Underemployment Intensty, ts Cost, and ther Consequences on the Value of Tme. Anl Alpman Pars School of Economcs and Unversty Pars 1 Pantheon-Sorbonne February 16, 2016 Abstract In ths paper, I propose a new method to estmate the cost of unemployment and underemployment by estmatng the value of the shadow prce nduced by a constrant on the market work hours. Ths method, by takng nto account the margnal utltes of the hours allocated to market work and to non-market work (such as home work and lesure), yelds a hghly accurate estmaton of the cost of unemployment. The emprcal analyss s performed on a mcro dataset, whch allows to explore the cost of unemployment as a functon of varous ndvdual characterstcs. The results show that the cost of unemployment represents roughly 65% of the real wage rate and that hgh underemployment levels are at least as costly as unemployment. Keywords: Labor Supply, Ratonng, Shadow Prce, Tme Allocaton, df d d d Underemployment, Unemployment, Unemployment Benefts. JEL: D12, J22, J65 1 Introducton An accurate estmaton of the cost of unemployment s an essental nformaton to determne the unemployment benefts that may be receved by unemployed ndvduals and to evaluate the opportunty cost of unemployment polces (Feldsten, 1
1978). However, unemployed ndvduals can allocate ther foregone market work hours to other actvtes such as home work or lesure. For nstance, Aguar et al. (2013) show that 50% and 30% of the forgone market work hours durng the Great Recesson were allocated to lesure and home work, respectvely, whle job search and other ncome generatng actvtes ncreased only by 10%. Therefore, the forgone earnngs or the potental earnngs of the unemployed overestmate the cost of unemployment by omttng the value, for example, of home work or lesure. The case of the unemployed can be generalzed to ndvduals that cannot work as much as they desre: constrants on the market work hours gve rse to a shadow prce as soon as the desred work tme s above the tme mposed by the constrant. These constrants may be specfc for each ndvdual and they may arse for varous reasons (e.g., msmatches and dsequlbrums on the labor market wll constran the market work tme of some ndvduals). In ths paper, I estmate the shadow prce of unemployment and underemployment: n a tme allocaton model where the market work tme may be ratoned (Prowse, 2009), the Lagrange multpler assocated wth the constrant on the market work tme gves the shadow prce of unemployment (f the constrant mples for an ndvdual that the market work tme must be zero) or underemployment (f the constrant mples a market work tme below the consumer s optmal choce). 1 In theory, ths approach yelds the exact cost of unemployment and underemployment; n practce however, t excludes the psychologcal effect of unemployment (Hellwell and Huang, 2014). I perform the estmates on the mcro dataset used by Alpman and Gardes (2015) who have matched, usng Rubn s statstcal matchng procedure (Rubn, 1986), the Amercan Tme Use Survey (ATUS) wth the Consumer Expendture (CE) survey over 2004 and 2012. The man results that I obtan ndcate that the shadow prce of unemployment s on average 63.69% of the real wage rate; the 1 In ths paper underemployment refers only to the stuaton where an ndvdual has to work less than her desred work tme. 2
average values of the shadow prces of unemployment and underemployment are hghly correlated to the unemployment rate and thus they have ncreased durng the Great Recesson; the shadow prces of unemployment and underemployment dffer consderably wth ncome, the number of chldren, the martal status, and other ndvdual characterstcs. These results may be useful for polcy makers who may wsh, fo example, to gve hgher unemployment benefts as the number of chldren n the famly ncreases. Fnally, the model used n ths paper shows that the shadow prce of tme (.e., the opportunty cost of tme) s negatvely related to the shadow prces of unemployment and underemployment. Therefore, the results obtaned n ths paper allows to explan the decrease of the opportunty cost of tme durng the Great Recesson despte the ncrease of the real wage rate. The theoretcal model used to estmate the shadow prces of unemployment and underemployment s presented n Secton 2. The estmaton procedure s dscussed n Secton 3. The dataset s presented n Secton 4. The results are dscussed n Secton 5. 