Essay: Agency theory – the Principal-Agent Model

1. Introduction

1.1 Behavioral Economics
Many of the assumptions on which economic models and theories are based are often wrong when we critically look at them from the perspective of social sciences. Thinking about yourself, do you always make fully rational choices that are only self-interested? Are your preferences time-consistent? Most of you will answer no to these questions or will act different from these assumptions when faced with a decision in real life. Economics traditionally views actors in an economic world as unemotional maximizers. Behavioral economics incorporates the facts, models and methods from neighboring sciences to provide a more realistic representation of human behavior into economics. It studies the findings about human behavior and social interaction of mainly psychology, but also sociology, anthropology and biology. Behavioral economics explores the implication that these findings from other sciences have for economic behavior. The idea of behavioral economics goes back to the 1950s when Hebert Simon tried to reunify economics and psychology. Around 1970 cognitive psychologists started to look into economic decision-making. The main essence of behavioral economics is that increasing the realism of the underlying assumption of economic models and theories will improve the field of economics. Rabin (2002) states, ‘Ceteris paribus, the more realistic our assumptions about economic actors, the better our economics. Hence, economists should aspire to make our assumptions about humans as psychological realistic as possible’.
Behavioral economists investigate in which ways the behavior of individuals differs from the assumptions made in the standard model and then show why this deviating behavior matters for economic theory. The last decade, the efforts of behavioral economists to incorporate more realistic notions of human nature into economics have increased remarkably. Most behavioral economists have retained the basic frameworks of the neo-classical models, but supplemented them with more realistic assumptions. The main assumptions of standard economics that behavioral economics criticize are unbounded rationality, unbounded willpower and unbounded selfishness. Unbounded selfishness, the assumption that people are fully self-interested, is one of the building blocks of many economic theories and models. Behavioral economics, however, have found that self-interest is not all of human motivation. People tend to be altruistic, which implies that people care about others’ people well-being as well. Moreover, it turns out that people do not only care about other people, but they also care about the fairness and distribution of resources and even about the intentions and motives of other people’s behavior.
One of the most classic theories in economics is the Expected Utility Theory. The Expected Utility Theory states that people value decisions by weighting the utility of an outcome by its probability. Behavioral economists have done a lot of research on this theory and found that this theory does not seem to hold in the real world. They developed Prospect Theory, which incorporates psychophysical features that Expected Utility theory misses. Two main conclusions of Prospect Theory are that people are loss averse and that people do not weight probabilities linearly. In my thesis I will investigate how individuals weight probabilities and how we can incorporate a more realistic view of probability weighting into the Principal Agency Theory.
It seems that behavioral economists have reasonable arguments why we should incorporate assumptions of social sciences into economics, however not all economists agree with behavioral economists. Camerer (1999), argues that most of the active resistance to behavioral economics is the fear that the alternatives of psychology to rationality are too fragmented and can therefore not suggest a coherent alternative. Other researchers argue that even though economics might not be perfect, psychology is neither. Levin (2009), argues that psychology is mainly focused on the behavior of individuals while economics is focused on the behavior of individuals in groups. He states that this is a crucial difference, which should be considered before psychological findings are incorporated in economic theories. Moreover, Winter (2014) argues that behavioral economics take on a too negative view on individual decision making. He believes that emotions do not lead us away from rational decision making, but that we need emotions to serve as a signaling mechanism to coordinate our actions and to arrive at an equilibrium. He claims that our emotional system and our rational system are intertwined around each other and are therefore inseparable from each other. However, most economist lately have agreed upon the main arguments of behavioral economists and believe that many of the assumptions on which economic theories and models are based are not consistent with reality.

1.2 Agency Theory
In my thesis I will investigate the Principal-Agent Model and I will try to tweak one of the assumptions on which the model is based according to the principles of Behavioral Economics. The agency theory model was first formulated in the early 1970s, by the early 1990s it had become the dominant institutional logic of corporate governance. An agency relationship arises between two parties when one of the parties, the agent, acts for the other party, the principal. The principal-agent framework is thus a method that we use to investigate problems in which the principal hires an agent to perform a certain task for him/her. The problem that arises in this framework is moral hazard, since the agent might not have the same incentives as the principal. Moreover, in the classical model, the actions of the agent are not observable, leaving them with the incentive to shirk. To elaborate on this, the model assumes that an agent chooses an action that influences the outcome, but does not determine the outcome. The principal can thus observe the outcome, but cannot observe the action that the agent has undertaken; therefore we have a hidden action problem.
Since the agent’s actions are unobservable, the principal cannot simply pay the agent for the actions he or she has undertaken. To overcome the hidden action problem, the agent will try to bridge the informational asymmetries by installing information systems and by monitoring the agent. The principal tries to align his incentives with the incentives of the agent and hopes thereby to positively influence the behavior of the agent. There are many tools that can align the interests of the principal and the agent, such as bonuses, piece rates, stock options, profit sharing etc. The problem with an incentive contract is that it will impose risk on the agent since the agent does not have full influence on the outcome.
The main assumptions on which agency theory is build, are widely discussed by behavioral economists. They state, for example, that agents are work averse and self-interested utility maximizers. Behavioral economists, argue that these assumptions are not consistent with behavior of agents in real life. Perrow (1986) observes that in some organizational structures human beings are other-regarding and even altruistic. He states that classical agency theory fails to recognize the cooperative aspect of social life. Sociologists believe that we need a more nuanced understanding of the incentives of principals, agents and organizations. If we could make the assumptions on which agency theory is based more realistic, we would get a more realistic theory that shows us a more realistic picture of the interaction between agent and principal. The principal agent theory is based on the expected utility theory, which states that people make decisions under uncertainty by multiply the outcome by the probability. Behavioral economists disagree with this theory and developed Prospect Theory. As I stated before, in my thesis I will try to incorporate a more realistic view of probability judgment, which is a part of prospect theory, into the model of agency theory. I will start by describing the classical model of agency theory. Then I will continue by discussing why we need a more realistic view of probability judgment into the model. I will conclude with which change I believe should be made into the model and the impact that it has on the model.

