The adverse e↵ect of fines on behavior
How psychological cost influences behavior
Susanna Teulings
Student number: 6137369 Date of final version: July 16, 2015 Master’s programme: Econometrics
Specialisation: Mathematical economics
Supervisor: mw. dr. M. M. J. W. van Rooij Second reader: Prof. dr. C. H. Hommes
Contents
1 Introduction 1
2 Literature Review 4
2.1 Adverse e↵ects of incentives in the literature . . . 4
2.2 The theory behind crowding-out . . . 5
2.3 Social-responsible according to Brekke et al. (2003) . . . 6
2.4 Social-acceptance by Lin and Yang (2006) . . . 7
3 The Model 10 3.1 Constructing the model . . . 10
3.2 Calibrating cost functions . . . 15
3.3 Extension: The dynamic model . . . 17
3.4 Extension: Internal mixed costs . . . 18
4 Results 19 4.1 The e↵ect of a fine estimated . . . 19
4.2 Extensions to the model estimated . . . 31
4.3 Comparing the results with reality . . . 37
5 Conclusion 41
A Proof of topological chaos 44
Bibliography 45
Introduction
At the heart of economic theory lies the relative price e↵ect, this e↵ect says that when prices increase, the demand decreases and vice versa. So for example when a good becomes more expensive we want less of it, and when a type of behavior becomes more expensive we will exhibit that behavior less (and of course vice versa). This implies that it is possible to change people’s behavior by rewarding or fining them and thus changing the profitability of their behavior. As early as in the 1970s this idea was questioned. Titmuss (as mentioned by Frey and Oberholzer-Gee, 1997) stated that the amount and quality of blood donation would actually decrease when a financial compensation was given to those who donate. He argued that the compensation would crowd out feelings of civic duty, and thus undermine people’s reason to donate. In other words, unlike classic economic theory predicts, a monetary incentive would have an adverse e↵ect on blood donation.
This theory generated quite some noise, since it contradicts the axioms of the Von Neumann-Morgenstern utility function (von Neumann and Neumann-Morgenstern, 1953), which is essential to eco-nomic theory. To see this, consider that people who donate blood prefer this above not donating blood, else they would not have done so in the first place. It is also assumed that people prefer receiving money over not receiving money. But somehow receiving money and doing the action is preferred less than not receiving the money and not taking the action. This suggests that people do not optimize their utility as the rational choice theory predicts. Although an inter-esting thought by Titmuss there was not enough empirical evidence to back up this claim, so the discussion quickly died out.
Since then however, there is substantial empirical evidence that indeed shows that monetary incentives can lead to unexpected adverse e↵ects. But how, when and why this phenomenon manifests itself is still not clear cut. Di↵erent models have been proposed, most of them based on the crowding-out theory stated by Titmuss (Frey and Jegen, 2001). This theory builds on the idea that there are two types of motivation, intrinsic and extrinsic. Intrinsic motivation comes from oneself and can be something like altruism, civic duty or pride. Or as Deci (1971, p.105) states ”One is said to be intrinsically motivated to perform an activity when one receives no apparent reward except the activity itself”. Extrinsic motivation on the other hand comes from
CHAPTER 1. INTRODUCTION 2 external incentives, such as financial rewards, and is thus the driving force behind the homo economicus who exhibits only rational behavior. The crowding-out theory proposes that when extrinsic motivation is increased, by for example placing a fine on bad behavior, this crowds-out the intrinsic motivation. So although extrinsic motivation increases, intrinsic motivation decreases, it is therefore possible that total motivation is decreasing when extrinsic motivation is increasing. This means that it is possible that an increase in extrinsic motivation has a negative e↵ect on behavior. However, this only happens when the decrease in intrinsic motivation is larger than the increase in extrinsic motivation. With the existing literature it is still hard to predict when the net e↵ect is negative and when it is positive. This is mainly because most studies focus on a specific situation and build a model for that purpose only. Therefore they lose generality and thus applicability in other cases. An other reason for this uncertainty is that most existing models are not very explicit, making it hard to estimate what the e↵ects of an incentive might be. In conclusion, the phenomenon of adverse e↵ects is still not well understood. Missing is a general model that is able to predict in which situations an adverse e↵ect is likely and in which cases it is not. In this thesis I therefore aim to build this general model to explain adverse e↵ect of monetary incentives.
Two interesting models have already been formulated by Brekke et al. (2003) and Lin and Yang (2006). Both papers make use of intrinsic motivation to (partially) explain the e↵ect a fine has on behavior, but they both use a di↵erent psychological base. Brekke et al. (2003) uses the idea that people experience psychological costs when their behavior di↵ers from the socially desired behavior, people thus have altruistic motivations. This model is called the social-responsible model. While Lin and Yang (2006) uses the idea that people experience psychological costs when their behavior di↵ers from the behavior of their peers, people are thus concerned with their image. This model is called the social-acceptance model. Both models have an interesting approach, since both the altruism motivator from Brekke et al. (2003) and the image motivator from Lin and Yang (2006) have shown to be the two most important intrinsic motivators (Carpenter and Myers, 2010; B´enabou and Tirole, 2006). But both models only use one of the two motivators, and are therefore still missing an important dimension. Both models are also still very vague, making it hard to predict what for e↵ect a fine will have on behavior. Combining the two models to form a general model could give a good approximation of reality, which makes it possible to make those predictions.
The objective of this thesis will be to show under which circumstances a fine leads to a negative e↵ect and under which it can actually stimulate the desired behavior1. In other words,
this thesis aims to predict how a fine influences peoples e↵ort in a society for a broad spectrum of situations. To do so a general model is built with di↵erent parameters, such as the importance of the public good and the value of spare time. These parameters can be set to di↵erent values to describe di↵erent situations, so that it can be predicted in which situations an increase in the
1Notice that I am only focusing on the e↵ect a fine has on behavior, and not the e↵ect a monetary reward has.
This is done to limit the scope of this thesis. However, in theory my model could be easily adapted to describe a reward situation.
fine leads to an increase in bad behavior. The results found by my model can subsequently be compared to results found by empirical research. To summarize, in this thesis I built an agent based model to answer the question in which situations an increase in the fine leads to adverse e↵ects, so that more people behave badly.
The remainder of this thesis is organized as follows. In Chapter 2 a literature review is given, here I will give an overview of papers in which a negative e↵ect of monetary incentives has been shown, examine more closely the psychological theory behind the crowding-out e↵ect and finally I will elaborate on the papers of Brekke et al. (2003) and Lin and Yang (2006), which will form the basis of this thesis. In Chapter 3 I will first discuss how I combined both models into one single model after which I will show what a possible calibrated cost function could look like. In the remainder of Chapter 3 I will give two possible extensions to my model. For this model and the proposed extensions I show, in Chapter 4 how to estimate the e↵ect a fine has on behavior in di↵erent situations. Next I will compare these results with the results found in the research of other scholars. At last Chapter 5 will conclude my findings.
Chapter 2
Literature Review
In this chapter I will elaborate more on the existing literature on the adverse e↵ect of a monetary incentive on behavior. To start I will first briefly state the results di↵erent scholars have found on the e↵ects of monetary incentives. This is done to show the variety in situations and results that have been found by scholars. Most of these scholars explain the found adverse e↵ects by using the Crowding-out theory, so after the di↵erent papers have been discussed, I will look closely to why intrinsic motivation is crowded out by extrinsic motivation according to the literature. Lastly I will more extensively discuss the models used by Lin and Yang (2006) and Brekke et al. (2003), since my model is based on these two models.
