Labor-Leisure Decisions and Labor Supply

Abhishek Dureja

Teaching Fellow: Shreya Kapoor

Plaksha University
 Labor-Leisure
Introduction

▶ Till now we have seen how the consumer maximizes utility

▶ The critical assumptions were:

i Utility depended only on the consumption of goods

ii The consumer income was exogenous and fixed

▶ But consumer’s income is never exogenous i.e. each consumer has

to earn his/her income
▶ How does the consumer earn his/her income?

▶ The consumer earns his/her income by exerting effort i.e. by

supplying labor
▶ Thus consumer’s income depends on the labor supplied by the
consumer

2 / 39
 Labor-Leisure
Introduction

▶ Further, if the consumer makes efforts

=⇒ The consumer loses utility. Why?

▶ Because consumer enjoys leisure time

▶ The more the effort consumer exerts =⇒ Lesser is the leisure time
available to the consumer
=⇒ Leisure gives positive utility to the consumer
=⇒ Consumer’s utility depends not only on consumption but
also on leisure
▶ More formally, individuals must make choices in deciding how
they want to spend their time apart from consumption
decisions

3 / 39
 Labor-Leisure
Introduction

▶ Assume that an individual’s utility during a typical day depends on:

i Consumption during that period (c)

ii Hours of leisure enjoyed (h)

Utility = U(c, h)

▶ Notice that the utility function has two composite goods:

Consumption and Leisure

▶ The consumer’s utility is derived from the income the consumer

earns

▶ The consumer faces two constraints

4 / 39
 Labor-Leisure
Introduction

▶ The first constraint relates to the time that is available to the

consumer

▶ If l represents the hours of work, then

l + h = 24 (1)

=⇒ The day’s time must be allocated either to work or to

leisure (nonwork)

5 / 39
 Labor-Leisure
Introduction

▶ The second constraint relates to the fact that an individual

can purchase consumption items only by working
→ Without loss of generality, we are currently assuming that
non-labor income is currently zero
→ Non-labor income (n) can include subsidies, transfer
payments, or even taxes (negative non-labor income)

▶ If the real hourly market wage rate the individual can earn is given
by w , then the income constraint is given by

c = wl (2)

6 / 39
 Labor-Leisure
Introduction

▶ Combining the two constraints we get

c = w (24 − h)

=⇒ c = 24w − wh

c + wh = 24w (3)

=⇒ Any consumer has a full income given by 24w

=⇒ A consumer who works all the time will have this much real
income for consumption of goods each day

7 / 39
 Labor-Leisure
Introduction

c + wh = 24w

▶ Each consumer can spend his/her full income either on

i Consumption (c)

ii Leisure i.e. by not working and enjoying leisure time (wh)

▶ Equation (3) suggests that the opportunity cost of consuming

leisure is w per hour
=⇒ Opportunity cost of consuming leisure is equal to earnings
forgone by not working

8 / 39
 Labor-Leisure
Utility Maximization

▶ Each consumer faces the problem of maximizing utility subject to

the full income constraint

Max. U(c, h)

subject to the full income constraint

c + wh = 24w

▶ Setting up the Lagrangian expression, we get

L = U(c, h) + λ(24w − c − wh)

9 / 39
 Labor-Leisure
Utility Maximization

L = U(c, h) + λ(24w − c − wh)

▶ The FOCs for the maximum are given by

∂L ∂U
= −λ=0 (4)
∂c ∂c

∂L ∂U
= − λw = 0 (5)
∂h ∂h

▶ Dividing equation 4 by 5, we get

∂U
∂h
∂U
= w = MRS (h for c) (6)
∂c

10 / 39
 Labor-Leisure
Utility Maximization

∂L
∂h
∂L
= w = MRS (h for c)
∂c

Consider leisure as good x and consumption as good y

=⇒ MRS (h for c) = w

▶ In equilibrium, if the consumer wants to increase his/her leisure by 1

hour
=⇒ The amount of goods the consumer can consume has to go
down by w

11 / 39
 Labor-Leisure
Utility Maximization

Utility-maximizing labor supply decision

To maximize utility given the real wage w, the individual should
choose to work that number of hours for which the marginal rate
of substitution of leisure for consumption is equal to w

12 / 39
 Labor-Leisure
Income and substitution effects of a change in w

▶ What will happen if the real wage rate increases?

