Handout 5th Lecture
ECON 4230/35
Kjell Arne Brekke
Note that the lecture will be in Georg Sverdrups house (library) Aud.
2.
1
Slutsky with Endowments
Consider the problem
max u(x)
x
subject to px = p!
that is
m = p! + m0 :
Now
@xj
@hj
=
+ (! i
@pi
@pi
xi )
@xj
@m
Problem 1 A worker is choosing between consumption c at prices p and
leasure L, and get an income wl from labour l, while the time budget is
L + l = L.
(a) Show that the consumers budget constraint can be written pc + wL =
wL.
(b) Use the Slutsky equation with endowment to analyze how the demand
for leisure respond to changes in wages w:
1
1.1
Aggregation
There is no such good as a jacket, or a car. There are thousands of models
of jackets, and the condition of a used car is more like a continuum. Still
to make analyses tractable we will consider things like the demand for
cars or the demand for clothes, the demand for food etc. Does this make
sense?
Assume that
p = tp0
and de…ne the commodity, e.g. food
X = p0 x
while the price is
P =t
De…ne the indirect utility function
V (P; q; m) = max u(x; z)
x;z
such that P p0 x + qz
m
Problem 2 V satisfy the conditions of an indirect utility function. How
do we then proceed to get the utility function U (X; z) taking the aggregate
commodity as an argument?
1.2
Aggregating across consumers
Given the individual demand
xi (p; mi )
we can de…ne an aggregate demand
x(p; m) =
I
X
xi (p; mi )
i=1
m = (m1 ; :::; mn )
2
It will inherit the structure from individual demand concerning prices.
but does it satsify the general properties of individual demand. Does
there exist a representative consumer? That is can we de…ne an income
M such that
x(p; m) = x(p; M )?
Due to Walras law, the only candidate is (why?):
M=
X
mi
Problem 3 Show that if
x(p; m) = x(p; M )
then redistribution of income cannot have an impact on aggregate consumption and moreover that this imply
@xkj
@xki
=
for all i and j
@mi
@mj
Do you think this is a reasonable assumption?
2
Expected utility
When grading exams I have often been surprised by how many students
that are unable to write down the equation for expected utility of a lottery. If you are unable to do this, it is unlikely that you will earn a single
point on the exam on questions about uncertainty. Before proceeding to
more subtle points about justi…cation of expected utility, risk aversion
etc, make sure that you know what expected utility is:
Suppose you got at lottery
x
probability p
y probability 1
3
p
If you get x you utility will be u(x) while if you get y your utility will
be u(y) and the expected value of your utility (before the outcome is
known) is
Expected Utility = pu(x) + (1
p)u(y):
Notation. Varian write
(x p
y (1
p))
for the lottery that I will write:
x
with probability p
y with probability 1
2.1
p
The independence axiom
Axiom 4 (Independence) Suppose
L1
L2
then
L1
probability p
y probability 1
L2
probability p
y probability 1
p
p
Theorem 5 The independence axiom (+ some less controversial assumption) implies expected utility
Basic argument: Assume there is a best, b, and a worst, w, outcome. For any x there is a probability u(x) such that
x
b
probability u(x)
w probability 1
u(x)
Do the same for y and we end up with a lottery where the probability
of winning (getting b) equals
Eu = pu(x) + (1
4
p)u(y)
We want this probability to be as high as possible, that we want to
maximize expected utility
Problem 6 Consider the two lotteries
4000 probability 80%
L1 =
0
probability 20%
L2 = 3000 for sure
and the lotteries
L3 =
4000 probability 20%
0
probability 80%
3000 probability 25%
L4 =
0
probability 75%
Show that
L3 =
L1 probability 25%
0 probability 75%
L4 =
L2 probability 25%
0 probability 75%
If L2
L1 what does the independence axiom imply about the ranking of
L3 versus L4 .
3
Only Linear transforms
Consider two lotteries
x1 with probability p1
Lx = ::
::
xn
pn
y1 with probability q1
Ly = ::
::
ym
qm
5
then
Lx
i¤ and only if
n
X
Ly
pi u(xi ) >
i=1
m
X
qj u(yj )
j=1
An alternative utilityfunction v(x) represents the same preferences if and
only if
v(x) = au(x) + b with a > 0
4
Risk aversion
Now consider gambles with monetary outcome. Consider a person with
wealth W who is o¤ered a lottery
100 probability 50%
100 probability 50%
Expected utility accepting the gamble is
1
1
u(W + 100) + u(W
2
2
100)
while declining yields
u(W ):
A risk seeker will accept the gamble
1
1
u(W + 100) + u(W
2
2
100) > u(W ):
while a risk averse person will decline:
1
1
u(W + 100) + u(W
2
2
100) < u(W ):
Graphically this depends on the shape of the utility function, e.g. for
risk aversion
6
y
x
The Arrow pratt measure of risk aversion
u00 (W )
u0 (W )
r(W ) =
We can prove that
1
probability p
1 probability 1
p
1
r(W )
2
More used is the relative risk aversion
=
5
u00 (W )W
u0 (W )
Subjective probability
If you ‡ip a choin it will sometimes land head and sometime tail. Doing
it over and over, we get a frequency. Some scholars would like to preserve
the idea of probability to such cases.
What is the probability that doubling CO2 concentration will increase temperature more more than 5o C? What is the probability that
the interest rate will have increased by the end of the year. Or even:
what is the probability that Zürich is the capital of Switzerland?
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Some claim that it makes no sense to talk about probabilities in
these cases. There are no frequencies. But you may still be faced with
gambles.
L1 = you win 1000 kr if Zürich is the capital
L2 = you win 1000 kr if Bern is the capital
L3 = you win 1000 kr if Wien is the capital
If your preferences over such lotteries satisfy cerntain conditions
they can be represented by a utility function and a probability distribution of cities that are candidates for being the capital of Switzerland. These probabilities are subjective probabilities. (Some will know
the capital for sure, others have no clue.)
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