Chapter 02 - Labor Supply
CHAPTER 2 2-1. How many hours will w ill a person allocate to leisure activities if her indifference curves etween consumption and !oods are concave to the ori!in"
A worker will either work all available time or will not work at all. As drawn in Figure A, A, point B is preerred to points A and C . !hus, the worker "hooses "hooses not to enter the the labor market. As drawn in Figure #, point C is is preerred to both points A and B. !hus, the worker "hooses not to "onsume any leisure and to work all available time. Figure A
Figure #
o od s
oods C
C
U 1
U 0
A
U 0 B
U 1
A
B
1ours o Leisure
1ours o Leisure
2-2. #hat is the effect of an increase in the price of mar$et !oods on a wor$er%s reservation wa!e& proaility of enterin! the laor force& and hours of wor$" ′
Suppose the pri"e o market goods in"reases rom p to p and the person$s non-labor in"ome is V . % she "hooses not to work, she "an pur"hase V/p units o "onsumption ater the pri"e "hange, whereas she "ould have "onsumed V/p units o "onsumption "onsumption prior to the pri"e in"rease. in"rease. !hus, her endowment point has moved rom E to to E in Figure A &on the ne't page(. )nder normal "onditions, in"luding that leisure is a normal good, the indieren"e "urve is steeper as we move up a verti"al line, indi"ating that the slope o the indieren"e "urve is steeper at E than than at E . !hus, an in"rease in the pri"e o goods lowers the reservation wage and makes the person more likely to work. ′
′ ′
′ ′
*Figure A is on the ne't page.+
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Chapter 02 - Labor Supply
Figure A. oods
E
V 6 p
V 6 p
′
E
′
0
1ours o Leisure
T
!o simpliy the illustration o the ee"t on hours o work, assume or simpli"ity that V 5 0. !he in"rease in the pri"e o goods shits the budget line rom FE to GE in Figure # below, moving the worker rom P to point R. !his shit indu"es both an in"ome ee"t and a substitution ee"t. !he pri"e in"rease lowers the person$s real wage rate, in"reasing the demand or leisure and leading to ewer hours o work &the substitution ee"t(. !his substitution ee"t is illustrated by the move rom point P to point Q in Figure #. !he pri"e in"rease also redu"es the worker$s wealth, lowering the demand or leisure and leading to more hours o work &the in"ome ee"t(. !his in"ome ee"t is illustrated by the move rom Q to R. As drawn the in"ome ee"t dominates the substitution ee"t and the pri"e in"rease lowers the demand or leisure and in"reases hours o work. %t is, o "ourse, possible or the substitution ee"t to dominate the in"ome ee"t ¬ pi"tured(, so that hours o work de"reases. !hus, without urther restri"tions on preeren"es, an in"rease in the pri"e o market goods has an ambiguous ee"t on hours worked. Figure #.
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Chapter 02 - Labor Supply
oods F
P Q
G
R E
1ours o Leisure
T
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Chapter 02 - Labor Supply
2-'. Tom earns (1) per hour for up to *+ hours of wor$ each wee$. He is paid ('+ per hour for every hour in e,cess of *+. Tom faces a 2+ percent ta, rate and pays (* per hour in child care e,penses for each hour he wor$s. Tom receives (+ in child support payments each wee$. There are 1 hours in the wee$. /raph Tom%s wee$ly ud!et line. • •
•
% !om does not work, he leisures or 78 hours and "onsumes 980. For all hours !om works up to his irst :0, his ater-ta' and ater-"hild "are wage e;uals &80 per"ent o 9<( = 9: 5 98 per hour. !hus, i he works or :0 hours, he will be able to leisure or 28 hours and "onsume 980 > 98&:0( 5 9:00. For all hours !om works over :0, his ater-ta' and ater-"hild "are wage e;uals &80 per"ent o 90( = 9: 5 920. !hus, i he works or 78 hours &28 hours at the overtime wage(, he will not leisure at all, but he will "onsume 980 > 98&:0( > 920&28( 5 92,?70.
!om$s weekly budget line is pi"tured below. ollars o Consumption 92,?70
9:00 980 28
78
1ours o Leisure
2-*. Cindy !ains utility from consumption C and leisure L. The most leisure she can consume in any !iven wee$ is 1 hours. Her utility function is U 0C & L = C L. This functional form implies that Cindy%s mar!inal rate of sustitution is C / L. Cindy receives ('+ each wee$ from her !reat-!randmotherre!ardless of how much Cindy wor$s. #hat is Cindy%s reservation wa!e"
!he reservation wage is the MRS when not working at all. !hus, w RES 5 MRS at ma'imum leisure e;uals C 6 L 5 970678 5 9.@<.
