Notes for Chapter 6

This week, we'll examine the determinants of labor supply. The framework for this discussion relies on the basic neoclassical model in which individuals are assumed to face a trade-off between labor and leisure time. In this model, it is assumed that there are only two possible uses of time: labor and leisure. Each individual is assumed to select the mix of time and purchased inputs that maximizes his or her level of satisfaction (utility).

Trends in labor force participation rates

During the past century, the labor force participation rate for males has declined, primarily for relatively young males and males aged 65 and older. These changes are primarily the result of increased education attainment (delaying entry into the labor force) and earlier retirement. (In fact, retirement is a relatively recent phenomenon, induced in part by the introduction of the Social Security system and increases in private pension plans.)

The labor force participation rate for females, on the other hand, have increased rather dramatically over the past century. This increase is most pronounced for married females (partly because this group began the century with very low labor force participation rates). Among the reasons for the increase in female labor force participation rates are:

Opportunity costs and the optimal allocation of time

Since there are only two uses of time in the basic neoclassical model, the opportunity cost of an additional hour of leisure time is the wage payment that is given up by choosing to not work. Individuals choose to not work an additional hours if the value of leisure time exceeds the market wage rate. Individuals will work an additional hour if the value of the products that can be purchased with the wage outweigh the benefits of an additional hour of leisure time.

Substitution and income effects of a wage change

A change in the wage results in two effects on an individual's labor supply: As the wage rate rises, the opportunity cost of leisure time rises. In response to this higher wage, individuals consume less leisure time and spend more time at work. This is the substitution effect resulting from a higher wage.

An increase in the wage, however, also raises an individual's real income. This leads to an increase in the consumption of all normal goods. Since leisure is expected to be a normal good for most individuals, a higher wage will generally induce individuals to consume more leisure time (and reduce hours of work). Individuals who receive a higher wage can afford to take more time off from work. This is the income effect resulting from a wage increase.

If we assume that leisure is a normal good, an increase in the wage will cause the quantity of labor supplied to:

This may result in a backward-bending labor supply curve (as illustrated below).

backward-bending labor supply curve

In the diagram above, it is suggested that, at relatively low wages, individuals respond to an increase in the wage by working additional hours (since the substitution effect exceeds the income effect). Eventually, though, when the wage becomes sufficiently high, individuals will begin to work less in response to a higher wage rate. (In practice, it appears that most labor supply curves are either upward sloping or vertical.)

Indifference curves and budget constraints

Let's examine how indifference curves and budget constraints may be used to illustrate the optimal combination of labor and leisure. An indifference curve is a graph of alternative combinations of goods that provide a given level of satisfaction (utility). In the simple neoclassical model of labor supply, it is assumed that the individual's utility level is a function of two goods: real income (Y), and leisure time (L). In mathematical terms, this utility function may be expressed as:

U = U(Y,L)

where U = the level of utility associated with alternative combinations of L and Y. In this case, an indifference curve provides a graph of all of the combinations of income and leisure that provides a given level of utility to an individual. The diagram below illustrates a possible indifference curve.

indifference curve diagram

This indifference curve is downward sloping because an individual is willing to give up some income to receive additional leisure (or vice versa).

A few points have been added to the diagram below. Due too the definition of the indifference curve, this individual would be just as happy with the combination of real income and leisure represented by point A as he or she would be with the combination of Y and L represented by point B. Points that lie above and to the right of the indifference curve, such as point C, provide a higher level of utility.

indifference curve diagram with points added

Each point in this diagram provides a particular level of utility. The indifference curve that passes through point C provides a higher level of utility. This means that U' corresponds to a higher level of utility in the diagram below.

indifference curve diagram w/ additional ind. curve added

Time and goods constraints

Individuals attempt to achieve the highest possible level of utility. The choice among alternative levels of Y and L, however, is restricted due to two constraints: The time constraint is given by:

H + L = T

where:

w = wage rate
H = hours of work
P = price index for real income
Y = real income

This time constraint simply notes that time spent at work plus time spent at leisure must add up to the total time available (since these are the only two uses of time in this model).

Using the definitions above, the goods constraint is given by:

pY = wH

This equation states that total spending (pY) must equal earnings (= wH). (Since this is a one-period model, saving and lending do not occur. More complex models that include this possibility have results that do not differ substantially from those derived below in this simpler model. The analysis of more complex multi-period models, however, requires mathematical tools that are beyond the scope of this course.)

