Econ 203: Topic 7
The choice between consumption and
leisure revisited
The Slutsky formulation
Reading: Varian Chapter 9
pages 166-171 and 174-175
Y
PxX= PxXc + PyYc
A
Y0
0
X0
X
Earnings
Leisure
=R
Total hours = T
W0
c
T
P
A
C0
0
R0
Leisure
Earnings plus Money (M/P)
W0
M
c
T
P
P
A
C0
0
R0
Leisure
Earnings plus Money
Price of leisure rises
from W0 to W1
Budget constraint
swivels in
W0
M
c
T
P
P
A
C0
0
R0
Leisure
Earnings plus Money
U1
W1
M
c T
P
P
But the reward for
work also rises
shifting the BC out
W0
M
c
T
P
P
A
0
RA
Leisure
Earnings plus Money
U1
W1
M
c T
P
P
U2
New Equilibrium at D
W0
M
c
T
P
P
D
A
0
RA
RD
Leisure
Earnings plus Money
U1
W1
M
c T
P
P
U2
Moving income back to
U1 (Hicks)
W0
M
c
T
P
P
D
A
0
RA
RD
Leisure
Earnings plus Money
U1 U2
W1
M
c T
P
P
Deriving equilibrium at new
prices on U1 we find point B.
RA-RB is the Hicksian substitution
effect {A-B on diagram}.
W0
M
c
T
P
P
D
B
0
A
RA
RB RD
Leisure
Earnings plus Money
U0
W1
M
c T
P
P
Can also look at what
demand would have
been on original leisureprice-rise line. That is
point C.
U1 U2
W0
M
T
P
P
D
B
A
C
0
RA
RB RD
Leisure
Earnings plus Money
U0
W1
M
c T
P
P
U1
U2
RB-RC is the standard
income effect {Point B
to C on the diagram}.
W0
M
T
P
P
D
B
A
C
0
RC
RA
RB RD
Leisure
Earnings plus Money
U0
W1
M
c T
P
P
U1
U2
RC-RD is the endowment
income effect {Point C
to D on the diagram}.
W0
M
T
P
P
D
B
A
C
0
RC
RA
RB RD
Leisure
U0
W1
M
T
P
P
U1
Substitution
Effect
A -> B
U2
W0
M
T
P
P
D
B
A
C
0
RC
RA
RB
Leisure
RD
U0
W1
M
T
P
P
U1
Income Effect
B -> C
U2
W0
M
T
P
P
D
B
A
C
0
RC
RA
RB
Leisure
RD
U0
W1
M
T
P
P
U1
Endowment
Effect C -> D
U2
W0
M
T
P
P
D
B
A
C
0
RC
RA
RB
Leisure
RD
Now we have a substitution effect from A to B,
causing the number of hours of leisure worked to
fall
from RA to RB
Next we have an income effect from B to C,
causing the number of hours of leisure to fall from
RB to RC
Finally we have an ENDOWMENT effect from C
to D,
causing the number of hours of leisure to rise
from RC to RD
• Remember that we usually assume that leisure is a
normal good. The rise in the price of leisure
causes us to buy less of it (from RA to RB).
•
• A rise in the price of any good makes us poorer,
and since ‘real income’ is lower (ignoring the
endowment effect) we buy less leisure (the fall
from RB to RC ).
•
• But now if we take account of the endowment
effect our total income has risen, since for every
hour of work we actually do, our pay has
increased, and so we will consume more leisure
(the move from RC to RD).
•
• Get the idea!
• Good, because now we are ready for the
nasty bit.
•
• The Endowment Slutsky Equation
•AAAGGGGHHH
Slutsky Endowment Effect
•
•
•
•
Varian Ch 9
And Especially
9.7 - 9.9
& the Appendix
Terminology:
•
•
•
•
•
•
•
•
I is Income
M is Money or unearned income
W is the nominal wage rate, (w real wage)
P is the price level
T is the total number of hours in the day
R is the number of hours of leisure taken
L is the hours of work done.
c is our actual or real consumption of goods
The Endowment Slutsky Effect
Our consumption of Leisure depends on the price of
R (W) and our overall level of Wealth. We are
endowed with M amount of Money and T units of
Time. So our Wealth (I) is:
I = M +WT = Pc+WR
or
c = (T-R)W/P +M/P
which in real terms
implies that
c = (T-R)w +m,
where w is the real wage rate and m is real money
balances
The Endowment Slutsky Effect
So R depends on w and I which is also a function of
w.
That is R[W, I(W)]
So
_
 R[W , I (W )]  R[W , I ]  R   I 




