WPS7598
Policy Research Working Paper 7598
The Week
Maya Eden
Development Research Group
Macroeconomics and Growth Team
March 2016
Policy Research Working Paper 7598
Abstract
Is a five-day workweek followed by a two-day weekend over the workweek. The structural parameters of the
a socially optimal schedule? This paper presents a model model are disciplined using daily variation in electricity
in which labor productivity and the marginal utility of usage per worker. The results suggest that increases in
leisure evolve endogenously over the workweek. Labor the ratio of vacation to workdays lead to output losses. A
productivity is shaped by two forces: restfulness, which calibration of the model suggests that a 2–3 day work-
decreases over the workweek, and memory, which improves week followed by a 1 day weekend can increase welfare.
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effort by the World Bank to provide open access to its research and make a contribution to development policy discussions
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may be contacted at meden@worldbank.org.
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The Week
Maya Eden∗
JEL Classiﬁcation: E23, E24, J08
Keywords: week, productivity, fatigue, memory, learning by doing,
leisure, religious institutions, calendar reform
∗
World Bank, Development Economics Research Group, Macroeconomics and Growth
Team. I thank Benjamin Eden, Aart Kraay, Valerie Ramey, Lael Schooler and Luis Serven
for helpful discussions. Contact: meden@worldbank.org.
1 Introduction
Since the 1940s, a ﬁve-day workweek followed by a two-day weekend has dic-
tated the rhythm of economic activity in the US.1 According to the Amer-
ican Time Use Survey, in 2013, 83% of employed persons worked on any
given weekday, whereas only 34% of employed persons worked on a given
weekend.2 This suggests that the structure of the week has real economic
consequences that are beyond the accounting of time. Yet, little is known
about its normative properties: does it maximize welfare, or can we improve
welfare by switching to a diﬀerent labor-leisure cycle? For example, it has
recently been proposed to transition to a four-day workweek, followed by
a three-day weekend.3 If adopted, how will this policy aﬀect output and
welfare?
The starting point of this paper is the observation that given the need for
wide-scale coordination in both production and leisure, observing an equilib-
rium outcome in which the current workweek is practiced is uninformative
of its optimality.4 Given the multiple-equilibria nature of the calendar, as-
sessing the optimality of the current workweek requires a more structural
approach.
To this end, this paper develops a simple representative agent framework
in which both labor productivity and the marginal utility of leisure evolve
1
For an excellent survey on the history of the week, see Zerubavel [1989].
2
See the American Time Use Survey Summary 2013, released on Wednesday, June 18,
2014.
3
See, for example, “Should Thursday Be the New Friday? The Environmental and
Economic Pluses of the 4-Day Workweek”, Scientiﬁc American, July 24, 2009, by Lynne
Peeples; or, “Where the Five-Day Workweek Came From: It’s a relatively new invention
- is it time to shave another day oﬀ?”, The Atlantic, August 21, 2014, by Philip Sopher.
4
For example, a factory worker will not be able to work on a weekend since his factory
will be closed. Similarly, on a workday, a worker on vacation would not derive the same
utility from leisure since his family and friends will be working. For a recent study in
psychology on the beneﬁts of coordinated leisure, see Hartig et al. [2013]. For further
reading on the coordination aspects of production and leisure, see Alesina et al. [2006],
Jenkins and Osberg [2004] and Pande and Pantano [2011].
2
endogenously over the workweek. Each day is designated as either a workday
or a rest day, and production occurs only during workdays. To capture the
eﬀects of fatigue, I assume that productivity depends on a stock of rest that
depreciates during workdays and is replenished during rest days. To capture
potential productivity gains from continual, uninterrupted work, I assume
that productivity depends also on a stock of knowledge that is accumulated
during workdays and depreciates during vacation days.
I restrict the structural parameters of the model based on characteristics
of the comovements between daily electricity usage and predetermined ﬂuctu-
ations in labor supply dictated by the calendar. Since direct output measures
on a suﬃciently high frequency are unavailable, electricity usage is used as
a measure of production intensity. I assume that, on a daily frequency, elec-
tricity and labor inputs are required in ﬁxed proportions. Fluctuations in
electricity reﬂect proportional variation in the utilization rate of capital and
in labor productivity, resulting in proportional variation in output.5
Additionally, I restrict the parameters of the model based on empirical
estimates of the wage penalty associated with recent prolonged absence from
work. In the model, absence from work is associated with a reduction in fa-
tigue, but also a depreciation of the stock of knowledge. The reentry penalty
disciplines the model by requiring that, after long vacations, the knowledge
eﬀect dominates the fatigue eﬀect.
For the set of admissible parameters, I compute the counterfactual pro-
ductivity gains associated with alternative weeks. The results suggest that,
despite the reduction in fatigue, increases in the proportion of vacation to
workdays tend to reduce both output and productivity. An important excep-
tion is the workweek consisting of two workdays followed by a single vacation
day. This schedule generates some productivity gains associated with more
and more-frequent leisure.
5
A similar methodology for recovering variations in eﬀective units of labor has been
previously used in the context of understanding business cycle ﬂuctuations (Basu [1996]).
3
To further explore the welfare implications of alternative schedules, I con-
sider a representative household that values both ﬁnal good consumption and
leisure. I follow Friedman [1969], Kydland and Prescott [1982] and Hotz et al.
[1988] and assume that the marginal utility of current leisure is decreasing
in recent leisure. I calibrate the agent’s relative preference towards leisure
to match a market overtime premium of between 0-50%, consistent with the
ﬁndings of Bell and Hart [2003]. The calibration suggests an optimal week
consisting of a 2-3 day workweek followed by a 1-day weekend. Depending on
the preference for leisure, the expected steady state welfare gains associated
with these alternatives are equivalent to a permanent increase in consump-
tion of between 3-7%. This analysis suggests that increasing the frequency of
leisure may be a more eﬃcient way to reduce worker fatigue than increasing
the proportion of rest to work days.
This paper is closely related to the literature assessing the relationship
between productivity and work hours. The conclusions of this literature are
mixed. Denison [1962], Leslie and Wise [1980], Hanna et al. [2005] Bourles
and Cette [2006], Bourles and Cette [2007], Cette et al. [2011] and Pencavel
[2014] present reduced-form evidence suggesting that, at least beyond a cer-
tain threshold, productivity is decreasing in hours. Feldstein [1967], Craine
[1973], Shapiro [1986] and Hart and McGregor [1988] present evidence that
productivity is increasing with working hours, or that there are higher re-
turns to hours per worker than to workers. Anxo and Bigsten [1989] ﬁnd that
productivity is roughly invariant to hours, and Garnero and Rycx [2014] and
Lee and Lim [2014] document a non-monotone relationship between produc-
tivity and hours.6 This paper departs from previous literature in adopting a
structural approach that implies that the relationship between work duration
and productivity depends not only on average hours but also on the spacing
of vacation time.
6
Similar to the analysis in this paper, Lee and Lim [2014] attribute the non-
monotonicity to the combination of a fatigue eﬀect and a leaning eﬀect.
4
It is worth noting that this paper studies the optimal division of time
into work and rest days, implicitly holding ﬁxed the number of hours per
workday. Pencavel [2014] presents evidence suggesting that variations in the
number of workdays (and, in particular, Sunday work) have a larger eﬀect on
output than variation in total hours worked. The related issue of the optimal
hours per workday is beyond the scope of this paper.
The idea that labor productivity may be aﬀected by fatigue appears else-
where in the literature, for example in Dixon and Freebairn [2009], Marchetti
and Nucci [2001] and Pencavel [2014]. The idea that productivity is increas-
ing with accumulated experience builds on Arrow [1962], Jovanovic [1979],
Lucas [1988] and Lucas [1993] (and a vast literature that followed). Con-
sistent with this view, there is evidence suggesting that part-time work-
ers are less productive than full-time workers (see Aaronson and French
[2004], Baﬀoe-Bonnie [2004] and Hirsch [2005]). The literature mostly treats
learning-by-doing as a stock of knowledge that does not depreciate. How-
ever, evidence from cognitive psychology suggests that it is more diﬃcult to
retrieve memories that have not been recently retrieved (see, for example,
Anderson and Schooler [2000]). In other words, workers may need time to
re-familiarize themselves with their tasks after returning from vacation, and
thus there may be some beneﬁts associated with consecutive, uninterrupted
work. This channel is related to the notion of human capital depreciation
during unemployment explored by the labor literature (see, for example,
Mincer and Ofek [1982], Keane and Wolpin [1997], and Pavoni [2009]).
Finally, this paper is related to the literature assessing the impact of re-
ligion on economic outcomes (see McCleary and Barro [2005] and references
therein). This literature has mostly focused on the role of religion in pro-
viding incentives for pro-social behavior, such as charity that can serve as
informal insurance. This paper contributes to this line of work by assess-
ing the economic impact of one particular religious institution, namely the
seven-day week. The results suggest that the optimal week departs from the
5
cycle of religious observance, which prescribes a single rest day in a seven-day
cycle. However, in section 5.1 I show that it is possible to generate welfare im-
provements under the restriction that the labor-leisure cycle is synchronized
with the cycle of religious worship.
The rest of the paper is organized as follows. Section 2 presents some
institutional background. Section 3 presents a model of production in which
labor productivity depends on the work-rest schedule. Section 4 discusses
restrictions on the parameterization of the model. Section 5 embeds the
production model in a representative agent framework, and discusses welfare
implications. Section 6 concludes and discusses avenues for further research.
2 Background
This section presents a brief survey of the week as an economic institution.
