The Economics of Employment and Hours Decisions of Firms

Transcription

1 The Economics of Employment and Hours Decisions of Firms Ronald L. Oaxaca January 2014 Abstract These notes introduce the basic concepts about employer choices over employment and the ork eek. The concepts are illustrated ith a simple Cobb- Douglas technology. The C-D example is used to demonstrate both the theoretical application and the empiricical estimation issues associated ith modeling labor demand. 1

2 I. INTRODUCTION Background reading: Cahuc, Pierre and André Zylberberg 2004, Labor Economics, MIT Press, Chapter 4, pp Hart, Robert A. 2004, The Economics of Working Overtime, Cambridge University Press. Bauer, Thomas and Klaus F. Zimmermann 1999, Overtime Work and Overtime Compensation in Germany, Scottish Journal of Political Economy, 464, pp ******************* When confronted ith setting the hours fohe orkeek and the number of orkers to employ, firms clearly are not indifferent in this choice. For example, imagine that a firm requires 400 person-hours per eek. This employment demand could be satisfied by employing 10 orkers for 40 hours each. But the same demand could be satisfied by employing 5 orkers for 80 hours each. Or even employing 400 orkers for 1 hour each, and several other combinations as ell. What is the source of an employer s lack of indifference beteen the number of orkers to be employed and the number of hours each individual orks? There are to reasons hy employers ould care about the mix of employment and hours. First of all, there are physical, legal, and logical limitations on the number of hours an individual can ork in a eek. Secondly, there are distinctly different costs associated ith an additional hour added to the ork eek versus adding another orkeo the payroll. Some of the cost differences may be induced by legal requirements regarding compensation for overtime ork. The table belo illustrates overtime hours las and agreements for different European countries. 2

3 Table 1: Principal features of overtime schemes Country Maximum orking time 1 minimum daily rest period, here no maximum daily hours Method of setting threshold 2 Threshold level Specific maximum overtime limits Conditions for use of overtime procedures, justifications Enhanced pay rate and/or time off in lieu Australia 38 hours per eek but not exceeding 152 hours over a 28 day cycle or an average of 38 ovehe period of an agreed roster cycle. Austria 10 hours per day, 50 hours per eek maximum under certain conditions. Belgium 8 hours per day, 40 hours per eek. Legislation 38 hours per eek. Legislation. 8 hours per day, 40 hours per eek, hich is above average collectively agreed orking time. Legislation and agreements at sector or company level. 8 hours per day, 40 hours per eek. None. No conditions. 50% pay rate fohe first to hours and 100% time thereafter, calculated on a daily basis or time off in lieu by the agreement. 5 hours per eek, and additional 60 hours per year. No conditions. None. May only be used on specific grounds - exceptional peaks of ork, force majeure, unforeseeable needs. Authorization procedures vary according to reason. 50% pay rate or 50% time off in lieu 50% pay rate 100% at eekends and public holidays - may be converted into time off in lieu if provided for by collective agreement. Continued on next page 1

4 continued Country Maximum orking time minimum daily rest period, here no maximum daily hours Method of setting threshold Threshold level Specific maximum overtime limits Conditions for use of overtime procedures, justifications Enhanced pay rate and/or time off in lieu Canada 8 hours per day, 48 hours per eek minimum daily rest period of 11 hours. Legislation and agreements 44 hours per eek. None. Individual agreement of the employee required for ork over 44 hours per eek. 50% pay rate or 50% time off in lieu. Paid time off must be taken ithin three months of the eek in hich the overtime as earned or, if the employee agrees in riting, it can be taken ithin 12 months. Denmark 48 hours per eek minimum daily rest period of 11 hours. Agreements at sector or company level. 37 hours per eek industry sector agreement. 12 hours over 4 eeks industry sector agreement. Notice period required industry sector agreement. Companies ith agreement - increased pay rate at %, then time off in lieu for overtime hours over a threshold 8 hours in 4 eeks in industry sector agreement. Companies ithout agreement - mostly time off in lieu. Continued on next page 2

5 continued Country Maximum orking time minimum daily rest period, here no maximum daily hours Method of setting threshold Threshold level Specific maximum overtime limits Conditions for use of overtime procedures, justifications Enhanced pay rate and/or time off in lieu Finland 8 hours per day, 40 hours per eek. Legislation or agreement. 40 hours or collectively agreed orking time. 138 hours over a 4-month period, 250 hours per year over statutory threshold of 40 hours, raised by 80 hours per year if the 138 hours over a 4-month period is complied ith. Individual agreement of the employee required for ork over 40 hours per eek. 50% pay rate fohe first 2 hours per day, 100% above that. May be converted into time off in lieu by agreement. France 10 hours per day, 48 hours per eek. Legislation. 35 hours per eek. 180 hours per year or set by collective agreement. No conditions. Permission from authorities required for exceeding annual limits. Beteen 35th and 43rd eekly hour - minimum pay rate of 10% 25% ithout agreement or time off in lieu by agreement. From 44th hour - 50% pay rate. Germany 8 hours per day, 48 hours per eek. Agreements at sector level. Varies beteen sectoral agreements. Varies beteen sectoral agreements. Agreement of orks council required, except here sectoral agreement includes specific provision. Increased pay rate and/or time off in lieu, by collective agreement. Appropriate are 25% on regular orking days, and 50% on Sundays and Holidays. Continued on next page 3