2.1
Adverse e↵ects of incentives in the literature
As mentioned before, several scholars have showed that monetary incentives can indeed have a negative e↵ect on behavior. When examining all these papers the di↵erences in situations and results stands out. For example Frey and Oberholzer-Gee (1997) research what happens when people have to vote for the placement of a nuclear waste facility in their neighborhood. Although such a project can have negative side e↵ects and people would not want it near their houses, fifty percent of the population still would vote in favor of the waste facility, according to the authors, out of civic duty. However, when a compensation is promised to all the residents near the facility only twenty-five percent is still willing to vote in favor of the facility, crowding-out the feeling of civic duty.
Such a negative e↵ect is also found in the research of Gneezy and Rustichini (2000), but in their study the incentive is not in the form of a compensation but in the form of a fine. They show that when a fine is introduced in a daycare center for every parent who picks up their kid to late, the number of parents coming to late actually increases. Even more remarkable is the fact that this higher level of late comers is sustained even after the removal of the fine. Showing that temporary incentives can have a lasting e↵ect.
This last result, however, is not found by Holm˚asa et al. (2010). They investigate how many days a patient has to wait to be transfered from a hospital to a long-term care institution.
They investigate this for two contrasting situations, when there is a fine put on the institution for every extra night the patient has to stay in the hospital versus when there is no fine for such behavior. They find that the hospital that gives a fine to the long-term care institutions has a significantly longer length of stay of patients than the hospital that does not fine the institutions, suggesting again a negative e↵ect of monetary incentives. However when some of the institutions changed hospitals the time they needed to accommodate new patients also changed. Institutions that now belong to the hospital with a fine increased their time as the theory would suggest, but institutions that now belong to the hospital without a fine actually decreased their time needed, which is contrary to the result of Gneezy and Rustichini (2000) who say that dropping the fine will not change behavior back to normal.
Although these previous papers suggest that monetary incentives always have adverse e↵ect on behavior, there are also case studies that do show that incentives can have a positive e↵ect. For example Strahilevitz (2000) shows that when in San Diego the (faster) carpool lane was also made available for solo drivers willing to pay a fee, so that time saving became commodified, the number of carpoolers actually increased. The idea behind this is that people felt that they were better of getting for free where others had to pay, therefore increasing the number of people who perform the desired action, namely carpooling. Such a scheme can be seen as a tax exemption for people who exhibit the desired behavior, and is thus an example where monetary incentives can have a positive e↵ect.
The above examples show that monetary incentives can indeed sometimes lead to adverse e↵ects. Although, as the last example has shown, there are also cases where a monetary incentive has the desired e↵ect. For even more examples of when a monetary incentive has a negative e↵ect I would like to direct the reader to the paper of Frey and Jegen (2001).
2.2
The theory behind crowding-out
The adverse e↵ect described above is often, as mentioned before, ascribed to the crowding-out e↵ect. There are multiple, psychological, theories why people lose their intrinsic motivation when confronted with an extrinsic one. In this section I will shortly mention some of these theories, however it will be beyond the scope of this paper to completely cover the basis of such a crowding-out.
According to Frey and Jegen (2001, p. 594) there are two psychological processes that lead to crowding-out of intrinsic motivation. The first one is the feeling of impaired self-determination, the extrinsic incentive is perceived as controlling and thus intrinsic motivation has to decline so as to not feel overjustified. The second one is the feeling of impaired self-esteem, people get the idea that their intrinsic motivation is not acknowledged and therefore their involvement is not valued as high as it should be, reducing their e↵ort.
Both processes can be seen as special cases of equity theory (Adams, 1996) which states that for someone to have the idea that he participates in a fair transaction his perceived inputs should be equal to his perceived outputs, compared with his peers. So when someone feels like
CHAPTER 2. LITERATURE REVIEW 6 his intrinsic motivation is not acknowledged because he gets the same output as someone who does not seem to have intrinsic motivation and thus has fewer input, he will perceive it as an unfair transaction and therefore lowers his own input.
This equity theory gives an idea why intrinsic motivation decreases when an extrinsic mo-tivation is added. However it still does not give an explanation how this a↵ects behavior and how this should be modeled. Numerous scholars have solved this by adding psychological costs to someones payo↵/utility function, so people incur psychological costs when their behavior is, for example, not altruistic enough, giving them an intrinsic motivation to be altruistic.
2.3
Social-responsible according to Brekke et al. (2003)
My model is based on two already existing models, namely that of Brekke et al. (2003) and of Lin and Yang (2006). In this section I will discuss the model of Brekke et al. (2003), the model of Lin and Yang (2006) follows in the next section. Recall that Brekke et al. (2003) built a model on the idea that people are social-responsible, in other words there intrinsic motivator is based on altruism, which is one of the two main intrinsic motivators. Like most scholars they model this motivator by constructing a utility function that has an extra psychological cost component. There (undefined) utility function looks as follows:
U = u(I, L, G, C) (2.1)
Here I represents someones disposable income, which is in most studies assumed equal to consumption, L is the value of leisure, G the value of public goods and C the psychological costs, which gives the intrinsic motivation1.
Brekke et al. (2003) let the psychological costs be based on altruism motivators by assuming that people incur psychological costs when their behavior is not socially ideal, this results in a built-in incentive in their utility function to act socially ideal. People determine the socially ideal level of e↵ort by calculating, for each situation, how everyone should behave for total welfare to be maximal. Deviating from this ideal level leads to psychological costs, this leads to the following costs function:
Ci= a(✏i ✏⇤i)2, a > 0 (2.2)
In this equation ✏i is the level of e↵ort performed by an individual, ✏⇤i the ideal level of e↵ort,
and a is a scalar which measures the sensitivity to psychological costs.
Next, Brekke et al. (2003) assume that the proceeds of a fine are directly invested in the public good. Hence, when an incentive is being emplaced on certain behavior it does not only change the disposable income of an individual but also the money available for the public good. Therefore an introduction of a fine changes total welfare and thus the psychological costs. What the e↵ect is of the introduction of a fine depends on how total welfare changes. Brekke et al.
1The attentive reader might notice that throughout this thesis I sometimes use slightly di↵erent notation then
(2003) come to the conclusion that when a symbolic fine, one that is not high enough to cover the cost the public good faces from bad behavior, is introduced, e↵ort weakly increases. However when a fine is no longer perceived as symbolic, e↵ort decreases to zero, since the fine can cover the cost to the public good and thus the ideal behavior can now be bad behavior.
This might seem to be in contrast with the crowding-out theory, since a symbolic fee does not lead to less e↵ort, but Brekke et al. (2003) explain this by saying that apparently a symbolic fee is not perceived as controlling because the responsibility for the public good still lays with the people and not the organization. Therefore there occurs no real feelings of impaired self-esteem or self-determination. When the fine is no longer perceived as symbolic the responsibility shifts to the organization, making the fine controlling and thus leading to a reduction in e↵ort. It is important to note however, that the perception of the fee is important and not the actual amount, meaning that a fee that is in fact symbolic can still lead to a decrease of e↵ort because it is not perceived as such.