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Chapter 02 - Labor Supply
2-). 3ou can either ta$e a us or drive your car to wor$. A us pass costs () per wee$& whereas drivin! your car to wor$ costs (+ wee$ly 0par$in!& tolls& !as& etc.. 3ou spend halfan-hour less on a one-way trip in your car than on a us. How would you prefer to travel to wor$ if your wa!e rate is (1+ per hour" #ill you chan!e your preferred mode of transportation if your wa!e rate rises to (2+ per hour" Assume you wor$ five days a wee$ and time spent ridin! on a us or drivin! a car does not directly enter your utility.
!aking a bus will save you 9<< a week, but it will "ost you < hours o leisure time due to the longer "ommute. Sin"e the pri"e o leisure is e;ual to the wage rate, the monetary value o the time lost is 9<0 when the hourly wage is 90 and 900 when the hourly wage is 920. !hereore, it makes sense or you to take a bus to work i you are paid 90 per hour, but you will swit"h to driving your "ar i your wage in"reases to 920 per hour.
2-. 4helly%s preferences for consumption and leisure can e e,pressed as U 0C & L
5 0C 2++
0 L + .
This utility function implies that 4helly%s mar!inal utility of leisure is C 2++ and her mar!inal utility of consumption is L +. There are 1 hours in the wee$ availale to split etween wor$ and leisure. 4helly earns () per hour after ta,es. 4he also receives ('2+ worth of welfare enefits each wee$ re!ardless of how much she wor$s. 0a /raph 4helly%s ud!et line.
% Shelly does not work, she leisures or 78 hours and "onsumes 920. % she does not leisure at all, she "onsumes 920 > 9<&78( 5 9,70. Shelly$s weekly budget line, thereore, is pi"tured below. ollars o Consumption 9,70
920 78
1ours o Leisure
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Chapter 02 - Labor Supply
0 #hat is 4helly%s mar!inal rate of sustitution when L 5 1++ and she is on her ud!et line"
% Shelly leisures or 00 hours, she works or 78 hours and "onsumes 920 > 9<&78( 5 9770. !hus, her /BS when doing this is
MRS
=
MU L MUc
C 200 L 80
770
−
=
=
,00
−
200
−
−
80
:70 =
20
=
9 2. .
0c #hat is 4helly%s reservation wa!e"
!he reservation wage is deined as the /BS when working no hours. Dhen working no hours, Shelly leisures or 78 hours and "onsumes 920. !hus, w RES
20 =
−
,78
200
−
,20 =
80
88
≈
9,.7
.
0d 6ind 4helly%s optimal amount of consumption and leisure.
1er optimal mi' o "onsumption and leisure is ound by setting her /BS e;ual to her wage and solving or hours o leisure given the budget line C 5 20 > <&78 = L(. w = MRS
<=
<=
20
+
C − 200 L − 80
<&,78 − L ( − 200 L − 80
< L − :00
=
?70
−
< L
L = ,7 . !hus, Shelly will "hoose to leisure 7 hours, work 2 hours, and "onsume 920 > 9<&2( 5 9:80 ea"h week.
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Chapter 02 - Labor Supply
2-7. E,plain why a lump sum !overnment transfer can entice some wor$ers to stop wor$in! 0and entices no one to start wor$in! while the earned income ta, credit can entice some people who otherwise would not wor$ to start wor$in! 0and entices no one to stop wor$in!.
A lump sum transer is asso"iated with an in"ome ee"t but not a substitution ee"t, be"ause it doesn$t ae"t the wage rate. !hus, i leisure is a normal good, a lump sum transer will likely "ause workers to work ewer hours &and "ertainly not "ause them to work more hours( while possibly enti"ing some workers to e'it the labor or"e all together. En the other hand, the arned %n"ome !a' Credit raises the ee"tive wage o low-in"ome workers by :0 per"ent &at least or the poorest workers(. !hus, someone who had not been working a"es a wage that is :0 per"ent higher than it otherwise was. !his in"rease may be enough to en"ourage the person to start working. For e'ample, i a worker$s reservation wage is 90.00 per hour but the only ob she "an ind pays 98.00 per hour, she will not work. )nder the earned in"ome ta' "redit, however, the worker views this same ob as paying 9.20 per hour, whi"h e'"eeds her reservation wage. Furthermore, the %!C "annot en"ourage a worker to e'it the labor or"e, as the beneits o the %!C are re"eived only by workers.
