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Advanced Microeconomics
The household optimum
Harald Wiese
University of Leipzig
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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Part B. Household theory and theory of the …rm
1
The household optimum
2
Comparative statics and duality theory
3
Production theory
4
Cost minimization and pro…t maximization
Harald Wiese (University of Leipzig)
Advanced Microeconomics
2 / 72
Nobel price 2015
In 2015, the Sveriges Riksbank Prize in Economic Sciences in Memory of
Alfred Nobel was awarded to the economist
Angus Deaton (Princeton University, NJ, USA)
for his analysis of consumption, poverty, and welfare
By linking detailed individual choices and aggregate outcomes, his
research has helped transform the …elds of microeconomics,
macroeconomics, and development economics.
How do consumers distribute their spending among di¤erent goods?
How much of society’s income is spent and how much is saved?
How do we best measure and analyze welfare and poverty?
Harald Wiese (University of Leipzig)
Advanced Microeconomics
3 / 72
The household optimum
Overview
1
Budget
2
The household optimum
3
Comparative statics and vocabulary
4
Applying the Lagrange method (recipe)
5
Indirect utility function
6
Consumer’s rent and Marshallian demand
Harald Wiese (University of Leipzig)
Advanced Microeconomics
4 / 72
Budget
Money budget and budget line
De…nition
The expenditure for a bundle of goods x = (x1 , x2 , ..., x` ) at a vector of
prices p = (p1 , p2 , ..., p` ) is the dot product (or the scalar product):
`
p x :=
∑ pg xg .
g =1
De…nition
The money budget:
n
B (p, m ) := x 2 R`+ : p x
The budget line:
Harald Wiese (University of Leipzig)
n
o
m , p 2 R` , m 2 R+
x 2 R`+ : p x = m
Advanced Microeconomics
o
5 / 72
Budget
Money budget: A two goods case
x2
Problem
m
p2
Assume that the household
consumes bundle A. Identify
the “left-over” in terms of
good 1, in terms of good 2
and in money terms.
C
B
A
m
p1
x1
Problem
What happens to the budget
line if
price p1 doubles;
if both prices double?
Harald Wiese (University of Leipzig)
Advanced Microeconomics
6 / 72
Budget
Money budget
Lemma
For any number α > 0:
B (αp, αm ) = B (p, m )
Problem
Fill in: For any number α > 0:
B (αp, m ) = B (p, ?) .
Harald Wiese (University of Leipzig)
Advanced Microeconomics
7 / 72
Budget
Marginal opportunity cost for two goods
Problem
Verify that the budget line’s slope is given by
p1
p2
(in case of p2 6= 0).
De…nition
If p1
0 and p2 > 0,
MOC (x1 ) =
p1
dx2
=
dx1
p2
– the marginal opportunity cost of consuming one unit of good 1 in terms
of good 2.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
8 / 72
Budget
Marginal opportunity cost
x2
m
p2
∆ x1
∆x 2 = MOC ∆x1
p
= 1 ∆ x1
p2
m
p1
Harald Wiese (University of Leipzig)
Advanced Microeconomics
x1
9 / 72
Endowment budget
De…nition
De…nition
For p 2 R` and an endowment ω 2 R`+ :
n
B (p, ω ) := x 2 R`+ : p x
– the endowment budget.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
p ω
o
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Endowment budget
A two goods case
budget line: p1 x1 + p2 x2 = p1 ω 1 + p2 ω 2
2
marginal opportunity cost: MOC = dx
dx1 =
p1
p2
x2
Problem
p1ω1 + p 2ω2
p2
What happens to the
budget line if
dx2
p
=− 1
dx1
p2
price p1 doubles;
ω2
both prices double?
ω1
Harald Wiese (University of Leipzig)
p1ω1 + p 2ω2
p1
x1
Advanced Microeconomics
11 / 72
Application 1
Intertemporal consumption
Notation:
ω 1 and ω 2 – monetary income in t1 and t2 ;
x1 and x2 – consumption in t1 and t2 ;
household can borrow (x1 > ω 1 ), lend (x1 < ω 1 ) or consume what it
earns (x1 = ω 1 );
r – rate of interest.