6 continued Country Maximum orking time minimum daily rest period, here no maximum daily hours Method of setting threshold Threshold level Specific maximum overtime limits Conditions for use of overtime procedures, justifications Enhanced pay rate and/or time off in lieu Greece* 9 hours per day, 45 hours per eek assuming fiveday eek. Legislation. 40 hours. Work beteen 41 and 45 hours a eek is called extra ork. Work exceeding 9 hours a day and/or 45 hours a eek is called overtime. Annual limits, varying by sector and region set every six months by Ministry of Labor. Overtime exceeding the laful limit or for hich the aforementioned procedures are not complied ith is unlaful overtime. Over 45 hours per eek requires justification, notification of authorities and record-keeping. Laful overtime is paid as follos: up to 120 hours a year by augmenting the hourly age paid by 40%, hile for overtime above the 120 hour limit, the augment is 60%. Unlaful overtime is paid ith an 80% augment. Hungary* 12 hours per day, 48 hours per eek. Legislation. 8 hours per day, 40 hours per eek. 250 hours per year, may be raised to 300 hours by agreement. Reasons required, notice to be given, record-keeping compulsory. 50% pay rare or time off in lieu by agreement, 100% pay rate for ork on a holiday or 50% if time off in lieu granted. Continued on next page 4

7 continued Country Maximum orking time minimum daily rest period, here no maximum daily hours Method of setting threshold Threshold level Specific maximum overtime limits Conditions for use of overtime procedures, justifications Enhanced pay rate and/or time off in lieu Ireland 48 hours per eek minimum daily rest period of 11 hours. Agreements. Varies beteen mainly company agreements average 39 hours. 2 hours per day, 12 hours per eek, 240 hours per year, or 36 hours over 4 consecutive eek. Limits can be exceeded ith permission from the authorities. No conditions. 25% pay rate agreements often lay don higher rates. Italy 48 hours per eek averaged over a 4-month period minimum daily rest period of 11 hours. Legislation and agreements at sector level. 40 hours per eek. 250 hours per year may be loer by agreement. Collective agreement required sector or company-level. 10% rate in absence of agreement on higher rate. Japan 8 hours per day, 48 hours per eek. Legislation. 8 hours per day, 48 hours per eek. 15 hours a eek, 27 hours in to eeks, 43 hours in 4 eeks, 45 hours in one month, 81 hours in to months, 120 hours in three months, 360 hours in one year. Any employer that requires orkers to ork in excess of statutory orking hours or on statutory days off must submit a Notification of Agreement on Overtime and Work on Days off to its local Labor Standards Inspection Office. 25% for additional ork on a orkday, 35% for holiday ork, an additional 25% for ork late at night usually defined as 10 PM to 5 AM, an additional 25% for ork exceeding 60 hours in a month. Continued on next page 5

8 continued Country Maximum orking time minimum daily rest period, here no maximum daily hours Method of setting threshold Threshold level Specific maximum overtime limits Conditions for use of overtime procedures, justifications Enhanced pay rate and/or time off in lieu Luxembourg* 10 hours per day, 48 hours per eek. Legislation. 8 hours per day, 40 hours per eek. None, but overall statutory daily and eekly orking time limits see first column. Permitted only on specific grounds e.g. exceptional cases, permission from the authorities required. 40% for all employees except for upper management, 50% time off in lieu by agreement. This additional hour is exempt from tax and social contributions. Netherlands 10 hours per day and 50 hours per eek ith an average of 40 hours per eek over a period of 13 eeks. Legislation and agreements. Varies beteen collective agreements no fixed level. None, but overall statutory daily, eekly and quarterly orking time limits including incidental hours, hich may be extended ithin limits by agreement see first column. Must be incidental and not structural. Collective agreements often require agreement of orks council and/or employees concerned. Increased pay rate 100%- 200% and/or time off in lieu, by collective agreement. Noray 13 hours per day or 48 hours per eek. Legislation. 10 hours per day, 25 hours per month, 200 hours per year 200 hours per year overtime beteen hours per year alloed by individual agreement. Permitted only 40% pay rate on specific usually 50% non-permanent by agreement, grounds e.g. and 100% after unforeseen 21:00. events or volume of ork. Subject if possible to discussion ith elected staff representatives and for overtime beteen hours to agreement ith employee. Continued on next page 6

9 continued Country Maximum orking time minimum daily rest period, here no maximum daily hours Method of setting threshold Threshold level Specific maximum overtime limits Conditions for use of overtime procedures, justifications Enhanced pay rate and/or time off in lieu Poland 10 hours per day, 40 hours per eek. Legislation. 8 hours per day, 40 hours per eek over 5-day eek. 4 hours per day, 150 hours per year. Permitted only on specific grounds e.g. employers special needs or rescue operations, monitored by the authorities. 50% pay rate for the first 2 hours, 100% for further hours and ork at night, on Sunday and holidays. May be converted into time off in lieu at request of employee and ith employers agreement. Portugal 8 hours per day, 40 hours per eek up to 10 hours per day and 50 hours per eek, by agreement. Legislation and agreements. 8 hours per day, 40 hours per eek up to 10 hours per day, 60 hours per eek by agreement. 2 hours per day, 200 hours per year. Permitted only on specific grounds e.g. unscheduled increased orkload or force majeure, record-keeping required. 50% pay rate for 1st hour, 75% thereaftehat, 100% on rest days and holidays. Plus time off in lieu at 25% of the hours orked. Slovakia* 48 hours per eek exemption available by collective agreement and permission from the authorities. Legislation. 40 hours per eek over 5- day eek regular orking schedule - daily minimum of 3 hours and maximum of 9 hours. 8 hours per eek in a period of at most 4 consecutive months, 150 hours per year excluding certain overtime, such as in the event of disasters. Up to 400 hours in special cases by companylevel agreement and ith authorities permission. No conditions for up to 150 hours per year. 25% pay rate higher by company-level agreement. Continued on next page 7