2. Agency Theory ‘ The Model
2.1 The standard model.
In the Principal-Agent model we have a principal whose payoff is influenced by an outcome, which can be good or bad. The principal hires an agent who helps him to achieve this outcome. The agent has to decide how much effort he/she will put in. The amount of effort that the agent will put in, influences the outcome. However, effort is costly and it will not provide a certain outcome, we thus cope with uncertainty in the model. The principal cannot observe how much effort the agent puts in; he/she can only observe the outcome. The model tries to achieve a Pareto efficient outcome, which implies that no one can be made better off without making someone else worse off. Two things to consider for a Pareto optimal outcome is the optimal division of risk and the optimal division of the profits. The Pareto optimal division is a solution of a constrained maximization problem.
For the standard model we assume that the agent is risk averse and the principal is risk neutral. The agent does not need to choose effort in this version of the model. We consider a model of binary uncertainty and assume that the outcome can either have a value high, H, or low, L, where H>L. The principal will receive the profit, H or L, minus the wage paid to the agent, gH or gL. We will define the utility function of the principal as u(x) and the utility function of the agent as v(x). Where v'(x) > 0 and v'(x) < 0, the utility function of the agent is thus concave, which implies that he/she is risk averse. The concave utility function has the effect that the agent will gain less utility from an uncertain wage then from a certain wage that is equal to the expected outcome of the uncertain wage. The probability distribution is characterized as P(H) = p and P(L) = 1-p. The maximization problem can now be defined as follow Max pu(H - gH) + (1-p)u(L - gL) (1) s.t. pv(gH) + (1-p)v(gL) ' U0 gH and gL represents the shares that the agent receives when the outcome is High or Low. Since we assumed the situation in which the principal is risk neutral and the agent is risk averse, it is most optimal to set gH = gL, meaning that the agent will get a fixed payment. The constraint represents the minimum utility, the reservation utility (U0), which the agent needs to receive in order to take on the job. If the expected utility of the agent will turn out lower than U0 the agent will refuse the job. In order to solve this maximization problem, the Lagrange method is used. 2.2 The 2x2 Model The 2x2 model is an extension of the standard model in which the agent now has to make the decision which effort level to choose. The agent can choose either a high, eH, or a low, eL, effort level. The effort level influences the outcome, but a high effort level does not necessary imply that the high outcome will be attained. The probability distribution now changes depending on the effort put in and the outcome attained. The probability distribution will be as following P(V=G|eH) = PH, P(V=G|eL) = PL, P(V=L|eH) = 1-PH and P(V=L|eL) = 1-PL. Where PH>PL, meaning that the more effort put in the higher the chance that the high outcome occurs. Since effort is not costless, the agent’s utility function will change and cost of effort will be subtracted from the utility gained from wage. We assume that the cost of low effort is zero. When the agent is risk averse and the principal is risk neutral, the agent will get a fixed wage. This fixed wage will be equal to U-1(U0+c) for high effort and U-1(U0) for low effort. The Pareto optimal outcome depends on which of the following is higher E(V|eH)-U-1(U0+c) or E(V|eL)-U-1(U0). We call this outcome the first best outcome. However, it can only be attained in two special cases. The first case is when effort is observable, since the principal can then set a wage according to the effort the agent has put in. The second case is when the agent is risk neutral, the principal can then transfer all risk towards the agent by giving himself a set wage. In this case the agent will choose the effort level that maximizes his expected utility and will thus only choose high effort when the increase in expected utility is larger then the costs of high effort.
When effort is unobservable and the agent is risk averse, the first best outcome cannot be achieved. Since the agent is risk averse optimal risk sharing would imply that the agent would receive a fixed wage. However, this will not incentivize the agent to put in more effort. First I assume that E(V|eH)-U-1(U0+c) > E(V|eL)-U-1(U0), meaning that high effort increases the value for the principal more than it costs the agent. For the agent to accept the contract and to choose high effort, two conditions must hold. The first condition is incentive compatibility, meaning that the agent will get a higher expected utility by choosing high effort than by choosing low effort. The IC condition can be written as, PHu(WG)+(1-PH)u(WB)-c’PLu(WG)+(1-PL)u(WB). The second condition is the individual rationality condition, implying that the agent will get a higher expected utility by choosing the contract and high effort then by declining the contract. This condition is written as PHu(WG)+(1-PH)u(WB)-c’U0. The principal needs to maximize E(V-W) subject two both constrains. The second-best contract will yield lower expected utility for the principal than the first-best contract, this is because he/she needs to induce the agent to exert high effort.
3. Decision Making Under Uncertainty
3.1 Probability valuation
‘Economists traditionally have assumed that, when faced with uncertainty, people correctly form their subjective probabilistic assessments according to the laws of probability. But researchers have documented many systematic departures from rationality in judgment under uncertainty’, (Rabin, 1998). Faced with decision under uncertainty, people depart from perfect rationality and seem to rely on some heuristic principles, which help them to make the complex task of assessing probabilities to a simpler judgmental operation. This can, however, lead to systematic errors. Hence, it seems that people are not following the principles of probability theory when they judge the likelihood of uncertain events. ‘Some deviations of subjective from objective probability seem reliable, systematic and difficult to eliminate’ (Kahneman & Tversky, 1974). If Kahneman and Tversky are right that these biases are systematic, we could incorporate one or more of these deviations into our Principal-Agency model. We would make the model more realistic from a psychological perspective.
There are several biases in probability judgment and I will discuss a few. One of the biases is the Law of Small Numbers, it implies that people exaggerate the chance that a small probability will occur, contrary they undervalue the probability that an event with a high probability will occur. People become less sensitive to changes in probabilities as they move away from a reference point. In probability distribution 0 and 1 can be seen as the reference points, which implies that the weighting function can be pictured as an inverse S-shape. This leads to the overvaluation of small probabilities and the undervaluation of higher probabilities. In section 4.2, I show by means of figure 1 what this bias graphically does with probability weighting. Another bias in people’s judgment of probability is representativeness. Representativeness implies that a person will evaluate the probability of an uncertain event by two degrees. First, the degree to which it is similar to other known events and second the degree to which it reflects similar features of the process by which it is generated. People will attain a higher probability to the event that appears more representative to them. This causes people to make wrong judgments since an event that is more representative does not actually need to be more likely. Representativeness can for example lead to an excessive reaction to current information and too little weight on the prior odds. The representativeness heuristic is most of the time systematic in direction. In section 4.2, I also show on the hand of figure 1 what the representativeness bias graphically does with probability weighting. Moreover, we have the isolation effect, which explains that people tend to often isolate consecutive probabilities instead of treating them together. This leads to inconsistent preferences when the same choice is presented in different forms. There are many more biases, many of them probably not even known yet. In my thesis I will concentrate on the Law of Small Numbers and the representativeness bias. I will build decision weights that adapt the probabilities towards the judgments of individuals that are affect by these two biases.

