Tempted to Overconsume: Explaining Overwork and
Underwork in Competitive Economies∗
Abhinash Borah, Christopher Kops, Michael Lamprecht†
Abstract
It is a well-known empirical phenomenon that a sizable fraction of
the workforce fails to obtain their desired working hours, i.e., overwork and underwork simultaneously exist in market economies. In this
paper, we provide a behavioral explanation for this phenomenon that
draws on two observations. First, worker’s consumption choices may
be influenced by an intrapersonal conflict between long-term rational
calculations and short-term visceral urges, and whereas some workers
(sophisticates) recognize this conflict, others (naives) might not. Second, the persistent accumulation of long working hours may adversely
impact a worker’s long-run productivity. We show, in the context of
a competitive economy, how these two features inexorably lead to the
endogenous existence of two competitive labor market segments, one
featuring overwork and the other underwork. In that respect our explanation departs from much of the literature that relies on market frictions, including informational ones, to explain the overwork-underwork
phenomenon.
JEL codes: D03, J22, J42, J81
Keywords: overwork and underwork, overconsumption, long-term
and short-term selves, naivety and sophistication, burnout, competitive labor markets with segmentation
∗
†
This draft: December 28, 2013
Borah: Johannes Gutenberg Universität Mainz (e-mail: [email protected]); Kops
and Lamprecht: Johannes Gutenberg Universität Mainz and Graduate School of Economics, Finance and Management (e-mails:
[email protected] and lamprecht@uni-
mainz.de). Correspondence address: Jakob-Welder Weg 4, 55128 Mainz, Germany
1
1
Introduction
There is a substantial body of evidence that a significant share of workers in
market economies is not satisfied with their working time. For instance, SousaPoza and Henneberger (2002) asked workers in 21 countries the following
question: “Think of the number of hours you work and the money you earn
in your main job, including regular overtime. If you only had one of theses
three choices which of the following would you prefer? (i) Work longer hours
and earn more money; (ii) Work the same number of hours and earn the
same money; (iii) Work fewer hours and earn less money.” They found that a
significant share of workers are not satisfied with their working hours: 21.9% in
Germany, 22.8% in the UK and 32.5% in the US said that they prefer option
(i), i. e., they consider themselves as underworked. At the same time, 9%,
6.3% and 10.1% in these respective countries said that they prefer option (iii),
i. e., they consider themselves as overworked. Evidence of this type naturally
raises the question: why does overwork and underwork simultaneously exist
in market economies? 1 In this paper, we propose an explanation for this
phenomenon.
Several papers in the literature have, of course, addressed the phenomena of
overwork and underwork. A predominant theme of these explanations has
been the reliance on exogenous market frictions to account for overwork or
underwork. That is, certain structural or informational impediments exist
in the labor market that prevents it from matching workers with their ideal
working time. For instance, several papers point out that, owing to the fixed
cost of adding employees, employers may have to impose a minimum workday
for employees.2 Other papers point out that if informational asymmetries
exist between employers and employees regarding effort and commitment to
1
For more evidence on overwork and underwork refer to Altonji and Paxson (1988), Bell
and Freeman (2001) Böheim and Taylor (2004), Dickens and Lundberg (1993), Euwals and
van Soest (1999), Stewart and Swaffield (1997), Wunder and Heineck (2012).
2
For example, Leete and Schor (1994); Golden (1996); Cutler and Madrian (1998).
2
the job, and these are correlated with hours worked, then long working hours
may be used by employers as a way of screening out less committed workers
(Rebitzer and Taylor 1995). At the same time, employees may use overwork as
a way of signaling commitment (Landers, Rebitzer, and Taylor 1996). These
explanations, no doubt, have merit, but our approach here is to abstract from
labor market frictions and conduct our analysis in the context of competitive
labor markets. The reason we do so is because we want to focus on a different explanation for overwork and underwork that is of a behavioral nature.
Abstracting from frictions allows us to highlight the scope of our explanation
in a more transparent manner.
The key behavioral insight that we draw on is that consumption decisions of
individuals are influenced both by long-term rational calculations as well as
short-term visceral urges (Hoch and Loewenstein 1991). In keeping with terminology from the literature, it is as if these consumption decisions are borne
out of an intrapersonal conflict between a patient long-term self—the planner —and an impulsive short-term self—the doer. The short-term self is often
confronted with spontaneous consumption urges that tempt her to deviate
from the plans conceived by the long-term self. What interests us here is the
ramification of this conflict on a worker’s consumption-leisure choice. Observe
that if such a conflict exists, then the marginal rate of substitution between
consumption and leisure from the perspective of the short-term self is strictly
greater than it is from the perspective of the long-term self. Alternatively
put, the short-term self values leisure relative to consumption much less than
the long-term self does. Accordingly, given that it is the short-term self who
makes the consumption-leisure choices, the number of hours that a worker
actually works has the potential to exceed the number of hours she originally
planned to work. That is, spontaneous consumption urges naturally biases
workers to overconsume and, hence, overwork. It is worth pointing out that
this connection that we are drawing between overconsumption and overwork
has featured prominently in the recent sociology and social economics literature. Some have considered the tendency to overconsume, overspend and
3
overwork to be a defining property of modern capitalist societies and have
characterized contemporary consumers as being tied up in a constant “work
and spend cycle” (Schor 1993, Schor 1999). Benhabib and Bisin (2010), building on insights from the sociology literature, have identified advertising as the
critical means through which firms create artificial and conspicuous consumption demands leading consumers to overconsume and hence overwork, while
they remain in psychological denial regarding their “true” consumption and
leisure habits.
