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The economics of staggered work hours
JOURNAL
OF URBAN
ECONOMICS
9, 349-364 (1981)
The Economics
of Staggered
Work Hours
J. VERNONHENDERSON’ Department of Economics,
Brown Universi~, Prwkience,
Rhode Island 02912
Received May 3 1, 1979
Until recently most firms in the Central Business District of a city tended to have identical work schedules. For example, in Lower Manhattan in 1970,over 75% of the l/2 million workers had starting times in the interval 9-9: 14 AM (O’Malley, 1974, exhibits 18). In this situation, transport planners are faced with the problem that tens or hundreds of thousands of workers want to arrive at the same location at the same time. It is physically impossible to accommodate the wishes of all commuters; and most workers must arrive early or late for work. For example, in Lower Manhattan in 1970, it would appear that at most l/3 of the people starting work at 9 AM arrived even within the 8:45-9:00 AM interval (O’Malley, 1974, exhibit 1). Vickrey (1969) and Henderson (1974, 1977) have modeled this type of situation. A primary result is that as early arrivals move their arrival times away from the starting time of work, their increasing waiting costs from arriving early at work are compensated by lower travel costs, given the reduced equilibrium levels of arrivals and congestion and hence higher travel speeds at earlier times. This traffic situation where so many people want to arrive at the same place at the same time seemsto present two potential sources of considerable resource wastage. First, there is the fact that since substantial numbers of people arrive early to work they must sit around “idle” for significant periods of time. While this idle period may not be especially unpleasant, most people may have better uses for their time. Second, transport facilities are heavily overburdened by the concentration of arrivals right around the single work starting time. This concentration plummets travel speeds and escalates congestion levels and capacity requirements. An obvious solution to this problem is for firms in a business district to have differing work schedules, or staggered starting and quitting times. Then desired arrival times and arrivals would be spread out. This would relieve the overburdening of transport facilities, reducing congestion and ‘In writing this paper, I have benefited from comments on the material by the members of the Urban Economics Seminar at Harvard University. I also had a very useful conversation with John Kennan several years ago concerning the nature of the staggered work hours externality. 349
0094-1190/81/030349-16$02.00/O copyrisht 0 1981 by Academic All rights
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travel times, and perhaps even reducing capacity requirements. Second, with appropriate scheduling, the period of idle waiting time involved with a single starting time would be entirely eliminated. People would start work when they arrived. This would shorten the amount of time people need to spend at work, without reducing time actually spent working. In Lower Manhattan within two years of the partial introduction of staggered work hours, about 25% of arrivals had been shifted off the heaviest peak time. While data on productivity and reduction in “idle time” (total hours in the business district) were not collected, information on employee morale, absenteeism and punctuality indicated significant improvements in those variables (O’Malley, 1974). Subway congestion was significantly reduced, although in New York no estimates of travel time savings were made. However, in a Toronto demonstration project, auto commuters who constituted over 50% of the participants experienced average travel time reductions of about 25% (Greenberg and Wright (1974), Guttman (1975)). Given these potential benefits of staggered work hours, why is this type of scheduling only a recent feature in business districts? Why is the degree of work hour staggering still very limited in many cities? To answer these questions, it is necessary to investigate the other effects of staggered work hours and then model an equilibrium staggered hours situation. Such modeling will reveal any problems in sustaining a staggered hours equilibrium and any economic problems involved in attaining an efficient degree of staggering. In developing a model, two other main features of staggered work hours must be incorporated. First, a person’s work schedule affects the scheduling of home activities. Commuters will have a most desired work schedule that best fits in with school hours of any children, work hours of a spouse, day light hours, T.V. and shopping hours, and scheduling of other family activities. Commuters could in general have the same most preferred schedule or range of equivalent schedules, although the data indicates that employees have differing most preferred schedules (mostly earlier than the traditional 9-5 schedule) depending on numbers of dependents, commuting times, and employment status (O’Malley (1974), Guttman (1974)). Regardless, within limits a choice and range of work schedules would result in staggered work hours as commuters shifted to a schedule that either or both involved better coordination with home activity and lower commuting costs. Second, staggered work hours may affect production activity. Neoclassical production analysis suggests two opposing effects. Within a firm, the concept of diminishing marginal returns indicates that it would be best to uniformly distribute employee work hours over the 24-hour day (Lucas, 1970). This would enhance labor marginal productivity and reduce capac-
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OF STAGGERED
WORK
HOURS
351
ity requirements. On the other hand, urban economic theory suggeststhat there must be external economies of scale in production in cities connected with the interaction of firms’ employees in business districts (Mills (1967), Henderson (1977)). White collar jobs involve extensive communications and interactions with corresponding people in opposing or complementary firms. The larger the mass of people at work in a business district the greater will be their individual productivity. Given the initial very negative reactions by firms to altering their starting times from the traditional schedule in their business district (O’Malley, 1974) it would appear that this latter productivity effect is perceived as being dominate. This perception, of course, would explain the persistence of traditional identical work schedules in business districts, despite the potential benefits of modifications. Interestingly enough, however, in some studies firms generally report either no productivity changes or productivity gains when they shift their starting times away from the norm. This would suggest that a simple “black-box” production function is not sufficiently sophisticated to capture some of the production effects involved in staggered work hours. These omitted effects would (a) include the productivity gains due to increased “morale” with the elimination of waiting times; (b) allow for the varying work activities of an individual and would distinguish between interactive activities (phone calls and meetings) and solo activities (writing up reports, forms, memos, and records and doing own research and planning); and (c) distinguish among occupations that are subject to varing degrees of diminishing returns within firms or scale externalities across firms or industries. Effect (b) might suggest that a most efficient work schedule from a production point of view might only involve a core period of, say, no more than 75% of the working day when all business district employees need be present at the same time. We now turn to a formal analysis of staggered work hours and the propositions that come out of such an analysis. MODELING
STAGGERED WORK HOURS
In a staggered work hour equilibrium, commuters in the business district choose a fixed starting time from the equilibrium distribution of starting times. We assume all commuters work the same fixed number of hours, although we will investigate the effects of relaxing this assumption. We will also comment on a flexitime model, where starting times are not fixed, but are chosen anew each day by commuters. This type of situation yields the same kind of results as a staggered work hour equilibrium, but must involve the complicating introduction of decision making under uncertainty.
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HENDERSON
COMMUTER DECISION MAKING Commuters seek to maximize the net benefits from working, which are their wages minus commuting cost and opportunity costs of work time. We start by assuming that all commuters have identical preferences and skills. Thus a representative commuter seeks to maximize W(s) - T(A(s)) - C(s - S).
(1)
s is the time this worker chooses to start work at and W(s) is the daily wage corresponding to that starting time. The commuter chooses s either by choosing to work in a firm with a starting time for all its employees of s or by choosing s from the range of starting times of a firm that offers its employees a choice of different work schedules. The endogenous starting and ending times of the morning commuting period in terms of arrivals in the business district are s, and s2, respectively. Thus s, < s I s2. Since all workers work the same number of hours, H, the evening range of departure times are s, + H to s2 + H. The fixed work hour assumption also implies that the morning pattern of arrivals between s, and s2 is identical to the evening pattern of departures between s, + H and s2 + H. A(s) is the total equilibrium number of arrivals in the business district at time s and is the number of departures at time s + H. Total daily (going and returning) travel costs for a commuter thus are T(W),
T’(1) = 0,
T’(A > 1) > 0,
T” 2 0.
