sleep pays

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Sleep PaysAn econometric analysis

12/30/2011

Lahore University of Management Sciences

Abstract: This paper uses the data in Survey 75 to build a relationship between

sleep and hourly wage. A number of variables are introduced to ensure unbiased

and consistent estimate of beta coefficients on sleep. The paper presents interesting

insights into the relationship between sleeps, naps, age, productivity, health and

hourly wage. The paper concludes with a result that daily hourly sleep has a

positive association with hourly wage. However, the authors do feel that there were

certain data limitations which severely constrained their ability for better analysis.

Group 10

Taimur Anwar Zuberi 2013-02-0316

Bheesham Lal 2013-02-0213

Qadeeruddin Ahmed 2013-02-0097


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Marx and Engels pointed out that labor is a commodity, like any other, and its price is thereforedetermined by exactly the same laws that apply to other commodities, which in effect meansthat labor sells labor-power on the market and is compensated for it in wages paid to it.The structure of the paper is such that the authors will firstly build upon microeconomic theoryof wages, and after establishing a theoretical basis for the paper will step by step build

econometric models. The paper is written in a manner such that the literature review, as well as,recommendations and limitation, along with the authors analysis are presented after eacheconometric model. The authors feel that this is a better way, since it helps maintain the flow ofthe paper, and the reader will not need to juggle through portions of the paper. Relevantdrawbacks and inspection of variables is done side-by-side.The data set being used for the paper is cross sectional, with a total of 706 observations.However, the hrwage of 174 or 24.6% of the total data is not available. Everybody who has notreported their hourly wage is not considered to be part of the labor force (inlf = 0). This is not alarge data sample, which is not good since higher n leads to lower variance, and also impliesasymptotic normality of the distribution of sample estimates under central limit theorem.The basic question of interest or the hypothesis that the author will try to answer by means of this

paper is, does sleep affect hourly wage.Henceforth, it follows that the population regression function of interest now becomes,

hrwage = 0+ 1a sleep* + u (1)

This is our basic simple regression model, where u is the error term containing all otherexplanatory variables that explain hrwage. 1a explains the marginal change in hrwage as sleep

increases by one minute.Before jumping to the data set or econometrics, the authors would like to give a reasoning basedon economic theory for the abovementioned relationship.Economic theory suggests that in a perfectly competitive

goods market MR = MC (marginal revenue equals marginalcost.) The diagram labeled Fig. 1 shows, the firm willcontinue to add units of labor until wage (w) = MRPL(marginal productivity of labor.) Since, MRPL = MR*MPLwhich is equal to w; it means that w and MPL are have apositive relationship. In other words, productivity affectswage.The authors recognize obvious constraints in the relationshipbetween productivity and wages. The most apparent one isthat wage data often only includes cash wages, and does not normally account for fringe benefitsand perks. Secondly, as Mankiw points out that often average productivity is compared with

median wage or average wage of production workers only, instead of average real wage; thelatter being the correct comparison from the standpoint of economic theory. Lastly, given the fastpace of technological development and the global wave of recent industrialization, the Cobb-Douglas assumption of constant factor shares is far from ideal. Therefore, given the fall inlabors share in income, and capital being the major input, marginal productivity is no longerequal to average productivity (the latter is what productivity data generally measure.)In short, sleep affects wages through productivity, i.e. to say sleep has an effect on productivityand productivity effects hourly wage. The paper by Biddle and Hamermesh, which states that

MRPL

Wage

Amountof Labor

Fig. 1


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sleep affects wages through its impact on labor productivity, adds weight to the claim thatsleep and productivity are related. Additional evidence comes from Rosekind et al, which reportsafter studying a large sample size that sleep loss related productivity losses compound to $1967per employee annually to the employers. Even from a medical standpoint, JAMA points out thatsleep loss is associated with significant neurobehavioral impairments, translating as loss in

productivity at the workplace.Moreover, the authors would like to introduce the reader to the idea of sleep debt and internalbody cycles here. The body follows a sleep and stomach cycle, and for individuals who do notmaintain a healthy sleep-food routine, they are prone to having sleep-associated problems. Toadd to that, accumulation of lost sleep diminishes ones ability to perform complex cognitivefunctions. Studies have shown that frontal lobe of the brain is particularly responsive tohomeostatic sleep pressure (Gottselig, 2006)Looking at Kernel density of hrwage(Graph 2a in the appendix), it is noted that the variable ispositively skewed with a considerably high peak value. Therefore, the log of hrwage is taken,which is more normally distributed than hrwage, as shown in Graph 2b in appendix B. The newvariable is called lhrwage. The log lessens the problems of positive variables being skewed or