Thus, the two equations that must be satisfied are:

  1. H + L = T
  2. pY = wH

Rewriting equation (1) as: H = T - L and substituting this into equation (2) results in:

pY = WT - wL

With a little algebraic manipulation, this becomes:

3. wT = pY + wL

This equation is called a "full-income constraint." Economists define full income as an individual's maximum earnings potential (= wT in this case). This equation states that full income equals the total explicit costs of goods and services (pY) plus the total implicit cost of leisure time (wL).

An alternative form of equation (3) is given by:

3'. Y = -(w/p)L + (w/p)T

This equation describes the relationship that exists between hours of leisure and real income. Equation (3') is the individual's budget constraint.

The intercept of the budget constraint on the horizontal axis equals T. This is the maximum amount of leisure time that an individual can receive. This is illustrated by the highlighted point in the diagram below. Notice that both H and L can be measured along the horizontal axis. The level of work effort decreases from T to 0 as the level of leisure time rises from 0 to T.

budget constraint

Noting that the budget constraint contained in equation (3') is expressed in slope-intercept form, the intercept of the budget constraint on the vertical axis equals wT/p (= the real value of full income). Using the slope-intercept form of the budget constraint in equation (3'), we can also see that the slope of the budget constraint equals -w/p. The diagram below illustrates the budget constraint facing this individual.

budget constraint

Utility maximization

In the diagram below, three indifference curves have been added to the diagram containing the budget constraint. Each point on the budget constraint is a feasible combination of income and leisure. It is assumed that the individual will select the combination of income and leisure that provides the highest possible level of utility. As indicated by the diagram below, this optimal combination of L and Y occurs at a point of tangency between the budget constraint and an indifference curve. In the diagram below, this optimal point occurs when real income equals Y* and hours of leisure equals L*. At this point, the individual chooses to work H* hours.

indifference curve diagram with equilibrium

Reservation wage

The absolute value of the slope of an indifference curve is a measure of the opportunity cost of time at that point. Note that the absolute value of the slope of an indifference curve serves as a measure of the amount of income that is required to induce the worker to give up an hour of leisure time. A steep indifference curve indicates that a large change in income is required to induce an additional hour of work; a relatively small increase of income can induce an additional hour of work when indifference curves are relatively flat. Thus, indifference curves are relatively steep when the value of time in nonmarket activities is relatively high. The diagram below contains a set of indifference curves for an individual who places a high value on nonmarket time.

indifference curve diagram with steep indifference curves

A corner solution occurs when the indifference curve is steeper than the budget constraint at the point corresponding to zero hours of work. This possibility is illustrated in the diagram below. A careful inspection of this diagram should indicate that the highest possible level of utility (given this budget constraint and these preferences) occurs at zero hours of work. An individual chooses to remain out of the labor force when a corner solution such as this occurs.

indifference curve diagram with equilibrium

A corner solution at zero hours of work will occur when the value of leisure time is relatively high and/or the market wage is relatively low. To see this, note that the absolute value of the slope of the indifference curve is a measure of the opportunity cost of leisure time while the absolute value of the slope of the budget constraint is the real wage. A corner solution occurs only if the value of leisure time (the absolute value of the slope of the indifference curve) exceeds the real wage (the absolute value of the slope of the budget constraint).

The absolute value of the slope of the indifference curve at the point corresponding to zero hours of work is the individual's reservation wage (expressed in real terms. This is illustrated in the diagram below.

indifference curve diagram with equilibrium

If the real wage in the labor market exceeds the reservation wage, the individual chooses to work. This possibility is illustrated in the diagram below. Notice that when the real wage exceeds the reservation wage, there are feasible points on the budget constraint that provides a higher level of utility than would occur at zero hours of work.

indifference curve diagram with equilibrium

If the real wage in the labor market is less than the reservation wage, the individual chooses to remain out of the labor force and a corner solution occurs. This possibility is illustrated in the diagram below.

indifference curve diagram with equilibrium

So, whenever the wage exceeds the reservation, an individual will chose to work. An individual will not work if the wage is below the reservation wage. The individual is indifferent between not working and working when the wage equals the reservation wage (since the opportunity cost of leisure time is just equal to the wage at this point).

Nonlabor income

Up to this point, we have assumed that all income is received in the form of labor income. Individuals, however, also receive income in the form of nonlabor income. As we noted at the beginning if this semester, income from nonlabor sources is referred to as "unearned income." This nonlabor income may be received in the form of interest payments, rent, dividends, profits, alimony payments, transfer payments, lottery winnings, lawsuit settlements, or as any other income that does not vary with hours worked.

