On the Welfare Effects of a Wage Floor - Application to Minimum

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On the Welfare Effects of a Wage Floor - Application to Minimum

    USBIG Discussion Paper No. 77, February 2004

    Work in progress, do not cite or quote without author’s permission

    On the Welfare Effects of a Wage Floor

     in a Two-Sector Labor Market with Impoverished Workers

    *Dr. David L. Wetzell


    That a wage floor, such as the minimum wage, lowers employment has been extensively empirically confirmed. However, among poorer workers the relatively inelastic nature of labor demand may permit an increase in the minimum wage to have a positive wage spill-over into the uncovered sector if covered sector employees' need to "moon-light" in the uncovered sector is reduced. The model indicates that wage floors may be a more appropriate instrument for reducing poverty when workers are impoverished and also affirms the need to empirically test for their impact on the income and hours worked by directly and indirectly affected workers.

     * Dr. David Wetzell is at Universidad de Las Americas-Puebla. He can be contacted through email address at and his address is Universidad de las Americas, Puebla; Departamento de Economia; Ex-hacienda de Santa Catarina Mertir; Cholula, Puebla, 72820.


    The following is an “ideal-type” situation where a wage floor, such as a minimum

    wage or a living wage, may be justified despite how it causes a reduction in the level of

    1employment in the covered sector. As with Mincer(1976), the model considers that

    displaced workers will be able to find work in the uncovered sector. Unlike Mincer (1976), the model abstracts from unemployment or search for covered-sector jobs and, instead, allows employed workers to supplement their income by “moon-lighting” in the

    2uncovered or informal sector. Also, unlike Mincer (1976), the model derives labor-

    supply directly from utility-maximizing labor supply decision. For simplicity, it is assumed that the hours of workers employed in the covered sector are determined exogenously by work-week restrictions. As such, the relevant choice is the time supplied to the uncovered sector. It is also assumed that workers are poor and that, as shown in

    3Wetzell(2001), “leisure” requires money or purchasable inputs.

    Here, the difference between consumption and leisure is that leisure consists of more time-intensive activities and consumption consists of more money-intensive activities. This is like the difference between reading a book and going to the movies. This alters the model by adding a pecuniary cost of leisure, p, to the budget constraint so L

     1 It is taken as given that an increase in the wage floor will induce a modest decline in the number of jobs as shown by an extensive empirical literature. 2 If worker is interpreted as a household then this could be interpreted as how other members of the household besides the head working for money in the uncovered sector. 3 For simplicity, for workers to be “poor” is taken to mean that they have no other source of income besides

    working. This assumption could be relaxed some without changing the results.

    If utility is defined over consumption and leisure then consumption can also have a time-cost. However, inasmuch as the time-cost of consumption complicates the analysis and tends to matter more for higher wage levels, the time-cost of consumption is omitted from the model. It is important to emphasize that from a time-allocation perspective, leisure is time-intensive consumption and consumption is money-intensive consumption.

    4now money is needed for any form of utility-producing activities. This implies workers

    need some source of money and if they lack non-earned income, they may need to work somehow regardless of their received wage offer. Or, they may not be able to afford a

    5positive reservation wage with respect to all forms of work. The absence of a positive

    reservation wage can make labor supply curves negatively-sloped. That is when they are offered a lower wage, the worker will likely respond by trying to work more hours. Generally, the introduction of a money/pecuniary cost of leisure into a labor-supply model tends to lower the observed elasticity of labor supply by increasing the need for money and correspondingly the hours worked. This alteration to the labor-supply model can help to explain the observed inelastic labor supply of welfare-recipients in many

    6studies, as is summarized in Moffit(2003). The vulnerability of workers in such a

    situation sets out labor-market regulations as ideally attempts to reduce the level of cut-throat competition between workers who may have very little bargaining power on their own accounts. However, regulations are usually imperfect and never cover all forms of work and, in reducing competition between workers they tend to favor some workers at the expense of others, potentially increasing inequality. If these short-comings are taken into account then do regulations still reduce poverty? This paper sets out an ideal-type

     4 The pecuniary or money cost of leisure is assumed a constant with the cost of consumption normalized for simplicity. Likewise, the time cost of leisure is normalized to keep the model simple; a virtue that will come into play through-out the model-building-process. 5 Work is here defined as activities engaged in primarily for pecuniary gain. 6 Moffit(2003) writes that substitution and income elasticities from the pre-1995 literature fell into the general range of those elasticities obtained from the literature on substitution and income effects estimated from wage and nonincome variation. However, as shown by Heckman (1993), the latter literature was flawed due to the negative bias caused by pervasive non-classical measurement error in reported hours of work. One would generally expect the non-adjusted labor supply elasticities to be more positive on average. A working paper of the author shows that when one removes observations likely to be measured with error with a quantile-based trim of a log-wage regression that the empirical labor-supply elasticities are uniformly higher. As such, the near zero and often negative labor-supply elasticities from policy-variation for poorer people probably genuinely reflect that this group tends to have less elastic labor supply than the majority of workers.

    situation where a wage floor and work-hour restrictions combine to improve the welfare of most workers in both covered and uncovered sectors.

