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ФЕДЕРАЛЬНОЕ ГОСУДАРСТВЕННОЕ АВТОНОМНОЕ ОБРАЗОВАТЕЛЬНОЕ УЧРЕЖДЕНИЕВЫСШЕГО ОБРАЗОВАНИЯ

«НАЦИОНАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ УНИВЕРСИТЕТ

«ВЫСШАЯ ШКОЛА ЭКОНОМИКИ»

Международный институт экономики и финансов

Выпускная квалификационная работа - БАКАЛАВРСКАЯ РАБОТА

по направлению подготовки 38.03.01 «Экономика»

образовательная программа «Программа двух дипломов по экономике НИУ ВШЭ и Лондонского университета»

Commuting costs and the choice of residence

Филюшкина Сона Александровна

Научный руководитель

Дементьев А.В.

Москва 2019



Abstract

Labour contract flexibility and labour force mobility has increased recently. The interaction between the worker's choice of residency and work place is largely shaped by commuting cost and preferences for the leisure time spent with family. The paper studies the effect of commuting cost on the choice of work week type once we allow the worker to avoid commuting and rent additional apartments in the location with high proximity to work place. In a game-theoretical model we show how the choice of 3, 4 or 5-day working week is affected by different parameters. The compressed work week and associated phenomena known as `weekday widowing' is discussed is discussed at the theoretical level. Our findings may shape the bargaining process and show which factors affect the parties' decisions. worker employer payoffs



Аннотация

Гибкость трудового договора и мобильность рабочей силы в последнее время возросли. Взаимосвязь между выбором места жительства работником и местом работы во многом определяется путевыми расходами и предпочтениями в отношении досуга, проведенного с семьей. В статье изучается влияние затрат на поездки на выбор типа рабочей недели, если мы позволяем работнику избегать поездок на работу и арендуем дополнительную квартиру в местах с высокой близостью к месту работы. В теоретико-игровой модели мы показываем, как на выбор 3, 4 или 5-дневной рабочей недели влияют различные параметры. Обсуждаемая рабочая неделя и связанные с ней явления, известные как «weekdaywidowing», обсуждаются на теоретическом уровне. Наши выводы могут сформировать процесс переговоров и показать, какие факторы влияют на решения сторон.



1. Introduction

Commuting has long become an integral part of working people's lives. The first association with commuting is the urgent problem of traffic jams on the occasion of the beginning of the working day and its ending.

Traffic congestion is one of the most common and acute problems in the urban and suburban areas as it affects many important aspects of human life (physical and emotional health, labour productivity, financial stance and other economic spheres). Every year road congestion during the peak hours increases due to commuting, attracting more attention of policymakers and scholars to development oftraffic demand management strategies (TDM), especially in the large and developed cities.

One of these strategies is the compressed working week (CWW). Such an arrangement of the working week allows the employee to work fewer days a week than traditional 5 working days. Correspondingly, more intensive working days require more working hours per day if the total number of weekly working hours remains unchanged. On the one hand, such a workload flexibility secures additional days off (normally one or two) which can be spent with family. On theother hand, greater number of working hours per day may undermine worker's productivity. In practice, such flexible arrangements take the form of a discrete choice of 5, 4 or 3-day workweek.

Generally, the employees' and managements' attitude towards such a shortened organization of the week is favorable, because of many benefits that outweigh any adverse effects. For example, the workers indicate that a very important advantage is an increase in the amount of free time that can be spent with the family and friends and a noticeable reduction in commuting time. The commonly mentioned problems are the reluctance to return to work after a long absence and employee dissatisfaction during the long working day. The motivation for the employers to apply such policy is the belief that the attractiveness of such workweek arrangement to the workers will stimulate effective organizational behavior and promote job satisfaction. However, regardless of the generally positive views of compressed workweeks, the total effect of the implementation of such workweek schedule is unclear.

The research question that we address in our paper is how the commuting cost affects the choice of work week type once we allow the worker to rent additional apartments. This phenomena is known as `weekday widowing' and is widely discussed in social media among people who have a well-paid job in a city but whose families live in a periphery. In this paper I employ a game-theoretic approach to endogenize the choice of 3, 4 or 5 workday week in a setup with sizeable commuting cost.At the methodological level I convert the integer programming model by Billionnet (1999) which is broadly discussed in the operational research literatureinto a simple sequential game with two players. A discrete choice of a compressed work week arrangements can be made either by the worker or by the employer depending on the relative bargaining power of the two. Our main results are largely driven by the commuting cost which we view as our main theoretical contribution to the emerging literature on CWW.

