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Labor Economics

by: Reba Terry

Labor Economics ECO 735

Reba Terry

GPA 3.77

David Ribar

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This 53 page Class Notes was uploaded by Reba Terry on Sunday October 25, 2015. The Class Notes belongs to ECO 735 at University of North Carolina at Greensboro taught by David Ribar in Fall. Since its upload, it has received 24 views. For similar materials see /class/229063/eco-735-university-of-north-carolina-at-greensboro in Economcs at University of North Carolina at Greensboro.


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Date Created: 10/25/15
A Trends Trends in Employment and Evidence based on the Static Model 1 Trends in men s and women s per capita employment Per capita employment by gender 1958 1968 Men 1978 Women 2 Agespeci c trends for men 1988 1993 2008 Year All 1624 2534 3544 4554 5564 65 1948 835 697 932 956 934 867 452 1958 785 615 908 929 912 830 337 1968 778 630 951 956 934 827 265 1978 738 661 911 930 888 713 195 1988 720 641 893 909 878 647 161 1998 716 608 895 898 867 662 159 2008 685 523 859 880 842 677 205 Men39s ageemp profiles over time 15724 25734 3544 45754 55754 657 1958 I 1978 quotmmquot1998 3 Agespeci c trends for women Year All 1624 2534 3544 4554 5564 65 1948 313 412 318 358 340 235 89 1958 345 381 330 407 455 336 99 1968 396 438 406 472 511 415 93 1978 464 538 581 586 547 400 80 1988 534 576 686 721 666 423 77 1998 571 572 726 742 741 500 83 2008 562 502 710 727 731 570 127 Women39s ageemp profiles 16724 25734 35744 45754 5764 1958 l 1978 quotmm199 B General Econometric Issues 1 Functional form of the labor supply function a How do we choose the appropriate functional form b Theory simply states that labor supply is a general function of wages and nonlabor income c One approach is to start with a known direct utility 9 function structural approach 1 For instance we could use a StoneGeary speci cation and let UC L aln CAc 171 1n LAL then the labor supply function will be HGKYL1lt1NYcW 2 approach is theoretically consistent can see how particular theoretical elements affect labor supply 3 requires a strong assumption regarding preferences An alternative is to choose an approximate representation e g linear function for the labor supply function 1 easier and more exible way to start 2 still implicitly imposes restrictions on preferences for instance a linear equation Hquot a BW 6N leads to a direct utility speci cation aK Lyyzx aaamp3 5 p 6K L 3 while this example is theoretically consistent other labor supply functions may not be unless they satisfy the integrability conditions UQLL 2 Sources of heterogeneity a b How do we account for the differences heterogeneity in labor supply across people There will be some observed heterogeneity in wages and nonlabor incomes that is variation in the budget constraints that people face Beyond this there may also be variation across people in their preferences 1 the sources of variation may depend on observed or unobserved variables 2 we could specify one or more of the preference parameters as a function of observed and unobserved variables in the StoneGeary specif1cation we could let 1K7YL X B a need to be mindful possible restrictions eg the above term would need to be positive represents a strong assumption 3 dif cult to assign heterogeneity speci cally to preferences if you start with an approximate representation in this case we usually just let the terms enter linearly interpretation is more difficult d measurement error in the labor supply outcome is another possible source of heterogeneity 1 could arise because people answer questions about average hours worked 2 could also re ect genuine recall errors 3 recall that measurement error in the explanatory variables will lead to other econometric problems not a great way to motivate your specification 3 How will the economic variables be measured a Labor supply variables 1 Will the study focus on the discrete employment decision or the continuous labor supply decision 2 For either outcome what is the appropriate dimension of time for an experiment a few hours or a day might be appropriate some analyses consider weekly data such data are relatively easy to obtain however people may have limited scope to vary their hours monthly or annual data allow more variation but also increase the chances of mismeasurement or recall errors problems also arise if people switch jobs b Wages 1 Are data on hourly wages available some people are only paid monthly or annually some people are paid on a commission or piece rate basis 2 There are obvious problems associated with using earnings data from longer periods such as weeks or years if we want to examine the effects of wages on labor supply earnings con ate wage and labor supply outcomes 3 May introduce bias spurious negative correlation if we divide earnings by hours consider equation HuBEH6N if earnings E WH the hours terms in the numerator and denominator cancel leaving W if E 7E WH some portion of hours will remain in the denominator leading to a negative correlation with the dependent hours variable 4 Are wages the same over the entire range of hours is the budget constrain