In the previous section, we derived a demand schedule for labor based on the marginal
product of labor. Profit-maximizing price taking firms will hire labor as long as marginal
product of labor is greater than the real wage. Marginal product of labor is a decreasing
function of the quantity of labor, so firms will hire workers until marginal product is
pushed down to the real wage. Thus, the marginal product of labor will be a labor
demand schedule mapping the real wage into the optimal quantity of labor.
Figure 1
W
P
MPL
LD
L
As economists we have a lot of experience examining the supply and demand framework.
Therefore it would be nice if we could posit an upward labor supply curve to match the
labor supply curve. An upward sloping labor supply curve would represent a positive
relationship between the real wage and the amount of work that workers would be willing
to do. Then there would be an equilibrium real wage, w*, such that labor supply equals
demand.
Figure 2
W
P
LS
MPL
w*
LD
L*
L
To fully understand the equilibrium, however, it would be helpful to formally derive the
labor supply curve from the behavior of the worker/consumer. Understanding the tradeoffs which lead to the labor supply function will be necessary to effectively describe the
outcomes in the labor market.
To understand these trade-offs, we might first ask why someone works in the first
place. The answer, clearly, is to earn wages that they can use to buy the consumer goods
that they would like to enjoy. Why might some one not work? The answer would be that
working causes people to sacrifice leisure time. Thus, we can see that the working
represents a trade-off between leisure time and consumer goods. Any consumer/worker
will be enjoying a given combination of consumption, denoted Ct, and leisure, denoted
lst. To find the bundle of consumer goods and leisure {Ct, lst} chosen by the
consumer/worker we should answer two questions. First, which combinations does the
consumer/worker want to choose, and second, from which combinations is it possible to
choose.
All possible positive combinations can be mapped as points on a quadrant of C and ls
Figure 3
C
×
×
×
×
×
×
×
×
×
×
×
ls
Where each point marked by × is a potential bundle of consumption {C,ls} amongst
which can be chosen.
Clearly some of the points are better than others and would be preferred by a
worker/consumer who was free to choose. We could rank all of the points in the quadrant
in order of how much the worker/consumer likes them. Unfortunately, as there is an
infinite number of points that need to be ranked, this would take a long time. We begin
our task by dividing the quadrant into groups of points called indifference sets. All of the
points in any indifferences sets are equally preferred by the consumer/worker who is
indifferent among them.
We will make certain assumptions about the preferences of consumer/workers.
The first axiom of consumer behavior is Axiom 1: More is Better. More is better means
that for any two points with the same level of leisure, the worker/consumer will prefer the
point with more consumption (preferring point A in Figure 4 to point B). It also means
that for any two points with the same level of consumption, the worker/consumer will
prefer the point with more leisure.(preferring point E to point D).
Figure 4
C
D E
× ×
A
×
B
×
ls
The implication of the first axiom will be that the indifferent sets must be one
dimensional lines rather than two dimensional shapes. If the indifference set was a shape
with the dimension of height, there would be two points in the set that had the same
leisure but different consumption levels. The worker/consumer must prefer the higher
point so they could not be indifferent between all the points in the set. If the indifference
set had the dimension of width, than there would be two points in the set with the same
level of consumption but different levels of leisure. Since the worker/consumer would
prefer the right most point with the most leisure, they could not be indifferent. We call
the one dimensional indifference sets, indifference curves.
Points on higher indifference curves and rightmost indifference curves are
preferred according to the more is better axiom. Consider indifference curves U1 and U2
C
D
×
U4
E
×
A
×
B
×
U3
U2
U1
ls
Because point A is higher than point B we can say that point A is prefrred to point point
B. Because all of the points on curve U2 are equally valued as point A, then all of the
points on curve U2 (for example point E) are equally preferred to point B. But as all of
the points on curve U1 are equally valued as point B, then all of the points on curve U2
are preferred to all the points on U1. So point E should be preferred to point D, even
though it has a little less consumer goods (along with quite a bit more leisure time).
Similarly, all of the points on U3 are preferred to points on U2; all of the points on U4 are
preferred to points on U3.
We can also say that these indifference curves would never cross each other. If
two indifference curves, called, say, U1 and U2. crossed each other, then in one part of the
quadrant U1 would be higher than U2 and on another point U2 is higher than U1. This
would mean that where U2 was higher, points on U2 would be preferred to being on U1.
But if U2 and U1 are indifferences curve then all points on U2 should be preferred to all
points on U1. But since the two curves cross, there are places where U1 is higher than U2.
This would contradict the principle of more is better.
.
Consider the slope of an indifference curve between points A and B, ΔC
Δls
C
ΔC
×
B
Δls
A
×
U1
ls
The slope is negative of course. A person would be indifferent between point B (which
has higher consumption) and point A only if point A has the compensatory feature of
more leisure. We can say the negative of the slope is the maximum amount of
consumption that the worker/consumer would be willing to give up in order to get more
leisure (per unit of leisure). If they do give up ΔC and get Δls, they will be just as happy.
If we assume the indifference curves are smooth, then we can refer to the pointwise slope
of the indifference curve as the negative of the marginal rate of substitution,
MRS = − ∂C
so called because it is the rate at which the worker/consumer will
∂ls
substitute leisure for consumption.
The second assumption we will make is Axiom 2: Each Good has Diminishing
Returns. We assume that when a worker/consumer is working all the time, the value of a
bit of leisure time would be greater than the value of some more leisure time to the
worker/consumer if they were sitting around the house all day. We also assume that the
value of an extra amount of consumer goods would be worth much more to someone who
was not enjoying very much of the fruits of wealth compared to how much they would be
worth to the same person if they were already awash in consumer goods.
The implication of the second axiom is that the slopes of the indifference curves
get flatter as you move toward the right. Remember, the slope of the indifference curve is
the amount of consumption you would be willing to give up to get some extra leisure
time and still be just as happy. Toward the left hand side of the graph, leisure time is
scarce, so the amount of consumption that the household is willing to give up to get some
would be great. On the other hand, toward the right hand side, the worker/consumer
would already have a lot of free time and would not be willing to give up a lot of
consumption to get some more.