Consumption in t2 :
x2 =
ω2
|{z}
+
second-period
income
= ω 2 + (1 + r ) ( ω 1
Harald Wiese (University of Leipzig)
(ω 1 x1 )
| {z }
amount borrowed (<0)
or lended (>0)
+
r (ω 1 x1 )
| {z }
interest payed (<0)
or earned (>0)
x1 )
Advanced Microeconomics
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Application 1
Borrow versus lend
borrow verwandt mit
borgen und
bergen („in Sicherheit bringen“) wie in
Herberge („ein das Heer bergender Ort“)
lend verwandt mit
Lehen („zur Nutzung verliehener Besitz“) und
leihen, verwandt mit
lateinischstämmig Relikt („Überrest“) und Reliquie („Überbleibsel oder
hochverehrte Gebeine von Heiligen“) und mit
griechischstämmig Eklipse („Ausbleiben der Sonne oder des Mondes“
> „Sonnen- bzw. Mond…nsternis“) und auch mit
griechischstämmig Ellipse (in der Geometrie ein Langkreis, bei dem die
Höhe geringer ist als die Breite und insofern ein Mangel im Vergleich
zum Kreis vorhanden ist – agr. elleipsis (έλλειψις) bedeutet
„Ausbleiben“ > „Mangel“
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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Application 1
Intertemporal consumption
2 ways to rewrite the budget equation:
in future-value terms:
(1 + r ) x1 + x2 = (1 + r ) ω 1 + ω 2 ,
in present-value terms:
x1 +
Harald Wiese (University of Leipzig)
x2
ω2
= ω1 +
.
1+r
1+r
Advanced Microeconomics
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Application 1
Intertemporal consumption
budget line: (1 + r ) x1 + x2 = (1 + r ) ω 1 + ω 2
dx2
dx1
marginal opportunity cost: MOC =
= 1+r
x2
(1 + r ) ω1 + ω2
Problem
dx2
= −(1 + r )
dx1
consumption point
of a lender
ω2
endowment
point
consumption point
of a borrower
ω1
Harald Wiese (University of Leipzig)
ω1 +
ω2
(1 + r )
Advanced Microeconomics
What happens to
the budget line if
the interest rate
decreases?
x1
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Application 2
Leisure versus consumption
Notation:
xR – recreational hours (0
household works 24
xR
xR hours;
24 = ω R ) ! good 1;
xC – real consumption ! good 2;
w – the wage rate;
ω C – the real non-labor income;
p – the price index.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
16 / 72
Application 2
Leisure versus consumption
Household’s consumption in nominal terms:
pxC = pω C + w (24
xR )
Household’s consumption in endowment-budget form:
wxR + pxC = w 24 + pω C
Harald Wiese (University of Leipzig)
Advanced Microeconomics
17 / 72
Application 2
Leisure versus consumption
budget line: wxR + pxC = w 24 + pω C
C
marginal opportunity cost: MOC = dx
dxR =
w
p
xC
Problem
w 24 + pωC
p
dxC
w
=−
dxR
p
What happens to the
budget line if the wage
rate increases?
consumption
point
endowment
point
ωC
recreational
time
Harald Wiese (University of Leipzig)
labor 24
time
Advanced Microeconomics
xR
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Application 3
Contingent consumption
Notation:
A – a household wealth;
L = [x1 , x2 ; p, 1 p ] – lottery
with insurance where
p – the probability of a
bad event;
x1 = A D + K γK =
A D + (1 γ ) K
! insured event
x2 = A γK
D – possible damage;
L = [A D, A; p, 1
– lottery without
insurance
p]
special cases
K – insurance sum;
γ – insurance rate
γK – insurance premium.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
x1 = A D, x2 = A
! no insurance (K := 0).
x1 = x2 = A γD
! full insurance (K := D).