linear overtime moonlighting or multiple jobs over time taxes c Nonlabor income 1 May observe only a fraction of nonlabor income may not observe certain types of retirement sav1ngs may not observe imputed returns from housing 2 Nonlabor income may depend on work effort direct relationships for sources such as unemployment insurance and welfare benefits may re ect previous work effort 3 Nonlabor income may be endogenous for other reasons could re ect earnings of other family members could depend on other behavior eg strategic bequests 4 Selectivity issues a Should we examine both workers and nonworkers 1 If nonworkers are included we face problems with censoring 2 If nonworkers are excluded we face problems with truncation and selectivity b How should we account for selectivity 1 Do we use a standard ML tobit approach 2 Do we use Heckman s twostage generalized tobit approach 3 Do we use a less restrictive speci cation like symmetricallycensored least squares How do we impute wages for nonworkers 1 Do we use modelbased imputations 2 Do we use hotdeck imputations 3 Do we use multiple imputation methods 4 How do we account for the errors in imputations C Summary of Empirical Evidence 1 Killingsworth 1983 distinguished between two generations of research a rst generation studies relied mostly on OLS recognized selection issues but did not use formal methods to address them second generation studies were much more concerned with selectivity and other econometric issues such as discontinuous labor supply schedules and nonlinear budget constraints used formal techniques to address these issues 2 Empirical ndings of rst generation studies results summarized in Tables 3135 of Killingsworth 1983 a uncompensated wage elasticities 1 men generally fell between 000 and 040 though positive elasticities appeared in some studies 2 women selectivity concerns led to fewer studies wider range of estimates 020 to 090 b income elasticities 1 men generally in the range 000 to 0 16 though some stronger results were reported 2 women generally in the range 010 to 020 c one general conclusion was that women s labor supply was more sensitive to wage and nonlabor income changes than men s 3 Income maintenance experiments a Description of experiments 1 different social experiments run in the late 1960s and 1970s in New Jersey amp Pennsylvania rural North Carolina Iowa Gary Seattle and Denver 2 designed to examine a negative income tax 3 changed effective wages and nonlabor incomes for some randomly chosen lowincome families experimental s 4 members of the control group faced their original budget constraint faced the existing tax and transfer system b Pencavel 1986 points out several limitations with the experiments 1 used selective lowincome samples 2 some nonrandom allocation between treatment and control groups 3 signi cant external changes in some locations 4 nonrandom attrition 5 some incentives to misreport income 6 experiment was only temporary 7 no saturation models were used Treatments which generally increased effective wage rates had negative effects on labor supply 1 range of uncompensated wage elasticities was 016 to 021 2 introduced many questions about selectivity and the empirical treatment of nonlinear budget constra1nts 3 impetus for second generation of studies 4 Empirical ndings from second generation studies see Tables 4244 in Killingsworth 1983 a b d As mentioned studies typically used formal methods to control for selectivity Killingsworth 1983 gives the following 3step approach as an example methodology 1 estimate reducedform probit model for employment 2 use results from this model to construct 9t and estimate a selectivitycorrected wage regression 3 use results from secondstage model to impute wages for nonworkers use imputed wages and 9 in a final generalized Tobit labor supply model In addition to employment selectivity models also considered other things like nonlinear budget constraints Studies led to a much wider range of ndings 5 Mroz 1987 reconciliation study a Wide range of second generation results 1 Studies were successful in demonstrating the sensitivity of labor supply estimates to alternative empirical techniques eg showing that corrections for selectivity and nonlinear budget constraints led to different findings 2 Studies however did not rigorously test their specif1cation assumptions and were not able to reconcile their different ndings 3 Mroz conducted an analysis of specification issues in the context of married women s labor supply b Key features of study 1 examined a simple model of married women s labor supply H do alan d2N 0L3Z a 2 considered the following speci cation issues exogeneity of wages nonwife income fertility and work experience selection into employment budget nonlinearities from taxes 3 examined annual hours for a single crosssection 1975 from the PSID sample consisted of 753 women 428 workers use of a single sample ruled out sampling differences C d Results 1 wage instruments using polynomials in experience substantially increased the estimated wage effect however models that used experience as an instrument appeared to be overidentif1ed little evidence of endogeneity bias in speci cations based on other