Updated: 29 May, 2007
MICRO ECONOMICS
(ECON 601)
Lecture 10
Topics to be covered:
a- Allocation of time
b- Two good models
c- Utility maximization
d- Income and substitution effects of a change in w
e- Analysis of labor supply
f- Dual statement of problem
g- Slutsky equation of labor supply
h- Cobb-Douglas labor supply
i- Effects of non-labor income
j- Individual supply curve of labor
k- Market supply curve
l- Job search theory
m-Economies of child bearing
n- Transportation choices
o- Modeling a union
p- Monopolist optimum
ECON 601: ADVANCED MICRO ECONOMICS
LABOUR MARKETS
Nicholson, Chapter 16
Allocation of Time
Fixed amount of time 24 hours of the day, the question is how it is allocated between
work, sleep, consumption and leisure.
A Two Good Model
Work, and not working
When working a wage of w per hour is earned. When not working the individual is
consuming (C) or enjoying leisure (H). Both are composite goods:
Utility = U (C, H)
Utility is a function of Consumption (C) and leisure (H). Utility is maximized subject to
the time constraint that the time spent working (L) plus leisure time (H) has to be equal to
24 hours a day.
L + H = 24
The second constraint is that
C = wL
Combining the two constraints, we have
C = w (24 – H)
OR,
C + wH = 24w
1
Full Income = 24 w
The opportunity cost of consuming leisure is w per hour. It is equal to earnings foregone
by not working.
Utility Maximization
Maximize utility subject to the full income constraint:
L = U (C, H) +  (24w – C – wH)
The first order conditions:
(1)
L U
=
–=0
C C
(2)
L U
=
– w = 0
H H
From (1) and (2)
U
(3)
U
H  w = MRS (H for C)
C
In order 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.
The MRS of leisure for consumption must be diminishing in order for utility to be
maximized according to equation 3.
2
Income and Substitution effects of a change in w
With a change in w, leisure becomes more expensive and people reduce their
consumption of it due to the substitution effect. At the same time as w increases there is
an income effect and as leisure is a normal good people will demand more of it.
Since leisure and work are mutually exclusive it is not clear if work will increase or
decrease if the wage rate (w) is increased. The substitution effect A to S tends to increase
the hours worked while the income effect reduces it.
Consumption
C = w1(24 – H)
B
C1
S
C = w0(24 – H)
C0
A
u1
u0
H1 H0
Leisure
In this case the individual works more (24 – H1) versus (24 – H0) and also consumes
more C0 to C1. Substitution effect is stronger than income effect.
3
Consumption
C = w1(24 – H)
C1
C0
S
C = w0(24 – H)
u1
u0
24
H0 H1
Leisure
In this case the increase in the wage rate from w0 to w1 causes an individual to work less
(24 – H1) from (24 – H0) and also to consume more C0 to C1. The income effect is
stronger on the labor-leisure choice than the substitution effect. It more than offsets the
substitution effect.
When the real wage rate increases, a utility maximizing individual may increase or
decrease hours worked. The substitution effect increases hours worked, the income effect
will tend to reduce hours worked as the individual uses his increased purchasing power to
buy more leisure hours.
Analysis of Labor Supply
Budget Constraint with Non-Labor Income
C = wL + N
L + H = 24
L = (24 – H)
C = w (24 – H) + N
C = Total Consumption
N is non-labor income.
L = U (C, H) +  (24w + N – C – wH)
4
(1)
L
U
=
– =0
C
C
(2)
L U
=
– w = 0
H H
From 2/1
U
H = w
U
C
Consumption
C = w0 (24 – H) + N
Leisure
C = w0 (24 – H)
Budget constraint shifts with changes in value of N.
Labor supply, L(W,N) will depend on real wage rate and non-wage income. If leisure is
normal good then
L
0
N
Dual Statement of Problem
Choose values for consumption (C) and Leisure time H = 24 – L so that amount
additional spending E = C – wL to achieve a level of utility U0 = U(C, H) is minimized.
(See Chap 5)
E
=–L
w
5
Each $1 increase in w reduces the value of additional spending (E) required $L because
this is the change in Labour’s earnings for a $1 change in the wage rate.
Slutsky Equation of Labor Supply
Equation reflects the substitution and income effects that result from changes in the real
wage. Expenditures (E) being minimized in the dual problem plays the role of non-labor
income (N) in the utility maximization problem.
By definition at the initial optimal point,
The combination of L, w, N, U for the substitution effect only labor supply function
determine for each.
= Lc(w, U) = L [w, E(w, U)] = L (w, N) where Lc(w,U) is the compensated (constant
utility) labor supply function and where [L(w, N)] is uncompensated labor supply
function.
Using the envelope relation.
L L E
Lc