10 continued Country Maximum orking time minimum daily rest period, here no maximum daily hours Method of setting threshold Threshold level Specific maximum overtime limits Conditions for use of overtime procedures, justifications Enhanced pay rate and/or time off in lieu Spain Seden* 9 hours per day, 40 hours per eek. 8 hours per day, 40 hours per eek. UK 48 hours per eek minimum daily rest period of 11 hours. Legislation. Legislation. 40 hours per eek, hich is above average collectively agreed orking time. Requires collective agreement or agreement by employee. 40 hours per eek, hich is above average collectively agreed orking time. Agreements companylevel. Varies beteen companylevel agreements. 80 hours per year. 48 hours over a period of four eeks or 50 hours over a calendar month, subject to a maximum of 200 hours per calendar year. None, but overall statutory eekly orking time limits from hich individuals may opt out. Must be justifiable e.g. special needs, or employers requirements and often subject to agreement company or orkplacelevel. Recordkeeping compulsory, monitoring by staff representatives. Increased pay rate average 18% oime off in lieu, by collective agreement. Increased pay rate usually 50% to 100% or time off in lieu, by collective agreement. No conditions. Increased pay rate oime off in lieu, by agreement. 1 Hoever it is described maximum or standard in the national regulations 2 Threshold beyond hich increased pay rate oime off in lieu for overtime begins, either called maximum orking time, ohe statutory period, or equivalent to the collectively agreed orking hours, depending on the country. Source: EIRO MALTA - Overtime Regulations 2012 hereinaftehe Regulations have been enacted to regulate overtime. The Regulations provide that full-time employees shall only ork overtime as required by their employer provided that their average eekly orking time, including overtime, does not exceed an average of 48 hours ovehe applicable reference period in terms of the Organization of Working Time Regulations, 2003 and that the employee can, hoever, give his/her consent in riting to ork more than this eekly average. The payment of overtime is regulated at the rate of one and a half times the normal rate for ork carried out in excess of a 40 hour eek, averaged 8

11 over a four-eek period or ovehe shift cycle at the discretion of the employer. The Regulations also allo the employeo introduce schemes to bank hours, i.e., for higher ork activity periods to be redeemed against loer activity periods. 9

12 A simple characterization of the firm s production technology is given by Q F E, h, K, t here Q is some measure of output, E is the number of orkers employed, h is the number of eekly hours per orker orkeek, K is some measure of the nonlabor inputs e ill use capital and nonlabor inputs interchangeably, and t is a linear time trend that reflects technological change. The standard restrictions hold fohe production technology: MP E F E > 0, 2 F E 2 < 0 MP h F h > 0, 2 F h < 0 2 MP K F K > 0, 2 F K < 0 2 F t > 0, here the MP s are the marginal productivities ith diminishing returns. For simplicity and convenience it is assumed that all orkers ork the same number of hours per eek, that there is only one type of labor, and that there is no economic distinction beteen the number or amount of capital/nonlabor inputs used and the utilization rate/machine hours for each unit of the nonlabor inputs. These restrictions can of course be relaxed to accommodate the employment of different occupational groups of orkers ith differing ork schedules as ell as an economic distinction beteen the level of nonlabor input usage and its utilization rate, e.g. machine hours. On the cost side of production there is an important distinction beteen the variable cost of labor, i.e. the hourly age and the quasi-fixed overhead labor costs B. Quasi-fixed labor costs are labor costs that vary ith the number of orkers employed but not ith hours orked. In other ords these nonage labor costs are incurred for each orker on the payroll regardless of the number of hours they ork. Some examples of quasi-fixed labor costs are listed belo: 3

13 Quasi-fixed labor costs B Hiring and Training Costs: advertising jobs, intervieing, recruiting orientation, specific-job training Employee Fringe Benefits: legally mandated and non-mandated benefits Administrative Costs: record keeping costs The firm s cost equation can be expressed as C {h + B 1 D + [h + B + λ h h D]} E + rk, 1 here h is a legally or contractually defined standard orkeek, e.g. 40 hours, 35 hours, etc., λ 1 is the overtime premium, r is the rental rate/user cost of capital, and D is an indicator variable defined as D 1h > h. In the presence of overtime hours h h > 0, D 1 and the cost equation becomes C {[h + B + λ h h ]} E + rk. In the absence of an overtime premium λ 1 or hen h < h, the cost equation simplifies to C h + B E + rk. 2 In the case in hich the actual ork eek equals the legally or contractually defined standard ork eek h h, the cost equation is simply C h + B E + rk. 3 4

14 Marginal input costs are obtained from MC E C E MC h C h MC K C K. The marginal input costs in each overtime hours regime are summarized belo. regime MC E MC h MC K h < h h + B E r h > h [h + B + λ h h ] λe r h h h + B r 5