The purpose of this survey is twofold: ﬁrst, to illustrate that the adoption of
the current ﬁve-day workweek was not an outcome of optimal policy; rather,
it was shaped by a variety of factors that are unrelated to its productivity
attributes. Second (and relatedly), the purpose of this survey is to illustrate
the importance of religious observance in shaping the pattern of the workweek
in order to motivate the discussion in section 5.1.
The Hebrew Bible, arguably the most inﬂuential book in history, opens
with a micro foundation of the seven day week.7 According to the Judeo-
Christian tradition, after spending six days creating the world, the Lord
rested on the seventh day. To honor this, Jewish religion places heavy restric-
tions on work during the Sabbath. While Christianity and Islam adopted the
custom of weekly religious observance during Sundays and Fridays, respec-
7
The seven-day week is also rooted in ancient traditions relating to the planetary cycle.
Their primary use was the accounting of time rather than dictating the work-rest schedule.
The Romans used an 8-day maket cycle, which was used as a device for coordinating on
market days. The adoption of the seven-day week came with the adoption of the Julian
calendar in 45 BC.
6
tively, there are mostly no restrictions on work. Nonetheless, most countries
adopted the day of religious observance as a day of rest.
The transition to a two-day weekend took place in the United States
during the beginning of the 20th century, and was legally instituted in 1940
(see Zerubavel [1989]). Prior to that, Sunday was the oﬃcial day of rest and
Saturday was a workday. In 1908, the New England Cotton Mill instituted
a two-day weekend to accommodate the religious observance of its many
Jewish workers. In 1926, Henry Ford began shutting down his factories on
Saturdays, in hope that a two-day weekend would allow suﬃcient time for
travel and increase the demand for cars.
The 1938 Fair Labor Standards Act (instituted in 1940) gave binding
legal status to the two-day weekend. Against the backdrop of the Great De-
pression, the legislation was motivated by the idea that reducing the number
of hours per worker would force companies to hire more workers and help
share employment. While this reasoning may have been appropriate at the
time, it is perhaps less relevant in environments with well-functioning labor
markets. Despite this, the two-day weekend has persisted to the current day.
Many countries followed suit and today, most countries have a two-day
weekend. Although the structure of the week is quite uniform across coun-
tries, consisting of a ﬁve-day workweek followed by a two-day weekend, there
are some exceptions. In some Muslim countries such as Iran or Afghanistan,
workers follow a six-day workweek with Fridays oﬀ. Brunei follows a schedule
with Fridays and Sundays oﬀ and work on Saturdays. Until very recently
(2013-2014), it was customary for schools in France to close on Wednesdays
(making up the hours either on other school days or with Saturday classes).8
There are some interesting historical examples of deviations from the
seven-day week cycle, which were, at least in part, explicitly targeted at
combating religious practice (see Zerubavel [1989] for details). The French
Revolutionary Calendar, instituted in France in 1773, consisted of a nine-
8
See The Economist, “Weird about Wednesday”, September 21st, 2013.
7
(a) The ﬁve-day week, 1930 (b) The six-day week, 1933
Figure 1: Calendars depicting the ﬁve-day week (1929-1931) and the six-day
week (1931-1940) in the Soviet Union. Under the ﬁve-day week, each worker
was assigned a color and took his day oﬀ on his color-coded day. In the
six-day week, every sixth day was a day oﬀ for the entire workforce. Source:
Foss [2004].
day workweek followed by a single vacation day. The conﬂict with religious
observance and the reduction in vacation time made it unpopular, and the
seven-day week was restored in 1805.
Another interesting example comes from the Soviet Union. Between 1929-
1931, the Soviet Union instituted a ﬁve-day workweek in which, on every day,
one-ﬁfth of the workforce had the day oﬀ (see ﬁgure 1a). While this arrange-
ment maximized the capital-labor ratio on any given day, and increased the
proportion of vacation days from one in seven to one in ﬁve, problems re-
sulting from lack of coordination at the workplace led to its abandonment.
In 1931, the ﬁve-day week was replaced by a six-day week with a uniform
day oﬀ for all workers (see ﬁgure 1b). In 1940, the seven-day week was re-
stored, with Sunday as the day of rest. The choice of Sunday as the day of
rest is interpreted by some as evidence that the failure of the six-day week
was rooted in the non-compliance of religious workers, even under the strict
Soviet regime.
8
In light of these historical experiences, it perhaps worth questioning the
implementability of policy prescriptions that require deviations from a seven-
day cycle. Most of the analysis in this paper ignores this constraint, and ﬁnds
that the optimal week consists of shorter cycles of 2-3 workdays followed by
a single vacation day. However, the paper concludes with a discussion on
how the results may be used to eﬃciently restructure a seven-day cycle that
is synchronized with religious worship.
3 Production model
Time is divided into discrete days indexed t ∈ {0, 1, 2...}. Each day is des-
ignated as either a “workday” or as a “rest day”. It is assumed that either
production or leisure (or both) require widespread coordination and that
agents coordinate to work on workdays and rest on rest days. Let χ(t) be an
indicator function that takes the value 1 if and only if day t is designated as
a rest day.
Labor productivity. Eﬀective units of labor are a function of the calen-
dar, the “stock of rest”, R, and the “stock of memory”, M :
αr
Lt = (1 − χ(t))Rt Mtαm (1)
The above speciﬁcation of labor productivity assumes that workers work only
on workdays (1 − χ(t) = 0 on rest days). The parameters αr , αm ∈ (0, 1)
govern the intensity in which rest and memory aﬀect labor productivity.
The stock of rest evolves as follows. Rest depreciates at a rate δr per day.
The stock of rest is replenished by rest days:
Rt+1 = (1 − δr )Rt + χ(t) (2)
where δr ∈ (0, 1). Formally, rest days constitute investment in the stock of
9
rest, helping workers reduce fatigue, ultimately making them more produc-
tive.9
In contrast, the stock of memory increases during workdays and depreci-
ates during rest days. During rest days, workers are engaged in other things
and forget some of their work-related memories. The depreciation rate of the
stock of memory is δm ∈ (0, 1):
Mt+1 = (1 − δm )Mt + (1 − χ(t)) (3)
In this formulation, the depreciation of the stock of memory is positively
related to the length of vacation. A worker returning to work after a single
vacation day will have an easier time reorienting himself than a worker re-
turning to work after a longer vacation. This model is in the spirit of the
cognitive psychology literature on memory (see, for example, Anderson and
Schooler [2000]), which ﬁnds that it is easier to recall information that has
been more recently recalled.
It is worth emphasizing that Rt and Mt capture notions of rest and mem-
ory as they relate to labor productivity, and are unrelated to the utility from
leisure. Preferences will be discussed in section 5.
Intuitively, at the beginning of the week, workers are well-rested but
perhaps somewhat disoriented. As the week progresses, memory improves
but fatigue increases. From a productivity standpoint, a very long workweek
is unlikely to be optimal because of the fatigue eﬀect. A very short workweek
may be suboptimal as well, as consecutive work increases productivity by
making work-related memories more readily available.
Steady state cycles. For a given cycle of i consecutive workdays followed
by j consecutive rest days, there are unique steady state cycles for Rt and
9
Note that this setup is a speciﬁc instance of a more general model, in which Rt+1 =
(1 − δ )Rt + κχ(t). However, as Rt is homogenous in κ (starting from R0 = 0), without
loss of generality I restrict attention to κ = 1.
10
Mt that can be solved recursively using equations 2 and 3. The subsequent
analysis will focus on comparing across diﬀerent steady states, ignoring the
transitions from one steady state to another.
Final good production. The ﬁnal good, Yt , is produced using a Leontief
production function with labor inputs and other underutilized factors, Xt :
Yt = min{Xt , Lt } (4)
This speciﬁcation assumes that, on a daily frequency, labor and other inputs
are required in ﬁxed proportions, and that other factors of production, such
as capital, are ﬁxed and underutilized. This assumption reﬂects the view
that substitutability between goods or inputs is lower at higher frequencies.
A given sewing machine requires ﬁxed amounts of labor, electricity and ma-
terials to produce a garment. It would be diﬃcult to ﬁnd productive uses for
more workers given a single sewing machine. Of course, depending on rel-
ative factor prices, ﬁrms will decide to install sewing machines that require
diﬀerent factor proportions, thus generating some substitutability between
inputs on longer horizons.10
Another crucial feature of equation 4 is that it embeds the assumption of
time-separability in production. While this assumption is commonly used in
the macro literature, it is likely somewhat restrictive when applied to a daily
frequency. For example, livestock that is left unfed for many days will die out,
implying that there are productivity gains associated with shorter vacations.
Alternatively, there are costs associated with sending and recalling long-range
ﬁshing boats, which would imply beneﬁts associated with longer periods of
10
See Gallaway et al. [2003] for a similar discussion regarding the higher substitutability
between consumption goods at longer horizons. More closely related, Griﬃn and Gregory
[1976] suggests that, for a given unit of installed capital, utilization is roughly proportional
to energy inputs, while the decision regarding which unit of capital to install may take
energy requirements into consideration. Eden and Griliches [1993] use a similar Leontief
assumption related to labor and capacity utilization.
11
consecutive work. In these examples, the productivities of non-labor inputs
depend on past values of χ(t). The assumption of time-separability is likely
to apply to most forms of industrial production and services, in which, at
least at high frequencies, the productivity of capital does not depend on its
past use.
Consider a week of length T , in which the ﬁrst (1 − λ)T days are workdays
and the subsequent λT days are rest days. Let Yd denote the steady state
value of output on workday d. The following lemma illustrates how the
evolution of labor productivity over the workweek changes with the model’s
parameters.
Lemma 1 1. If αr = 0 then Yi+1 − Yi ≥ 0 for 1 ≤ i < (1 − λ)T .
