) Total labor input, and the incentive constraints: 11

12 if if These can be obtained through first order conditions. We can see that these mean that if the attention to task k is positive then marginal private cost to the agent will be equal to the cost of risk-bearing. Proposition 5: In the model described above, it is never optimal for two agents to be jointly responsible for any task k. For the mathematical proof see page 45 of Holmstrom and Milgrom s paper. Some intuition we should take away from this is that the principal incurs some fixed cost as the agent assumes some fraction of the risk associated with that task and/or that task s measurement. Assigning joint responsibility for any task would incur two fixed costs (one for each agent) for the principal. This is simply unnecessary. Thus, for every one task there should only be one agent assigned to it. Having established this fact, we now turn to how the tasks will be grouped. Holmstrom and Milgrom redefine some variables for ease of interpretation: Now, is the hypothetical commission rate that the principal would need to pay the agent to elicit some level of effort from agent i if he were assigned task k (see equation 17 which is a constraint in the minimization function). Task assignment variable: =1 if agent is assigned to task k, 0 otherwise. Thus, actual commission rate paid to agent i for task k is: if agent i is assigned to task k =0 otherwise Proposition 3 implies that at the optimum,. This means that the task assignment to agent i times the time spent on that assignment will be equal to the attention agent i devotes to k. Either the agent is assigned to that task or not, so this makes sense. Now, the principal s task assignment problem can be stated as it is in equation 16 of the text (page 45) with respect to constraints (17)-(20). Holmstrom and Milgrom give a detailed explanation of how these equations work together so that all feasible assignments yield the same total CE wealth. Since we are interested in the asymmetric case, Holmstrom and Milgrom let. This means that agent 1 devotes less attention to her tasks than agent 2. Relaxing constraint (20), they allow for the task assignment variable to simply be greater than 1 (recall it used to be either 0 or 1) this new equation is (21). Since we have two agents we will have two Lagrange multipliers 12

13 (associated with constraint in equation (18)). After optimizing equation (16) subject to (17) (19) and equation (21), Holmstrom and Milgrom obtain equation (22). and if if We know that given their assumptions:. Equations in (22) characterize the solution to the original problem (the relaxation of (20) to (21) is not very different) and identify the marginal tasks. Marginal task is defined as a task where the advantage of assigning the task to agent 1, in terms of lower risk premium required, is just offset by the higher marginal value of agent 1 s time. There are costs here related to measurement error attached to the task and the amount of time the task requires. For this reason, Holmstrom and Milgrom define the following The noise-to-signal ratio of task k is defined as: Information coefficient is defined as: Let -1 Note that a higher means that it will be harder to measure performance in task k. The equations in (22) can be restated as the following proposition: Proposition 6: Suppose that the two agents devote different amounts of total attention to their tasks (i.e. ). Then, tasks are optimally assigned in this model so that all the hardest-tomonitor tasks are undertaken by agent 1 and all the easiest-to-monitor tasks are undertaken by agent 2. That is, agent 1 is assigned all the tasks k for which, and agent 2 is assigned all those with. This states that tasks which are harder to measure are grouped together, while tasks which are easier to measure are grouped together. This idea is a little simplistic and doesn t encompass the idea of the piece-rate example where we might want the agent to produce quality and quantity items. Holmstrom and Milgrom then go onto giving normalized performance measure and they determine that the normalized commissions must all be equal for an agent. This follows from the fact that all attention to various tasks are perfect substitutes in the cost function (one of the assumptions made earlier in the paper. Even though the two agents in the model are identical ex 13