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Application 3
Contingent consumption
budget line: x1 +
1 γ
γ x2
= (A
1 γ
γ A
γ
dx2
dx1 = 1 γ
D) +
marginal opportunity cost: MOC =
x1 = A − D + K − γK
x 2 = A − γK
x2
(prob. 1 − p )
no insurance
A
A − γD
dx2
γ
=−
dx1
1− γ
Problem
Interpret the part of the
budget line
full insurance
right of the
full-insurance point;
45°
A − D A − γD
Harald Wiese (University of Leipzig)
x1
(prob. p )
Advanced Microeconomics
left of the
no-insurance point!
20 / 72
The household optimum
1
Budget
2
The household optimum
3
Comparative statics and vocabulary
4
Applying the Lagrange method (recipe)
5
Indirect utility function
6
Consumer’s rent and Marshallian demand
Harald Wiese (University of Leipzig)
Advanced Microeconomics
21 / 72
The household’s decision problem
De…nition
Best response function x R :
x R (B ) : = x 2 B: there is no x 0 2 B with x 0
x
or
R
x (B ) : = arg max U (x )
x 2B
Any x from x R (B ) – a household optimum.
Notation: also x R (p, m ) or x R (p, ω ) or just x (p )
Lemma
For any number α > 0:
x R (αp, αm ) = x R (p, m )
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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The household’s decision problem
Exercise 1
x2
x2
Problem
A
A
indifference
curve
budget
line
B
budget
line
indifference
curve
(a)
x1
(b)
x1
(d)
x1
x2
x2
Assume
monotonicity of
preferences. Are
the highlighted
points A or B
optima?
budget
line
A
budget
line
indifference
curve
B
(c)
Harald Wiese (University of Leipzig)
x1
A
indifference
curve
Advanced Microeconomics
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The household’s decision problem
Exercise 2
Problem
Assume a household’s decision problem (B (p, ω ) , -).
x R (B ) consists of the bundles x that ful…ll the two conditions:
1
The household can a¤ord x:
p x
2
p ω
There is no other bundle y that the household can a¤ord and that he
prefers to x:
y x )??
Substitute the question marks by an inequality.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
24 / 72
MRS versus MOC
dx2
dx1
Marginal willingness to pay:
MRS =
If the household consumes
one additional unit of good 1,
how many units of good 2
can he forgo so as to remain indi¤erent.
movement on the
indi¤erence curve
Marginal opportunity cost:
MOC =
If the household consumes
one additional unit of good 1,
how many units of good 2
does he have to forgo so as to remain
within his budget.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
dx2
dx1
movement on the
budget line
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MRS versus MOC
dx2
dx1
| {z }
MRS =
dx2
dx1
| {z }
>
absolute value
absolute value
of the slope of
of the slope of
the indi¤erence curve
the budget line
= MOC
) increase x1 (if possible)
x2
indifference
curve
budget
line
1 unit
of good 1
MOC
MRS
x1
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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MRS versus MOC
MRS > MOC ) increase x1 (if possible)
x2
budget line
indifference curves
household
optimum
Harald Wiese (University of Leipzig)
Advanced Microeconomics
x1
27 / 72
MRS versus MOC
Alternatively: the household tries to maximize U x1 , pm2
p1
p 2 x1
.
Consume 1 additional unit of good 1
utility increases by
∂U
∂x 1
reduction in x2 by MOC =
utility decrease by
∂U
∂x 2
dx2
dx1
dx2
dx1
=
p1
p2
and hence
(chain rule)
Thus, increase consumption of good 1 as long as
∂U
∂x1
|{z}
marginal bene…t
of increasing x1
or MRS
Harald Wiese (University of Leipzig)
∂U dx2
∂x2 dx1
| {z }
>
marginal cost
of increasing x1
=
∂U
∂x1
∂U
∂x2
>
Advanced Microeconomics
dx2
= MOC
dx1
28 / 72
Household optimum
Cobb-Douglas utility function
U (x1 , x2 ) = x1a x21
a
with 0 < a < 1
The two optimality conditions
MRS =
∂U
∂x1
∂U
∂x2
=
a x2
1 a x1
! p1
p 2 and
=
!
p1 x1 + p2 x2 = m
yield the household optimum
x1 (m, p ) = a
m
,
p1
x2 (m, p ) = (1
Harald Wiese (University of Leipzig)
a)
Advanced Microeconomics
m
.
