Topic 7 Econ 203: The choice between consumption and leisure revisited: The Slutsky formulation
Reading: Varian Chapter 9 pages 166-171 and 174-175
y2
y1
U2
U1
x2
x1
If x is a Normal good,
then if Px   demand for x 
Copyright ©Dr. Cillian Ryan 1998 ECON203, Topic 7 Unemployment and the Consumption/Leisure Choice
When we studied the conventional Slutsky equation we saw that the Slutsky substitution
effect could be derived by subtracting the income effect from the overall change in demand:
So
 x
x
x
x

 
I
  px  U  px
where I is income
Really this is an exercise in reverse engineering: We actually observe points 1 and 2 below
and derive point 3 using the equation above.
3
y2
y1
2

1
U2
U1
x2
x1
Copyright ©Dr. Cillian Ryan 1998 ECON203, Topic 7 Unemployment and the Consumption/Leisure Choice
Our usual budget constraint is
I 0  p x x  p y y , where I0 is our initial income
In this case when px rises, the budget constraint moves in as in the last example.
_
But now suppose that you are a producer of the good x and that you produce
x units of x.
Then as the price of x rises will cause your income to rise also.
_
I 1  p 1x x  p 1x x  p y y
That is, income has risen to I1. This is sometimes called the Endowment Income effect.
_
Note that the maximum x you can consume will not change, that will still be, x . The rationale is obvious if
you produce and consume all of it, then the market price of x, and whether it goes up or down, doesn’t affect
you. However, if you sell all your production of x and consume only y then you are definitely better off since
you now have higher income and can buy more y.
So now the budget constraint swivels out around the point xmax instead of in around the original ymax point.
New ymax
Original ymax
xmax =
_
x
Copyright ©Dr. Cillian Ryan 1998 ECON203, Topic 7 Unemployment and the Consumption/Leisure Choice
Suppose that the agent gets a sum of money M whether s/he works or not. Its real value is M/P. In addition
s/he gets paid W for every hour s/he works, up to a maximum of T hours per day (T=24 hours for example).
Earnings
plus
M
W0
T
P
P
M/P
Leisure
0
T hours
As usual we can think of W/P as the reward for working or as the ‘price of leisure’ - the income we
forego by taking an hour off!
Copyright ©Dr. Cillian Ryan 1998 ECON203, Topic 7 Unemployment and the Consumption/Leisure Choice
What happens here if Wages increase, or in other words, the price of leisure rises?
The usual effect of a rise in price, as we know, is that the budget constraint swings in. But here we have an
endowment of T hours which can be used for work or leisure. So the rise in the price of leisure is effectively
an increase in the value of our endowment of T. So instead of swinging in, our endowment swings out. That’s
the ‘endowment’ intuition, but the economics of the case are straightforward to see. If we take no leisure, and
work T hours then our total income will rise to
W1
M
T
P
P
Earnings
plus
W1
M
T
P
P
W0
M
T
P
P
M/P
T hours
0
In the analysis which follows suppose that agents can choose either consumption equal to their earnings or R
(for hours of Relaxation) which must be less than T. The T-R = the number of hours worked. And
Consumption will be c 
W
M
(T  R) 
P
P
Copyright ©Dr. Cillian Ryan 1998 ECON203, Topic 7 Unemployment and the Consumption/Leisure Choice
Earnings plus Money
W1
M
T
P
P
1
D
B
W0
M
T
P
P
A
C
RC
0
RA
RC
RB
In the standard analysis we saw in Econ 201 a change in price resulted in a substitution and an income effect.
Here the story is a bit more complicated.
Now we have a substitution effect from
A to B,
causing the number of hours of leisure to fall (work to rise) from R A to RB.
Next we have an income effect from
B to C,
causing the number of hours of leisure to fall (work to rise) from R B to RC
Finally we have an ENDOWMENT effect from
C to D,
causing the number of hours of leisure to rise (work to fall) from R C 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 R C to RD).
Get the idea! Good, because now we are ready for the nasty bit.
The Endowment Slutsky Equation
AAAGGGGHHH!
Copyright ©Dr. Cillian Ryan 1998 ECON203, Topic 7 Unemployment and the Consumption/Leisure Choice
Now lets go back to our Macroeconomics and our employment/hours worked story.
When we did our original analysis agents choose between consumption and hours worked. Implicitly if there
was a shock to the economy which effected agents wealth or the wage rate agents would respond by changing
the hours they worked. But we saw in the stylised facts that the number of hours per worker varies over the
cycle but by less than the variation in output and that most of the variation in total hours comes about as a
result of a variation in the number of people employed. Thus, up to now variations in employment have been
variations in the number of hours per worker rather than a variation in employment.
In the next few lectures Dr. Dickinson will discuss various models of unemployment with you. These will be
models of involuntary unemployment, that is, given the existing wage rate and unemployment benefits agents
would rather work but cannot find employment. However, in this lecture we want to look at a simpler case,
where agents choose either to work or not depending on the wage rate and unemployment benefits. That is,
this is a model of voluntary unemployment. Barro & Grilli Chapter 13 provides a detailed account of how
agents may vary in their willingness to take the first job that comes along at the going wage rate over the
business cycle from a macroeconomic perspective. This chapter provides a useful introduction to Dr.
Dickinson’s material. However, here we are going to concentrate on the microeconomics of the agent’s
decision.
We can do this by using the story we have developed above. In the normal model we employed up to now the
consumer would choose to R* hours of leisure, T-R* hours of work and consume c* when wages are W0 ,
unearned income is M and Prices are P.
Earnings
plus
W0
M
T
P
P
C*
M/P
T hours
0
*
R
Copyright ©Dr. Cillian Ryan 1998 ECON203, Topic 7 Unemployment and the Consumption/Leisure Choice
However, this is the ideal world. In reality agents cannot freely choose their hours worked. Instead typically
they have to work a minimum of an 8 hour day, and then if they wish, do additional overtime.
_
So if we impose a minimum number of hours worked of
welfare from U2 = U(C*,R*) to U1=
R in the diagram below, this reduces the consumer
Earnings
plus
W0
M
T
P
P
U1
U2