2 The Cost of Unemployment and Underemployment As n the tme allocaton model (Becker, 1965), I assume that the consumer s utlty, denoted U, s a functon of the quanttes of n actvtes, denoted Z 1, Z 2,..., Z n : U = u(z 1, Z 2,..., Z n ) (1) It s assumed that dfferent actvtes are mperfect substtutes and the margnal utlty of each actvty s decreasng but always postve. The consumer produces each actvty by combnng her tme wth market goods: Z = z(x, t ) (2) where x s the quantty of market goods and t s the amount of the consumer s tme used n the producton of actvty. I assume that nputs are mperfect sub- 3
sttutes and that the productvty of each nput s decreasng but always postve. In the tme allocaton model the consumer faces an ncome constrant (gven by equaton [3]) and a tme constrant (gven by equaton [4]): n p x = wt w + V (3) t w + n t = T (4) where p s the prce of the market good, w s the wage rate, t w s the tme allocated to the market work, V s other ncome, and T s total tme. Wth the assumptons made so far and wthout further assumptons, the shadow prce of tme would be equal to the consumer s wage rate. Ths equalty occurs because t has been mplctly assumed, among others, that the consumer can allocate any amount of tme to the labor market provded that t w T. I drop ths assumpton by ntroducng a constrant on the market wok tme: t w R (5) where R T. A constrant exsts when the market work tme of an ndvdual cannot ncrease above R even f that ndvdual s wllng to work more at the prevalng wage rate. For nstance, R = 0 for an ndvdual who s nvoluntarly unemployed, R = 3 per day for an ndvdual who desres to work full-tme but fnds only a part-tme contract where he works 3 hours per day, R equals to the legal lmt of market work hours f such a lmt exsts. These examples show that R may be specfc for each ndvdual and t s the greatest constrant (.e., the smallest R) to whch the ndvdual s subjected to (e.g., f an ndvdual s nvoluntarly unemployed n a country where a legal lmt of market work hours exsts, then that ndvdual s subject to t w = 0). The examples above show that ndvduals market work tmes may be restrcted for varous reasons regardless of ther actual market work tmes. As t wll be shown below, t s the dfference between the desred market work tme and 4
the actual market work tme that matters for the cost nduced by a constrant on the market work hours (f the actual market work tme s lower than the desred market work tme, the ndvdual s assumed to be subject to a market work tme constrant). Therefore, I defne the underemployment ntensty, denoted I, as I = td w t a w t d w (6) where t d w s the desred market work tme and t a w s the actual market work tme. The term underemployment n ths paper refers only to the dfference between t d w and t a w. For example, f an ndvdual desres to work 50 hours but the legal lmt on workng hours s 40, the underemployment ntensty of that ndvdual would be I = 20%. When t d w > 0 and t a w = 0, the ndvdual s unemployed and the underemployment ntensty s 100%. Note that the underemployment ntensty may be negatve f the ndvdual work more than her desred work tme. Combnng equatons (1) and (2) allows to express the utlty as a functon of tme and market goods, and thus the consumer problem consst of maxmzng that utlty functon subject to equatons (3) to (5). The Lagrangan s gven by L = u ( z 1 (x 1, t 1 ),..., z n (x n, t n ) ) n + λ(wt w + V p x ) + µ(t t w n t ) + φ u (R t w ) (7) The soluton of ths consumer problem s found usng the Kuhn-Tucker theorem (Prowse, 2009). The partal dervatves of L wth respect to the choce varables are L t = u z z t µ = 0 u z z t = µ L x = u z z x λp = 0 ( u z z x ) /p = λ (8) L t w = λw µ φ u = 0 ω µ λ = w φu λ 5