W
W
 I  W 
The Endowment Slutsky Effect
_
 R[W , I (W )]  R[W , I ]  R   I 




W
W
 I  W 
But if we differentiate I=M +WT with respect to W
we get T
_
 R[W , I (W )]  R[W , I ]  R


T
W
W
I
The Endowment Slutsky Effect
_
 R[W , I (W )]  R[W , I ]  R


T
W
W
I
What about the first term on the RHS?
It is the change in R when Income cannot change
-that is, it’s the usual Slutsky effect
_
 R[W , I ]  R
R

R
W
W
I
S
The Endowment Slutsky Effect
_
 R[W , I (W )]  R[W , I ]  R


T
W
W
I
_
 R[W , I ]  R
R

R
W
W
I
S
The Endowment Slutsky Effect
 R[W , I (W )]

W
R

T
I
R
R
R
W
I
S
The Endowment Slutsky Effect
 R[W , I (W )]  R
R
R

R
T
W
W
I
I
S
Substitution
Effect
RA to RB
Income Effect
RB to RC
Endowment Effect RC
to RD
or in other words, the overall effect is:
the substitution effect (from RA to RB),
minus the usual income effect (from RB to RC)
plus the endowment effect (from RC to RD).
The Endowment Slutsky Effect
 R[W , I (W )]  R
R
R

R
T
W
W
I
I
S
 R[W , I (W )]  R
R

 (T  R)
W
W
I
S
The Endowment Slutsky Effect
 R[W , I (W )]  R
R

 (T  R)
W
W
I
S
Differentiating c = (T-R)w +m, with
respect to the real wage yields
 R
 R  R  I 

 



  w U  w  I   w 
  R
R
R
  R  T

 
I
  w U  w
Hours worked
when Free to
Choose
Earnings plus Money
Holdings
U2
C
*
M/P
0
R*
T hours
Earnings plus Money
Holdings
But what if you have
to work a 40 hour
week minimum?
U2
C
*
M/P
_
0
R
R*
T hours
Utility falls to U1
and work and
consumption rise
Earnings plus Money
Holdings
U1
_
U2
C
C
*
M/P
_
0
R
R*
T hours
Earnings plus Money
Holdings
Note if choose not to
work, get U0
but since U1 is greater
will prefer work.
U0 U1
_
U2
C
C
*
M/P
_
0
R
R*
T hours
But if like leisure more
you have steeper U
curves and so choose not
working U1 to work U0
Earnings plus Money
Holdings
U0
U1
U2
C
*
M/P
0
R*
T hours
Exercise Topic 7
• Bottom Line:
• Changes in w/p over the cycle will not
induce much variation in hours worked
• However, changes in overtime rates will
(pure substitution effect)
• Main effect will be on marginal individual
making the work/no-work decision
• In particular for higher w/p people are more
likely to make the effort to be in
employment
What are the implications of this for our
Labour Demand & Supply diagrams?
W/P
lD
lS
W/P
lD
lS
W/P
lD
lS
lS’
ll
lu
W/P
lD
lS
lS’
ll
lu
W/P
lD
lS
lS’
ll
lu
W/P
lD
lS
lS’
ll
lu
Non-Assessed test 1998/99
Frequency
40
30
20
10
0
29.9
39.9
49.9
59.9
69.9
More
Non-Assessed test 1998/99 &
Non-Assessed test 2000/1
Frequency
40
30
20
10
0
29.9
39.9
49.9
59.9
69.9
Better, but still not very good
More
203 Assessed test
98/99
99/00
50
40
30
20
10
0
90-100
80-89
70-79
60-69
50-59
40-49
35-39
30-34
<30