We consider two regions: a central business district (CBD) and a periphery. There are two active agents: a worker and an employer. The work place is located in the CBD and the employee lives in the periphery, therefore, he bears time and monetary costs of commuting. In addition, he has an option to rent a second apartment next to the work place in order to save on commuting.

The rest of the paper has the following structure. In Section 2 there is a literature review and the description of the relevant modelling approach that subsequently helps us determine the optimal number of working days under different conditions. Section 3 presents the model setup according to which in Section 4 the payoffs for the employer and the worker are derived. Section 5 considers a sequential-move game between two agents. The analysis depends on who is the leader of the game, that is chooses the desired number of working days.Section 6 presents our main findings and concludes the work.



2. LiteratureReview

The current research on the compressed workweek and hence the commuting patterns and costs are mostly concentrated on empirical analysis. Theyexamine various commuting and non-commuting issues and problems following the shift from the traditional 5-day workweek to the compressed workweek schedule.

In (Ho, A., & Stewart, J. (1992)) studyit was found that the employees decrease the number of their trips per week and the travelled distance per week after shifting to the compressed workweek schedule.

In (Sundo, M. B., &Fujii, S. (2005))paper the purpose was to assess the changes in activity-time lifestyle patterns. The main finding was that the shift to the 4-day compressed workweekhad a great and noticeable impacton activity-travel patterns. The duration of commuters' household activities was reduced by approximately one hour, sleeping time decreased by 20 minutesand pre-work preparation time by half an hour. Moreover, it was found that the CWW schedule had a negative impact on commuting times, as it led to a decline in commuting duration.

Pierce, J. L., & Dunham, R. B. (1992)focused on the individual and organizational effects. In this paper the compressed workweek is analyzed in a form of four-days-on, four-days-off pattern. Results showed that the adoption of such workweek organization allows workers to “better harmonize their nonwork activities with family and community rhythms” (Pierce, J. L., & Dunham, R. B. (1992), p.), therefore, overpassing many of the adverse effects of the CWW. Besides the significant increase in satisfaction with the hours of work there were an improvement in job, life, and leisure time satisfaction.

In the study of Vega, A., & Gilbert, M. J. (1997)the impacts of a three-day work week schedule on workers' productivity and the overall satisfaction with different fields of lifeare evaluated. It was observed that there arepositive effects on productivity indicators and the self-reported overall life satisfaction.Most officers reported that the compressed work week schedule does not affect on-the-job fatigue. Nearly all officers argued that the three-day compressed work week effectively met their requirements regarding the working aspects and the personal life aspects.

The aim of The Mitchell, R. J., & Williamson, A. M. (2000) paper was to examine the change of the work schedule from an 8 h roster to a 12 h roster in a power station. The evaluation of the resulting effects was donethrough the monitoring of the after-shift performance, the general health indicators, sleep mode and mood behavior. Results revealed that personal life of workers got betterafter the shift, that is under the 12 h roster. There also were improvements in all employee's indicators mentioned above.

Important and decisive articles become the papers of Helminen, V., &Ristimдki, M. (2007) and the work of Billionnet (1999).

According to Helminen, V., &Ristimдki, M. (2007), most of the commuters (nearly 65%) who used to spend on one-way commuting trip more thantwo hours had a second apartment near the workplace in order to save on commuting time and monetary costs. In this article the important assumption is thatthe workerswho have a second one-room apartmentwill make only one round-trip during theworkweek. Thus, this imply that the employee stays in the apartment during the working days. Hence, in my analysis, this assumption also takes place.

As I have mentioned above, I use a game-theoretic approach and I transform the integer programming model by Billionnet (1999), which is also discussed by Seзkiner, S. U., Gцkзen, H., & Kurt, M. (2007), into a extensive-form game with two players: the employer and the worker. A discrete choice of the number of working days per week can be made either by the worker or by the employer.

So, I have added the idea about the second apartment into the sequential game between the employee and the employer in which they decide on the various possible workweeks arrangements.