2. If αm = 0 then Yi+1 − Yi ≤ 0 for 1 ≤ i < (1 − λ)T .
3. For 1 ≤ i < (1 − λ)T , (Yi+1 − Yi )/Yi is increasing in αr , decreasing in
αm , and decreasing in δr .
4. (Y(1−λ)T − Y1 )/Y1 is decreasing in δm for T = 7 and λ = 2/7 (though
not necessarily for other values of T and λ).
The proof, together with other omitted proofs, is in the appendix. This
lemma illustrates the ﬂexibility of the production structure in generating
diﬀerent evolutions of labor productivity over the workweek. If productivity
depends only on the stock of memory (αr = 0), then it is increasing over
the workweek. If, in contrast, productivity depends only on the stock of
rest (αm = 0), then it is decreasing over the workweek. In general, the
productivity increase between two consecutive workdays is positively related
to the intensity of memory, αm , and negatively related to the intensity of rest,
αr , and to the depreciation rate of rest, δr . While the productivity increase
over the workweek (Monday to Friday) is generally a non monotone function
of δm , numerical results conﬁrm that for the 5-2 workweek, it is decreasing
in δm .
12
To develop intuition for how the optimal week varies with the model’s
parameters, the following lemma illustrates the comparative statics of the
model with respect to diﬀerent schedules:
Lemma 2 Let Y ¯ denote the average output per workday given λ and T :
¯ = (1−λ)T Yd /((1 − λ)T ).
Y d=1
¯ is
1. Assume that only rest is important: αr > 0 and αm = 0. Then, Y
increasing in λ and decreasing in T .
¯
2. Assume that only memory is important: αm > 0 and αr = 0. Then, Y
is decreasing in λ and increasing in T .
The fatigue eﬀect will tend to imply productivity gains associated with
more and more-frequent vacation. Vacation days increase the stock of rest,
which makes labor more productive during workdays. The lemma illustrates
that it is possible to reduce the adverse eﬀects of fatigue not only by increas-
ing the proportion of vacation days (λ), but also by increasing the frequency
of vacation days: for a given λ, a lower value of T would imply productivity
gains from reducing fatigue.
The memory eﬀect works in the opposite direction. More vacation is as-
sociated with lower productivity, because the stock of memory depreciates
during vacation. Moreover, for a given λ, there are productivity gains associ-
ated with “bunching” vacation days to allow for longer periods of consecutive
work. Interruptions from work are costly in terms of productivity, because
workers need to rebuild their stock of memory after it depreciates during the
break.
This analysis illustrates that the productivity gains associated with diﬀer-
ent work patterns depend crucially on the relative importance of the fatigue
and memory eﬀects. The main result of this paper is that more-frequent
leisure increases productivity and welfare, reﬂecting gains from reducing fa-
tigue.
13
4 Parameterization
In this model, labor productivity is a function of the calendar, as well as
two depreciation rates, δr and δm , and two intensity parameters, αr and αm .
In this section, I narrow down the parameter space based on three criteria.
First, the parameters must imply labor productivity that peaks during the
workweek, consistent with the patterns of electricity usage documented be-
low. Second, the variation in productivity over the workweek must be within
the range of predicted variation in electricity usage per worker. Finally, the
parameters must be such that a year of absence from work is associated
with a reentry penalty, consistent with magnitudes estimated in the labor
literature.
The ﬁrst two criteria are based on predicted variation in electricity per-
worker over the workweek. Variation in electricity usage is informative of
variation in the utilization rate of capital, which, in the context of the model,
is proportional to variations in Lt (equation 4). I use data on electricity loads
from seven regional transmission organizations (RTOs): ERCOT, PJM, ISO-
NE, CAISO, MISO, NYISO and SPP.11 The aggregate generating capacity of
these RTOs corresponds to roughly 60% of US totals, though coverage varies
by RTO.12 Observations range from January 1st, 2003 and February 15th,
2015. Data are available on an hourly frequency (and on a higher frequency
for some RTOs) and are aggregated to a daily frequency. To remove long-run
and seasonal trends, the data is de-trended using a weekly moving average.13
To account for potential variation in the number of workers over the
workweek, I use the American Time Use Survey (ATUS) for the years 2003-
11
Data for PJM, ERCOT and SPP are taken from their websites. Data for ISO-NE,
CAISO, MISO and NYISO are taken from energyonline.com.
12
Unfortunately, load data for the remaining RTOs is not readily available. In particular,
the data does not cover most of the southeast, and parts of the northwest and the southwest
of the United States.
13
Speciﬁcally, I construct percent deviations from the weekly average as et = t =E t
t+3 ,
t =t−3
Et
where Et denotes the electricity load data.
14
2013. Survey respondents are asked to report their time usage by minute
for a particular diary day. The data are aggregated to obtain daily working
hours, averaging respondents by their sample weights.14 Note that ATUS
respondents are a random sample of the population aged 15 or older, and in-
clude many respondents that are not full-time employed. Thus, average daily
working hours during the workweek vary between 4-4.5 hours, signiﬁcantly
lower than full-employment hours.
I generate predicted weekday values of electricity and hours based on a
regression that includes weekday ﬁxed eﬀects. Days that fall within a week
of a holiday are omitted, as the dynamics of labor productivity are likely
diﬀerent during those weeks. For example, workers are likely to be more
well-rested on a Tuesday that follows a Monday holiday than on an ordi-
nary Tuesday. Though the model does not include any anticipation eﬀects,
I omit all 7-day periods preceding holidays to allow for such eﬀects (for ex-
ample, people taking time oﬀ before a holiday), and to account for potential
error resulting from the moving-average ﬁltering of the electricity series. Af-
ter omitting holidays and the days surrounding them, the predicted values
of electricity and hours capture “normal” weekly variation induced by the
structure of the week, provided that the dynamic eﬀects of holidays on labor
productivity are contained within a 2-week period.
Figure 2 plots the resulting weekday coeﬃcients for working hours, and
Figure 3 plots the resulting weekday coeﬃcients for electricity usage in four
out of the seven RTOs (the ﬁgures for the remaining RTOs are in Appendix
D). Both hours and electricity exhibit weekly cyclicality, and, consistent with
the model, are signiﬁcantly higher during workdays than during weekends.
Note that both hours and electricity peak during the workweek.
Not surprisingly, the results indicate positive levels of work and electric-
ity usage during weekends, reﬂecting the reality that some people work on
14
Total minutes worked correspond to the sum of categories 050101, 050103 and 050189,
which represent work related activities.
15
4.5
4
3.5
3
Average work hours
2.5
2
1.5
1
0.5
Sun Mon Tue Wed Thu Fri Sat
Day of the week
Figure 2: Predictable weekly variation in working hours over the week.
Dashed lines represent the associated 95% conﬁdence intervals.
weekends, and some electricity is used (for production or otherwise). The
model can accommodate this by assuming some constant amount of hours,
¯ , and some constant amount of electricity, e
H ¯, that do not vary over the
workweek. Using e
ˆd and Hˆ d to denote the predicted values of electricity and
hours on weekday d, I construct the following proxy for output per worker:
eˆd −e
¯
ˆ d −H
¯ if d is a workday (Mon-Fri)
ˆd =
Y H
(5)
0 if d is a weekend (Saturday-Sunday)
where e ¯ = mind {e
ˆd } and H¯ = mind {H ˆ d } are the predicted Sunday values of
electricity and hours, respectively. Note that e ˆd − e
¯ is the “extra” electricity
used on weekday d, beyond the amount that is used during the weekend,
and similarly, Hˆd − H ¯ is the “extra” hours worked on weekday d, beyond
the constant amount worked during the weekend. The proxies Y ˆd capture
the variation in electricity usage per-worker over the workweek, which is
interpreted as variation in labor productivity given the Leontief production
structure.
ˆd , I consider the (log) deviations of Y
To interpret the units of Y ˆd from its
16
1.03 1.06
1.02 1.04
1.01
1.02
1
1
Average electricity
Average electricity
0.99
0.98
0.98
0.96
0.97
0.94
0.96
0.92
0.95
0.94 0.9
Sun Mon Tue Wed Thu Fri Sat Sun Mon Tue Wed Thu Fri Sat
Day of the week Day of the week
(a) (b)
1.08 1.06
1.06
1.04
1.04
1.02
1.02
1
Average electricity
Average electricity
1
0.98 0.98
0.96
0.96
0.94
0.94
0.92
0.92
0.9
0.88 0.9
Sun Mon Tue Wed Thu Fri Sat Sun Mon Tue Wed Thu Fri Sat
Day of the week Day of the week
(c) (d)
Figure 3: Average electricity usage in diﬀerent RTOs over the week. Dashed
lines represent the associated 95% conﬁdence intervals. See ﬁgure 6 for the
remaining three RTOs.
minimal workday value:
ˆd − mind {Y
Y ˆd |d is a weekday}
Yd = (6)
ˆd |d is a workday}
mind {Y
This normalization maps into the range of variation in Yd . For example,
ˆT ue = 0.1 and Y
Y ˆF ri = 0 implies that electricity per hour is, on average, 10%
higher on a Tuesday than on a Friday.