Sometimes economists put another axiom on preferences which is Axiom 3:
Consumption and leisure are normal goods. This axiom says that if we compare two
points on different indifference curves where both indifference curves have the same
slope, then the point that is on the higher indifference curve will feature both more
consumption and leisure. Intuitively, this means that all else equal we need both more
consumption and leisure to make us happier.
Though the indifference curve framework offers many intuitive ways of thinking
about preferences that we can draw on a graph and observe. However, the geometric
approach can be limited. First, if we want to examine labor numerically, the geometric
approach may be non-specific. Moreover, graphs are inherently limited to two
dimensions and we may want to consider cases where agents value more than two
commodities. To deal with these cases, we often use an algebraic tool to rank the
different consumption-leisure combinations which we call the Utility function. The utility
function is an algebraic function which maps consumption leisure combinations into an
actual number.
U t = U (Ct , lst )
(1.1)
A utility function will represent preferences if consumption and leisure combinations that
are preferred get higher utility scores.
One question about the utility function is, “what is the unit of measure of utility?”
For our purposes, however, we will not need to answer this question. We can think of the
utility function as simply a scoring function, giving scores for different combinations of
consumption and labor. As long as the function gives higher scores, it won’t matter what
those scores mean. In this way, utility is just like the score of a basketball game. A
basketball team gets a certain number of points as a function of the number of field goals,
long range field goals, and free throws that they made. It does really matter what units
“points” are measured in, all that matters is that better combinations of shots get more
points. Some economists like to measure utility in something called “utils” which they
think of as some sort of measure of happiness. That is not necessary for our purposes.
We can easily described the first two axioms of preferences with the utility
function. Thinking of the utility function as smooth and continuous, we can examine the
partial slopes of the function. The first axiom, more is better, means that these slopes are
positive. An increase in consumption or leisure (holding the other constant) always
increases utility.
∂U
>0
MU C ,t ≡ U1 (Ct , lst ) ≡
∂C
(1.2)
∂U
>0
MU ls ,t ≡ U 2 (Ct , lst ) ≡
∂ls
Call the partial slope with respect to consumption, the marginal utility of consumption
and the partial slope with respect to leisure, the marginal utility of leisure. More is better
means that both marginal utilities are always positive.
The second axiom of diminishing returns means that the extra benefit of both
consumption and leisure in terms of their impact on utility decreases as consumption and
leisure, respectively, gets larger.
∂MU C ,t ∂ ∂U ∂C ∂ 2U
=
≡
<0
∂C
∂C
∂C 2
(1.3)
U
∂
2
∂
∂MU ls ,t
∂ls ≡ ∂ U < 0
=
∂ls
∂ls
∂ls 2
The marginal utility of consumption and the marginal utility of leisure are both positive
but diminishing functions.
U( C ,ls)
U(C, ls )
MUls
MUC
C
ls
One of the easiest utility functions to work with is the log-log utility function
U t = ln Ct + Γ ln lst
(1.4)
Where the parameter Γ> 0 simply weights the importance of leisure relative to
consumption. This function meets the axioms remember when f(x) = ln(x),
2
f '( x) = df
= 1 and also f ''( x) = d f 2 = −1 2 So we can write:
dx
x
dx
x
∂U
= 1 > 0; ∂U
=Γ >0
∂C
C
∂ls
ls
(1.5)
2
∂U
∂
U
−
1
−Γ
0;
0
=
<
=
<
∂ls
C2
ls 2
∂C 2
The second part of calculating the optimal amount of labor will be to decide
which are the set of consumption leisure combinations that the worker consumer might
choose from. For any worker, the amount of leisure that they might consume is limited by
the number of hours in the day. Refer to the endowment of time in a day as TIME. This
time may be used for either labor or leisure
Lt + lst = TIME
(1.6)
For any consumer, the amount of consumption is limited by budget. Consumption cannot
exceed income. Divide real income into two parts: 1) Labor income; and 2) non-labor
Wt
⋅L
Pt t
Call non-labor income which might include interest income or other types of financial
income plus any social welfare payments from the government less taxes. Denote nonletter income with the greek letter Πt.
W
Ct = t ⋅ Lt + Π t
(1.7)
Pt
Combining (1.6) and (1.7) we get a budget equation that shows directly the trade-off
between leisure and consumption.
W
Ct = t ⋅ (TIME − lst ) + Π t
(1.8)
Pt
For each level of leisure, there is a level of consumption which could be affordable. The
household could use all of its time for leisure, in which case consumption would be
limited to non-labor income: Πt. For each hour of leisure that the worker/consumer was
W
willing to give up to work, they will get an additional unit of consumption t . If they
Pt
W
spend all their time in work and have lst = 0, then they can consume t ⋅ TIME + Π t .
Pt
The levels of consumption which are consistent with given levels of leisure are a group of
combinations of {Ct,lst} which we can map on the quadrant.
income. Labor income is the real wage multiplied by the total quantity of labor,
C
Wt
Pt
⋅ TIME + Π t
Wt
Pt
1
Πt.
TIME
ls
The worker/consumer must choose from the point on this set. Which of these points is
most preferred? The point which is on the highest indifference curve will be the first
choice. What would that indifference curve look like. First, of all, it could not be an
indifference curve that never touches the budget line. No point on such an indifference
curve would be feasible as a choice. Second, we can rule out any indifference curve that
crosses the budget line. Any indifference curve that crosses the budget line will have two
points on the budget line, like A and B, which might be potential choices. However, it
would also have a point like D directly under the budget line. Since this would be directly
below a point like E which features the same leisure as D and more consumption.
Therefore, E must be better than D. But since D is equally preferred as A and B, E must
be better than A and B. Therefore, any points on an indifference curve that crosses the
budget curve cannot be the best choice.
C
Wt
Pt
⋅ TIME + Π t
B
×
E
×
×
D
×A
Πt.
TIME
ls
If the optimal point cannot be on an indifference curve that does not touch the budget
line and it cannot be on a curve that crosses the line, it can only be on a curve that
touches but does not cross the line. When two curves, touch but do not cross they are said
to be tangent. The optimal consumption leisure combination {C*,ls*} would be on such a
tangent point.
C
Wt
Pt
⋅ TIME + Π t
{C*,ls*}
×
Πt.