.
=
W E W
W
Using,
E
L L
  L and

W
E N
L
L
L
L
Lc
L

L
=
W
E W
N
W
L
L
Lc
L
=
N
W W
Defining compensated labor supply function as,
Lc
L
=
W
W
u u0
6
L
L
=
W
W
L
u u0
(+)
L
W
L
L
N
(–)
>0
u u0
L
<0
N
The negative income effect works in the opposite direction to the substitution effect and
may often more than offset the positive substitution effect.
Labor Supply Functions
Cobb-Douglas Utility:
U(c,h) = c h
 +  =1
Constrained by two equations
1. Consumption is financed by
c = w l + N, where N is non-labor income.
2. Total time constraint, set total time available = 1
l+h=1
L  U (c, h)   wl  N  c 
L  U (c, h)   w(1  h)  N  c 
L  U (c, h)   w  wh  N  c 
L  c h    w  N  wh  c 
L
(1)
 c   h     0
c
L
( 2)
  c  h   w  0
h
L
(3)
 w  N  wh  c  0

7
Ls = Ld
a + b (w + k) = c – d (w + t)
Net wage is now:
w** = w* -
bk  dt
bd
w**= w* - t, w** is net wage, w* is wage inclusive of t*
if k=0 workers derive no benefits.
Therefore, w** = w* -
dt
bd
Employees will pay a share of the cost (tax) = d / b + d, the equilibrium quantity of labor
hired falls. This held as
w
S
(K= 0)
w*
w**
d
d-t
L1
L
L0
Labor supply function is therefore
L (w , N ) = 1 – h = ( 1 -  ) -
N
w
Labor supply function:
L (w, N) = (1 - ) -
N
w
8
Properties of the Cobb-Douglass Labor Supply Function
1) If N = 0
l
0
w
Person always spends (1 -  ) proportion of the time for working no matter what the wage
rate.
2) If N  0 then
l
 0 because the person spends n of it on leisure.
w
Leisure “costs” w per hour so an increase in the wage means fewer hours of leisure can
be bought with non-wage income N. A rise in w increases labor supply.
3)
l
 0 An increase in non-labor income allow a person to buy more leisure so labor
w
supply falls.
Cobb-Douglas Labor Supply
Example 16.1
U=
CH
Budget constraint,
C = wL + N
and by a time constraint
H=1–L
To simplify matters the total time available is set equal to 1. Substituting for C and H, we
have,
U2 = CH = (wL + N) (1 – L)
U2 = CH = wL – wL2 + N – NL
U 2
L
= w – 2wL – N = 0
If U2 is maximized then U will also be maximized.
L=
w N 1 N
 
2w
2 2w
9
If, N = 0, L = 1/2
Person will work ½ of the time no matter what the wage is. The income and substitution
effects offset each other. Consider the income effect and substitution effect separately.
The income effect in the Slutsky equation is in the case of this example,
L
L
1
N
 1 N  1 

= 
 

N
4w 4w 2
 2 2 w  2 w 
If N = 0, income effect will be,
L
L
1
=
N
4w
Since leisure is a normal good the income effect of an increase in w on amount of labor
supplied is negative. To derive the substitution effect if the Slutsky equation
L
W
we
u u0
need to derive the indirect utility function as a function of w and N. The indirect utility
function is given to us as:
U=
U-
w
N

2
2 w
w
N 
w w

U2 w 2
N
2
2
2 w
 N= 2 wU  w
Solving for N from indirect utility function, we have: N= 2 wU  w and we use this to
eliminate N in the equation for optimal labor choice L =
Lc (w, U) =
1 2 wU  w

2
2w
10
1 N

, giving us
2 2w
1
 2 wU
w 
1
U
1

Lc (w, U) = 2   2w  2w   2 
w 2


Lc (w, U) = 1 –
U
w
Lc
U

W 2w 3 2
Replacing U with indirect utility function, U =
w
N

2
2 w
1
N
Lc

=
4w 4w 2
W
If N = 0,
Lc
1

w
4w
Hence, the Slutsky equation,
L Lc
L
1
1

L


0
w w
N 4w 4w
The substitution effect completely cancels the income effect of an increase in wage rate.
Effects of Non-Labor Income
If N is present the precise offsetting of income and substitution effects would not occur.
From L 
1 N

, the individual will always choose to spend half of his or her non-labor
2 2w
income on leisure. Leisure “costs” w per hour and a rise in w will mean that less leisure
can be “bought” with a fixed number of N dollars. If, for example, N = $2 per hour and w
is $10 per hour, from equation L =
1 N
1
2
4

we derive L = 
.