14 ante, an optimal solution necessarily treats them asymmetrically requiring them to specialize in different tasks. More information is given in the following proposition: Proposition 7: Suppose that the information coefficients are not all identical and consider the variant of program (16)-(20) in which the variables. are added to the list of choice variables. This program has no symmetric optimal solution. There is an optimum at which agent 1 is assigned less strenuous work (, takes responsibility for the hard-to-measure tasks and receives lower normalized commissions The mathematical proof of proposition 7 is found in page 48 of Holmstrom and Milgrom s paper. Since we know agents are not allowed to work on the same task then it makes sense that there is no optimum where there is a symmetric solution. Since we know by proposition 6 that there needs to be grouping involved, it will turn out that one agent will be assigned the harder to measure tasks. By proposition 6, then the other agent will be given the remaining tasks which are easier to measure. Caveats: 1. The assumption that tasks are small and that the principal has perfect freedom to group them together in any way to form a job are two assumptions which are particularly attractive. a. There could be a finite number of tasks (this reverses some of the conclusions that Holmstrom and Milgrom make). b. Tasks cannot always be separated (such as maintaining quality and producing a product). This caricatures the problem of how jobs are constructed. 2. The assumption that the errors of measurement are independent is not valid. a. According to a previous paper by Milgrom he had found that the errors were positively correlated to each other and that separating the tasks allows use of comparative performance evaluation. Also, grouping tasks in which performance is negatively correlated reduces the agent s risk premium. Present model, according to Holmstrom and Milgrom, is incomplete. 3. The attention allocation model is a simplication which forces all activities to be equal subsitutes in the agents cost function. This excludes the possibility that some activities may be complementary. a. There were varying findings regarding complementaries. These distinctions with resepct to attention allocation are things that the theory cannot address. 4. Model does not allow for issues of job rotations (like in real world). The models they have studied assume that the agents focus their attention on the same tasks for all time. 14

15 Conclusion With an incomplete set of performance measures and a complex set of potential responses from the agent, how does one incentivize the agent in a way to attain maximum utility from them? The problem of providing incentives to agents is far more complex than what the standard principal-agent model examines. The performance measure that is used to reward agents could aggregate highly disparate aspects of performance into a one and cause agents to ignore the other aspects of performance that are also essential to the firm. Holmstrom and Milgrom approach the principal-agent model with a more holistic view. They address the fact that to control an agent s performance in one task requires more than just deciding how to pay for their performance. Their analysis extends to change of asset ownership, restrictions on ways a job is conducted, and changing the limits and incentives of other activities. Along with these extensions, the principal-agent can also be related to other papers we have studied. Mortensen and Vishwanath s model on personal contacts proposes that the wage offered by an employer will be higher if the employee applies for the position through a contact. Incorporating this idea into the model would suggest that if an agent were to apply for a job through a contact the principal would offer them a higher wage. In this case the wage will be determined by information signals and contact probability, w(x,p). At the same time the amount of effort that an agent puts into his task changes with the contact probability. If the agent is hired through a contact, the amount of effort he puts into his task will more higher. Another paper we could relate this model to is Akerlof s Market for Lemons which deals with cases of asymmetric information. For a task where performance cannot be measured effectively a situation of asymmetric information arises. Here only the agent is aware of the actual effort he is putting into the task since the principal has no direct way of measuring the performance of the agent. This affects the way incentives are structured for the agent. The concept of outsourcing can also be drawn from this model. In the section on job design, tasks that are easy to measure are assigned to one agent while the difficult to measure tasks are assigned to another agent. In the case of a firm, they separate operations according to how easy it is to measure them and they are then assigned to different departments. The reason sales calls for companies have been outsourced to call centers in countries like India is because the task is to answer a certain number of calls or make a certain amount of calls per day and can be easy measured. Although Holmstrom and Milgrom have extended the standard principal-agent model, there are many more variables that could affect the way incentives are structured for agents by the principal. 15

16 Appendix Derivation of function (5): Differentiate =0 subject to t and set it equal to 0, then Q.E.D Derivation of function (7): Function (5) is the set of, so you can easily derivate function (7) just by open the matrix and with the assumption: is infinite Derivation of (10) and (11) Derive wrt t: α- C (t+ Atk)=0, α= C (t+ Atk) (10) Derive wrt tk: Avk (tk)- C (t+ Atk)=0, Avk (tk)= C (t+ Atk), Using (10) we can simply the equation to get: Avk (tk)= Aα, α=vk (tk) (11) 16