and the complementary slackness condton s: φ u (R t w ) = 0 φ u = 0 or R t w = 0 (9) In equaton (8), φ u s the shadow prce of strengthenng the constrant on the market work tme and ω denotes the shadow prce of tme. Suppose that φ u = 0: f the optmal market work hours s lower than R, then the assumpton that φ u = 0 s true and ( u/ z )( z / t )(1/w) = ( u/ z )( z / x )(1/p ). Otherwse, that s, f the optmal market work hours s hgher than R under the assumpton that φ u = 0, then ths assumpton s false and t must be concluded that φ u > 0 and R t w = 0. A constrant preventng the ndvdual to work as much as she desres mples that the ndvdual has to allocate more tme than desred to non-market work actvtes ([ u/ z ][ z / t ] decreases) and the consumer s not able to buy as much as desred market goods ([ u/ z ][ z / x ] ncreases). As a result, the greater s the dfference between the desred market work tme and the value of the constrant, the greater s φ u = λw µ: a postve relaton exsts between the underemployment ntensty and φ u. The shadow prce of strengthenng the constrant on the market work tme n monetary unts s gven by φ m φ u λ = w ( / u u z z t z z x p ) (10) Equaton (10) shows that φ m decreases as the margnal utlty of tme nput ncreases and φ m ncreases wth the wage rate and wth the margnal utlty of market goods dvded by ther prces. As prevously dscussed, φ m ncreases as the value of R decreases (.e., the market work tme gets more restrcted) because a lower t w and a hgher t lowers the margnal productvty of t and ncreases the margnal productvty of x (under the assumpton that t w = R and φ m > 0). Equatons (8) show that the shadow prce of tme s gven by ω = w φ m (11) 6
When the market work tme s constraned, equaton (11) shows that the shadow prce of tme may dffer from the wage rate: ω decreases as φ m ncreases. Durng the Great Recesson, the real wage rate has ncreased whle Alpman and Gardes (2015) have found that the opportunty cost of tme has decreased. The results of Secton 4 show that the ncrease of the average value of φ m durng the Great Recesson has been mportant enough to nduce a decrease of the average value of the shadow prce of tme. 3 Emprcal Specfcaton Assumng that the utlty and the producton functons take the Cobb-Douglas form (.e., U = Π n Zγ utlty functon as and Z = m α t β ), Gardes (2014) proposes to wrte the U = m α γ t β γ (12) where m = Π m (α γ )/( α γ ) and t = Π t (β γ )/( β γ ) are the the geometrc weghted means of monetary and tme nputs, respectvely, and m = p x. 2 When equaton (12) s maxmzed subject to equatons (3), (4), and (5), the shadow prce of the constrant on market work hours s gven by 3 φ m = w µ λ (13) where µ/λ = ω and t s gven by: ω = µ λ = U/ t U/ m = β m α t (14) Snce each producton functon requres two nputs and takes the Cobb-Douglas 2 The utlty s a functon of expendtures rather than quanttes because most datasets nform only the expendtures. To neutralze the effect of prces, expendtures are dvded by a prce ndex and the estmatons are performed on repeated cross-sectons. 3 The Lagrangan s now gven by L = (Π m α γ α γ ) α γ (Π t β γ β γ ) β γ + λ(wt w n m ) + µ(t t w n t ) + φ u(l t w) 7
form, β = 1 α. Thus, α and β can be obtaned from equaton (14): α = m ωt + m and β = ωt ωt + m (15) The optmal allocaton of the same factor among varous actvtes mples that µ = U/ t = U/ t j : β γ t j = β j γ j t (16) Combnng equatons (15) and (16) yelds the followng system: m 2 γ 1 = m 1 γ 2 + ωγ 2 t 1 ωγ 1 t 2 m 3 γ 1 = m 1 γ 3 + ωγ 3 t 1 ωγ 1 t 3... m n γ 1 = m 1 γ n + ωγ n t 1 ωγ 1 t n (17) By fxng γ 1 as the full-ncome share of one of the actvtes such as food, the system of equatons (17) yelds the estmates ˆγ 1, ˆγ 2,..., ˆγ n and ˆω (Gardes, 2014). Usng ˆω, equaton (15) gves ˆα 1, ˆα 2,..., ˆα n, and ˆβ 1, ˆβ 2,..., ˆβ n. When ω s estmated by equaton (14) usng ˆα and ˆβ, t wll be, by constructon, the same for each ndvdual and equal to ˆω (.e., ω = ( ˆβ m )/(ˆα t ) = ˆω ˆω 1 ). Therefore, the shadow prce of the constrant on the market work hours can be estmated as ˆφ m1 = w ˆω 1 (18) Whle ω n equaton (18) s a pont estmate that has the same value for each ndvdual, t s possble to dfferentate t across observatons by substtutng U/ t wth U/ ˆt and U/ m wth U/ ˆm where ˆt = Π t ( ˆβ ˆγ )/( ˆβˆγ ) and ˆm = Π m (ˆα ˆγ )/( ˆα ˆγ ). The shadow prce of tme thus wrtes 4 ˆφ m2 = w ˆω 2 = w U/ t U/ m = w ( ˆβˆγ ) ˆm ( (19) ˆα ˆγ )ˆt 4 Equaton (19) contans the expressons ˆβˆγ and ˆα ˆγ because ˆm and ˆt are the geometrc weghted means of the monetary and the tme nputs used n all the actvtes. 8