instruments 2 Similarly could not reject exogeneity of nonwife income and fertility 3 Selection compared standard ML Tobit truncated regression and generalized twostage Tobit V also examined relaxed distributional assumptions in standard Tobit rejected standard Tobit found little evidence of selectivity in other models 4 Found that the treatment of taxes had little effect on estimation results Final results 1 among speci cations that could not be rejected Wage effects maximum of 150 hours per year per extra dollar in wages Income effects maximum of 22 hours per year per extra 1000 in income 2 overall results suggested that married women s work behavior was very similar to married men s that is relatively insensitive to wage and nonlabor income changes Simple Static Labor Supply Model A Quick Review of Consumer Demand Theory 1 Rationale a the rst couple of labor supply models that we will consider are simple extensions of consumer demand models b before looking at the labor supply models it s useful to review the some of the general properties of consumer demand 2 General model a consider a person with preferences over N goods x1 x2 xN b assume that preferences can be described by a utility function Ux1 x2 xN where gt O and U1 lt O for j 1 N assume that the person has a total income Y and faces prices for each of the goods p the budget constraint can be written Y 2 219 x 0 P the person s problem is to choose quantities of the goods to maximize the utility function subject to the budget constraint e we can incorporate the budget constraint by rewriting the maximization problem as a Lagrangian Mix UxlxN 1Y 2 pjxj f for an interior solution the rst order conditions are U hp for all j the second order conditions are guaranteed if the budget constraint is convex and the utility function is strictly concave g the interior rstorder conditions imply that UjUk pjpk for all and k h we obtain the direct demand functions x jp1 pN Y for all j a great amount of empirical work I focuses on estimating the parameters of these demand functions 3 Properties of demand functions and related functions a direct demand functions satisfy the following propert1es 1 adding up 2 homogeneity of degree 0 in prices and income 3 symmetric Slutsky matrix matrix of substitution terms 4 negative semidef1nite Slutsky matrix b recall that x 6h 6x J J J xk apk apk aY c indirect utility function Vp17 7pN7 Y Uflp17 7pN7 Y 7fNp17 7pN7 l Vis nonincreasing in prices and nondecreasing in income 2 Vis quasiconvex in p 3 Vis homogeneous of degree 0 in prices and income d expenditure function ep1 19M 1 is the solution to subject to Ux1xN 2 H l e is homogeneous of degree 1 in prices 2 e is concave in prices e Hicksian compensated demand functions hjp1 19M 1 are the set of demands which satisfy expenditure minimization 4 functional forms a CobbDouglas utility specif1cation U X 3 In x I demand functions are x B Y pj 2 convenient but very restrictive b StoneGeary Linear Expenditure System utility specif1cation U X B lnx Otj I demand functions are xj 1 Bocj Bj9j jl310 kj 2 still relatively convenient 3 less restrictive than CobbDouglas though still relatively restrictive c there are lots of other functional forms d in general functional form speci cations involve tradeoffs of convenience and exibility e a substantial amount of empirical work in consumer demand theory focuses on testing the restrictions inherent in both the theory and these speci c functional forms B Static LaborLeisure Model 1 Assumptions a assume that preferences are defined over two goods consumption C and time spent outside the labor market leisure L we can write the utility function U UC L b assume that there is a maXimum total amount of time available K and that time is divided between two mutually exclusive uses market work H and time outside the labor market we can write the time constraint K L H c assume that the individual receives an hourly wage W has nonlabor income N and faces a price on consumption goods P 1 budget constraint can be written WH N 2 PC 2 the budget constraint does not exactly conform to our earlier consumer demand constraint because total income is no longer exogenous depends on the amount of work 3 note however that we can rewrite the budget constraint as WK N 2 WL PC let WK N Fbe full income this is similar to the budget constraint from consumer demand theory 4 for simplicity we C normalize prices at F P l 5 a graph of the budget constraint appears on the right N 2 IndiVidual s choice a indiVidual solves hgax U C L subjectto F 2 WL C b obtain demandtype solutions C CW F and L LW P c these demand functions are subject to the standard restrictions 1 homogeneity imposed by the price normalization 2 V adding up imposed if we de ne the consumption function as CVKF F W39LWF symmetry of the Slutsky matrix also imposed by construction 3 V 4 Slutsky matrix negative semide nite ie dLdW lu lt O testable restriction 3 Testing the Slutsky restriction a recall 6L 6W aL LaL U W FaF W want to test whether this is negative b practical problems 1 how do we de ne L and F ie how do we de ne 10 2 what does it mean to hold F constant while changing W c note that if we differentiate LW F wrt Wholding N constant we obtain 