Given the intrapersonal conflict that we hypothesize workers being faced with,
a key consideration that emerges is whether they identify and acknowledge
this conflict. We model two types of workers, naive types who fail to acknowledge this intrapersonal conflict and sophisticated types who do so. Whereas
workers are susceptible to overwork, in principle, a labor market could endogenously emerge that provides a working hour restriction as a commitment
device to address this intrapersonal conflict. This is indeed an equilibrium
implication of our model. A sophisticated type sees the intrinsic merit of
this commitment device, but a naive type does not. So, whereas a sophisticated type is inclined to go to such a restricted labor market that curtails her
choice, a naive type would much rather go to an unrestricted one that does
not do so. Indeed, this is a robust implication of the set of equilibria that
we characterize. In any such equilibrium the sophisticated types participate
in a restricted market and the naive types in an unrestricted one. But this
means that ex post, any naive type finds herself consuming more and consequently working more than what she originally planned. This explains why
we find a fraction of the workers overworking. But, how does this explain the
simultaneous existence of underwork?
To explain this, we appeal to another known empirical fact about the relationship between hours worked and productivity. It is widely recognized that
the persistent accumulation of long working hours may have an adverse impact on a worker’s long-run productivity, a phenomenon that in its extreme
4
form is often referred to as burnout.3 In our model, because of such a (gradual) burnout syndrome, the productivity of anyone who persistently works in
the unrestricted labor market falls over time. Further, if the sets of workers
participating in the restricted and unrestricted labor market segments do not
change over time, as is the case in equilibrium, then a wage differential (corresponding to the productivity differential) opens up between the two segments.
That is, if all naive types go to the unrestricted market and all sophisticated
types to the restricted one, in all periods up to and including period t, then
clearly the average productivity of workers in period t in the restricted segment is strictly greater than that of the workers in the unrestricted one. If
wages, in turn, are determined competitively, then it follows that the wage
rate in the restricted market segment in period t is greater than that in the
unrestricted one. Accordingly, although a priori naive types do not see the
merit of participating in the restricted market, over time, this market may
become more appealing to them because of the higher wages. As such, the existence of the two labor market segments, predicated as it is on the continued
participation of a typical naive type in the unrestricted market, necessitates
that the restriction in the restricted market on working hours is low enough so
as to dissuade her from participating in it. As a consequence, the participants
in the restricted market only get to avail a restriction on their working time
that is less than their desired working time and end up underworking. So,
to summarize, the equilibrium reasoning that we pursue establishes that the
interaction of competitive market forces with naivety and sophistication in
decision making results in the endogenous functioning of two labor markets
segments. First, an unrestricted one in which the participating workers end
up overworking. Second, a restricted one with a work restriction that is less
than the desired working time of workers, so that the participating workers
3
An abundance of evidence shows that workers’ health is impaired by consistently
working long hours (see Carusco (2006); Harrington (2001); Van der Hulst (2003);
Sparks, Cooper, Fried, and Shirom (1997); Spence and Robbins (1992); (Nishiyama and
Johnson 1997)) leading to a variety of health problems, ranging from exhaustion, high
blood pressure to even death from overwork. The driving force for these negative health
effects is a mismatch between actual and desired working time (Barnett 1998)
5
end up underworking.
The fact that we can simultaneously explain both overwork and underwork
allows us to distinguish our paper from others in the literature which can either
(i) explain only one of the two or (ii) even when they can explain both cannot
account for them simultaneously. Few models in the literature provide a
framework that can simultaneously account for both overwork and underwork.