3.2 Implication for the Principal Agent Model.
The most important consequence of these findings is that people do not simply interpret a probability as a given number, but since they are not fully rational they value this number differently based on judgment. The beliefs and expectations of an individual about the probability of a certain event will differ from the actual probability of that event. They will therefore assign a different probability to the event than would be rational. Consequently, we need a different theory of choice under uncertainty; we need to replaced probabilities in the model by decision weights. Decision weights take into account the beliefs and expectations that the individual has about the probability of a certain event. Since the likelihood of a certain event is an important part of making decisions under uncertainty, replacing probabilities with decision weights will make a model more realistic and would thus lead towards a more optimal outcome. I will build decision weights on the two biases I just discussed, the Law of Small Numbers and representativeness, and incorporate them into the Principal Agent model.
Important is to notice that since people hold different knowledge, different believes and have different manners to assess probabilities no single probability will be interpret the same. Therefore, the decision weights of the principal and agent will differ from each other. Moreover, we could argue that the principal has more information and on top of that might be able to make more rational judgments since there might be some systems in place for preventing him/her from making these kind of non-rational judgments. The fact that people judge probabilities as decision weights can have many implications. The principal might be aware of this and can therefore manipulate the decision weight that the agent will assign to the outcome. If we take for example the representativeness bias, the principal might manipulate the representativeness of the outcome he prefers, making that outcome seem more salient to the agent will cause the agent to put a higher decision weight on the preferred outcome of the principal. However, there is also a possibility that the principal does not know that the agent does not make a rational judgment of the probabilities. The principal will then make a wrong assessment of the valuation process of the agent and this will then lead to an outcome that is sub-optimal for one or both parties. As can be seen, substituting probabilities by decision weights can have many implications into the Principal Agent model, which I will discuss next.