2.4
Social-acceptance by Lin and Yang (2006)
The other paper I base my model on is from Lin and Yang (2006). Just like Brekke et al. (2003) they use an extra psychological cost component in an utility function to model intrinsic motivation. However, their intrinsic motivation is not altruism but image, which means that people are motivated to behave in a certain manner in order to be perceived as ”good” people. This image motivator is, according to Carpenter and Myers (2010) more susceptible to crowding out by extrinsic motivation than for example the altruism motivator. Lin and Yang (2006) implement this by saying that the psychological costs of not adhering to social norms diminishes when a monetary incentive is introduced because of the crowding-out e↵ect. They also assume that social norms erode when less people adhere to them. If a lot of people behave badly it reflects less on their image when others do so as well, therefore they will experience less psychological costs. These assumptions gives them the following cost function:
C = · c(f, x); cf =
@c
@f < 0, cx @c
@x < 0 (2.3) In this model f stands for the fine and x for the percentage of people violating the social norm. is an idiosyncratic sensitivity indicator, which says that some people are more sensitive for a certain social norm than others. This sensitivity indicator is uniformly distributed between 0 and 1 over the population.
Lin and Yang (2006) next calculate ˆ, this is the for which people are indi↵erent between good and bad behavior, so that Ubad = Ugood. This means that for everyone with a smaller
than ˆ, bad behavior yields a higher utility than good behavior, hence they exhibit bad behavior. Note that since is assumed to be uniformly distributed with a support on [0,1], ˆ is equal to the percentage of bad behavior, so ˆ = x, which can be used to find the steady states to which the percentage of bad behavior converges. To calculate ˆ Lin and Yang (2006) specify
CHAPTER 2. LITERATURE REVIEW 8 the following utility function:
U = I + vt C (2.4)
In this I is again income, which decreases by a fine, f , when someone exhibits bad behavior. v is the value of leisure and t is the amount of free time, which is 0 when e↵ort is high and 1 when e↵ort is low. Note that Lin and Yang (2006) do not take into account the e↵ect a public good might have on the utility and behavior of an individual.
Next calculating ˆ and its derivatives to the percentage of bad behavior, x, and the fine, f , gives the following formula’s:
ˆ = v f c(f, x) ˆf = @ˆ @f = 1 c (v f )cf c2 R 0 ˆx= @ˆ @x = (v f )cx c2 > 0 ˆxx= @ˆx @x = (vf )cxx c2 + 2(vf )c2 x c3 R 0 (2.5)
Lin and Yang (2006) then assume that ˆ, for which people are indi↵erent between good and bad behavior, has the properties of a production function, with x an input variable. This means that ˆ first has increasing marginal returns until a tipping point after which it has diminishing returns. Hence, ˆ is an inversed tangent shaped function with two stable and one unstable steady state, as shown in Figure 2.1 where ˆ is plotted. Note that this figure, as given by Lin and Yang (2006), does not stem from any specified formulas but is just an abstract representation of their model.
Furthermore Lin and Yang (2006) assume that if the fine, f , is small the derivative of ˆ to f is greater than zero. This means that instating a small fine will lead to an upward shift of the function, possibly even leading to the disappearance of the lower two steady states, leaving only the upper stable steady state, as can be seen in Figure 2.1, where the dotted line represents ˆ when there is no fine and the solid line when there is a small fine instated.
However, Lin and Yang (2006) also show that if the fine, f , is sufficiently large the derivative of ˆ to f is smaller than zero. This means that the graph shifts downward when the fine is increased by a large amount. Which ultimately can lead to the disappearance of the upper two steady states, only leaving the lower stable steady state.
Summarizing the results, the model of Lin and Yang (2006) has two stable steady states, one with a low violation of the norm and one with a high violation, and also one non-stable steady state in between the stable steady states when there is no fine instated. When introducing a fine smaller than a certain value f1 the graph shift upwards, increasing the number of people
exhibiting bad behavior for each steady state. At some point the graph shifts up by so much that the lower two steady states disappear, only leaving the higher level of bad behavior as a stable steady state, therefore drastically increasing the number of people that exhibit bad behavior. Even more important however is that when the fine is revoked the steady states go
Figure 2.1: ˆ = c(f,x)v f as assumed by Lin and Yang (2006, p. 205)
back to the position they had before the fine was introduced. But since society is now in the higher steady state the number of people exhibiting bad behavior stays high. That means that the only way to get the percentage of bad behavior back down is to actually further increase the fine above f1, shifting the graph downwards until the upper two steady states disappear, in
order for society to shift back to the lower stable steady state. Put di↵erently, the only way to make people perform better is to set the fine so high that the costs of bad behavior o↵set the benefits.
Chapter 3
The Model
3.1
Constructing the model
The models of Brekke et al. (2003) and Lin and Yang (2006), as described in the previous chapter are both based on common psychological e↵ects (Carpenter and Myers, 2010), namely altruism and image motivators respectively. Since it is reasonably to assume that di↵erent people have di↵erent motivations, it would be interesting to see what happens when both models are combined. This requires the model of Lin and Yang (2006) to be more precisely specified and the model of Brekke et al. (2003) to become compatible with the model of Lin and Yang (2006). When this is done it is possible to construct a model in which some people are of the social-responsible type, and thus have altruism as a motivator, and some people are of the social-acceptance type, and thus have image as a motivator. With this model a prediction can be made in which situations a fine has a positive e↵ect and in which situations it has a negative e↵ect.
E↵ort
First o↵, it is important to define e↵ort more precisely since it is the input variable of the model. E↵ort is the use of energy to do something, and in this model putting in e↵ort is always a personal burden because it takes up someones free time. However the public good benefits from e↵ort, therefore society benefits from e↵ort, which is thus labeled as good behavior. For simplicity reason I will assume that the level of e↵ort is not an interval of possible values, so that it is impossible to put in just a little bit of e↵ort. But rather that someone either exhibits full e↵ort (good behavior), in that case ✏ is 1 or no e↵ort at all (bad behavior), in that case ✏ is 0. This gives the advantage that the utility function for bad and good behavior become relatively simple and are thus easier to compare.
Another advantage of making e↵ort a dummy variable is that the average level of e↵ort, ¯✏, is equal to 1 x, where x is the percentage of bad behavior. So that I can use the same idea as in the model of Lin and Yang (2006) in which ˆ, the sensitivity indicator for which people are indi↵erent between good and bad behavior, is equal to the percentage of bad behavior, x.
Variable Description Domain
✏ Level of e↵ort 0[ 1
¯
✏ Average level of e↵ort [0, 1] x Percentage of bad behavior (x = 1 ¯✏) [0, 1]
f Fine level [0, v]
v Value of spare time [0, 1] Value of the public good [0, 1] m Diminishing return factor of e↵ort > 1
n Population size ! 1
p Percentage of people of social-acceptance type [0, 1] Table 3.1: Variable description
Utility
Next, recall that both original models are also based on an utility function with an extra psychological cost component to model the intrinsic motivation (see (2.1) and (2.4)). The utility functions are unfortunately not identical, so a common utility function first needs to be defined. Since Brekke et al. (2003) did not concretely specify their utility function, I will use the utility function given by Lin and Yang (2006), see (2.4). This utility function is still lacking a public good component, so this needs to be added. To do so, I use the properties given by Brekke et al. (2003) for the value of the public good. They assume that, just like most production functions, e↵ort has a diminishing e↵ect on the value of the public good. They also assume that every fine that is paid directly flows back into the public good. This leads to the following utility function:
U = i f (1 ✏) + v(1 ✏) + (ggov+ f (1 ¯✏) + ¯✏
1 2m✏¯
2) C (3.1)
The first part of this utility function consists of personal benefits/costs such as income and free time, the middle part is the value of the public good and the last component is the psychological cost denoted with C. The personal benefits consists of someones income, i, the fine, f , that has to be paid when e↵ort, ✏, is low, and the value of the extra amount of free time that is gained from having a low level of e↵ort, v. The value of the public good is determined by , which gives the importance of a public good, the monetary input to the public good, which is the input of the government ggovand the average payed fines (f (1 ¯✏)), and the average level of
e↵ort ¯✏ 2m1 ¯✏2 that is put into the public good. Here m is just a constant larger than 1, which establishes that the average level of e↵ort has a diminishing return, which is in line with most production functions and the assumptions of Brekke et al. (2003). For a complete overview of all the parameters used in this thesis see Table 3.1.