3. TheModelsetup

There are two regions: a central business district (CBD) and a periphery. We will concentrate on a single individual who is assumed to work in the CBD and permanently lives with his family in the periphery. Therefore, each working day the employee must spend time and money on commuting. We assume that the employee has the opportunity to reduce these commuting costs (time and monetary parts) by purchasing the second apartment, which is located opposite the place of work. The employee must work a fixed number of hours per week regardless of the number of working days. That is, he can work according to the traditional schedule (5 working days, 8 hours each) or have a compressed workweek, that is to work 4 or 3 days a week, provided that he works 10 or 13 hours respectively to compensate for the reduction in number of working days. As it is assumed that in any case the worker has at least two weekends, we exclude these two days from our analysis and, therefore, the word `week' implies only 5 days in total.

3.1 Worker

First, we consider the behavior patterns of a representative worker. The salary per week (and thus the wage per hour) constitutes a non-negotiable part of the labour contract which the worker always accepts. Thus, the labour supply measured in working-hours per week is fixed and there is no difference between a salaried person and a wage earner.Without loss of generality, we may assume that the total number of working hours per week is fixed and equal to 40. The employer pays a fixed weekly wage (salary) for 40 hours of work.

There are people in a family that lives in the periphery. Hence, . A person derives some kind of emotional benefit from spending time with his family, let parameter represents family time utility factor. Denote total number of hours in a week by . Therefore,



where is total working time for one week; is total leisure time and is total commuting time. Total leisure time represents a sum of free time spent with the family ( and free time spent alone in the second apartment (. Total commuting time is equal to the product of the number of commuting trips ( and commuting time per one trip . Hence,

The worker can eliminate commuting by renting the second apartment in the CBD, therefore, if the agent stays in the center, the commuting cost is 0.

Let be the rent for the house where the family lives and be the rent for the one-room apartment in the center. The utility of the representative worker is

where t is the total commuting costs (travel time costs and monetary costs).

Therefore, the aim of the employee is to maximize his utility by choosing whether to commute each working day or stay in the center and selecting the optimal number of working days according to his personal characteristics.

3.2 Employer

Employer hires a worker for a fixed number of hours a week at a fixed wage. However, the number of working days per week can vary and can be chosen either by the employer (in the first variant of the game) or the worker (in the second variant of the game). The employer may induce the desirable commuting pattern of the worker (i.e. to stay in the CBD and not to commute) by compensating his rental cost in kind. Obviously, the employer's payoff negatively depends on the workers' wage. It is also positively dependent on worker's productivity.

The crucial assumption that substantially drives the results of our further analysis is varying (and non-monotonous) labour productivity. According to Barzel (1973), marginal labour productivity per hour within a working day increases with the number of working hours per day from zero, then it reaches its maximum at a certain point and declines back to zero when the worker works `too much'. The corresponding marginal product curve is depicted in Figure1. We build our further analysis on the previous studies on compressed workweek documenting that the daily labour output is the highest in the case of 10 working hours per day (Alyssa M. Freas and Stuart M. Anderson (1990)).

Naturally, since the number of working hours per week is fixed (by law) the employer opts for maximizing the total labour output per week. Since the maximum daily labour productivity is reached for 10 hour working day, the highest weekly output is obtained for 4 working days. We denote the corresponding average output per week by . Therefore, and .

The utility of the employer is

where the last term represents a disutility for the employer if the worker chooses to commute each day and not to rent the second apartment (the coefficient is responsible for the employee fatigue due to commuting that worsens his performance).



4. Payoffs

Before proceeding to the solution of the sequential move game, we have to derive the payoffs for two players: the employer and the employee.

First, we will consider payoffs for the employer. There are three possible work week schedule: 3,4 and 5 working days for each of which the worker has to choose between purchasing the second apartment close to a work spot and stay there during the whole work week in order to save on commuting and commuting from the house located in the periphery and save on renting a one-room flat. Therefore, the employer ends up with six possible outcomes presented in Table 1. In `no commuting' case the payoffs depend only on average output per week and fixed wage rate. In `commuting' case there is also additional negative impact from commuting on worker's performance. As in any work week schedule the employee must to make at least one commuting trip, we exclude that trip from each payoff. Hence, if the worker faces a traditional work week schedule (5 working days) and decides to commute, then the number of commuting trips ( is equal to 4. Analogically for compressed work week schedules.