▶ Will the consumer increase labor supply (i.e. reduce leisure) or

decrease labor supply (i.e. increase leisure)

▶ When w increases, the price of leisure becomes higher

=⇒ Consumer must give up more in lost wages for each hour of

leisure consumed
=⇒ The substitution effect of an increase in w on the hours of
leisure will be negative

▶ As leisure becomes more expensive =⇒ consumer will

consume less of it

13 / 39
 Labor-Leisure
Income and substitution effects of a change in w

▶ However, because the real wage rate w has increased

=⇒ For each hour of labor supplied the compensation has increased

=⇒ For any given level of labor supply the consumer’s income

increases

▶ Since leisure is a normal good

=⇒ The higher income resulting from a higher w will increase

the demand for leisure
=⇒ The income effect will be positive

14 / 39
 Labor-Leisure
Income and substitution effects of a change in w

▶ Due to an increase in the real wage rate (w ),

▶ Substitution effect predicts that the consumer will decrease
leisure (h) and increase labor supply (l)
▶ Income effect predicts that the consumer will increase leisure (h)
and decrease labor supply (l)
▶ The income and substitution effects work in opposite
directions
▶ It is impossible to predict on a prior theoretical grounds whether an
increase in w will increase or decrease the demand for leisure time
▶ The substitution effect tends to increase hours worked when w
increases
▶ Whereas the income effect (because it increases the demand
for leisure time) tends to decrease the number of hours worked

15 / 39
 Labor-Leisure
Income and substitution effects of a change in w

▶ It is also impossible to predict what will happen to the number of

hours worked

▶ Whether the substitution effect is stronger or the income effect – It

is an important empirical question

▶ Only through data can we then empirically check what happens

▶ If substitution effect dominates the income effect

=⇒ Labor supply increases as real wage rate increases

▶ If income effect dominates the substitution effect

=⇒ Labor supply decreases as real wage rate increases

▶ Let’s see this graphically

16 / 39
 Labor-Leisure
Income and substitution effects of a change in w

When Substitution Effect dominates Income Effect

17 / 39
 Labor-Leisure
Income and substitution effects of a change in w

When Income Effect dominates Substitution Effect

18 / 39
 Labor-Leisure
Income and substitution effects of a change in w

When Income Effect dominates Substitution Effect

▶ When the wage rate increases from w to w1

=⇒ The optimal combination moves from (c0 , h0 ) to point (c1 , h1 )

▶ This movement consists of two effects
▶ The substitution effect is represented by the movement of the
optimal point from (c0 , h0 ) to S
▶ The income effect is represented by the movement from S to (c1 , h1 )

▶ The substitution effect of an increase in w outweighs the income

effect
=⇒ Individual demands less leisure (h1 < h0 )

=⇒ Individual will work longer hours when w increases

19 / 39
 Labor-Leisure
Income and substitution effects of a change in w

▶ The same two effects take place in this case as well

▶ But the effect on leisure is reversed

▶ The income effect of an increase in w more than offsets the

substitution effect
=⇒ The demand for leisure increases (h1 > h0 )

▶ The individual works shorter hours when w increases

▶ The result appears unusual at first

20 / 39
 Labor-Leisure
Income and substitution effects of a change in w

▶ But the reason is that in the case of leisure and labor – the income
and substitution effects always work in opposite directions

▶ An increase in w makes an individual better-off because he or she is

a supplier of labor
=⇒ Increased income will increase leisure (and decrease labor
supply)