Unfortunately an increase in m does not only mean that an extra doses of e↵ort adds more to the public good, but it also means that the value of the public good increases (especially for high levels of e↵ort). A function that would not have this disadvantages is ¯✏m1, however this
CHAPTER 3. THE MODEL 12 function has an even bigger disadvantage, namely that the first derivative is very complicated, making it difficult to estimate the optimal level of e↵ort and to analyze the model. Therefore I have chosen to work with ¯✏ 2m1 ¯✏2, despite its disadvantage.
Social-acceptance type
Now having defined the basis of my model it is possible to construct the model when everyone is of the social-acceptance type (and thus motivated by image). Recall from Chapter 2.4 that Lin and Yang (2006) calculate ˆ, which is the sensitivity indicator for which people are indi↵erent between good and bad behavior. Also recall that ˆ is equal to the percentage of bad behavior, x, because the sensitivity indicator, is uniformly distributed between 0 and 1. Lin and Yang (2006) show that for there (simpler) utility function ˆ is equal to c(f,x)v f .
Calculating ˆ for the more extensive utility function I have defined in (3.1) actually still gives the same equality. To see this, note that ¯✏1, which gives the average level of e↵ort when
ones own e↵ort is high, is equal to ¯✏0+n1. Where n is the population size and ¯✏0 the average
level of e↵ort when your e↵ort is low. Now lets set the utility function for good behavior equal to the utility function of bad behavior.
v f + (f· x0 f· x1+ 1 x0 1 + x1 1 2m(1 x0) 2+ 1 2m(1 x1) 2) C accept= 0 (3.2)
When the population size goes to infinity ¯✏1 converges to ¯✏0 (and so does x1 to x0). This means
that, in the limit of n going to infinity, the e↵ect the public good has on ˆ goes to zero, so that ˆ is equal to v f
Caccept, which is the same formula given by Lin and Yang (2006) in (2.5).
Social-responsible
Next lets look at the social-responsible model. Brekke et al. (2003) state that when people give less e↵ort than the morally ideal amount, they experience psychological costs, as can be seen in (2.2). So before determining their behavior people first calculate the ideal level of e↵ort. Brekke et al. (2003) state that people calculate this by looking at which level of e↵ort maximizes total welfare when everyone is equal, so that everyone will make this calculation and act upon it. However in a society where not everyone is social-responsible not everyone will do this and thus assuming that everyone is equal is even more far stretched. Therefore in my model people calculate which level of their e↵ort maximizes total welfare, W , given the average level of e↵ort of the other people in society, ¯✏. Social-responsible people thus maximize the following equation to get their optimal level of e↵ort.
W = f (1 ✏) + v(1 ✏) + n ✓ f·⇣1 ¯✏ ✏ n ⌘ + ¯✏ + ✏ n 1 2m ⇣ ¯ ✏ + ✏ n ⌘2◆ ✏opt= n(¯✏ + m (v f ) + m(f 1)) (3.3)
Here n is the population size, which I assume goes to infinity1. This means that the calculated optimal level of e↵ort goes from minus infinity to plus infinity, while real e↵ort can only be zero
or one. This is solved by rescaling the optimal level of e↵ort. ✏⇤ = 1
1 + e ✏opt (3.4)
Recall that Brekke et al. (2003), using the optimal level of e↵ort, then say that the psycho-logical cost function has the following form:
Cresp = a· (✏ ✏⇤)2 (3.5)
This cost function has a couple of problems. First and foremost exhibiting to much e↵ort actually leads to psychological costs. In a model where everyone either exhibits an e↵ort of 0 or 1 this means that almost no one will actually perform good behavior, since e↵ort leads to a reduction in the normal utility components and too high an e↵ort also leads to an increase in the psychological costs. This is not realistic. Just like it is not realistic that people will feel bad about themselves when performing too good. For these reasons I believe it to be more realistic when the psychological cost function looks like the equation given in (3.6), in the sense that the psychological costs are zero when to much e↵ort is exhibited.
Secondly it is important to note that in the paper of Brekke et al. (2003) a is not an idiosyncratic sensitivity indicator that is uniformly distributed between 0 and 1. Rather, it is an arbitrary constant (that can be larger than 1). This means that it is not possible to construct an ˆa which is equal to x, as is done in the social-acceptance model. So to be able to solve my model a is changed from a constant to an idiosyncratic sensitivity indicator which is uniformly distributed between 0 and 1. However, the constant that was in the model from Brekke et al. (2003) also had its advantages, making it possible to rescale the model. Because this is not longer possible with the new defined a, I also add a constant, c, to manually boost up the psychological costs.
Cresp= 8 < : a· c(✏ ✏⇤)2 if ✏✏⇤ 0 if ✏ > ✏⇤ (3.6)
With this cost function I am then able to calculate for which a people are indi↵erent between good and bad behavior, defined as ˆa, so that it can be seen which steady states exist. Just like in the social-acceptance case, I set the utility when exhibiting good behavior equal to the utility when exhibiting bad behavior. Again I assume that the population goes to infinity which leads to the equation for ˆa below. Note that ˆa is, just like with ˆ is the social-acceptance case, is equal to the percentage of bad behavior, x.
ˆ
a = v f
ce⇤2 (3.7)
However, with this function a new problem arises. When the (rescaled) optimal level of e↵ort, ✏⇤, is equal to zero ˆa goes to infinity. Now recall that everyone with an a lower than ˆa exhibits bad behavior and that a only takes values between 0 and 1. This means that when ˆa exceeds one, everyone will exhibit bad behavior, because everyone has an a lower than ˆa. For calculation
CHAPTER 3. THE MODEL 14 advantages I therefore set ˆa equal to one, when it exceeds this value. Because in the limit ✏⇤ is equal to 0 when ✏opt is smaller than zero, so that ˆa goes to infinity, I can set ˆa equal to one when ✏opt < 0 for a good approximation.
ˆ a = 8 < : v f c if ✏opt 0 1 if ✏opt < 0 (3.8)
Next I calculate the derivatives of ˆa, given that ✏opt 0, and of ✏opt itself to the fine, f , and the percentage of bad behavior, x, to get a better sense of how the model will look like.
ˆ af = 1 c 0 ˆ ax= 0 ✏optf = nm ( 1) 0 ✏optx = n > 0 (3.9)
Note that when ✏opt decreases, the likelihood of ✏opt < 0 increases and thus does the likelihood
that ˆa = 1 and everyone behaves badly. So when the percentage of bad behavior, x, increases, the changes of a social-responsible person exhibiting bad behavior decreases, even though it does not e↵ect ˆa directly. When the fine increases, this leads to a decrease in ˆa, and thus to less bad behavior, until the fine increases so much that ✏opt becomes smaller than zero, after which the percentage of bad behavior becomes equal to 1. This e↵ect is also found by Brekke et al. (2003) in their original model.