p2
29 / 72
Household optimum
Perfect substitutes
U (x1 , x2 ) = ax1 + bx2 with a > 0 and b > 0
An increase of good 1 enhances utility if
p1
a
= MRS > MOC =
b
p2
holds. Therefore
8
m
>
,0 ,
>
>
< n p1
x1 , pm2
x (m, p ) =
>
>
>
: 0, m
p2
Harald Wiese (University of Leipzig)
p1
p 2 x1
h
2 R2+ : x1 2 0, pm1
Advanced Microeconomics
io
a
b
a
b
a
b
>
=
<
p1
p2
p1
p2
p1
p2
30 / 72
Household optimum
Concave preferences
U (x1 , x2 ) = x12 + x22
An increase of good 1 enhances utility if
x1
2x1
=
=
x2
2x2
∂U
∂x1
∂U
∂x2
= MRS > MOC =
p1
p2
holds. Therefore, corner solutions:
8
m
>
p1 < p2
>
> n p1 , 0 ,
<
o
m
m
x (m, p ) =
p1 = p2
p 1 , 0 , 0, p 2
>
>
>
m
: 0,
p1 > p2
p2
Harald Wiese (University of Leipzig)
Advanced Microeconomics
31 / 72
Household optimum
Dixit-Stiglitz preferences for love of variety
`
∑ xj
U (x1 , ..., x` ) =
ε 1
ε
j =1
! ε ε1
with ε > 1
with
∂U (x1 ,x2 ,...,x ` )
∂xg
∂U (x1 ,x2 ,...,x ` )
∂xk
ε
=
ε 1
ε
ε 1
ε 1
=
xg ε
ε 1
ε
xk
∑
ε 1
`
ε
x
j =1 j
ε
ε 1
1
∑
ε 1
`
ε
x
j
j =1
ε
ε 1
1
1
1
1
=
xg ε
xk
1
ε
=
xg
xk
ε 1
1
ε 1
ε
1
ε 1
ε
ε xg
ε 1
ε xk
1
ε
=
xk
xg
1
ε
!
=
pg
pk
and, in the case of two goods (` = 2), we obtain
m
m
m
x1 (m, p ) =
, x2 (m, p ) =
ε =
1
ε
ε
p1 + p2 p1
p2 + p11 ε p2ε
p1 + p2 pp21
Harald Wiese (University of Leipzig)
Advanced Microeconomics
32 / 72
Household optimum and monotonicity I
Lemma
Let x (p, m ) be a household optimum. Then
local nonsatiation implies p x = m (Walras’law); and ...
Proof.
p x
x2
m (why?).
Assume: p x < m
) contradiction!
m
p2
x*
y
m
p1
Harald Wiese (University of Leipzig)
Advanced Microeconomics
x1
33 / 72
Household optimum and monotonicity II
Lemma
Let x (p, m ) be a household optimum. Then
...
strict monotonicity implies p >> 0;
local nonsatiation and weak monotonicity imply p
0.
Proof.
Assume pg
0 ) household can be made better o¤ by consuming more of
good g (strict monotonicity). Contradiction!
Assume pg < 0 ) household can “buy” additional units of g without
being worse o¤ (weak monotonicity). Household has additional funding for
preferred bundles (nonsatiation). Contradiction!
Harald Wiese (University of Leipzig)
Advanced Microeconomics
34 / 72
Comparative statics and vocabulary
1
Budget
2
The household optimum
3
Comparative statics and vocabulary
4
Applying the Lagrange method (recipe)
5
Indirect utility function
6
Consumer’s rent and Marshallian demand
Harald Wiese (University of Leipzig)
Advanced Microeconomics
35 / 72
Vocabulary
De…nition
The (Marshallian) demand function for good g ! xg (pg );
The cross demand function for good g with respect to pk of good
k 6= g ! xg (pk ) ;
The Engel function for good g ! xg (m ) .
In case of an endowment budget the household is called
a net supplier of good g if xg (p, ω ) < ω g and
a net demander if xg (p, ω ) > ω g .