4. The Adapted Model with Binary Uncertainty, without effort.
4.1. Adapting the standard Principal Agent Model
In its simplest version the Principal Agent model starts with 2 persons, a principal that hires an agent. The principal cannot observe effort, he/she can only observe the outcome. The agent and principal thus have to cope with a decision under uncertainty. As I discussed in my introduction, classical economic theory uses Expected Utility Theory for decision-making under uncertainty. Agency theory is also based on Expected Utility Theory. Behavioral economists, however, invented Prospect Theory. Prospect Theory introduces a more realistic view of decision making under uncertainty and I will change the Principal-Agent model by incorporating some of the findings of this theory. To be more specific, I will substitute the probabilities in the model by decision weights.
When we consider binary uncertainty, as discussed in section 2.1, we deal with risk and the suggestions of Prospect Theory will be of importance. One of the assumptions on which the binary model is build is the probability distribution P(H) = p and P(L) = 1 ‘ p. This assumption is one of the basics of the qualitative law of probability; it is called the conjunction rule. The conjunction rule states that the probability of a conjunction P(A&B), cannot exceed the probabilities off either P(A) or P(B), since it is an extension of these elements. Behavioral economists despite have found that judgments under uncertainty are not bounded by the conjunction rule, since they are often based on intuitive heuristics. A violation of the conjunction rule is called the conjunction fallacy and often occurs because of the representativeness bias.
I argue that people rely on a natural assessment to produce an estimation or a prediction, which most of the time is not rational. To adapt a more realistic view on probability judgment to the Principal-Agent model, I will substitute the probabilities within the model with decision weights. There are many biases in probability judgment and to make decision weights as realistic as possible, all these biases would need to be incorporated. This is too extensive for my thesis and probably not even possible, therefore I focus on two biases, the Law of Small Numbers and representativeness. Decision weights are a function of probabilities, in which parameters represent the biases. The probabilities will, thus, be affected by the degree of these biases.

4.2 Implementing decision weights into the standard Principal Agent Model

Lattimore, Baker and Witte (1992) suggest that decision weights should be specified by the following formula:
w(p)= (??p^??)/(??p^??+'(1-p)’^?? ) (2)
The parameter ?? represents the Law of Small Numbers, it establishes whether ‘small’ probabilities will be under- or over-weighted. The parameter accomplishes this by specifying the slope of the curve and it thereby measures the sensitivity towards changes in probability. ?? can take on values between 0 and 1, so 0”??1. The smaller ??, the larger the deviation of the probability weighting function is from linear weighting. When ?? = 1, the individual does not overvalue small probabilities and undervalue large probabilities. The parameter ?? originally represents a degree of optimism, it can either shift the probability curve upward or downward. Originally it can take on values below or above 1, presenting an optimistic or pessimistic view on the probability of the outcome. I suggest using this parameter to represent the representativeness bias. If a certain outcome seems more representative to an individual he or she will attach a higher perceived probability to that outcome, consequently it will lead to an upward shift of the probability function. Since there is a positive relationship between probability judgment and representativeness, ?? can only take on the value 1 or higher, so ‘?? 1. If a certain outcome does not seem representative to the individual, ?? will be equal to 1. Moreover, if ?? = 1 the conjunction rule will hold. If ??>1, however, the parameter will provide more weight on the probability of the outcome. Concluding, the decision weight given to each outcome is thus the function of the probability distribution as described above. When ??=??=1, Expected Utility Theory holds. However, most individual behavior deviates from this situation.
Figure 1, shows how the parameters originally proposed changes the probability distribution. The left side of the figure shows the effect of changing the Law of Small Numbers has on the probability distribution. The lower, ??, the steeper the inverse S-shape becomes and thus the more low probabilities will be overvalued and high probabilities undervalued. The right side shows the representativeness bias, keeping ?? constant. In my case, ?? would only lead to an upward shift of the probability distribution. Causing the point at which p = 0.5 to only shift above the linear line.

Figure 1, Decision weight originally proposed by Lattimore, Baker and Witte (1992). Notice that in my case ??>1, and not below 1.
Retained from Gonzalez, R., & Wu, G. (1999). On the shape of the probability weighting function. Cognitive psychology, 38(1), 129-166.