(2)
People’s cost of travel is related to their speed of travel, where declining speedsimply higher costs due to higher time and “frustration” costs for a given trip distance. Speed of travel for a commuter in turn is related to the number of people on the road at the same time as the commuter, where an increase in the flow (measured by arrivals) of traffic increases congestion and reduces travel speed. Following Henderson (1977), we specifically assume a single entry-exit point transport system where the speed of travellers entering a system is solely a function of the number of travellers entering with them. Their speed of travel is constant throughout the journey, implying that entrants do not draw apart, earlier entrants are not overtaken by later entrants, and thus the travellers you enter the system with are the same ones you arrive with at the business district. This model of a transport system is easy to work and seems“more realistic” than the alternatives.’ The restriction that T” 2 0 is consistent with empirically 2An additional implication at time t + dt cannot have overtake people who left at equilibrium. With respect to speeds for a group entrants
of our assumption should be noted. People entering the system sufficiently higher travel speeds in equilibrium so that they t. That is, if T is travel time in the system ISr/dtl < 1 in alternative models, the only way in which the effect on travel by people entering before and after has been captured is to
ECONOMICS
OF STAGGERED
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HOURS
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estimated congestion functions (Inman, 1978). The implicit assumptions that all commuters travel on one system (or are distributed in proportion to capacity over an existing set of systems) and that they all travel the same distance can be relaxed in applying our model, but this relaxation generates no new principles. The C(s - S) function in (1) is the opportunity cost of working time. This opportunity cost has its lowest value for work schedules with a starting time of 5, this being the schedule that is best coordinated with family activities, daylight hours, etc.3 As starting times move away from S in either direction opportunity costs rise, perhaps imperceptibly at first and then more rapidly as inconveniences grow. For simplicity, we assume the C( 0) function is symmetrical about S or C(t-s;t-s= dC/dsr
-x
C(s - S; s - S = x > 0), all x if s( :I.?,
(3)
c > 0. COMMUTER
EQUILIBRIUM: NO PRODUCTION EFFECTS
For our identical commuters to be in equilibrium they must all have equal net benefits from working in (1). Therefore, gLy&&(), k = dA/a!s. FkzdW/ds, (4) In equilibrium k varies according to how equilibrium productivity of workers varies by work schedule. Assume for the minute that k = 0, or there are no production effects. If J$’= 0, equilibrium is determined entirely by looking at commuters, where from (4) arrivals must vary such that
assume that the speed of any traveller anywhere in the system is the ssme and determined solely by the total “load” (vehicles) in the system (Agnew, 1973).This implies that, for people about to exit, their speed is determined by the number of people behind them. This is a reasonable model for computer systems but perhaps not roads. 31t would really be more accurate to view J as the most desired time to leave home at (this lewing time in equilibrium then directly implies an arrival time back home). Then we would write the function as C((S - T(A(s)) - Q. For technical reasons it is beat to base the analysis either on CBD times or home times but not both. Since we have chosen CBD arrival times, we use Eq. (3).
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HENDERSON
FIGURE 1
An equilibrium pattern of arrivals is depicted in Fig. 1. The symmetrical pattern of arrivals about S is necessary for equilibrium.4 Since C(m)has the same value at S - d and S + 6 for all 6, in equilibrium in (l), T(A(S - 6)) = T(A(3 + 6)) for all 6, implying that A(5 - 6) = A(S + 6). For this reason S - s, = s2 - S; and from (2) A(s,) = A(s,) -+ 1. For A > 1, T’ > 0, or T increases. Thus, if A > 1 at s, [s,], one commuter could arrive a fraction of time earlier [later] and for an infinitesimal change in C( -) significantly reduce his travel costs. (If the onset of congestion only occurs for A > 112 commuters, then A(s,) = A(+) -+ m.) While Fig. 1 may represent a stable equilibrium under appropriate assumptions once it is achieved, there is some question about how the equilibrium is attained. In postulating (l), (4) and (5) the implication is that commuters know the distribution of arrivals and the set of travel costs facing them. While this is a strong information requirement, presumably a similar equilibrium would arise in a model where commuters in choosing s are maximizing expected benefits, given a perceived distribution of possible commuting costs associated with each s and ways of acquiring information about commuting costs at different times. Secondly, with no production affects, firms presumably are willing to stagger work hours in response to employee demands. This can involve internal staggered hours or staggering across firms. To move from a nonstaggered equilibrium to the one in Fig. 1, firms would have to be willing to adjust starting times in response to employee demands. The inducement to adjust would be workers’ willingness to shift to firms offering better work schedules in a non-equilibrium situation. ‘To achieve a more typical pattern of arrivals with a relatively slow build-up to the peak, in comparison to a fast drop-off after the peak, we would have to make C(e) asymmetrical with j&s
-S)l
< ld(5 +&)I.
ECONOMICS OF STAGGERED WORK HOURS
355
We can incorporate differences in preferences into this model by, say, allowing the shape of the C( .) function to vary, but maintaining the same best starting time, S, for all commuters, (We can vary S’s also but this raises the messy possibility of the equilibrium involving a multi-nodal distribution of arrivals.) Suppose there are two groups of people B and D where IC,(x)( > IC,(x)(, a11x 3 0. Equilibrium is pictured in Fig. 2. Each group of people has an Eq. (1) and a corresponding equilibrium condition in Eqs. (4) and (5). At any point of overlap of these arrivals, in Eq. (5) both groups will have the same T’ (we could, of course, allow this to differ also). Thus people B with their steeper C(a) function will require a higher k for equilibrium-hence the shape of the A curves in Fig. 2. The equilibrium configuration in Fig. 2 with D people commuting on the tails is the only one that satisfies the necessary global equilibrium condition that, if people alter their s’s, they cannot be made better off. For example, if D people move from the interval s,,I, to s’,,S (or vice versa for B people), they are worse off. The dashed extension of the A curve for D people as we move beyond f, indicates from (5) the level of arrivals necessary to yield D people equal benefits with an arrival time in s,, j,. Since equilibrium arrivals for B people are higher than this, D people would be worse off if they moved into the interval s’,,S. No other equilibrium configuration of arrivals meets this criterion. For example, if we put B people on the tails in a redrawn Fig. 2, they would benefit by moving from their s,, 3, interval to the i,, S interval. Note that for such an equilibrium if firms employ both B and D people they will have to offer differing starting times for their two groups of employers. There is also the empirically testable hypothesis, that, if the C(a) functions vary across people in some identifiable way (e.g., by
A T
356
J. VERNON HENDERSON
number and age of family members), we should expect to see similar people having two arrival intervals-one before and one after the height of peak arrivals. COMMUTER
OPTIMUM:
NO PRODUCTION
EFFECTS
Is the degree of staggering of work hours implicit in (5) optimal? When commuters, through their firms, adopt work schedules they do not account for the effect of their decisions on other people’s travel costs-raising costs for those with the same schedules and lowering costs for all other schedules. This externality, of course, is a version of the traditional congestion externality. A social planner’s problem is to minimize the opportunity costs of work, or to minimize
S2A(S)(T(A(S)) + C(s- S))ak I SI Rearranging the Euler equation yields
-d > A=2T+T”A i =< 1 0
as
s
< i => Is.