15 II. THEORETICAL COBB-DOUGLAS EXAMPLE To make things concrete in terms of input demand, a simple Cobb-Douglas technology is assumed: Q Ae gt E α h β K γ here A, g > 0, 0 < β < α < 1, 0 < γ < 1, and 0 < α + β + γ < 1. The parameter g is the percentage groth rate in output arising from neutral technological change. The marginal productivities fohe inputs are given by MP E Q E αae gt E α 1 h β K γ 4 MP h Q h βae gt E α h β 1 K γ 5 MP K Q K γae gt E α h β K γ 1. 6 Later it ill be easieo ork ith the Cobb-Douglas production function expressed in logs: lnq lna + gt + αlne + βlnh + γlnk. There are four distinct economic objectives that can be considered in obtaining the derived demand for inputs: cost minimization, long-run profit maximization, shortrun profit maximization, and output maximization subject to a budget constraint. 6

16 III. COBB-DOUGLAS COST-MINIMIZATION EXAMPLE - THEORY Under cost minimization, the employer seeks to produce a given level of output at the loest possible cost. An example of such an objective is a branch plant in hich production orders are exogenously determined by company headquarters. The plant manager s objective is to produce the assigned level of output at the loest possible cost. The fundamental cost-minimizing conditions are described by It is cleahat satisfaction of 7 and 8 implies MP E MC E MP h MC h 7 MP E MC E. MP K MC K 8 MP h MP K MC h MC K. 9 The conditional input demand functions corresponding to h, E, and K are derived belo for each of the three overtime hours regimes. h < h In this regime it is assumed that the ork eek is less than the standard ork eek ohat there are no overtime las or contractual obligations regarding overtime. We first apply the efficiency condition corresponding to 7 : MP E αaegt E α 1 h β K γ MP h βae gt E α h β 1 K α h γ β E MC E MC h h + B E. The employment level E cancels out hich allos us to solve fohe demand for hours per orker ork eek: or in logs h β B, 10 α β β B lnh ln + ln. α β 7

17 Note that the demand for hours per orker does not depend on Q or r. All that matters is the ratio of overhead labor costs to the straight-time hourly age rate B. In fact the optimal eekly labor cost per orker is proportional to the quasifixed labor costs: β B h + B + B α β α α β B. To schedule a ork eek less than the standard ork eek in the presence of overtime requirements, it must be the case that β B h < h α β 0 < B < α β 1 h. Employment and nonlabor input demands can be obtained from the efficiency condition corresponding to 8: MP E αaegt E α 1 h β K γ MP K γae gt E α h β K α K γ 1 γ E α K γ E MC E h + B MC K r α B. α β r We arrive at the optimal capital/employment ratio: K γ B E α β r or K α B. α β r γ B E. 11 α β r In terms of logs, e have γ B ln K ln + ln + ln E. 12 α β r 8

18 The conditional input demand function for E can no be derived from the production function after substituting 10 and 11 for h and K: [ β Q Ae gt E α α β ] β [ B γ α β ] γ B E. r Afteaking logs, collecting terms, and solving for ln E, e obtain the conditional input demand function for employment in logs: 1 ln E [β + γ ln α β lna βlnβ γlnγ] β β + γ B 1 g + ln ln + lnq t. r r The conditional input demand function for K is obtained by substituting 13 for lne in 12 and collecting terms: 1 ln K β + ln + r γ [β + γ ln α β lna βlnβ γlnγ] + ln B α β ln r + 1 lnq α β g The partial effects of the variables, B, r, Q, and t on the input demands are derived as follos. Starting ith the demand fohe ork eek, e see that lnh ln 1 < 0 lnh lnb 1 > 0 lnh lnr lnh lnq lnh t t. 14 9

19 Next, e considehe conditional input demand for employment: lne ln β > 0 lne β + γ < 0 lnb lne lnr γ > 0 lne lnq 1 > 0 lne g t < 0. In the case of the conditional input demand for capital, e have: lnk ln β > 0 lnk lnb α β > 0 lnk lnr α < 0 lnk lnq 1 > 0 lnk g t < 0. 10

20 We can also examine the partial effects of, B, r, Q, and t on aggregate/total hours demanded H defined by H he: lnh ln lnh ln + lne ln 1 + β < 0 lnh lnb lnh lnb + lne lnb β + γ 1 α β > 0 lnh lnr lnh lnr + lne lnr 0 + γ γ > 0 lnh lnq lnh lnq + lne lnq > 0 lnh t lnh t 0 g g < 0. + lne t The signs of the partial effects of the variables, B, r, Q, and t on the input demands are summarized in the table belo. 11

21 h < h h E K H B r Q t Perhaps the positive effect of the age rate on employment E may seem counterintuitive. Since output is held constant, an increase in the age hile reducing the demand fohe ork eek is also leading to the substitution of employment and nonlabor inputs for hours per orker. Nevertheless, total hours of labor demanded decline ith an increase in the age rate. Next e consider conditional input labor demands in the overtime hours regime. h > h We apply the efficiency condition corresponding to 7 in the case of overtime hours: MP E α h MP h β E MC E [h + B + λ h h ]. MC h λe The employment level E cancels out hich allos us to solve fohe demand for hours per orker ork eek: β 1 λ h α β λ 1 β λ α β h + [ 1 λ h + 1 β λ B α β ] > h > 0. B 15 B α > β λ 1 h. 12