In equaton (19), ˆγ, ˆα, and ˆβ for all dffer between the ndvduals of the dataset. It s possble to explore φ m by assumng that ˆγ and the technology avalable to the consumers (.e., ˆα and ˆβ ) are dentcal across observatons and equal to those of the average ndvdual: ˆφ m3 = w ˆω 3 = w U/ t U/ m = w ( ˆβˆγ ) ˆm ( (20) ˆα ˆγ )ˆt where ˆβˆγ and ˆα ˆγ are the averages of ˆβˆγ and ˆα ˆγ, respectvely, over the entre dataset. ˆφm1, ˆφ m2, and ˆφ m3 provdes three alternatve measures of φ m that can be used for robustness checks. 4 The Data The estmaton procedure descrbed above requres data on tme and ncome allocatons. The Amercan Tme Use Survey (ATUS) reports the tme allocated to varous actvtes that the respondents have carred out durng 24 hours, startng at 4 a.m. the day before the ntervew and endng at 4 a.m. the day of the ntervew. The Consumer Expendture (CE) survey nforms about monthly expendtures, ncome, and households characterstcs. Let X denote ndvdual characterstcs observed n the ATUS and the CE survey. Thus, the ATUS nforms (T, X) and the CE survey nforms (M, X), where T = {t 1,..., t n } and M = {m 1,..., m n }. Wth usual matchng procedure, the mssng values of T are mputed accordng to the relaton of T wth X, and the mssng values of M are mputed accordng to the relaton of M wth X. Therefore, the same T values, for example, would be mputed to unts wth dfferent M and dentcal X when n fact the mputed T values should dffer wth M for a gven X f the partal correlaton between T and M gven X s dfferent than zero. When the partal correlaton value between T and M gven X s mplctely and wrongly assumed to be zero, an emprcal applcaton of the home producton 9
Table 1: Descrptve Statstcs (Alpman and Gardes, 2015) Averages Lesure Housng Food Transportaton Other γ 0.223 0.239 0.121 0.244 0.243 α 0.397 0.724 0.551 0.656 0.600 β 0.603 0.276 0.449 0.344 0.400 m 147.878 243.670 104.560 275.187 251.604 t 30.767 13.722 14.395 9.742 19.259 model lose ts nterest and the resultng coeffcents are hghly based (Alpman, 2015). To overcome ths ssue Alpman and Gardes (2015) have matched the ATUS and the CE survey usng the statstcal matchng procedure proposed by Rubn (1986). The procedure of Rubn predcts T as a functon of X and M, and M as a functon of T and X by assumng a partal correlaton value between T and M gven X. Alpman and Gardes (2015) have calbrated the partal correlaton value as 0.2. Alpman (2015) shows that when the correct partal correlaton value s assumed, the procedure of Rubn (1986) allows to recover accurately the dstrbutons of T and M and the statstcal analyss produces unbased regresson coeffcents. To estmate the shadow prce of unemployment and underemployment, I extract w, γ, α, β, m, t, m,and t from the dataset used by Alpman and Gardes (2015). Table 1 presents the descrptve statstcs of these varables. 5 Results Table 2 presents the yearly averages of ˆφ m1, ˆφ m2, ˆφ m3, ˆω 1, ˆω 2, and ˆω 3 per workng hours estmated by equatons (18), (19), and (20). All shadow prces are n dollars. The costs of unemployment n Table 2 are the yearly averages of ˆφ m1, ˆφ m2, and ˆφ m3 for ndvduals that declared themselves as unemployed. The results n Table 2 show that ˆφ m1, ˆφ m2, and ˆφ m3 are hghly correlated: wth ρ denotng the correlaton coeffcent, ρ( ˆφ m1, ˆφ m2 ) = 0.97, ρ( ˆφ m1, ˆφ m3 ) = 0.81, ρ( ˆφ m2, ˆφ m3 ) = 0.81. When the observatons n the dataset are restrcted to ndvduals workng 50 hours or less, the average values of ˆφ m1, ˆφ m2, and ˆφ m3 are much more smlar to each other: ˆφm1 = 10.2, ˆφ m2 = 9.4, and ˆφ m3 = 10.2. 10