Figure 4 plots the resulting estimates of Yd for four out of the seven RTOs
17
(the ﬁgures for the remaining three RTOs are in Appendix D). The weekly
patterns of Yd seem to diﬀer substantially across RTOs, reﬂecting perhaps
diﬀerences in industrial composition. However, one commonality is that, in
all seven RTOs, Yd is neither monotonically increasing nor monotonically
decreasing over the workweek. This suggests that, at the beginning of the
workweek, there are gains from the reconstruction of M , and towards the
end of the week the fatigue eﬀect dominates. For six out of seven RTOs,
Yd peaks during the workweek. Thus, to account for the pattern in weekly
variation in Yd in most RTOs, it is necessary to consider parameters that
0.14 0.09
0.08
0.12
0.07
0.1
Average electricity per worker (net)
Average electricity per worker (net)
0.06
0.08 0.05
0.04
0.06
0.03
0.04
0.02
0.02
0.01
0 0
Mon Tue Wed Thu Fri Mon Tue Wed Thu Fri
Day of the week Day of the week
(a) (b)
0.12 0.09
0.08
0.1
0.07
Average electricity per worker (net)
Average electricity per worker (net)
0.08 0.06
0.05
0.06
0.04
0.04 0.03
0.02
0.02
0.01
0 0
Mon Tue Wed Thu Fri Mon Tue Wed Thu Fri
Day of the week Day of the week
(c) (d)
Figure 4: Net electricity per worker (Yd ) over the workweek in diﬀerent RTOs
(see also ﬁgure 7).
18
generate productivity peaks during the workweek. This motivates the ﬁrst
criterion, which is that the model’s parameters must be consistent with an
interior maximum of productivity over the workweek.
This ﬁrst restriction is consistent with Brogmus [2007], who documents
that the occupational injury rate in the US is minimized on Wednesdays. If
the weekly variation in the occupational injury rate reﬂects weekly variation
in fatigue and in absent-mindedness that is monotonically related to variation
in productivity, this ﬁnding suggests that productivity peaks mid-week.
The second criterion relates to the range of variation, measured as the
maximum level of Yd . The estimates for diﬀerent RTOs suggest that output
per worker varies by between 8.6% and 22% over the workweek. This range
is inconsistent with a “naive” model in which output per worker is constant
over the workweek; nor is it consistent with extreme parameters which imply
wild variation in output per-worker.
The last criterion is based on empirical estimates of human capital de-
preciation due to absence from work. Mincer and Ofek [1982] and Keane
and Wolpin [1997] (among others) document that recent unemployment or
non-participation in the labor market is associated with lower wages. With
the usual caveats, unemployment or non-participation can be thought of as
a long vacation. Their ﬁndings can therefore be interpreted as suggesting
that, for relatively large increases in vacation, the “learning by doing” eﬀect
dominates the fatigue eﬀect, resulting in an overall loss in worker produc-
tivity. The empirical estimates suggest that a year of absence is associated
with a reentry wage that is 3.5-30.5% lower than the continual employment
wage. These estimates are inconsistent with parameters that imply long-run
gains from rest that outweigh the losses from the depreciation of the stock
of memory.
To isolate the set of admissible parameters, I evaluate the criteria on
1004 points in the parameter space (αr , αm , δr , δm ) ∈ (0, 1)4 . I compute
the steady state productivity cycle implied by each vector of parameters,
19
ysun , ..., ysat . Consistent with the empirical counterparts, I compute the range
as the maximal value of (yd − ymin )/ymin , where ymin is the minimal value of
y during the workweek (Monday through Friday). To compute the reentry
penalty, I assume that the annual wage is equalized with the net present
value of output over the following year, and compare the wage of an agent at
the steady state with the wage of an agent that reenters the workforce after
one year of absence.15
Table 1 summarizes the results. While the criteria are insuﬃcient to
uniquely pin down the model’s structural parameters, they are suﬃcient to
narrow down the set of admissible parameters to 0.25% of the parameter
space. The most binding criterion is the restriction that productivity peaks
during the workweek, which is satisﬁed by only 16.5% of the parameter space.
This suggests that most parameter choices imply either a dominant fatigue
eﬀect or a dominant learning-by-doing eﬀect, generating weekly productivity
patterns that peak either on Monday or on Friday. The second-most binding
criteria are the upper-bound on the range of variation, and the minimal
reentry penalty. This implies that most parameters generate wild variation
in weekly labor productivity, and that many parameters generate only a small
(or even negative) reentry penalty. The restrictions on the maximum reentry
penalty and on the maximum productivity range do not seem to be very
binding.
Table 1: Criteria for admissible parameters
Criterion Percent satisfying criterion
Productivity peak during the workweek 16.5
Productivity range > 8.6% 92
Productivity range < 22% 20.9
Reentry penalty > 3.5% 20.9
Reentry penalty < 30.5% 96
All criteria 0.25
15
To compute the net present value of output, I use an annual depreciation rate of 3 pp.
20
Table 2: Summary statistics of admissible parameters
Mean Standard deviation Range
αr 0.12 0.07 [0.01,0.5]
αm 0.84 0.12 [0.37,1]
δr 0.55 0.2 [0.12,1]
δm 0.27 0.04 [0.16,0.39]
Range 13% 4% [8.6%,22%]
Reentry penalty 5.75% 2% [3.5%,16.6%]]
Table 2 presents some summary statistics for the set of admissible pa-
rameters. It may be interesting to note that while the criteria substantially
narrow down the set of admissible parameters, there is a rather large range of
individual parameters satisfying the criteria; the criteria impose relatively mi-
nor restrictions on the values of particular parameters, and relatively strong
restrictions on the admissible combinations of parameters. The high mean
depreciation rates should be viewed through this lens: while, on their own,
these values might imply implausibly large variation in labor productivity
over the workweek, the restrictions on the set of admissible parameters guar-
antee that any admissible combination of parameters implies weekly variation
in labor productivity that is within the range of predicted electricity usage
per-worker. The mean range of variation is substantially lower than the
upper-bound of 22%, and is consistent with the magnitudes in ﬁgure 4. The
average reentry penalty is 5.75%, substantially lower than the upper bound
of 30.5%.
Figure 5a presents the weekly productivity cycle implied by the mean
values of admissible parameters, which are reported in table 2 (the mean
vector happens to constitute an admissible vector of parameters). For these
parameters, output peaks on Tuesday and declines thereafter, suggesting that
the fatigue eﬀect dominates during most of the week, but that the learning
eﬀect plays a dominant role at the beginning of the week. Figure 5b presents
the distribution of productivity implied by the set of admissible parameters.
21
0.12 0.25
0.1
0.2
Output (relative to minimum weekday value)
Output (relative to minimum weekday value)
0.08
0.15
0.06
0.1
0.04
0.05
0.02
0 0
Mon Tue Wed Thu Fri Mon Tue Wed Thu Fri
Day of the week Day of the week
(a) Mean vector (b) Distribution
Figure 5: Figure 5a presents simulated weekly variation in output for the
mean vector of admissible parameters. Output is normalized as deviations
of the workday minimum (in this case, Friday): (yd − ymin )/ymin . Figure 5b
presents the mean values of (yd − ymin )/ymin and their associated 95 inter-
percentile ranges implied by the set of admissible parameters.
For all admissible parameters, productivity is minimized either on Monday or
on Friday. The admissible parameters also include parameter values in which
either Monday or Friday are highly productive days, with productivity up to
20 percent higher than the minimum.
I proceed by using the set of admissible parameters to evaluate diﬀerent
week structures. Table 3 illustrates the range of output gains (relative to the
current workweek 5-2). The output gain associated with a week consisting
of i workdays and j rest days is computed based on average daily output,
including vacations days. Using yd (i, j ) to denote the output on workday d
given an i-j week, the reported output gains are:
1 i
i+j d=1 yd (i, j )
Output gain = 1 5 (7)
7 d=1 yd (5, 2)
The results reveal that, unfortunately, almost any schedule that has a
22
Table 3: Range of output gains
Rest days
1 2 3
Work 1 [0.55,0.74] [0.27,0.43] [0.17,0.3]
days 2 [0.91,1.07] [0.55,0.71] [0.37,0.53]
3 [1.04,1.23] [0.75,0.87] [0.55,0.7]
4 [1.05,1.3] [0.9,0.96] [0.68,0.81]
5 [1.03,1.33] [1,1] [0.78,0.89]
6 [1,1.36] [1.01,1.07]] [0.84,0.94]
Highlighted cells are schedules in which the ratio of rest days to work days exceed that
of a 5-2 schedule. Output gains are computed as (average daily output)/(average daily
output in the 5-2 week) (equation 7).
Table 4: Range of productivity gains
Rest days
1 2 3
Work 1 [0.79,1.05] [0.58,0.92] [0.48,0.85]
days 2 [0.98,1.15] [0.79,1.01] [0.66,0.94]
3 [0.99,1.18] [0.9,1.04] [0.78,1]
4 [0.93,1.16] [0.96,1.03] [0.85,1.02]
5 [0.88,1.14] [1,1] [0.89,1.02]
6 [0.83,1.14] [0.96,1.02] [0.9,1.01]
Highlighted cells are schedules in which the ratio of rest days to work days exceed that of
a 5-2 schedule. Productivity gains are computed as Y ¯5,2 =(average workday output
¯i,j /Y
in week (i, j ))/(average workday output in the 5-2 week).
higher ratio of vacation to work days generates an output loss. The one pos-
sible exception is the week 2-1 (two workdays followed by one rest day). The
ratio of rest to work days in this schedule is 0.5, which is higher than 2/5=0.4.
Some admissible parameters imply that this alternative is associated with up
to a 7 percentage point increase in output.
While most schedules that increase leisure generate output losses, some
are consistent with productivity gains at least for some admissible parame-
ters. Table 4 presents the average workday productivities relative to the 5-2
benchmark. The 2-1 schedule may be associated with productivity gains of
up to 15 percent. The productivity gains from switching to 4-3 are bounded
23
Table 5: Expected output gains
Rest days
1 2 3
Work 1 0.64 0.35 0.24
days [0.58, 0.7] [0.3, 0.41] [0.19, 0.28]
2 0.97 0.64 0.46
[0.91, 1.04] [0.58, 0.69] [0.41, 0.51]
3 1.12 0.82 0.63
[1.07, 1.19] [0.0.78, 0.86] [0.58, 0.68]
4 1.19 0.93 0.75
[1.12, 1.26] [0.91, 0.95] [0.71, 0.79]
5 1.23 1 0.83
[1.12, 1.3] [1, 1] [0.8, 0.86]
6 1.23 1.04 0.89
[1.11, 1.32] [1.02, 1.06] [0.85, 0.92]
See notes in Table 3. In each cell, the ﬁrst row contains the mean value generated by the
set of admissible parameters, and the second row corresponds to the 95 inter-percentile
range.
above by 2 percent, and bounded below by -15 percent.