The decision weights I will use are defined as follows
w(pH) = (??_(i,H) p^??)/(‘??_(i,H) p’^??+'(1-p)’^?? ) w(pL) =(??_(i,L) ‘(1-p)’^??)/(??_(i,L) ‘(1-p)’^??+p^?? ) (3)
Where 0 ‘ ‘?? 1, ‘?? 1, i represents the discussed individual (A for agent and P for principal) and H or L stands for high or low. For example ??_(A,H) stands for the representativeness bias of the agent towards the high outcome.
Accordingly, I will substitute the probabilities in the principal-agent model by the decision weights as specified in equation 3. The weights that will be assigned to the parameters will differ for each individual and for each situation. Some people might be very sensitive to the Law of Small probabilities and therefore will have a low value for ?? in there decision weight equation. The weight that is attached to ??, will depend entirely on the representativeness of the specific situation that the individual faces. In section 2.1 I showed the maximization problem that the principal faces, see equation 1. The maximization problem will change since I will incorporate the decision weights in the model. In section 3.2, I discussed the different scenarios that could occur. For now I will discuss the scenario in which the principal is rational and does not use decision weights, the agent however judges probabilities according to judgmental heuristics. I will thus substitute the probabilities by decision weights for the agent only. The model can now be defined as follows
Max pu(H – gH) + (1-p)u(L – gL) (4)
s.t. ((??_(A,H) p^??)/(‘??_(A,H) p’^??+'(1-p)’^?? ))v(gH) + ((??_(A,L) ‘(1-p)’^??)/(??_(A,L) ‘(1-p)’^??+p^?? ))v(gL) ‘ U0
The optimization problem can be solved by using the Lagrangian method. The first step in this method is rewriting the constraint to
U0 – ((??_(A,H) p^??)/(‘??_(A,H) p’^??+'(1-p)’^?? ))v(gH) + ((??_(A,L) ‘(1-p)’^??)/(??_(A,L) ‘(1-p)’^??+p^?? ))v(gL) = 0 (5)
We can then rewrite the problem according to the Langrangian method as follows
pu(H – gH) + (1-p)u(L – gL) + ??(U0 – ((??_(A,H) p^??)/(‘??_(A,H) p’^??+'(1-p)’^?? ))v(gH) – ((??_(A,L) ‘(1-p)’^??)/(??_(A,L) ‘(1-p)’^??+p^?? ))v(gL)) (6)
The first order conditions will now be
‘L/ ‘gH = pu'(H-gH) – ??((??_(A,H) p^??)/(‘??_(A,H) p’^??+'(1-p)’^?? ))v'(gH) = 0 (7)
‘L/ ‘gL = (1-p)u'(L-gL) ‘ ??((??_(A,L) ‘(1-p)’^??)/(‘??_(A,L) (1-p)’^??+p^?? ))v'(gL) = 0 (8)
‘L/ ‘?? = U0 – ((??_(A,H) p^??)/(‘??_(A,H) p’^??+'(1-p)’^?? ))v(gH) – ((??_(A,L) ‘(1-p)’^??)/(‘??_(A,L) (1-p)’^??+p^?? ))v(gL) (9)

4.3 Exploring the effect of decision weights on the standard Principal Agent model
4.3.1 Effect on the participation constraint
Now that we implemented the decision weights into the Principal Agent model we can explore the effect that it has on the model. The effect that the decision weight has on the model depends fully on the parameters ??, the representativeness bias, and ??, the Law of Small Numbers. If we assume that ??=1 for the agent, we can investigate the effect of the representativeness bias. When one of the outcomes seems more representative to the agent, ?? will be greater than 1. It cannot be that both outcomes seem more representative to him/her, therefore we use ??H and ??L, representing either the high outcome or the low outcome. If the agent feels that the high outcome is more representative, ??H > 1 and ??L = 1. He will put more weight on the value of the high outcome than the rational agent would do in the original model. Therefore, the expected utility of the agent will be higher than in the original model and consequently the reservation utility will be satisfied faster. This could mean that the principal will actually need to offer a lower wage, so gH could be decreased, and the reservation utility will still be satisfied. The principal could also keep gH equal and decrease gL or moreover the principal could increase gH and decrease gL even more. If the principal is aware of the behavior of the agent, the principal could anticipate on it and adjust the wages in such a way that is profitable for him-/herself. The optimal choice depends on the condition of optimal risk sharing, as I will calculate later on. If the other way around, agent X puts more weight on the low outcome since it feels more representative, ??H = 1 and ??L > 1. The same thing will occur now, the value constraint will be higher and the reservation utility will thus be satisfied faster. The principal could thus offer a different waging scheme. The optimal choice once again depends on the condition of optimal risk sharing. Concluding, if the agent puts more weight on one of the outcomes, the value of the participation constraint will be higher than in the original model. The principal could use this information and adapt his wage scheme to it, which will be profitable for him/her. However, the effect will be smaller for the low outcome than for the high outcome, since gH>gL.
We move on by investigating the situation in which ??H=??L=1 and ?? < 1 for the agent, so we examine the effect that the Law of Small Numbers has on the new model. The Law of Small Numbers implies that ?? < 1 and will have the effect that more weight will be put on the smallest probability but less weight on the high probability. The lower ?? and the larger the difference between the probabilities the larger the effect of the Law of Small Numbers is. When the agent is risk averse and the principal is risk neutral we saw in the original model that gH = gL. There are two different scenarios in this case. The first one is when the high outcome is more likely than the low outcome, the Law of Small Numbers will then imply that the high outcome will be undervalued and the low outcome will be overvalued. The second scenario is when the high outcome is less likely then the low outcome, the Law of Small Number will then imply that the high outcome will be overvalued and the low outcome undervalued. Since w(pL) + w(pH) = 1 in this case and gH=gL, the change in probabilities does not affect the expected utility of the agent. In both scenarios the expected utility of the agent does not change and the principal does not need to change the wage and can keep gH equal to gL. However, the condition of optimal risk sharing will show if this is still the optimal choice of the wages. The total effect that the decision weight has on the adjusted model depends on the specific situation. If for example ??=0.75, the high outcome has a higher probability than the low outcome, p > 1-p, and on top of that the agent feels like the high outcome is more representative, so ??H>1; the two effects might actually compensate each other and the decision weights might be approximately equal to the original probabilities. The principal might not change anything in the wage scheme in this case. However, if for example ??=0.75, the low outcome has a higher probability than the high outcome, 1-p > p, and the agent feels like the high outcome is more representative, so ??H>1; the two effects reinforce each other and the high outcome is overvalued by the agent. Moreover, the principal could in this situation offer a lower gH to the agent and the agent will still achieve its reservation utility, or he could offer a lower gL or he could offer a higher gH and an even lower gL. The optimal choice depends on the condition of optimal risk sharing.