heteroskedastic. Moreover, taking logs usually narrows the range of the variable, which makesthe data less sensitive to outliers. (Wooldridge)A major problem with sleepvariable is that it is self reported. In other words, minutes of sleep isa variable that is not easily observable; unlike wages, ethnicity and marital status. Data on thesevariables is likely to be self reported, which is prone to lead to a measurement error (actual valueof sleep*icould be equal tosleepi+ i.) This measurement error could lead to an attenuation bias,

i.e. if 1a is positive, 1a will tend to underestimate 1a. This error can be mitigated by takinganother measurement of sleep- sleepi**- and making it an IV for sleepi in the above equation.However, due to unavailability of data, the authors have to proceed with reported sleep. Theauthors have scaled the variable, sleep, for cosmetic purposes, by introducing another variablehrsleep, described in appendix A, table 1.

log (hrwage)= 0 + 1b hrsleep+ u (2)

After running the regression of equation (2), the coefficient on 1b is -0.0413 (0.0267), which isstatistically insignificant at the 5% confidence interval. The sign of the beta coefficient isnegative, which is counter intuitive, and goes against the reasoning presented above. As seen inthe literature above, wage and sleep have a positive relationship, unlike the one shown in ourresults from running this simple regression model in (2). The interpretation from the coefficienton sleep is that with an additional one-hour increase in sleep, wage decreases by 4.1%. Thecontradictory results might be due to an omitted variable bias, and by conducting the RamseysRESET, this claim of the authors is confirmed.

Intuitively, sleep has a diminishing effect on wage; simply put, too much sleep could lead to adecrease in wage. To put that econometrically, a quadratic function of hrsleepshould be added tothe right hand side of equation (2). (The square of hrsleep is added to equation (2), to account forthis.) Thus, the new equation becomes,

log (hrwage)= 0 + 1chrsleep+ 2a sqsleep+ u (3)


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The coefficients on 1c and 2a are 0.355 (0.231) and -0.0257 (0.0149), respectively. The results

of the regression are as expected, with the coefficient of 1cpositive and 2a negative. The dataaffirms the assertion that sleep indeed has a diminishing effect on wage. The maximum ofequation (3) is at 6.89 hours. This suggests, that up to around 7 hours sleep per night, additionalsleep has a positive impact on wage, after which more sleep results in lower productivity or

lesser wage. A modest review of the literature did not yield any study about the relationshipmentioned in equation (3), and the authors feel that there is a great scope for research here.It must be noted that sleep in all the above equations only refers to sleep at night, and not naps.Therefore, the authors feel that the impact of naps on hrwagewould be interesting. Colloquiallyspeaking, this also gives us a chance to put the adage of power-naps to econometric testing. Inother words, whereby much has been hypothesized above about the link between sleep at night,and hourly wage, the authors enhance their understanding by adding naps to the right hand sideof equation (3), thereby studying the impact of naps on wage, too. For cosmetic reasons, napshave been scaled as daily hourly nap- hrdnaps. Therefore, the regression equation becomes,

log (hrwage)= 0 + 1d hrsleep+ 2b sqsleep+ 3a hrdnaps+ u (4)

hrdnaps, has the same measurement error problems, as does sleep. Since naps will also be self-reported the odds of people in the survey of exactly knowing their amount of naps is even lessthan that for sleep. Despite this drawback, the authors feel that it is quite important that naps be

included in the regression. The coefficient on 3ais quite interesting. A negative coefficient, with-0.0950 (.0489) economic significant and a p-value of 0.052, tell us that a half-hour nap,decreases expected hourly wage by around 4.75%, keeping night sleep constant.This occurrence can be understood by means of sleep disturbances i.e. those who may be unableto sleep well during the night have to take naps to compensate for lost sleep during the night. Forthose suffering with sleep disturbances like insomnia or other sleep related disorders, or evenfatigue and/or stress could cause one to take naps, since the individual is for medical and