A model for the mixed society
At this point I have built two compatible models. The reason for doing so is to be able to combine them in one model and make it possible for a society to have di↵erent motivators for di↵erent people. To do so I assume that a percentage, p, is of the social-acceptance type, and thus has an image motivator, while 1 p percent is of the social-responsible type and has an altruism motivator. The total percentage of bad behavior, x, can thus be described as a linear combination of the percentage of bad behavior by the social-acceptance type, xaccept, and the
percentage of bad behavior by the social-responsible type, xresp.
x = p· xaccept+ (1 p)· xresp (3.10)
Recall that and a are both uniformly distributed over 0 and 1 and therefore ˆ and ˆa are equal to the percentage of bad behavior of their respective population group. So xaccept and
xrespare equal to ˆ and ˆa respectively. Since ˆ and ˆa both (indirectly) depend on the percentage
of people misbehaving in the whole society, this results in an expression which on both sides depends on x.
Note that in reality someone can not know beforehand what the average level of bad behavior is going to be. Therefore I assume that people use the bad behavior of last period as their
reference point to base their behavior upon for this period, this gives the following model: xt= p· ˆ(xt 1, f ) + (1 p)· ˆa(xt 1, ✏⇤) (3.11)
Solving this model for xt= xt 1 gives all the possible steady states, which can be used to find
out to which percentages of bad behavior society converges.
3.2
Calibrating cost functions
In the previous section I have given a general model for the psychological cost functions of both types of people. However, to be able to perform calculations and generate output it is necessary to give a possible specified cost function for both types. These models are not based on any data, since that is outside the scope of this thesis, instead they chosen in such a manner that the models have the same properties as in the articles on which they are based.
For the social-acceptance type this means that I sought for a function that satisfies all the properties that are given in (2.3) and (2.5) and that is shaped like an inverse tangent, giving two stable and one unstable steady state. This function also should, for some values of the parameters v, and m, shift upward for a small fine, thereby losing it’s lower stable (and unstable) steady state. An example of such a function is:
Caccept= 1.5· 1 + e 12(x 0.22) e3f cf = 4.5· 1 + e 12(x 0.22) e3f < 0 cx = 18· e 12(x 0.22) e3f < 0 ˆ = e3f 1 + e 12(x 0.22) · (v f ) ˆf = (3(v f ) 1)· e3f 1 + e 12(x 0.22) > 0 i↵ v f > 1 3 (3.12)
The social-responsible model is already more defined than the social-acceptance model, and only the constant, c, needs to be specified. In the rest of this paper I will assume that c = 3. Again this constant is arbitrarily chosen, but has been chosen in such a manner that the e↵ort levels are realistically low. For a good approximation of the true value of the constant real world data should be used.
Now lets assume that the value of spare time, v, is 0.7, the importance of the public good, , is 0.9 and the diminishing return factor, m, is 3, so that ˆ and ˆa can be plotted for di↵erent values of the fine, f , as shown in Figure 3.1. It can be seen that for the social-acceptance type the graph indeed shifts upwards, so that the two upper steady state disappear, just like Lin and Yang (2006) assume in their paper. For the social responsible model it can be seen that
CHAPTER 3. THE MODEL 16
(a) Social-acceptance (b) Social-responsible
Figure 3.1: ˆ and ˆa for f = 0 (blue), f = 0.175 (green), f = 0.35 (red), f = 0.525 (light blue) and f = 0.7 (purple)
when a fine increases the graph shifts downwards, leading to a decline in bad behavior. However when an increase in good behavior leads to a decrease in total welfare, so that shirking becomes the ideal level of e↵ort, it can be seen that the graph jumps upward to ˆa = 1, in other words everyone shirks. Again this is in line with the results found by Brekke et al. (2003).
It also shows that in a society with only social-responsible type of people, no stable steady states exists, instead a stable 2-cycle exists. This is logical since if there is a high level of shirkers, putting in e↵ort has a great e↵ect on total welfare. This leads to a large portion of people switching from not giving any e↵ort to putting in e↵ort. But then the average level of e↵ort becomes high, and total welfare is more served with someone paying the fine and enjoying his free time than putting in e↵ort. So next period again everyone switches, leading to a never ending cycle. This result is not found by Brekke et al. (2003), since they assume that people calculate the ideal level of behavior as a level everyone should adhere to in order to maximize total welfare, and not as a level that is ideal given the average level of behavior of others. Therefore the average level of behavior does not influence an individuals behavior in the model of Brekke et al. (2003) and thus people do not change their behavior when the average level of e↵ort changes.
Now that there is a specified ˆ and ˆa, they can be implemented in the mixed society model given in (3.11). This results in a model with one variable, namely x, and five parameters, namely f , v, , m and p2. xt= 8 > < > : p· (v f ) e3f 1.5(1+e 12(xt 1 .22)) + (1 p) if ✏opt < 0 p· (v f ) e3f 1.5(1+e 12(xt 1 .22)) + 1 3(1 p)· (v f ) if ✏opt > 0 with ✏opt= n⇣1 x +m (v f ) + m(f 1) ⌘ (3.13)
3.3
Extension: The dynamic model
The model described in the previous sections is very static, since the population does not change over time nor does their decision making process. This assumption is made to simplify the model and make it easier to study the e↵ects of an increase of the fine. This assumption can be made because it seems not logical for individuals to easily switch back and forth between di↵erent type of psychological costs, because psychological costs are not in their control. However, although people can not freely switch between the two types of psychological costs, it is not completely out of the question that the population composition, p, changes over time. In reality we observe that young people are often more idealistic and politically left wing oriented, and thus are more interested in that everyone is well o↵. While older people tend to lean more to the right. This would suggest that it is likely in this model that young people are more likely to be of the social-responsible type, while older people are more likely to be of the social-acceptance type.
An other reason why the composition of society might change over time is that people who are happy are more likely to live longer and reproduce more. This means that the type of person that has a higher utility, and is thus happier, is less likely to die and more likely to reproduce and get the same type of children. For this reason it is likely that the number of people, that on average have a higher utility, will increase. Note that such a change is only small from period to period, but can have large e↵ects in the long run.
Because the composition of society can change over time, it is useful to extent my model, to a model where the percentage of people of the social-acceptance type, p, changes over time. To model this I first assume that peoples sensitivity to psychological costs, and a, do not change over time, even when their type of costs do change. Now lets assume that the psychological costs of the acceptance type are larger than that of the responsible type and the social-responsible type thus have a higher utility, so that 1 p is expected to grow over time. If this is the case then ˆ < ˆa3, in other words the social-responsible type of people are still behaving badly
while the social-acceptance type of people with the same level of sensitivity to psychological costs are not. This means that everyone with a sensitivity indicator, , smaller than ˆa would benefit from switching, since behaving badly becomes then (more) profitable thus increasing the utility. Note that everyone with a or a larger than ˆa would be indi↵erent to switching, since they will exhibit good behavior anyway and thus do not experience psychological costs.