Cooper (1982) shows that under asymmetric information about the firm’s
technology, overemployment (resp., underemployment) may result if leisure
is a normal (resp., inferior) good. This suggests that in a similar framework
with heterogeneous workers the simultaneous existence of overemployment
and underemployment may be possible, although to the best of our knowledge
this hasn’t been formally worked out. When it comes to papers that can
explain one of the two, the predominant interest in the literature has been to
explain overwork. For example, in Ashworth, McGlone, and Scotland (1977),
the firm is faced with random demand for its output along with large hiring
and firing costs. Because of this, a tradeoff emerges between costly overtime
for a smaller workforce and higher fixed costs for a larger workforce that
may result in workers overworking. Landers, Rebitzer, and Taylor (1996)
highlight the phenomenon of overwork in a model of adverse selection. They
show that when a worker’s record of hours worked is used as an indicator in
promotion decisions, a separating equilibrium may arise under which actual
working hours exceed desired ones. Sousa-Poza and Ziegler (2003) present a
model in which a worker’s productivity is negatively correlated with her desire
for leisure. Under asymmetric information about productivity, firms use long
working hours as a mechanism to sort productive workers which may induce
such workers to overwork. In Rebitzer and Taylor (1995), firms use dismissal
threats to elicit high levels of work effort. Under asymmetric information,
a labor market pooling equilibrium may result that entails fewer short-hour
jobs than is optimal which may explain the phenomenon of overwork.
Our paper also relates to behavioral explanations that have been provided for
6
underwork or overwork. Akerlof (1982) provides an explanation for underwork
in terms of involuntary unemployment which is based on norms of behavior
that are endogenously determined. Benabou and Tirole (2004) assume timeinconsistent preferences to account for overwork. In particular, imperfect
recall of past working experiences may drive some types of agents to work
long hours in order to improve the signal they send to future incarnations of
the self.
The structure of the paper is as follows. In Section 2, we present the framework. In Section 3, we present our equilibrium concept. Finally, in Section
4, we characterize overwork and underwork. The proof of our main result is
provided in the appendix.
2
The Framework
We consider an economy over an infinite horizon with time denoted by t =
0, 1, . . . The economy consists of a unit mass of capitalists and a unit mass
of workers, both distributed over the interval [0, 1] with known distributions.
There is one good that is produced and consumed in each period. This good
is produced using a technology with labor as the only factor of production.
Each capitalist owns such a technology. Each worker is endowed with one unit
of time in every period and she can allot any amount of that time to labor.
Only workers supply labor. We assume that there are no saving opportunities
in this economy.
We allow for the possibility that the labor market may be (endogenously)
segmented into two segments. Under such a segmentation, the first segment,
denoted by k = 1, features a restriction ¯l ∈ (0, 1) on working hours with
the interpretation that ¯l is the maximum amount of time per period that a
worker in that segment can work. The second segment, denoted by k = 2, is
an unrestricted one in which workers can work any amount of time between
7
zero and one. The exact scope of the first market segment shall become
clearer in the subsequent analysis. For now, one may think of this segment as
an institutional commitment to a better work-life balance in the sense that
workers in this segment shall in no period find themselves working more than
¯l units of time. Of course, whether such a market segmentation exists and,
if so, what the exact restriction ¯l is in the restricted market segment are not
immediately obvious. In our analysis, the answer to these questions will be
determined in equilibrium.
Each time period t is split up into two subperiods. In the first subperiod,
capitalists and workers choose which labor market segment to participate in.
We assume that in each period any capitalist and worker can participate in
only one market segment. In the second subperiod, each capitalist chooses
labor demand and each worker chooses labor supply in their chosen segments.
Market clearing wages are determined competitively in each market segment
and production and consumption take place.
Workers In each period t, any worker values current consumption and current leisure. As suggested in the introduction, we think of the consumptionleisure decision of a worker as being borne out of an intrapersonal conflict
between a patient long-term self and an impulsive short-term self. The shortterm self is confronted with spontaneous consumption urges that cause her to
value consumption (relative to leisure) more dearly than the long-term self.
We use the two subperiods structure within a period to model this conflict
thus. In the first subperiod, we assume that the preferences of any worker are
represented by the utility function:
U1 (ct , lt ) = β ln ct + ln (1 − lt ),
(1)
where 0 < β < 1 and ct , lt denote this worker’s consumption and labor
supply in period t, respectively, so that 1 − lt denotes her leisure in this
period. We think of this utility function as representing the preferences of the
long-term self. It underlies the consumption-leisure choices that the worker
8
plans or desires to make. However, in the second subperiod, we assume that
the worker’s preferences are represented by the utility function:
U2 (ct , lt ) = ln ct + ln (1 − lt ).
(2)
We think of this utility function as representing the preferences of the shortterm self. It reflects the consumption-leisure choices that the worker actually
makes. Observe that, whereas the worker’s marginal rate of substitution
between consumption and leisure is β(1 − lt )/ct from the perspective of the
first subperiod (long-term self), it is (1 − lt )/ct from the perspective of the
second (short-term self). That is, the short-term self values consumption
higher than the long-term self does.
Workers take wages as given. Observe that for any prevailing wage rate a
worker will inelastically supply 1/2 units of labor in the second subperiod
if her working time is not regulated.4 However, from the perspective of the
first subperiod, she prefers to work only β/(1 + β) < 1/2. Therefore, workers
who acknowledge the intra-personal conflict in their preferences may have an
incentive to accept a regulation in their working time.