22 The demand for overtime hours is easily obtained as [ ] β 1 λ 1 β B h h 1 h + > α β λ λ α β As in the case hen h < h, the demand for hours per orker does not depend on Q nor does it depend on r. Upon substitution of 16 for overtime hours in the expression for marginal cost in the overtime regime one obtains [ α MC E 1 λ h + B ]. α β Employment and nonlabor input demands can be obtained from the efficiency condition corresponding to 8: MP E α K MP K γ E MC E [h + B + λ h h ] MC K r α K γ E α [ 1 λ h + B ]. α β r We next arrive at the optimal capital/employment ratio: K γ [ 1 λ h + B ], E α β r or K In terms of logs e see that γ ln K ln α β α [ 1 λ h + B ]. α β r γ [ 1 λ h + B ] E. 17 α β r [ + ln + ln 1 λ h + B ] + ln E. 18 r In ordeo derive expressions for conditional input demands for labor in the overtime regime, it is easieo ork in terms of logs. The demand fohe orkeek in an overtime situation is obtained as β lnh lnλ + ln α β 13 [ + ln 1 λ h + ] B. 19

23 The conditional input demand function for lne can no be derived from the production function in logs after substituting 19 and 21 for lnh and lnk: lnq lna + gt + αlne { β + β lnλ + ln α β { γ + γ ln + ln + ln α β r [ + ln 1 λ h + B [ 1 λ h + B ] ]} } + ln E. After collecting terms and solving for ln E, e obtain the conditional input demand function for employment in logs: [ 1 β lne lna + βln α β γ β + γ ln ln r 1 g + lnq γ + γln α β [ 1 λ h + B ] βlnλ ] t. 20 The conditional input demand function for K in logs is obtained by substituting 20 for lne in 18, and collecting terms: [ 1 β lnk lna + βln α β α α β ln + ln r 1 g + lnq γ + γln α β [ 1 λ h + B ] ] γ βlnλ + ln α β t

24 The partial effects of the variables, B, r, Q, t, λ, and h on the conditional input demands in an overtime regime are derived belo. Starting ith the demand fohe ork eek, e see that lnh ln lnh 1 B 1 λ h + B < 0 lnh lnb B lnh B 1 1 λ h + B > 0 lnh lnr lnh lnq lnh 0 t lnh lnλ λ lnh λh 1 + λ 1 λ h + B < 0 lnh lnh 1 λ h lnh h h 1 λ h + B < 0 h h β 1 λ 1 β αλ h α β λ α β λ < 0. 15

25 Next, e considehe conditional input demand for employment: lne ln lne 1 [ β + γ 1 λ h + B lne lnb B lne B [ 1 β + γ 1 λ h + B lne lnr γ > 0 lne lnq 1 > 0 lne g t < 0 lne lnλ λ lne λ lne lnh β 1 + h lne h ] B γ > 0 ] B < 0 β + γ λh β 1 λ h + B > 0 β + γ λ 1 h 1 λ h + B > 0. 16

26 In the case of the conditional input demand for capital, e have: lnk ln lnk α [ ] β α 1 α 1 λ h + B + 1 > 0 lnk lnb B lnk B [ ] α β 1 1 λ h + B > 0 lnk lnr α < 0 lnk lnq 1 > 0 lnk g t < 0 lnk lnλ λ lnk λ β [ ] β α λh β 1 λ h + B + 1 > 0 since B α > β λ 1 h lnk lnh lnk h h [ ] α β 1 λ h 1 λ h + B < 0 17

27 We next examine the partial effects of, B, r, Q, t, λ, and h on aggregate/total hours demanded H he: lnh ln lnh ln + lne ln 1 1 λ h + B λ h + B B 1 + B [ β + γ 1 λ h + B β α γ < 0 ] B γ lnh lnb lnh lnb + lne lnb 1 B 1 λ h + B 1 1 λ h + B B [ 1 1 [ β + γ β + γ 1 λ h + B ] > 0 ] B lnh lnr lnh lnr + lne lnr 0 + γ γ > 0 lnh lnq lnh lnq + lne lnq > 0 18

28 lnh t lnh t 0 g g < 0 + lne t lnh lnλ lnh ln λ + lne ln λ λh λ h + B β β + γ λh β 1 λ h + B 1 λh 1 λ h + B + β 1 < 0 lnh lnh lnh ln h + lne ln h [ ] 1 λ h 1 λ h + B [ 1 λ h 1 λ h + B β + γ 1 λ h 1 λ h + B ] α β < 0. 19

29 The signs of the partial effects of the variables on input demands are summarized in the table belo. h > h h E K H B r Q t λ h The last case to be considered is that of the standard orkeek. h h The standard orkeek is efficient over a range of values fohe ratio B/W. To be α precise, for β 1 h B αβ λ 1 h. This can be illustrated in a graph. 20

30 When hours are set at the standard orkeek, the production function is given by Q Ae gt E α h β K γ, or in logs by lnq lna + gt + αlne + βlnh + γlnk. 22 The corresponding marginal products fohe variable inputs are as follos: MP E Q E αae gt E α 1 h β K γ 23 MP K Q K γae gt E α h β K γ The derived conditional input demands for employment and capital are obtained from the cost minimizing efficiency conditions: MP E αaegt E α 1 h β K γ αk MP K γae gt E α h β Kγ 1 γe αk γe h + B. r The optimal capital/employment ratio is therefore given by K γ E h + B. α r Solving for K in terms of E, e have γ h + B K E, α r or in logs MC E h + B MC K r γ h lnk ln + ln α + B + lne. 25 r 21

31 The conditional input demand function for employment in logs can no be derived from the production function after substituting 25 for lnk in 22 and solving for lne: lnq lna+gt +αlne+βlnh +γ [ γ ln + ln + ln h α + B ] + lne r 1 lne 1 + [ γ lna + γln α lnq ] γ + βlnh ln h + B r g t. 26 In the case of the nonlabor inputs in the standard orkeek regime, the conditional input demand function in logs is obtained by substituting 26 for lne in 25 and collecting terms: 1 lnk 1 + [ γ αln α lnq ] lna βlnh g α ln h + B + r t