2 2w
2 2(10) 10
11
This person spends $1 of his or non-labor income on leisure each hour. At a wage of $10
this $1 will buy
buy
1
of an hour of leisure. If, on the other hand, if w = $5, the $1 would
10
2
1 2
3
of an hour, and L =  
10
2 10 10
From the labor supply function L 
1 N
L  1


, we have
2 2w
N 2 w
With non-labor income, therefore, the income and substitution effects of a change in
wage rate are not now exactly offsetting the substitution effect dominates and a fall in
wages reduces hours of work.
An increase in N in this problem will reduce the hours of work. With N = $4 and w = 10.
L
1 N
1
4
3

. Then L= 
. 3/10 of an hour of work supplied. For N = $10,

2 2w
2 2(10) 10
hours of work would fall to zero. If N is interpreted as a subsidy – this explains the
negative labor supply effects of income maintenance programs.
Individual Supply Curve of Labor
Real
Wage
(w)
Real
Wage
(w)
S
S
S
S
Hours of work
Income effect less than
substitution effect
Hours of work
Income effect greater than
substitution effect
Short-run supply curves seem to be positively sloped e.g. higher overtime pay rates. In
long term labor supply curves seem to be backward lending.
12
Market Supply Curve
Market supply is the sum of individual supply curves – Higher wage rates increase labor
force participation.
Real
Wage
S2(W)
Real
Wage
Real
Wage
S1(W)
S=S1(W)+S2(W)
Wmin2
Wmin1
Individual (1)
Individual (2)
The Market
Wm in this minimum wage for person to enter the labor force.
Mandated Benefits
Suppose employees are required to give their workers certain benefits. The effect on the
labor market is determined by how the benefits are valued by the workers.
ls = a + b w
ld = c – d w
Ls = Ld
w
ca
bd
Suppose benefit costs t per unit of labor hired. Unit labor costs are therefore now w + t.
suppose benefit has a monetary value to workers of k per unit of labor supplied.
13
c   h   
 
 w
(1/2)  c h
 h 1

 c w
nwh 
c (1   )

c


or
c
wh
(1   )
Full income constraint
c = w + N– w h
1  
c=w+N- 
 c
  
c c 

)  w  N
c  ( 
  

 c  c  c 

  w N



1
c  w  N  c   (w  N )

  
wh
  w  N  wh
1  
  
wh
  wh  w  N
1  
   

h w
  w   w  N
 1  

 w w(1   ) 
h

  w N
1 
1 
 w 
h
  w N
1  
 w
h   w  N
 
 (w  N )
h
w
Individual spends  and  shares of full income on consumption and leisure respectively.
If k = t the new wage will fall by exactly the amount of the cost.
14
w** = w* -
tk  dt
 w* - t
bd
If k  t the equilibrium wage will fall by more than the benefit cost and equilibrium
employment will rise.
S
w
S+k
w*0
w*1
d
d-t
L0 L1
L
For CES Utility Function:
(Ch. 4)
L (w, n) = 1 – h =
k
wl  k  n
1
1
w