Under the minimal assumption that the true vector of parameters is in
the admissible set, the only informative statistics are the ranges of estimates:
for example, we can rule out that the productivity gains from switching
to a particular schedule are lower than the minimal gains implied by the
admissible set, or higher than the maximum gains implied by the admissible
set. As an alternative approach, it is possible to consider a uniform prior on
the admissible set. This prior implies that, absent additional information,
each vector of admissible parameters is equally likely. Under this prior, the
expected gain from switching to a certain schedule is the average gain implied
by the admissible set. Similarly, the 95 inter-percentile range corresponds to
the 95% conﬁdence interval.
Tables 5 and 6 present the expected output and productivity gains from
switching to diﬀerent schedules, together with their associated 95% conﬁ-
dence intervals. Note that all schedules that increase leisure, including 2-1,
are associated with expected output losses. Under the uniform prior, we can
24
Table 6: Expected productivity gains
Rest days
1 2 3
Work 1 0.92 0.76 0.67
days [0.83, 1] [0.65, 0.87] [0.55, 0.8]
2 1.04 0.91 0.82
[0.98, 1.11] [0.83, 0.98] [0.73, 0.92]
3 1.07 0.97 0.9
[1.02, 1.14] [0.92, 1.02] [0.83, 0.96]
4 1.07 1 0.94
[1, 1.13] [0.97, 1.02] [0.89, 0.98]
5 1.05 1 0.95
[0.96, 1.11] [1, 1] [0.92, 0.99]
6 1.03 0.99 0.95
[0.93, 1.1] [0.97, 1.01] [0.91, 0.98]
See notes in Table 4. In each cell, the ﬁrst row contains the mean value generated by the
set of admissible parameters, and the second row corresponds to the 95 inter-percentile
range.
reject any productivity gains associated with moving to a four-day workweek
followed by a three-day weekend at the 5% level. Most schedules that increase
leisure are expected to reduce productivity, while most schedules that reduce
leisure are expected to increase productivity. This suggests that, for most
schedules, the memory eﬀect dominates the fatigue eﬀect in shaping produc-
tivity patterns. However, an important exception is 2-1, which increases both
leisure and expected productivity. While this schedule is associated with an
increase in leisure of 25 percent,16 the expected output loss is only 3 percent.
5 Welfare implications
The analysis in the previous section suggests that there are no “free lunches”
associated with switching to diﬀerent calendars, as any increase in the propor-
tion of vacation days is likely to be met with a drop in output. Nonetheless,
16
Under 2-1, the ratio of vacation to workdays is 0.5. Under 5-2, the ratio is 2/5=0.4.
The diﬀerence between the two is 0.1 = 0.25 ∗ 0.4.
25
depending on the preferences towards ﬁnal good consumption and leisure,
there may be some gains associated with changing the structure of the week.
To evaluate welfare, it is necessary to take a stance on agents’ preferences
towards leisure. I therefore begin by discussing how to introduce utility
from leisure into this framework. Consider a representative household, whose
preferences are given by:
∞ 1−φ φ γ
t ( ct lt )
U ({ct , lt }∞
t=0 ) = β where lt = ft ({χ(τ )}∞
τ =0 ) (8)
t=0
γ
In this environment, β ∈ (0, 1) is the discount factor; γ governs the in-
tertemporal elasticity of substitution; and φ ∈ (0, 1) is the preference towards
leisure.
Leisure is denoted by lt , where lt is a function of the calendar. The
conventional approach of introducing leisure as a ﬂow (e.g., one minus the
share of time spent on work) and assuming time-separable utility from leisure
does not seem appropriate for evaluating leisure at a daily frequency: it
does not capture the plausible notion that the marginal utility of a vacation
depends on past vacation. I therefore follow Kydland and Prescott [1982]
and assume the following utility function:
∞
lt = α0 χ(t) + α1 (1 − η )τ χ(t − τ ) (9)
τ =0
The variable lt is a linear combination of current vacation time, χ(t), and a
discounted sum of past vacation time, where the discount rate is 1 − η (and
η ∈ (0, 1)). The parameters α0 and α1 capture the relative importance of
current and past vacation, respectively. The case α1 = 0 is the time-separable
speciﬁcation, in which only current vacation enters into the utility function,
whereas α1 → ∞ implies that past vacation is more relevant for evaluating
the marginal utility of leisure. Note that when η → 0, the utility from leisure
is roughly proportional to the share of rest days in the week.
26
This speciﬁcation of preferences assumes that the marginal utility of va-
cation is higher after a long period of consecutive work than after a recent
vacation.17 While this concavity is commonly assumed in the macro litera-
ture, one might question its applicability at a daily frequency. For example,
if a longer vacation allows for more weekend travel, agents might have a pref-
erence for leisure bunching over leisure smoothing. This preference cannot
be incorporated in the functional form in equation 9. However, there are per-
haps also good reasons for assuming a preference towards leisure smoothing
at a daily frequency - Milton Friedman addresses this issue in the Appendix
to his seminal essay “The Optimum Quantity of Money” (Friedman [1969]):
“Expenditure on going to the movie is regarded as expenditure
on the maintenance or building up of capital in the form of a
stock of memories of movies seen. The stock may depreciate very
rapidly, in which case, for example, for some individuals, it may
require going to one movie a week to keep the stock constant, but
the utility derived from the stock is regarded as not concentrated
at the moment of paying for the movie ticket, or even during the
time of seeing the movie, but as derived at a steady rate so long
as the stock is maintained.”
In equation 9, ∞ τ
τ =0 (1 − η ) χ(t − τ ) can be interpreted as a stock of leisure
along the lines suggested in Friedman [1969]. The stock depreciates at a daily
rate of η and is maintained by vacation days, which constitute investment in
the stock.
I assume that agents can smooth the consumption of the ﬁnal good by
buying and selling bonds (bt ). The interest rate r satisﬁes β (1 + r) = 1.
This assumption abstracts from consumption smoothing motives in shaping
17
Note that this result follows from a relabelling of Lemma 2, part 1: if output depends
only on the stock of rest, there are productivity gains associated with shorter weeks. Simi-
larly, since utility depends only on a stock of leisure (and on time-separable consumption),
there are utility gains from more-frequent leisure.
27
the optimal calendar. Given reasonable discount rates, β → 1 on a daily
frequency, and thus r → 0; the assumption β (1 + r) = 1 is thus similar to
assuming a storage technology for goods. In contrast, leisure is non-tradable
and non-storable.
The social planner’s problem takes the following form:
max U ({ct , lt }∞
t=0 ) (10)
{{χ(t)}∞ ∞
t=0 ∈Γ,bt+1 ,ct ,lt ,Mt ,Rt ,Lt ,Yt }t=0
s.t. b0 = 0, R0 , M0 , {χ(−t)}∞t=1 , equations 1, 2, 3, 8 and 9, Yt = Lt and the
budget constraints:
ct + bt+1 = Yt + (1 + r)bt (11)
∞
bt
≥0 (12)
t=0
(1 + r)t
The budget constraints (equations 11-12) imply that the net present value
of consumption equals the net present value of output. Since output is as-
sumed to be proportional to Lt (Yt = Lt ), this speciﬁcation abstracts from
costs associated with producing non-labor inputs.
To discuss the implications of this omission, it is useful to distinguish
between capital and materials (which include capital utilization costs such
as electricity). The omission of materials will not aﬀect the results under
the assumption that the unit cost of producing materials is constant.18 The
omission of payments to ﬁxed capital will tend to overstate the consumption
gains associated with productivity gains, as, in the model, more productive
18
To illustrate, assume that the cost of producing electricity (e) is p, and that output
is given by Yt = {et , Lt }. Optimality requires that et = Lt , and thus the expenditure
on electricity is proportional to output. Since ignoring these costs amounts to a propor-
tional change in output, a percent change in output will map into a percent change in
consumption. However, if there are decreasing returns in the production of materials, the
price of materials will be increasing with et , and the expenditure share on material will be
increasing with Lt . In this case, the model over-states the consumption gains associated
with schedules that imply productivity gains.
28
labor requires more capital.19 However, this may not be a realistic feature
of the model. It is possible to think of circumstances in which the eﬃcient
level of capital is proportional to the number of workers, and is invariant to
labor productivity. For example, even the most productive textile worker
uses one sewing machine at a time. Highly productive researchers may use
the same amount of computers as less productive researchers, etc. If the
optimal capital stock depends on the number of workers rather than on labor
productivity, any gains in output will map into consumption gains and will
not necessitate additional investment.
The set Γ (equation 10) represents the planner’s choice set of calendars.
As a starting point, I assume that Γ consists of a set of weeks, where a week
is deﬁned as a calendar consisting of a recurring cycle of work and vacation
days. I restrict attention to weeks that have no more than 6 consecutive work
days and 3 consecutive rest days. These restrictions are made primarily for
computational reasons.20 Later, I will consider the alternative restriction
that calendars must conform to a seven-day cycle.