We assume that the proportion of workers who are aware of and acknowledge
this intra-personal conflict is µ, where 0 < µ < 1. We call this type of workers
sophisticated. On the other hand, we refer to the workers who are either
unaware of or fail to acknowledge this conflict as naive. The key difference
between sophisticated and naive types is that whereas sophisticated types
realize that they will choose their working time in the second subperiod of
any period t according to the utility function U2 , naive types mistakenly
believe that they will make this choice based on the utility function U1 .
Overwork and Underwork Since, under our specification of a worker’s
preferences, the desired labor supply decision is independent of the wage
4
Observe that if wt is the prevailing wage rate, then the optimization problem of the
worker in the second subperiod is max{ln wt lt + ln(1 − lt )}
lt
9
rate, we can provide a definition of overwork and underwork that is purely
preference-based. We say that a worker overworks in any period if she is
working more than her desired working time of β/(1 + β). On the other hand,
if she works less than this threshold value, we say that she underworks in that
period.
Worker Productivity We allow for the possibility that the number of
hours that a worker has worked in the past may affect her current productivity.
We model this as follows. Let le ∈ (0, 1) be a threshold such that if a worker
works more than le in a period, it negatively affects her productivity. Let
et denote this worker’s stock, at time t, of all past working hours beyond le .
Then, the evolution of the stock et can be described by the difference equation
et+1 = et + max{0, lt − le }
(3)
with the initial value e0 = 0. We assume that a worker’s productivity at time
t is a function, θ : R+ → [0, 1], of et . We make the following three assumptions
about the function θ:
1. θ(0) = 1 (Normalization)
2. θ is differentiable and θ0 < 0 (Gradual Burnout Effect)
3. lime→∞ θ(e) = θ > 0 (Maximal Burnout Effect)
In other words, although, by condition 2, the stock et adversely affects productivity, condition 3 puts a bound on how detrimental this effect is.
Capitalists In any period t, a capitalist, participating in market segment
k ∈ {1, 2}, produces the single homogeneous good using the technology
F (l, θt (k)) = θt (k)l1/2 , where θt (k) is the average productivity level of the
workers participating in this market segment. We assume that each marginal
10
unit of labor offered in this segment is randomly distributed among the capitalists in it. Hence, each capitalist in a segment produces with the same
productivity as the average productivity level of the workers, θt (k), in that
segment. Capitalists also take wages as given. Accordingly, the profit function
of a capitalist in period t, participating in market segment k, is
1/2
πt = θt (k)lt
− wt (k)lt ,
(4)
where wt (k) is the wage rate in market segment k at time t. We assume that
in any period, a capitalist’s objective is to maximize his consumption and,
hence, profits.
3
Competitive Equilibrium with Labor Market Segmentation
We first analyze the optimal decisions of workers and capitalists. In the way
of notation, for any time period t, γt : [0, 1] −→ {1, 2} and λt : [0, 1] −→
{1, 2} shall denote, respectively, the market segment participation decisions
of workers and capitalists in this period. That is, γt (i) ∈ {1, 2} specifies the
market segment that worker i ∈ [0, 1], in period t, participates in. Likewise,
λt (j) ∈ {1, 2} specifies the market segment that capitalist j ∈ [0, 1], in period
t, participates in.
Labor Supply Decision It is straightforward to verify that for any period
t and any wage rates in market segments 1 and 2, the labor supply decision
of any worker in these segments is given, respectively, by:
(
¯l,
if ¯l ≤ 12
ls (1) =
1
,
otherwise
2
ls (2) =
1
2
11
(5)
Labor Demand Decision In any period t, it is straightforward to verify
that for any capitalist j participating in market segment k ∈ {1, 2}, the
optimal labor demand is given by
ltd (k) =
2
θ̄t (k)
4 [wt (k)]2
(6)
Market Segment Participation Decision of Workers In the first subperiod of any period t, since any sophisticated worker i can correctly predict
her labor supply decision in subperiod 2, optimally choosing which market
segment to participate in requires that γt (i) = k satisfies:
β ln wt (k)ls (k) + ln(1 − ls (k)) ≥ β ln wt (k 0 )ls (k 0 ) + ln(1 − ls (k 0 ))
(7)
for k 0 6= k. On the other hand, since any naive worker i believes that she will be
making her labor supply decision in subperiod 2 based on the utility function
Ut,1 , optimally choosing which market segment to participate in requires that
γt (i) = k satisfies:
max {β ln wt (k)l + ln(1 − l)} ≥ max {β ln wt (k 0 )l + ln(1 − l)}
l∈[0,l(k)]
l∈[0,l(k0 )]
(8)
for k 0 6= k, with ¯l(1) = ¯l and ¯l(2) = 1 denoting the maximum working time
in market segment 1 and 2, respectively. To simplify the characterization of
equilibria, we assume that whenever a sophisticated type is indifferent between
participating in the two market segments, she chooses the market segment
with restriction. On the other hand, for a naive type faced with the same
situation, we assume that she chooses the segment without restriction. Below,
we will refer to this simplifying assumption as the tie-breaking assumption.