    The model is a simple one with two sectors, one covered by a wage-floor and work-hours restriction. To simplify the model, the number of workers, or households, is

    7normalized to one with a time endowment of one. A worker can only work half of their

    time-endowment in the covered sector and is assumed to be working half of their time-endowment in the covered-sector by the model. Additionally, it is also assumed that the wage floor in the covered-sector is so low that covered workers work additional hours in the uncovered sector.

    The model then proceeds as follows: there are no skill-differences between workers, but a fraction of workers work half their endowment in the covered sector and, taking their covered sector income as exogenous, then choose how many hours to supply to the uncovered sector. Similarly, the other workers choose how many hours they will supply to the covered sector. The fraction of workers in each group is exogenously determined by labor demand in the covered sector and the wage floor value, or minimum wage. Labor demand in the covered sector has a negative elasticity proportional to the minimum wage level, or

    cwme8L(1) . The uncovered sector wage is inversely proportional to the total d2

    hours worked in the uncovered sector, or

     7 It is trivial to show that assuming these values do not affect the comparative-statics of the model. 8 This assumption guarantees that Labor Demand will never exceed the amount of labor legally available, given the work-week restriction.

    (2)wH, where is the total hours worked in the uncovered sector, and u,tuHu,t

    (3)H2LH;(12L)HL, where is the formal sector’s labor demand u,tdu,cdu,ud

    HHand and are the hours worked in the uncovered sector by covered sector and u,cu,u

    9non-covered sector workers, respectively.

    Hence, it follows that

    w(4) . ucwcwmm He+ H(1- e )u,cu,u

    1CLFor simplicity, Workers’ have uniform Cobb-Douglass preferences, U(C, L) = ee

    and take as exogenous the uncovered-sector wage, w. The only difference between the u

    two groups is the constraint they face for their uncovered-sector labor-supply decision.

    1CLCovered-sector workers maximize subject to H+ Le=1/2 and pLe+C=w u,cl ue

    1CLH+w/2. Uncovered-sector workers, on the other hand, must maximize subject u,cme

    to H+ Le=1 and pLe+C = w H. u,ul uu,u

    The above two maximization problem give us:

    p;(1)wwp;w(w;w)(w;w)1LumLuumum(5) H== for u,c2(p;w)2(p;w)22(p;w)LuLuLu

    wwworkers employed in the covered sector, so long as H>0 and , u,cuc

    p;(1)wwpLuuL11;(6) H= and u,u(p;w)(p;w)(p;w)LuLuLu

    cwcwmm(p;(1)ww)e;2(p;(1)w)(1e))LumLuH(7) == u,t2(p;w)Lu

     9 For convenience, it is assumed that all uncovered sector work renders the same wage at no risk to the

    worker and that there are no negative externalities from uncovered-sector work, such as the weakening of

    property-rights or an increase in the mortality rate.

    Then, combining equation 4 and 7, the resultant equation simplifies to a quadratic formula in terms of the uncovered sector wage.

    2((;;4(1)(1L)p11dL(8)wu2(1)(1L) d


    Noticeably, an increase in the minimum wage will have both a positive effect and a negative effect on the wage in the uncovered sector. The negative effect is through the reduction in the quantity of labor demanded in the covered sector. The positive effect is through the reduction of hours supplied by covered-sector workers shown in equation 5. If the labor demand is inelastic then the reduction of hours by covered-sector workers

    10from an increase in the wage floor is more likely to prevail. The reduction in the

    uncovered-sector wage will then reduce the hours supplied by uncovered-sector workers, which will further reduce their wages. The increase in the money for workers creates a virtuous circle that reduces the level of cut-throat competition among workers.

    If, alternatively, all the workers have an unearned income of p, sufficient money L

    to potentially consume leisure with the entirety of their time-endowment, then it can be shown that uncovered-sector only workers’ labor supply will be inelastic at H =, 1u,u

10w(w)*H(w,w(w)) If one totally differentiates with respect to the covered-sector umu,tmum

    EH,wwwwmuutuwage then we get then we get . If the total uncovered-sector www(1;E)mmmH,wut

    hours worked fall as the covered sector wage rises then the numerator is negative and if the total covered-

    wusector hours is less than unitary elastic with respect to the uncovered sector wage then >0. It can also wm

    be shown that there exists a covered-sector wage where the uncovered-sector wage is equal to the covered-sector wage. As stated earlier, the model assumes the covered-sector wage is less than the minimum wage. If that wasn’t the case then the two-sector’s wages would be equal to each other.

instead of negatively elastic as shown by equation six. Covered-sector workers’

    uncovered sector labor supply will be positively-sloped with

    (ww)(w;w)p11umumL(10)H(); if, or workers .5u,c22(w;p)22(w;p)(w;p)uLuLuL

    prefer consumption over leisure. Otherwise, if they prefer leisure then they will not work at all in the uncovered sector. In both cases, a covered-sector worker will work less hours in the uncovered sector and are more likely to not work at all in the uncovered-sector. This increases the likelihood that an increase in the wage-floor value will have a negative spill-over for uncovered-sector workers.