psychological reasons- unable to sleep well at night. Also naps could, as mentioned above, beassociated with bad allocation of time and sleep cycle. Simply put, voluntary irregular sleeppatterns could be associated with sleep deficiencies at night, and resultantly, naps duringdaytime. Instead of these naps boosting productivity, they just ensure that the body is able tomaintain minimal brain and physiological functions. Moreover, even greater detrimental effectsof restricted night time sleeps including mood disturbances have been studied by Park, Dinges,William and others. Walker in his trial reports that the detrimental effects of restrictive nightsleep build up over time.The negative effect of naps could also be due to data limitations. By data limitations, the authorsrefer to the fact that a mere 38% of the data (268 out of 706) takes naps and is part of the laborforce. A larger sample could really aid in coming to a better understanding of the relationship

between naps and wage, but for now the authors report a reserved judgment that naps have anegative effect (on productivity and) wage.Apart from that, the authors also feel that there are even greater problems in the data. Keepingaside the already touched upon issue of self reported data bias, the respondents are not askedabout sleep disorders or other stress-related sleep dysfunctions. Similarly, medical conditionslike narcolepsy, which cause individual to take excess daytime sleep have not been specificallyaccounted for. A deeper analysis of such conditions could be very helpful in ensuring a betterunderstanding of the relationship between wage, naps and sleep. Such medical conditions that


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cause lack of sleep at night could also have effects on labor productivity. For instance, stresscould affect ones sleep, as well as ones ability to perform task that require high concentration atthe workplace, and henceforth, the impact of stress enhance loss of productivity directly as wellas indirectly, through sleep-loss.After considering sleep and sleep-related medical conditions, the authors shall proceed to

studying the impact of age. Age, the authors consider, affects both sleep and wage. Moreover,the marginal affect of age on the productivity is a function of age itself, hence, the authors shallbe considering the age2term as well. Age even affects health. Physical strength and health arereduced as workers grow older. Therefore, additionally, because of the preceding discussion onhealth and medical factors affecting wage and productivity, the authors have also decided toinclude the dummy of good health as explanatory variable in the next regression. The equationnow becomes,

log (hrwage) = 0+ 1dhrsleep+ 2bsqsleep+ 3ahrdnaps+ 4aage

+ 5a age2 +6a gdhlth + u (5)

Intuitively speaking, 4a and6a should be positive, and 5a should be negative- meaning age has adiminishing effect on wage, whereby, good health has the obvious positive effect on wage. Tocheck this intuitive model, the authors run the abovementioned regression, which confirms ourmodel. The coefficients of age and age2 give around 44 years of age as the turning point forwage. The coefficient on gdhealth of 0.0673 (0.0866) can be graphically explained as an

intercept value that is 0.0673 higher than the 0 intercept.A careful review of literature tells us the relationship between age, wage and productivity; andage and sleep. The discussion follows. The relationship between age and sleep is further backedby the study by J. C. Marque et al. whereas, the relationship between sleep and wage issupported by the study by Jan C. van Ours.The study by Marque et al, (for a sample size of 2767 healthy wage earners in the age group of

32 to 52) finds age and stress to be associated with sleep complaints. The results drawn by meansof an ANOVA test with sleep difficulty index as the dependent variable, found age and stress tobe very significant (p0.0001for both).The study by Jan C. van Ours is based on panel data of 4,437 observations. The study finds thatage has a positive association with wage, as well as, productivity. Furthermore, this study findsthat wage and productivity are affected by age in a similar manner. This finding is incontradiction to earlier studies where productivity was either unaffected by wage or there was aconsiderable gap between the graph of wage and productivity plotted against age. The study alsofinds a clear hump-shape relationship between age and productivity. The workers agedbetween 35 and 45 are shown to have the maximum productivity, while the productivity ofyounger and older workers is lower.After having discussed age in sheer detail, the authors focus on good health. Good health hasbeen discussed earlier in the paper and the discussion here follows from that. The data does nottell us as to how the data on good health was calculated, and hence, severely limits ourunderstanding. The basic problem is that good health is binary variable, and therefore, it takes noaccount as to the severity of disease one suffers. From a terminally ill chronic leukemia patient toa mere influenza-virus affected individual shall both be considered to be in ill-health (gdhlth =0).Moreover, as mentioned above, health is not only physiological but also mental. However, we donot know if the data has been collected through household surveys only, or were physicians