2-. 8n 1999& *&+ TA:6 recipients were as$ed how many hours they wor$ed in the previous wee$. 8n 2+++& *&'92 of these recipients were a!ain su;ect to the same TA:6 rules and were a!ain as$ed their hours of wor$ durin! the previous wee$. The remainin! * individuals were randomly assi!ned to a <:e!ative 8ncome Ta,= 0:8T e,periment which !ave out financial incentives for welfare recipients to wor$ and were su;ect to its rules. >i$e the other !roup& they were as$ed aout their hours of wor$ durin! the previous wee$. The data from the e,periment are contained in the tale elow.
Total :umer ?f Recipients
TA:6 :8T Total
*&'92 * *&+
:umer of Recipients #ho #or$ed At 4ome Time in the 4urvey #ee$
Total Hours ?f #or$ @y All Recipients in the 4urvey #ee$
1999
2+++
1999
2+++
1&217 1'1 1&'*
1&) 21' 1&71
1)&)7 1&' 17&21
2+&9 2&)') 2'&2''
0a #hat effect did the :8T e,periment have on the employment rate of pulic assistance recipients" evelop a standard difference-in-differences tale to support your answer.
mployment Bate !A4F 4%!
??? 2@.@G 28.0G
2000 <.@G :<.
i 8.0G @.
i-in-i ?.
!he 4%! in"reased the probability o employment by ?.< per"entage points. 4ote that the per"ent numbers are ound by dividing the H4umber o Be"ipientI "olumns &2 nd and rd o original( by the 4umber o Be"ipients "olumn & st o original(.
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Chapter 02 - Labor Supply
0 #hat effect did the :8T e,periment have on the wee$ly hours wor$ed of pulic assistance recipients who wor$ed positive hours durin! the survey wee$" evelop a standard difference-in-differences tale to support your answer.
Deekly 1ours Dorked Jer Dorking Jerson !A4F 4%!
??? 2.8 2.<
2000 .2 .?
i 0.: -0.7
i-in-i -.0
!he 4%! de"reased weekly hours worked, o those working, by hour. 4ote that the average weekly hours o work per persons is ound by the H!otal 1ours o DorkI "olumns &: th and
2-9. Consider two wor$ers with identical preferences& Phil and @ill. @oth wor$ers have the same life cycle wa!e path in that they face the same wa!e at every a!e& and they $now what their future wa!es will e. >eisure and consumption are oth normal !oods. 0a Compare the life cycle path of hours of wor$ etween the two wor$ers if @ill receives a one-time& une,pected inheritance at the a!e of ').
#e"ause the workers have the same lie "y"le wage path and the same preeren"es, they will have the same lie "y"le path o hours o work up to the une'pe"ted event. An inheritan"e provides an in"ome ee"t or #ill with no substitution ee"t, and thus, he will work ewer hours &or at least not more hours( than Jhil rom the age o < orward.
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Chapter 02 - Labor Supply
1ours Dorked
Lie Cy"le Jath o 1ours Dorked Ater Age <
#eore Age <
Jhil
#ill and Jhil
<
#ill
Age
0 Compare the life cycle path of hours of wor$ etween the tw o wor$ers if @ill had always $nown he would receive 0and& in fact& does receive a one-time inheritance at the a!e of ').
%n this "ase, be"ause the inheritan"e is ully anti"ipated, and be"ause it oers the same in"ome ee"t with no substitution ee"t, #ill will work ewer hours &or at least not more hours( than Jhil over their entire work lives. &!he graph is on the ne't page.(
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Chapter 02 - Labor Supply
1ours Dorked
Lie Cy"le Jath o 1ours Dorked
Jhil
#ill
<
Age
2-1+. Bnder current law& most 4ocial 4ecurity recipients do not pay federal or state income ta,es on their 4ocial 4ecurity enefits. 4uppose the !overnment proposes to ta, these enefits at the same rate as other types of income. #hat is the impact of the proposed ta, on the optimal retirement a!e"
Suppose so"ial se"urity beneits are the only pension beneits available to a retiree. !he ta', thereore, "an be interpreted as a "ut in pension beneits. !he "ut in pension beneits shits the budget line rom FH to FE in the igure below, shiting the worker rom point P to point R. &4ote that FE and FH are both downward sloping, indi"ating that total retirement "onsumption is greater the later in lie one retires.( !his shit generates both in"ome and substitution ee"ts. #oth o these ee"ts, however, work in the same dire"tion. First, the ta' redu"es the retiree$s wealth, redu"ing her demand or leisure, and leading her to retire later &the in"ome ee"t(. At the same time, the ta' redu"es the HwageI that retirees re"eive when retired, ee"tively in"reasing &in relative terms( the wage they earn while working and generating a substitution ee"t that leads to more work hours, thus urther delaying retirement. )nder normal "onditions, thereore, a ta' on pension beneits will redu"e the optimal retirement age &i.e., workers will delay retirement(.