Now, we will look at the worker's payoffs. As for the employer, six payoffs are possible. Remember that the utility function is given by the equation . Thus, firstly, we need to analyze the values for each component of the equation for the total time (1). The results are shown in Table 2. For example, if there is a compressed work week case when the individual has to work 3 days and he prefers not to commute, then total commuting time will be zero. As another two days will be weekends, time spent with the family will be equal to two fifth of the total number of hours in a week. The remaining hours represent leisure time that the individual spend alone in his second apartment far from his family. Therefore, we have



If there is a compressed work week case when the individual has to work 3 days and he prefers to commute, then there will be no free time that he spends alone and the total number of hours will be distributed among commuting time, leisure time spent with the family and working time. According to our discussion above, the commuting time is double the commuting time per one trip. The remaining time is just the free time during which the employee stays with his family. Thus,

.

Plugging (4) and (5) into the utility function (2), we obtain payoffs for the worker if under the contract he works 3 days a week for 10 hours.

• No commuting:

• Commuting:

Similarly, other payoffs are obtained and presented in Table 3.

It should be noted that I do not include into payoffs the term that indicates the rent for the house where the whole family lives ( ) because the agent incurs these costs in any circumstances. In addition, in equation (6) , in equation (7) .



Table 1. Employer's payoffs

Payoffs
5 Working Days	4 Working Days	3 Working Days
No commuting	Commuting	No commuting	Commuting	No commuting	Commuting

Table 2. Time distribution for the worker

Time Distribution
	5 Working Days	4 Working Days	3 Working Days
	No commuting	Commuting	No commuting	Commuting	No commuting	Commuting
	0		0		0
	0
		0		0		0

Table 3. Worker's payoffs

Payoffs
5 Working Days	4 Working Days	3 Working Days
No commuting	Commuting	No commuting	Commuting	No commuting	Commuting



5. Sequential-move game

This section is focused on the derivation of the Subgame perfect Nash equilibrium. We consider separately the two alternative timelines in this sequential game exploring the first-mover advantage in the wage-bargaining process. The working conditions have additional dimension - the choice of the number of working days - which allows us to study the problem of compressed work week.

There are two possible versionsof the game: the first one is when the leader is the employer and the second one is when the leader is the worker. According to the former case, the employer assigns the number of working days per week, that isthe number of working hours per day. The reservation wage is normalized to zero thus the employee accepts a proposed job offer by choosing the highest value of utility function among the three alternative work schedules - 3, 4 or 5 working days a week. In the latter case, when the worker is the first-mover, the employee has the advantage in choosing the number of working days.

At the second stage when the employer is a leader the worker chooses between commuting every day from a house that is not in the CBD and not commuting thus renting the apartment close to the workplace so as not to waste time and money on commuting. When the worker is the leader, at the second stage the employer sets the wage given the number of working hours per day individual is ready to work. At the last stage, the employee faces the choice whether to commute or not.

We consider a perfect information game in which each player can observe the moves of the other player.This sequential-move game can be solved by implementing backward induction.

Regardless of who is the leader, at the last stage of the game the employee always raises the question of the commuting. Worker picks “not commute” option if payoff from that option is higher than payoff derived from commuting each working day. Hence, depending on the value of the second apartment's rent worker will prefer to commute or to stay at the second apartment located in the center during the work week. Derivation of rent values is presented in Appendix.

· If, then the worker chooses to stay at the second apartment during 5,4 and 3 working days. Call this range for the rent values by the `small rent'.

· If, then for worker it will be optimal to commute only if there is a “3-day” work week, otherwise (in the case of 4 and 5 working days) he will prefer to not commute and have the second apartment in CBD. Let this range represents the `relatively small rent'.

· If , then worker will prefer to have the apartment near to his workplace only in the case of “5-day” work week. When there is a compressed work week (3 or 4 working days), employee chooses to commute each day. The rent within these limits will be called `relatively high rent'.

· If, then due to very high cost of the second apartment it will be more beneficial for the worker to commute during usual “5-day” work week and the compressed work week. For these rent values we will use the expression the `high rent'.

Forthe sake of tractability of our analysis and easy reading, we describe only non-trivial solutions where different payoff comparison is meaningful. Thus, in a game tree representation we omit expressions for some payoffs if they are not crucial, I depicted not all payoffs on the charts, but only those that are needed for this particular case.