TIME
ls
When two lines are tangent, they have the same slope at the point of tangency. This
makes sense as the slope of the indifference curve is the maximum amount of
consumption that the consumer/worker would be willing to give up to get more leisure
while the slope of the budget line is the amount of extra consumption you would actually
have to give up to get more leisure. Only when these too are equal, could we be at an
optimal point. If the household chose a point to the left of {C*,ls*} the slope of the
Wt
. In other words, the amount of
Pt
consumption you would have been willing to give up to get more leisure would be more
than the amount you would need to give up to get the leisure. You could improve your
situation by taking more leisure. To the right of {C*,ls*}, the slope of the indifference
W
curve would be flatter than t , so the amount of consumption you would be willing to
Pt
give up to get leisure would have been less the what you would need to give up. In that
situation, you should give up some leisure. Only when the slopes are equal can you be at
the right amount of leisure.
We can also use calculus to solve for the optimal amount of leisure. The utility
function itself U t = U (Ct , lst ) has no maximum. The worker/consumer would, if given a
choice, have an infinite amount of consumption and leisure. However, the
worker\consumer can only choose from amongst those points such that T. Sometimes we
write the problem of an agent trying to maximize one function while maintaining equality
of another function in the form
max U t = U (Ct , lst )
(1.9)
W
subject to Ct = t ⋅ (TIME − lst ) + Π t
Pt
There are different ways that we might maximize utility subject to the budget constraint.
The first is to directly insert the budget constraint function into the utility function. That
is, if the worker/consumer honors the budget constraint, we can write consumption
directly as a function of leisure Ct = C(lst) where the function C(.) is the budget
constraint. Then we would write the utility function as
U t = U (C (lst ), lst )
(1.10)
So that utility is a function of leisure both directly and indirectly through its impact on
the amount of consumption that can be done. Leisure has positive direct impacts but
negative direct impacts so once the budget constraint is honored, the worker/consumer
would not want to choose optimal leisure. Thus, equation (1.10) is a function of ls which
looks like a hill where the slope is only flat right at the very peak. We need to solve for
the level of lst at that peak. When a variable affects a function directly and indirectly, we
need to examine the both the direct and indirect slope to find the total slope. If y =
dy ∂h df ∂h
h(f(x),x) then
= ⋅ + . In this situation h( ) is the utility function, f is
dx ∂f dx ∂x
consumption in the budget constraint and x is equivalent to leisure. So
dU ∂U dC ∂U
(1.11)
=
⋅
+
dls ∂C dls ∂ls
dC
W
Note the slope of the budget constraint with respect to leisure is
The
=− t
Pt
dls
dU
optimum is found where
=0. So the first order condition is
dls
indifference curve would have been above
∂U ∂U
+
=0→
Pt ∂C ∂ls
(1.12)
Wt
MU C ⋅
= MU ls
Pt
We can think of the right hand side of this equation as the marginal benefit of leisure. The
left hand side is the marginal cost of leisure. Every unit of leisure taken means that the
worker/consumer must give up a unit worth of real wages. But the value of real wages is
weighted by the value of the extra consumer goods that might be purchased with the extra
wages. If the marginal utility of leisure exceeded the marginal cost, the worker would
improve outcomes by working less and taking more free time. But since there are
diminishing returns, taking more free time will push down the marginal utility of free
time. The worker/consumer should, however, keep taking more free time until they have
pushed down the extra benefit of another unit of leisure to the marginal cost in terms of
the value of wages.
We could use the example of the log-log utility function to examine this technique.
max U t = ln Ct + Γ ln lst
−
s.t. Ct =
Wt
Pt
Wt
⋅
⋅ (TIME − lst ) + Π t
W
max U t = ln ⎡ t ⋅ (TIME − lst ) + Π t ⎤ + Γ ln lst
⎣⎢ Pt
⎦⎥
dU
=
dls Wt
−
Pt
Wt
Pt
⋅ (TIME − lst ) + Π t
+
(1.13)
1
Γ
Γ
W
→ t ⋅
=
Pt C
lst
lst
Nt N
MU C
MU ls
Another way of maximizing one function (referred to as the objective function)
subject to some constraint is the Lagrangian multiplier method. The Lagrangian method
is to maximize the objective function but impose some penalty if you overshoot the
constraint. In this particular case, we know that a worker/consumer left to his own
devices would want to consume an infinite amount or at the very least, they would like to
consume more than their budget constraint would allow. Any amount of consumption
beyond the budget constraint would be overspending.
W
(1.14)
Overspendingt = Ct − ⎡ t ⋅ (TIME − lst ) + Π t ⎤
⎢⎣ Pt
⎥⎦
The utility function is a scoring function that gives some scores for different levels of
consumption and leisure. The Lagrangian method simply modifies the scoring function to
subtract a penalty for cheating by overspending. The size of the penalty for each dollar of
over-spending is λt which is called the multiplier. Maximize:
max Lt = U t − λt ⋅ Overspendingt
(1.15)
W
max Lt = L(Ct , lst ; λt ) = U (Ct , lst ) − λt Ct − ⎡ t ⋅ (TIME − lst ) + Π t ⎤
⎣⎢ Pt
⎦⎥
{
}
Applying a penalty for overspending will limit the overspending the consumer/worker
would be willing to do. We would find the penalty adjusted maximum
∂L
∂U
∂Overspending
∂U
=0→
− λt ⋅
=0→
= λt
∂C
∂C
∂C
∂C
∂L
∂U
∂Overspending
∂U Wt
=0→
− λt ⋅
=0→
=
⋅λ
Pt t
∂ls
∂ls
∂ls
∂ls
(1.16)
The first part of (1.16) says that the level of consumption that maximizes L will be set so
that the worker/consumer keeps increasing their consumption until the marginal benefit
of consumption is pushed down to the marginal cost which in this case is the penalty for
overspending. The second part says that the level of leisure is set so that leisure will be
pushed up until the marginal benefit equals the marginal cost which is the penalty for
overspending multiplied by the impact of leisure on overspending which is the real wage.