6L 6L 6L 6F6L 6L 6WN6WF 6FW6W6WF ava d rearranging terms 6L 6L 6L 6WF6WN ava e substituting into the Slutsky equation L L L 6 a Ka 6L H6L aWU aWN aNW aWN aNW f nally note that dH dL so that 5H 5H H LH aWU aWN 6N W g can obtain the two partial derivatives on the RHS from a regression of hours on wages and nonlabor income hours wages and nonlabor income can all be directly observed 4 Comparative statics of the labor supply function a de ne the labor supply function as H HW N b income effect if nonmarket time is a normal good an increase in nonlabor income leads to an increase in nonmarket time and a decrease in labor supply c wage increase leads to two types of effects 1 income effect increases income and decreases labor supply 2 substitution effect increases the effective price of leisure increases labor supply 3 effects are offsetting the net effect is ambiguous 5 Shape of the labor supply curve a labor supply curve maps different wages to different levels of labor supply holding all other characteristics nonlabor income preferences etc constant b for very low wages W no labor is supplied H 0 c at some wage the reservation wage WR positive hours are W supplied d initially the labor supply curve slopes l upward and bends forward at H 0 there is no income effect near there the income effect is small the substitution effect dominates the income effect e thereafter the labor supply curve may bend backward if the income effect dominates the substitution effect f in the diagram the solid line depicts a labor supply function that is forward bending at all wages the dotted line depicts a labor supply function that is backward bending for some wages 6 Determination of the reservation wage a consider the following Lagrangian problem hgax UCL MF WL C b the KuhnTucker conditions imply that UL LW 2 O c evaluated atL K ie H O the condition is ULN K 9t 2 W the LHS of this expression defines the reservation wage d reservation wage is wage at which person is indifferent between working and not working e intuitively the reservation wage is the slope of the indifference curve evaluated at H O Nonlinear Budget Constraints Introduction 1 So far we have considered a very simple description of the budget constraint a wages are constant as hours increase b nonlabor income is xed and does not depend on other characteristics such as income In this section we are going to consider more realistic budget constraints that account for taxes government subsidies work costs and the like A common feature of these modi cations is that they lead to nonlinear budget constraints which in turn lead to estimation issues Examples of nonlinear budget constraints 1 Fixed money costs of work a consider several possible expenses associated with work such as child care union dues uniforms b sometimes these expenses do not vary with the number of hours worked ie are lumpsum expenses c effect on budget constraint 1 let the initial budget constraint without xed costs ofwork be Y WH N Q 2 with xed costs the new budget constraint is Y N Y WH N 7F 3 the resulting budget C constraint has a notch at the no work point ifthe person does not work ifthe person works effect on labor supply 1 compare behavior for people who do and do not face xed costs of work 2 for people who work many hours C like person A on right the imposition of xed costs is similar to a drop in non labor income it will cause these people to work more 3 for people who work few hours like person B above it may be optimal to stop working 4 imposition ofa xed cost increases the reservation wage and results in a discontinuous labor supply function 2 Effect of taxes on labor supply a effect ofa at proportional taX on earnings 1 consider an individual with a pretaX budget constraint Y WH N 2 assume that the government imposes a at proportional taX of t on earnings the aftertaX budget constraint is Y WH N 7 tWH litWH N graphically proportional C taX decreases S39f39 39 W the effective wage from W x to 170W x 4 equivalent to Slope1IW a wage decrease 5 will cause L L substitution DJ V and income effects net effect on labor supply will be ambiguous 6 is this a realistic description of the tax system describes social security amp some state systems poor description of federal system progressive income tax 1 the federal income tax is mostly progressive 2 progressive means that the marginal tax rate increases with income what would regressive mean 3 V in 2009 the marginal tax rates for a single taxpayer are 10 for adjusted incomes below 8350 15 on the next increment up to 33950 25 on the next increment up to 82250 28 on the next increment up to 171550 33 on the next increment up to 372950 35 on adjusted incomes thereafter 4 tax rates only apply to adjusted income income less deductions exemptions 5 initial tax rate is 0 up to adjustment amount 6 the effect of a progressive tax on the budget constraint is shown below slope lt85W Adj rst segment corresponds to income below adjusted amount next segment corresponds to rst taxable bracket 10 marginal tax rate nal segment corresponds to second taxable bracket 15 marginal tax rate 7 general concave shape c 2001 Tax ReliefAct 1 cut tax rates over several years 2 introduced the 10 bracket into the 