φ m Over the Years 2004-2012 Table 2 show that ˆφ m1, ˆφ m2, and ˆφ m3 are closely related to the busness cycle: durng the sustaned growth perod (2004-2006), the estmates of φ m are decreasng. As the economc actvty slows down n 2007 (the GDP per capta growth rate that year was only 0.8%), the estmates of φ m ncrease slghtly. Wth an average GDP per capta growth rate of 1.2% n 2008, the estmates of φ m contnue to ncrease (except a small decrease of 2.2% for ˆφ m1 ) and all three estmates reach ther heghest values durng the year of the Great Recesson (2009). Durng the recovery years (2010-2012) the estmates of φ m are lower than ther 2009 values but they reman hgher than ther values durng the sustaned growth perod. When the busness cycle s observed through the unemployment rate, the relatons between the former and ˆφ m1, ˆφm2, and ˆφ m3 become even more obvous: the correlaton coeffcent between the yearly averages of the estmates of φ m and the unemployment rate are ρ( ˆφ m1, u) = 0.80, ρ( ˆφ m2, u) = 0.81, and ρ( ˆφ m3, u) = 0.76. These results match the theoretcal predctons: durng recesson the yearly average value of φ m ncreases because the market work hours are more ratoned (.e., R decreases) for a greater number of ndvduals. Table 2: Estmated Shadow Prce of Unemployment and Tme 2004 2005 2006 2007 2008 2009 2010 2011 2012 ˆφ m1 9.96 9.99 8.93 10.48 10.25 11.45 11.30 10.49 10.95 ˆφ m2 8.94 8.91 7.96 9.65 9.68 10.64 10.34 9.82 10.23 ˆφ m3 7.84 7.34 6.49 7.34 7.90 8.48 8.11 7.79 8.15 Cost of unemployment (wth ˆφ m1 ) 10.04 9.80 9.39 10.92 10.20 11.16 11.38 10.21 10.94 Cost of unemployment (wth ˆφ m2 ) 9.12 8.84 8.50 10.25 9.85 10.45 10.49 9.67 10.67 Cost of unemployment (wth ˆφ m3 ) 11.47 10.66 10.74 11.21 11.26 11.55 11.71 10.79 11.84 ˆω 1 6.06 5.69 6.23 4.86 5.27 4.77 4.86 4.87 4.92 ˆω 2 7.36 7.01 7.49 5.89 6.13 5.70 5.98 5.75 5.83 ˆω 3 8.46 8.59 8.96 8.20 7.91 7.85 8.22 7.78 7.91 unemployment rate 5.5 5.1 4.6 4.6 5.8 9.3 9.6 8.9 8.1 Notes: The unts of the values n the Table 2 are per workng hours 11
0 5 12.5 grp2!m3<10% of the!m3>90% of the wage rate wage rate [51-56[ 20 2,1505376344086 7.50 [56-61[ 14,7272727272727 5,37634408602151 61 and more 14,5454545454545 20,4301075268817 ]0,6[ 0 [6,11[ [11,16[ [16-21[ [21-26[ [26-31[ [31-36[ [36-41[ [41-46[ [46-51[ [51-56[ [56-61[ 61 and more 0 6.25 10.00 8.50 7.00 5.50 4.00 11.6 Note that the costs of unemployment and the unemployment rate are also hghly ph_1 ph_2 ph_3 10.8 correlated: for nstance, the correlaton coeffcent between the cost of unemploy- 2004 9.748794 8.699058 10.30316 ment (wth ˆφ 2006 9.04851 8.116326 9.495161 2007 10.59346 m1 9.839089 ) and10.06353 the unemployment 10 rate s 0.76. φ m Cost of underemployment wthout ndvduals workng more than 50 hours 11.2 2005 9.69986 8.637979 9.806407 10.4 2008 9.930046 9.511211 10.30465 9.6 2009 11.28378 10.52826 10.93355 2010 11.1758 10.12171 10.65364 9.2 2011 10.01762 9.349038 9.844046 2012 10.85138 10.29292 10.86012 8.8 φ m as a Functon of the Underemployment Intensty 5.00 2004 2005 2006 2007 2008 2009 2010 2011 12 8.4 8 2004 2005 2006 2007 2008 2009 2010 2011 2012 As mentoned n Secton 2, φ m s postve as soon as an ndvdual s desred market work tme s above the tme mposed by the constrant. Furthermore, φ m should ncrease, Cost of unemployment everythng (ndvduals wth market else work tme=0 equal, wth the underemployment ntensty. year ph_22 ph_4 ph_m unemployment corr ph22 u corr ph4 u corr hm u 2004 10.04156 9.119686 11.4693 5.50 0.76 0.74 0.54 10.21 11 The2005underemployment 9.801778 8.844592 10.66208ntensty 5.10 s unobserved snce 9.77 the desred market work 2006 9.392154 8.504533 10.73655 4.60 9.54 2007 10.917 10.24983 11.20659 5.60 10.79 9 2008 10.2011 9.846041 11.2636 5.80 10.44 tme s not nformed n the dataset. Nevertheless, t can be 8assumed that the 2009 11.1642 10.44521 11.55456 9.30 11.05 2010 11.37804 10.49496 11.70821 9.60 11.19 2011 10.20889 9.670876 10.7897 8.90 10.22 underemployment ntensty