Harald Wiese (University of Leipzig)
Advanced Microeconomics
36 / 72
Vocabulary
De…nition
A good g is:
ordinary if
∂xg
∂pg
0
(non-ordinary otherwise) ! slope of demand curve;
normal if
∂xg
0
∂m
(inferior otherwise) ! slope of Engel curve;
Problem
Consider the demand function x = a mp , a > 0.
Is the good an ordinary and/or a normal good?
Harald Wiese (University of Leipzig)
Advanced Microeconomics
37 / 72
Vocabulary
De…nition
A good g is:
a substitute of good k if
∂xg
∂pk
0;
∂xg
∂pk
0.
a complement of good k if
Problem
Consider the demand functions x1 = a pm1 and x2 = (1 a) pm2 , 0 < a < 1,
and …nd out whether good 1 is a substitute or a complement of good 2!
Harald Wiese (University of Leipzig)
Advanced Microeconomics
38 / 72
Price-consumption curve and demand curve
Deriving the demand curve graphically
x2
dx 2
pD
=− 1
dx1
p2
B
p1B
x1B
x1C
priceconsumption curve
D
C
Problem
p1C
p1D
x1D
Assuming that good 1 and
good 2 are complements,
sketch a price-consumption
curve and the associated
demand curve for good 1.
x1
p1
p1B
demand curve
p1C
p1D
x1B
x1C
x1D
Harald Wiese (University of Leipzig)
x1
Advanced Microeconomics
39 / 72
Price-consumption curve and demand curve
Deriving the demand curve analytically
1
2
Assume the utility function U (x1 , x2 ) = x13 x23 with household optimum
x1 =
2m
1m
, x =
.
3 p1 2
3 p2
The demand curve for good 1 is x1 = f (p1 ) =
1 m
3 p1 .
x2 = h (x1 ) = 23 pm2 is already the price-consumption curve–x2 is a
constant (boring) function x1 .
Note: It is important that h is not a function of p1 .
Problem
Determine, analytically, the price-consumption curve for the the case of
perfect complements, U (x1 , x2 ) = min (x1 , 2x2 )! Can you also …nd the
demand function for good 2? Assume p1 > 0 and p2 > 0!
Harald Wiese (University of Leipzig)
Advanced Microeconomics
40 / 72
Vocabulary
Saturation quantity and prohibitive price
De…nition
Let xg (pg ) be the quantity demanded
for any pg
0. )
xgsat
pg
p gproh
:= xg (0)
demand curve
– the saturation quantity;
p1proh := min fpg
0 : xg (pg ) = 0g
xsat
g
xg
– the prohibitive price.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
41 / 72
Income-consumption curve and Engel curve
Deriving the Engel curve graphically
x2
mD
income-consumption curve
mC
D
m
B
C
B
x1B x1C x1D
Problem
Assuming that good 1 and
good 2 are complements,
sketch an income-consumption
curve and the associated Engel
curve for good 1!
x1
m
Engel curve
mD
mC
mB
x1B x1C x1D
Harald Wiese (University of Leipzig)
x1
Advanced Microeconomics
42 / 72
Income-consumption curve and Engel curve
Deriving the Engel curve analytically
Assume the household optimum x1 =
The Engel curve for good 1 is
1 m
3 p1 ,
x1 = q (m ) =
x2 =
2 m
3 p2 .
1m
.
3 p1
Income-consumption curve:
Solve good 1’s demand for m and obtain m = 3p1 x1 . Subsituting in x2
3p x
yields x2 = 32 pm2 = 23 p12 1 = 2 pp21 x1 and hence the
income-consumption curve
p
x2 = g ( x1 ) = 2 1 x1 .
p2
Note: It is important that g is not a function of m.
Problem
Determine, analytically, the income-consumption curve and the
Engel-curve function for U (x1 , x2 ) = min (x1 , 2x2 )!
Harald Wiese (University of Leipzig)
Advanced Microeconomics
43 / 72
De…ning substitutes and complements
Exercise
Problem
Determine
∂x1 (p,m )
∂p 2
and
∂x2 (p,m )
∂p 1
for the quasi-linear utility function:
U (x1 , x2 ) = ln x1 + x2
Assume positive prices and
m
p2
( x1 > 0 ) !