Solving equation (1.16) we get
∂U Wt
∂U
=
⋅
(1.17)
Pt ∂C
∂ls
Simply by setting some penalty for overspending, we have forced the consumer worker to
take into account the trade-off between leisure and consumption and to find the proper
first order condition. However, if we set λt too small, the consumer/worker will have an
incentive to overspend. If we set λt too large, they will have an incentive to underspend
relative to the budget constraint. If we set it exactly correct, then they will neither
overspend nor underspend and they will both set the marginal cost of leisure equal to the
marginal benefit and choose a level of consumption and leisure consistent with the
budget constraint. We can solve for the appropriate level of the penalty by solving three
equations for three unknowns (Ct, lst, λt ):
∂U
= λt
∂C
∂U Wt
=
⋅λ
Pt t
∂ls
W
Ct = ⎡ t ⋅ (TIME − lst ) + Π t ⎤
⎢⎣ Pt
⎥⎦
As an example, we can assume a log log utility function.
(1.18)
Wt
Γ
1
W
= & Ct = ⎡ t ⋅ (TIME − lst ) + Π t ⎤ →
Pt C ls
⎢⎣ Pt
⎥⎦
t
t
⋅
Wt
Pt
=
Γ
→
lst
⎡Wt ⋅ TIME − ls + Π ⎤
t)
t⎥
⎢⎣ Pt (
⎦
Wt
W
lst = Γ ⎡ t ⋅ (TIME − lst ) + Π t ⎤ →
Pt
⎣⎢ Pt
⎦⎥
⎡
⎤
Πt ⎥
⎢
→
lst = Γ ⎢(TIME − lst ) +
Wt ⎥
⎢
⎥
Pt ⎦
⎣
⎡
⎤
⎡
⎤
Πt ⎥
Πt ⎥
⎢
Γ ⎢
→ lst =
(1 + Γ) ⋅ lst = Γ ⎢TIME +
TIME +
Wt ⎥
Wt ⎥
1+ Γ ⎢
⎢
⎥
⎢
Pt ⎦
Pt ⎥⎦
⎣
⎣
W
C = ⎡ t ⋅ (TIME − lst ) + Π t ⎤
⎣⎢ Pt
⎦⎥
(1.19)
⎡
⎤
⎛
⎡
⎤⎞
⎢W
⎥ W
⎜
⎟
Π
⎢
⎥
TIME Π t
Γ
(1.20)
TIME + t ⎥ ⎟ + Π t ⎥ = t ⋅
= ⎢ t ⋅ ⎜ TIME −
+
⎢
Pt
Pt 1 + Γ 1 + Γ
W
+
Γ
1
t
⎢
⎥
⎜
⎟
⎢
Pt ⎥⎦ ⎠
⎢⎣
⎥⎦
⎣
⎝
1
1
λt = =
Π
TIME
Ct Wt ⋅
+ t
Pt 1 + Γ 1 + Γ
Wages and Labor Supply
Consider the first order condition.
W
MU ls (ls ) = t ⋅ MU C (C )
(1.21)
Pt
−
−
The marginal utility of leisure is a negative function of leisure and the marginal utility of
consumption is a negative function of leisure. If the real wage drops, then the marginal
utility of leisure must rise relative to the marginal utility of consumption. Holding
consumption constant, a rise in the marginal utility of leisure will come about only if
leisure falls. The worker is taking leisure up until the point where the extra benefit of
some more free time equals the cost. If the cost of taking an extra unit of leisure goes up,
the point of at which they will stop taking free time will come sooner. If leisure falls, then
labor rises. We could think of this positive relationship between labor and real wages,
holding consumption constant, as a labor supply curve.
W
LS|C
P
w*
LD
L*
L
One way to think about this is to review our example log-log utility function. We would
write the first order condition as
−1
ls
1
Γ
Γ
Wt
W
(1.22)
⋅ =
→⋅ t =
= Γ ⋅ ⎛⎜ t ⎞⎟
Pt C ls
Ct Wt
⎝ Pt ⎠
t
t
Pt
The demand for leisure relative to the demand for consumption is a negative function of
the price of leisure. Wt, relative to the price of consumption Pt. When the relative price of
leisure goes up, the worker/consumer substitutes away from expensive leisure and toward
cheaper consumption. This effect of the real wage on leisure and labor is called the
substitution effect.
W
On the other hand, consider the budget constraint Ct = ⎡ t ⋅ (TIME − lst ) + Π t ⎤ .
⎢⎣ Pt
⎥⎦
Again holding the level of consumption constant, a rise in the level of real wages means
that leisure can go up. The amount of work needed to hit any given level of consumption
is reduced when the wage for that work goes up. This means that there is a counterveiling effect of the real wage on leisure, you simply don’t need to work as hard to earn
income when real wages are high. This counter-veiling channel is called the income effect.
Another way of thinking about this is that when real wages goes up, at any given level of
leisure, consumption will rise. This means that the marginal utility of consumption falls.
W
Recall the marginal cost of leisure is t ⋅ MU C (C ) . Increasing consumption means that
Pt
−
the value of that wage to the consumer is reduced. Therefore, the rising real wage will
directly push up the marginal cost of leisure, but also indirectly reduce the price of leisure
by making the consumption basket more affordable. Whether the income or substitution
effect is stronger is not obvious.
We might think about what this would mean for the equilibrium labor market.
Consider what would happen if technology increased the marginal product of labor
shifting labor demand outward. Higher labor productivity would increase the wages that
firms were willing to offer. Holding consumption constant, the higher wages would cause
the workers to substitute leisure for labor. However, the higher wages would also
increase consumption, reducing the amount of labor that the agents would want to
provide. The labor supply curve would shift inward. The equilibrium wage would rise,
but the effect on labor would be ambiguous.