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Chapter 02 - Labor Supply
Consumption uring Betirement F
)
J
B
1
)0
20
Kears o Betirement
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Chapter 02 - Labor Supply
2-11. A wor$er plans to retire at the a!e of )& at which time he will start collectin! his retirement enefits. Then there is a sudden chan!e in the forecast of inflation when the wor$er is ' years old. 8n particular& inflation is now predicted to e hi!her than it had een e,pected so that the avera!e price level of mar$et !oods and wa!es is now e,pected to e hi!her. #hat effect does this announcement have on the person%s preferred retirement a!e 0a if retirement enefits are fully ad;usted for inflation"
!here will be no ee"t on the person$s retirement de"ision i retirement beneits are ully adusted or inlation as nothing "hanges in the person$s "al"ulations in real terms the relative magnitudes o pri"es, wages and retirement beneits are the same with or without inlation. !he person a"es the same "hoi"e, so his de"ision does not "hange. 0 if retirement enefits are not fully ad;usted for inflation"
% retirement beneits are not adusted or inlation, the pur"hasing power o retirement beneits alls. % the person does not retire, he "an enoy the same "onsumption as he would without inlation as wages are assumed to ully adust or inlation. % he retires at 7<, his beneits are worth less in real terms &they "an buy him less "onsumption( with inlation than without, so he "annot aord the same "onsumption path as beore. 1en"e, his "hoi"e set over the years o retirement and "onsumption lies below the original &pre-inlation( "hoi"e set e'"ept at one point where he does not retire at all. !hus, as long as leisure &i.e., years o retirement( and "onsumption are normal goods, the in"ome and substitution ee"ts both lead to the individual retiring later in lie.
2-12. Presently& there is a minimum and ma,imum social security enefit paid to retirees. @etween these two ounds& a retiree%s enefit level depends on how much she contriuted to the system over her wor$ life. 4uppose 4ocial 4ecurity was chan!ed so that everyone a!ed ) or older was paid (12&+++ per year re!ardless of how much she earned over her wor$in! life or whether she continued to wor$ after the a!e of ). How would this li$ely affect hours wor$ed of retirees"
Labor or"e parti"ipation is likely greatest or those retirees whose so"ial se"urity in"ome is low &below 92,000 per year(. !hus, the "hange in beneits oers these retirees a pure &positive( in"ome ee"t. !hese retirees should redu"e their hours worked i not leave the labor or"e all together ater the age o 7<. %n "ontrast, the poli"y "hange oers all retirees who would have earned more than 92,000 per month a pure &negative( in"ome ee"t. !hese retirees will be"ome more likely to work, or, i already working, more likely to work more hours ater the age o 7<.
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Chapter 02 - Labor Supply
2-1'. ?ver the last 1++ years& real household income and standards of livin! have increased sustantially in the Bnited 4tates. At the same time& the total fertility rate& the avera!e numer of children orn to a woman durin! her lifetime& has fallen in the Bnited 4tates from aout three children per woman in the early twentieth century to aout two children per woman in the early twenty-first century. oes this su!!est that children are inferior !oods"
!he "onventional wisdom &and empiri"al eviden"e( suggests that "hildren are normal goods. "onomi"ally, "hildren are a lot more e'pensive today than they were 00 years ago &"onsider edu"ation, housing, "lothing, entertainment e'penses(. Children also produ"e less or the household in the 2 st "entury than they did 00 years ago. !he biology6evolution argument is that inant mortality rates have also allen dramati"ally over the last 00 years, so a woman needs to have ewer "hildren to be more "onident that some o her "hildren will rea"h adulthood.
2-1*. Consider a person who can wor$ up to + hours each wee$ at a pre-ta, wa!e of (2+ per hour ut faces a constant 2+D payroll ta,. Bnder these conditions& the wor$er ma,imies her utility y choosin! to wor$ )+ hours each wee$. The !overnment proposes a ne!ative income ta, wherey everyone is !iven ('++ each wee$ and anyone can supplement her income further y wor$in!. To pay for the ne!ative income ta,& the payroll ta, rate will e increased to )+D. 0a ?n a sin!le !raph& draw the wor$er%s ori!inal ud!et line and her ud!et line under the ne!ative income ta,.
