Handout 5th Lecture
ECON 4230/35
Kjell Arne Brekke
I expect to cover the topics at least until the slutsky equation with
endowment in class, but maybe not all of it.
1
The Slutsky Equation
Di¤erentiate the identity
hi (p; u)
xi (p; e(p; u))
with respect to pi , using shepards lemma and denote m = e(p; u); and
solve for
@xi (p;m)
@pj
give the Slutsky equation
@xi (p; m)
@hi (p; u)
=
@pj
@pj
xj (p; m)
@xi (p; m)
@m
Interpretation: When we change prices and move alond the indi¤ernce curve
@hi (p;u)
@pj
we do two things: Change prices
(p;m)
sates income xj (p; m) @xi@m
.
1.0.1
Classi…cation
Necessary goods:
@xi (p; m) m
<1
@m
xi
Luxury good
@xi (p; m) m
>1
@m
xi
1
@xi (p;m)
@pj
and compen-
Inferior goods
@xi (p; m)
<0
@m
Normal goods
@xi (p; m)
>0
@m
Gi¤en good
@xi (p; m)
>0
@pi
Problem 1 Use the Slutsky equation to explain why normal goods are
not Gi¤en goods.
1.0.2
Implications of Shepards lemma and Slutsky
Next note that
@hi (p; y)
@ 2 e(p; u)
@ 2 e(p; u)
@hj (p; y)
=
=
=
@pj
@pj @pi
@pi @pj
@pi
Two goods are complements if
@hj (p;y)
@pi
@hj (p;y)
@pi
< 0 and substitutes if
> 0.
Note that this de…nition would not work with Marshalliand demand
where it is conceivable that
@xi (p; m)
@xj (p; m)
<0<
@pj
@pi
1.1
Properties of the demand function
Symmetry and the negative diagonal elements
@hi (p; y)
@pi
2
0
2
Revealed preferences
Problem 2 If we invoke the assumptions about preferences above (including monotonicity) and observe that the person choose x at prices p
then
x
y for all y such that py < px
Why do we know this?
De…nition 3 If we observer xt and pt then for all x such that
pt xt > p t x
we say xt P D x that xt is strictly directly revealed preferred to x. While if
pt xt
pt x
then xt RD x
De…nition 4 If there is a sequence xt RD xs RD xa RD ::::RD x then xt Rx
and we say that xt is weakly revealed preferred to x.
For this to be preferences we cannot be able to generate cycles with
at least one strict preference:
Theorem 5 (GARP) xt Rxs implies NOT xs P D xt
2.1
Afriats teorem (is utility theory empty?)
Is there any dataset that in principle would rejects consumer theory?
There is:
Theorem 6 (Afriat) Let (pt ; xt ) be a …nite set of observations. Then
the following is equivalent.
(1) There exist a locally nonsatiated utility function that rationalizes
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data;
(2) The data satis…es GARP
(3) There exist numbers (ut ;
us
t
) such that
ut +
t t
p (xs
xt )
for all t and s.
Problem 7 Assess the following claim: "It would be a good thing if no
dataset could possibly reject economic consumer theory, since then we
could be sure that the theory can help explain any consumer behavior we
may observe". Do you agree?
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Recoverability
We have one observation x1 what can we for an arbitrary x say about
the sets
P = fy : y
xg
Problem 8 Make a diagram for the case of two commdities. Plot an
x1 that is chosen at prices p1 , and pick a point x with x1 P D x . (x1
x
by revealed preference.) Assume preferences are monotone and convex.
(a) Mark in the …gure the largest possible area that you know must be
contained in P
(b) Mark in the …gure the larges possible area that you know are not
intersecting with P .
4
Duality
Suppose that you are given an expenditure function e(p; u), can we …gure
out the underlying preferences? Let us …x a utility level u and try to
…nd the set
Pu = fx : u(x)
4
ug
Pick any price vector p with all prices positive. Now, if for a bundle
x0
px0 < e(p; u)
we know that
u(x0 ) < u
otherwise, if u(x0 )
u we would reach the utility level at a lower cost
than e(p; u).
Problem 9 Illustrate the argument above in a …gure (two commodities)
and show graphically how you could …nd the entire set Pu by varying
prices, provided preferences are convex and monotone.
Formally
Pu = fx : px
e(p; u) for all pricevectors p >>0g
and for convex monotone preferences
Pu = P u
Thus we do not have to starte with the preferences at all, but we can
starte with the expenditure function. The problem is: If we are given
a candidate for an expenditure function, a function (p; u), how do we
know that it is in fact an expenditure function and that there are some
preferences behind to be recovered. This is the theorem in section 6.2
of the textbook. The candidate
must be non-negative, hmogenous in
degree one, non-decreasing and concave in prices.
Problem 10 Why would be ever want to start with the expenditure function and not the preferences? Can you see any advantages?
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5
Slutsky with Endowments
Consider the problem
max u(x)
x
subject to px = p!
that is
m = p! + m0 :
Now
@hj
@xj
=
+ (! i
@pi
@pi
xi )
@xj
@m
Problem 11 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:
5.1
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
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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 12 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?
5.2
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 )
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
7
mi
Problem 13 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?
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