C
C*
M/P
_
0
R*
R
T hours
Now note that the consumer still chooses to work here as the utility from working U 1 is greater than the utility
from not working, U0 , as is shown in the diagram below where consumption is M/P and leisure =T.
Earnings
plus
W0
M
T
P
P
U1
U0
U2
C*
M/P
_
0
R R*
Copyright ©Dr. Cillian Ryan 1998 ECON203, Topic 7 Unemployment and the Consumption/Leisure Choice
However, this doesn’t have to be the case. It could be the case that the consumer might prefer not to work. In
the example below this is the case, now the utility associated with the consumption of the unearned income
_
alone is preferable to working and consuming
c.
Earnings
plus
U1
W0
M
T
P
P
U2
_
c
C*
M/P
T hours
_
*
R R
0
_
_
The diagram below includes the indifference curve through ( c, R ) which has a lower level of utility than the
point (c=M/P, R=T). That is,…
Earnings
U1  U (
U1
W0
M
T
P
P
_ _
M
, T )  U 0  U (c , R)
P
U2
U0
_
c
C*
M/P
0
_
R*
T hours
R
Copyright ©Dr. Cillian Ryan 1998 ECON203, Topic 7 Unemployment and the Consumption/Leisure Choice
Where does that leave us as regards unemployment as compared with hours worked. In this analysis more
people are likely to be up against the minimum number of hours worked constraint and so are unlikely to vary
their hours worked in response to a change in wages. However, a change in the wage rate now alters the value
_
_
of the ( c, R ) combination and thus the utility associated with it. You can try this for yourself. As a
consequence changes in W/P over the cycle will cause some people to choose unemployment rather than
employment. As the Barro & Grilli chapter 13 points out, we can think of this as being variations in the
willingness to take jobs as the wage rate and opportunities vary over the cycle. Thus, people who have been
laid off or who are considering entering or re-entering the work force are less likely to search for new jobs or
to take the first job that comes along if it doesn’t match their ideal job specification since the opportunity cost
of doing so is low compared to when the economy is buoyant and there are many attractive, well-paying jobs
and overtime opportunities.
Copyright ©Dr. Cillian Ryan 1998 ECON203, Topic 7 Unemployment and the Consumption/Leisure Choice
Copyright ©Dr. Cillian Ryan 1998 ECON203, Topic 7 Unemployment and the Consumption/Leisure Choice