Bank of Canada Research Department

Standard Theories of Working Time over the Long Run Alexander Ueberfeldt∗ Bank of Canada Research Department July, 2005

ABSTRACT Over the last 130 years weekly hours worked per employed person, the workweek length, decreased considerably in today’s advanced industrialized countries. The number of employed persons in the population 15 or older, the employment rate, displays large movements but no clear secular pattern. These facts motivated the question: What accounts for the large decrease in the workweek length and the development of the employment rate? This paper presents ex-ante promising theories. It considers among others: technological progress, proportional taxes, workweek length constraints, and changes in the sectoral composition of the economy. The main finding is that all these theories fail to account for the facts. Some of the theories predict a constant workweek length; others that allow for a decrease of the workweek length have counterfactual implications.

∗

I thank Edward C. Prescott for his advice, support, and encouragement, Ellen R. McGrattan for advice and guidance. Furthermore, I thank Simona Cociuba, Sami Alpanda, and Suquin Ge for helpful comments. The views expressed herein are those of the author and not necessarily those of the Bank of Canada or the Federal Reserve System. The financial support of the Doctoral Dissertation Fellowship of the University of Minnesota is acknowledged.

1. Introduction Over the last 130 years, working time in today’s industrialized countries has decreased. The common component of this change is a strong decline in the weekly hours worked per employed person, the workweek length. Figure 1 illustrates this fact for three countries: Germany, the United Kingdom, and the United States. The most pronounced movement is observable in Germany, where employed persons in 1870 used to work twice as much in a given week as they work today. Figure 1: The workweek length over the 20th Century. 80

Workweek length

70

Germany

60

50

U.K.

40

U.S. 30 1870

1900

1930

1960

1990

The other component of the working time decline is the employment rate, which refers to the number of persons employed relative to the population age 15 and older. The employment rate follows no clear pattern comparable to that of the workweek length.1 Figure 2 illustrates this fact for the same three countries considered before. The United Kingdom experienced a decrease, the United States an increase, and Germany a fairly constant development of the employment rate in the long run. Any theory of working time used to evaluate the long run consequences of labor market policies has to capture these long run observations. 1

This statement still holds, if I redefine employment rate to be the number of employed persons relative to the population age 15 to 64.

Figure 2: The employment rate over the 20th Century.

Employment rate (in %)

80

70

U.K.

60

50

Germany

U.S. 40

30 1870

1900

1930

1960

1990

In this paper, I present the main model used in the applied general equilibrium literature to consider questions regarding changes in the workweek length and the employment rate. The paper shows that the model is not able to account for the secular changes in the working time per employed person. I then consider extensions of the model and find that these extensions are either not improving the model’s performance, or are in violation with some fact found in the data. Among the variations considered are changes in Total Factor Productivity, effective taxes on labor income, government spending, commuting time, workweek length constraints, home production, and multiple sectors of market production. The first generation of Real Business Cycle (RBC) models such as Kydland and Prescott (1982) does not differentiate between hours per employed person and employment, but only concentrates on the variation of total labor hours over the business cycle and therefore is silent about variations in the workweek. Hansen (1985) introduces a non-convexity to the standard model by requiring labor to be indivisible, i.e., workers either work a prespecified number of hours or not at all. The economy is convexified through employment lotteries. This model generates no variation in the so called ”intensive margin” (changes in the workweek length) by construction and all variation in aggregate labor hours is due to the ”extensive margin” (changes in the employment rate).

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Hornstein and Prescott (1993) builds a model where both margins can be utilized, but Kydland and Prescott (1991) shows numerically that this model generates no variation in the workweek length and only the extensive margin is utilized in equilibrium unless some friction, in the form of costs to changing the level of employment, is introduced to the model. Fitzgerald (1998) and Osuna and Ríos-Rull (2001) both use this model to assess government policies aiming to reduce the workweek length. To generate variation in the workweek Fitzgerald (1998) introduces costs of employment, and Osuna and Ríos-Rull (2001) introduces an employment externality to the utility function of the agents. This paper shows analytically that the standard model (Kydland and Prescott, 1991) with slight variations does not generate any change in the workweek length. This paper relies on work done by Alpanda and Ueberfeldt (2003) and Prescott (2003). The paper adds to the literature by incorporating home production and multiple sectors into the basic framework and explores resulting implications. I will proceed as follows. In the next section, I present the standard model used in the Real Business Cycle literature and show that it is not able to capture the secular decline in the workweek length as observed in the data. I also consider variations of the standard model and show that these variations do not resolve the inconsistency between the model and the data. More extensive changes are made in section two. In particular, the impact of home production and multiple sectors on the workweek length are considered. The paper concludes with section three.

2. Standard model In this section, I present the standard model for analyzing working time to consider long-term developments. The model’s main advantage as pointed out among others by Kydland and Prescott (1991) is its conformity with business cycle properties of the workweek length and the employment rate. For the purpose of this paper, the main advantage of this model is that it allows for a decomposition of average hours worked into hours worked per employed persons and the fraction of employed persons in the working age population. This decomposition is needed to account for the long run decline in the workweek length. In the appendix A. , I present an alternative theory of the working time development.