15 bracket 3 28 bracket from 2000 was reduced to 25 4 highest tax bracket was reduced from 40 to 35 5 graphically budget constraint has swivelled out 6 what are the effects on labor supply 1 other features of the federal income tax system 1 these rates aren t the only feature of the federal system there are other exclusions credits etc 2 V retirement related features exclusion of pension contributions from income deferment of earnings on pension savings exclusion of some individual retirement savings and Social Security bene ts 3 4 V poverty related Earned Income Tax Credit V health related exclusion of employer contributions for insurance and medical care exclusion of Medicare bene ts deductibility of large medical expenses 5 V employment related dependent care tax credit exclusion of employerprovided child care targeted jobs tax credits exclusion of certain employee bene ts 6 housing related mortgage interest deduction 7 these features make the actual tax system much more complicated to model e state and local taxes add to the complexity 3 Meanstested transfers a consider a transfer program program that transfers money from the government to people that 1 provides a guaranteed bene t G for people without income and 2 reduces the bene t by a rate BRR for each dollar of income they receive 0 0 0 3 total bene ts would be B MaXGiBRRXY 0 graphically I assume pretransfer budget constraint is Y WH ie no nonlabor income also ignore other taxes 2 people with no income receive G 3 earnings are effectively taxed at a rate BRR until bene ts are exhausted big work disincentive what effect would an increase in the benefit reduction rate to 100 have on labor supply and program participation 39D 1 people who don t work or C work little initially like a and b will not work 2 people were participating but working like c near the cutoff will work more 3 people who were not participating like d will not be affected AFDC and TANF l AFDC 7 Aid to Families with Dependent Children old welfare program 2 TANF 7 Temporary Assistance for Needy Families program after 1996 reforms 3 trends in programs real value of bene ts peaked in the 1970s beneflts steadily eroded thereafter when Food Stamps and Medicaid are added benefits still fall over time bene ts have been relatively constant since 2000 recipiency rose sharply during the late 1960s and early 1970s recipiency also rose sharply during the late 1980s and early 1990s but has fallen dramatically since recipiency still low despite recession Flgurc YANF 1 AFDCJTANF Famllles Rncolvllm Incumv Animus m mquot Iquot wyu m mm m lm mm mm mum m 13mm m m m m m v16 m n m m w W m 1m mu m ayIn nm vumbwuh vluu39w rwlm39 nownrwxmwmavameapuhM39amm mmwmufmwamtv lwlwtnsummmwl m va acMhnawawawmum mm nw lmrmn mukummwmmemmmuimmmatm swmvmunmauawpmgs Muxmrmw a mmmnnwumnmvwmxmmmamI slumvmlhmmsuwnmgaln wlmnwmwzrmmwdmu ml m mmmmm wufmlu39mmumrmaxszmlu my w y Figure TANF 2 Mly amll E 5 E a was mu ms ma was was ma mm ms nu sav wlmravmngau Wmvmamuawmnnm m VN wzwwaunmmzw m umhwnpmmwmvwmmwnnmm ammm Kmmmmmmmudm ummmnmm quotmy barmanmmrmmv nmv mdmmnnhmmzmmr nmVn a namzmmmmwammn metamu as rm 9 n39hm gum mm mm sum Differences between AFDC and TANF AFDC TANF Federal funding Unlimited matching Fixed grant State funding Matching required Maintenance of effort requirement Eligibility Single parent families Set by state Income limits Benefit levels Entitlement Work requirement Exemptions from work requirement Work trigger Time limits with children two parent families with unemployed parent Set by state Set by state States required to assist all families eligible under income standards Participation in JOBS program Parents with children under 3 or 1 at state s discretion None None Set by state Set by state No entitlement By 2002 half of state s caseload must be in speci ed work activities None though states may exempt single parents with children under 1 Required after 2 years 5 year limit 20 hardship exemptions Source 1998 Green Book Table 71 Statebystate Welfare Benefits 2000 FOOd pov FOOd pov State TANF Stamps stand State TANF Stamps stand Ala 164 490 42 Mont 469 703 61 Alaska 923 1101 95 Neb 364 630 54 Arizona 347 618 53 Nevada 348 618 53 Ark 204 518 45 N Ham 575 777 67 Calif 626 813 70 N Jer 424 672 58 Colo 357 625 54 N Mex 439 682 59 Conn 636 820 71 N York 577 779 67 Del 338 611 53 N Car 272 565 49 DC 379 640 55 N Dak 457 695 60 Florida 303 587 51 Ohio 373 636 55 Georgia 280 571 49 Okla 292 579 50 Hawaii 712 1061 92 Oregon 460 697 60 Idaho 293 580 50 Penn 421 669 58 Illinois 377 639 55 R I 554 763 66 Indiana 288 576 50 S Car 204 518 45 Iowa 426 673 58 S Dak 430 676 58 Kansas 429 675 58 Tenn 185 504 44 Kent 262 55 8 48 Texas 201 515 45 Louis 190 508 44 Utah 451 690 60 Maine 461 697 60 Ver 708 870 75 Mary 417 667 58 Virginia 354 623 54 Mass 579 780 67 Wash 546 757 65 Mich 459 696 60 W Va 328 604 52 Minn 532 789 68 Wisc 673 846 73 Missi 170 494 43 Wyom 340 613 53 Misso 292 579 50 Source 2000 Green Book Tables77 78 and US Census Bureau f Deficit Reduction Act of 2005 1 The general provisions of TANF were renewed in 2005 through the DRA 2 The primary change from the DRA was to strengthen the work requirements of TANF resets the caseload reduction credit