Market Segment Participation Decision of Capitalists In any period
t, it is straightforward to verify that for any capitalist j, optimally choosing
which market segment to participate in requires that λt (j) = k satisfies
1/2
1/2
θt (k) ltd (k)
− wt (k)ltd (k) ≥ θt (k 0 ) ltd (k 0 )
− wt (k 0 )ltd (k 0 )
for k 0 6= k.
12
(9)
The Equilibrium We can now define our equilibrium concept. We assume
that there is public record keeping of which labor market segment a worker
has chosen to participate in, in the past. Therefore, how much any worker has
worked in the past can be inferred based on her optimal labor supply decision
(5) and her current productivity can be inferred based on the function θ.
Consequently, the average productivity θ̄t (k) for each market segment k in
period t can be inferred based on the functions γt . In the way of notation, for
any A ⊂ [0, 1], m(A) will denote its measure. In particular, m(γt−1 (k)) and
m(λ−1
t (k)) denote the measure of workers and capitalists in market segment
k in period t, respectively.
Definition 3.1. A competitive equilibrium with labor market segmentation
(CMS) consists of a work restriction ¯l along with a list ls (1), ls (2),{γt , λt ,ltd (1),
ltd (2),θ̄t (1), θ̄t (2), wt (1), wt (2)}∞
t=0 such that the following conditions are satisfied:
1. In each period t, ls (1), ls (2), ltd (1), ltd (2), γt and λt are chosen optimally
given θ̄t (1), θ̄t (2), wt (1) and wt (2) (as specified in equations (5) to (9)).
2. For each t, m(γt−1 (k)), m(λ−1
t (k)) > 0, for k = 1, 2.
3. In each period t, both market segments k = 1, 2, clear,5 i. e.,
d
m(γt−1 (k))ls (k) = m(λ−1
t (k))lt (k).
For ease of notation we will refer to a CMS with working restriction at ¯l as
a CMS-¯l. Condition 1 of the definition requires that in any period t and any
market segment k = 1, 2 of that period, workers and capitalists optimally
choose their labor supply and labor demand, respectively, given wages, wt (k),
in that market segment. This, of course, requires that capitalists correctly
5
It is straightforward to verify that Walras’ law holds in each period, i. e., market clearing
in the two labor market segments implies market clearing in the goods market. Therefore,
we do not specify separately a goods market clearing condition.
13
infer the average productivity level θ̄t (k) in that segment k. Furthermore,
this condition requires that in the first subperiod of any period t, workers
and capitalists optimally decide on which market segment to participate in,
given their beliefs about their behavior in the second subperiod. The second
condition requires that in each period there is a positive measure of capitalists
and workers in each market segment. In other words, the condition requires
that in any period, both the restricted and unrestricted market segments
exist. Finally, the third condition requires that the wage rate is determined
competitively in each period, in each market segment. Of the three conditions,
the second is the most demanding. A priori, it is not obvious that the existence
of the two labor market segments in every period, one with a restriction
on working time at ¯l, is consistent with individual optimization and market
clearing. However, if condition 2 is satisfied and a CMS-¯l exists, it means that
this economy has an institutional arrangement that, in any period, provides
any worker with the ability to commit to working no more than ¯l units of
time. At the same time, it does not force workers to participate in this
institutional arrangement as, in any period, any such worker has the option
to go to the unrestricted labor market where there is no restriction on her
working time. It is also worth noting that because ¯l is an equilibrium object,
any restriction on working time in our model is not exogenously imposed, but
rather endogenously determined.
We will call a CMS-¯l non-trivial if ¯l < 12 . Observe that a CMS-¯l, with ¯l ≥ 12 ,
is equivalent in terms of outcomes to a situation in which a single unrestricted
labor market exists. Therefore, in such a CMS, the segmentation of the labor
market has no content. That is why we focus on the set of non-trivial CMS
in the subsequent analysis.
Next, we relate the issue of overwork-underwork to the concept of a CMS that
is our focus here. We say that a CMS-¯l features overwork and underwork if
¯l < β/(1 + β). For such a CMS, note that ls (1) = ¯l < β/(1 + β), i.e., workers
who participate in the restricted segment work below their desired working
14
time. At the same time, ls (2) =
1
2
> β/(1 + β), i.e., workers who participate
in the unrestricted segment work above their desired working time.
4
Characterizing Overwork and Underwork
We can now state our central result. It establishes that in our framework the
simultaneous existence of overwork and underwork is a robust phenomenon.