Given T” 2 0, all other necessary conditions are satisfied. Boundary conditions require that, at si and s2, -(A(s,))*T’(A(s,)) = -(A(s2))*T’(A(s2)) = 0. Thus A(s,) = A(s2) = 1 (minimal flows), given T’(1) = 0. The optimal pattern of arrivals in depicted in Fig. 1. A sufficient but not necessary condition for k’ < 0 is T”’ 2 0. Since for any A and s, 1&,I < 1Acquil 1from Eqs. (5) and (7) at any crossover point of the curves the A, curve must cut the Aequil curve from above as depicted. For this reason, given the same total flows (areas under the A curves), sy < s, and sz > s2. To achieve such an optimal solution, we could impose the usual congestion toll on commuters, here equal to AT’ which is the measure of the congestion externality. Adding AT’ to (1) yields the equilibrium flows depicted by (7). The tolls which are heaviest at S eliminate the heavy congestion around 3 and induce commuters to spread out their travel and work starting/quitting times to s’s where tolls are lighter. In this case, commuters in choosing starting times are made to account for the external effects of their decisions. In this simple model, an optimal solution could also be attained by taxing firm starting times. Firms would be taxed A(s)T’(A(s)) per worker for workers starting at time s. The tax would be passed onto workers through wage adjustments (B’ in (4)) relative to workers starting at times s,
ECONOMICS OF STAGGERED WORK HOURS
357
and s2, where AT’ = 0. The institutional advantage of imposing the tax this way is that collection is easy, compared to the economic and political problems of taxing commuters directly. The viability of this second taxation procedure depends on three assumptions. Everyone travels on the same mode so that AT’ is equal for all commuters at any time; and car pooling does not occur. These first two assumptions are simply a statement that taxing starting times does not provide the correct incentives to adjust modes and occupants per car on roads. The final assumption is either that all workers in a firm start at the same time or else that wages may differ for similar workers in the same firm. Clearly, within a firm, for there to be an equilibrium with similar workers starting at different times wages must vary by starting time as taxes vary. This could be institutionally difficult, although wages do vary by eight-hour shifts in a variety of industries. At this point in the paper, it is easiest to assume that all workers in a firm start at the same time, so equilibrium under taxation only requires some wage adjustment across firms. This wage difference problem will be raised again later. Finally we note that the revenue raised from congestion tolls is the key to demonstrating how in practical examples the imposition of a toll improves the welfare of commuters on the tails of the arrival distribution. For example, in Fig. 1, people at SF and S: relative to si and s2 have unchanged travel costs but higher C(s) costs. To show that they actually gain from imposing congestion tolls, it is necessary to bring in revenue considerations. Following Henderson’s (1977) modeling of the waiting time problem, commuters on the tails of the distribution (and, in fact, at all starting times) gain in net once the toll revenue raised is uniformly distributed back to the commuters in a lump gift form. Another way to view the problem is to bring in capacity cost considerations. The collection of toll revenue reduces or eliminates other levies imposed prior to the introduction of tolls that were needed to finance capital costs. If the prior levy was an annual or daily lump fee on users only covering capital costs, people throughout the arrival distribution benefit in net from congestion tolls because the reduction in lump fees for users exceeds any increase in day-to-day costs (additional C( .) costs for people on the tails and the advent of high tolls for people nearer the peak of arrivals). Of course, the move from an equilibrium to an optimum will also alter capacity requirements. If capacity is denoted by K and the price per unit of capacity is $1, adding K to Eq. (6) yields an optimal capacity condition of -
‘*A aT/aKds
I SI
= 1.
(8)
Note also that, if T is homogeneous of degree 0 in A and K (i.e., doubling
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J. VERNON HENDERSON
capacity and flows leaves travel speeds and costs unchanged), tolls cover capacity costs. Capacity costs equal K( /ST- &Z/X) dr) if (8) is satisfied. Tolls equal J,“12A( A aT/&4) dr. Thus tolls equal capacity costs when T is homogeneous of degree 0 which implies - KaT/iK = A aT/aA. If a transport planner satisfied (8) for the equilibrium prior to tolls, then in general, the optimal capacity will be less than the equilibrium, because the elimination of the heavy arrivals around S reduces the need for capacity. COMMUTER
EQUILIBRIUM: PRODUCTION PRESENT
EFFECTS
The introduction of production effects serves two purposes. One is to illustrate the nature and characteristics of possible production equilibria under staggered work hours and the second is to raise certain institutional problems involved in sustaining a staggered work hours equilibrium. We assume each firm has an instantaneous production function at time t of the form
n(s) is the firm’s own work force and the coefficient a represents its internal technology. Thus for simplcity we are ignoring diminishing returns within the firm-a factor viewed by firms as being unimportant in staggered work hours. g(N(t)) is an external economy of scale shift factor where N(t) is the total number of employees working in the business district at time t. g’ > 0,g” < 0. This specification captures the primary production problem associated by firms with staggered work hours. If a firm shifts to an early schedule where N(t) is very low, productivity will be low because its employees will have fewer employees in other firms in the business district to interact and do business with. Firm profits per employee starting work at time s are
H is the fixed length of work day, s, is the beginning time of morning arrivals and s2 the ending time. Given a commuting period of s2 - si, any firm has (H - (s2 - s,)) hours when all workers are present in the business district and thus production by one of its employees during this interval is (H - (s2 - si))g(N)a, where # is total business district work force. Moving to the second expression on the RHS of (9), N(t) is the number of employees at work in the business district at time t. Thus N(t) is
ECONOMICS OF STAGGERED WORK HOURS
359
accumulated arrivals, or
N(f)=p4(s)d.Y, ni= A(s). SI
(10)
Therefore, a worker’s production from the time he arrives, s, until all workers have arrived is J,S2g,,(N(t))adt. Turning to the third expression, in the evening people exit from the business district in the same order that they arrived. Total employees at time t + Zf are
IT- / ‘+HA(s)d.v= F- N(t). s,+lY
(11)
Equations (10) and (11) are illustrated for an equilibrium (see later) in Fig. 3. Given (1 I), a worker’s production during the evening rush period before he leaves is /,“,:z g ,(c- N(t))adt. Note g,(x) z g,(x) all x, or the “0” and “1” subscripts only designate morning vs evening production, not differences in functional form. Given how wages vary in equilibrium with starting time S, firms choose s to maximize profits. Alternatively, given identical production functions and workers, da/& = 0, for all s in any stable equilibrium. Otherwise, further shifting among starting times will occur. Differentiating (9), we see that equilibrium requires that
4g, - go) = *.
(12)
360
J. VERNON HENDERSON
Combining this with the consumer equilibrium condition in (4), we find that for a work staggering equilibrium with production effects
k _ 4& - go) - d T’ ’
(13)
k’ < 0. An equilibrium solution, such as depicted in Fig. 1, where the arrival pattern is symmetric about 5 satisfies (13) and equilibrium conditions on boundary points, where n(s,) = n(s2) given W(S) from (1). If arrivals are symmetric about S then, from Fig. 1, N(5 - 6) = ” - N(S + 6) all 6; and thus g,(% N(S - 6)) - ge(N(S - 8)) = -[g,(N - N(S + s>) - g,(N(3 + S)J > 0. Note fo_r a symmetric pattern g,(N(S)) = g,(N - N(S)) = g(f N) so that (g, N - N( S - 8)) 7 gO(N(S - 6)) > 0. Combining these results with Eq. (3), in Eq. (13), N=A > [ <]0 as s < [ >]O. This pattern of N(t) and N - N(t) is illustrated in Fig. 3. With a symmetrical arrival pattern, s2 - 5 = S - s,, where n(sl) - n(s2) = ( lsyg,adt - lsff++HH8,adt) - (T(A(s,)) - T(A(s,)) - (C(s, - S) - C(s, - S)) = 0 if S - s, = Q - S. Note that as before equilibrium at s, and s2 requires congestion to approach zero, which given our functional assumptions implies that A(s,) = A(s,) + 1. The new feature of equilibrium is that wages must vary according to starting time. The instantaneous marginal product of a worker is illustrated in Fig. 4a, where dg, /drj > 0 and dg, /dr = (dg, /d( g - N(s))( - A(s)) < 0; the curves are not necessarily concave throughout. A worker starting at S has the highest total productivity during the peak periods-operating on the S to s2 segment of MP(s) and the s, + H to S + H segment of MP(s + H). Total productivity of a worker declines as he moves away
ECONOMICS OF STAGGERED WORK HOURS
361
from S in either direction to lowest productivity at si and s2. Thus total daily wages should vary qualitatively as illustrated in Figure 4b. Such required wage variation for identical skill workers may present an institutional impediment to attaining and sustaining staggered work hour equilibria. This may be a partial explanation of why staggered work hours was a non-existent or minor feature of scheduling in business districts for so long. If we assume all similar skill workers of each firm start at the same time, that at least would eliminate the need for wage variation for identical skill workers in the same firm and would only require wage variation across firms. If we reintroduce scheduling taste (C( *) function) differences for identical skill workers, then in Fig. 2, workers will in part sort themselves out across firms by their tastes given the firms’ starting times. If we allow skill differences among workers, we will get the same type of equilibrium as illustrated in Fig. 2. For example, suppose there are two groups of workers with productivity gu, and ga,, where a, > a2. Then, by similar arguments as for taste differences, in equilibrium a, people will arrive at times near S and a, people on the tails. This, of course, makes sense because in general total productivity in Fig. 4a is enhanced for starting times around S and it is most profitable to enhance (by a multiplicative term) the productivity of the most productive. If firms use both types of workers in production, then from Fig. 2 their different types of workers must start work at different times and receive different wages on the basis of both basic productivity differences (a, > u2) and starting time differences. Two additional points should be noted. This requirement that wages differ across identical skill individuals according to their work schedules holds regardless of the specification of production effects (e.g., the introduction of diminishing returns), as long as production effects are present. Second, our results are consistent with the results to be obtained from flexitime models, where people must work the same number of hours but choose their starting time on a day-to-day basis. Flexitime would introduce the complication that the distribution of 7’(A)‘s is uncertain on a day-to-day basis, as is the distribution of cumulated arrivals. Thus everyone operates on expected values, eliminating the deterministic nature of the model but not the basic principles. However, if the number of hours worked becomes a free choice variable for everyone, our model is more complicated. For example, commuters might maximize W(s, e)Zf - T( A(s)) - C( H, e, s - S,s + H - i), where e is (uniform but endogenous) effort per instant worked, H is variable hours worked, Sis desired starting time, and tis desired quitting time. aW/ae > 0, X/aH > 0, K/se > 0 and X/as ‘, 0. There are many interesting types of equilibria that could arise from this specification, especially if the cross partial derivatives of C(e) are nonzero.
