Utility. UTILITY FUNCTIONS u A preference relation that is complete, reflexive, transitive and...
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Transcript of Utility. UTILITY FUNCTIONS u A preference relation that is complete, reflexive, transitive and...
UTILITY FUNCTIONS
A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function (as an alternative, or as a complement, to the indifference “map” of the previous lecture).
Continuity means that small changes to a consumption bundle cause only small changes to the preference (utility) level.
UTILITY FUNCTIONS
A utility function U(x) represents a preference relation if and only if:
x0 x1 U(x0) > U(x1)
x0 x1 U(x0) < U(x1)
x0 x1 U(x0) = U(x1)
~
UTILITY FUNCTIONS
Utility is an ordinal (i.e. ordering or ranking) concept.
For example, if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y. However, x is not necessarily “three times better” than y.
UTILITY FUNCTIONSand INDIFFERENCE CURVES
Consider the bundles (4,1), (2,3) and (2,2).
Suppose (2,3) > (4,1) (2,2). Assign to these bundles any
numbers that preserve the preference ordering;e.g. U(2,3) = 6 > U(4,1) = U(2,2) = 4.
Call these numbers utility levels.
UTILITY FUNCTIONSand INDIFFERENCE CURVES
An indifference curve contains equally preferred bundles.
Equal preference same utility level. Therefore, all bundles on an
indifference curve have the same utility level.
UTILITY FUNCTIONSand INDIFFERENCE CURVES
So the bundles (4,1) and (2,2) are on the indifference curve with utility level U
But the bundle (2,3) is on the indifference curve with utility level U 6
UTILITY FUNCTIONSand INDIFFERENCE CURVES
Comparing more bundles will create a larger collection of all indifference curves and a better description of the consumer’s preferences.
UTILITY FUNCTIONSand INDIFFERENCE CURVES
Comparing all possible consumption bundles gives the complete collection of the consumer’s indifference curves, each with its assigned utility level.
This complete collection of indifference curves completely represents the consumer’s preferences.
UTILITY FUNCTIONSand INDIFFERENCE CURVES
The collection of all indifference curves for a given preference relation is an indifference map.
An indifference map is equivalent to a utility function; each is the other.
UTILITY FUNCTIONS
If (i) U is a utility function that represents a preference relation; and (ii) f is a strictly increasing function,then V = f(U) is also a utility functionrepresenting the original preference function.Example? V = 2.U
GOODS, BADS and NEUTRALS
A good is a commodity unit which increases utility (gives a more preferred bundle).
A bad is a commodity unit which decreases utility (gives a less preferred bundle).
A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).
GOODS, BADS and NEUTRALS
Utility
Waterx’
Units ofwater aregoods
Units ofwater arebads
Around x’ units, a little extra water is a neutral.
Utilityfunction
UTILITY FUNCTIONS
0)b 0,(a , 2121 baxxxxU
bx, 2121 axxxU
min, 2,121 xxxxU
Cobb-Douglas Utility Function
Perfect Substitutes Utility Function
Perfect Complements Utility Function
Note: MRS = (-)a/b
Note: MRS = ?
UTILITYPreferences can be represented by a utility function if the functional form has certain “nice” properties
Example: Consider U(x1,x2)= x1.x2
1. u/x1>0 and u/x2>0
2. Along a particular indifference curve
x1.x2 = constant x2=c/x1
As x1 x2i.e. downward sloping indifference curve
UTILITY
Example U(x1,x2)= x1.x2 =16
X1 X2 MRS
1 16
2 8 (-) 8
3 5.3 (-) 2.7
4 4 (-) 1.3
5 3.2 (-) 0.8
3. As X1 MRS (in absolute terms), i.e convex preferences
COBB DOUGLAS UTILITY FUNCTION
Any utility function of the form
U(x1,x2) = x1a x2
b
with a > 0 and b > 0 is called a Cobb-Douglas utility function.
Examples
U(x1,x2) = x11/2 x2
1/2 (a = b = 1/2)V(x1,x2) = x1 x2
3 (a = 1, b = 3)
COBB DOUBLAS INDIFFERENCE CURVES
x2
x1
All curves are “hyperbolic”,asymptoting to, but nevertouching any axis.
PERFECT SUBSITITUTES
5
5
9
9
13
13
x1
x2
x1 + x2 = 5
x1 + x2 = 9
x1 + x2 = 13
All are linear and parallel.
V(x1,x2) = x1 + x2
PERFECT COMPLEMENTS
Instead of U(x1,x2) = x1x2 or V(x1,x2) = x1 + x2, consider
W(x1,x2) = min{x1,x2}.
PERFECT COMPLEMENTSx2
x1
45o
min{x1,x2} = 8
3 5 8
35
8
min{x1,x2} = 5
min{x1,x2} = 3
All are right-angled with vertices/cornerson a ray from the origin.
W(x1,x2) = min{x1,x2}
MARGINAL UTILITY
Marginal means “incremental”. The marginal utility of product i is
the rate-of-change of total utility as the quantity of product i consumed changes by one unit; i.e.
MUUxii
MARGINAL UTILITY
U=(x1,x2)
MU1=U/x1 U=MU1.x1
MU2=U/x2 U=MU2.x2
Along a particular indifference curve
U = 0 = MU1(x1) + MU2(x2)
x2/x1 {= MRS} = (-)MU1/MU2
MARGINAL UTLITIES AND MARGINAL RATE OF SUBISITUTION
The general equation for an indifference curve is U(x1,x2) k, a constant
Totally differentiating this identity gives
Uxdx
Uxdx
11
22 0
MARGINAL UTLITIES AND MARGINAL RATE OF SUBISITUTION
Uxdx
Uxdx
22
11
rearranging
2
1
1
2
/
/
xU
xU
xd
xd
This is the MRS.
MU’s and MRS: An example
Suppose U(x1,x2) = x1x2. Then
Ux
x x
Ux
x x
12 2
21 1
1
1
( )( )
( )( )
1
2
2
1
1
2
/
/
x
x
xU
xU
xd
xdMRS
so
MU’s and MRS: An example
MRSxx
2
1
MRS(1,8) = - 8/1 = -8 MRS(6,6) = - 6/6 = -1.
x1
x2
8
6
1 6U = 8
U = 36
U(x1,x2) = x1x2;
MONOTONIC TRANSFORMATIONS AND MRS
Applying a monotonic transformation to a utility function representing a preference relation simply creates another utility function representing the same preference relation.
What happens to marginal rates-of-substitution when a monotonic transformation is applied? (Hopefully, nothing)
MONOTONIC TRANSFORMATIONS AND MRS
For U(x1,x2) = x1x2 the MRS = (-) x2/x1
Create V = U2; i.e. V(x1,x2) = x12x2
2 What is the MRS for V?
which is the same as the MRS for U.
MRSV xV x
x x
x x
xx
//
1
2
1 22
12
2
2
1
2
2