> 1, in order to avoid a corner solution!
Conclusion: good g can be the substitute of good k while k is not a
(strict) substitute of g !
Harald Wiese (University of Leipzig)
Advanced Microeconomics
44 / 72
Vocabulary
Price elasticity of demand
De…nition
Let xg (pg ) be the demand at price pg (other prices are held constant). )
dxg
xg
dp g
pg
εxg ,pg :=
=
dxg pg
dpg xg
|{z}
mathematically doubtful
– the price elasticity of demand.
Problem
Calculate the price elasticities of demand for the demand functions:
xg (pg ) = 100
Harald Wiese (University of Leipzig)
pg and xk (pk ) =
Advanced Microeconomics
1
.
pk
45 / 72
Vocabulary
Price elasticity of demand - application
Drug users have inelastic demand: jεx ,p j < 1 or εx ,p >
1
Then a price increase increases expenditure:
d (px (p ))
dp
dx
dp
p dx
1+
x dp
= x +p
= x
= x (1 + εx ,p ) = x (1
jεx ,p j) > 0.
Political implication:
Making drugs expensive
by taxing them or
by criminalizing selling or buying
increases expenditure and hence drug-related crime (stealing money in
order to …nance the addiction).
Harald Wiese (University of Leipzig)
Advanced Microeconomics
46 / 72
Vocabulary
Income elasticity of demand
De…nition
Let xg (m ) be the demand at income m. The income elasticity (of
demand) is denoted by εxg ,m and given by
εxg ,m :=
dxg
xg
dm
m
=
dxg m
.
dm xg
De…nition
We call a good g
a luxury good (such as caviar) if εxg ,m
a necessity good (such as oat groats) if 0
Harald Wiese (University of Leipzig)
Advanced Microeconomics
1 holds;
εxg ,m
1 holds.
47 / 72
Vocabulary
Income elasticity of demand - exercise
inferior goods
normal goods
0
necessity
goods
1
luxury
goods
ε x ,m
1
Problem
Calculate the income elasticity of demand for the Cobb-Douglas utility
1
2
function U (x1 , x2 ) = x13 x23 ! How do you classify (demand for) good 1?
Harald Wiese (University of Leipzig)
Advanced Microeconomics
48 / 72
Vocabulary
Average income elasticity of demand - lemma
Lemma
Assume local nonsatiation and the household optimum x . Then the
average income elasticity is 1:
`
∑ sg εx ,m = 1
g
g =1
where the weights are the relative expenditures, sg :=
Harald Wiese (University of Leipzig)
Advanced Microeconomics
pg xg
m
.
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Vocabulary
Average income elasticity of demand - proof
According to Walras’law, the household chooses x on the budget
line, p x (m ) = m. (How about ` = 1?)
Find the derivative of the budget equation m =
respect to m to obtain
`
dxg
1 = ∑ pg
dm
g =1
`
∑g =1 pg xg (m) with
and,
by multiplying the summands with
`
1=
∑
pg
g =1
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xg m
m xg
= 1,
` p x dx
`
dxg xg m
g g
g m
= ∑
= ∑ sg εxg ,m
dm m xg
g =1 m dm xg
g =1
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Vocabulary
Aggregate demand
De…nition
Let x i (p ) be the demand functions of individuals i = 1, ..., n. )
n
x (p ) := ∑i =1 x i (p ) – aggregate demand.
p
p
x1 (p )
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p
x2 (p )
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x(p )
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Vocabulary
Aggregate demand
Problem
xg1 (pg ) = max (0, 100
pg ) ,
xg2 (pg ) = max (0, 50
2pg ) and
xg3
3pg ) .
(pg ) = max (0, 60
Find the aggregate demand function!
Hint: Find the prohibitive prices …rst!
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Advanced Microeconomics
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Vocabulary
Inverse demand function
De…nition
h
i
Let x1 : 0, p1proh ! [0, x1sat ] be an injective demand function. )
p1 = x1
1
h
i
: 0, x1sat ! 0, p1proh
x1 7! p1 (x1 ) where p1 (x1 ) is the unique price resulting in x1 .