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CONSUMER OPTIMUM: PRODUCTION PRESENT
EFFECTS
With production effects, a social planner’s objective is to maximize the sum of production in all firms minus the opportunity and travel costs of all workers. Thus the planner seeks to maximize
J = g( N)uN( H - (s* - s,)) + ~~*&)(N(s))uN(s) dr SI + J:‘,“pl(
-
F - N(s))u( l7 - N(s))ds - jS2A(s)T(A(s))h SI
S2A(s)C(s - i) ds I $1
subject to ni = A(s), W,)
= 0,
N(S*) = Iv. The first three expressions in J are, respectively, production during the noncommuting period, production during the morning peak period, and production during the evening peak period. Note that production in (14) is consistent with (9) where total morning peak period production of all people (A(s)) arriving at time s is J~‘zgc(N( t ))aA (s) dt and total production over all s’s is ~,“,z(J~zga(N(t))aA(s) dt) ds = integrating /,‘,‘(jlA(s)dr)ug,(N(t))dt = l:g,$N(t))uN(t)dt. The last two terms of (14) are opportunity costs for all people starting at time s integrated over all s ‘s. The Euler equation implies that
/Liz 4g, - &I)(1 + cl - d 2T’ + T”A
*
(15)
Without loss of generality, to simplify the discussion of (15), we have defined c = (&o/dW~MWO/go) =-C&,/d(F - NtNM@ - Wt))/g,), where e is constant for 0 < N(t) I N. The other necessary conditions for maximizing J in (14) are satisfied (but not necessarily sufficiency conditions given production is convex in N(t)). Boundary conditions require that T’(A(s,))(A(s,))* = T’(A(s,))(A(s,))* = O;_orth_atA(s,) = A(s2) = 1. (If Z is the sum of integrands and F = g(.N)aN(N - (s2 - si)) then (Z - A aZ//aA),, + aF/b, = 0 for si = s,,s2.) A < 0, where T”’ 2 0 is sufficient but not necessary for this.
ECONOMICS OF STAGGERED WORK HOURS
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FIGURE 5
As in Fig. 1 and Eq. (13), a symmetrical arrival pattern satisfies all equations for the solution. Equation (15) relative to (13) indicates the presence of two conflicting externalities. Congestion externalities raise the denominator in (15) suggesting a slower build-up (k) of traffic for any A, the intuitive notion being one of spreading out arrivals to reduce peak congestion levels as in Fig. 1. However, the introduction of the (1 + C) term indicates a need to speed-up arrivals to accumulate people and exploit the scale economies in production that are present. In Fig. 5, we present two alternative optimal solutions relative to the equilibrium solution, either of which is possible given the conflicting forces on arrival rates. To achieve the optimal solution, we could impose a tax per worker arriving at time s equal to Tax (.s) = T’A(s) - a,[ ~s2godt + $pIt]
;O.
(16)
Note, as follows from the discussion above, arrival times are taxed for congestion and subsidized for scale economy benefits. At s, and s2 since 7” = 0, Tax (s,) = Tax (sz) < 0; or, we are subsidizing arrivals to encourage accumulation of arrivals and scale benefits, given congestion is miniscule. As we move towards S both expressions in (16) increase in absolute value (see Fig. 4a with respect to the second expression), although presumably the first escalates in value, so that around S Tax (s) > 0. Institutionally, the tax could be imposed on firms according to when each worker starts. If we add (16) to (9) and solve for market equilibrium, the solution is characterized by (15), and the same boundary conditions. However, imposing the congestion part of the tax on firms in a more general model does not allow for the optimal encouragement of car-pooling
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and allocation of commuters across other modes of travel. In a more general context T/A(s) should be imposed directly on travellers as a congestion toll. The externality subsidy could be offered directly to firms to encourage shifting of starting times; or, equivalently it could be offered directly to workers as a function of their starting time. CONCLUSIONS In this paper, the nature of equilibrium in a staggered work hours situation has been examined, and divergences between equilibrium and optimum solutions investigated. While the modeling in itself is conceptually instructive, what is also very interesting is the multitude of empirically testable hypotheses that spring directly from the model. In a staggered work hours equilibrium wages should vary across similar occupation workers by work starting time. Identifying commuters by skill and scheduling taste (family characteristics) differences, we would expect to find each category of commuters arriving in two different time intervals, one before and one after the peak of arrivals. Presumably also there will be a link between the composition of workers arriving on the tails of the distribution as opposed to at the peak and the nature of their job tasks. For example, workers who interact directly with others to a relatively small degree in their particular jobs should be found on the tails. In examining business districts that have shifted from the traditional single work schedule to staggered hours, we would expect to seepeak congestion eliminated, congestion on the tails of the arrival distribution raised, and time per commuter spent in the business district (holding work hours fixed) reduced. REFERENCES Agnew, C. E. (1973), “The Dynamic Control of Congested Systems Through Pricing,” Report No. 6, Stanford University Center for Interdisciplinary Research. Greenberg, M. A. and D. W. Wright (1974) “Staggered Hours Demonstration: Final Evaluation Report,” Ontario Ministry of Transportation and Communications, mimeo. Guttman, J. (1975), “Predicting the Effect of Variable Working Hours on Peak Period Congestion,” University of Chicago, mimeo. Henderson, J. V. (1974), Road congestion: A reconsideration of pricing theory, J. Urban Econ., 1, 346-365. Henderson, J. V. (1977), “Economic Z%eov and the Cities,” Academic Press, New York. Inman, R. P. (1978), A generalized congestion function for highway travel, J. &ban &on., 5, 21-34. Lucas, R. E. (1970), Capacity, overtime, and empirical production functions, Amer. Econ. Assoc. Papers Proc., 60, 23-27.
Mills, E. S. (1967), An aggregative model of resource allocation in a metropolitan area, Amer. Econ. Rev., 57, 197-210. O’Malley, B. W. (1974), “Work Schedule Changes to Reduce Peak Transportation Demand,” Port Authority of New York and New Jersey, mimeo. Vickrey, W. S. (1969) Congestion theory and transport investment, Amer. Econ. Reo., 59, 251-260.