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A. The model and its implications The model economy described below is similar to the one used by Kydland and Prescott (1991). There is a continuum of infinitely lived identical families of measure one, each endowed with a unit of time each period and k0 units of initial capital. The utility function representing a family’s preferences is given by: E

∞ X

β t u (ct , ht )

t=0

1−σ

( cα (1 − h)1−α ) with u(c, h) = 1−σ

−1

where c is consumption and h is the time spend working in the market. The parameters satisfy the following conditions: β ∈ [0, 1), α ∈ (0, 1) , σ > 0. Each family’s consumption, labor income and capital income is taxed at the respective proportional rates2 : τ c,t , τ e,t , and τ k,t . The collected tax revenue is used to fund government spending Gt . Each period the government balances its budget using lump sum transfers, Tt . Production is carried out in plants. A plant is defined by the amount of hours it is operated, h, the amount of capital used, k, and number of workers employed, e. The plant technology is given by:

   A hφ kθ e e ≤ e¯  0,t if y=  A hφ k¯θ e e > e¯  0,t

where y is output, which can be used either for consumption or investment. The output also depends upon the level of economy wide knowledge for a given time period t, A0,t . Concerning the parameters, I require that the parameters satisfy φ, θ ∈ [0, 1] , and θ + φ > 1.

3

For this plant technology and a given workweek length, only one plant will be operated 2

Deviating from the proportionality assumption would make some difference for later results. In particular for the standard framework with multiple income groups (for example, due to differences in labor productivity) an increase in the convexity of the income tax schedule would lead to a decrease in the workweek length. This decrease would be of minor compared to the long run picture. 3 In the case of φ + θ ≤ 1, the economy reduces to a standard economy with all persons employed and working the same length of hours.

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in equilibrium and its aggregate output is: Yh = At hφ K θ E 1−θ with At = (¯ e)θ−1 A0,t The law of motion of capital is: kt+1 = xt + (1 − δ) kt where x denotes investment. Let the consumption possibility set of a family be denoted by S which is a convex, 2 compact subset of R+ with a generic element s = (c, h).

4

Let B (S) be the Borel σ-algebra

of S and M be the set of signed measures over B (S) . A family chooses probabilities η s over R B (S) , hence η s ∈ M such that η s ≥ 0 and η s = 1. In the aggregate η s will also denote the measure of family member with s = (c, h).

In general a family’s problem is given by:

max

{µt ,kt+1 ,xt }

∞ X

β

t

t=0

Z

u (ct , ht ) dη t

s.t. ∀t :

Z

((1 + τ c,t ) ct − (1 − τ e,t ) wt (ht )) dη t + xt − (1 − τ k,t ) rt kt − Tt ≤ 0

∀t : xt − kt+1 + (1 − δ) kt ≥ 0 Z ∀t : dη t = 1

∀t : η t ∈ M (<+ × [0, 1]) , kt+1 , xt ≥ 0 k0 given Given that the firm’s- and the family’s-problem are time separable, if I take the optimal 4

Note that I have assumed an upper bound for consumption. This is only for technical purposes and will not bind in equilibrium.

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choice of capital as given, I can reduce the choice of optimal bundles {st } into a sequence of problems of the form:

P : max

Z

u (c, h) dη

s.t. Z

((1 + τ c ) c − (1 − τ e ) w (h)) dη + Z ≤ 0 Z dη ≤ 1

η ∈ L (<+ × [0, 1])

Then the time index can be dropped since it is not important for the results. Furthermore, Z = x − rk − T with Z taken as given by a family. Here are several noteworthy observation: first, persons working in the market with different workweek lengths get different wages due to the nature of the plant technology. Second, working or not working has no implication on the investment behavior of the family, and hence investment - capital decisions are not ordered by the workweek length.

5

And

third, this specification assumes complete insurance for the family members. Several decentralizations with explicit insurance markets have been proposed by Hansen (1985) and Osuna and Rios-Rull (2003) but the nature of the insurance does not matter due to complete markets.

Characterization of the solution to the family problem

A family’s problem (P ) has at most two points on which there will be positive mass. This follows from basic linear programming results. Furthermore the solution is characterized by the following first order necessary condition: u (c, h) − λ ((1 + τ c ) c − (1 − τ e ) w (h)) − µ ≤ 0 5

Hansen (1985) establishes this result.

6

(1)

with equality if η(c, h) > 0. Now I will use the properties of the technology and the utility function. 1−σ

Proposition 1. Let u(c, h) =

(cα (1−h)1−α ) 1−σ

ª © and w(h) = maxk≥0 Ahφ k θ 11−θ − rk , then

every solution to the above problem has to be of the form: χ(c0 , 0) + (1 − χ) (c1 , h1 ) for χ ∈ [0, 1), h1 ∈ (0, 1) , c0 , c1 ≥ 0. Proof. See the appendix.

This result is crucial, since it allows me to easily characterize the solution. Given the finding, I can reduce the problem P considerably, such that it takes the following form: max eu (c1 , h) + (1 − e) u (c0 , 0) s.t.