W
LS|C
P
w**
w*
LD
L*
L
To see the determinants of the outcome, we might explore some examples. The
first example will assume zero non-labor incomeΠt = 0. WhenΠt.=0, a 1% increase in the
real wage rate will lead to a 1% income holding leisure constant. Assume the preferences
are of the form:
U t = ( Ctψ + Γ1−ψ lstψ )
1
ψ
ψ <1
(1.23)
Holding leisure constant, we can write utility as a function of a function of a function U =
1
f(g(h(C))) where f ( g ) = g ψ , g (h) = A + h, A ≡ Γ1−ψ lsψ , and h(C ) = Cψ . The chain rule
of calculus says that the slope of a function of a function is the product of the slope of
1 1−ψ
∂U df
df dg dh
df 1 1ψ−ψ 1
dg
= ψ ⋅ g = ψ ⋅ gψ
and
=1
each function
=
=
⋅ ⋅
. Going back
∂C dC dg dh dC
dg
dh
( )
df
dg
dh
dC
1−ψ
dh
∂U df
ψ −1
ψ −1
ψ
1
=
= ⋅ g ⋅ 1N ⋅ψ C . Note that
and
= ψ C . So
∂C dC ψ
dC
dg
dh
1
ψ
1−ψ
( )
1
⋅ g ψ = ψ1 ⋅ g ψ
1−ψ
= ψ1 ⋅ ( f )
1−ψ
= ψ1 ⋅ (U )
1−ψ
. Cancelling out we derive
∂U
= Ctψ −1 ⋅ U t1−ψ .By a similar logic, holding C constant we can write utility as a
∂C
function of a function of leisure, f(g(j(ls))) where j (ls ) = Γ1−ψ lsψ and the constant A is
dj
redefined as A ≡ Γ1−ψ lsψ so
= Γ1−ψψ lsψ −1 we can write
dls
MU C =
df
dj
dg
dls
1−ψ
∂U df
df dg dj 1 ψ
1−ψ
ψ −1
=
=
⋅ ⋅
= ⋅ g ⋅ 1N ⋅ Γ ψ ls . Canceling, we can write
∂ls dls dg dh dls ψ
dg
dh
∂U
∂U
∂U
> 0 and
= Γ1−ψ lstψ −1 ⋅ U t1−ψ . Note, in each case, more is better
>0
∂ls
∂ls
∂C
∂MU ls
1−ψ
ψ −2
and in each case we see diminishing returns
=Γ
ψ − 1) ⋅ ls
⋅ U t1−ψ < 0 and
N ⋅ (
t
N
N
∂ls
+
MU ls =
+
−
+
∂MU C
1−ψ
ψ −2
1−ψ
ψ − 1) ⋅ C
=Γ
⋅U
<0
N ⋅ (
t
t
N
N
∂C
+
+
+
−
For this equation we write the first order condition as
W
W
W
MU C ,t ⋅ t = MU ls ,t → Ctψ −1 ⋅ U t1−ψ ⋅ t = Γ1−ψ ⋅ lstψ −1 ⋅ U t1−ψ → Ctψ −1 ⋅ t = Γ1−ψ lstψ −1 (1.24)
Pt
Pt
Pt
1
Raise both sides of this equation to power
1 −ψ
1
1
1
⎡ ψ −1 Wt ⎤ψ −1
⎡Wt ⎤ψ −1
ψ −1 ψ −1
−1
1−ψ
⎡
⎤
C
ls
C
⋅
=
Γ
→
⋅
⎢ t
⎥
⎥ = Γ lst →
t
t ⎢
⎣
⎦
P
P
t ⎦
⎣
⎣ t⎦
1
ψ −1
(1.25)
⎡W ⎤
ls
Γ⋅⎢ t ⎥ = t
Ct
⎣ Pt ⎦
So the level of leisure demanded relative to consumption is a function of the price of
leisure relative to the price of consumption. Call the ratio of leisure to consumption lcratt
1
1
⎡W ⎤ψ −1
ls
dlcratt
1 ⎡ Wt ⎤ψ −1
= Γ⋅
lcratt ≡ t = Γ ⋅ ⎢ t ⎥ →
⎢ ⎥
W
ψ − 1 ⎣ Pt ⎦
Ct
⎣ Pt ⎦
d t
Pt
Wt
d lcratt
Pt
Convert
into elasticity form by multiplying by
W
lcratt
d t
Pt
ls
Wt
d lcratt
%Change in t
d lcratt
Pt
lcratt
Ct
⋅
=
=
Wt
W
W
lcratt
d
d t
%Change in t
Pt
Pt
Pt
Wt
Pt
Γ⋅
=
1
−1
ψ −1
1 ⎡Wt ⎤
ψ − 1 ⎢⎣ Pt ⎥⎦
lcratt
N
1
W
ψ −1
Γ⋅⎡⎢ t ⎤⎥
⎣ Pt ⎦
⋅
Wt
Pt
Γ⋅
=
1 ⎡Wt ⎤
ψ − 1 ⎢⎣ Pt ⎥⎦
W
Γ⎡ t ⎤
⎢⎣ Pt ⎥⎦
1
ψ −1
−1
1
ψ −1
=
(1.26)
(1.27)
1
ψ −1
We call the %drop (the negative of the % change) in relative demand for two goods with
respect to the %change in the relative price, the price elasticity of substation of the two
1
goods . In this case, the elasticity of substitution between the two goods is
This is a
1 −ψ
useful measure of how good a substitutes two goods are. Consider if ψ = 1, then
U t = ( Ct + lst ) , utility would be only the sum of leisure plus consumption. A unit of
consumption and a unit of leisure are always perfectly interchangeable. The elasticity of
1
substitution is
=∞. If the two goods are perfect substitutes, then the
1 −ψ
consumer\worker will put all of their effort into whichever activity has the lowest price.
As ψ gets lower, the price sensitive of the relative demand is reduced. Note, that when ψ
= 0, then the first order condition is equivalent to the log-log utility function which is
sometimes referred to as the unit elasticity of substitution function.