OF URBAN
ECONOMICS
9, 349-364 (1981)
The Economics
of Staggered
Work Hours
J. VERNONHENDERSON’ Department of Economics,
Brown Universi~, Prwkience,
Rhode Island 02912
Received May 3 1, 1979
Until recently most firms in the Central Business District of a city tended to have identical work schedules. For example, in Lower Manhattan in 1970,over 75% of the l/2 million workers had starting times in the interval 9-9: 14 AM (O’Malley, 1974, exhibits 18). In this situation, transport planners are faced with the problem that tens or hundreds of thousands of workers want to arrive at the same location at the same time. It is physically impossible to accommodate the wishes of all commuters; and most workers must arrive early or late for work. For example, in Lower Manhattan in 1970, it would appear that at most l/3 of the people starting work at 9 AM arrived even within the 8:45-9:00 AM interval (O’Malley, 1974, exhibit 1). Vickrey (1969) and Henderson (1974, 1977) have modeled this type of situation. A primary result is that as early arrivals move their arrival times away from the starting time of work, their increasing waiting costs from arriving early at work are compensated by lower travel costs, given the reduced equilibrium levels of arrivals and congestion and hence higher travel speeds at earlier times. This traffic situation where so many people want to arrive at the same place at the same time seemsto present two potential sources of considerable resource wastage. First, there is the fact that since substantial numbers of people arrive early to work they must sit around “idle” for significant periods of time. While this idle period may not be especially unpleasant, most people may have better uses for their time. Second, transport facilities are heavily overburdened by the concentration of arrivals right around the single work starting time. This concentration plummets travel speeds and escalates congestion levels and capacity requirements. An obvious solution to this problem is for firms in a business district to have differing work schedules, or staggered starting and quitting times. Then desired arrival times and arrivals would be spread out. This would relieve the overburdening of transport facilities, reducing congestion and ‘In writing this paper, I have benefited from comments on the material by the members of the Urban Economics Seminar at Harvard University. I also had a very useful conversation with John Kennan several years ago concerning the nature of the staggered work hours externality. 349
0094-1190/81/030349-16$02.00/O copyrisht 0 1981 by Academic All rights
of reproduction
in any form
Press. Inc. -cd.
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travel times, and perhaps even reducing capacity requirements. Second, with appropriate scheduling, the period of idle waiting time involved with a single starting time would be entirely eliminated. People would start work when they arrived. This would shorten the amount of time people need to spend at work, without reducing time actually spent working. In Lower Manhattan within two years of the partial introduction of staggered work hours, about 25% of arrivals had been shifted off the heaviest peak time. While data on productivity and reduction in “idle time” (total hours in the business district) were not collected, information on employee morale, absenteeism and punctuality indicated significant improvements in those variables (O’Malley, 1974). Subway congestion was significantly reduced, although in New York no estimates of travel time savings were made. However, in a Toronto demonstration project, auto commuters who constituted over 50% of the participants experienced average travel time reductions of about 25% (Greenberg and Wright (1974), Guttman (1975)). Given these potential benefits of staggered work hours, why is this type of scheduling only a recent feature in business districts? Why is the degree of work hour staggering still very limited in many cities? To answer these questions, it is necessary to investigate the other effects of staggered work hours and then model an equilibrium staggered hours situation. Such modeling will reveal any problems in sustaining a staggered hours equilibrium and any economic problems involved in attaining an efficient degree of staggering. In developing a model, two other main features of staggered work hours must be incorporated. First, a person’s work schedule affects the scheduling of home activities. Commuters will have a most desired work schedule that best fits in with school hours of any children, work hours of a spouse, day light hours, T.V. and shopping hours, and scheduling of other family activities. Commuters could in general have the same most preferred schedule or range of equivalent schedules, although the data indicates that employees have differing most preferred schedules (mostly earlier than the traditional 9-5 schedule) depending on numbers of dependents, commuting times, and employment status (O’Malley (1974), Guttman (1974)). Regardless, within limits a choice and range of work schedules would result in staggered work hours as commuters shifted to a schedule that either or both involved better coordination with home activity and lower commuting costs. Second, staggered work hours may affect production activity. Neoclassical production analysis suggests two opposing effects. Within a firm, the concept of diminishing marginal returns indicates that it would be best to uniformly distribute employee work hours over the 24-hour day (Lucas, 1970). This would enhance labor marginal productivity and reduce capac-
ECONOMICS
OF STAGGERED
WORK
HOURS
351
ity requirements. On the other hand, urban economic theory suggeststhat there must be external economies of scale in production in cities connected with the interaction of firms’ employees in business districts (Mills (1967), Henderson (1977)). White collar jobs involve extensive communications and interactions with corresponding people in opposing or complementary firms. The larger the mass of people at work in a business district the greater will be their individual productivity. Given the initial very negative reactions by firms to altering their starting times from the traditional schedule in their business district (O’Malley, 1974) it would appear that this latter productivity effect is perceived as being dominate. This perception, of course, would explain the persistence of traditional identical work schedules in business districts, despite the potential benefits of modifications. Interestingly enough, however, in some studies firms generally report either no productivity changes or productivity gains when they shift their starting times away from the norm. This would suggest that a simple “black-box” production function is not sufficiently sophisticated to capture some of the production effects involved in staggered work hours. These omitted effects would (a) include the productivity gains due to increased “morale” with the elimination of waiting times; (b) allow for the varying work activities of an individual and would distinguish between interactive activities (phone calls and meetings) and solo activities (writing up reports, forms, memos, and records and doing own research and planning); and (c) distinguish among occupations that are subject to varing degrees of diminishing returns within firms or scale externalities across firms or industries. Effect (b) might suggest that a most efficient work schedule from a production point of view might only involve a core period of, say, no more than 75% of the working day when all business district employees need be present at the same time. We now turn to a formal analysis of staggered work hours and the propositions that come out of such an analysis. MODELING
STAGGERED WORK HOURS
In a staggered work hour equilibrium, commuters in the business district choose a fixed starting time from the equilibrium distribution of starting times. We assume all commuters work the same fixed number of hours, although we will investigate the effects of relaxing this assumption. We will also comment on a flexitime model, where starting times are not fixed, but are chosen anew each day by commuters. This type of situation yields the same kind of results as a staggered work hour equilibrium, but must involve the complicating introduction of decision making under uncertainty.
352
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HENDERSON
COMMUTER DECISION MAKING Commuters seek to maximize the net benefits from working, which are their wages minus commuting cost and opportunity costs of work time. We start by assuming that all commuters have identical preferences and skills. Thus a representative commuter seeks to maximize W(s) - T(A(s)) - C(s - S).
(1)
s is the time this worker chooses to start work at and W(s) is the daily wage corresponding to that starting time. The commuter chooses s either by choosing to work in a firm with a starting time for all its employees of s or by choosing s from the range of starting times of a firm that offers its employees a choice of different work schedules. The endogenous starting and ending times of the morning commuting period in terms of arrivals in the business district are s, and s2, respectively. Thus s, < s I s2. Since all workers work the same number of hours, H, the evening range of departure times are s, + H to s2 + H. The fixed work hour assumption also implies that the morning pattern of arrivals between s, and s2 is identical to the evening pattern of departures between s, + H and s2 + H. A(s) is the total equilibrium number of arrivals in the business district at time s and is the number of departures at time s + H. Total daily (going and returning) travel costs for a commuter thus are T(W),
T’(1) = 0,
T’(A > 1) > 0,
T” 2 0.