4.3.2 Effect on the condition of optimal risk sharing
The condition of optimal risk sharing can be calculated by the first order conditions. We could rewrite both equation 7 and equation 8 in the form ?? is equal to. ?? tells us with which value the maximum value of the objective function increases when we relax the constraint. Equation 7 and 8 are rewritten as follows
pu'(H-gH) – ??((??_(A,H) p^??)/(‘??_(A,H) p’^??+'(1-p)’^?? ))v'(gH) = 0 (7)
‘ ?? = (pu'(H-gH) )/(((??_(A,H) p^??)/(‘??_(A,H) p’^??+'(1-p)’^?? ))v'(gH) ) (9)
‘L/ ‘gL = (1-p)u'(L-gL) ‘ ??((??_(A,L) ‘(1-p)’^??)/(‘??_(A,L) (1-p)’^??+p^?? ))v'(gL) = 0 (8)
‘ ?? = ((1-p)u'(L-gL) )/(((??_(A,L) ‘(1-p)’^??)/(‘??_(A,L) (1-p)’^??+p^?? ))v'(gL)) (10)
Since ?? is equal to ??, we will get the following condition of optimal risk sharing
(pu'(H-g_H))/(((??_(i,H) p^??)/(‘??_(i,H) p’^??+(1-p)^?? ))v'(g_H)) = ((1-p)u'(L-g_L))/(((??_(i,L) (1-p)^??)/(‘??_(i,L) (1-p)’^??+p^?? ))v'(g_L)) (11)
The left hand side of the equation tells us how the utility of the principal changes when gH increases by 1 unit relative to the change in the utility of the agent with 1 unit increase in gH. The right hand side tells us the same but then for gL. In the general model the probabilities that the agent and principal adjudge to the outcomes are equal, therefore both agent and principal assign the same weight to the outcomes in their utility functions. Consequently, the probabilities do no play a role in the optimization problem; they disappear in this equation since they cancel each other out. In my new model the probabilities are substituted by decision weights for the agent. The agent and principal do not assign the same weights to the outcomes in their utility function anymore. Therefore, the probabilities do play a role in the optimal solution. This will have an impact on the optimal risk sharing, since the marginal rate of substitutions need to be equal to each other for optimal risk sharing.
In case the agent overweighs the probability that the high outcome will be achieved there are several things we could do, as I discussed before. The most optimal choice depends on the optimal risk sharing condition. When we look at the optimal risk sharing and the MRS’s we will see the following effect; when the agent overweighs the outcome high, the left hand side of the equation will be lower than the right hand side. This occurs since the denominator will be higher than the numerator on the left side of the equation. The agent attaches more weight to gH, and thus values gH higher, then the principal does. Therefore, it is optimal for the principal to give the agent more gH and less gL. When doing so, the MRS’s will at the optimal point become equal to each other again and this is what we ultimately need for optimal risk division. More gH will cause a decrease in v'(gH) since the utility curve of the agent is convex, he/she is risk averse, u'(H-gH) will stay the same since the principal is risk neutral and thus has a linear utility curve. For the same reason less gL will cause an increase in v'(gL) and u'(L-gL) will stay the same. Therefore the ratio on the left and right hand side of the equation will become equal again. Noteworthy is that in the original model gH=gL since the agent is risk averse and the principal is risk neutral, meaning that it is optimal for the principal to carry all risk. In my new model this is not the case anymore, the agent will get more gH and less gL than in the original case. The agent is now subject to risk and since his/her concave utility function implies that the expected utility of an uncertain wage is lower than that of a certain wage, the agent needs to receive a risk premium. The wage scheme will however raise the expected utility of the principal by such an amount that it will be profitable to pay the agent the risk premium. Moreover, Gonzalez and Wu (1999) argue that when w(p) + w(1-p) > 1, the individual tends to embrace risk in that specific situation. Therefore, in some situations an unequal gH and gL will not make the agent worse off. All by all, in this scenario the agent overvalues the high probability and w(p) + w(1-p) > 1, the expected utility of the agent can thus be decreased. The optimal way to do this is thus by increasing gH and decreasing gL in such a way that the total sum of wages is lower than in the original case, since the overall expected utility of the agent is then still equal to the reservation utility. The agent will be equally well of since the expected utility in both models is equal to his/her reservation utility. The principal however is better off since the total sum of wages is lower than in the original case and this will increase his expected utility.
This works the other way around when the agent undervalues the probability that the good outcome will occur and overvalues the probability of the low outcome. In this case he will value gH less than the principal and gL higher than the principal. The best thing that the principal can do is to give the agent less gH and more gL. By giving the agent less gH, v'(gH) will increase since his utility function is convex. On the other hand, v'(gL) will decrease since we increase gL. The ratio on the left side of the equation will then decrease and the ratio on the right hand side of the equation will increase. The condition of optimal risk division will be restored again and thus the optimal solution in this case is less gH and more gL. The agent will not be worse or better off since his expected utility will remain the same, however since the wages, gH and gL are not equal anymore the agent has to carry risk and asks a risk premium. The principal however has different beliefs about the probabilities of the outcomes high and low and this makes it profitable in terms of expected utility for the principal to increase gL and decrease gH. Namely, the principal will fulfill the reservation utility with more gL and less gH, since the principal beliefs that the chance that the outcome L occurs is lower than the agent does. The principal is therefore willing to pay this risk premium.