Using the definition:

A = total amount of nonlabor income

the time and goods constraints that we derived above become:

  1. Time constraint: H + L = T
  2. Goods constraint: wH + A = pY
Note that the time constraint is the same as that discussed earlier (those with more nonlabor income do not have any additional hours in the day....). The goods constraint is modified to account for two source of income: earned income (wH) and unearned income (A).

Solving equation (1) for H:

H = T-L

Substituting this for Y in equation (2) results in:

Y = -(w/p)L + (wT+A)/p

An inspection of this budget constraint equation indicates that the slope equals -w/p (as in the simpler model) and the intercept on the vertical axis equals (wT+A)/p

Three budget constraints corresponding to alternative levels of nonlabor income (A) appear in the diagram below. As the level of nonlabor income rises, the budget shifts vertically in an upward direction. The slope remains constant at -w/p.

budget constraints with nonlabor income

Notice that the slope of the budget constraint stays the same when nonlabor income changes. While the budget constraint shifts upward as nonlabor income rises, it still terminates at T hours of leisure. No matter how wealthy you are, there are still only 24 hours in your day.

If leisure is a normal good, an increase in nonlabor income results in an increase in leisure time and a reduction in hours worked (as illustrated below).

budget constraints  and indifference curves with nonlabor income

The change in hours worked that results from a change in real income, holding relative prices constant, is called a "pure income effect." When leisure is a normal good, this income effect reduces hours worked when income rises.

Substitution and income effects

As the wage rate rises from wo to w', the budget constraint pivots upward. The diagram below illustrates this possibility. In response to this increase in the wage, the equilibrium shifts from point A to point C. In this example, the quantity of labor supplied has decreased in response to this higher wage. As noted earlier, this suggests that the income effect must be larger than the substitution effect for this individual (i.e., this person is operating on the backward-bending portion of his or her labor supply curve).

substitution effect

In the diagram below, the effect of the wage increase has been decomposed into separate substitution and income effects. The substitution effect is the change in the mix of L and Y that results from a change in the relative price of leisure (the real wage), holding utility constant. This is represented by the movement from point A to point B in this diagram. The budget constraint that is tangent to the indifference curve at point B is a hypothetical budget constraint. It is constructed so that it has a slope equal to the slope of the new budget constraint (-w'/p) and is tangent to the initial indifference curve. Notice that, as expected, the quantity of leisure consumed declines when the relative price of leisure rises.

substitution and income effects

The shift from point B to C is a pure income effect (and is equivalent to the pure income effect discussed above). Since leisure is a normal good for this individual, the quantity of leisure consumed rises (and hours worked declines) as real income rises in response to the higher wage.

When the income effect is smaller than the income effect, an increase in the wage will result in an increase in hours worked and a reduction in leisure time (as illustrated below). In this example, the substitution effect is again illustrated by the shift from point A to B; the shift from B to C is the income effect. The net effect in this case is an increase in hours worked and a reduction in leisure time.

substitution and income effects

It is important to note that we never observe separate income and substitution effects when the wage rate changes. Instead, we only observe the combined substitution and income effects (represented by a movement from point A to point C). The decomposition of this change into substitution and income effects, however, explains why a backward-bending labor supply curve may exist.

Unemployment compensation and full disability

The diagram below represents an optimal combination of leisure and work for an individual facing a wage rate equal to w. This individual will work Ho hours (leisure time = Lo) and will receive an income of Yo.

unemployment compensation

Suppose that an unemployment compensation program is introduced that provides a replacement for all lost income. If this individual becomes unemployed, he or she would shift from point A to point B (as illustrated in the diagram below).

unemployment compensation

Since point B lies above the original indifference curve, however, this individual would receive a higher level of utility if he or she were unemployed. This occurs because leisure time is valued by the worker. For this reason, unemployment compensation systems do not generally provide complete replacement of lost income.

To maintain the worker at the original level of utility, an appropriate level of compensation would be equal to Y' in the diagram below. The problem, of course, is that Y' cannot be determined by the government. In the U.S., unemployment benefits are typically equal to approximately 1/2 of an individual's lost income.

unemployment compensation

A similar argument can be applied to disability insurance programs. If disable workers receive the same level of income after an injury as before and receive more leisure time, their level of utility would increase (assuming that "pain and suffering" and medical expenses are fully compensated). Disability insurance programs require medical examinations by approved physicians to reduce the possibility that workers will file fraudulent disability claims.