32 The partial effects of the variables, B, r, Q, t, and h on the input demands for E and K as ell as on aggregate hours H are derived as follos. For employment demand e have lne ln lne γ h < 0 h + B lne lnb B lne B γ B < 0 h + B lne lnr γ > 0 lne lnq 1 > 0 lne g t < 0 lne lne lnh h h γ h < 0. h + B 23

33 In the case of the nonlabor inputs, the partial effects of, B, r, Q, t, and h are given by lnk ln lnk α h > 0 h + B lnk lnb B lnk B α B > 0 h + B lnk lnr α < 0 lnk lnq 1 > 0 lnk g t < 0 lnk lnk lnh h h α h > 0. h + B 24

34 Finally, the partial effects of, B, r, Q, t, and h on aggregate hours demanded H h xe assuming that the actual ork eek changes to match the changed ne standard orkeek are given by: lnh ln lne ln γ h < 0 h + B lnh lnb lne lnb γ B < 0 h + B lnh lnr lne lnr γ > 0 lnh lnq lne lnq 1 > 0 lnh t lne t g < 0 lnh lnh lnh lnh + lne lnh γ h 1 > 0. h + B 25

35 The signs of the partial effects of the variables on input demands are summarized in the table belo. h h E K H B r Q t h It should be cleahat changes in the ratio B/ can lead to a regime sitch. Which regime is cost minimizing is determined according to min [C, B, r, Q, h h < h, C, B, r, Q, h h > h, C, B, r, Q, h h h ], hich corresponds to B α < β 1 h B α > β λ 1 h α β 1 h B αβ λ 1 h. 26

36

37 IV. COBB-DOUGLAS COST-MINIMIZATION EXAMPLE - EMPIRICAL We no considehe empirical estimation of the conditional input demand functions under cost-minimization. We begin ith the empirical specification of the conditional input demand functions hen there is no overtime. As ill be shon belo, as long as e have data on, r, B, and Q it is possible to estimate all of the parameters of the CD technology and the conditional input demand function parameters from data on employment alone. Although the overtime premium λ is treated as a knon parameter it could be treated as a variable if it ere to change. h < h lnh t ln Bt t a 01 + ε h1t ln E t b 01 + b 11 ln ln K t c 01 + c 11 ln t r t t + b 21 ln + c 21 ln Bt r t Bt + b 31 lnq t + b 41 t + ε E1t + c 31 lnq t + c 41 t + ε K1t. From the theoretical model e kno that there are ithin and cross-equation restrictions on the parameters: 1 b 01 [β + γ ln α β lna βlnβ γlnγ] b 11 β α + γ > 0 β + γ b 21 < 0 b 31 1 > 0 b 41 g < 0 27

38 γ c 01 b 01 + ln c 11 β b 11 > 0 [ ] b11 + b 21 b 01 + ln α β 1 + b 21 c 21 α β 1 + b 21 > 0 c 31 1 b 31 > 0 c 41 g b 41 < 0 β b11 a 01 ln ln α β 1 + b 21 The resulting estimating equations incorporating these restrictions are as follos: Bt b11 lnh t ln ln + ε h1t 28 t 1 + b 21 t Bt ln E t b 01 + b 11 ln + b 21 ln + b 31 lnq t + b 41 t + ε E1t 29 ln K t ln Bt [ ] b11 + b 21 b 01 + ln + b 11 ln 1 + b 21 t + b 21 ln Bt + b 31 lnq t + b 41 t + ε K1t. 30 The model could be estimated by nonlinear seemingly unrelated regression NLSUR ith cross-equation restrictions. The parameters of the underlying CD technology are identified and can be recovered from the estimated conditional input demand function parameters: { 1 Ã exp [ˆb11 + ˆb [ 21 ln ˆb11 + ˆb ] 21 ˆb 21 ln 1 + ˆb 21 ˆb31 α 1 + ˆb 11 + ˆb 21 ˆb31 β ˆb 11 γ ˆb31 ˆb11 + ˆb 21 g ˆb 41 ˆb31. ˆb31 ˆb 11 ln ˆb11 ˆb ] } 01 28

39 It is evident that all of the parameters of the conditional input demand functions as ell as of the CD production function can be estimated from the demand for employment equation alone 29. While not fully efficient, this strategy ould be elcome if data on h t and K t ere unavailable. An alternative estimation strategy is to directly estimate the CD production function parameters by NLSUR and recovehe conditional input demand function parameters: Bt β lnh t ln ln + ε h1t t α β 1 ln E t [β + γ ln α β lna βlnβ γlnγ] β t β + γ Bt + ln ln r t 1 g + lnq t t + ε E1t 1 lnk t [β + γ ln α β lna βlnβ γlnγ] γ β t α β Bt + ln + ln + ln α β r t 1 g + lnq t t + ε K1t. 1 [ b01 ˆβ + ˆγ ln ˆα ˆα + ˆγ ˆβ lnâ βln ˆβ ] ˆγlnˆγ ˆβ b11 ˆα + ˆγ ˆβ + ˆγ b21 ˆα + ˆγ b31 1 ˆα + ˆγ b41 ĝ ˆα + ˆγ 29