– the inverse demand function.
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Advanced Microeconomics
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Vocabulary
Inverse demand function
pg
p gproh
p g (xˆg )
inverse
demand curve
demand curve
pˆg
xˆg
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x g (pˆg )
xsat
g
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xg
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The Lagrange method
1
Budget
2
The household optimum
3
Comparative statics and vocabulary
4
Applying the Lagrange method (recipe)
5
Indirect utility function
6
Consumer’s rent and Marshallian demand
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Advanced Microeconomics
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Applying the Lagrange method (recipe)
"
L (x, λ) = U (x ) + λ m
`
∑ pg xg
g =1
#
with:
U – strictly quasi-concave and strictly monotonic utility function;
prices p >> 0;
λ > 0 – Lagrange multiplier - translates a budget surplus
m ∑g` =1 pg xg > 0 into utility
Increasing consumption has
a positive e¤ect via U, but
a negative e¤ect via decreasing budget and λ
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Advanced Microeconomics
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Applying the Lagrange method (recipe)
Di¤erentiate L with respect to xg :
∂U (x1 , x2 , ..., x` )
∂L (x1 , x2 , ..., λ)
=
∂xg
∂xg
!
λpg = 0 or
∂U (x1 , x2 , ..., x` ) !
= λpg
∂xg
and hence, for two goods g and k
∂U (x1 ,x2 ,...,x ` )
∂xg
∂U (x1 ,x2 ,...,x ` )
∂xk
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!
=
pg
!
or MRS = MOC
pk
Advanced Microeconomics
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Applying the Lagrange method (recipe)
Problem
Set the derivative of L with respect to λ equal to 0. What do you …nd?
λ – the shadow price of the restriction:
λ=
dU
.
dm
But: U does not have m as an argument so that
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Advanced Microeconomics
dU
dm
is not quite correct.
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Indirect utility function
1
Budget
2
The household optimum
3
Comparative statics and vocabulary
4
Applying the Lagrange method (recipe)
5
Indirect utility function
6
Consumer’s rent and Marshallian demand
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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Indirect utility function
De…nition
De…nition
Consider a household with utility function U. )
V
: R`
R+ ! R,
(p, m) 7! V (p, m) := U (x (p, m))
– indirect utility function.
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Advanced Microeconomics
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Utility function versus indirect utility function
function
arguments
utility
function
indirect
utility
function
x 2 R`+
optimal bundles
m and p 2 R`
x (p, m )
x (p, m )
Problem
Determine the indirect utility function for the Cobb-Douglas utility
function U (x1 , x2 ) = x1a x21 a (0 < a < 1)!
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Revisiting the Lagrange multiplier
Aim: λ =
dU
dm
!λ=
dV
dm
Di¤erentiating the budget function with respect to m yields
∂x
∑g` =1 pg ∂mg = dm
dm = 1 (as we know from the the proof about the
average income elasticity)
Di¤erentiating the indirect utility function V (p, m ) = U (x (p, m ))
with respect to m leads to
`
`
`
∂xg
∂xg
∂U ∂xg
∂V
= ∑
= ∑ (λpg )
= λ ∑ pg
=λ
∂m
∂x
∂m
∂m
∂m
g
g =1
g =1
g =1
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Revisiting the Lagrange multiplier
The optimization condition can be rewritten as:
∂U (x1 , x2 , ..., x` ) !
= λpg =
∂xg
|
{z
}
marginal utility
∂V
pg
{z }
|∂m
marginal cost
Interpreting the marginal cost of consuming one additional unit of good g :
You consume one additional unit of good g ,
your expenditure increases by pg so that
the income left for other goods decreases by pg and hence
utility decreases by
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∂V
∂m pg .
Advanced Microeconomics
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Consumer’s rent and Marshallian demand
1
Budget
2
The household optimum
3
Comparative statics and vocabulary
4
Applying the Lagrange method (recipe)
5
Indirect utility function
6
Consumer’s rent and Marshallian demand
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Advanced Microeconomics
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Marginal willingness to pay
Assume:
x2 – “all the other goods” (money);
p2 = 1.