to 2005 rede nes work activities expands work participation calculations to include some adults funded entirely through state funds 4 The Earned Income Credit a refundable tax credit available to people with low to moderate levels of earned income b background 1 rst implemented in 1975 to reduce tax burden associated with social security 2 expanded several times most significantly in 1990 and 1993 now viewed as a major antipoverty program 3 costs increased from 125 billion 62 million families in 1975 to 30 billion 19 million families in 1999 4 some states have similar credits in their tax systems 5 important and oftenoverlooked component of welfare reform one of the kinder provisions C 0 general structure 1 percentage supplement to earnings up to some maximum bene t bene t increases with earnings 2 bene t is xed for some intermediate earnings 3 bene t is phased out over some other range 4 provisions vary across family size 5 limitations on unearned income and other factors C ZL siX schedules as of 2008 three for married three for single for single people 1 one child 7 34 of earnings up to 8550 maX credit 2917 phaseout rate 16 after 15750 2 two children 7 40 of earnings up to 12050 maX credit 4824 phaseout rate 21 after 15750 3 no children 7 765 of earnings up to 5700 maX credit 412 phaseout rate 765 after 7200 e American Recovery and Reinvestment Act of 2009 made temporary changes to the EIC 1 adds new schedules for three or more children 2 extends start of phaseout points 5 Food Stamp Program Supplemental Nutrition Assistance Program a Meanstested program to provide food assistance to needy families Federal government funds bene ts states administer program and share administrative expenses Bene ts based on Thrifty Food Plan households expected to devote 30 percent of their net income to food purchases Eligibility generally depends on 1 Gross income test 130 of poverty threshold 2 Net income test 100 of poverty threshold 3 Assets Benef1ts reduced with income Very large program served 267 million people in 2006 at a cost of 328 billion Trends shown on next page 6 Supplemental Security Income a Cash assistance to needy aged blind and disabled children and adults b Uniform nationwide eligibility criteria c Some states supplement bene ts some states administer their own supplemental progmms d In 2006 served 72 million people including 11 million children total bene ts Were 389 billion C Virtual prices and incomes 1 Consider a general budget constraint Y EI1 N Where EI1 is a nonlinear earnings mction 2 Given this constmint suppose that a person chooses some level of Work HE 3 A U1 6 At this level ofwork we can de ne a virtual or effective wage W E H0 this is simply the slope ofthe budget constraint at H0 We can also de ne a virtual nonlabor income M EH0 N7 WHO Graphically C Y EH N islope W NN N With the virtual wage and virtual nonlabor income we have linearized the budget constraint at H0 a consider an indirect utility function v and a labor supply function h de ned for the general linear budget constraint problem ie the same functions that we have considered previously at H0 the person s indirect utility is V vVV M optimal hours are H0 MW M that is at H0 the solution to the nonlinear budget constraint problem is the same as the linearized one CLO U D Estimation issues for a twopart convex budget set 1 Consider a person who faces a budget constraint in which a N is the nonlabor income b W1 is the wage for hours worked up to H0 and c W2 lt W1 is the wage for hours worked beyond HC 2 Assume that the person has a utility function UC L such that the optimal labor supply in the presence of a linear budget constraint with wage W and nonlabor income M is H h W M 3 Graphically a for rst segment C virtual wage is i W1 and virtual skips W2 nojnllabor 1ncome N Is j k slope Wl b for second I N segment v1rtual wage is W2 and 39 C virtual nonlabor K H K L income is N NW1 W2HC 4 What would happen if we just regressed observed hours on observed virtual wages and nonlabor incomes a note that the person s choice follows 1 segment 1 H hW1N ifhW1N ltHC 2 segment 2 H hW2 N2 ifhW2 N2 gt HC 3 at H0 ifhW1 N 2 Ho and hW2 NJ Ho b consider a simple linear speci cation for h hW N ocBW yN c assume that 0L varies across individuals but that B and y do not d run OLS on observations drawn from segment 1 EHHltHc ZBW1YNEXIHltHC BW1 YN E0l l 0 lt HLBWFYN e Clearly OLS leads to biased results 1 0L is truncated from above 2 the truncation depends on the independent variables 5 general problem with nonlinear budget constraints is that people selfselect onto particular segments of the budget constraint observed virtual wages and nonlabor incomes depend on budget factors but also preference factors 6 instead of OLS could use a technique similar to the endogenous switching regression method either MLE or 2step to estimate this model Moffltt 1986 describes methods for twopart conveX and concave budget constraints E Estimation issues for more compleX budget constraints 1 twopart budget constraint is a restrictive and special case in general posttax posttransfer budget constraints are much more complicated similar set of selection issues arise with more complicated models however the