40 [ c 01 b b ln b ] b 21 c 11 b 11 c b 21 c 31 b 31 c 41 b 41 ã 01 ln b b 21. Next, e consider estimation of the conditional input labor demands in the overtime hours regime. h > h [ lnh t ln 1 λ h + Bt t ] + lnλ a 02 + ε h2t lne t b 02 + b 12 ln t [ + b 22 ln 1 λ h + B ] t t + b 32 lnq t + b 42 t + ε E2t [ t lnk t c 02 + c 12 ln + c 22 ln 1 λ h + B ] t t + c 32 lnq t + c 42 t + ε K2t. From the theoretical model there are again ithin and cross-equation restrictions on the parameters: 30

41 [ ] 1 β γ b 02 lna + βln + γln βlnλ α β α β b 12 γ α + γ < 0 β + γ b 22 < 0 b 32 1 > 0 b 42 g < 0 γ c 02 b 02 + ln b 02 + ln b 12 c 12 α α β 1 + b 12 > 0 c 22 α β 1 + b 22 > 0 c 32 1 b 32 > 0 c 42 g b 42 < 0 β a 02 ln ln b 12 b 22. α β The resulting estimating equations incorporating the above restrictions are as follos: [ lnh t ln 1 λ h + ln E t b 02 + b 12 ln ln K t ln t t Bt t ] + lnλ ln b 12 b 22 + ε h2t 31 [ + b 22 ln 1 λ h + B ] t t [ ln 1 λ h + B ] + b 32 lnq t + b 42 t + ε E2t 32 b 02 + ln b 12 + b 12 ln [ + b 22 ln 1 λ h + B ] t t t + b 32 lnq t + b 42 t + ε K2t. 33 The model could be estimated by nonlinear seemingly unrelated regression NLSUR ith cross-equation restrictions. 31

42 The parameters of the underlying CD technology are identified and can be recovered from the estimated conditional input demand function parameters: [ Ã exp ˆb12 ˆb 22 lnλ ˆb12 ˆb ˆb12 22 ln ˆb ] ˆb + ˆb ˆb12 12 ln ˆb ˆb α 1 + ˆb 12 ˆb32 β ˆb 12 ˆb 22 ˆb32 γ ˆb 12 ˆb32 g ˆb 42 ˆb32. All of the parameters of the conditional input demand functions as ell as of the CD production function could be estimated from the demand for employment equation alone 32. While not fully efficient, this strategy ould be feasible if data on h t and K t ere unavailable. An alternative estimation strategy is to directly estimate the CD production function parameters by NLSUR and recovehe conditional input demand function parameters: [ lnh t ln 1 λ h + Bt t ] [ 1 β lne t lna + βln α β γ t β + γ ln ln 1 g + lnq t t + ε E2t β + lnλ ln + ε h2t α β γ + γln α β [ 1 λ h + B t t ] βlnλ ] 32

43 1 lnk t b02 b12 b22 [ β lna + βln α β α t α β + ln + ln + 1 g lnq t t + ε K2t. [ 1 1 ˆβ lnâ + ˆβln ˆα + ˆγ ˆα + ˆγ ˆα ˆβ ˆγ ˆα + ˆγ ˆβ + ˆγ ˆα + ˆγ b32 1 ˆα + ˆγ b42 ĝ ˆα + ˆγ c 02 b 02 + ln b 12 c b 12 c b 22 c 32 b 32 c 42 b 42 ã 02 ln b12 b 22. γ + γln α β [ 1 λ h + B t t ] βlnλ ] γ + ln α β ] γ + γln ˆα ˆβ ˆβlnλ We last considehe estimation of the conditional input demand functions corresponding the standard orkeek. h h t h + B t lne t b 03 + b 13 ln + b 23 lnq t + b 33 t + ε E3t r t t h + B t lnk t c 03 + c 13 ln + c 23 lnq t + c 33 t + ε K3t 33

44 From the theoretical model there are again ithin and cross-equation restrictions on the parameters: 1 b 03 [ γ ] lna + γln + βlnh α here φ lna + βlnh b 13 γ < 0 1 b 23 > 0 b 33 g < 0 γ b13 c 03 b 03 + ln b 03 + ln α 1 + b 13 c 13 α 1 + b 13 > 0 1 c 23 b 23 > 0 c 33 g b 33 < 0. 1 [ γ φ + γln, α] The resulting estimating equations incorporating the above restrictions are as follos: t h + B t lne t b 03 + b 13 ln + b 23 lnq t + b 33 t + ε E3t 34 t h + B t lnk t ln b13 b 03 + ln + b 13 ln 1 + b 13 t h + B t + b 23 lnq t + b 33 t + ε K3t. 35 As in the other hours regimes, the model could be estimated by nonlinear seemingly unrelated regression NLSUR ith cross-equation restrictions. The folloing parameters of the underlying CD technology are identified and can be recovered from the estimated conditional input demand function parameters: 34

45 α 1 + ˆb 13 ˆb23 γ ˆb 13 ˆb23 g ˆb 33 φ ˆb23 [ 1 ˆb03 + ˆb ˆb13 13 ln ] ˆb ˆb. 13 Perhaps not surprisingly, the CD production function parameters A and β are not identified from data on just the invariant standard ork eek. One ould need variation in h to identify these parameters. Without such variation, all that can be identified is φ lna + βlnh. All of the identified parameters of the conditional input demand functions as ell as of the CD production function could be estimated from the demand for employment equation alone 34. Although not fully efficient, this strategy ould be feasible if data on K t ere unavailable. An alternative estimation strategy is to directly estimate the identified CD production function parameters by NLSUR and recovehe conditional input demand function parameters: 1 lne t lnk t 1 + [ γ φ + γln α] lnq t [ γ αln α lnq t g ] φ g γ ln t + ε E3t. α ln + t + ε K3t. t h + B t t h + B t 35