(1 + τ c ) (ec1 + (1 − e) c0 ) = e (1 − τ e ) w (h) + Z 0 ≤ e≤1 Here e is the measure of family members working in the market, h is the number of hours they work in the market, and (c1 , c0 ) are the amounts of the consumption good the family members receive if they work in the market, respectively, if they don’t work in the market. I derive the following first order necessary and sufficient conditions: (2)

c : eu1 (c1 , h) = λ (1 + τ c ) e c0 : (1 − e) u1 (c0 , 0) = (1 − e) (1 + τ c ) λ

(3)

e : u (c1 , h) − u (c0 , 0) + λ ((1 − τ e ) w (h) − (1 + τ c ) (c1 − c0 )) − µ = 0

(4)

h : eu2 (c1 , h) = λe (1 − τ e ) w0 (h)

(5)

λ : (1 + τ c ) (ec1 + (1 − e) c0 ) ≤ e (1 − τ e ) w (h) + T

(6)

µ : µ (1 − e) = 0; µ ≥ 0; e ≤ 1

(7) 7

In the case of an interior solution the problem’s first order conditions reduce to a very simple form. Proposition 2. The unique interior optimal workweek length in the case of e < 1 in any given period is constant and given as the implicit non-zero solution to the following equation: 1−θ 1 − (1 − h − (1 − h)1−ε ) = h, ε φ with ε =

(1−α)(1−σ) 6 . 1−σ(1−α)

Proof. See the appendix. This result has several important implications: First: The workweek length is not changing along a transition path toward the steady state of the economy. Second: The workweek length is not affected by any changes in: • Total Factor Productivity (TFP), At • Proportional tax rates, (τ c,t , τ e,t , τ k,t ) • Government spending, Gt Third: The employment rate is affected by changes in TFP, tax rates, or government spending, even if the workweek length is not. The second implication might seem surprising, since in the standard one sector growth model with an endogenous labor supply decision TFP, taxes and government spending have a profound impacts on the hours worked. In this model the workweek does not change at all in response to changes in TFP, taxes, or government spending. The third implication points out, that this the constancy of the workweek length does not mean that working time does not react. It is rather the case that the employment rate picks up all changes in TFP, taxes and government spending. 6

This result relies on the interiority of the solution. It requires that employment is below the full employment level. If the employment rate was one, then the workweek length might increase beyond the level, defined by the equation in this proposition.

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Figure 3: TFP and the workweek length, USA 1870 to 2002.

Workweek length

60 1878

50 1950

40 2002

30 0

200

400

600

800

TFP, Base 1878 = 100

With regard to the long run decline in the workweek length the findings suggest that the model can’t account for this phenomenon, even if augmented with TFP, taxes, and government spending.

Now that I have a theoretical result, I will consider data for the workweek length and each of the three factors, starting with TFP. Figure 3 shows a plot of TFP and the workweek length for the United States from 1870 to 2000. TFP is measured using the aggregate production function Yh with θ = 0.33 and φ = 0.84. The plot is robust to parameter changes. I have normalized the TFP of 1878 to 100.

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The resulting figure looks as if a regime switch with respect to the workweek length occurred. From 1870 to about 1950 the picture displays a negative correlation between TFP and the workweek length, but then from 1950 till 2002 TFP more than doubled, but the workweek length did not change significantly. The model, I presented so far, is consistent with the US working time experience for the later period, i.e., the period from 1950 to 7

I choose the year 1878, because it is associated with the lowest measured TFP level in the period 1870 to 2002.

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Figure 4: Effective tax on labor income and the workweek length , USA 1960 to 2002.

Workweek length

60

50

40

30 95 105 115 125 Effective labor income tax, Base 2002 = 100

2002. Another aspect of the data which the model can capture is the large variability of the employment rate and the low variability of the workweek length as observed in the later period over the U.S. business cycle. What about taxes and the workweek length? There is an extensive literature that shows that taxes on labor income have a negative impact on the working time. A recent example is Prescott (2004). The main finding is, that there is a significant negative impact of labor taxes on working time. The model preserves this result but qualifies it by saying that in the case of proportional tax rates all the labor adjustment to changes in the labor income tax will be made with respect to the employment rate. Figure 4 shows a plot of the effective labor income tax rate against the workweek length, for the United States from 1960 to 2002. Here the effective tax rate on labor income is measures as: (τ e + τ c ) / (1 + τ c ). The main implication of this plot is that the effective tax on labor income varied a lot from 1960 to 2002, but the workweek length did not vary much at all in the case of the United States. This observation is consistent with the theoretical implication of the model, i.e., the workweek length is independent of changes in the effective labor income tax rate.

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B. Variations of the model Now that I have shown the limitations of the standard model, I will turn to variations that seem promising. The first two variations I consider are: a commuting externality and working time constraints. One of the main changes that has accompanied industrialization is the movement from the country side to the cities. This urbanization of the population has brought a large structural change in the living conditions of persons living in today’s industrialized countries. As part of urbanization commuting time has changed. The argument then is, that an increase in commuting time has led to a decrease in the workweek through decreasing the time available for work and a negative impact on an employed person’s utility. Another possible theory of the working time decline is that constraints on the workweek have forced employed persons to work a different working time, from what they would have chosen in an unconstraint environment. The main candidates for such constraints are government laws and regulations. Examples of government constraints on the workweek length are the Fair Labor Standards Act (FLSA) of the United States from 1938 and the overtime taxation imposed by the French government in 1998.

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I will consider such constraints in a

simplified form. Commuting externality Suppose there is a negative externality to agent’s utility from aggregate employment due to congestion during the commute to work as in Osuna and Ríos-Rull (2003). The question asked here is: In how far are changes in the commuting time able to account for the decline in the working time per employed person? To consider this question, I modify the utility function of the representative family by introducing a commuting externality factor, η (E). This factor is zero, if a person is not working, and a function of aggregate employment otherwise. ¡ α ¢1−σ c [1 − h − η (E)]1−α −1 U (c, h) = 1−σ 8

In 1998, the French government introduced a tax on overtime, payable to the government. This tax becomes payable by a firm, if an employee works more than 35 hours per week. Averaging over the year is possible.