Indeed, every non-trivial CMS features overwork and underwork. Furthermore, in each such equilibrium it is the naive types who overwork and the
sophisticated types who underwork.
h
i1
(1+β)1+β 2β
and le < 12 , then the set of non-trivial
Proposition 4.1. If θ ≥ 21+β β β
CMS is non-empty and each such equilibrium features overwork and underwork with the naive types overworking and the sophisticated types underworking in all periods.
h
(1+β)1+β
21+β β β
i 2β1
If θ =
, there is a unique non-trivial CMS-¯l. On the other hand, if
h
i1
1+β 2β
θ > (1+β)
, then there exists a continuum of such equilibria, with work
21+β β β
restriction ¯l in some interval [L, L]. The bounds L and L are determined by
the market segment participation decisions of the naives and sophisticates and
the property that any such equilibrium separates them into different market
segments—the naives participate in the unrestricted whereas the sophisticates
in the restricted market segment.6 The lower bound L is determined by the
fact that if ¯l happens to be too low, then the sophisticated types would prefer
to go to the unrestricted market. On the other hand, L is determined by
the fact that if ¯l happens to be too high, the naive types have an incentive
to switch to the regulated market. This is because, since le is less than 1/2,
the productivity of the naive types falls over time and with it the wages
6
We show this in the proof.
15
in the unregulated segment. As such, what prevents the naive types from
switching to the regulated market, where wages are comparatively higher, is
the fact that they consider the working restrictions to be significantly lower
than their (perceived) desired working time of β/(1+β). For the same reasons
θ cannot be too low, so as to not make the wage differential between the
market segments too high.
To summarize, the structure of incentives that simultaneously sustains overwork and underwork is as follows. A certain fraction of workers, i. e., the naive
types, are inclined to overwork as they do not acknowledge the inconsistency
involved between their desired plans and actual choices. Such overwork reduces their long-run productivities. Further, because the labor market is
competitive and interactions are impersonal, markets do not have the incentive to address the productivity losses associated with persistent overwork.
Together, this sustains overwork. The interesting insight that emerges from
our analysis is that the same mechanism also sustains underwork. This is
because as productivities of the naive types fall and, with it, the wages in the
unrestricted market, what sustains their overwork and dissuades them from
switching to the restricted market is that the restriction in this market is
sufficiently unattractive to them. This of course means that even the sophisticated types, who recognize the intrapersonal conflict in their preferences,
are unable to work at their desired level and are forced to underwork.
5
Conclusion
Several empirical papers point out that there is a misalignment between actual and desired working time of workers. This effect is not only one-sided
but goes in both directions, i.e., overwork and underwork simultaneously exist
in market economies. In our paper, we present a behavioral explanation of
this phenomenon. We show that an intra-personal conflict in preferences that
16
biases workers towards overconsumption along with an adverse impact of long
working hours on workers’ productivity can result in the simultaneous existence of overwork and underwork. The proportion of workers who overwork
and underwork depend on how sophisticated individuals in the economy are.
We have shown that a higher degree of sophistication can increase underwork
and decrease overwork in the economy.
A
Proof of proposition 4.1
Existence of non-trivial CMS
We first show the existence of a non-trivial CMS. To that end, consider any
market segment restriction ¯l < 1 . In any period t, if there exists a positive
2
measure of workers and capitalists participating in each market segment, there
exists a market clearing wage in each of these segments. In any segment k,
this market clearing wage satisfies
d
m(γt−1 (k))ls (k) = m(λ−1
t (k))lt (k)
[θt (k)]2
, it follows that
4[wt (k)]2
−1
1
1 m(γt (k)) s
lt (k)]− 2 θt (k).
[
2 m(λ−1
(k))
t
Given that optimal labor demand is given by ltd (k) =
the wage rate in each segment k is given by wt (k) =
Accordingly, the profits for any capitalist participating in market segment k
in period t is:
1
1
1
πt (k) = θt (k)[ltd (k)] 2 − [ltd (k)]− 2 θt (k)ltd (k)
2
1
1
= θt (k)[ltd (k)] 2
2
21
1
m(γt−1 (k)) s
= θt (k)
lt (k) .
2
m(λ−1
t (k))
If a positive measure of capitalists are to participate in both segments, their
profits have to be equal across segments. That is, πt (1) = πt (2) or ln πt (1) =
17
ln πt (2). Hence
1
m(γt−1 (1))
1
m(γt−1 (2))
s
s
ln θt (1) +
+ ln l (1) = ln θt (2) +
+ ln l (2)
ln
ln
2
2
m(λ−1
m(λ−1
t (1))
t (2))
Accordingly, the distribution of average productivities across segments have
to satisfy
θt (1)
θt (2)
2
=
−1
s
[m(λ−1
t (1))/m(γt (1))]/l (1)
.