(2)
People’s cost of travel is related to their speed of travel, where declining speedsimply higher costs due to higher time and “frustration” costs for a given trip distance. Speed of travel for a commuter in turn is related to the number of people on the road at the same time as the commuter, where an increase in the flow (measured by arrivals) of traffic increases congestion and reduces travel speed. Following Henderson (1977), we specifically assume a single entry-exit point transport system where the speed of travellers entering a system is solely a function of the number of travellers entering with them. Their speed of travel is constant throughout the journey, implying that entrants do not draw apart, earlier entrants are not overtaken by later entrants, and thus the travellers you enter the system with are the same ones you arrive with at the business district. This model of a transport system is easy to work and seems“more realistic” than the alternatives.’ The restriction that T” 2 0 is consistent with empirically 2An additional implication at time t + dt cannot have overtake people who left at equilibrium. With respect to speeds for a group entrants
of our assumption should be noted. People entering the system sufficiently higher travel speeds in equilibrium so that they t. That is, if T is travel time in the system ISr/dtl < 1 in alternative models, the only way in which the effect on travel by people entering before and after has been captured is to
ECONOMICS
OF STAGGERED
WORK
HOURS
353
estimated congestion functions (Inman, 1978). The implicit assumptions that all commuters travel on one system (or are distributed in proportion to capacity over an existing set of systems) and that they all travel the same distance can be relaxed in applying our model, but this relaxation generates no new principles. The C(s - S) function in (1) is the opportunity cost of working time. This opportunity cost has its lowest value for work schedules with a starting time of 5, this being the schedule that is best coordinated with family activities, daylight hours, etc.3 As starting times move away from S in either direction opportunity costs rise, perhaps imperceptibly at first and then more rapidly as inconveniences grow. For simplicity, we assume the C( 0) function is symmetrical about S or C(t-s;t-s= dC/dsr
-x
C(s - S; s - S = x > 0), all x if s( :I.?,
(3)
c > 0. COMMUTER
EQUILIBRIUM: NO PRODUCTION EFFECTS
For our identical commuters to be in equilibrium they must all have equal net benefits from working in (1). Therefore, gLy&&(), k = dA/a!s. FkzdW/ds, (4) In equilibrium k varies according to how equilibrium productivity of workers varies by work schedule. Assume for the minute that k = 0, or there are no production effects. If J$’= 0, equilibrium is determined entirely by looking at commuters, where from (4) arrivals must vary such that
assume that the speed of any traveller anywhere in the system is the ssme and determined solely by the total “load” (vehicles) in the system (Agnew, 1973).This implies that, for people about to exit, their speed is determined by the number of people behind them. This is a reasonable model for computer systems but perhaps not roads. 31t would really be more accurate to view J as the most desired time to leave home at (this lewing time in equilibrium then directly implies an arrival time back home). Then we would write the function as C((S - T(A(s)) - Q. For technical reasons it is beat to base the analysis either on CBD times or home times but not both. Since we have chosen CBD arrival times, we use Eq. (3).
354
J. VERNON
HENDERSON
FIGURE 1
An equilibrium pattern of arrivals is depicted in Fig. 1. The symmetrical pattern of arrivals about S is necessary for equilibrium.4 Since C(m)has the same value at S - d and S + 6 for all 6, in equilibrium in (l), T(A(S - 6)) = T(A(3 + 6)) for all 6, implying that A(5 - 6) = A(S + 6). For this reason S - s, = s2 - S; and from (2) A(s,) = A(s,) -+ 1. For A > 1, T’ > 0, or T increases. Thus, if A > 1 at s, [s,], one commuter could arrive a fraction of time earlier [later] and for an infinitesimal change in C( -) significantly reduce his travel costs. (If the onset of congestion only occurs for A > 112 commuters, then A(s,) = A(+) -+ m.) While Fig. 1 may represent a stable equilibrium under appropriate assumptions once it is achieved, there is some question about how the equilibrium is attained. In postulating (l), (4) and (5) the implication is that commuters know the distribution of arrivals and the set of travel costs facing them. While this is a strong information requirement, presumably a similar equilibrium would arise in a model where commuters in choosing s are maximizing expected benefits, given a perceived distribution of possible commuting costs associated with each s and ways of acquiring information about commuting costs at different times. Secondly, with no production affects, firms presumably are willing to stagger work hours in response to employee demands. This can involve internal staggered hours or staggering across firms. To move from a nonstaggered equilibrium to the one in Fig. 1, firms would have to be willing to adjust starting times in response to employee demands. The inducement to adjust would be workers’ willingness to shift to firms offering better work schedules in a non-equilibrium situation. ‘To achieve a more typical pattern of arrivals with a relatively slow build-up to the peak, in comparison to a fast drop-off after the peak, we would have to make C(e) asymmetrical with j&s
-S)l
< ld(5 +&)I.
ECONOMICS OF STAGGERED WORK HOURS
355
We can incorporate differences in preferences into this model by, say, allowing the shape of the C( .) function to vary, but maintaining the same best starting time, S, for all commuters, (We can vary S’s also but this raises the messy possibility of the equilibrium involving a multi-nodal distribution of arrivals.) Suppose there are two groups of people B and D where IC,(x)( > IC,(x)(, a11x 3 0. Equilibrium is pictured in Fig. 2. Each group of people has an Eq. (1) and a corresponding equilibrium condition in Eqs. (4) and (5). At any point of overlap of these arrivals, in Eq. (5) both groups will have the same T’ (we could, of course, allow this to differ also). Thus people B with their steeper C(a) function will require a higher k for equilibrium-hence the shape of the A curves in Fig. 2. The equilibrium configuration in Fig. 2 with D people commuting on the tails is the only one that satisfies the necessary global equilibrium condition that, if people alter their s’s, they cannot be made better off. For example, if D people move from the interval s,,I, to s’,,S (or vice versa for B people), they are worse off. The dashed extension of the A curve for D people as we move beyond f, indicates from (5) the level of arrivals necessary to yield D people equal benefits with an arrival time in s,, j,. Since equilibrium arrivals for B people are higher than this, D people would be worse off if they moved into the interval s’,,S. No other equilibrium configuration of arrivals meets this criterion. For example, if we put B people on the tails in a redrawn Fig. 2, they would benefit by moving from their s,, 3, interval to the i,, S interval. Note that for such an equilibrium if firms employ both B and D people they will have to offer differing starting times for their two groups of employers. There is also the empirically testable hypothesis, that, if the C(a) functions vary across people in some identifiable way (e.g., by
A T
356
J. VERNON HENDERSON
number and age of family members), we should expect to see similar people having two arrival intervals-one before and one after the height of peak arrivals. COMMUTER
OPTIMUM:
NO PRODUCTION
EFFECTS
Is the degree of staggering of work hours implicit in (5) optimal? When commuters, through their firms, adopt work schedules they do not account for the effect of their decisions on other people’s travel costs-raising costs for those with the same schedules and lowering costs for all other schedules. This externality, of course, is a version of the traditional congestion externality. A social planner’s problem is to minimize the opportunity costs of work, or to minimize
S2A(S)(T(A(S)) + C(s- S))ak I SI Rearranging the Euler equation yields
-d > A=2T+T”A i =< 1 0
as
s
< i => Is.