4.3.3 Total effect on the maximization problem
The new version of the general model that I suggest thus substitutes probabilities by decisions weights for the agent. The decision weights depend on two variables that represent the Law of Small Numbers and the representativeness bias and can have different effects on the model. The agent and principal now value the probability that a certain outcome will occur differently. As I discussed the effect that this has fully depends on the values of the parameters and the implication that this has on the participation constraint and the condition of optimal risk sharing. Moreover, the condition of optimal risk sharing ultimately decides what the most optimal wage scheme is and whether this is profitable or unprofitable for the principal.
When the principal is fully aware of the behavior of the agent he/she could try to manipulate the decision weight of the agent for his/her own sake. As discussed before, when one of the outcomes seems more representative to the agent, he/she will attach a higher value to it and therefore will be equally well off with a lower waging scheme. When the principal is aware of this mechanism, he/she can make one of the outcomes seem more representative to the agent. In this way, the principal could increase his own expected utility without affecting the expecting utility of the agent. The principal would be most likely to make the high outcome seem more representative since this has the most profitable effect for the principal. Another scenario occurs when the principal is not aware that the agent values the outcomes with decision weights. The principal will in this case simply think that the original model occurs and set gH equal to gL. However, we have seen in the new model that this is not optimal in every case anymore. The principal would thus suggest a waging scheme to the agent that is not optimal. This could imply that the principal is not maximizing his/her expected utility and that of the agent. Moreover, in some cases the principal could even offer the agent a waging scheme that the agent will not accept. This shows the importance of the new model, the more realistic view helps to attain the optimal situation; the maximization of the utility of both the agent and principal.

5. The Adapted 2×2 Model
5.1 Implementing decision weights into the 2×2 model
In section 2.2 I discussed the 2×2 model, which is an extension of the general model. I will focus on the general case with the second best result, since this is the most likely case. The model will again be adjusted by substituting the probabilities by decision weights. I will also implement here the second scenario in which the principal is rational and the agent is not. Since the probabilities in the 2×2 model change, the decision weights will look different as well. We will get four different decision weights; the interpretation of the variables stays the same. ?? presents the representativeness of the outcome, so it depends on G or B. Consequently, ?? has the same value in PH as in PL, ??H, and ?? has the same value in (1-PH) and (1-PL), ??L. ?? represents the Law of Small Numbers; this variable stays the same across the different outcomes. The decision weights are as follows
w(PH) = (??_(i,H) ‘P_H’^??)/(‘??_(i,H) P_H’^??+'(1-P_H)’^?? ) (12)
w(PL) =(??_(i,H) ‘P_L’^??)/(??_(i,H) ‘P_L’^??+'(1-P_L)’^?? ) (13)
w(1-PH) = (??_(i,L) ‘(1-P_H)’^??)/(??_(i,L) ‘(1-P_H)’^??+’P_H’^?? ) (14)
w(1-PL) =(??_(i,L) ‘(1-P_L)’^??)/(??_(i,L) ‘(1-P_L)’^??+’P_L’^?? ) (15)
The principal tries to maximize E(V-W) subject to the two conditions explained in section 2. There is one critical assumption made; the expected outcome minus wage is higher when high effort is put in than with low effort. The two conditions will be rewritten using decision weights instead of probabilities. The incentive compatibility condition will change to
((??_(i,H) ‘P_H’^??)/(‘??_(i,H) P_H’^??+'(1-P_H)’^?? ))v(WG)+( (??_(i,L) ‘(1-P_H)’^??)/(??_(i,L) ‘(1-P_H)’^??+’P_H’^?? ))v(WB)-c ‘ ((??_(i,H) ‘P_L’^??)/(??_(i,H) ‘P_L’^??+'(1-P_L)’^?? ))v(WG)+( (??_(i,L) ‘(1-P_L)’^??)/(??_(i,L) ‘(1-P_L)’^??+’P_L’^?? ))v(WB) (16)
The second condition, the individual rationality condition, will change to
((??_(i,H) ‘P_H’^??)/(‘??_(i,H) P_H’^??+'(1-P_H)’^?? ))v(WG)+( (??_(i,L) ‘(1-P_H)’^??)/(??_(i,L) ‘(1-P_H)’^??+’P_H’^?? ))v(WB)-c’U0 (17)