Welfare gains, denoted wi,j , are computed as the compensating increase
in consumption that leaves the representative agent indiﬀerent between a
week consisting of i workdays and j rest days and the 5-2 week:
U ({(1 + wi,j )ct (5, 2), lt (5, 2)}∞ ∞
t=0 ) = U ({ct (i, j ), lt (i, j )}t=0 ) (13)
where ct (i, j ) and lt (i, j ) denote equilibrium values corresponding to an i-
j week, at the steady state. Note that this exercise abstracts away from
19
Given the Leontieﬀ production structure, the eﬃcient level of the capital stock is
given by the maximal level of Lt : Kt = max{Lt }. If capital depreciates at a rate δ ,
steady state investment is given by δ max{Lt }, and steady state consumption is given by
c = rN P V (Y ) − δ max{Lt } (where N P V (Y ) is the net present value of output). Thus, the
omission of capital accumulation would tend to overstate the consumption gains associated
with schedules that increase max{Lt }.
20
The set of all possible calendars between dates 0 and t is 2t , since each day is asso-
ciated with a binary indicator variable. The restriction to a limited set of weeks reduces
dimensionality substantially.
29
transitional dynamics, and compares diﬀerent steady-state welfare levels.
Calibration. Table 7 summarizes the calibration of the preference param-
1
eters. I assign β = 0.97 365 to capture an annual discount rate of 3 percent
adjusted for a daily frequency, and, following Kydland and Prescott [1982],
set γ = −0.5. As a baseline speciﬁcation, the parameters α0 , α1 and η
are chosen to correspond to those in Kydland and Prescott [1982]. As an
alternative speciﬁcation, α0 , α1 and η are calibrated instead to match the
ﬁndings in Hotz et al. [1988]. Appendix A describes the procedure for ad-
justing these parameters for a daily frequency (Kydland and Prescott [1982]
calibrate their model for quarterly data, and Hotz et al. [1988] estimate a
model using annual data). The two speciﬁcations yield highly similar re-
sults, which are distinguishable only at the third decimal point.21 Since the
reported results are rounded at the 2 decimal point level, they are consistent
with both speciﬁcations.
I calibrate φ so that, given a 5-2 workweek, an agent is indiﬀerent with
respect to working on a Saturday given an overtime premium of s.22 Bell
and Hart [2003] estimate the overtime premium to be between 0-50%. This
maps into a range of φ ∈ (0.3, 0.4). Within this range, I consider the two
extreme values φ = 0.3, 0.4.
Table 8 presents the range of welfare gains implied by the set of admis-
sible parameters, and Table 9 presents the corresponding means with their
associated 95% conﬁdence intervals. The results suggest that the 5-2 week
is not easily dominated by any other week. When φ = 0.3, for any other
21
Given η → 0 and β → 1, both speciﬁcations are approximately equal to a model in
which l is proportional to the average ratio of rest and work days.
22
In Kydland and Prescott [1982], the parameter φ is calibrated as 2 3 to match the
steady state fraction of time spent on work. The calculation is roughly as follows: people
spend around 40 hours a week at work. Assuming that 8 hours a day are necessary for
−40
sleep, there are 16x7=112 available hours a week: 112 112 ≈ 23 . When the unit of account
is days rather than hours, the calculation changes, as the steady state number of workdays
2
is approximately 5 days a week, or 7 ≈ 0.3. This value is consistent with the lower end of
the calibrated range of φ.
30
Table 7: Simulation parameters - preferences
Paramter Value Source Original
β 0.9999 Kydland and Prescott [1982] 0.99
γ -0.5 Kydland and Prescott [1982] -0.5
φ 0.3,0.4 Bell and Hart [2003] s ∈ (0, 50%)
η 0.0011 Kydland and Prescott [1982] 0.1
α0 0.0057 Kydland and Prescott [1982] 0.5
α1 0.00057 Kydland and Prescott [1982] 0.05
η 0.0014 Hotz et al. [1988] 0.4
α0 0 Hotz et al. [1988] 0
α1 0.0035 Hotz et al. [1988] 1
Parameters are speciﬁed in a daily frequency. The original parameters are speciﬁed for
a quarterly frequency in Kydland and Prescott [1982] and an annual frequency in Hotz
et al. [1988].
week, there is some vector of admissible parameters that would suggest wel-
fare losses from that week. When φ = 0.4, there are two alternatives that
strictly dominate the 5-2 workweek: 4-2 and 2-1.
Table 9 suggests that expected welfare is maximized by a 2 or 3 day
workweek followed by a 1 day weekend. Expected steady state welfare gains
vary between 3-7%. Note that 3-1 is associated with an expected 12% gain
in output, while the expected output loss from 2-1 is only 3% (see Table
5). The increase in leisure is suﬃcient to compensate for this loss, resulting
in a welfare gain. Whether the optimal schedule is 2-1 or 3-1 depends on
the preference parameter φ. However, conditional on φ, the welfare gains
associated with one of these alternatives are statistically signiﬁcant at the
5% level.
The results are also useful for evaluating alternative schedules that have
been practiced or proposed. For example, the analysis suggests that the
transition from 6-1 to 5-2 during the 1940s was almost certainly welfare
improving. However, current proposals to transition to a 4-3 workweek are
likely to be associated with welfare losses.
31
Table 8: Range of welfare gains (wi,j )
Rest days
φ = 0.3 φ = 0.4
1 2 3 1 2 3
1 [-0.29,-0.06] [-0.61,-0.38] [-0.75,-0.55] [-0.19,0.07] [-0.52,-0.25] [-0.68,-0.43]
Work days
2 [-0.03,0.15] [-0.3,-0.1] [-0.49,-0.27] [0.01,0.19] [-0.2,0.03] [-0.39,-0.13]
3 [-0.02,0.17] [-0.13,0.01] [-0.31,-0.11] [-0.05,0.13] [-0.06,0.09] [-0.21,0.01]
4 [-0.1,0.12] [-0.04,0.03] [-0.19,-0.03] [-0.17,0.03] [0,0.06] [-0.11,0.06]
5 [-0.18,0.06] [0,0] [-0.12,0] [-0.28,-0.07] [0,0] [-0.06,0.07]
6 [-0.26,0.02] [-0.04,0.01] [-0.11,0] [-0.37,-0.14] [-0.07,-0.02] [-0.07,0.04]
Highlighted cells are schedules in which the ratio of rest days to work days exceed that of
a 5-2 schedule. Welfare gains are computed using equation 13.
Table 9: Expected welfare gains (wi,j )
Rest days
φ = 0.3 φ = 0.4
1 2 3 1 2 3
1 -0.18 -0.49 -0.64 -0.07 -0.38 -0.55
[-0.26,-0.11] [-0.57,-0.41] [-0.71,-0.58] [-0.15,0.02] [-0.47,-0.28] [-0.63,-0.47]
2 0.04 -0.19 -0.37 0.07 -0.08 -0.24
[-0.02,0.11] [-0.26,-0.13] [-0.44,-0.3] [0.02,0.15] [-0.15,0] [-0.33,-0.16]
Work days
3 0.06 -0.06 -0.2 0.03 0.02 -0.09
[0.01,0.13] [-0.1,-0.01] [-0.26,-0.14] [-0.02,0.09] [-0.03,0.07] [-0.16,-0.02]
4 0.02 -0.01 -0.11 -0.06 0.03 -0.02
[-0.04,0.08] [-0.03,0.02] [-0.16,-0.06] [-0.12,0] [0.01,0.05] [-0.07,0.03]
5 -0.03 0 -0.07 -0.14 0 0
[-0.11,0.03] [0,0] [-0.1,-0.03] [-0.22,-0.1] [0,0] [-0.04,0.03]
6 -0.08 -0.02 -0.05 -0.22 -0.05 -0.02
[-0.17,-0.02] [-0.04,0] [-0.09,-0.02] [-0.3,-0.17] [-0.07,-0.03] [-0.05,0.02]
See notes in Table 8. In each cell, the top row is the average welfare gain and the bottom
row is the 95 inter-percentile range.
32
5.1 The optimal seven-day cycle
The historical experience described in section 2 suggests that there may be
diﬃculties in implementing calendars that are unsynchronized with the cycle
of religious worship. In light of this, I consider the optimal calendar under
the restriction of a seven-day cycle. The set of calendars that satisfy this
restriction is only partially included in the previous analysis, as it includes
some cycles with non-consecutive vacation time (e.g., a week with Tuesday
and Saturday oﬀ). I compute output and welfare for all possible seven-day
cycles beginning with a workday and ending with a rest day (there are 25
such cycles, many of which are equivalent).
It turns out that in all speciﬁcations, the seven-day cycle that maximizes
expected welfare is 3-1-2-1: three workdays followed by a single vacation
day, followed by two additional workdays and ending with a vacation day.
For example, in Christian countries, this cycle could consist of a week in
which Sundays and Wednesdays are days oﬀ; in Muslim countries, the cycle
could consist of a week in which Fridays and Tuesdays days oﬀ; and, in
Jewish countries, the cycle could consist of a week in which Saturdays and
Wednesdays are days oﬀ.23 The expected welfare gains from this cycle are
equivalent to a 5.6 percent permanent increase in consumption.
The ﬁrst column of Table 10 summarizes the distribution of steady state
welfare gains implied by the set of admissible parameters. Given that η → 0
in both the Kydland and Prescott [1982] and Hotz et al. [1988] speciﬁcations,
utility from leisure is approximately proportional to the share of vacation
days. Thus, since the proportion of vacation days in 3-1-2-1 remains 2/7,
the welfare gains reﬂect gains in steady state output: w = (y (3 − 1 − 2 −
1) − y (5 − 2))/y (5 − 2), where y (·) is the average output level induced by
the calendar. Consequently, across all speciﬁcations, the steady state welfare
23
It is easy to see that there are other equivalent options, for example, in a Christian
country, having Sunday and Wednesday oﬀ is the same as having Sunday and Thursday
oﬀ.