)nder the original s"enario, let I be total weekly in"ome, L be hours o leisure, and H be hours worked. !he worker$s ater-ta' wage rate is 80G o 920 whi"h e;uals 97 per hour. !hus, when the worker works all 80 hours in the week, she earns 97 ' 80 5 9,280 and her budget line is des"ribed by I 5 280 = 7 L. 4oti"e that when L 5 80, the worker earns 90. And when L 5 0, the worker earns 97 M <0 5 9800. )nder the negative in"ome ta', the worker is given 900 ea"h week, but now her ater-ta' wage rate is <0G o 920 whi"h e;uals 90 per hour. %n this "ase, when the worker works all 80 hours in the week, she earns 90 M 80 > 900 5 9,00 and her budget line is properly des"ribed by I 5 00 = 0 L. 4oti"e that when L 5 80, the worker re"eives 900. And when L 5 0, the worker re"eives 900 > 90 ' <0 5 9800. !he two budget lines or both s"enarios are graphed on the ne't page.
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Chapter 02 - Labor Supply
Deekly #udget Lines Deekly 9,280 %n"ome 9,00
Eriginal S"enario
9800
4egative %n"ome !a' 900
0
80
1ours o Leisure
0 4how that the wor$er will choose to wor$ fewer hours if the ne!ative income ta, is adopted.
!o answer this ;uestion, one needs to ind where the budget lines interse"t. Setting the budget lines e;ual and solving or L reveals that the budget lines interse"t at L 5 0. !hus, the indieren"e "urve that is tangent to the original budget line at L 5 0 must not be tangent to the budget line under the negative in"ome ta' &be"ause L 5 0 was the optimal "hoi"e without the negative in"ome ta'(. %n parti"ular, the worker$s original indieren"e "urve must be below the new budget line to the right o L 5 0. !hereore, when a"ed with the negative in"ome ta', the worker will move in that dire"tion, whi"h re;uires her to in"rease L &hours o leisure( and "on"urrently de"rease H &hours o work(. 0c #ill the wor$er%s utility e !reater under the ne!ative income ta,"
%n this parti"ular "ase, the worker$s utility will in"rease under the negative in"ome ta' be"ause she "ould "ontinue to leisure 0 hours ea"h week and re"eive 9800 &whi"h was her out"ome beore the negative in"ome ta'( but instead the worker decides to leisure more &and "onsume less(. !his "hange in behavior must in"rease her utility.
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Chapter 02 - Labor Supply
2-1). The asolute value of the slope of the consumption-leisure ud!et line is the after-ta, wa!e& w. 4uppose some wor$ers earn w for up to *+ hours of wor$ each wee$& and then earn 2w for any hours wor$ed thereafter 0called overtime. ?ther wor$ers may earn w for up to *+ hours of wor$ each wee$& and then only earn +.)w thereafter as wor$in! more than *+ hours reFuires !ettin! a second ;o which pays an hourly wa!e less than their primary ;o. @oth types of wor$ers e,perience a <$in$= in their consumption-leisure ud!et line.
0a /raph in !eneral terms the ud!et line for each type of wor$er. Deekly #udget Lines Deekly %n"ome !he wage in"reases &to 2 w( or hours o work in e'"ess o :0.
!he wage alls &to 0.
#oth a"e a wage o w or :0 hours
T - :0
T
1ours o Leisure
0 #hich type of wor$er is li$ely to wor$ up to the point of the $in$& and which type of wor$er is li$ely to choose a consumption-leisure undle far away from the $in$" !he worker who e'perien"es a de"rease in her wage ater working :0 hours is &more( likely to work e'a"tly :0 hours as the marginal beneit o working e'perien"es a negative ump down at this point. %n "ontrast, the worker who e'perien"es an overtime premium ater working :0 hours is &more( likely to not work e'a"tly :0 hours. #e"ause o the overtime premium, on"e the worker hits :0 hours o work, the worker e'perien"es a positive ump up in the marginal beneit o working. Jut dierently, this worker may opt to only work 20 or 0 hours, but i she inds hersel having worked :0 hours be"ause the T = :0th hour o leisure was not as valuable as w, then it is very likely that she will also ind that the T = :st hour o leisure is not as valuable as 2 w, and thereore she works the : st hour &and possibly ;uite more(.
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