number of segments and kinks makes it impractical to model all of the possible choices standard approach has been to discretize the budget constraint that is to restrict the number of choices of hours eg no work parttime fulltime and consider discrete decisions to participate in programs or takeup particular sets of bene ts with just a few programs or elective taX features the number of choices can still grow quickly researchers typically specify a utility function so approach is structural this is an advantage as reactions to new types of budget constraints can be simulated utility function typically includes heterogeneity in tastes and perhaps program stigma a error terms are sometimes given a simple and easily computed distribution like multivariate independent extreme value distribution b alternatively more compleX distributions can be used however this increases the computational burden of the model see eg Keane amp Mof tt 1998 Lifecycle Labor Supply A Elementary discretetime perfect foresight model 1 initial assumptions a b 0 P D 3quot consider a model with T discrete periods I O 7 there is a maximum amount of time in each period which can be divided between market and nonmarket activities L H L for l O T wages in each period W and the person s initial endowment of wealth A are given exogenously there is perfect foresight in the model the values for all economic variables are known in period 0 there are perfect capital markets the person can borrow and save any amount at the market interest rate there is a constraint on ending wealth at the end of period T the person s wealth must equal zero note could easily extend the model to include bequests 2 utility function a b person has preferences defined over consumption and nonmarket time in all periods such that U UC0CCL0LL person chooses consumption and hours of work in all periods to maximize this utility specif1cation subject to the time constraints described above and the lifetime budget constraint described below 3 lifetime budget constraint a let Zt denote the person s net worth in period I b net worth evolves according to Ztl Zt th VVth PtCt where r denotes the interest rate assumed constant over time and Pt denotes the periodspeci c price of consumption goods c given the initial and terminal wealth conditions the lifetime budget constraint can be expressed T T A 21rt Wth 2lrquot 13 C t0 t0 d a fullincome equivalent can be written T T AZ1r39tWL Z1r t13 Ct WLt t0 t0 T F Z1r t13 Ct WLt t0 4 hours functions a this is a standard consumer choice problem we will obtain demandtype solutions b leth 1 r t c the solutions can be written H HR0WoRTWTRO1RTPTA Ct CIR0W0RTWTRO13RTPTA P these solutions will satisfy standard demand conditions e consider the Slutsky equation 8H BH BR WT A H1 BA T 3H U BR WT T compensated own wage effect should be positive could examine other properties of Slutsky matrix 3quot increase in A income effect which leads to decreases in labor supply in all periods g increase in VK leads to income effect decrease in labor supply which is spread diluted across all periods standard substitution effect change in relative price of consumption and leisure in period lleads to an increase in labor supply in period I intertemporal substitution effects work in period I becomes more valuable leisure in lbecomes more expensive relative to work in other periods substitute away from work in other periods and toward work in period I 5 empirical issues a what happened to nonlabor income Nonlabor income in each period is represented by th which is clearly endogenous Exogenous equivalent to nonlabor income is the initial endowment A b perperiod labor supply and consumption are now functions of current wages and prices but also of wages and prices in all other periods this represents a tremendous data burden c for estimation we need to consider restrictions that reduce the data burden B Timeseparable utility speci cations 1 utility speci cation a let the utility function be given by T T U Z1ptUCtHt EDI UCIHI 10 t0 where DI lp t b can rewrite the maximization problem as a Lagrangian T Max L ZDtUHtCt t0 C0 CTH0HT 2AiRIVHt PtCt t0 f1rstorder conditions are Ht DI 2RVK 0 Ct Dthli ARB 0 T 2 Az01zltWth Bc0 the solutions are mzmgw px I t j QQWPM 9 UI 0 these are referred to as lconstant supply and demand functions or Frischtype functions a note that the labor supply and consumption functions depend on adjusted withinperiod prices and wages and A b everything from other periods is summarized by A if we could obtain an estimate for A or control for A this would greatly simplify the empirical requirements interpretation of wage effects for these functions can decompose effects of a wage change as follows aHt aHt ax aHt am W W W 374 x a evolutionary wage changes represented by second term above this corresponds to a change in the wage profile that leaves A unchanged eg planned wage growth this effect should be positive implies hours pro le and wage pro le should have same general shape similar to ownperiod substitution effect from earlier discussion but holding marginal utility of wealth constant b parametric wage changes this is a shift in the entire wage pro le it produces evolutionary effects as well