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where E is aggregate employment which is taken as given by the agents. The commuting externality factor, η (E), is an increasing function of total employment, E. Of course in equilibrium E = e and hence I get an equation analogous to the one in the standard model: −

i (1 − θ) (1−α)(1−σ) 1 − α (1 − σ) h 1 − ht − η (et ) − [1 − ht − η (et )]1− 1−α(1−σ) = ht . (1 − α) (1 − σ) φ

It is obvious that in this setup working time is not longer dependent on the parameters alone. To assess the explanatory power of this working time externality, I perform a numerical experiment. For this experiment, I set η (Et ) = η. This experiment will provide an idea of the magnitude of the externality needed to account for the secular decline in the workweek length. Consider an economy defined by the parameters in Table 2.1. Table 1: Parameters for numerical example.

Parameters Values

α

σ

0.3

2

θ

φ

0.3 0.8

I solve for the optimal workweek length as a function of the commuting time η. The result is displayed in Figure 5. The qualitative features of the working time - commuting time relationship, as displayed in the figure, are robust to parameter changes. In the figure, I have normalized the optimal workweek length at η = 20 to 100. This allows for an easy comparison with the observed decline in the workweek length. From 1870 to 2002, the workweek in the U.S. to a level of 66 percent of its 1870 level. Two things are noteworthy in this plot. First, a large change in commuting time is able to account for the large decline in the workweek length. Second, to get the large decline in the workweek length the commuting time has to either decrease by 20 hours per week to zero, or to increase from 20 hours per week to 50 hours per week. Both of these extreme changes seem to be unrealistic. A recent paper by Francis and Ramey (2005) presents data suggesting a 35 percent decrease in the commuting time of employed persons from 1900 to 12

Figure 5: Commuting time and the optimal workweek length.

Optimal workweek

100 90 80 70 60 0

10

20

30

40

50

Commuting time

2000. In the given framework their numbers would suggest a decrease of the workweek length due to commuting time of 2 percent. This implies that commuting time changes play a minor role in accounting for the secular decline in the workweek length. Workweek length constraints To assess the implications of workweek length constraints, I use the following model. A family solves the problem: max log (w (h) e) + αe log (1 − h) s.t. ¯ h≤h where w (h) = hλ , λ > 1. The model I focus on is static and has no capital. The basic results generalize to the fully fledged model considered so far. The family sends a fraction e of its members to work h hours in the market and receive w (h) e income from this work. This income is used for consumption. The family is constraint in its choice of the workweek length by an upper ¯ bound h. 13

This setup helps to focus the discussion on the main implications for the working time. The solution to the problem is:

h

e

   h ¯ ¯ ho > h  = if  h ¯  ho ≤ h o  ¡ ¢¢  −α log ¡1 − h ¯ −1 =  (−α log (1 − h ))−1 o

if

 ¯ ho > h  ¯  h ≤h o

where ho is the solution to this equation: −(1 − ho ) log (1 − ho ) = ho /λ.

¯ Two things are noteworthy: first, the employment rate is monotone decreasing in h. This means that a forced decrease in working time per employed person is at least partially offset by an increase in the employment rate. Second, average hours worked, e h , are de¯ Hence, if the workweek length is binding and h = h ¯ < ho , then the change creasing in h. ¯ is more then offset by the implied change in the emin the workweek length, from ho to h, ployment rate. This leads in total to an increase in average hours worked relative to the unconstraint problem’s solution. These results are hard to evaluate in the data. For some countries the introduction of working time constraints were followed by an increase in the employment rate (e.g., United States 1938, Japan 1987), but it is not clear if these increases were due to the imposed constraints or something else. Goldin (1988) finds that the introduction of working time constraints for women in the 1920s although leading to a decrease in the workweek length had only a minimal impact. In many cases working time constraints are not necessarily binding, at least in the long run. The FLSA for example imposes a 50 percent overtime pay in the case of a weekly working time above 40 hours. A coalition of employers and employees can easily circumvent this constraint by rewriting the contracts. Even employers on their own can use inflation and constant nominal wages to undermine the effectiveness of this workweek length constraint. There is some econometric evidence that the FLSA, at least in the short run, was successful in reducing the workweek length. A study by Costas (1998 (a)) finds that the FLSA led to a five percent decrease in the standard workweek length in the retail and wholesale trade from

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1938 to 1950 and a considerable adjustment in the wages. The impact on employment is not reported. More important than the possibility of circumventing workweek length constraints is the fact that a large number of working time constraints with a broad coverage were imposed after the workweek length had already decreased considerably. For example, Germany and the United States in the 1930s passed laws regulating the working time of employees. If you look at Figure 1, you can see that for both countries most of the decrease in the workweek length had already occurred by then. Thus it seems as if working time constraints, although potentially important in the short run, can not account for the long run decrease in the working time per employed person.