33
Table 10: Welfare-improving seven day cycles
3-1-2-1 4-1-1-1
Mean output gains 1.056 1.035
Mean welfare gains 0.056 0.035
95 percent inter-percentile range: output gains [1.01,1.12] [1.01,1.08]
95 percent inter-percentile range: welfare gains [0.01,0.12] [0.01,0.08]
Range of output gains [1,1.16] [1,1.11]
Range of welfare gains [0,0.16] [0,0.11]
Output gains and welfare gains are computed relative to a 5-2 workweek. The schedule 3-
1-2-1 (ﬁrst column) corresponds to three workdays, followed by one vacation day, followed
by two additional workdays and ending with a vacation day. The schedule 4-1-1-1 (second
column) corresponds to the Brunei calendar: four workdays followed by a vacation day, a
workday and a vacation day. While there are other seven-day cycles that generate welfare
gains for some parameters, these two schedules are the only ones that create welfare
improvements for all parameter values.
gains are the same and equal to the output gains. Further, it is worth noting
that the steady state welfare gains are positive for all admissible parameters.
While 3-1-2-1 is able to accommodate weekly religious worship, it can-
not simultaneously accommodate the Christian, Muslim and Jewish worship
days, and is thus perhaps more feasible in religiously-homogeneous countries.
However, it should be noted that the current 5-2 schedule can only accommo-
date two out of the three major religious worship days. For example, in the
United States and in Europe, the convention of having Saturday and Sunday
oﬀ accommodates Christians and Jews but not Muslims.
In many predominantly Christian countries in Europe, the Muslim pop-
ulation far exceeds the Jewish population. In these countries, one might
consider the adoption of the Brunei calendar, that has Fridays and Sundays
oﬀ (however, one should also consider the fact that the “cost” of working on
the Sabbath may be larger for Jewish people, since Islam does not prohibit
Friday work while Judaism places heavy restrictions on Saturday work). To
evaluate the economic implications of this alternative, the second column of
34
Table 10 presents the welfare statistics associated with the 4-1-1-1 schedule.
The analysis suggests that there are positive welfare gains from adopting the
Brunei calendar, with expected welfare gains of 3.5 percent.
6 Conclusion
There are several challenges associated with assessing the eﬃciency properties
of the 5-2 week and quantitatively evaluating the tradeoﬀs associated with
altering its structure. This paper attempts to formalize these challenges and
make preliminary steps towards addressing them.
The ﬁrst challenge relates to evaluating the relationship between work
duration and productivity on a daily frequency. Conceptually, increasing the
duration of consecutive work periods may be associated with some produc-
tivity losses due to fatigue, or with some productivity gains due to learning-
by-doing. Empirically evaluating the quantitative relevance of each of these
channels presents a challenge given that daily output measures are unavail-
able. Under the assumption that, on a daily frequency, electricity and labor
are required in ﬁxed proportions, the structural parameters of the model can
be estimated based on movements in electricity usage. The ﬁndings suggest
that both channels are quantitatively relevant for the evolution of labor pro-
ductivity over the workweek. However, a permanent reduction in the ratio
of workdays to rest days unambiguously results in lower output.
A second challenge relates to calibrating the preference towards leisure.
Evaluating alternative schedules requires taking a stance towards the pref-
erence for leisure smoothing on a relatively high frequency. Here, I follow
the macro literature and assume that agents have a preference for smoothing
leisure. However, the institutional practice of “long weekend” holidays (e.g.
Labor Day, Presidents’ Day, Memorial Day etc) suggests that there may be
some preference for leisure-bunching at high frequencies, or some ﬁxed costs
associated with vacation. These plausible channels have the potential to
35
eliminate the gains from switching to a shorter week cycle. Further analysis
regarding the preference towards leisure smoothing is necessary in order to
reﬁne the welfare analysis in this paper.
Finally, there may be scope for introducing heterogeneity into this frame-
work. Here, I consider the simplest case of a representative agent. The min-
imal criteria on the set of admissible parameters allows for some diﬀerences
in parameters across occupations. However, the structural parameters may
systematically diﬀer across occupations, and therefore the optimal structure
of the week may vary with occupational composition. For example, cogni-
tive tasks may be more intensive in memory and less subject to fatigue than
physical labor. Additionally, a richer framework may be able to incorporate
the reality of many people working during weekends.
The ﬁndings of this paper suggest that there are potentially large welfare
gains associated with restructuring the week. In light of this, there is scope
for further research geared at meeting the above challenges.
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40
A Calibration details
This section documents the process of adjusting α0 , α1 and η for an annual
frequency. Let α ˆ 1 and η
ˆ0, α ˆ denote estimated parameters which were derived
for a unit of time consisting to T days.
The discount rate η is calculated in the standard way, so that (1 − η )T =
1−η ˆ. The remaining parameters, α0 and α1 , are more problematic: their
relative magnitudes governs the relative importance of “current” leisure rel-
ative to “past” leisure. When a time period consists of T days, some “past”
days are regarded by the agent as “current”. I therefore transform α0 and
α1 so that the end-of-period utility is maintained. Formally, I choose α0 and
α1 so that for every L1 and L2 :
T −1 ∞
ˆ1 (1 − η
α ˆ) t
(ˆ ˆ 1 )L1 +
α0 + α L2 = (α0 + α1 ) (1 − η ) L1 + α1 (1 − η )t L2 (14)
ˆ
η t=0 t=T
The expressions correspond the value of lt that is constructed from L1 units
of leisure for the past T days (including the current day), and L2 units of
leisure forever before that. This implies that:
T −1
1 − (1 − η )T
(ˆ
α0 + α
ˆ 1 ) = (α0 + α1 ) (1 − η )t = (α0 + α1 ) (15)
t=0
η
η
⇒ α0 = (ˆ
α0 + α
ˆ 1 )( ) − α1
1 − (1 − η )T
and:
∞
ˆ1 (1 − η
α ˆ) (1 − η )T ˆ1 (1 − η
α ˆ) η
= α1 (1 − η )t = α1 ⇒ α1 = ( T
) (16)
η
ˆ t=T
η η
ˆ (1 − η )
Substituting into equation 15 generates the value for α0 .
In Hotz et al. [1988], the utility function is speciﬁed slightly diﬀerently,
41
as:
∞ ∞
τ −1 α
lt = χ(t) + α (1 − η ) χ(t − τ ) = χ(t) + (1 − η )τ χ(t − τ ) (17)
τ =1
1−η τ =1
(where, in their framework, χ(t) is a continuous variable representing leisure
at time t). Rewriting the above to correspond to the speciﬁcation in Kydland
and Prescott [1982] yields:
∞
α α
lt = χ(t) − χ(t) + (1 − η )t χ(t − τ ) (18)
1−η 1−η t=0
α α
ˆ 0 = 1 − 1−
In this framework, α η
ˆ1 =
and α 1−η
. For the simulation, I use the
estimated values of η = 0.4 and α = 0.6.
B Proof of Lemma 1
To prove the ﬁrst part of the Lemma, let M0 = MT denote the steady state
stock of memory on the last vacation day (Sunday in the current workweek).
On workday 1 ≤ i ≤ (1 − λ)T , the stock of memory Mi is given by:
M1 = (1 − δm )M0 + 1 (19)
M2 = (1 − δm )M1 + 1 = (1 − δm )2 M0 + (1 − δm ) + 1 (20)
.
.
. (21)
i− 1
i 1 − (1 − δm )i
Mi = (1 − δm ) M0 + (1 − δm )t = (1 − δm )i M0 + (22)
t=0
δm
42
Since there is no accumulation of memory during the weekend, M0 = (1 −
δm )λT M(1−λ)T , and hence:
1 − (1 − δm )(1−λ)T
M0 = (1 − δm )λT ((1 − δm )(1−λ)T M0 + ) (23)
δm
(1 − δm )λT − (1 − δm )T
= (1 − δm )T M0 +
δm
Solving for M0 yields:
(1 − δm )λT − (1 − δm )T
M0 = (24)
δm (1 − (1 − δm )T )
To show that Mi is increasing in i for 1 ≤ i ≤ (1 − λ)T , note that the
derivative of Mi (equation 22) with respect to i is:
∂Mi ∂ (1 − δm )i 1
= (M0 − ) (25)
∂i ∂i δm
∂ (1−δm )i
Since ∂i
< 0, the above is positive if and only if:
1
M0 < (26)
δm
which follows from equation 24. It follows that M(1−λ)T > M1 , and hence, if
αr = 0 and αm > 0, Yi is decreasing in i.
To prove the second part of the lemma, let R0 = RT and note that
for 1 ≤ i ≤ (1 − λ)T , Ri = (1 − δ )i R0 . Thus, Ri is decreasing in i for
1 ≤ i ≤ (1 − λ)T , and hence, if αm = 0, Yi is decreasing in i.
To prove the third part of the lemma, note that:
αr αm αr αm
Yi+1 − Yi Ri +1 Mi+1 − Ri Mi
= αr (27)
Yi Ri Miαm
Ri+1 αr Mi+1 αm
=( ) ( ) −1 (28)
Ri Mi
43
The derivative with respect to αr is negative because Ri+1 < Ri . The
derivative with respect to αm is positive because Mi+1 > Mi .
To show that (Yi+1 − Yi )/Yi is decreasing in δr , note that for 1 ≤ i ≤
(1 − λ)T , Ri = (1 − δr )i R0 and hence (for i < (1 − λ)T ):
Ri+1 (1 − δr )i+1 R0
= = (1 − δr ) (29)
Ri (1 − δr )i R0
It follows from equation 27 that (Yi+1 − Yi )/Yi is decreasing in δr .