Given T” 2 0, all other necessary conditions are satisfied. Boundary conditions require that, at si and s2, -(A(s,))*T’(A(s,)) = -(A(s2))*T’(A(s2)) = 0. Thus A(s,) = A(s2) = 1 (minimal flows), given T’(1) = 0. The optimal pattern of arrivals in depicted in Fig. 1. A sufficient but not necessary condition for k’ < 0 is T”’ 2 0. Since for any A and s, 1&,I < 1Acquil 1from Eqs. (5) and (7) at any crossover point of the curves the A, curve must cut the Aequil curve from above as depicted. For this reason, given the same total flows (areas under the A curves), sy < s, and sz > s2. To achieve such an optimal solution, we could impose the usual congestion toll on commuters, here equal to AT’ which is the measure of the congestion externality. Adding AT’ to (1) yields the equilibrium flows depicted by (7). The tolls which are heaviest at S eliminate the heavy congestion around 3 and induce commuters to spread out their travel and work starting/quitting times to s’s where tolls are lighter. In this case, commuters in choosing starting times are made to account for the external effects of their decisions. In this simple model, an optimal solution could also be attained by taxing firm starting times. Firms would be taxed A(s)T’(A(s)) per worker for workers starting at time s. The tax would be passed onto workers through wage adjustments (B’ in (4)) relative to workers starting at times s,
ECONOMICS OF STAGGERED WORK HOURS
357
and s2, where AT’ = 0. The institutional advantage of imposing the tax this way is that collection is easy, compared to the economic and political problems of taxing commuters directly. The viability of this second taxation procedure depends on three assumptions. Everyone travels on the same mode so that AT’ is equal for all commuters at any time; and car pooling does not occur. These first two assumptions are simply a statement that taxing starting times does not provide the correct incentives to adjust modes and occupants per car on roads. The final assumption is either that all workers in a firm start at the same time or else that wages may differ for similar workers in the same firm. Clearly, within a firm, for there to be an equilibrium with similar workers starting at different times wages must vary by starting time as taxes vary. This could be institutionally difficult, although wages do vary by eight-hour shifts in a variety of industries. At this point in the paper, it is easiest to assume that all workers in a firm start at the same time, so equilibrium under taxation only requires some wage adjustment across firms. This wage difference problem will be raised again later. Finally we note that the revenue raised from congestion tolls is the key to demonstrating how in practical examples the imposition of a toll improves the welfare of commuters on the tails of the arrival distribution. For example, in Fig. 1, people at SF and S: relative to si and s2 have unchanged travel costs but higher C(s) costs. To show that they actually gain from imposing congestion tolls, it is necessary to bring in revenue considerations. Following Henderson’s (1977) modeling of the waiting time problem, commuters on the tails of the distribution (and, in fact, at all starting times) gain in net once the toll revenue raised is uniformly distributed back to the commuters in a lump gift form. Another way to view the problem is to bring in capacity cost considerations. The collection of toll revenue reduces or eliminates other levies imposed prior to the introduction of tolls that were needed to finance capital costs. If the prior levy was an annual or daily lump fee on users only covering capital costs, people throughout the arrival distribution benefit in net from congestion tolls because the reduction in lump fees for users exceeds any increase in day-to-day costs (additional C( .) costs for people on the tails and the advent of high tolls for people nearer the peak of arrivals). Of course, the move from an equilibrium to an optimum will also alter capacity requirements. If capacity is denoted by K and the price per unit of capacity is $1, adding K to Eq. (6) yields an optimal capacity condition of -
‘*A aT/aKds
I SI
= 1.
(8)
Note also that, if T is homogeneous of degree 0 in A and K (i.e., doubling
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J. VERNON HENDERSON
capacity and flows leaves travel speeds and costs unchanged), tolls cover capacity costs. Capacity costs equal K( /ST- &Z/X) dr) if (8) is satisfied. Tolls equal J,“12A( A aT/&4) dr. Thus tolls equal capacity costs when T is homogeneous of degree 0 which implies - KaT/iK = A aT/aA. If a transport planner satisfied (8) for the equilibrium prior to tolls, then in general, the optimal capacity will be less than the equilibrium, because the elimination of the heavy arrivals around S reduces the need for capacity. COMMUTER
EQUILIBRIUM: PRODUCTION PRESENT
EFFECTS
The introduction of production effects serves two purposes. One is to illustrate the nature and characteristics of possible production equilibria under staggered work hours and the second is to raise certain institutional problems involved in sustaining a staggered work hours equilibrium. We assume each firm has an instantaneous production function at time t of the form
n(s) is the firm’s own work force and the coefficient a represents its internal technology. Thus for simplcity we are ignoring diminishing returns within the firm-a factor viewed by firms as being unimportant in staggered work hours. g(N(t)) is an external economy of scale shift factor where N(t) is the total number of employees working in the business district at time t. g’ > 0,g” < 0. This specification captures the primary production problem associated by firms with staggered work hours. If a firm shifts to an early schedule where N(t) is very low, productivity will be low because its employees will have fewer employees in other firms in the business district to interact and do business with. Firm profits per employee starting work at time s are
H is the fixed length of work day, s, is the beginning time of morning arrivals and s2 the ending time. Given a commuting period of s2 - si, any firm has (H - (s2 - s,)) hours when all workers are present in the business district and thus production by one of its employees during this interval is (H - (s2 - si))g(N)a, where # is total business district work force. Moving to the second expression on the RHS of (9), N(t) is the number of employees at work in the business district at time t. Thus N(t) is
ECONOMICS OF STAGGERED WORK HOURS
359
accumulated arrivals, or
N(f)=p4(s)d.Y, ni= A(s). SI
(10)
Therefore, a worker’s production from the time he arrives, s, until all workers have arrived is J,S2g,,(N(t))adt. Turning to the third expression, in the evening people exit from the business district in the same order that they arrived. Total employees at time t + Zf are
IT- / ‘+HA(s)d.v= F- N(t). s,+lY
(11)
Equations (10) and (11) are illustrated for an equilibrium (see later) in Fig. 3. Given (1 I), a worker’s production during the evening rush period before he leaves is /,“,:z g ,(c- N(t))adt. Note g,(x) z g,(x) all x, or the “0” and “1” subscripts only designate morning vs evening production, not differences in functional form. Given how wages vary in equilibrium with starting time S, firms choose s to maximize profits. Alternatively, given identical production functions and workers, da/& = 0, for all s in any stable equilibrium. Otherwise, further shifting among starting times will occur. Differentiating (9), we see that equilibrium requires that
4g, - go) = *.
(12)
360
J. VERNON HENDERSON
Combining this with the consumer equilibrium condition in (4), we find that for a work staggering equilibrium with production effects
k _ 4& - go) - d T’ ’
(13)
k’ < 0. An equilibrium solution, such as depicted in Fig. 1, where the arrival pattern is symmetric about 5 satisfies (13) and equilibrium conditions on boundary points, where n(s,) = n(s2) given W(S) from (1). If arrivals are symmetric about S then, from Fig. 1, N(5 - 6) = ” - N(S + 6) all 6; and thus g,(% N(S - 6)) - ge(N(S - 8)) = -[g,(N - N(S + s>) - g,(N(3 + S)J > 0. Note fo_r a symmetric pattern g,(N(S)) = g,(N - N(S)) = g(f N) so that (g, N - N( S - 8)) 7 gO(N(S - 6)) > 0. Combining these results with Eq. (3), in Eq. (13), N=A > [ <]0 as s < [ >]O. This pattern of N(t) and N - N(t) is illustrated in Fig. 3. With a symmetrical arrival pattern, s2 - 5 = S - s,, where n(sl) - n(s2) = ( lsyg,adt - lsff++HH8,adt) - (T(A(s,)) - T(A(s,)) - (C(s, - S) - C(s, - S)) = 0 if S - s, = Q - S. Note that as before equilibrium at s, and s2 requires congestion to approach zero, which given our functional assumptions implies that A(s,) = A(s,) + 1. The new feature of equilibrium is that wages must vary according to starting time. The instantaneous marginal product of a worker is illustrated in Fig. 4a, where dg, /drj > 0 and dg, /dr = (dg, /d( g - N(s))( - A(s)) < 0; the curves are not necessarily concave throughout. A worker starting at S has the highest total productivity during the peak periods-operating on the S to s2 segment of MP(s) and the s, + H to S + H segment of MP(s + H). Total productivity of a worker declines as he moves away
ECONOMICS OF STAGGERED WORK HOURS
361
from S in either direction to lowest productivity at si and s2. Thus total daily wages should vary qualitatively as illustrated in Figure 4b. Such required wage variation for identical skill workers may present an institutional impediment to attaining and sustaining staggered work hour equilibria. This may be a partial explanation of why staggered work hours was a non-existent or minor feature of scheduling in business districts for so long. If we assume all similar skill workers of each firm start at the same time, that at least would eliminate the need for wage variation for identical skill workers in the same firm and would only require wage variation across firms. If we reintroduce scheduling taste (C( *) function) differences for identical skill workers, then in Fig. 2, workers will in part sort themselves out across firms by their tastes given the firms’ starting times. If we allow skill differences among workers, we will get the same type of equilibrium as illustrated in Fig. 2. For example, suppose there are two groups of workers with productivity gu, and ga,, where a, > a2. Then, by similar arguments as for taste differences, in equilibrium a, people will arrive at times near S and a, people on the tails. This, of course, makes sense because in general total productivity in Fig. 4a is enhanced for starting times around S and it is most profitable to enhance (by a multiplicative term) the productivity of the most productive. If firms use both types of workers in production, then from Fig. 2 their different types of workers must start work at different times and receive different wages on the basis of both basic productivity differences (a, > u2) and starting time differences. Two additional points should be noted. This requirement that wages differ across identical skill individuals according to their work schedules holds regardless of the specification of production effects (e.g., the introduction of diminishing returns), as long as production effects are present. Second, our results are consistent with the results to be obtained from flexitime models, where people must work the same number of hours but choose their starting time on a day-to-day basis. Flexitime would introduce the complication that the distribution of 7’(A)‘s is uncertain on a day-to-day basis, as is the distribution of cumulated arrivals. Thus everyone operates on expected values, eliminating the deterministic nature of the model but not the basic principles. However, if the number of hours worked becomes a free choice variable for everyone, our model is more complicated. For example, commuters might maximize W(s, e)Zf - T( A(s)) - C( H, e, s - S,s + H - i), where e is (uniform but endogenous) effort per instant worked, H is variable hours worked, Sis desired starting time, and tis desired quitting time. aW/ae > 0, X/aH > 0, K/se > 0 and X/as ‘, 0. There are many interesting types of equilibria that could arise from this specification, especially if the cross partial derivatives of C(e) are nonzero.