21 / 39
 Labor-Leisure
Income and substitution effects of a change in w

Income and substitution effects of a change in the real wage

▶ When the real wage rate increases, a utility-maximizing individual

may increase or decrease hours worked

▶ The substitution effect will tend to increase hours worked as

the individual substitutes earnings for leisure, which is now
relatively more costly

▶ On the other hand, the income effect will tend to reduce hours
worked as the individual uses his or her increased purchasing
power to buy more leisure hours

▶ The net effect is theoretically unknown and is an empirical question

22 / 39
 Labor-Leisure
Empirical Possibility

▶ An empirical possibility is that the labor supply curve is

backward-bending

▶ When the wage rate is small, the substitution effect is larger than
the income effect,
=⇒ An increase in the wage rate will decrease the demand for
leisure and hence increase the supply of labor

▶ When wage rates are high, the income effect may outweigh the
substitution effect
=⇒ An increase in the wage will reduce the supply of labor

23 / 39
 Labor-Leisure
Empirical Possibility

24 / 39
 Labor-Leisure
Mathematical Analysis of Labor Supply

▶ Let us derive a mathematical statement of labor supply decisions

▶ Let’s first make the consumer’s budget constraint more general

▶ Let us amend the budget constraint slightly to allow for the presence
of nonlabor income
c = wl + n

▶ where n is real nonlabor income

▶ n may include government transfer benefits, lump-sum taxes paid

(then n is negative), dividend and interest income, or any other gifts

25 / 39
 Labor-Leisure
Mathematical Analysis of Labor Supply

▶ Maximization of utility subject to this new budget constraint would

yield results virtually identical to those we have already derived
=⇒ Optimum condition is still given by equation 6
=⇒ Adding non-labor income n does not make a difference in the
optimum condition
▶ Because the value of n does not affect the labor-leisure choices
being made as n is a lump-sum receipt or loss of income
▶ Introducing nonlabor income n into the analysis shifts the
budget constraints outward or inward in a parallel manner
=⇒ It does not affect the trade-off rate between earnings and
leisure
=⇒ The labor supply of an individual depends upon the wage rate,
w , and the non-labor income n

26 / 39
 Labor-Leisure
Mathematical Analysis of Labor Supply

▶ We can write the individual’s labor supply function as l(w , n)

▶ l(w , n) indicates that the number of hours worked will depend
on
i Real wage rate (w )

ii The amount of real nonlabor income received (n)

▶ We assume that leisure (h) is a normal good

∂h
>0
∂n

∂l
=⇒ <0
∂n

=⇒ An increase in n will increase the demand for leisure (h)

and reduce labor supply (l)

27 / 39
 Labor-Leisure
Dual statement of the problem

▶ An individual’s primal problem of utility maximization given a

budget constraint is the dual problem of minimizing the
expenditures necessary to attain a given utility level

▶ In the context of labor-leisure decisions, this problem can be phrased

as choosing values for consumption (c) and leisure time (h) (i.e.
24 − l) such that the amount of spending given by

E = c − wl − n

required to attain a given utility level [say, U0 = U(c, h)] is as small

as possible

28 / 39
 Labor-Leisure
Dual statement of the problem

▶ The dual problem is

Min.E = c − wl − n

subject to U0 = U(c, h)
▶ Now to examine the change in the minimum value for these
extra expenditures due to a change in real wage rate w , we can
use the Envelope theorem

∂E
=⇒ = −l (7)
∂w

=⇒ The labor supply function can be calculated from the

expenditure function by partial differentiation

29 / 39
 Labor-Leisure
Dual statement of the problem

∂E
= −l (7)
∂w

▶ Since utility is held constant in the dual expenditure minimization

approach (in the above equation; equation 7)
=⇒ This labor supply function should be interpreted as a
compensated (constant utility) labor supply function
=⇒ The compensated labor supply function is denoted by l c (w , U)

▶ The uncompensated labor supply function was given by l(w , n)