Consider the solution for the effect of real wages on labor. Solve the equations:
1
⎡W ⎤ψ −1
lst
W
& Ct = t ⋅ (TIME − lst ) →
= Γ⋅⎢ t ⎥
Pt
Ct
⎣ Pt ⎦
1
1
⎡W ⎤ψ −1
⎡ W ⎤ψ −1 Wt
= Γ ⋅ ⎢ t ⎥ → lst = Γ ⋅ ⎢ t ⎥
⋅ TIME − lst ) →
Pt (
⎣ Pt ⎦
⎣ Pt ⎦
⋅ (TIME − lst )
lst
Wt
Pt
ψ
ψ
⎡ W ⎤ψ −1
⎡W ⎤ψ −1
lst = Γ ⋅ ⎢ t ⎥ ⋅ (TIME − lst ) Γt * ≡ Γ ⋅ ⎢ t ⎥
⎣ Pt ⎦
⎣ Pt ⎦
Γt *
*
*
+
Γ
⋅
=
Γ
⋅
→
=
⋅ TIME →
1
ls
TIME
ls
( t) t t
t
1 + Γt *
Γt*
1
⋅ TIME =
⋅ TIME
Lt = TIME − lst = TIME −
*
1 + Γt
1 + Γt *
Lt =
1
ψ
ψ −1
⋅ TIME
W
1+ Γ ⋅ ⎡ t ⎤
⎢⎣ Pt ⎥⎦
(1.28)
So we see that whether a change in real wages, increases the optimal labor supply or
reduces it depends on the sign of
ψ
ψ
If 0 < ψ < 1, then
< 0 . In this case, a rise
ψ −1
ψ −1
ψ
ψ −1
W
in real wages makes ⎡ t ⎤ smaller and makes the denominator smaller and makes the
⎢⎣ Pt ⎥⎦
1
to be larger. On the other hand, if ψ < 0, then
the fraction of time spent working
1 + Γt *
ψ
> 0 so , a rise in real wages makes
ψ −1
ψ
⎡Wt ⎤ψ −1 larger and makes the denominator
⎢⎣ Pt ⎥⎦
1
to be smaller. Intuitively,
smaller and makes the fraction of time spent working
1 + Γt *
when the substitution effect is strong, (i.e. when ψ is relatively close to 1 and the
elasticity of substitution is high), it will outweigh the income effect.
An increase in the real wage will lead the worker/consumer to so greatly
substitute consumption for leisure, that the household will want to work harder when the
real wage is high. On the other hand, when the substitution effect is weak (i.e. when ψ is
relatively low and the elasticity of substitution is near to zero) the household doesn’t want
to substitute consumption for leisure very much when real wages rise. Combine the weak
substitution with the fact that the income effect is raising consumption anyway, reducing
the value of extra wages on the margin, and we see why an increase in real wages might
reduce the optimal labor supply.
1
Note when ψ = 0 and the elasticity of substitution is
=1, then Γt * ≡ Γ and
1 −ψ
the real wage has zero effect on labor supply. Remember, if leisure is held constant, a 1%
rise in the real wage will raise consumption by 1%. In the unit elasticity case, the leisure
to consumption ratio must fall by 1% if the real wage goes up by 1%, but even without
leisure changing at all, consumption will rise by 1% which would reduce the leisureconsumption ratio by 1%. Thus, the 1% income effect exactly matches the 1%
substitution effect and no change in labor is indicated.
When goods are better than unit substitutes on the other hand, a 1% rise in the real
wage will require more than a 1% decline in the leisure-consumption ratio. With a 1%
income effect, this cannot be achieved without reducing leisure. When goods are worse
than unit substitutes on the other hand, a 1% rise in the real wage rate will cause a less
than 1% decline in the leisure-consumption ratio. Since consumption would rise by 1% if
leisure did not increase, a smaller than 1% decline in the leisure-consumption ratio
requires leisure to actually rise.
Lets consider another case, that in which ψ =0 and non-labor income Π t ≠ 0 . In
this case, a 1% rise in real wages will have (holding leisure constant) increase income by
less than 1% if Π t > 0 and by more than 1% if Π t < 0 . Conceivably, if taxes were
sufficiently large, we could see Π t < 0 . Solving for the labor supply as a function of real
wage
⎡W ⎤
lst
= Γ⋅⎢ t ⎥
Ct
⎣ Pt ⎦
−1
& Ct =
Wt
Pt
⋅ (TIME − lst ) + Π t →
−1
⎡W ⎤
= Γ⋅⎢ t ⎥ →
⎣ Pt ⎦
⋅ (TIME − lst ) + Π t
lst
Wt
Pt
−1
⎡W ⎤
lst = Γ ⋅ (TIME − lst ) + Γ ⋅ ⎢ t ⎥ ⋅ Π t →
⎣ Pt ⎦
Π
Π
Γ
Γ
⋅ TIME +
⋅ t →
(1 + Γ ) ⋅ lst = Γ ⋅ TIME + Γ ⋅ W t → lst =
(1 + Γ )
(1 + Γ ) Wt
t
Pt
Pt
Lt = TIME − lst = TIME −
Π
Π
1
Γ
Γ
Γ
⋅ TIME −
⋅ t =
⋅ TIME −
⋅ t
W
W
1+ Γ
(1 + Γ )
(1 + Γ ) t
(1 + Γ ) t
Pt
Pt
An increase in
Wt
Pt
makes
Πt
Πt
closer to zero. If
< 0, then labor supply will be
Wt
Wt
Pt
Pt
Πt
> 0, then labor supply will increase.
Wt
Pt
Note also that Π t has a negative effect. The more positive is Π t , the lower is
leisure. Non-labor income has a pure income effect. A higher level of non-labor income
increases consumption affordability which would reduce the marginal utility of another
hours worth of wages. This has the effect of reducing the marginal cost of leisure.
Intuitively, if Π t < 0 then the income effect is more than 1%, which is greater than
the 1% substitution effect. When the income effect (which increases the amount of leisure
desired) is stronger than the substitution effect (which reduces the amount of leisure
desired), than a rise in the real wage will increase the amount of leisure and reduce the
amount of labor. When Π t > 0 , the income effect is less than 1% and the substitution
effect is stronger. So, a rise in the real wage will increase the optimal amount of labor.
Think about the impact of taxation on labor supply. Consider two kinds of tax,
labor taxes and non-labor taxes. So write the income as:
W
Ct = t ⋅ Lt + NonLabor Incomet − TAX t
Pt
(1.29)
TAX t = Labor Income Taxt + Net NonLabor Taxt
Typically, income taxes increase in the amount of income that is earned. Simplify
the tax code, and assume that labor income taxes are proportional to labor income:
W
(1.30)
Labor Income Taxt = τ ⋅ t ⋅ Lt
Pt
= Non labor Income − Nonlabor Tax so we can
Redefine after-tax non-labor income, Π
smaller. If
t
t
t
write the after tax income as
Wt
⋅ TIME − lst ) + Π
(1.31)
t
Pt (
To find the optimal level of consumption-leisure, maximize the Lagrangian function
W
⎤
(1.32)
Lt = U (Ct , lst ) − λt Ct − ⎡ (1 − τ ) ⋅ t ⋅ (TIME − lst ) + Π
t⎥
Pt
⎣⎢
⎦
Ct = (1 − τ ) ⋅
{
The first order conditions are:
∂U
∂U
W
= λt
= (1 − τ ) ⋅ t ⋅ λt
Pt
∂C
∂ls
W
⎤
Ct = ⎡(1 − τ ) ⋅ t ⋅ (TIME − lst ) + Π
t⎥
Pt
⎢⎣
⎦
We can write the first order condition
W
MU ls ,t = (1 − τ ) ⋅ t ⋅ MU C ,t
Pt
}
(1.33)
(1.34)
Wt
W
≡ (1 − τ ) ⋅ t as the after tax wage rate. We can see that the first order
Pt
Pt
conditions in terms of the after-tax wage rate.