3. Extensions This section extends the basic model by introducing home production and multiple market sectors. These extensions are done in two separate steps. A. Home production Greenwood et al. (2002) have reported, that over the course of the 20th century a broad variety of household appliances were introduced into American homes. One possible force behind the decline in the workweek length could be that these appliances allowed women to join the market. This led to an increase in the work participation which then allowed for a substitution from hours per employed person to more persons working shorter hours. The model I consider is closest to a model considered by Benhabib et al. (1990). The main difference is that I have endogenized the household’s decision of how many hours to supply to the market. As the basis of my analysis, I consider the following household problem: max eu (ψ1 c1,m , c1,d , h1,d , hm ) + (1 − e) u (ψ0 c0,m , c0,d , h0,d , 0)

15

s.t. ec1,m + (1 − e)c0,m ≤ ew (hm ) + R ci,d ≤ f (hi,d , (1 − ψi ) ci,m )

i = 0, 1

It is assumed, that a fraction e of household members work hm hours in the market and h1,d hours at home. This work in the market generates a total wage income of ew (hm ). In addition the household has some external funds R.9 The household income is used to buy consumption goods, ci,m , in the market. A fraction ψi of these market goods is consumed and enters the utility of the household, while the remainder is combined with working time at home, hi,d , to produce a home good, ci,d . This home good enters the utility of the household and is produced by persons working in the market and persons not working in the market. In order to get analytic solutions, I restrict my attention to the following functional forms: ¡ α γ ¢1−σ ci,m ci,d (1 − hi,d − hi,m )1−α−γ u (i) = , 1−σ f (i) = Bhεi,d c1−ε i,m , i = 0, 1 w (h) = Ahλ . I assume parameters satisfy: α, γ, ε, 1 − α − γ ∈ (0, 1) , A, B, σ, λ − 1 > 0. Proposition 3. If the parameters of the model satisfy the condition σ < 1 +

λ−1 , α(λ−1)+γ(1−ε)

then an interior equilibrium exists and the following holds: 1. Working time in the market and at home are independent of the Total Factor Productivities in the economy, A, B. 2. Persons not working in the market work longer hours at home, than persons working in the market. Proof. Using the first order necessary conditions and some algebra, I find the following 9

Introducing capital into this economy and making it dynamic does not change the key results.

16

relationships: −Q1 (1 − hm − (1 − hm )1+Q2 ) = with Q1 =

1−α(1−σ) , Q2 (1−α−γ(1−ε))(1−σ)

hm , λ

= − (1−σ)(1−α−γ(1−ε)) . 1−(α+γ(1−ε))(1−σ)

This equation has one interior solution, which gives us the market hours as a function of the fundamental parameters and thus part 1 of the proposition. The existence of the interior solution is ensured by the fact that the left-hand-side as a function of h is strictly concave, has zero value at h = 0 and h = 1, and has a slope greater than 1/λ at the point h = 0. Concerning the second result, the first order conditions imply the following relationship: h1,d = D (1 − hm ) , h0,d = D, with D = 1 −

1−α−γ . 1−α−γ+εγ

The fact that hm is in the interval (0, 1), then implies the result. Note that the model reduces to the standard model, if I assume that γ = 0. I conclude that for a basic model the introduction of a household sector is not able to account for the decline in the workweek length. It also does not matter for the workweek length how productive the household sector is either by itself or relative to the market sector. Note that the introduction of a household sector with a varying productivity level has an impact on the employment rate. An empirical study by Boden and Avner (1994) uses time diary data from the 1960s to the 1980s for the United Kingdom and the United States. They look at the diffusion of home appliances and entertainment goods and the effect of this diffusion on domestic working time and leisure time. Their study suggests that the adoption of home appliances was accompanied by a decrease in hours worked at home and an increase in leisure time. This implies that technological progress in home production may only lead to a leisure effect and not a workweek length effect in regard to market hours.

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B. Multiple sectors The idea behind multiple sectors is, that there was a shift in the sectoral composition of the economy. Employment shifted from sectors with a long workweek to length sectors with a short workweek. This shift has led to a decrease in the average hours worked per employed person by increasing the number of persons that work a shorter workweek. The following two sector model is capturing the main aspects of this idea. Of particular interest is, that the model allows for two different interior workweek lengths in the economy. Representative family solve the problem: max

∞ X t=0

β t [log (c1,t ) + log (c2,t ) + α (e1,t log (1 − h1,t ) + e2,t log (1 − h2,t ))]

s.t. c1,t + pt (c2,t + kt+1 − (1 − δ) kt ) ≤ w1,t (h1,t ) + w2,t (h2,t ) + rt kt Here ci,t is the consumption of good i = 1, 2, hi,t , ei,t are the the hours worked, respectively, the number of family members employed in the sector i. The only other new element is pt , which is the price of the second consumption good and of the investment good relative to the price of consumption good one. There are two sectors, each producing one good. Their respective problems are given by: φ

θ1 1−θ1 max A1,t h1,t1 K1,t E1,t − rt K1,t − w1,t (h1,t ) e1,t

φ

θ2 1−θ2 max pt A2,t h2,t2 K2,t E2,t − rt K2,t − w2,t (h2,t ) e2,t

18

Finally, the resource constraints are given by: φ

θ1 1−θ1 E1,t c1,t = A1,t h1,t1 K1,t φ

θ2 1−θ2 c2,t + kt+1 − (1 − δ) kt = A2,t h2,t2 K2,t E2,t

e1,t = E1,t e2,t = E2,t kt = K1,t + K2,t If I make the same parameter restrictions as in the case of the standard model, then there are two main implications from this model.10 On the one hand, I find, that there are two different workweek lengths prevailing in the economy. Each is defined as the solution for the respective equation i: − log (1 − hi,t ) (1 − hi,t ) =

1 − θi hi,t φi

These optimal workweek lengths are time invariant and independent of the technology levels, {Ai,t }2i=1 . On the other hand, I find, that the fraction of family members employed in sector one is constant over time. It only depends on the optimal workweek length, h1 , of that sector, e1,t =

φ1 1 − h1 . α h1

All of these implications are counterfactual with respect to the long run data. In the data, for all the standard sectors (agriculture, manufacturing, services) the working time per employed person in the past was long and has declined considerably. Also, there is no standard sector for which the employment rate has remained constant. These two facts are illustrated in the Figures 6 and 7 for France since 1950. The figures present the working time data for the following sectors: agriculture, forestry, and fishing (ag), manufacturing (man), construction (con), wholesale and retail (w&r), transportation and communication (t&c), and services (serv). The data are in striking 10

The implications below remain unaltered, if I add land as an input factor for one or both of the two sectors.