decreases wth the market work tme. 6 Moreover, an 2012 10.94454 10.67015 11.83586 8.10 11.15 ntensty of 1 and 0 wll be n the neghborhoods of φ m 4 w and φ m 0, unemployment repsectvely. The left graph n Fgure 1 shows ˆφ m1, ˆφ m2, and ˆφ m3 as a functon of the market work tme whereas the rght graph shows, as a functon of the market work tme, the proporton of ndvduals for whom ˆφ m > w 10%w (.e., φ m w) and ˆφ m < w 90%w (.e., φ m 0). Accordng to the rght graph n Fgure 1, 40% of 9 9.75 10.5 11.25 12 the ndvduals for whom ˆφ m < w 90%w work zero hours. In addton, ˆφ m1, ˆφ m2, and ˆφ m3 have a decreasng trend for employed ndvduals workng less than 45 12 10 7 5 ph_22 ph_4 ph_m unemployment 2004 2005 2006 2007 2008 2009 2010 2011 2012 16.00!m1!m2!m3 40!m w!m 0 30 12.00 30 22.5 8.00 20 15 4.00 10 7.5 0.00 0 ]0,6[ [6,11[ [11,16[ [16-21[ [21-26[ [26-31[ [31-36[ [36-41[ [41-46[ [46-51[ [51-56[ [56-61[ 61 and more ]0,6[ [6,11[ [11,16[ [16-21[ [21-26[ [26-31[ [31-36[ [36-41[ [41-46[ [46-51[ [51-56[ [56-61[ 61 and more 0 Fgure 1: φ m and Underemployment Intensty 12 2
hours on the labor market. As expected by the predctons of Secton 2, these results suggest that the underemployment ntensty s the hghest for those who cannot work on the labor market and t decreases wth market work hours. However, the left graph shows that above 45 hours, ˆφ m1 and ˆφ m2 ncrease wth market work hours whle ˆφ m3 contnues to decrease and t approaches 0 around 61 hours or more. Accordng to these results, the ncreasng trend of ˆφ m1 and ˆφ m2 can be explaned by partcular home producton technologes or partcular preferences (e.g., less dsutlty from workng on the labor market) specfc to ndvduals workng more than 45 hours. Note that the rght graph also shows a small ncrease n the proporton of ndvduals for whom φ m w when the market work tme s above 45 hours. Fnally, the shadow prce faced by employed ndvduals workng less than 16 hours appears to be hgher than the shadow prce of unemployment (left graph n Fgure 1). Smlarly, there s an mportant fracton of unemployed ndvduals for whom φ m 0. These result may be explaned by the unemployment benefts receved and the lower taxes pad (Feldsten, 1978) by unemployed ndvduals, whch compensate partally the cost of unemployment. φ m as a Functon of Income Fgure 2 shows a postve and almost lnear relaton between ˆφ m1, ˆφ m2, and ˆφ m3 and the ncome, whch s proxed by the total expendtures. Ths relaton was theoretcally predcted by equaton (10): frst, the loss of ncome nduced by a margnal ncrease of the market work tme constrant ncreases wth ncome, everythng else equal. Second, f two ndvduals face the same market work tme constrant, the underemployment ntensty of the ndvdual that has a hgher wage rate wll be hgher than the other, everythng else constant, because the hgher wage rate ncreases the desred market work tme (assumng that the supply curve s not backward sloppng). The relaton between ncome and φ m s not verfed for the hghest ncome category because the data show that ndvduals belongng to ths category do 13
not perceve the essental part of ther ncome from the labor market. Therefore ths observaton s consstent wth the theory. φ m as a Functon of Age Fgure 3 plots φ 1, φ 2, and φ 3 over the lfe cycle. I dentfy 5 groups: ndvduals n group 1 (18-21 years old) have a very low and stable φ m, roughly 1.75$. Ths group s lkely to contan an mportant proporton of students lvng n ther parents house and gettng fnancal support from them. Moreover, only a small fracton of the ndvduals n group 1 have chldren and mortgages. As a result, the expendtures and the desred work tme n ths group should be relatvely lower than the others, explanng thereby the low values of φ m. For ndvduals n group 2 (22-30 years old), φ m s ncreasng wth age and reaches on average 9.75$ when the age reaches 30 years old. Ths group can be seen as a transton group from the frst one to the thrd. The ncrease of expendtures and the labor supply leads φ m to ncrease gradually. Group 3 (31-56 years old) covers roughly 25 years (.e., most of the professonal lfe). φ m s stable and hgh (around 10.75$). Wthn ths range, the famly 40.00!m1!m2!m3 16.00!m1!m2 30.00 12.00 20.00 8.00 10.00 4.00 0.00 [0,100[ [100,200[ [200,300[ [300,400[ [400,500[ [500,600[ [600,700[ [700,800[ [800,900[ [900,1000[ [1000,1100[ [1100,1200[ [1200,1300[ [1300,1400[ [1400,1500[ [1500,1600[ [1600,1700[ [1700,1800[ [1800,1900[ [1900,2000[ 2000 and more 0.00 0 ]0,6[ [6,11[ [11,16[ [16-21[ [21-26[ [26-31[ [31-36 14.00!m1!m2!m3 Fgure 2: φ m vs Income!m1!m2 14 12.00 10.50 9.00 6.00 7.00 3.00