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asked to evaluate on a persons health. If the data on good health is self reported it could lead toa biased coefficient since, as it would be based on ones perception of ones health rather than ona robust medical report. It is not easy for one to quantify or make a sure shot judgment call ontheir mental health; rather a psychiatrist could make that decision. Lastly, it does not account fordisabilities- which could greatly impair ones productivity. A person who has lost a limb in an

accident would be less productive than the average laborer, but would not be classified in poor-health. The authors strongly feel that all these issues need to be taken into account.The total income of the household or the wealth of the individual also affects both hourly wageand sleep. It affects the former through the unobservable factor of motivation (if someone hasenough family earning, his/her motivation for earning more would be lesser); it affects sleepsince poorer people are likely to be victims of financial worry, low job satisfaction and greaterdaily stress. A study by Gallup-Healthways reports that 35% of those with incomes of less than$24,000- the highest among all income and demographic categories- did not sleep well comparedto 28% of individuals with incomes of $90,000 or higher.The equation now becomes,

log (hrwage) = 0+ 1ehrsleep+ 2csqsleep + 3bhrdnaps+ 4bage+ 5bage2 +6b gdhlth + 7bltotinc+ u (6)

Therefore, we have added another explanatory variable ltotinc, in equation (6), which is the log

of the sum of spousal income and other income, given in the data. 7b measures elasticity, whoseinterpretation is that a 1%-increase in total other income decreases expected daily hourly wage

by 0.11%, keeping all other factors constant. A negative coefficient on 7bties in well with ourmodel, since higher household incomes lessens incentive/ enthusiasm to earn.Next, the authors assess the impact of dummy variables:

log (hrwage) = 0+ 1fhrsleep+ 2dsqsleep+ 3chrdnaps+ 4cage+ 5cage2 +6c gdhlth+

+7cltotinc+ 8afemale+ 9aclerical +10aconstruc + 11aselfe+ u (7)

The above mentioned equation (6) has four dummy variables for gender, clerical or constructionworker and self employment. The authors shall proceed with a systematic description of eachdummy variable, but before that tests are conducted to check for heteroskedasticity andfunctional form misspecifications.Ramsey RESET tells us that the model has no omitted variables. The special case of the whitetest for heteroskedasticity rejects H0: the model is heteroskedastic. Since, equation (7) isheteroskedastic, OLS is no longer the best linear unbiased estimate, and therefore we shall usethe robust standard errors. Furthermore, the model as a whole is jointly highly significant (F(11,520) = 15.72, p-value= 0.000)

Using these robust standard errors, hrsleepand sqsleepare both statistically significant at the10% significance level, however, not at the 5% level.The authors calculate the ideal amount of sleep at night to be 7.32 hours (7 hours 19 minutes) i.e.after this additional sleep is expected to lead to a loss in productivity, and hence, lower hourlywage. The most productive age is found to be 44.74 years (44 years 9 months), and after thatexpected hourly wage will tend to fall. These results, however, are based on the limited dataavailable, and generalizations to the populations must be made in a calculated manner. It must be


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kept in mind that the data was collected a little less than four decades ago, and many things havechanged since then, including the society.Lastly, a negative coefficient on female means that females tend to earn less hourly wage thanmen and a major reason for that could be household work and childcare, which are generallyconsidered to be a females responsibility in division of labor at the home. This puts into

perspective the idea that household work, which is generally done by females, affects both,female productivity and sleep. The positive coefficient on clerical and negative coefficient onconstruccreates a distinction between manual, physically tiring work and office work, which isless strenuous in nature. Since, construcworkers are likely to get more tired due to the physicalexertion in their occupation, compared to clerical workers working the same number of hours ina day, the former will tend to accumulate sleep debt, lethargy, fatigue and resultantly, be lessproductive than their peers in the labor force. Self employment also has a negative coefficientand probably has the most interesting interpretation. Those who are self employed generally willnot tend to have fixed office hours, and it is possible that there would be less distinction betweentheir work and personal life. Therefore, a self employed individual would have additional stressand an irregular sleep cycle.

All the results presented, however, are based on the limited data available, and generalizations tothe populations must be made in a calculated manner. It must be kept in mind that the data wascollected a little less than four decades ago, and much has changed since then.


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Appendix A

Table 1: Data Description

Variable Description

hrwage Hourly wage per person, in dollars

lhrwage Log(hrwage)sleep* sleep per week at night, in minutes in the population (unobserved)

sleep sleep per week at night, in minutes (Self Reported)

hrsleep ; sleep per day at night, in hours

sqhrsleep (hrsleep)2

slpnaps sleep including naps per week (sleep + naps), in minutes

hrdnaps ; naps per day, in hours

age In years

sqage (Age)2

othinc Annual income from other source other than spouse pay, in dollarsspspay Annual spouse pay, in dollars

ltotothinc Log(spouse pay +othinc)

female =1 if person is female; 0= if person is male

clerical 1= if person is a clerical worker; 0= otherwise

construc 1= if person is a construction worker; 0=otherwise

selfe 1= if person is self-employed; 0=otherwise

gdhlth 1= if person is in good or excel health; 0= not in good health


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Table 2: Regression Table (Dependent variable is log(hrwage))