19

Figure 6: Average annual hours worked per employed person by private sector, France 1950 to 1996. 2,100 2,000 1,900 1,800 1,700 1,600 1,500 1,400 1950

1960

ag

man

1970

const

1980

w&r

1990

t&c

serv

contrast to the model. The workweek length is not constant in any of the sectors. Moreover, regarding the idea that the workweek length decline is due to a shift in employment toward sectors with a shorter workweek length, I find that employment has not always shifted toward the sectors with the shortest working time.

11

4. Conclusion This paper is motivated by the decline in the working time per employed person, the workweek length, over the long run. Over the last 130 years, the workweek length in today’s industrialized countries has decreased considerably. Employed persons used to work at least 50 percent more in 1870, than they do today. This paper considers various theories that have ex ante the potential to account for this phenomenon. The analysis is conducted within the framework of a two labor margin general equilibrium model. One margin is the workweek length and the other margin is the employment per person age 15 and older, the employment rate. Within the class of these models, neither taxes, nor Total Factor Productivity, nor government spending can account for the workweek length decline. 11

Wherever data are available with respect to hours worked and employment by sector, the same picture of a long run decline in the working time per employed person with shifts in the employment distribution across sectors emerges.

20

Figure 7: Employment by sector relative to total employment, France 1950 to 1996.

Base total employment = 100

30 25 20 15 10 5 0 1950

1960

ag

man

1970

const

1980

w&r

1990

t&c

serv

As extensions, I consider commuting time and workweek length constraints and show that neither of those extensions can credibly address the secular decline in the workweek length. Similarly unsuccessful are the introduction of home production and multiple sectors. One interesting result based on a multiple sector version of the basic model is the possibility of multiple interior workweek lengths.

21

5. Appendix The appendix is subdivided into several different parts. Part one presents an alternative approach to address the secular decline in the workweek length and the change in the employment rate. It also shows that the main alternative theory has problems of its own. The next part consists of some of the proofs from the main text. Finally, I give references to the data sources used for the figures in the text. A. An alternative theory A popular theory of the workweek length decline is that households like leisure and as they get richer they substitute away from consumption toward more leisure. In that view the workweek length decline is driven by a strong wealth effect. I will formalize this idea below in a simple model. Consider an economy populated by a continuum of households. Each household has preferences ordered by: u({ct , ht }∞ t=0 )

=

∞ X t=0

β

t

·

¸ hνt cαt − , α ν

where c stands for consumption and h for the workweek length and the parameters are such that α, ν ≤ 1; α, ν 6= 0. The households are endowed with one unit of labor supply in each period and face the following budget constraint: ct = (1 − τ t ) wt ht Here, I assume for simplicity that the wage sequence, {wt } and the tax sequence, {τ t }, are given exogenously. In this model the working hours of a household are given by: ht =

³ ν ´1/(α−ν) α

((1 − τ t ) wt )−α/(α−ν) .

Thus I have that for α, ν > 0, α − ν > 0 or α, ν < 0, α − ν < 0, an increasing income sequence will lead to a decrease in the working time. 12

12

Alternative preferences that will give a wealth effect that can drive down the workweek length are of the Stone-Geary form, e.g., u (c, h) = log (c − ψ) + α log (1 − h) ; ψ > 0. One might think that the standard

22

One paper that is suggesting a wealth theoretical approach to the workweek length decline is Greenwood and Vanderbroucke (2005). They find that for a reasonably determined size of the minimum consumption requirement of a Stone-Geary utility function the wealth effect is strong enough to account for the observed long run decline in the workweek length of the United States. However, their model abstracts form very important facts, namely that taxes on labor income increased a lot over the considered time horizon, that the workweek length in the United States did stabilize after the 1960s, and that the employment rate in such a model would be one. In general the income effect - preference oriented theory of the working time decline has multiple problems. First, in most models of this type all households work in equilibrium. Thus by solving the problem of the workweek length decline the theory creates the new problem: What determines the employment relative to the working age population? Second, in the United States over the course of the 20th century taxes on labor income have increased a lot, but the workweek length decreased a kit before the major tax increases and barely changed afterwards. In the simple model a tax increase would lead to a workweek length increase and a tax decrease would lead to a workweek length decrease. This is inconsistent with the data on the workweek length. Finally, at least for the United States there was a large increase in the income of households after the 1960s, but there was little or no change in the workweek. Again, the simple preference theory would predict a steady decrease in the workweek length. These three observations have to be addressed by a preference oriented working time theory. B. Proofs This section contains the proofs of the results in the main text. Proof of Proposition 1 I know from basic linear programming, that P can have at most two points with positive mass. The goal is to characterize the points on which to put positive mass. The model equipped with these preferences would give us the desired decline in the workweek length. It turns out that neither of the two preference formulations would change the result of a constant workweek if they were used in a standard model.