)
Marginal willingness to pay for one extra unit of x1 is
MRS =
p1
= p1 .
p2
Inverse demand function measures (cum grano salis) the marginal
willingness to pay for one extra unit of a good.
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Marginal willingness to pay
pg
p gproh
p hg
consumer`s
rent
inverse demand curve
= willingness to pay
pˆg
aggregate willingness
to pay
xˆg
xsat
g
xg
Problem
p (q ) = 20
4q, p = 4
aggregate willingness to pay? consumer’s rent?
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Advanced Microeconomics
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The Marshallian willingness to pay and the Marshallian
consumer’s rent
De…nition
Let pg be an inverse demand function.
The Marshallian willingness to pay
R x̂
for the quantity x̂g : 0 g pg (xg ) dxg
for a price decrease from pgh to p̂g < pgh :
for a price decrease from pgproh to p̂g <
CR
Marshall
(p̂g ) =
=
Z xg (p̂g )
0
Z xg (p̂g )
0
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R pgh
p̂ g
proh
pg
pg (xg ) dxg
(pg (xg )
Advanced Microeconomics
xg (pg ) dpg
(consumer’s rent)
p̂g xg (p̂g )
p̂g ) dxg =
Z pgproh
p̂ g
xg (pg ) dpg
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The diamond-water paradox
Why are diamonds more expensive than water
although water is more „useful“?
pD
pW
water
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diamants
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The Marshallian willingness to pay
An imperfect concept
Marshallian willingness to pay for the quantity x̂g
— > “sum” the prices for all the units from 0 up to x̂g
But:
According to Marshallian demand, the consumers pay one price for all
the units.
R x̂
In contrast, the integral 0 g pg (xg ) dxg presupposes that the
consumer pays (in general) di¤erent prices for the …rst, the second,
and so on, units.
Now, if higher prices are to be paid for the …rst units, the household has
less income available to spend on the additional units:
demand is lowered
consumer’s willingness to pay or consumer’s rent is lower
Marshallian willingness to pay exaggerates “real” consumers’rent
— > next chapter
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Advanced Microeconomics
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Further exercises
Problem 1
Discuss the units in which to measure price, quantity, expenditure. If you
are right, expenditure should be measured in the same units as the product
of price and quantity.
Problem 2
Sketch budget lines:
Time T = 18 and money m = 50 for football F (good 1) or basket
ball B (good 2) with prices
pF = 5, pB = 10 in monetary terms,
tF = 3, tB = 2 and temporary terms
Two goods, bread (good 1) and other goods (good 2). Transfer in
kind with and without probhibition to sell:
m = 300, pB = 2, pother = 1
Transfer in kind: B = 50
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Further exercises
Problem 3
Assume two goods 1 and 2. Abba faces a price p for good 1 in terms of
good 2. Think of good 2 as the numeraire good with price 1. Abba’s utility
p
functions U is given by U (x1 , x2 ) = x1 + x2 . His endowment is
1
ω = (25, 0) . Find Abba’s optimal bundle. Hint: Distinguish p
10 and
1
!
p < 10
Problem 4
∂x (p,m )
Show by way of example that 1∂p1 < 0 and
happen. Hint: use perfect complements.
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Advanced Microeconomics
∂x1 (p,ω )
∂p 1
> 0 may well
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Further exercises
Problem 5
Derive the indirect utility functions of the following utility functions:
(a) U (x1 , x2 ) = x1 x2 ,
(b) U (x1 , x2 ) = min fa x1 , b x2 g where a, b > 0 holds,
(c) U (x1 , x2 ) = a x1 + b x2 where a, b > 0 holds.
Problem 6
Consider the preferences given by the utility function
U (x1 , x2 ) = x1 + 2x2 . Find x (p, m ) . Sketch the demand function for
good 1. Sketch the Engel curve for good 1 for the case of p1 < 21 p2 while
observing the usual convention that the x1 -axis is the abscissa!
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