Equation No. 2 3 4 5 6 7 7(Robust)

Intercept 1.752 0.255 0.317 -0.94 -0.906 -0.505

(0.862)

-0.505

(0.772)-0.208 -0.893 -0.89 -0.961 -0.96

hrsleep -0.041 0.354 0.344 0.322 0.3268 0.302(0.206)

0.302(0.177)-0.026 -0.231 -0.23 -0.229 -0.229

sqsleep -0.0257 -0.025 -0.023 -0.0233 -0.206 -0.206

-0.0149 -0.015 -0.014 -0.015 -0.013 -0.011

hrdnaps -0.095 -0.103 -0.1084 -0.102 -0.102

-0.049 -0.048 -0.049 -0.044 -0.042

age 0.0625 0.063 0.054 0.054

-0.019 -0.019 -0.017 -0.018

age2 -0.0007 -0.0007 -0.0006 -0.0006

-0.0002 -0.0002 -0.0002 -0.0002

gdhlth 0.0673 0.075 0.0364 0.0364

-0.086 -0.086 -0.078 -0.057

ltotinc -0.011 0.0005 0.0005

-0.007 -0.006 -0.006

female -0.577 -0.577

-0.053 -0.06

clerical 0.032 0.032

-0.068 -0.068

construc -0.023 -0.023

-0.143 -0.092

selfe -0.192 -0.192

-0.058 -0.132

Total Obs 532 532 532 532 532 532 532

R2 0.45% 1.01% 1.71% 4.09% 4.57% 24.21% 24.21%


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Appendix B

Kdensity Curve of hrwage(1a) and lwage(1b) as compared to normal distribution

Graph 1a

Graph 1b

0

.05

.1

.15

.2

Density

0 10 20 30 40hourly wage

Kernel density estimate

Normal density

kernel = epanechnikov, bandwidth = 0.6312


0

.2

.4

.6

.8

Density

-1 0 1 2 3 4lwage


Normal density

kernel = epanechnikov, bandwidth = 0.1454



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Appendix C

1) Ramsey RESET test using powers of the fitted values of log(wage) of equation ()Ho: model has no omitted variables

F(3, 516) = 0.53

Prob > F = 0.6641

Fail to reject Hoat 5% significance level. It gives no evidence of omitted variable bias.

2) Breusch-Pagan / Cook-Weisberg using the fitted values log(wage) of equation()Ho: Constant variance

chi2(1) = 9.20

Prob > chi2 = 0.0024

Hois rejected at 5% significance level. It gives evidence of heteroskedasticity in the model.

3) Special case of White test using the fitted values log(wage) of equation()Ho: homoskedasticity

chi2(68) = 94.95

Prob > chi2 = 0.0171

Hois rejected at 5% significance level. It gives evidence of heteroskedasticity in the model.


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Sleep Research15.1 (2006): 31-40. Print.

Greg, Mankiw. "How Are Wages and Productivity Related?" Web log post. Greg Mankiw's Blog -

Random Observations for Students of Economics. 29 Aug. 2006. Web. 29 Dec. 2011.

.

Hamermesh, Daniel S., and Jeff E. Biddle. "Sleep and the Allocation of Time." The National

Bureau of Economic Research. Journal of Political Economy, Vol. 98, No. 5, Part 1, Oct.

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Marquie, J. C., J. Foret, and Y. Queinnec. "Effects of Age, Working Hours, and Job Content on

Sleep: A Pilot Study." Experimental Aging Research25.4 (1999): 421-27. Print.


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National Institute of Neurological Disorders and Stroke. "Narcolepsy Fact Sheet." National

Institute of Neurological Disorders and Stroke (NINDS). 28 Feb. 2007. Web. 29 Dec. 2011.

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Ours, Jan C., and Lenny Stoeldraijer. "Age, Wage and Productivity." (2010). Web. 30 Dec. 2011.

Pelham, Brett W. "Rest Eludes Nearly 30% of Americans." (2010). Gallup.Com - Daily News,

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Lecture.


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Wooldridge, Jeffrey M. "Chapter 6." Introductory Econometrics: a Modern Approach. Australia:

South-Western College Pub., 2003. Print.

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