23

proof uses the envelope theorem with respect to equation (1), to characterize its solutions. Let R (c, h; λ, µ, τ c , τ e ) = u (c, h) − λ ((1 + τ c ) c − (1 − τ e ) w (h)) − µ Note that given the assumptions, R (c, h; λ, µ) is strictly concave in c. I define: c (h; λ, τ c ) ≡ arg max R(c, h; λ, µ, τ c , τ e ) c≥0

A solution to the problem will be interior given the Inada condition on consumption. This solution satisfies the following first order necessary and sufficient condition: u1 − λ ≤ 0 with strict equality, if c > 0.

(8)

This condition implicitly defines a function c (h; λ, τ c ) , and since u1 is continuous I know that the function c (h; λ, τ c ) is continuous in h. For c (h; λ, τ c , τ e ) ∈ <++ using the implicit function theorem: dc u12 = dh u11 for u11 (c (h; λ, τ c ) , h) 6= 0. Plugging the function c (h; λ) back into R, I get a function G: G (h; λ, µ, τ c , τ e ) = R (c (h) , h; λ, µ, τ c , τ e ) . A solution to the following problem exists for each set of parameters (λ, µ, τ c , τ e ) by the Weierstrass Theorem. max G (h; λ, µ, τ c , τ e )

h∈[0,1]

The solution might not be unique, since the function is not concave in h.

24

Figure 8: Qualitative features of workweek length defining function G (h; λ, µ, τ c , τ e ).

1 h

G(h;*,*)

I get as the FONC, for interior solutions: G0 (h; λ, µ, τ c , τ e ) = c0 (u1 − λ (1 − τ c )) + λ (1 − τ e ) w0 − u2

8

⇒ G0 (h; λ, µ, τ c , τ e ) = λ (1 − τ e ) w0 − u2 µ ¶ 1 − τ e 0 u2 0 G (h; λ, µ, τ c , τ e ) = u1 w − 1 + τc u1 Note that u1 (c(1), 1) → ∞, which implies that: lim G0 (h; λ, µ, τ c , τ e ) =

h→1

1 − τe 0 λw (1) − lim u2 (c, h) = −∞. h→1 1 + τc

(9)

Thus there can’t be a global maximizer at the points (c, 1). Now I establish that there can not be two interior solutions. To do this it is sufficient to show that the function G(h; λ, µ, τ c , τ e ) has at most one inflection point. The idea is illustrated in Figure 8.

25

Given the particular functional forms of the problem at hand the following is known: (1 − α) c (h, λ) u2 = u1 α (1 − h) 1 σ (1 − α) ³ α ´ 1−α(1−σ) (1 − h) 1−α(1−σ) . = α λ

Note that

(1−α) α

1 ¡ α ¢ 1−α(1−σ)

λ

−σ 1−α(1−σ)

> 0 and

< 0. Aside from this I know that

w(h) = A1/(1−θ) (θ/r)θ/(1−θ) hφ/(1−θ) is a strictly increasing strictly convex function. In summary: As functions of the workweek length

u2 u1

and w0 cross at most once, which implies that

G(h; λ, µ, τ c , τ e ) has at most one inflection point. This completes the proof. ¥ Proof of Proposition 2 The proof follows from a couple of algebraic operations. Given the specific utility function I get: α(1 − σ)u (c1 , h) = λc1 α(1 − σ)u (c0 , 0) = λc0 This implies: −λ (c1 − c0 ) = −α(1 − σ) (u (c1 , h) − u (c0 , 0)) Thus I get from the other first order conditions: (1 − α(1 − σ)) (u (c1 , h) − u (c0 , 1)) = −λw (h) u2 (c1 , h) = λw0 (h)

(10) (11)

Furthermore, I have: −(1−α)(1−σ) c0 = (1 − h) 1−α(1−σ) c1

I combine this with equation (10)/(11) to find: ´ σ 1 − α (1 − σ) ³ w (h) 1−α(1−σ) (1 − h) − (1 − h) = 0 (1 − σ) (1 − α) w (h) 26

(12)

From the firms problem, the following price restrictions arise: Aφhφ−1 kθ e1−θ µ ¶θ k φ A (1 − θ) h e ³ e ´1−θ Aθhφ k φ θ 1−θ Ah k e

= w0 (h) e = w (h) = r = w (h) e + rk

Note that: w (h; r) = (1 − θ) A w0 (h; r) = φA

1 1−θ

1 1−θ

θ µ ¶ 1−θ φ θ h 1−θ , r

θ µ ¶ 1−θ φ θ h( 1−θ −1) . r

also observe that w and r are homogenous of degree zero in (k, e). I combine the last two relationships to get: w (h; r) 1−θ = h 0 w (h; r) φ

(13)

Combining 13 and 12 gives me one equation in one unknown.

−

with ε =

´ 1−θ σ 1 − α (1 − σ) ³ (1 − h) − (1 − h) 1−α(1−σ) = h (1 − σ) (1 − α) φ ¢ 1−θ 1¡ ⇔ − 1 − h − (1 − h)1−ε = h ε φ

(1−α)(1−σ) . 1−α(1−σ)

Now, I show that this equation has one interior solution. As a first step, I will consider the left hand side of this equation.

W (h) = − Note that: W (0) = W (1) = 0.