5.1 Employer is a leader

In the first instance, I will examine the case when the worker makes his decision after the employer. We define the game as a graph: it has decision nodes and branches from decision nodes to successor nodes. Hence, in the Figure 2. you can see a game tree that help us to depict the sequence of actions.

As was noted above, the optimal actions of the individual who offers his services are directly dependent on the size of the additional rent that has to be paid if the worker wishes not to spend time on everyday commuting. Considering the previous analysis, we have four segments for the rent values on which the employee will optimally choose different strategies.

Worker must plan for three different contingencies. He must decide what to do if the employer plays `5 working days', how to respond if the employer picks `4 working days' and what is optimal to choose if the employer acts `3 working days'. Denote `commuting' option by , `no commuting'option by .

I. Employer is a leader

If, then no matter what work schedule the employer offers, the worker's optimal response will be to rent the second apartment and save on commuting costs. Hence, the employer does not have to compensate for the rental costs, and he compares his payoffs: , and . As average output per week is the highest when the employee works 10 hours a day, the employer selects `4 working days'.The choices obtained by backward induction are shown in figure below.

High and relatively high rent

i. If, the rent is quite high and it exceeds the cost of commuting in the case of compressed work week. Thus, the optimal responses for the worker are: play C if a shorter working week is offered, play if there is a traditional work week schedule.The employer needs to choose the utility maximizing work week arrangement, i.e. need to decide on the number of working days in one week. However, he can affect the decision of the worker to commute in the case of the compressed work weeks through renting him an apartment close to the workplace.

When the employer is better off in case of compensating for the rental costs when there are 4-day and 3-day work weeks. The employer's payoffs areand . The last payoff is smaller than the second one, hence, the employer chooses between 5 and 4 days of work. If , 5-day workweek is chosen, otherwise 4 working days will be preferred.

When the employer provides the compensation in kindonly in case of 4-day work week. Payoffs are equal toand. Equivalently, and .

· Employer chooses 5 working days if .

· Employer chooses 4 working days if .

· Employer chooses 3 working days if

When the employer will not compensate for the rental costs in kind. Thus, comparing the employer's payoffsand , we can conclude that

· Employer chooses 5 working days if .

· Employer chooses 4 working days if .

· Employer chooses 3 working days if

.

ii. If , the second apartment is too expensive and under any circumstances it is more profitable for the employee to spend his time and money on commuting.

When the employer provides the compensation in kind in all cases as the costs associated with commuting are greater than the rent costs. The employer's payoffs are , and thus, `4 working days' schedule is chosen.

When the employer chooses to compensate in the case of 4 and 5 working days and his payoffs are, and .. Therefore, the employer choosesthe compressed work week options. If , the 4-day work week will be selected; otherwise, the 3-day work week will be chosen.

When it is beneficial for the employer to compensate only in the case of 5 working days. Hence, his payoffs are, and .

· Employer never chooses 5 working days as we obtain that The left part of inequality is negative; however, the right part is positive. Contradiction is obtained.

· Employer chooses 4 working days if , i.e. if (as is always negative and ).

· Employer chooses 3 working days if .

When the compensation does not make the employer better off, hence, the employer chooses between , and . Again, his choice depends on parameter .

· Employer never chooses 5 working days as we obtain a contradiction with our assumption that (

· Employer chooses 4 working days if , i.e. if (as is always negative and ).

· Employer chooses 3 working days if .

· The higher the rent for the second apartment, the higher the probability that the employee will not prefer to stay in the center during the work week.Therefore, with the really high rent, the more days the employee has to work, the more commuting trips he takes, which negatively affects the worker's contribution. Depending on how much the particular employee gets tired of commuting, the employer sets the optimal schedule for the work week.

Proposition 2:When the rent is high and relatively high the probability that the employee chooses a 4-day compressed work week increases with the value of the rent for the second apartment and decreases withthe loss of productivity due to commuting.

Also, it should be noted that when the rent and the loss of productivity due to commuting both increases, the likelihood that the employer will choose 3-day compressed work week increases.

5.2 Worker is a leader

Now we will consider extensive-form game in which the worker enjoys the advantage in choosing the preferred number of working and rest days.As in the previous section, we have to look at four ranges for rent values. In the Figure 7. you can see a graph that represents this sequential-move game.