−1
s
[m(λ−1
t (2))/m(γt (2))]/l (2)
(10)
Now, consider the market segment participation decision of a sophisticated
type. In any period t, a sophisticate will participate in market segment 1 if
and only if
⇔
β ln wt (1)ls (1) + ln(1 − ls (1)) ≥ β ln wt (2)ls (2) + ln(1 − ls (2)
− 21
1 m(γt−1 (1)) s
β ln
l (1)
θt (1)ls (1) + ln(1 − ls (1))
−1
2 m(λt (1))
− 12
1 m(γt−1 (2)) s
θt (2)ls (2) + ln(1 − ls (2))
≥ β ln
l (2)
2 m(λ−1
(2))
t
12
m(λ−1
t (1))
β ln
θt (1)ls (1) + ln(1 − ls (1))
m(γt−1 (1))ls (1)
12
m(λ−1
t (2))
−β ln
θt (2)ls (2) − ln(1 − ls (2)) ≥ 0
m(γt−1 (2))ls (2)
21
−1
s
[m(λ−1
θt (1)ls (1)
(1 − ls (1))
t (1))/m(γt (1)]/l (1)
β ln
+
β
ln
+
ln
≥0
−1
s
(1 − ls (2))
[m(λ−1
θt (2)ls (2)
t (2))/m(γt (2)]/l (2)
θt (1)
θt (1)ls (1)
(1 − ls (1))
β ln
+ β ln
+ ln
≥ 0 (using equation (10))
(1 − ls (2))
θt (2)
θt (2)ls (2)
⇔
⇔
⇔
⇔
[θ̄t (1)]2β [ls (1)]β (1 − ls (1)) ≥ [θ̄t (2)]2β [ls (2)]β (1 − ls (2))
Defining the function f : [0, 1] × [0, 1] → R by f (l, θ) = θ2β lβ (1 − l) and noting
that ls (2) = 1 and ls (1) = ¯l for ¯l < 1 , we can rewrite the above inequality as
2
2
f (¯l, θ̄t (1)) ≥ f (1/2, θ̄t (2)).
18
On the other hand, a naive type participates in market segment 2 if and only
if
max {β ln wt (2)l + ln(1 − l)} ≥ max{β ln wt (1)l + ln(1 − l)}.
l∈[0,1]
l∈[0,l]
Letting l∗ = argmaxl∈[0,l] {β ln wt (1)l + ln(1 − l)} and noting that
β
1+β
=
argmaxl∈[0,1] {β ln wt (2)l + ln(1 − l)} we have
β
β
β ln wt (2)
+ ln 1 −
≥ β ln wt (1)l∗ + ln(1 − l∗ )
1+β
1+β
Similar calculations as above show that a naive type participates in market
segment 2 if and only if
f (β/(1 + β), θ̄t (2)) ≥ f (l∗ , θ̄t (1))
It is worth noting, that for any fixed θ the function f is strictly concave and
has its maximum at
β
.
1+β
Further, for fixed l the function f is increasing in θ.
To complete the proof of existence of a non-trivial CMS, we need to find a ¯l
such that a positive measure of workers participate in each market segment, in
every period t. We will do so by finding a ¯l for which all sophisticated types
participate in market segment 1 and all naive types participate in market
segment 2. Consider any ¯l in the set
(
1+β )
β
1
1
β
β
≥ lβ (1 − l) ≥
: [θ]2β
(11)
Λ = l ∈ 0,
1+β
1+β 1+β
2
We know that Λ is non-empty, because of our maintained assumption that
h
i1
1+β 2β
.7
θ ≥ (1+β)
21+β β β
Note that in period zero, a naive type weakly prefers segment 2 over segment
1, since f (β/(1 + β), 1) ≥ f (l∗ , 1). Drawing on the tie-breaking assumption,
7
Note that, f (l, θ) = θ2β lβ (1 − l) is continuous in both of its arguments, attains its
maximum in the first argument at β/(1 + β) and satisfies f (0, 1) = 0. Therefore, f (0, 1) <
f (1/2, 1) < f (β/(1+β), 1). By the intermediate value theorem ∃ˆl ∈ [0, β/(1+β)) such that
f (ˆl, 1) = ˆlβ (1 − ˆl) = (1/2)1+β = f (1/2, 1) and for l ∈ (ˆl, β/(1 + β)): lβ (1 − l) > ˆlβ (1 − ˆl).
1
h
i 2β
h
iβ
1+β
β
1
2β
1+β
Further, for θ ≥ (1+β)
,
we
have
[θ]
.
1+β
β
1+β
1+β ≥ (1/2)
2
β
19
it follows that, in period zero, all naive types participate in market segment
2. On the other hand, a sophisticated type participates in segment 1, since
¯lβ (1 − ¯l) = f (¯l, 1) ≥ f (1/2, 1) = 1 1+β , which is true for ¯l ∈ Λ.
2
Next, consider any t ≥ 1 and a sophisticated type i and a naive type j. We
now establish that if all other sophisticated types participate in segment 1
and all other naive types in segment 2 in all periods τ ∈ {0, . . . , t}, then i
participates in segment 1 and j participates in segment 2 in period t. Note
that, if the antecedent of the above statement is true, then in all periods
τ ∈ {0, . . . , t − 1} all (other) naive types have worked 1/2 units and all
(other) sophisticated types strictly less than that. Accordingly, since le < 21 ,
the productivity of all (other) sophisticated types is strictly greater than that
of all (other) naive types in period t. This, in turn, implies that θ̄t (1) > θ̄t (2).