 1
If δ = 0.5 and k = -1
Supply function is l (m, n) =
If n = 0
l  n / w2
1
1
w
l
 0 because of high degree of substitutability between consumption and
w
leisure.
15
If δ = -1 and k = 0.5 labor supply is l (m, n) =
n=0
l  n / w0.5
1  w0.5
l
 0 because of smaller degree of substitutability in the utility function.
w
Job Search Theory
Wage
Rate
W
W
W
S
P
F
O
m
O
S
T
G
Demand for labor in
open sector
E
O
D
S
Q
O
Open Market
Employment
QS
A B
QQV
QPr
C
Quasi-voluntary
Unemployed
D
Number employed
Protected Sector
Search
Unemployment
Wm = minimum wage for people who will otherwise search
WP = Protected sector (unionized) wage
C – B = Quasi – Voluntary Unemployed
B – A = Search Unemployment
16
Figure above depicts a labor market in which both search unemployment and the standard
type of quasi-voluntary unemployment coexist. The curve SST shows the total supply of
labor to this market. There are two employment opportunities the protected sector jobs
paying high wages (WP), but they are restricted in quantity. Only the quantity QPr are
available. There are plenty of jobs available at a wage of W0. However, if one is working
at W0 it lowers the probability of getting a high paying job. Hence, some people would
m O
prefer to search for a high paying job than work for W0. The curve W S is the supply
curve, inclusive of the effect of searching that faces the open market. The lateral distance
O
between this supply curve and the prior supply curve, SS , is the quantity of search
m
unemployment corresponding to a given open market wage. When the wage is W , the
m
number of workers who opt for search unemployment is equal to the distance W E,
O
whereas it is the difference between F and G at the open market wage W . This distance
m
is greatest at the wage W , the minimum wage at which all those not working for the
protected sector would prefer to remain unemployed while searching for protected sector
jobs instead of accepting open sector jobs. As the open market wage rises, fewer and
fewer workers are willing to forgo open market earnings in order to seek protected sector
P
jobs, until, finally, as the open market wage approaches the protected sector wage, W ,
the quantity of search unemployment approaches zero.
Economics of Child Bearing
1.
Cost of child raising is primarily the foregone wages of the care givens:
as w there is an income effect that suggests that more children would be
demanded, at the same time the wage rate will increase the “price” of children
leading to a substitution effect to cause people to have fewer children.
17
Transportation Choices
Time savings are the principal benefit of transportation investments. –time saved is
valued at between 50 and 100 percent of the wage they earn while working.
Compensating Differentials
Differences in location, job characteristics, may cause large differences in wage rates for
same skill. For example, truck drivers wages in Florida $100/day versus Alaska $300/day
concept of supply price of labor very important.
–
The competitive price at which a sufficient number of workers will make
themselves available to work in a given occupation, at a given place, under given
working conditions.
Compensating Difference
Real
Wage
Sunpleasant
SP
wu
wP
D
Lu
18
LP
Quantity of labor
Labor Unions
Unions as a monopoly facing a demand curve for Labor.
Wage
S
w2
E2
S2
E1
w1
S1
w3
S3
D
O
L2
L1
L3
MR
(1)
Quantity of labor
per period
Union may try to maximize the wage bill wL. This is at L1 where MR = 0 and the
wage rate will be w1. At w1 there will be an excess supply of labor of S1 – E1 that
must not be allowed to enter the job market and bid down the wage.
(2)
Union may try to maximize the total economic rent accruing to its members. This
will occur at the point where MR = Supply price or L2 and a wage of w2. This
generates an excess supply of S2 – E2. Union must restrict entry e.g. in Germany
a person must be an apprentice for 10 years to be a plumber. Doctor’s in U.S.
control entry by exams for entry, but no reexamination.
(3)
The third possibility is to maximize the employment of its members. This
involves the choosing of w3L3 which is the perfectly competitive solution.
19
Modeling a Union
1. Supply curve of workers L = 50 w
Coal mine is a monopsonist.
Assume:
2. MRP = 70 – 0.1 L
From (1)
3. w =
L
50
L2
Total wage bill Lw =
50
TC 2 L L


= ME
L
50 25
Monopsony position of coal mining company:
ME = MRP
L
= 70 – 0.1 L
25
0.1 L +
L
= 70
25
1 
1
L    = 70
 10 25 
 7
L   = 70
 50 
 50 
L =   70 = 500
 7
w=
L
500
=
= 10
50
50
Monopsonist solution is L = 500 and w = 10.
If mine hired people competitively.
20
If equated demand and supply and behaved like a competition.
Then MRPL = w
 70 – 0.1L = w
and L = 50 w
Hence, 70 – 0.1 (50) w = w
6w = 70
w = 11.66
If L = 50 w
then, L = 50 (11.66) = 583
Therefore, L = 583 and w = 11.66
Monopolist optimum
If union behaved like a monopolist then it would set marginal increment yield to union
members equal to the supply price of union or
L
.
50
Income to union members = L(MRP)
 ( L.MRP )  ( L(70  0.1L))

= 70 – 0.2L
dL
dL
w=
L
50
Hence,
L
= 70 – 0.2L
50
L = 3500 – 10L
11L = 3500
L = 318
at L = 318, then MRPL = 70 – 0.1 (318) = $38.2.
21
The wage would be set at $38.2.
Monopolist position L = 318, w = 38.2
Monopsonist position L = 500, w = 10
Competitive position L = 583, w = 11.66
As the various positions are very different, the final outcome will have to come about by
bilateral bargaining.
The End
22