    In the case where covered-sector workers do not work in the uncovered sector, it is easy to show that an increase in the wage floor will reduce the uncovered sector wage fall by reducing the number of covered-sector workers. In the case where covered-sector workers may work in the covered sector then w will be equal to the same formula as (8), u

    but with instead of, where ((21

    . (11)(((1)p);L(p;w)2LdLm

    It is trivial to show that the uncovered sector wage is now higher. If one

    pLconsiders how for both groups, the hours supplied to the uncovered sector is p;wLU

    less than their previous value, it is apparent that the total hours supplied to the uncovered sector is the same value. Also apparent is that the covered-sector wage where covered-sector workers no longer supply any hours to the uncovered-sector is now lower in both

    11cases. The implication is that when workers are in a better situation to exert an exit-threat then an increase in the wage floor is more likely to have a negative-spill-over for workers in the uncovered sector.

    Now, the usefulness of the above model is an empirical question. Empirical studies that evaluate the impact of changes in the minimum wage or living wages on income, such as Adams and Neumark(2004), are in order. For, as Adams and Neumark(2004) says in their introduction, “while economic theory predicts that raising mandated wage floors will lead to some employment reductions, it makes no predictions whatsoever regarding the effects on the distribution of family incomes, or on poverty specifically. The distributional effects depend on both the magnitudes of the wage and employment effects, and on their incidence throughout the family income distribution.” One can also add that looking at the impact of these changes on the hours worked would

    12also matter. This paper adds that when many workers, or households, exhibit uncompetitive negatively-sloped labor supply behavior due to their low wages and lack

    wu11 It can be shown that will now tend to be higher. If one totally differentiates wm

    pLw(w)*(H(w,w(w))) in the same way as before, it can be shown that umu,tmumpw(;)Lu

    EH,wwwmutuHin the first case and =, where is the change u,tHpwwmmu,tL(1;E)H,wup;wHtuu,tL

    in the total hours worked in the uncovered sector from the first to the second scenario. Then, since the uncovered sector wage is higher with no change to the covered sector wage, their ratio should be higher and the denominator’s value should be lower in the second scenario. This implies that, so long as the model’s conditions still hold, the uncovered-sector wage will be more responsive to an increase in the covered-sector wage. This is likely because covered-sector workers now work less hours in the uncovered-sector and so when the covered-sector wage increases their uncovered-sector hours are reduced by a greater percentage. 12 Although, it has been shown by Bound et. Al (1994) that reported hours of work was by far more corrupted with measurement error than reported yearly earnings based on the PSID Validation Study.

    of resources and find it necessary to work in “informal” markets that are not coved by labor market regulations then an increase in the wage floor may improve the economic situation for both covered and un-covered workers. It also points to the need to measure the impact of programs like “the living wage” on labor supply, since reduced hours of work would be an additional criterion to a higher income for evaluating the altered welfare of workers impacted by the program. This possibility is granted some additional empirical plausibility by Adams and Neumark (2004) that finds increased incomes for semi-poor workers and the absence of reduced incomes for the poorest workers impacted by living-wage programs.


    Adams, Scott and Neumark, David. 2004. “Living Wage Effects: New and Improved

    . Evidence” forthcoming in Economic Development Quarterly

    Bound, John, Brown, Charles, Duncan, Greg and Rodgers, Willard L. 1994. “Evidence on the Validity of Cross-Sectional and Longitudinal Labor Market Data,” Journal of

    Labor Economics, vol. 12, no. 3, pp. 345-368.

    Heckman, James J. 1993. “What Has Been Learned About Labor Supply in the Past Twenty Years?” In “Lessons from Empirical Labor Economics: 1972-1992”, American

    Economic Review, 83 (2) pg. 116-121.

    Mincer, Jacob. 1976. “Unemployment Effects of Minimum Wages” Journal of Political Economy” 84, 87-104.

    Moffit, Robert. 2003. “Welfare and Labor Supply.” NBER Working Paper 9168 forthcoming in the Handbook of Public Economics.

    Wetzell, David. 2002. “On some Unappreciated Implications of Becker’s Time allocation Model of Labor Supply.” Economic Letters 75, 219-225.

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