30 / 39
 Labor-Leisure
Slutsky equation of labor supply

▶ We will now use these concepts to derive a Slutsky-type equation

to reflect the substitution and income effects that result from
changes in the real wage
∂l(w ,n)
=⇒ We can to derive a Mathematical expression for ∂w (as
∂x
previous we did ∂p x

▶ l(w , n) is the labor supply function and w (real wage rate) is the
price of labor

▶ As the price of the labor varies (i.e. w varies), we want to see how
the labor supply varies

▶ As before, we will decompose this effect into two effects:

Substitution effect and Income effect

31 / 39
 Labor-Leisure
Slutsky equation of labor supply

▶ We know from the consumer theory that at the initial

utility-maximizing choice and prices (here wage rate w is the price)
=⇒ Uncompensated and compensated demand is the same
=⇒ Uncompensated demand and compensated demand curves
intersect each other
▶ The reason is that the optimal consumption bundle is the same
in the both cases:
i Optimal bundle by using minimum expenditure required for achieving
a given level of utility

ii Given that minimum level of expenditure as income and then the

consumer maximises utility

=⇒ Expenditure required for achieving a given of utility

would give the same demand bundle if the consumer is given
that level of expenditure as income and then the consumer
maximises the level of utility
32 / 39
 Labor-Leisure
Slutsky equation of labor supply

▶ Similarly, for any given wage rate, the level of labor supply at the
minimum expenditure level needed to achieve a level of utility would
be the same as the labor supply if the individual had income exactly
equal to that minimum expenditure level

▶ Mathematically this means that:

l c (w , U) = l(w , E (w , U)) = l(w , n)

▶ Taking the partial derivative both sides with respect to the wage
rate w
∂l c (w , U) ∂l ∂l ∂E
= +
∂w ∂w ∂E ∂w

33 / 39
 Labor-Leisure
Slutsky equation of labor supply

∂l c ∂l ∂l ∂E
= +
∂w ∂w ∂E ∂w
▶ Using the envelope theorem result from equation 7,

∂E
= −l c (w , U)
∂w

∂l c ∂l ∂l
=⇒ = − lc
∂w ∂w ∂E

▶ Since ∂l c ∂l
∂w = ∂w |U=U0

∂l ∂l ∂l
=⇒ = U=U0 + lc
∂w ∂w ∂E

34 / 39
 Labor-Leisure
Slutsky equation of labor supply

∂l ∂l ∂l
= |U=U0 + lc
∂w ∂w
| {z } ∂E
| {z }
Substitution effect Income effect

▶ The change in labor supplied in response to a change in the real

wage can be disaggregated into the sum of

i A substitution effect in which utility is held constant

ii An income effect that is analytically equivalent to an appropriate

change in nonlabor income

▶ The substitution effect (of change in w on labor supply such that

utility is held constant) is always positive

35 / 39
 Labor-Leisure
Slutsky equation of labor supply

▶ The substitution effect (of change in w on labor supply such that

utility is held constant) is always positive

▶ An increase in water rate will leisure more costly =⇒ Individual will

reduce leisure
=⇒ Individual will increase labor supply

∂l
=⇒ |U=U0 > 0
∂w

36 / 39
 Labor-Leisure
Slutsky equation of labor supply

∂l ∂l ∂l
= |U=U0 + lc
∂w |∂w {z } ∂E}
| {z
Substitution effect Income effect

▶ Since leisure is a normal good, i.e. ∂h

∂E >0

∂(24 − l) ∂l
> 0 =⇒ <0
∂E ∂E
∂l
▶ Therefore, l c ∂E <0
=⇒ The income effect is negative
=⇒ The substitution and income effects work in the opposite
direction

37 / 39
 Labor-Leisure
Slutsky equation of labor supply

▶ Mathematical development supports the earlier conclusions from our

graphical analysis

▶ Mathematical formulation suggests at least the theoretical

possibility that labor supply might respond negatively to
increases in the real wage

38 / 39
 References

Chapter 16
Snyder, Christopher and Nicholson, Walter. (2012). Microeconomic
Theory: Basic Principles and Extensions

39 / 39