In mathematical economics it is often assumed that of those people who are indi↵erent only half of them switches. Applying this to my model will result in an unsolvable model. To see this recall that a en are uniformly distributed so that ˆa and ˆ are equal to the percentage of bad behavior in their respective groups. Now when a larger percentage of people with a low than a high switches, and a are both no longer uniform distributed. This makes it extremely difficult to calculate the percentage of bad behavior. However, when the same percentage of people switch in every cohort of , then both and a stay uniformly distributed and thus ˆ and ˆa stay equal to the percentage of bad behavior in their respective groups. For
3To see this recall that both ˆ and ˆa are of the shape v f
CHAPTER 3. THE MODEL 18 this reason I will assume that when the social-acceptance type people have larger psychological costs a percentage z will switch from the social-acceptance type to the social-responsible type and no one will switch the other way around, even if they are indi↵erent. This means that the percentage of people who are of the social-acceptance type, p, will decrease with z% and that the percentage of people who are of the social-responsible type, 1 p, increases with z% of p. This gives an equation of p over time, given in (3.14). Combining this with the static model of the mixed society model, (3.13), gives a dynamic model where both the percentage of bad behavior,x, and the percentage of social-acceptance type, p, change over time.
pt= 8 > > > < > > > : (1 z)pt 1 if ˆa > ˆ (1 z)pt 1+ 12z if ˆa = ˆ pt 1+ z(1 pt 1) if ˆa < ˆ (3.14)
3.4
Extension: Internal mixed costs
An other extension that is likely to bring my model closer to reality is if there is no longer a strict devision between the two types of people, but everyone is a little bit of both. So people are both susceptible to image motivators and altruism motivators. To model such a thing it is assumed that people have the same division between image and altruism motivators. This is of course a simplification of reality, but this makes the model a lot simpler to work with. The model is, however, postulated in such a manner that it is not difficult to extend the model further so people do have a di↵erent division between the two motivators. The general cost function then has to following outlook, where Caccept and Crespons are equal to their respective
calibrated cost function, as formulated in Chapter 3.2:
C = µ(q· Caccpet+ (1 q·Crespons)) (3.15)
Here q gives the division between the two motivators, so that when q is high, image is a more important motivator and when q is low, altruism is a more important motivator. µ is again an idiosyncratic sensitivity indicator which is uniformly distributed, just like and a in the main model, in order to get ˆµ, which is again the µ for people that are indi↵erent between good and bad behavior, equal to the percentage of people who behave badly. Calculating ˆµ, for a population that goes to infinity, gives the following model:
ˆ µ = 8 < : v f q·Caccept if ✏ opt 0 v f q·Caccept+3·(1 q) if ✏ opt> 0 (3.16)
Results
4.1
The e↵ect of a fine estimated
Now that a model has been postulated in (3.13), the next step is analyzing it. Recall that for this thesis I am interested in how the percentage of bad behavior changes in a society when a fine is introduced or raised. To analysis this, I will first investigate under which conditions the model is stable and when more complicated dynamics occur. Next I will investigate how an increase in the fine influences the percentage of bad behavior when the model is stable. To do so the e↵ect of the parameters on the model are analyzed, in order to be able to determine in which situations an increase of a fine has its desired e↵ects. Next the e↵ect of the parameters on the stability of the model is analyzed. After which the parameters are set to specific values to analyze the model and its dynamics further. Finally the possible bifurcation diagrams of the model are investigated.
Stability of the model
Normally stability of the model is investigated by first calculating the steady states. Unfortu-nately this model can not be solved grammatically for xt= xt 1, so that the percentage of bad
behavior is constant over time, so a less straight forward way has to be used to determine the existence of steady states.
First it is important to note that xt is bounded. This can intuitively be understand by the
fact that it is not possible to have more than 100% and less than 0% of bad behavior in a society. More formally the bounded character of xt can be seen by realizing that xt is composed of a
linear combination of two bounded functions. Therefore xtis also bounded by the lowest lower
bound and the highest upper bound of the two functions it is a linear combination of. Because xt is bounded there exists at least one steady state (that is not necessarily stable) and xt can
not diverge. So the model then either converges to a steady state or to a k-cycle or the model gives chaos1.
1For now I will not distinguish between whether the model gives a stable k-cycle or if the dynamic is chaotic.
I will only focus on if the model converges to a stable steady state or if there is complicated dynamics.
CHAPTER 4. RESULTS 20 If the percentage of bad behavior in period t, xt, is not only bounded but also increasing
for every value of xt 1, then there always exists at least one stable steady state. To see this
remember that a steady state is stable when the derivative is (in absolute value) smaller than 1, in other words an increasing graph crosses the x = y line at the steady state from above (Hommes, 2013). Now suppose that l is the lower bound of the model and u is the upper bound. Then it is automatically the case that for xt 1= l the graph is above (or equal to) the x = y
line, while for xt 1 = u the graph is below (or equal to) the x = y line. Since the graph is
continuous and increasing, the graph should then have crossed the x = y line from above at least once, thus leading to at least one stable steady state.
So to investigate whether xtis always increasing with xt 1, the derivative of xtis calculated.
Note that the derivative is equal to p·(v f)cx
C2
accept > 0 for all values of the optimal level of e↵ort,
✏opt, except when ✏opt= 0 and the derivative is equal to minus infinity. The percentage of bad behavior, x, for which ✏opt = 0 is called xopt and has the following function:
xopt = 1 +m
(v f ) + mf m (4.1)
The fact that there is a point on the graph for which the derivative goes to minus infinity means that there is not always a stable steady state. This is only the case when there is just one steady state and it is equal to xopt. To see this consider first that there is also a steady state with a lower percentage of bad behavior than xopt, this steady state is then always stable, since the
graph is above the x = y line for an xt 1equal to the lower bound and thus the first steady state
then cuts the x = y line from above. Note that if there is a steady state smaller than xopt and equal to xopt, that there is also a third (unstable) steady steady in between the two. Secondly
consider that there exists a steady state equal to and larger than xopt. Since the steady state at xopt has cut the x = y line from above, the graph is immediately after xopt below the x = y
line. An extra steady state thus means that the graph crosses the x = y line from below and the steady state is thus unstable. However, the graph is now above the x = y line and since the upper bound is always below the x = y line, the graph has to cut the x = y line again and this results in a stable steady state. This means that the model either has one, three or five steady states. These results can be summarized as follows:
• if x0t(x⇤) < 1 the steady state is stable, more steady state might exist
• if x0t(x⇤) > 1 the steady state is unstable, but there also exists at least one stable steady
state
• if x0t(x⇤)! 1 the steady state is unstable and there exists complicated dynamics if this
is the only steady state
Note that when there is complicated dynamics there automatically is an alteration between ✏opt > 0 and ✏opt < 0, so between the first and second part of the model, since x0t > 0 every where except for the drop at ✏opt= 0.
The e↵ects of a fine in case of stability
Now lets first assume that there is a stable steady state, and there is thus no alteration between ✏opt> 0 and ✏opt < 0. To see what the e↵ect is of a fine on the level of e↵ort, the derivative of xt to f is calculated. If the derivative is smaller than zero an increase of the fine will lead to
more e↵ort. Deriving the first and second derivative of xtgives us:
xf = 8 > < > : e3f 1.5(1+e 12(x .22))(3(v f ) 1) p if ✏ opt< 0 e3f 1.5(1+e 12(x .22))(3(v f ) 1) p 1 3(1 p) if ✏opt> 0 xf f = p· 2e3f 1 + e 12(x .22)(3(v f ) 2) 8 ✏ opt (4.2)
Two situations can be distinguished in which the derivative of xt is smaller than zero.