¢ 1¡ 1 − h − (1 − h)1−ε ε

27

(14)

Furthermore, I find for the function W (h) : ¢ 1¡ −1 + (1 − ε) (1 − h)−ε ε 00 W (h) = − (1 − ε) (1 − h)−ε−1 W 0 (h) = −

Note that for all values of σ > 0 and α ∈ (0, 1): ε < 1. Thus we have that W (h) is a strictly concave function on the interval [0, 1]. This in turn with the boundary values implies that W (h) is hump-shaped with zero values at h being zero and one. Moreover observe that W 0 (0) = 1. As a second step, I now consider the right hand side of the above equation, namely G (h) =

1−θ h. φ

Here you should note that G0 (h) > 0 and G (0) =

1−θ φ

< 1.

This together with the first step establishes that there exists a unique interior optimal solution to the analyzed equation. This finishes the proof. ¥ C. Data This part of the appendix lists the different data sources used to generate the presented figures. First I make a list of the data sources for the time series that have multiple sources and which I use in multiple figures. Then I list the figures and sources used to prepare them. Workweek length The following data sources were used to construct the weekly hours worked for the respective countries: U.S.A.: • Robert M. Waples, The Shortening of the American Work Week: An Economic and Historical Analysis, Dissertation University of Pennsylvania, 1990. • U.S. Department of Labor, Bureau of Labor Statistics, Employment and Earning, Publications 1959.1-2004 U.K.:

28

• Hans-Joachim Voth “The Longest Year: New Estimates of Labor Input in England, 1760-1830,” The Journal of Economic History, 61(4), 2001. • Thelma Liesner, One Hundred Years of Economic Statistics, The Economist Publications, Fact on File, New York and Oxford, 1989. • Department of Employment and Productivity, British Labour Statistics: Historical Abstract 1886-1968, London 1971. • Michael Huberman “Working Hours of the world unite? New international evidence of working time, 1870-2000, Part 1 and 2,” Working Papers at the University of Montreal, March 2003. Germany: • Michael Schneider, Streit um Arbeitszeit, Geschichte des Kampfes um Arbeitszeitverkuerzung in Deutschland, Bund Verlag, Koeln, 1984. • International

Labor

Organization,

Geneva:

Laborsta

Internet

Database,

http://laborsta.ilo.org, downloaded June, 2004. Employment rates For the construction of the population of age 15 and older and the number of employed persons various sources were used: 1. Organization for Economic Co-operation and Development: OECD Statistics Portal, download 05/03/2004, http://www.oecd.org/statsportal/0,2639,en_2825_293564_1_1_1_1_1,00.html 2. B.R. Mitchell, International Historical Statistics, Europe 1750-2000, ed 5, Palgrave Macmillan, 2003 3. U.S. Department of Labor, Bureau of Labor Statistics, Employment and Earning, Publications 1959.1-2004 4. Ed. William Lerner, U.S. Department of Commerce and Bureau of the Census, Historical Statistics of the United States, Colonial Times to 1970, Part 1 and 2, 1975. 5. Thelma Liesner, One Hundred Years of Economic Statistics, The Economist Publications, Fact on File, New York and Oxford, 1989. 29

6. Bundesamt fuer Statistik, Statistisches Jahrbuch 2003, Wiesbaden, pg. 44. 7. Editor: C.F. Feinstein, National income, expenditure and output of the United Kingdom, 1855-1965, Cambridge: Cambridge University Press, 1972. 8. Bairoch, P., Deldycke, T., Gelders, H., and Limbor, J.-M., La Population active et sa Structure, The Working Population and its Structure, Universite Libre de Bruxelles, Insitute de Sociologie, Centre d’Economie Politique, 1968. For notational convenience, below I use E for the number of employed persons and N for the population of age 15 or older. Germany E: 1 and 5. N: 1, 2, 5, and 6. United Kingdom E: 1, 5, and 7. N: 1, 5, and 7. United States of America E: 1, 3, and 5. N: 3 and 4. To cross-check the correctness of some employment and working age data I made use of source 8. National accounts for the U.S.A. For the early period I used Kendrick, Productivity Trends in the U.S. and for the later period I used data from the Bureau of Economic Analysis. Effective marginal tax rate on labor For the average marginal tax rate on labor income I used Joines (1981) for the time period 1929 to 1975. For the later period, I used the trend from Marion and Mulligan (2004). The tax on consumption is found using the "Indirect business taxes less subsidies" and the "private consumption expenditures" from the National Income and Product Accounts of the United States.

30

The effective marginal tax rate on labor is defined as: τ=

τc + τe , 1 + τc

where as in the paper τ c is the tax on consumption and τ e is the tax on labor income. Figures Figure 1: Use the workweek length data for the respective three countries. Figure 2: Use the employment rate data for the respective countries. Figure 3: Use the national accounts data for the United States, listed above, to get TFP via the formula: At =

GDPt φ θ 1−θ ht Kt et

for all t = 1870, ..., 2002,

with θ = 0.33 (the capital income share for the United States post WWII) and φ = 1 (following Hornstein and Prescott (1993)). Also, using the workweek length data for the United States listed above. Figure 4: The tax rate data and the workweek length data for the United States are taken from the sources mentioned above. Figure 5: As described in the paper. Figure 6 and 6: From "Groningen Growth and Development Centre and The Conference Board, 10-Sector Database, June 2004, http://www.ggdc.net/dseries/totecon.html"

31

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