W
MU ls ,t = t ⋅ MU C ,t
Pt
(1.35)
⎡Wt
⎤
Ct = ⎢
⋅ (TIME − lst ) + Π t ⎥
⎣ Pt
⎦
A change in the labor tax rate, τ, will affect labor supply directly through its effect
on after-tax wages. Thus, its impact will depend on whether the income or substitution
effect is stronger. Non-labor income taxes will have only an income effect. A cut in nonlabor income taxes financed by a rise in labor income taxes should reduce labor NEEDS
MORE.
Define
Search Models
The supply-demand model of the labor market imagines that there is some price (i.e. the
real wage) at which supply is equal to demand. In this paradigm, only workers who
would be unwilling to work at rates above the going rate in the labor market would not
work. However, one aspect of the labor market that we think we observe is
unemployment. There are, at any time, workers who are looking for jobs, who would be
willing to work at the going rate in the labor market.
Unemployment is measured using survey methods. In any month or week,
statistical authorities call a large number of adults and ask them 2 questions. The first
question is “Are you gainfully employed?” If the answer is yes, they count you as
employed (Et) and hang up1. If the answer is no, they ask “Have you taken actual steps
to seek work in the last week or month?” If you answer no, they will categorize you as
out of the labor force and if you say yes, they will categorize you as unemployed (Ut).
The labor force is the sum of the employed and the unemployed LFt = Ut + Et. and the
Ut
unemployment rate is the ratio of the unemployed to the labor force urt ≡
.
U t + Et
Given the cross-country variation in unemployment rates that we observe, it
would be useful to model unemployment. One explanation for unemployment is that it is
frictional in nature. For various reasons, people leave jobs or rejoin the labor force. When
they do so, their first step will be to find a job. Given that most people will want to find a
position that will be a good match for their particular skills and personality, they will
likely want to check out a variety of employers to find that match. This will take some
time, so at any point in time when government statisticians call, there will be some
people in this process and therefore unemployed.
We project a simple model in which there are two states of the world, employed
and unemployed. In any period, a fraction of the people will be in the process of moving
back and forth between the two groups. In a more complicated and realistic model, we
would have people moving in and out of the labor force. For simplification, we
concentrate on two. We assume that in every period a fraction of the employed, 0 < s < 1
lose their job for random, idiosyncratic reasons (such as a fight with their boss or a desire
1
In actuality, they will request a wide range of demographic information.
to change geographical locations, etc.). Unemployed people are, by definition, trying to
find a job. The fraction of unemployed people that find a job in a given period is f. Since
the fraction of the population that is unemployed is ur, therefore the fraction of the total
population that finds a job is f×ur. The fraction of the population that is employed is 1-ur,
so the fraction of the population that loses a job is s×ur.
st×(1-urt) %
Job Losers
s×Et
Unemployed: Ut
Employed: Et
Job Finders
f·Ut
ft×urt %
If the number of people who find a job exceeds the number of people who lose a job, then
the unemployment rate will shrink. If the number of people who lose a job exceeds the
number who find a job, then the unemployment rate will grow. Draw a phase diagram
showing the path of unemployment.
s×(1-ur) < f×ur
0
We might draw these
s×(1-ur) < f×ur
ur
s
f×ur
s×(1-ur)
ur
urSS
1
When s ⋅ (1 − ur ) = f ⋅ ur the unemployment rate is neither growing nor shrinking. We
could call this the steady state unemployment rate. Solving for the steady state, it is at:
s ⋅ (1 − ur SS ) = f ⋅ ur SS → s = s ⋅ ur SS + f ⋅ ur SS →
s
1
(1.36)
=
ur SS =
s + f 1+ f
s
The unemployment rate in steady state is increasing in the job separation rate but
decreasing in the job finding rate. The more people that are losing their job, the more will
be in a state of unemployment at any given time. The faster that people find a job, the less
will be the unemployment rate.
Imagine a country is in steady state at time 0 and the job finding rate goes up.
Then, the number of unemployed people finding jobs would rise. The unemployment rate
would fall, meaning that fewer people will lose jobs and more people will find them. This
process continues until the new, lower steady state unemployment would be reached.
s
s×(1-ur)
f×ur
ur
ur0
1
If the job separation rate goes up, on the other hand, we would see an immediate jump in
the number of people losing their jobs. This would lead to a higher unemployment rate.
s
f×ur
s×(1-ur)
ur
urSS
1
The above describes a fairly generic set of dynamics. To make this richer, we might add a
theory explaining the job finding rate. Assume that jobs have a fixed number of hours, H ,
so the only labor supply decision that workers make is whether to work or not to work.
The income level of the worker will depend on whether or not they are employed and on
the real wage they receive.
U E (W )
P
U U (b)
W
P
w
Assume that the an employed worker receives zero non-labor income so their
W
W
consumption is Ct = t × H when their wage is t . The utility of the worker would
Pt
Pt
R
be given by U tE = U( Ct, lst) = U(wt× H , TIME- H ) . The utility of the employed worker
depends on the real wage as shown above. This utility function would be increasing in the
wage, but also diminishing. Assume that an unemployed person receives an
unemployment benefit, b, from the government so that the utility of an unemployed
person U tU = U(b,TIME). We could draw this level to compare with the utility of the
worker. There will be a level of the real wage, wR, at which the utility of the unemployed
W
person is exactly equal to the utility of an employed person with t = wR. We call wR
Pt
the reservation wage as any worker who received a wage offer below wR would reserve
their labor, preferring to remain unemployed rather than accept a job that did not offer a
worthwhile wage.
