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# AP ECON 2300 F2012 Session 3.doc

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York University

Economics

ECON 2300

Wai Ming Ho

Fall

Description

AP ECON 2300 F2012 Session 3
Department of Economics
Topic: Utility and Choice
Reading: Chapter 4 and 5.
Ch 4: Utility
Preferences - A Reminder
• x y: x is preferred strictly to y.
• x ~ y: x and y are equally preferred.
• x ≥ y: x is preferred at least as much as is y.
Completeness: For any two bundles x and y it is always possible to state either that
x ≥ y
or that
y ≥ x.
Reflexivity: Any bundle x is always at least as preferred as itself; i.e.
x ≥ x.
Transitivity: If
x is at least as preferred as y, and
y is at least as preferred as z, then
x is at least as preferred as z; i.e.
x ≥ y and y ≥ z x ≥ z.
Utility Functions
• A preference relation that is complete, reflexive, transitive and continuous can be represented
by a continuous utility function.
• Continuity means that small changes to a consumption bundle cause only small changes to the
preference level.
• A utility function U(x) represents a preference relation if and only if:
x’ > x” U(x’) > U(x”)
x’ < x” U(x’) < U(x”)
x’ ~ x” U(x’) = U(x”).
• Utility is an ordinal (i.e. ordering) concept.
• E.g. if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y. But x is not
preferred three times as much as is y.
Utility Functions & Indiff. 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.
• An indifference curve contains equally preferred bundles.
• Equal preference ⇒ same utility level.
Page 1 of 13 • Therefore, all bundles in an indifference curve have the same utility level.
• So the bundles (4,1) and (2,2) are in the indiff. curve with utility level U º 4
• But the bundle (2,3) is in the indiff. curve with utility level U º 6.
• On an indifference curve diagram, this preference information looks as follows:
• 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.
• 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.
• There is no unique utility function representation of a preference relation.
• Suppose U(x ,x 1 =2x x r1p2esents a preference relation.
• Again consider the bundles (4,1), (2,3) and (2,2).
U(x 1x 2 = x x1,2so
U(2,3) = 6 > U(4,1) = U(2,2) = 4;
that is, (2,3) > (4,1) ~ (2,2).
U(x 1x 2 = x 1 2 (2,3) > (4,1) ~ (2,2).
• Define V = U . 2
• Then V(x ,x1) 2 x x 12 22 and
V(2,3) = 36 > V(4,1) = V(2,2) = 16
so again
(2,3) > (4,1) ~ (2,2).
• V preserves the same order as U and so represents the same preferences.
U(x 1x 2 = x 1 2 (2,3) (4,1) ~ (2,2).
• Define W = 2U + 10.
Page 2 of 13 • U(x ,x ) = x x (2,3) > (4,1) ~ (2,2).
1 2 1 2
• Define W = 2U + 10.
• Then W(x ,1 )2= 2x x1+20 so
W(2,3) = 22 > W(4,1) = W(2,2) = 18. Again,
(2,3) > (4,1) ~ (2,2).
• W preserves the same order as U and V and so represents the same preferences.
• Utility Functions
• If
o U is a utility function that represents a preference relation ≥ and
o f is a strictly increasing function,
• then V = f(U) is also a utility function representing ≥ .
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
Utility
function
Units of Units of
water are water are
goods bads
x’ Water
Around x’ units, a little extra water is a neutral.
© 2010 W. W. Norton & Company, Inc. 47
Some Other Utility Functions and Their Indifference Curves
• Instead of U(x 1x 2 = x 1 2onsider
V(x 1x 2 = x 1 x .2
What do the indifference curves for this “perfect substitution” utility function look like?
Perfect Substitution Indifference Curves
Page 3 of 13 Perfect Substitution Indifference
Curves
x
2
x1+ x =25
13
x + x = 9
1 2
9
x + x = 13
1 2
5
V(x 1x )2= x +1x . 2
5 9 13 x
1
All are linear and parallel.
© 2010 W. W. Norton & Company, Inc. 50
Instead of U(x1,2 ) =1x2x or
V(x ,x ) = x + x , consider
1 2 1 2
W(x 1x2) = min{x 1x2}.
What do the indifference curves for this “perfect complementarity” utility function look like?
• Perfect Complementarity Indifference Curves
Perfect Complementarity
Indifference Curves
x2
45 o
W(x ,x ) = min{x ,x }
1 2 1 2
8 min{x ,x1} 2 8
min{x ,x } = 5
5 1 2
3 min{x ,x } = 3
1 2
3 5 8
x 1
All are right -angled with vertices on a ray
© 2010 W. W. Norton & Company, Inc. 53
from the origin.
A utility function of the form: 1(x2,x ) 1 f(x 2 + x is linear i2 just x and is called quasi-linear.
1/2
• E.g. U(x1,x2) = 2x1 + x2.
Page 4 of 13 • Quasi-linear Indifference Curves
Quasi-linear Indifference Curves
x 2 Each curve is a vertically shifted
copy of the others.
x
1
© 2010 W. W. Norton & Company, Inc. 55
a b
Any utility function of the form U(x ,1 2 = x 1 2
with a > 0 and b > 0 is called a Cobb-Douglas utility function.
1/2 1/2
• E.g. U(x 1x2) = x1 x2 (a = b = 1/2)
V(x1,x2) = x1x 2 (a = 1, b = 3)
• Cobb-Douglas Indifference Curves
Cobb -Douglas Indifference
x 2 Curves
All curves are hyperbolic,
asymptoting to, but never
touching any axis.
x 1
© 2010 W. W. Norton & Company, Inc. 57
Marginal Utilities
• Marginal means “incremental”.
• The marginal utility of commodity i is the rate-of-change of total utility as the quantity of
commodity i consumed changes; i.e.
∂ U
MU i=
∂ x i
Page 5 of 13 1/2 2
• E.g. if U(x1,x2) = x 1 x 2 then
∂ U 1
MU 1= = x11 /x22
∂ x 1 2
MU = ∂ U = 2 x1 /x2
2 ∂ x 1 2
2
Marginal Utilities and Marginal Rates-of-Substitution
• The general equation for an indifference curve is
U(x 1x 2 ≡ k, a constant.
Totally differentiating this identity gives
∂ U ∂ U
dx 1+ dx 2 = 0
∂ x 1 ∂ x 2
∂ U dx = − ∂ U dx
∂ x 2 ∂ x 1
2 1
d x ∂ U / ∂ x
2 = − 1 .
d x ∂ U / ∂ x
1 2
Suppose U(x ,x 1 =2x x . 1h2n
∂ U
= (1)( x 2 = x 2
∂ x 1
∂ U
= ( x )(1) = x
∂ x 2 1 1
d x ∂ U / ∂ x x
MRS = 2 = − 1 = − 2 .
d x ∂ U / ∂ x x
1 2 1
Page 6 of 13 Marg. Utilities & Marg. Rates -of -
Substitution; An example
U(x ,1 )2= x 1x 2
x 2
8
MRS(1,8) = - 8/1 = -8
MRS(6,6) = - 6/6 = -1.
6
U = 36
U = 8
1 6 x 1
© 2010 W. W. Norton & Company, Inc. 68
Marg. Rates-of-Substitution for Quasi-linear Utility Functions
• A quasi-linear utility function is of the form U(x1,x2) = f(x1) + x2.
∂ U = f ( x1) ∂ U
∂ x 1

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