as wealth effects the net effect is ambiguous c oneperiod parametric wage change this is special type of parametric shift in which the wage change is limited to one period it leads to effects that are similar to a parametric change but with diminished wealth effects similar to income ownperiod and intertemporal substitution effects described earlier because of separability all intertemporal effects operate through A 5 MaCurdy 1981 a utility speci cation CES perperiod utility functions T Ui 212th it t0 where0ltl1 s l andoo2gtl b speci cation for hours function 1 assuming an interior solution the labor supply function is lnH ln7ci lnlt lnp2 ln anlt D l t 2 2 let ln bit n gm 5 m21717 and b 5 ln p r 3 then we can write an empirical counterpart lnH E bt 81an a c estimation procedure 1 estimate b and 5 using xed effects procedure ie differencing or meandifferencing observations for the same individual across years to condition out F advantage can obtain estimates with as few as two longitudinal observations disadvantage this only gives us estimates of the evolutionary effect we cannot examine general effects or make predictions without an estimate of F1 2 approximate Fl by regressing estimated fixed effects on lifecycle variables d data 1 196776 annual labor supply data on 513 prime age white married males from the Panel Study of Income Dynamics 2 men were 2546 in 1967 continuously married to the same spouse and had labor supply data available for all ten years e results 1 evolutionary elasticity 5 10 to 23 2 own period elasticity 10 to 50 3 crossperiod elasticity nearly zero C Extensions in the timeseparable framework 1 Heckman and MaCurdy 1980 a important extensions to MaCurdy 1981 1 consider female labor supply 2 consider attendant problems of zero hours and missing wages b employ xed effects Tobit procedure 1 because of the Tobit procedure is nonlinear the fixed effects cannot be conditioned out of the estimating equation this leads to the incidental parameters problem 2 estimates of the fixed effects m the other coefficients are only consistent as T n 00 instead of N gt 3 some other Monte Carlo work by Heckman suggests that resulting bias is likely to be small c estimation of conditional fixed effects procedure l procedure requires at least one working observation per individual 2 total sample of 672 white women 30 to 65 years 452 23 worked at least once in eight years 3 adopt a simple procedure to account for sample selection bias from everworked criteria 2 Incorporating uncertainty into a family lifecycle model Browning Deaton and Irish 1985 a utility speci cation 1 household chooses leisure for two individuals In and I and consumption qt in period I 2 to maximize T I th 9 21 7qt 2 El Zuk lk 9 12k aqk ktl where E is the expectations operator and the sub period utilities incorporate the effects of discounting 3 subject to a per period budget process Atl Al wlthlt WZcht ptqt 1 and terminal conditions for A0 and AT b let vtAt w Wm 19 be the indirect subperiod utility function at I and let vtAt be the sum of current and expected future utilities as perceived at t given assets A c optimization gives 1 at I T VTAT VTAT7 WIT wlTapT 2 at Zlt T v At maxvtAt wlt W21 pt Etvt1At wlthlt W2th2t ptqt 3 with labor supply solutions h1t f1t39ta wlta wlta pt hlt f2t39t7 wlta wlta pt where it is the expected marginal utility of wealth at I very similar to lconstant demands from MaCurdy but with stochastic marginal wealth controls 4 V estimate model using an approximate technique that essentially comes down to appending an error term to A C Intertemporally nonseparable preferences Hotz Kydland and Sedlacek 1988 1 Theoretical model a indiVidual maximizes E EftBTquotUZT CT where CT is consumption at 17 and Zr is a distributed lag in nonmarket time b numerous studies suggest that preferences for leisure may be intertemporally nonseparable eg current bene ts from previous nonmarket production and experience alternatively could represent habit formation and consumption capital c speci cally assume Zt evolves as Zt It ocat where at lt11 39rat1 c individuals face a perperiod budget constraint At1Atwtlll Ct1rt with terminal conditions on A0 and A T d leads to a dynamic programming problem V A a w maxUz 0 BEtVt1YtAt Wtht Ct 1nat 1 wt1i e can show that the solutions satisfy EUct pyUCt1 o E U2t wUctgt BaUzt11nUzt1 Wr1Uct1l 0 f actual realizations will follow U0 ByrUcU ur1 Uzt wUCt BccUZt1 1 nUZtl w1UCt1 um1 g rational expectations implies Etum O and that these errors are orthogonal to all information available at and up to time I 2 Generalized Method of Moments procedure a parameterize utility speci cation Hotz et al use translog speci cation form sample moments that are the product of the realization errors and the information set variables minimize a weighted quadratic 0N0 WN 0N6 where the optimal most ef cient weighting matriX is chosen according to Hansen 1982 note that Hotz et al base most of their results on the rst optimality condition in consumption structural estimates and tests for nonseparability here are independent of the precise evolution of wages not sensitive to assumptions about exogeneity of wages


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