1. 3(v f ) 1 < 0, v f < 13 2. ✏opt > 0 and e3f 1.5(1+e 12(x .22))(3(v f ) 1) p < 1 3(1 p), p < Caccpet 9(v f ) 3+Caccept
These two scenarios can be explained by the fact that for xf to be smaller than zero the benefits
of shirking, namely v f , have to decrease more than the psychological cost. Recall from the properties of the calibrated cost function in (3.12) that for the social-acceptance type this is the case when v f < 13, since then ˆf < 0. But when v f > 13 the psychological cost decrease
faster than the benefits, making it more lucrative to exhibit bad behavior. Note, however, that for a di↵erent calibrated cost function ˆf may be equal to zero for a di↵erent value of v f .
For the social-responsible type the psychological cost are not (directly) a↵ected by f . So when ✏opt> 0 the derivative is always smaller than zero, as can be seen in (3.9), and when ✏opt< 0 an
increase in the fine does not have an e↵ect. Therefore when v f < 13 an increase in f always leads to a decrease in bad behavior, since both ˆa and ˆ decrease in value. But when v f > 13 it can only lead to a decrease in bad behavior when ✏opt > 0, so that ˆa decreases, and ˆ does not increase by much, so that every increase by ˆ is counterbalanced by a decrease in ˆa.
I am interested in which situations an increase in the fine leads to more bad behavior and in which it does not. Therefore I need to know how the parameters influence the e↵ect a fine has, and thus how parameters influence the sign of the derivative of x to the fine, f . To analyze for which parameters the first scenario, for which the derivative of xf is smaller than zero, occurs
is pretty straight forward. The second scenario, however, is more difficult. To see in which situations this scenario takes place let’s look at how all the parameters influence the e↵ect a fine has on the average level of behavior. In table 4.1 all the e↵ects that I describe below are summarized. I will start with the most important one, namely the fine itself. In (3.9) it is shown that when the fine increases, the optimal level of e↵ort, ✏opt, also increases. This means that it becomes more likely that ✏opt is larger than zero, and thus more likely for scenario 2 to occur. The reason that a larger fine leads to a higher level of optimal behavior is that when the fine increases, the personal gain of behaving badly decreases, since a (larger) fine has to be paid. For that reason total welfare, which is also partly people their own welfare, increases
CHAPTER 4. RESULTS 22 when you behave better. An attentive reader might note that on the other hand the value of the public good also increases with a larger fine making it not as bad to shirk. Although this is true this has a smaller e↵ect than the e↵ect on the personal gain, since the value someone attaches to the public good, , is smaller than one. As stated before, a higher optimal level of e↵ort leads to higher psychological costs and thus it becomes more likely that an increase in the fine, f , does not scare people o↵ from bad behavior.
The fine does not only a↵ect the optimal level e↵ort, but also more directly the likelihood of scenario 2. It is found that when the net personal benefit, v f , is smaller than 23 the first derivative of x to f , xf, decreases. This means that it is even more likely for scenario 2 to exist.
In other words when the net personal benefit of bad behavior is between 13 and 23, an increase in the fine leads to a increase in the likelihood of scenario 2 to occur. However, when the net benefit, v f , is larger than 23 xf increases, so that it becomes less likely that the derivative is
smaller than zero, decreasing likelihood of scenario 2. But since eopt increases, which increases
the likelihood of scenario 2, it is impossible to say how a small increase in the fine a↵ects the likelihood of scenario 2 to exist. This switch at v f = 23 can be attributed to the fact that the behavior of social-acceptance type of people is influenced by two counterbalancing e↵ects. On the one hand psychological costs decrease when the fine increases, making bad behavior more profitable. On the other hand the benefits also decrease, making the advantages of bad behavior smaller. When v f = 23 the decrease of benefits exactly neutralizes the decrease in costs.
The parameters , which is the importance of the public good, and m, which is the dimin-ishing return factor of e↵ort on the public good, only influence the derivative of x to the fine, f , by changing the size of ✏opt and are thus influencing the sign of x
f indirectly. It can be easily
seen that an increase in the importance of the public good, , leads to an increase of ✏opt and thus making it more likely for scenario 2 to occur. That positively e↵ect the optimal behavior resonates with reality, because a higher means that people value the public good more, and putting in e↵ort to sustain the value of the public good is thus more important.
How changes in the diminishing return factor, m, change the size of the optimal level of e↵ort, ✏opt, is less clear cut. Remember that the lower m is, the less e↵ect an extra doses of e↵ort has on the value of the public good. So let’s assume m goes to infinity, if that is the case it is always better for the public good to put in e↵ort, because there is no diminishing return. Having said that, it would seems logical for m to have a positive e↵ect on ✏opt. However it can be found that the diminishing return factor only has a positive e↵ect when v +f ( 1) < 0. This is due to the fact that, as mentioned before, m does not only influence the level of diminishing return, but also the overall value of the public good. In other words a high diminishing return factor (so that there is almost no diminishing return) means that the e↵ort can lead to a greater value of the public good.
The next parameter of which I will examine the e↵ect is p, the percentage of people of the social-acceptance type, this parameter only influences xf directly and not also through optimal
level of e↵ort, ✏opt. It can be seen that if v f > 1
3 an increase in p will lead to an increase in the
derivative of x to the fine, f , thus making it less likely for scenario 2 to occur. Again this is logic since with a higher p, more people are of the social-acceptance type. Since for v f > 13 the social acceptance type are actually more likely to exhibit bad behavior when the fine increases, it is obvious that it becomes less likely for scenario 2 to exist.
When looking at how the value of extra spare time, v, influences xf, we have to look at both
✏opt and xf. It can be seen that an increase in v leads to a higher value of xf and lower value
of ✏opt. This means that an increase in v decreases the likelihood of scenario 2 in two di↵erent
ways. It is therefore very likely that a high value of v means that it is not possible for scenario 2 to occur, or only under very strict circumstances. Again this is as expected, because when the value of spare time increases the personal gain of low e↵ort increases. This means that the benefits increase, leading to the fact that xf becomes larger. But next to that total welfare also
gains more (or loses less) by someone shirking than when the value of spare time, v, is low, so that the optimal level of e↵ort also becomes lower.
Lastly I examine the e↵ect the of the percentage of bad behavior, x, has on xf. Just like
the value of spare time, v, this variable a↵ects both xf and ✏opt. When x increases the optimal
level of e↵ort decreases, having thus a negative e↵ect on the likelihood of scenario 2. This is logical since a high percentage of bad behavior means that putting in extra e↵ort leads to large gains in the value of the public good, because so little e↵ort has been put in by others. Next it follows from the properties of the social-acceptance cost function that its derivative to x is negative, and since the derivative of xf to x is equal to C2cx
accept(3(v f ) 1)p an increase in bad
behavior leads to an increase in xf when v f > 13. Making it even less likely for scenario 2 to
exist. ✏opt x f xf < 0 f " # if v f < 2 3 " if v f > 23 + ± x # " p · " v # " " · + m # if v f < (1 f ) " if v f > (1 f ) · + Table 4.1: E↵ect the increase of a parameter has on ✏opt, x
f and the likelihood that xf < 0
E↵ect of the parameters on complicated dynamics
Until now I have assumed that there is a stable steady state, and that an upward shift thus leads to higher values of the steady state and vice versa. In other words that an upwards shift
CHAPTER 4. RESULTS 24 thus means that an increase in the fine leads to an increase in bad behavior and vice versa. So now lets look at when there is no stable steady state. Recall that this is the case when xopt, which is the percentage of bad behavior for which the optimal level of e↵ort is zero, is the only steady state so that there is an alteration between ✏opt > 0 and ✏opt < 0. Because it is not possible to calculate the steady states exactly it is impossible to say under which circumstances xopt is the only steady state. However, it is possible to determine under which circumstances it
is more likely that this is the case.