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CONSUMER OPTIMUM: PRODUCTION PRESENT
EFFECTS
With production effects, a social planner’s objective is to maximize the sum of production in all firms minus the opportunity and travel costs of all workers. Thus the planner seeks to maximize
J = g( N)uN( H - (s* - s,)) + ~~*&)(N(s))uN(s) dr SI + J:‘,“pl(
-
F - N(s))u( l7 - N(s))ds - jS2A(s)T(A(s))h SI
S2A(s)C(s - i) ds I $1
subject to ni = A(s), W,)
= 0,
N(S*) = Iv. The first three expressions in J are, respectively, production during the noncommuting period, production during the morning peak period, and production during the evening peak period. Note that production in (14) is consistent with (9) where total morning peak period production of all people (A(s)) arriving at time s is J~‘zgc(N( t ))aA (s) dt and total production over all s’s is ~,“,z(J~zga(N(t))aA(s) dt) ds = integrating /,‘,‘(jlA(s)dr)ug,(N(t))dt = l:g,$N(t))uN(t)dt. The last two terms of (14) are opportunity costs for all people starting at time s integrated over all s ‘s. The Euler equation implies that
/Liz 4g, - &I)(1 + cl - d 2T’ + T”A
*
(15)
Without loss of generality, to simplify the discussion of (15), we have defined c = (&o/dW~MWO/go) =-C&,/d(F - NtNM@ - Wt))/g,), where e is constant for 0 < N(t) I N. The other necessary conditions for maximizing J in (14) are satisfied (but not necessarily sufficiency conditions given production is convex in N(t)). Boundary conditions require that T’(A(s,))(A(s,))* = T’(A(s,))(A(s,))* = O;_orth_atA(s,) = A(s2) = 1. (If Z is the sum of integrands and F = g(.N)aN(N - (s2 - si)) then (Z - A aZ//aA),, + aF/b, = 0 for si = s,,s2.) A < 0, where T”’ 2 0 is sufficient but not necessary for this.
ECONOMICS OF STAGGERED WORK HOURS
363
FIGURE 5
As in Fig. 1 and Eq. (13), a symmetrical arrival pattern satisfies all equations for the solution. Equation (15) relative to (13) indicates the presence of two conflicting externalities. Congestion externalities raise the denominator in (15) suggesting a slower build-up (k) of traffic for any A, the intuitive notion being one of spreading out arrivals to reduce peak congestion levels as in Fig. 1. However, the introduction of the (1 + C) term indicates a need to speed-up arrivals to accumulate people and exploit the scale economies in production that are present. In Fig. 5, we present two alternative optimal solutions relative to the equilibrium solution, either of which is possible given the conflicting forces on arrival rates. To achieve the optimal solution, we could impose a tax per worker arriving at time s equal to Tax (.s) = T’A(s) - a,[ ~s2godt + $pIt]
;O.
(16)
Note, as follows from the discussion above, arrival times are taxed for congestion and subsidized for scale economy benefits. At s, and s2 since 7” = 0, Tax (s,) = Tax (sz) < 0; or, we are subsidizing arrivals to encourage accumulation of arrivals and scale benefits, given congestion is miniscule. As we move towards S both expressions in (16) increase in absolute value (see Fig. 4a with respect to the second expression), although presumably the first escalates in value, so that around S Tax (s) > 0. Institutionally, the tax could be imposed on firms according to when each worker starts. If we add (16) to (9) and solve for market equilibrium, the solution is characterized by (15), and the same boundary conditions. However, imposing the congestion part of the tax on firms in a more general model does not allow for the optimal encouragement of car-pooling
364
J. VERNON HENDERSON
and allocation of commuters across other modes of travel. In a more general context T/A(s) should be imposed directly on travellers as a congestion toll. The externality subsidy could be offered directly to firms to encourage shifting of starting times; or, equivalently it could be offered directly to workers as a function of their starting time. CONCLUSIONS In this paper, the nature of equilibrium in a staggered work hours situation has been examined, and divergences between equilibrium and optimum solutions investigated. While the modeling in itself is conceptually instructive, what is also very interesting is the multitude of empirically testable hypotheses that spring directly from the model. In a staggered work hours equilibrium wages should vary across similar occupation workers by work starting time. Identifying commuters by skill and scheduling taste (family characteristics) differences, we would expect to find each category of commuters arriving in two different time intervals, one before and one after the peak of arrivals. Presumably also there will be a link between the composition of workers arriving on the tails of the distribution as opposed to at the peak and the nature of their job tasks. For example, workers who interact directly with others to a relatively small degree in their particular jobs should be found on the tails. In examining business districts that have shifted from the traditional single work schedule to staggered hours, we would expect to seepeak congestion eliminated, congestion on the tails of the arrival distribution raised, and time per commuter spent in the business district (holding work hours fixed) reduced. REFERENCES Agnew, C. E. (1973), “The Dynamic Control of Congested Systems Through Pricing,” Report No. 6, Stanford University Center for Interdisciplinary Research. Greenberg, M. A. and D. W. Wright (1974) “Staggered Hours Demonstration: Final Evaluation Report,” Ontario Ministry of Transportation and Communications, mimeo. Guttman, J. (1975), “Predicting the Effect of Variable Working Hours on Peak Period Congestion,” University of Chicago, mimeo. Henderson, J. V. (1974), Road congestion: A reconsideration of pricing theory, J. Urban Econ., 1, 346-365. Henderson, J. V. (1977), “Economic Z%eov and the Cities,” Academic Press, New York. Inman, R. P. (1978), A generalized congestion function for highway travel, J. &ban &on., 5, 21-34. Lucas, R. E. (1970), Capacity, overtime, and empirical production functions, Amer. Econ. Assoc. Papers Proc., 60, 23-27.
Mills, E. S. (1967), An aggregative model of resource allocation in a metropolitan area, Amer. Econ. Rev., 57, 197-210. O’Malley, B. W. (1974), “Work Schedule Changes to Reduce Peak Transportation Demand,” Port Authority of New York and New Jersey, mimeo. Vickrey, W. S. (1969) Congestion theory and transport investment, Amer. Econ. Reo., 59, 251-260.