5.2 Exploring the effect of decision weights on the 2×2 model
I will concentrate on the effect that the implementation of decision weights has on the IC condition. The decision weight is, as discussed, based on two biases and I will first investigate the effect of both biases separately. When ??H=??L=1 and ??<1 we can investigate the Law of Small Numbers, it has the effect that it brings the probabilities of both outcomes actually closer to each other. The greater probability will be undervalued and the lower probability will be overvalued. This effect is the greatest when the probabilities lie further apart. The incentive compatibility condition states that the expected utility with high effort minus the costs should be higher than the expected utility with low effort. When the probabilities will come closer to each other because of the Law of Small Numbers, this will actually be harder to achieve. Since pH > pL the effect of the Law of Small Numbers will be greater on the left hand side of the equation than on the right hand side. Therefore, the value of the left hand side will become closer to the value of the right hand side and it will thus be harder to achieve the incentive compatibility condition.
Moreover, the Law of Small Numbers has another effect. As you can see in figure 1, when the probabilities are very small or very high, the curve is almost flat. Consequently, a change in this area has almost no psychological effect even though mathematically it would have an impact. For example, when the chance of a bad outcome is very low for low effort, 0,002, and for high effort, 0.001. Since both probabilities are so small the individual will attach the same weight to both probabilities, even though the bad outcome for low effort is actually twice as high as for high effort. This would imply that the value on the left hand side and right hand side of the equation become closer to each other and it would be harder to fulfill the condition. Hence, both effects of the Law of Small Numbers have the same effect. The two effects together, would thus enhance each other.
When ??=1 and ??H>1, ??L=1 o r ??L>1, ??H=1 we can investigate the effect of the representativeness bias. When ??H>1 and ??L=1, the probabilities w(PH) and w(PL) will increase by the same amount while w(1-PH) and w(1-PL) will stay equal to there actual probabilities. Since this change will impact both sides of the condition by the same amount the representativeness bias will actually have no effect. The exact same situation occurs when ??L>1, ??H=1. Hence, only the Law of Small Numbers will have an effect on the incentive compatibility condition and once ??<1 it will negatively effect the condition. The Law of Small Numbers makes it harder for the principal to provide the agent with an incentive to choose high effort and this influences the tradeoff between efficient choice of action and efficient risk sharing. The principal needs to pay the agent a higher gH and lower gL than in the original model. Hence, it might not be profitable anymore for the principal to provide this incentive. However, if we analyze the IR condition it also suggests that gH and gL should depart from each other under the Law of Small Numbers. Therefore, it might still be optimal for the principal to induce the agent to provide high effort. 6. Conclusion Behavioral economics stresses the importance of incorporating the findings of other social sciences into economic theories en models. One of these finding is that when faced with a decision under uncertainty people depart from perfect rationality. They seem to rely on judgmental heuristics to assess the probability of an event. I incorporated this finding into the Principal-Agent model by substituting probabilities for decision weights. Decision weights change probabilities based on several heuristics. I focused on two of them, the Law of Small Numbers and representativeness, and implemented them in the form of parameters into the decision weights. The substitution of probabilities by decision weights for the agent into the general Principal-Agent model has several implications. The most important effect is that the agent and principal now weight the outcomes high and low differently. Both variables in the decision weight, representing the Law of Small Numbers and representativeness, can have a different impact. The effects can work in the same direction or in the opposite. Depending on the value of both parameters, the participation constraint and the condition of optimal risk sharing change. When the principal is aware of the decision weights that the agent assigns to the outcomes, he/she can adapt his waging scheme to it. Interestingly enough, in most situations it is now not optimal anymore to set gH equal to gL even though the agent is risk averse and the principal risk neutral. The principal can increase his/her expected utility by adapting his waging scheme based on the ratio's in the condition of optimal risk sharing. When the principal would not be aware of these decision weights, we would get a suboptimal wage scheme. This shows the importance of the adaption to the Principal Agent Model.

Source: Essay UK - http://turkiyegoz.com/essays/economics/essay-agency-theory-the-principal-agent-model/


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