5.2.1 Small and relatively small rent

i. First, we will analyze the case in which the rent is small enough that the employee prefers to stay in the CBD and live far from the family house during the whole work week regardless of the number of working days. Solving the game backwards the worker always chooses to avoid commuting. Consequently, the employer has no incentive to compensate the worker for the rental cost in order to influence the decision not to commute (clearly, for the employer `no commuting' option is preferable in every such a subgame). Therefore, he offers a wage, which provides non-negative payoffs for a fixed total number of working hours per week. However, the final decision on the number of working days is made by the worker.He compares payoffs from three available options and selects the highest one: , and . Reducing the same components, we transform these expressions: 0, and .

· If , thus, almost always the employee chooses 3-day working week.

· If , . Hence, for the worker it will be preferable to work 5 days a week.

ii. If , we face the situation in which the worker prefer to commute only during a 3-day working week. Therefore, the employercan compensate the worker for the rental cost so that the worker will be indifferent between `commuting' and `no commuting'. The employer has to compare his gains in each case: `no compensation' and `compensation'. If the worker commute, the gain is ; if the worker stays in the apartment, the gain is

II. Worker is a leaderWhen then the employer chooses to compensate the worker for the rental costs to eliminate his costs associated with the worker's tiredness after commuting trips. Thus, the payoffs that have to be compared by the worker are , and . Reducing the same components, we obtain expressions 0, and .

· If , therefore, the worker chooses a compressed work week option with 3 working days.

· If ,we have that , hence, 4 working days can never be the optimal choice. However, the relationship among 0, and is unclear. If , then the 3-day work week will be chosen; otherwise, the worker prefers the traditional 5-day work week.

When , the employer does not want to provide the compensation in kind because it will reduce his final payoff. Hence, the workers must choose the greatest payoff among, and . Transforming the expressions, we have to compare 0, and . Hence, according to the payoffs, we will obtain that

· If , 0 is always less than or equal to, hence, the worker will not select the traditional work week and choose between different arrangements of the compressed work week.

· If , 0 is always greater than , from this we can conclude that the individual will never prefer to work 4 days a week.If , the 3-day compressed work week will be preferred. Otherwise the traditional 5-day work week is selected be the employee.The choice of the worker depends on the commuting parameters. The probability that the employee chooses 5-day work week increases when the commuting costs (time and monetary) rise.

Proposition 3: When the rent is low and relatively low and the loss of productivity due to commuting is high the likelihood that the employee chooses a 3-day compressed work week increases with the family size and family time utility factor.

Proposition 4: When the rentis relatively low and the loss of productivity due to commuting does not exceed the cost of the compensation incurred by the employer, the choice of the worker depends not only on the family size and family time utility factor, but also on the user's commuting costs. If the user's commuting costs rise, the probability of CWW decreases.

5.2.2 High and relatively high rent

iii. If , at the last stage of the game the worker will always commute if there is a compressed work week (3- or 4-day work week). Hence, the employer may have incentive to provide the compensation in order to make the worker not spend his forces on commuting.

When , the costs associated with the compensation for the rental costin kind is less than those associated with the employee's tiredness. Thus, the employer chooses to `compensate' changing the decision of the worker from `commuting' to `no commuting' choice in the case of compressed work week. Therefore, now the worker has to choose the optimal work week schedule by comparing the payoffs , and . With the help of some manipulations we consider these expressions: 0, and .

· If , therefore, the worker chooses a compressed work week option with 3 working days.

· If ,, thus, 3-day compressed work week will never be optimal choice. If , then the employee plays `4 working days'. Otherwise he selects 5-day work week.

When , the employee chooses to compensate for the rental cost in kind only in the case of 4 working days; when there are 3 working days, he is better off withoutdoing it. The resulting final payoffs for the worker are , and . Transforming expressions, we obtain 0, and. General results are not easily obtained, however, from payoffs we can conclude that the likelihood of the compressed work week increases if the family size and family time utility factor rise and if the user's commuting costs decrease.

When , the compensation in kind have a greater negative effect on the employer's payoffs than worker's fatigue after commuting, thus, the employer does not wish to compensate. The payoffs of the worker from 5-, 4- and 3-day work week are , and . By subtracting the first expression from each of the three given above, we obtain 0, and . The second expression is always less than the third payoff, therefore, the worker faces the choice between the traditional work week and 3 working days schedule.

• If and if , the worker selects 3 working days schedule.

• If , , therefore, there will be a traditional work week.