46 1 b03 ˆα + ˆγ b13 ˆγ ˆα + ˆγ b23 1 ˆα + ˆγ b33 ĝ ˆα + ˆγ [ ˆφ + ˆγln c 03 b b ln 1 + b 13 c b 13 c 23 b 23 c 33 b 33. ] ˆγ ˆα If the available data spans all three hours regimes and for each observation it is knon hich regime is in effect, then all of the conditional input demand functions could be jointly estimated ith cross-equation restrictions. Joint Estimation Across Hours Regimes We note belo the cross-equation, conditional input demand function parameter restrictions across hours regimes. b 02 [ ] 1 β γ lna + βln + γln βlnλ α β α β b 01 b 11 lnλ b 12 γ b 11 + b 21 β + γ b 22 b 21 b 32 1 b 31 b 42 g b 41 36

47 c 02 b 02 + ln b 12 b 01 b 11 lnλ + ln [ b 11 + b 21 ] c b b 11 + b 21 c b b 21 c 32 b 32 b 31 c 42 b 42 b 41 a 02 ln b 12 b 22 ln b 11 b 03 1 [ γ lna + γln α b 01 + b 21 ln 1 + b 21 + b 11 ln ] + βlnh b11 h b 11 + b 21 ln 1 + b 11 + b 21 b 13 γ b 11 + b 21 b 23 1 b 31 b 33 g b 41 37

48 b13 c 03 b 03 + ln 1 + b 13 b 01 + b 21 ln 1 + b 21 + b 11 ln b11 + b 21 + ln 1 + b 11 + b 21 b 01 + b 21 ln 1 + b 21 + b 11 ln + ln [ b 11 + b 21 ] b11 h b11 h b 11 + b 21 ln 1 + b 11 + b b 11 + b 21 ln 1 + b 11 + b 21 c b b 11 + b 21 c 23 b 23 b 31 c 33 b 33 b 41 We define indicator variables for each hours regime as follos: D 1t 1h t < h, D 2t 1h t > h, and D 3t 1 D 1t D 2t 1h t h. The demand system fohe inputs is specified belo. ] b11 Bt lnh t D 1t [ln + ln 1 + b 21 [ t + D 2t {ln b 11 + ln 1 λ h + + D 3t lnh + ε ht Bt t ] } lnλ 38

49 ln E t D 1t [b 01 + b 11 ln t + b 21 ln Bt + D 2t {b 01 b 11 lnλ + b 11 + b 21 ln ] + b 31 lnq t + b 41 t [ + b 21 ln 1 λ h + B ] t t t +b 31 lnq t + b 41 t} b11 + D 3t [b 01 + b 21 ln 1 + b 21 + b 11 ln b h 11 + b 21 ln 1 + b 11 + b 21 ] t h + B t + b 11 + b 21 ln + b 31 lnq t + b 41 t + ε Et [ ] b11 + b 21 ln K t D 1t {b 01 + ln + b 11 ln 1 + b 21 t b 21 ln Bt +b 31 lnq t + b 41 t} t + D 2t {b 01 b 11 lnλ b 11 + b 21 ln r [ t b 21 ln 1 λ h + B ] } t + b 31 lnq t + b 41 t + ε Kt t b11 + D 3t {b 01 + b 21 ln 1 + b 21 + b 11 ln 1 + b h 11 + b 21 ln 1 + b 11 + b 21 t h + B t +ln [ b 11 + b 21 ] b 11 + b 21 ln +b 31 lnq t + b 41 t} + ε Kt. As before, another strategy is to jointly estimate the conditional input demand system through direct estimation of the CD production function parameters. ] β Bt lnh t D 1t [ln + ln α β [ t ] } β + D 2t {ln + ln 1 λ h Bt + lnλ α β + D 3t lnh + ε ht t 39

50 { 1 ln E t D 1t [β + γ ln α β lna βlnβ γlnγ] + β ln t β + γ ln Bt + 1 lnq t g } t { [ ] 1 β γ + D 2t lna + βln + γln βlnλ α β α β γ [ ln t β + γ ln 1 λ h + B ] t t + 1 lnq t g } t { 1 [ γ γ + D 3t lna + βlnh t h + γln ln α] + B t r } t 1 g + lnq t t + ε Et { 1 lnk t D 1t [β + γ ln α β lna βlnβ γlnγ] γ +ln + β α β ln t α β Bt + ln + 1 lnq t g } t { [ ] 1 β γ γ + D 2t lna + βln + γln βlnλ + ln α β α β α β + α [ ln t α β + ln 1 λ h + B ] t t + 1 lnq t g } t { 1 [ γ ] α + D 3t αln lna βlnh α t h + B t + ln r } t 1 g + lnq t t + ε Kt. As long as one knos hich regime corresponds to each period in the data, the CD production function parameters and the conditional input demand function parame- 40

51 ters can be estimated from data on employment alone. Furthermore, if t,, B t, and Q t are exogenous, the NLSUR estimator is consistent. For example, these variables could plausibly be assumed to be exogenous for a branch plant of a multi-plant firm. If Q t ere endogenous, then a nonlinear 3SLS estimator ould be consistent. 41