Handout 4th Lecture
ECON 4230/35
Kjell Arne Brekke
1
Utility
The consumption set is denoted X and is usually the non-negative ortant
of Rn .
Preferences, is a relation between two objects x and y in X.
x
y
means "The consumer think that x is at least as good as y."
We assume preferences are
Complete: x
y or y
Re‡exive: x
x
Transitive: x
x.
y and y
z implies x
z.
Note that we can construct strict preferences and indi¤erences from
the basic preferences:
x
y if x
y and y
x
y if it is not the case that y
x
y means y
x
x (similar for
1
x (by completness x
)
y)
Convexity
Preferences are convex if for any x; y and z 2 X; such that x
and y
z then tx + (1
t)y
z
z for t 2 [0; 1]
Preferences are strictly convex if for any x; y and z 2 X; such that
x
z and y
z then tx + (1
z for t 2 (0; 1)
t)y
A technical condition called continuity:
fx 2 X : x
yg and fx 2 X : x
yg are closed sets.
Two assumptions that makes sure that a consumer spend the entire
budget:
Monotonicity
If x
y and x 6= y then x
y:
Local nonsatiation:
for any x and " > 0 there is an y such that jy
xj < " and y
x
Problem 1 Explain in word (understandable to a non-economist) what
monotonicity means. Still in words understandable to a non-economist,
give an example of preferences that are nocally nonsatiated but not monotone.
Now we are able to state the main theorem:
Theorem 2 (Existence of a utility function) Suppose that preferences are complete, re‡exive, transitive, continuous and strongly monotonic,
then there exist a continuous utility function u such that
(x
y) () (u(x)
2
u(y))
1.0.1
What is utility
The important message to take home from this is that even when we use
a utility function, we really talk about preferences. Utilities are only a
way of expressing prefenrences.
1.1
Utility maximization
Budget set
B = fx 2 X : px
mg
Assuming monotonicity or local nonsatiation
px = m
Thus the utility maximization may be written
v(p; m) = max u(x)
such that px = m
The function v(p; m) is called the indirect utility function.
1.2
The social animal
Humans are social animals, we like to hang together. (Franz de Waal,
in ’The age of empathy’quote an aboriginal: "It is bad to die, because
when you die you are alone.")
Problem 3 To what extent does the framework above alow for a description of humans as social animals?
1.3
Properties of the indirect utility
1) Non-increasing in p and non-decreasing in m. (look at the budget set)
2) Homogenous of degree zero in (p; m) :
v(tp; tm) = v(p; m)
3
(obvious property of the budget set)
3) Quasiconvex in p, that is
fp :v(p; m)
kg is convex for all k
4) Continuous in (p; m) for positive prices and income.
1.4
The expenditure function
De…ned as
e(p; u) = min px
such that u(x)
u
Note that this is just like a cost function
c(w; y) = min px
such that f (x)
y
so the properties are the same.
1.5
Marshallian and Hicksian demand
The marshallian demand x(p; m) is the solution to the utility maximization problem
v(p; m) = max u(x)
such that px = m
while the Hicksian demand h(p; y)is the solution to expenditure minimization problem
e(p; u) = min px
such that u(x)
4
u
Note that the hicksian demand can be derived from the expenditure
function using Shepards lemma
hi (p; y) =
@e(p; u)
@pi
Problem 4 If the price of co¤e increases by 1 kroner per cup, and the
income Bob need to maintain his utility inceases by 20 kroner. How
many cups do Bob buy per month? Use the above theory to answer.
1.6
Some identities
Both problem get the same solution, where the indi¤erence curve and
the budget line are in tangent.
u0i
pi
= 0
pj
uj
They only take income or utility as given respectively, but are intimately
related:
e(p; v(p; m)) = m
v(p; e(p; u)) = u
xi (p; m) = hi (p; v(p; m))
hi (p; u) = xi (p; e(p; u))
1.7
Roys Identity
xi (p; m) =
1.8
@v(p;m)
@pi
@v(p;m)
@m
Moneymetric utility
Moneymetric utility is
m(p; x) =e(p; u(x))
Moneymetric indirect utility
(p; q;m) =e(p; v(q;m))
5
2
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
@xi (p;m)
@pj
and compen-
(p;m)
sates income xj (p; m) @xi@m
.
2.0.1
Classi…cation
Necessary goods:
@xi (p; m) m
<1
@m
xi
Luxury good
@xi (p; m) m
>1
@m
xi
Inferior goods
@xi (p; m)
>0
@m
Normal goods
@xi (p; m)
<0
@m
Gi¤en good
@xi (p; m)
>0
@pi
Problem 5 Use the Slutsky equation to explain why normal goods are
not Gi¤en goods.
6
2.0.2
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
@xj (p; m)
@xi (p; m)
<0<
@pj
@pi
2.1
Properties of the demand function
Symmetry and the negative diagonal elements
@hi (p; y)
@pi
3
0
Revealed preferences
Problem 6 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?
7