Note that if the unemployment benefit rises, then the utility gained from being
unemployed is higher, thus, the level of wage that the worker would require to make
them better off taking a job than being unemployed would be higher. Compare the utility
of an unemployed worker at two levels of benefits, b' and b where b' > b
U E (W )
P
U U (b′)
U U (b)
W
P
w
w '
Naturally, UU(b') > UU(b), since a larger unemployment benefit will mean more
consumption without any extra work. This will likewise imply a higher reservation wage.
Since greater unemployment benefits makes unemployment more palatable, the wages
that would be required to make employment even more attractive would be all the greater.
One of the reasons that an unemployed worker might not immediately find a job
is that the labor market is something of a matching process. Specific jobs require specific
skills and each worker comes equipped with a specific skill set. Depending on the match
between workers skills and the needs of a specific job, the productivity of each worker at
any given job may not be particularly high. If the match is poor, the productivity of the
worker might be low and a potential employer might not be willing to pay a very high
wage, if they were willing to make any offer at all. Conversely, if the match was
particularly good, then the firm might be willing to make a high wage offer. If we think
of job search as a hit-or-miss process of matching between workers and firms, then the
degree to which workers are able to find a good match in any period will be, to an extent,
a matter of luck.
Suppose that a certain fraction, 0 < p < 1, of unemployed workers get a job offer
in any period. The wage that they will be offered should depend on the luckiness of the
match of the worker with the job. For a given worker, then, we assume that the wage
offer they receive is drawn at random from a set of possible outcomes, each of which will
have a given likelihood.
When we think about the likely outcomes of a randomly drawn number, it is
natural to think of games of chance. For example, consider the likeliness of outcomes of
rolling a dice. For a single die, there are six possible outcomes, each occurring with equal
frequency. Thus, we could think of their being a probability of each outcome which
would be equal to 1 6 .
R
R
1/6
1
1/6
2
Probability
1/6
1/6
3
4
1/6
5
1/6
6
However, it seems that in a broader economy that we would be unlikely to see such a
limited number of possible wage outcomes that a worker might receive. Rather, we might
think that possible wages fall into a range with a minimum w and a maximum w . Within
that range, it seems possible for any value in the range to be offered. When the random
number might possibly take on any number within a range, rather than a small number of
discrete values, then describing the probability of a given outcome is slightly more
complicated. After all, given that the random variable might take on an infinite number of
values within a range, the actual probability of the variable equals any given level is
infintitely small.
In the case that the random variable might take on a continuum of values, we
measure likelihood with what we call a cumulative distribution function or c.d.f. The c.d.f.
for a given number, v, is not the probability that the random variable equals that number
(which would be infinitely unlikely) but rather the probability that the variable would be
less than v. Call the c.d.f. for a random variable w, as Fw(v) and define it as Fw(v) =
Prob(w ≤ v). For example, in the case of the dice, F(3) = 3 6 = ½ , since the likelihood
that the dice will be rolled to a level less than or equal to 3 is the likelihood the dice
would be equal to 1 or 2 or 3, which would happen half the time.
The c.d.f. is a probability so it must always be greater than equal to zero and less
than equal to 1. In the case of a random variable over a range Fw(w) = 0 and F w ( w) = 1 .
In other words, the probability that the random variable would be less than the minimum
level would be zero and the probability that the random variable would be less than or
equal to the maximum would be 1. For any number v, the draw should be either greater
than or equal to or less than v, so Prob(w ≤ v) +Prob(w > v) = 1. So Prob(w > v) = 1 –
Fw(v). We can also use the c.d.f. to divide up the probability within any sub-range. For
instance, consider two numbers y and v such that w < y < v < w . We can say that the
probability that the draw of the variable is less than v is equal to the probability that the
variable is less than y Prob(w ≤ y) and the probability that the variable is between y and v,
Prob(y < w ≤ v) so
Prob ( w ≤ y ) + Prob ( y < w ≤ v ) = Prob ( w ≤ v ) →
F w ( y ) + Prob ( y < w ≤ v ) = F w (v) →
(1.37)
Prob ( y < w ≤ v ) = F w (v) − F w ( y )
Prob(w ≤ y) + Prob(y < w ≤ v) = Prob(w ≤ v). In the example of our dice, the
probability that the roll of the dice would be less than or equal to 3 is ½ while the
probability that the roll is less than or equal to 1 is 1 6 so the probability that the roll is
either 3 or 2 is 1 − 1 = 1
2
6
3
Near point v we can see that the slope of the c.d.f. can be thought of as measuring
a likelihood that the random variable will be in some range near v . The slope is
F w (v + Δ ) − F w (v )
where the numerator
Δ
F (v + Δ) − F (v) = Pr( w ≤ v + Δ ) − Pr( w ≤ v) = Pr(v < w ≤ v + Δ ) is the probability that the
variable is in a range between v and v+Δ and the slope is the size of that probability
relative to the size of the range Δ. When the c.d.f. is smooth enough to have a pointwise
slope, we can evaluate the slope at an infinitely small level of Δ. This pointwise slope,
dF
f (v) ≡
measures the likelihood weight that we would put on the random variable
dv
appearing in a very narrow range around v.
The simplest example of a c.d.f. would be one in which the probability that the
random variable is less than some level v grow proportionally to the size of v itself.
( v − w) . The slope is constant, f w (v) = 1 , so this is called the uniform
F w (v ) =
w−w
w−w
distribution.
Suppose that a fraction of workers, 0 < p < 1 will receive a job offer in every
period. Given that wage offers are randomly distributed only some fraction of those wage
offers will be above the reservation wage, Hw(wR) = 1- Fw(wR). Thus the fraction of
unemployed workers that are offered a job that is a sufficiently good match to be worth
taking is f = p×Hw(wR).
This offers one reason explaining cross-country differences in unemployment is
cross-country differences in unemployment benefits. A country with a relatively high
unemployment benefit will have a relatively high reservation wage which means the
fraction of accepted job offers will be reduced. The higher unemployment benefit leads to
a lower job finding rate and thus a higher steady state unemployment rate.
1
p
1 + ⋅ H w ( wR )
s