To prove the ﬁnal part of the lemma, using equation 27, rewrite the
expression as:
Y(1−λ)T − Y1 R(1−λ)T αr M(1−λ)T αm
=( ) ( ) −1 (30)
Y1 R1 M1
Thus, the derivative with respect to δm depends on the derivative of the ratio
M(1−λ)T /M1 with respect to δm . Using the above derivations,
(1−λ)T
M(1−λ)T (1 − δm )(1−λ)T M0 + 1−(1−δδm
m
)
= (31)
M1 (1 − δm )M0 + 1
Substituting for M0 yields:
λT T
M(1−λ)T (1 − δm )(1−λ)T (1−δ1
m) −(1−δm )
−(1−δm )T
+ 1 − (1 − δm )(1−λ)T
= λT δm )T
(32)
M1 (1 − δm ) (1−δm ) −(1−T
1−(1−δm )
+ δm
(1−δm )T −(1−δm )(2−λ)T
1−(1−δm )T
+ 1 − (1 − δm )(1−λ)T
= (1−δm )λT +1 −(1−δm )T +1
1−(1−δm )T
+ δm
(1 − δm )T − (1 − δm )(2−λ)T + (1 − (1 − δm )(1−λ)T )(1 − (1 − δm )T )
=
(1 − δm )λT +1 − (1 − δm )T +1 + δm (1 − (1 − δm )T )
(1 − δm )T + 1 + (1 − δm )(1−λ)T
=
(1 − δm )λT +1 − (1 − δm )T +1 + δm (1 − (1 − δm )T )
44
It is possible to conﬁrm numerically that this expression is decreasing in
δm for T = 7 and λ = 2/7; in general, this expression is non-monotone.
For example, T = 70 and λ = 2/7 generates a non-monotone relationship
between M(1−λ)T /M1 and δm .
C Proof of Lemma 2
To prove the ﬁrst part of the lemma, solving recursively for R0 yields:
λT −1
T 1 − (1 − δr )λT
R0 = RT = (1 − δr ) R0 + (1 − δr )t = (1 − δr )T R0 + (33)
t=0
δr
1 − (1 − δr )λT
⇒ R0 =
δr (1 − (1 − δr )T )
Thus, R0 is decreasing in λ. Let R denote the stock of rest under an alter-
native schedule λ , T , where λ > λ and T = T . Using the above, R0 > R0 ,
and hence Ri = (1 − δr )i R0 > (1 − δr )i R0 = Ri for all 1 ≤ i ≤ (1 − λ)T . If
αm = 0,
(1−λ )T (1−λ)T
¯ = 1 αr 1
Y (Ri ) ≥ (Ri )αr ≥ (34)
(1 − λ )T i=1
(1 − λ)T i=1
(1−λ)T
1 αr ¯
Ri =Y
(1 − λ)T i=1
where the ﬁrst inequality follows from the fact that Ri is decreasing in i,
and the second inequality follows from the fact that Ri > Ri . Thus Y ¯ is
increasing in λ.
To show that Y¯ is decreasing in T , note that, if αm = 0, Y ¯ can be written
as:
(1−λ)T (1−λ)T
¯ 1 αr 1 αr
Y = R = (1 − δr )αr d R0 = (35)
(1 − λ)T d=1 d (1 − λ)T d=1
45
1 − (1 − δr )λT αr (1 − δr )αr − (1 − δr )αr ((1−λ)T +1)
( )
δr (1 − (1 − δr )T ) (1 − λ)T (1 − (1 − δr )αr )
Note that Y ¯ ) is decreasing in T . Using
¯ is decreasing in T if and only if ln(Y
the above, ln(Y¯ ) can be written as:
¯ ) = αr (ln(1 − (1 − δr )λT ) − ln(1 − (1 − δr )T ))
ln(Y (36)
+ ln(1 − (1 − δr )αr (1−λ)T ) − ln(T ) + c
where c = c(δr , λ, αr ) is a function that does not depend on T . Taking the
¯ ) with respect to T yields:
derivative of ln(Y
∂ ln(Y¯) ln(1 − δr )λ(1 − δr )λT ln(1 − δr )(1 − δr )T
= αr (− + ) (37)
∂T 1 − (1 − δr )λT 1 − (1 − δr )T
ln(1 − δr )(1 − λ)αr (1 − δr )αr (1−λ)T 1
− ) −
1 − (1 − δr )αr (1−λ)T T
To show that the above expression is negative, I multiply it by T and establish
the following inequality:
ln(1 − δr )λT (1 − δr )λT ln(1 − δr )T (1 − δr )T
αr (− + ) (38)
1 − (1 − δr )λT 1 − (1 − δr )T
ln(1 − δr )(1 − λ)T αr (1 − δr )αr (1−λ)T
− ≤1
1 − (1 − δr )αr (1−λ)T
Introducing the notation h = (1 − δr )T , the above inequality can be rewritten
as:
ln(hλ )hλ ln(h)h ln(hαr (1−λ) )hαr (1−λ)
αr (− + ) − ≤1 (39)
1 − hλ 1−h 1 − hαr (1−λ)
I verify the above inequality numerically for all (h, λ, αr ) ∈ (0, 1)3 (note that
h = (1 − δr )T ∈ (0, 1)). This establishes that Y ¯ is decreasing in T , and
concludes the proof of the ﬁrst part of the lemma.
To prove the second part of the lemma, assume that αr = 0 and αm > 0.
46
Using equation 24, it is immediate that M0 is decreasing in λ. Thus, if λ > λ
and T = T , then M0 < M0 . Using the recursive formula for Mi , it follows
that Mi < Mi for all 1 ≤ i ≤ (1 − λ )T . Since Mi is increasing in i, it follows
that:
(1−λ )T (1−λ)T
¯ =
¯ −Y 1 1
Y (Mi ) αm
− Miαm ≤ (40)
(1 − λ )T i=1
(1 − λ)T i=1
(1−λ )T
1
((Mi )αm − Miαm ) ≤ 0
(1 − λ )T i=1
¯ is decreasing in λ.
Which establishes that, in this case, Y
To show that Y ¯ , the
¯ is increasing in T , using equations 22 and 24, Y
stock Mi can be written as:
(1 − δm )λT − (1 − δm )T 1 − (1 − δm )i
Mi (T ) = (1 − δm )i ( ) + = (41)
δm (1 − (1 − δm )T ) δm
1 (1 − δm )λT +i − (1 − δm )T +i + (1 − (1 − δm )i )(1 − (1 − δm )T )
( )=
δm 1 − (1 − δm )T
1 (1 − δm )λT +i − (1 − δm )T +i + 1 − (1 − δm )T − (1 − δm )i + (1 − δm )i+T
( )=
δm 1 − (1 − δm )T
1 (1 − δm )λT +i + 1 − (1 − δm )T − (1 − δm )i 1 i 1 − (1 − δm )
λT
( ) = (1−(1 −δ m ) )
δm 1 − (1 − δm )T δm 1 − (1 − δm )T
To show that Mi (T ) is increasing in T , it is enough to show that (1 − (1 −
δm )λT )/(1 − (1 − δm )T ) is decreasing in T , or that ln((1 − (1 − δm )λT )/(1 −
(1 − δm )T )) is decreasing in T . To show this:
λT
∂ ln( 11−(1−δm )
−(1−δm )T
) ln(1 − δm )λ(1 − δm )λT ln(1 − δm )(1 − δm )T
=− + (42)
∂T 1 − (1 − δm )λT 1 − (1 − δm )T
To show that this expression is negative, I multiply it by T > 0 and arrive
47
at:
ln((1 − δm )λT )(1 − δm )λT ln((1 − δm )T )(1 − δm )T
− + <0 (43)
1 − (1 − δm )λT 1 − (1 − δm )T
where the inequality follows from the fact that x ln(x)/(1 − x) is increasing
in x, and (1 − δm )λT > (1 − δm )T .
To conclude the proof, consider an alternative schedule with T > T and
λ = λ. If αr = 0,
(1−λ)T (1−λ)T
¯ −Y
¯ = 1 1 1
Y ( Mi (T ) − Mi (T )) (44)
(1 − λ) T i=1
T i=1
Since Mi is increasing in i, it follows that:
(1−λ)T
¯ −Y
¯ ≥ 1
Y (Mi (T ) − Mi (T )) ≥ 0 (45)
(1 − λ)T i=1
where the inequality follows from the fact that Mi (T ) is increasing in T . This
¯ is increasing in T when αr = 0 and αm > 0.
establishes that Y
48
D Additional ﬁgures
1.08 1.06
1.06
1.04
1.04
1.02
1.02
1
Average electricity
Average electricity
1
0.98 0.98
0.96
0.96
0.94
0.94
0.92
0.92
0.9
0.88 0.9
Sun Mon Tue Wed Thu Fri Sat Sun Mon Tue Wed Thu Fri Sat
Day of the week Day of the week
(a) (b)
1.04
1.02
1
Average electricity
0.98
0.96
0.94
0.92
Sun Mon Tue Wed Thu Fri Sat
Day of the week
(c)
Figure 6: Average electricity usage in diﬀerent RTOs over the week. Dashed
lines represent the associated 95% conﬁdence intervals. See ﬁgure 3 for the
remaining four RTOs.
49
0.16 0.12
0.14
0.1
0.12
Average electricity per worker (net)
Average electricity per worker (net)
0.08
0.1
0.08 0.06
0.06
0.04
0.04
0.02
0.02
0 0
Mon Tue Wed Thu Fri Mon Tue Wed Thu Fri
Day of the week Day of the week
(a) (b)
0.25
0.2
Average electricity per worker (net)
0.15
0.1
0.05
0
Mon Tue Wed Thu Fri
Day of the week
(c)
Figure 7: Net electricity per worker (Yd ) over the workweek in diﬀerent RTOs
(see also ﬁgure 4).
50