The choice of the worker depends on the commuting parameters. The probability that the employee chooses 5-day work week increases when the commuting costs (time and monetary) rise. If , the rent for the second apartment is too high and outweighs the costs associated with everyday commuting, thus, regardless of the number of working days the employee will choose the `commute' option. As in the analysis above, the employer can influence the worker's commuting choice. Whether he will apply the tactic affecting the decision of the employee depends on the relationship between the values of the rent and coefficient representing worker's fatigue.

When , employer is readyto compensatein order to enjoy higher profits by getting rid of costs of weariness of the employee. Consequently, worker plays in any case. The worker has to choose the optimal work week schedule. Comparing the payoffs , and that are equivalent to 0, and , we can conclude that

• If , thus, the employee selects 3-day working week.

• If , . Hence, for the worker it will be optimal to work 5 days a week.

• The worker never selects the 4-day work week.

hen the employer will be better off if he provides the compensation in kind in the case of 5 and 4 working days. Hence, the employee's payoffs are , and or equivalently 0, and .

• If meaning that the traditional 5-day work week will be not selected in almost all cases. The worker chooses the compressed work week.

• If , payoff representing a 4-day work week is negative and so less preferred than 5-day work week. is also negative, thus, `5 working days' is the optimal decision.

When the employer optimally decides to give a bonus when there is a 5-day work week, but he does not benefit from such strategy in the case of compressed work week. Hence, when the worker decides on the optimal number of working days, he chooses the best outcome from and . Equivalently, compare 0,and . The second expression is always smaller than the last one. Therefore,

• The worker never selects the 4-day work week.

• If , , therefore, there will be a traditional work week.

If and if , 3 working days schedule is preferred by the employee.The choice of the worker depends on the commuting parameters. The probability that the employee chooses 5-day work week increases when the commuting costs (time and monetary) rise.

When it is too costly for the employer to make the worker to change his mind, hence, the employer does not want to compensate for the rental costs. Hence, the workercompares payoffs from three available work week arrangements and play according to the utility maximizing principle. The payoffs are , and for 5, 4 and 3 working days respectively. The largest payoff is in the case of 3 days of work, hence, for that values of rent for the second apartment the employee will always provide his services for a 3-day work week.

Proposition 5: When the rent is high and relatively high and the loss of productivity due to commuting is high the probability that the employee chooses a compressed work week increases with the family size and family time utility factor.

Proposition 6: When the rent is relatively high and the loss of productivity due to commuting does not exceed the cost of the compensation incurred by the employer, the choice of the worker depends not only on the family size and family time utility factor, but also on the user's commuting costs. If the user's commuting costs rise, the probability of CWW decreases.



6. Conclusion

We have developed a game-theoretic approach to endogenize the choice of 3, 4 or 5-day workweek. The results that we obtained are consistent with the empirical findings discussed in the literature review and the common sense logic.

When the employer dictates the number of days that he is willing the employee to work, we get quite a logical result that with a small and average values of the rent for the second apartment the worker in most cases chooses to save on commuting costs and not to commute each day and just make one round-trip during the workweek. Hence, there will be 4-day compressed workweek schedule established by the employer as the worker's contribution in this scenario is maximum.

The higher the rent for the second apartment, the higher the likelihood of everyday commuting. Hence, we obtained that with the really high rent, the more days the employee has to work the more commuting trips he takes, which has a negative impact on the worker's contribution. Depending on how much the particular employee gets tired of commuting, the employer sets the optimal schedule for the work week.

When the worker has the right to choose the preferred schedule of the workweek, we concluded that in the case of really high rents, 3-day CWW will always be chosen by the worker, which is consistent with logic as the he prefers to decrease his commuting costs. In addition, when the rent is small the worker also chooses 3 working days option if he is very attached to his family and wants to spend as much time with them as possible.

In the case of average rent level in the market, the choice of the worker depends on the trade off between family factors and commuting costs. Thus, depending on worker's personal preferences and the market conditions, any of the 5, 4 or 3-day work week can be chosen by the worker.



References

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Appendix

Derivation of rent values.

• 5-day work week

The worker chooses `no commuting' if payoff from that option is higher than payoff from `commuting' option. Therefore, we get

which is equivalent to

Transforming this inequality, we obtain

• 4-day work week

Equivalently

Finally, we have

• 3-day work week

Equivalently

Finally, we obtain

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