First of all for xopt to be a steady state, xopt should lie between 0 and 1. Note that when
xopt > 1 every social-responsible person always exhibit bad behavior, and under such circum-stances it is more likely that an increase in the fine leads to more bad behavior. When xopt< 0 the social-responsible people will decrease the percentage of bad behavior for an increase in the fine, therefore making it more likely that a fine leads to a decrease in bad behavior.
Now recall from (4.1) that xopt= 1 +m
(v f ) + mf m. So for xopt to be larger than zero
but smaller than one, there are two requirements that need to be fulfilled. First > v f1 f, so that
m
(v f )+mf m < 0 and xoptis thus smaller than one. Second, the diminishing return factor,
m, should be small, so that the negative value of the second part does not become to negative, making xopt < 0. From all these requirements it follows that the diminishing return factor, m, and the value of spare time, v, should be small and that the importance of the public good, , and the fine, f , should be large, to ensure that complicated dynamics can arise. Especially the requirements for m is important, since this can not be counter balanced by other parameters.
Secondly when xopt is between zero and one, it should also be the only steady state. Now recall that xoptis the x for which the model makes a drop in value, so that the derivative is equal to minus infinity. Also remember that every other part of the model has a positive derivative. This means that when the drop is large, the change of the graph crossing the x = y line on an other point than at xoptis smaller than when the drop is small. This drop becomes larger when
the first part of the graph increases more than the second part of the graph or when the first part of the graph decreases with less than the second part. Therefore it is again useful to look at the derivatives of the model to di↵erent parameters.
When looking at the derivative of xt to the fine, f , in (4.2), it can be seen that the second
part of the model has an extra negative component. Therefore the derivative of the model is always lower for the second part of the model. That means that when the personal benefit of shirking, v f , is smaller than 13 both parts of the model are decreasing, but the second part is decreasing more, therefore increasing the gap between the two. When v f > 13 there are two options. Either both derivatives are positive but the first part of the model has a higher derivative, so again the gap is increasing. Or the first part of the model has a positive derivative, while the second part has a negative derivative. If the last option is true, which is usually the case for a benefit, v f , close to 13, the e↵ect of an increase in f on the gap between the two parts of the model is very large. For any other f an increase in the fine also leads to an increase in the gap, but not by as much.
When decreasing the value of spare time, v, both parts of the model will decrease. However, the second part of the model will decrease with a slightly larger bit, namely with 13(1 p) more, therefore a decrease in v increases the changes of complicated dynamics. Note that a large f and a small v positively a↵ect the likelihood of complicated dynamics in two ways, because they do not only increase the gap at xopt, but they also increase the probability that xopt is between zero and one.
Last, but not least, a decrease of the percentage of social-acceptance type, p, would lead to a greater gap between the two parts of the model. This is quite logical since the social-responsible part of the model, ˆa, is the part that is responsible for the gap in the model. So when more weight is put on this, it will lead to a bigger gap. This is by far the most important way in which the gap between the two parts of the model can increase. To summarize the above results have been put in Table 4.2.
prob complicated dynamic " f "
" v # " p # " m # " "
Table 4.2: E↵ect the increase of a parameter has on the likelihood of complicated dynamics
An example of possible dynamics
Until now I have estimated how the value of di↵erent parameters influence the e↵ect that an increase of the fine has on behavior. Will this lead to lower or higher percentage of bad behavior or will it lead to chaotic behavior. Next I set all the parameters to a fixed value, as to get a better image of how the dynamics works in this model. To do so lets assume the same values for the parameters as in chapter 3.2, so v = 0.7, = 0.9 and m = 3, and lets also assume that p = 0.7. Recall that in Figure 3.1 ˆ and ˆa are plotted for di↵erent values of the fine, f . Now I also plot the mixed society model with the given parameters in Figure 4.1, in Figure 4.2 also its bifurcation diagram is given.
Since the value of spare time is set to 0.7, it is known that for a fine larger than 0.3667 an increase in the fine always leads to a decrease in bad behavior. Because in that case the net personal benefit, v f , is smaller than 13 and the model is in scenario 1, and thus the steady state decreases in value. Indeed it can be seen that for an increase of the fine, f , from 0.35 to 0.525 the graph has shifted down. Looking closely at the bifurcation diagram it can be seen that the level of bad behavior is already declining for a fine larger than 0.34. This implies that for a fine between 0.34 and 0.3667 scenario 2 occurs in the model. This means that only for a very small interval scenario 2 exists, this can be explained by the fact that both the value of
CHAPTER 4. RESULTS 26
Figure 4.1: xt for f = 0 (blue), f = 0.175 (green), f = 0.35 (red), f = 0.525 (light blue) and
f = 0.7 (purple)
spare time, v, and the percentage of social-acceptance type, p, have a high value, while both of them have a direct negative e↵ect on the likelihood of scenario 2 (see Table 4.1). For every fine lower than 0.34 an increase in the fine actually leads to more bad behavior, so that a fine has an adverse e↵ect.
In Figure 4.1 it can be seen that for f = 0.525 the model has three steady states, of which only the upper one is stable. Increasing the fine even further we see that the upper two steady states disappear, creating chaos. This type of bifurcation is called a tangent bifurcation (Hommes, 2013), and results in only one unstable steady state. In Figure 4.2 it can indeed be seen that for a fine larger than 0.53 the steady state disappears and a more complicated pattern emerges. It is not surprising that the chaos only arises for a fine larger than 0.53, since, as mentioned before, a large fine has a positive e↵ect on the likelihood that complicated dynamics arise. And because both the value of spare time, v, and the percentage of social-acceptance type, p, whom both have a negative e↵ect on the likelihood, are also large a very high fine is needed to counterbalance those e↵ects. For a formal proof that chaos exists for a fine larger than 0.53 see Appendix A.
The dynamics compared with a Cobweb model
The shape of the model as shown in Figure 4.1 is quite similar to the shape of a nonlinear Cobweb model with adaptive expectations (Hommes, 1994), as can be seen in Figure 4.3. It is therefore interesting to compare both models, to better understand the dynamics of my model. The Cobweb model with adaptive expectations predicts the price of a good that arises through supply and demand. The special part of the model is that producers determine their supply by a price prediction. This price prediction is a linear combination of price realized in the last period and expected price of last period, pet = wpt 1+ (1 w)pet 1. When the combination is
strongly influenced by the expected price, so w is low, there exists a stable steady state, when producers are influenced by both expected price and realized price there is chaotic behavior and when they are heavily influenced by realized price, so w is high, there is a stable 2-cycle. It can be seen in Figure 4.3a that when w is low the function has a similar shape as when the fine is small in my model. When w increases the shape of the model changes in a similar fashion as my behavior model. The reason for these similarities is that both models are based on an increasing and a decreasing function that are combined in one model, namely the supply and demand function and the social-acceptance and social-responsible model.
This similarity in shape leads to believe that the dynamics are also similar, however because in my model the changes are so abrupt and thus all the di↵erent bifurcations happen on such a short interval of the fine, f , this is hard to see. Therefore I relax the assumption that the population size goes to infinity and instead plot the model for n = 100, this makes the jump in the graph more gradual so that the bifurcations are further apart. Again it can then be seen in Figure 4.4 that at f = 0.53 the upper two steady states lose stability because a tangent bifurcation occurs. However, what now can be noticed, in comparison with the bifurcation