1+β
Consider i’s participation decision. Since ¯l is such that ¯lβ (1 − ¯l) ≥ 12
, it
1+β
follows that [θ̄t (1)]2β ¯lβ (1− ¯l) > [θ̄t (2)]2β 12
, i.e., f (¯l, θ̄t (1)) > f (1/2, θ̄t (2)).
Accordingly, i participates in segment 1 in period t. Next, consider j’s partich
iβ
1
≥ ¯lβ (1 − ¯l). That is,
ipation decision. Note that ¯l is such that [θ]2β β
1+β
1+β
f (β/(1 + β), θ) ≥ f (¯l, 1). Further, since θτ (2) ≥ θ and θ̄τ (1) ≤ 1 for all τ , it
follows that f (β/(1+β), θ̄t (2)) ≥ f (¯l, θ̄t (1)) = f (l∗ , θ̄t (1)), since ¯l < β/(1+β).
Therefore, j participates in segment 2 in period t.
Hence, we have established that in any time period τ ≥ 0 all sophisticated
types participate in market segment 1 and naive types in market segment 2.
Therefore a non-trivial CMS-¯l exists for ¯l ∈ Λ.
All non-trivial CMS feature overwork and underwork.
Next, we show that every non-trivial CMS features overwork and underwork
with the sophisticated types underworking and the naive types overworking.
To do so, we introduce the following lemma.
Lemma A.1. In any non-trivial CMS, if in all periods τ ∈ {0, . . . , t − 1},
20
t ≥ 1, all sophisticates participate in segment 1 and all naives in segment 2,
then they do likewise in period t.
Proof. First, note that, under any non-trivial CMS, if in all periods τ ∈
{0, . . . , t − 1}, t ≥ 1, all sophisticates participate in segment 1 and all naives
in segment 2, then we can show that in period t, it has to be the case
that θ̄t (1) > θ̄t (2). To do so, assume otherwise. If θ̄t (1) ≤ θ̄t (2), then
f (β/(1 + β), θ̄t (2)) ≥ f (β/(1 + β), θ̄t (1)) ≥ f (l∗ , θ̄t (1)). Given the tie-breaking
assumption, this means that all naive types go to market segment 2 and,
hence, market segment 1 is populated only by sophisticated types (since the
measure of workers participating in these markets has to be positive in any
CMS). But, note that in all previous periods the naive types have worked 1/2
units and the sophisticated types strictly less than that. Accordingly, since
le < 12 , the productivity of any sophisticated type is strictly greater than that
of a naive type in period t. This, in turn, implies that θ̄t (1) > θ̄t (2) and brings
us to our desired contradiction.
Next, note that, in any CMS, ¯l satisfies ¯lβ (1 − ¯l) ≥
1 1+β
2
. This is because
as we have shown above, the naives always participate in market segment
2 in period 0. Therefore, for a CMS-¯l to exist, sophisticates have to participate in segment 1 which implies the above condition. When combined
1+β
, i.e.,
with θ̄t (1) > θ̄t (2), this implies that [θ̄t (1)]2β ¯lβ (1 − ¯l) > [θ̄t (2)]2β 1
2
f (¯l, θ̄t (1)) > f (1/2, θ̄t (2)). It, therefore, follows that sophisticates strictly
prefer to participate in segment 1 in period t. In turn, this means that the
naive types must at least weakly prefer to participate in segment 2 over 1
or otherwise, the measure of workers participating in segment 2 will be zero,
contradicting the second equilibrium condition. The tie-breaking assumption
then guarantees that all naive types participate in segment 2 in period t.
We know that in any CMS, in period 0, the sophisticated types participate
in segment 1 and the naive types in segment 2. Therefore, an immediate
corollary of the above lemma is that in any non-trivial CMS, in all periods, the
21
sophisticated types participate in segment 1 and the naive types in segment
2.
Finally, we show that in any non-trivial CMS, ¯l < β/(1+β). To do so, assume
otherwise. Then, it follows that l∗ = β/(1 + β). We have shown above that in
any non-trivial CMS the naive types always participate in segment 2. That is,
in any period t, f (β/(1 + β), θ̄t (2)) ≥ f (l∗ , θ̄t (1)) = f (β/(1 + β), θ̄t (1)). This
necessitates that in any period t, θ̄t (2) ≥ θ̄t (1). But, we know from above that
in any non-trivial CMS for all t ≥ 1, θ̄t (2) < θ̄t (1), which brings us to our
desired contradiction. Therefore, every non-trivial CMS features overwork
and underwork with the naive types overworking and the sophisticated types
underworking.
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