Partial disability

A work-related injury that results in a partial disability reduces the wage that the affected worker will receive. This reduction in the wage generates both substitution and income effects on the quantity of labor supplied. If the goal is to adequately compensate the worker, however, an appropriate income replacement scheme would be to provide a payment that is just large enough to offset the income effect resulting from the reduction in the wage (since it is only the income effect that involves a loss is utility).

Welfare system

Let's examine the effect of the U.S. welfare system on labor supply.

The first major national attempt at providing aid to low-income households in the U.S. occurred during the Great Depression. Most of the relief programs developed during this period, however, were temporary programs designed to deal with the problems resulting from the Depression.

The modern U.S. welfare system was introduced in the early 1960s as part of the War on Poverty during the Johnson administration. Under this welfare system, a poverty level was established based upon studies that attempted to determine the amount of income needed to provide households with an adequate level of nutrition and basic necessities. Under this system, it is assumed that a household of a given size in a particular geographical area must receive an appropriate level of income (Yt) to ensure that these basic needs could be satisfied. (This level of income is higher for larger households and for residents in geographical regions where the cost of living is higher.)

Under this welfare system, the government provides welfare benefits to those households in which the level of income falls below the target level (Yt). These welfare benefits may take the form of monetary payments or subsidies for food, housing, medical care, or other basic commodities. The goal is to provide a level of welfare benefits that brings the level of household income up to the target level.

The diagram below illustrates the budget constraint that results from the introduction of such a welfare system. If the individual does not work at all, the level of welfare benefits equals Yt. If the individual is working, but receives a level of income that falls below Yt, the government provides enough welfare benefits to provide a total income of Yt.

basic welfare system

Thus, as illustrated above, the budget constraint becomes horizontal at an income level of Yt. In this portion of the budget constraint, the marginal wage (the additional income resulting from an additional hour of work) equals zero. If a welfare recipient works an additional hour and receives a wage of $6, welfare benefits are reduced by $6, leaving total income unchanged.

In the situation illustrated below, it can be seen that an individual who, in the absence of a welfare system, has a level of income that lies below the target level of income would always prefer to leave the labor force when such a welfare system is available (since the level of income and leisure both increase in this case).

basic welfare system

Some individuals who, in the absence of this welfare system, would have received a level of income that exceeds Yt, would also choose to leave the labor force (as illustrated below).

basic welfare system II

Note that the substitution effect that results from lowering the marginal wage to zero reduces the quantity of labor supplied, as does the income effect resulting from the provision of welfare benefits.

Because of the labor supply disincentives effects resulting from this type of welfare system, this system was replaced in 1967 with a welfare system that allowed individuals to keep a small amount of monthly earned income ($30) without giving up any welfare benefits. Beyond this point, welfare benefits were reduced by $2 for every $3 earned (as compared to a $1 reduction for every $1 earned under the earlier system). This system reduced the labor supply disincentive effects resulting from the earlier system.

This revised system remained in effect until the early 1980s. During the Reagan administration, the welfare system was restored to a form that was essentially equivalent to that of the early 1960s. The reason for this change was a desire to reduce welfare benefits for higher income welfare recipients while preserving benefits for the "truly needy."

With the restoration of a system that provides a marginal wage of zero, the number of "working poor" declined and a larger share of welfare recipients left the labor force.

To deal with the labor supply disincentives that were reintroduced during the Reagan administration, a "workfare" system has been adopted. Under this system, welfare recipients are required to work a minimum number of hours to qualify for welfare benefits. Welfare benefits are zero unless welfare recipients work the minimum number of hours (or are engaged in approved job training or educational programs). Individuals are also restricted to receiving welfare benefits for a maximum of five years under this system.

The budget constraint below illustrates the effect of a workfare requirement. Individuals receive no benefits if they work fewer than the minimum number of hours (Hm) in this example). If they work Hm or more hours, they receive the same level of benefits as under the earlier system.

workfare

Under such a system, it would be expected that welfare recipients would choose to work Hm hours. If they work more than this, the marginal wage falls to zero (until they work enough hours so that all benefits are eliminated). This outcome us illustrated in the diagram below.

workfare

The earned-income tax credit is an alternative method of providing increased income to low-income households. Under an earned-income tax credit, a tax credit is provided that rises with income up to a point and then gradually declines as income rises beyond this point. The earned-income tax credit generates a smaller labor supply disincentive effect than the current welfare system. The diagram below illustrates how an earned-income tax credit alters the shape of an individual's budget constraint.

earned income tax credit