expands (not only the number of chldren ncreases but also the expenses related to them) and mortages as well as the savngs for retrment ncrease. In addton, t s wthn these ages that ndvduals advance n ther carrear so that gettng unemployed can have long-term consequences by mpedng the potental professonal achvement. Thus,!m1 the cost!m2 of gettng unemployed!m3 s hgh. 40.00 Group 4 (56-62 years old) are characterzed by a slghtly lower φ m than group 3. Whle most of the dscusson made for group 3 s relevant for group 4, the 30.00 greater proporton of ndvduals n ths group that have a stable job wth a hgher wage rate (relatve 20.00 to group 3) may decrease the desred market work tme, 8.00 whch would decrease φ m. 10.00 4.00 In the fnal group (63-65 years old) φ m s decreasng between 61 and 65 snce ths group s already near retrment (long term unemployment s less costly), wth 0.00 [0,100[ [100,200[ [200,300[ [300,400[ [400,500[ [500,600[ [600,700[ [700,800[ less expenses related to ther chldren, and wth lttle more to acheve professonally. As a result, the desred market work and φ m should decrease. [800,900[ [900,1000[ [1000,1100[ [1100,1200[ [1200,1300[ [1300,1400[ [1400,1500[ [1500,1600[ [1600,1700[ [1700,1800[ [1800,1900[ [1900,2000[ 2000 and more 16.00 12.00 0.00 0 ]0,6[ [6,11[!m1!m2 [11,16[ [16-21[ [21-26[ [26-31[ [31-36[ [36!m1!m2!m3 14.00!m1!m2 12.00 10.50 9.00 6.00 7.00 3.00 3.50 0.00 2004 2005 2006 2007 2008 0.00 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 10.00!m3 ω3 Fgure 3: φ m vs Age 8.75 7.50 15 6.25 5.00 2004 2005 2006 2007 2008 2009 2010 2011 2012
φ m as a Functon of Other Indvdual Characterstcs Table 3 presents the regresson coeffcents of varous ndvdual characterstcs on ˆφ m1, ˆφ m2, and ˆφ m3. The regresson coeffcents of the market work tme, total expendtures, and age yeld the same result as dscussed above. Among the most nterestng results, Table 3 reveals that φ m ncreases wth the number of chldren (except the regresson coeffcent assocated wth 6 chldren whch s not sgnfcantly dfferent than 0 chld). As mentoned above, each addtonal chld necesstates addtonal expendtures that can be bought only wth a greater market work tme. Thus, an ndentcal market work tme constrant on two households would be more costly to the household wth the hgher number of chldren. On the other hand, φ m s lower for marred ndvduals than for sngles. The exstence of equvalence scales and publc goods wthn the household may decrease the households overall desred work tme. Everythng else constant, the underemployment ntensty, and thus φ m, of a household wth two adults would be lower than that of a household wth a sngle adult for an dentcal market work tme constrant. Fnally, φ m s ncreasng wth the educaton level: the accumulaton of human captal not only necesstates mportant expendtures but t may also lead to hgher expendtures under the lfe-cycle hypothess because the expected ncome ncreases wth the educaton level. A hgher expected ncome could lead to a hgher labor supply. Thus, as n the prevous cases, for a gven market work tme constrant, the underemployment ntensty and φ m would be hgher for ndvduals wth hgher educaton levels, everythng else constant. 6 Concluson In ths paper I estmate the cost of the constrants that prevent ndvduals to work as much as they desre. When ndvduals cannot work at all (.e., when ndvduals are nvoluntarly unemployed), the estmatons yeld the cost of unem- 16
