View
243
Download
1
Embed Size (px)
Citation preview
2
Topics covered
1. Budget Constraint
2. Axioms of Choice & Indifference Curve
3. Utility Function
4. Consumer Optimum
3
Y
X
YA
YB
XA XB
A
B
Bundle of goods
• A is a bundle of goods consisting of XA units of good X (say food) and YA units of good Y (say clothing).
• A is also represented by (XA,YA)
5
BBA
BBA
xxttx
yytty
))1((
))1((CB of Slope
C is on the st. line linking A & B
AB of Slope
BA
BA
BA
BA
xx
yy
txtx
tyty
Conversely, any point on AB can be written as
[0,1] t where))1(,)1(( BABA yttyxttx
Convex Combination
6
Slope of budget line
y
x
P
P
dx
dy (market rate of substitution)
Unit: $) ofnt (independejar per loaves loafper $
jarper $
7
Example:
jar of beer Px=$4loaf of bread Py=$2
jarper loaves 2loafper 2$
jarper 4$
y
x
P
P
Both Px and Py double,
xP
I0
feasible consumption set
yP
I0|Slope|=
y
x
P
P
jarper loaves 2loafper 4$
jarper 8$'
'
y
x
P
PNo change in market rate of substitution
8
Tax: a $2 levy per unit is imposed for each good
2
3
2$2$
2$4$
tP
tP
y
x Slope of budget line changes
y
x
20I
40I
40I
60I
after levy is imposed
80I
After doubling the prices
10
Axioms of Choice
• Nomenclature: : “is preferred to” : “is strictly preferred to” : “is indifferent to”
• Completeness (Comparison)– Any two bundles can be compared and one of the
following holds: AB, B A, or both ( A~B)• Transitivity (Consistency)
– If A, B, C are 3 alternatives and AB, B C, then A C; – Also If AB, BC, then A C.
11
Axioms of choice• Continuity
– AB and B is sufficiently close to C, then A C.
• Strong Monotonicity (more is better)– A=(XA , YA), B=(XB , YB) and XA≥XB, YA≥YB with at
least one is strict, then A>B.
• Convexity– If AB, then any convex combination of A& B is
preferred to A and to B, that is, for all 0 t <1,– (t XA+(1-t)XB, tYA+(1-t)YB) (Xi , Yi), i=A or B.– If the inequality is always strict, we have strict
convexity.
12
Indifference Curve• When goods are divisible and there are only
two types of goods, an individual’s preferences can be conveniently represented using indifference curve map.
• An indifference curve for the individual passing through bundle A connects all bundles so that the individual is indifferent between A and these bundles.
13
Properties of Indifference Curves
• Negative slopes• ICs farther away from
origin means higher satisfaction
X
Y
A
I
II
Not preferred bundles
Preferred bundles
14
• Non-intersection– Two indifference
curves cannot intersect
• Coverage– For any bundle, there
is an indifference curve passing through it. X
Y
AQ
P
Properties of Indifference Curves
15
• Bending towards Origin– It arises from
convexity axiom
– The right-hand- side IC is not allowed
X
Y
Properties of Indifference Curves
17
Utility Function
• Level of satisfaction depends on the amount consumed: U=U(x,y)
• U0 =U(x,y)– All the combination of x & y that yield U0 (all
the alternatives along an indifference curve)
• y=V(x,U0), an indifference curve U(x,y)/x, marginal utility respect to x,
written as MUx.
18
X
Y
A
B
YA
YB
XA XB
U0
dyy
yxUdx
x
yxUdU
),(),(
0
( , ) ( , )0
U x y U x ydU dx dy
x y
(by construction)
Slope: /
0/
xU U
y
dy U x MU
dx U y MU
(if strong monotonicity
holds)
19
Bxy
Axy MRSMRS A
B
X
Y
0U
The MRS is the max amount of good y a consumer would willingly forgo for one more unit of x, holding utility constant (relative value of x expressed in unit of y)
20
• Marginal rate of substitution
0
/0
/x
xyU U y
dy dy U x MUMRS
dx dx U y MU
DMRS: 0 constant 0U
dMRS
dx
21
U=10
U=20
U=30
V=100
V=200V=2001
An order-preserving re-labeling of ICs does not alter the preference ordering.
Measurability of Utility
22
11
22
66
222''''
'''
''
222'
yxUU
xyUU
xyUU
yxUU
xyU
Positive monotonic (order-preserving) transformation
• They are called positive monotonic transformation
23
Positive Monotonic Transformation
What is the MRS of U at (x,y)?
How about U’?
xy
U
yx
U
/MRS
/
U x y
U y x
yxy
U
xyx
U
2'
2'
2
2
2
2
'/ 2MRS'
'/ 2
U x xy y
U y x y x
24
Positive Monotonic Transformation
• IC’s of order-preserving transformation U’ overlap those of U.
• However, we have to make sure that the numbering of the IC must be in same order before & after the transformation.
25
Positive Monotonic Transformation
• Theorem: Let U=U(X,Y) be any utility function. Let V=F(U(X,Y)) be an order-preserving transformation, i.e., F(.) is a strictly increasing function, or dF/dU>0 for all U. Then V and U represent the same preferences.
26
Proof
Consider any two bundles and
Then we have:
( , )A AA x y
( , ).B BB x y
( , ) ( , )
( ( , )) ( ( , ))
( , ) ( , )
U
A A B B
A A B B
A A B B
V
A B
U x y U x y
F U x y F U x y
V x y V x y
A B
Q.E.D.
28
Constrained Consumer Choice Problem
• Preferences: represented by indifference curve map, or utility function U(.)
• Constraint: budget constraint-fixed amount of money to be used for purchase
• Assume there are two types of goods x and y, and they are divisible
29
Consumption problem• Budget constraint
– I0= given money income in $– Px= given price of good x– Py= given price of good y
• Budget constraint: I0Pxx+Pyy• Or, I0= Pxx+Pyy (strong monotonicity)
dI0= Pxdx+Pydy=0 (by construction)
Pxdx=-Pydy
30
D
BYB
YD
XB XA XD
C
A
BA
DBxy XX
yyMRS
Psychic willingness to substitute
At B, my MRS is very high for X. I’m willing to substitute XA-XB forYB-YD. But the market provides me more X to point D!
y
xxy P
PMRS
31
Consumer Optimum
• Normally, two conditions for consumer optimum:
• MRSxy = Px/Py (1)
• No budget left unused (2)
33
coffee
teaU0
U1
U2
Generally low MRS
tea
coffee
Generally high MRS
y)(x, allfor y
xxy P
PMRS y)(x, allfor
y
xxy P
PMRS
Special Cases
35
y
x
(1)
(2)
(3)
(4)
(1) Corner at x=0(2) Interior solution 0<x<R(3) “corner” at R(4) “corner” at R
36
,max
subject to (1)
,
MRS market rate of sub. (2)
or
Hence ,
x y
x y
x y
x x
y y
y x
x y
U xy
I P x P y
MU y MU x
MU y P
MU x P
P y P x
I Ix y
P P
( (1))
An Example: U(x,y)=xy
37
C
D
B
A
A satisfies (1) but not (2)B, C satisfy (2) but not (1)Only D satisfies both (1) &(2)
1 11
2 2 / 2x x x
x I I I
I x P x P I P
39
An application: Intertemporal Choice
• Our framework is flexible enough to deal with questions such as savings decisions and intertemporal choice.
40
Intertemporal choice problem
C1
C2
1600
1000
500 Slope = -1.1
u(c1,c2)=const
Income in period 2
Income in period 2