ployment, whch s on average 10.05$ per workng hours or 63.69% of the average real wage rate. The shadow prce of unemployment s slghtly lower than the shadow prce of underemployment esmated for ndvduals who work less than 6 hours per week: the estmates show that ther shadow prce s roughly 14$, whch corresponds approxmately to 74% of ther real wage rate. The dfference s most lkely nduced by the unemployment benefts that the unemployed receve and, wthout the unemployment benefts, the shadow prce of unemployment may be close to 74%. If polcy makers whsh to fully compensate the cost of unemployment, they do not have to compensate fully the forgone wage rate of the unemployed snce the dfference between the wage rate and the cost of unemployment s compensated by the value of the tme allocated to non-market work actvtes such as home work or lesure. However, polcy makers should be aware that a full compensaton of the cost of unemployment (whch s lower than the wage rate) would remove the economc ncentves to work. Based on the results of these paper, polcy makers may consder to dfferentate the unemployment benefts accordng to some ndvdual characterstcs, especally regardng the number of chldren and the martal status. On the other hand, the estmates show that between 2% and 7% of the populaton have a negatve shadow prce of unemployment and underemployment. The work tme of these ndvduals are normally dstrbuted around 50 hours, whch s above the average work tme. These ndvduals would be better off by workng less. Fnally, ths paper shows that the opportunty cost of tme dffers from the wage rate as soon as the cost of underemployment s postve. Gven the relatve stablty of the real wage rate over the years 2004-2012, the results n ths paper suggest that the evoluton of the opportunty cost of tme s more related to the shadow prce of unemployment and underemployment than to the wage rate. As a result, the evoluton of the opportunty cost of tme s also more related to the unemployment rate than to the evoluton of the wage rate. 17
Work n progress: - Calculatng the unemployment benefts receved by the unemployed and substractng them from the estmates. - Predctng the value of the cost of unemployment and underemployment from Table 3. - A more torough analyss of the relaton between the underemployment ntensty and ts cost. - Recoverng the desred market work tme from the estmated shadow prce - Comparng my results wth the lterature 18
Table 3: ˆφm1, ˆφ m2, and ˆφ m3 Regressed on Indvdual Characterstcs Varable ˆφm1 ˆφm2 ˆφm3 Market work tme -0.038*** -0.036*** -0.014 (0.008) (0.009) (0.015) (Market work tme) 2 0.001*** 0.001*** -0.002*** (0.000) (0.000) (0.000) Total Expendtures 0.018*** 0.018*** 0.018*** (0.001) (0.001) (0.001) (Total Expendtures) 2-0.000*** -0.000*** -0.000*** (0.000) (0.000) (0.000) Male 2.716*** 3.084*** 3.295*** (0.148) (0.158) (0.185) Age 0.347*** 0.404*** 0.310*** (0.040) (0.044) (0.052) (Age) 2-0.003*** -0.004*** -0.003*** (0.000) (0.001) (0.001) Marred -0.278* -0.450*** -1.519*** (0.148) (0.160) (0.186) Black 0.363 1.284*** 3.812*** (0.239) (0.253) (0.301) Number of chldren: 0 Reference level Number of chldren: 1 0.956*** 1.667*** 3.450*** (0.181) (0.195) (0.227) Number of chldren: 2 1.501*** 2.090*** 3.432*** (0.197) (0.214) (0.253) Number of chldren: 3 2.441*** 2.890*** 5.515*** (0.284) (0.306) (0.356) Number of chldren: 4 2.473*** 3.180*** 6.929*** (0.572) (0.612) (0.639) Number of chldren: 5 7.523*** 8.638*** 12.766*** (2.725) (2.845) (2.824) Number of chldren: 6-0.673-0.278 3.839 (3.225) (3.417) (3.302) Educaton level: 1 Reference level Educaton level: 2 1.860*** 1.531*** 0.114 (0.179) (0.223) (0.267) Educaton level: 3 3.728*** 3.086*** 0.159 (0.182) (0.223) (0.271) Educaton level: 4 6.144*** 5.419*** 1.650*** (0.251) (0.288) (0.338) Educaton level: 5 10.865*** 10.335*** 5.772*** (0.622) (0.643) (0.717) R 2 0.583 0.540 0.498 No. of observatons 9084 9084 9084 Notes: Robust standard errors are n brackets.,, ndcate sgnfcance dfferent than zero at 90%, 95%, and 99% confdence, respectvely. 19
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