Preferences, Budget Constraint and Optimal Choices
ECON 212 Lecture 5
Tianyi Wang
Queen’s Univerisity
Winter 2013
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
1 / 32
Preferences
3 axioms of preference (basics of consumer theory)
I
I
I
Completeness
Transitivity
Monotonicity
Ordinal ranking VS Cardinal ranking
Utility Functions: math expressions of preferences
I
I
e.g. U (x ) = x 1 /2
e.g. U (x, y ) = x 1 /2 y 1 /2
Hard to draw on blackboard (3-D)
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
2 / 32
Indi¤erence Curves
2-D representation of 3-D Utility surface.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
3 / 32
Indi¤erence Curves
2-D representation of 3-D Utility surface.
I
Combination of bundles that give the same level of utility
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
3 / 32
Indi¤erence Curves
2-D representation of 3-D Utility surface.
I
I
Combination of bundles that give the same level of utility
Contour plot (mountain), easy to draw
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
3 / 32
Indi¤erence Curves
2-D representation of 3-D Utility surface.
I
I
I
Combination of bundles that give the same level of utility
Contour plot (mountain), easy to draw
Have nice properties
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
3 / 32
Indi¤erence Curves
2-D representation of 3-D Utility surface.
I
I
I
Combination of bundles that give the same level of utility
Contour plot (mountain), easy to draw
Have nice properties
"well-behaved" ICs have 4 properties:
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
3 / 32
Indi¤erence Curves
2-D representation of 3-D Utility surface.
I
I
I
Combination of bundles that give the same level of utility
Contour plot (mountain), easy to draw
Have nice properties
"well-behaved" ICs have 4 properties:
1
Slope is negative (when you like both goods)
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
3 / 32
Indi¤erence Curves
2-D representation of 3-D Utility surface.
I
I
I
Combination of bundles that give the same level of utility
Contour plot (mountain), easy to draw
Have nice properties
"well-behaved" ICs have 4 properties:
1
2
Slope is negative (when you like both goods)
ICs cannot intersect
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
3 / 32
Indi¤erence Curves
2-D representation of 3-D Utility surface.
I
I
I
Combination of bundles that give the same level of utility
Contour plot (mountain), easy to draw
Have nice properties
"well-behaved" ICs have 4 properties:
1
2
3
Slope is negative (when you like both goods)
ICs cannot intersect
Every bundle lies on one & only one IC
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
3 / 32
Indi¤erence Curves
2-D representation of 3-D Utility surface.
I
I
I
Combination of bundles that give the same level of utility
Contour plot (mountain), easy to draw
Have nice properties
"well-behaved" ICs have 4 properties:
1
2
3
4
Slope is negative (when you like both goods)
ICs cannot intersect
Every bundle lies on one & only one IC
ICs are not thick (nonsatiation)
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
3 / 32
100
80
60
40
20
100
0
0
80
60
20
40
40
60
20
80
100
0
Figure: Utility Surface in 3D
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
4 / 32
90
80
70
60
50
40
30
20
100
10
0
20
40
50
60
80
100
0
Figure: Utility Surface (IC Skeleton)
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
5 / 32
100
90
80
70
60
50
40
30
20
10
0
0
20
40
60
80
100
Figure: Indi¤erence Map
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
6 / 32
Marginal rate of substitution (MRS)
De…ned as the rate at which consumers is willing to substitute one
good for another, while holding utility level constant.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
7 / 32
Marginal rate of substitution (MRS)
De…ned as the rate at which consumers is willing to substitute one
good for another, while holding utility level constant.
I
negative slope of indi¤erence curve, since MRS usually de…ned as a
positive number.
MRSx ,y = ∆∆Y
X
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
7 / 32
Marginal rate of substitution (MRS)
De…ned as the rate at which consumers is willing to substitute one
good for another, while holding utility level constant.
I
I
negative slope of indi¤erence curve, since MRS usually de…ned as a
positive number.
MRSx ,y = ∆∆Y
X
Notice ∆U = MUx ∆X + MUy ∆Y
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
7 / 32
Marginal rate of substitution (MRS)
De…ned as the rate at which consumers is willing to substitute one
good for another, while holding utility level constant.
I
I
I
negative slope of indi¤erence curve, since MRS usually de…ned as a
positive number.
MRSx ,y = ∆∆Y
X
Notice ∆U = MUx ∆X + MUy ∆Y
set ∆U = 0, so that stay on the same IC.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
7 / 32
Marginal rate of substitution (MRS)
De…ned as the rate at which consumers is willing to substitute one
good for another, while holding utility level constant.
I
I
I
I
negative slope of indi¤erence curve, since MRS usually de…ned as a
positive number.
MRSx ,y = ∆∆Y
X
Notice ∆U = MUx ∆X + MUy ∆Y
set ∆U = 0, so that stay on the same IC.
MU x
get MU
= ∆∆YX = MRSx ,y
y
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
7 / 32
Marginal rate of substitution (MRS)
De…ned as the rate at which consumers is willing to substitute one
good for another, while holding utility level constant.
I
I
I
I
negative slope of indi¤erence curve, since MRS usually de…ned as a
positive number.
MRSx ,y = ∆∆Y
X
Notice ∆U = MUx ∆X + MUy ∆Y
set ∆U = 0, so that stay on the same IC.
MU x
get MU
= ∆∆YX = MRSx ,y
y
Diminishing MRS
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
7 / 32
Marginal rate of substitution (MRS)
De…ned as the rate at which consumers is willing to substitute one
good for another, while holding utility level constant.
I
I
I
I
negative slope of indi¤erence curve, since MRS usually de…ned as a
positive number.
MRSx ,y = ∆∆Y
X
Notice ∆U = MUx ∆X + MUy ∆Y
set ∆U = 0, so that stay on the same IC.
MU x
get MU
= ∆∆YX = MRSx ,y
y
Diminishing MRS
I
MRS decreases as consumption of x increases.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
7 / 32
Marginal rate of substitution (MRS)
De…ned as the rate at which consumers is willing to substitute one
good for another, while holding utility level constant.
I
I
I
I
negative slope of indi¤erence curve, since MRS usually de…ned as a
positive number.
MRSx ,y = ∆∆Y
X
Notice ∆U = MUx ∆X + MUy ∆Y
set ∆U = 0, so that stay on the same IC.
MU x
get MU
= ∆∆YX = MRSx ,y
y
Diminishing MRS
I
I
MRS decreases as consumption of x increases.
direct consequence of Diminishing Marginal Utility
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
7 / 32
Marginal rate of substitution (MRS)
De…ned as the rate at which consumers is willing to substitute one
good for another, while holding utility level constant.
I
I
I
I
negative slope of indi¤erence curve, since MRS usually de…ned as a
positive number.
MRSx ,y = ∆∆Y
X
Notice ∆U = MUx ∆X + MUy ∆Y
set ∆U = 0, so that stay on the same IC.
MU x
get MU
= ∆∆YX = MRSx ,y
y
Diminishing MRS
I
I
MRS decreases as consumption of x increases.
direct consequence of Diminishing Marginal Utility
Example, see class notes
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
7 / 32
Several Utility Functions
Cobb-Douglas utility function
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
I U (X , Y ) = aX + bY , where a, b are positive constants
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
I U (X , Y ) = aX + bY , where a, b are positive constants
I MRSx ,y = a = CONSTANT
b
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
I U (X , Y ) = aX + bY , where a, b are positive constants
I MRSx ,y = a = CONSTANT
b
I
always substitute at the same same rate (perfect subs).
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
I U (X , Y ) = aX + bY , where a, b are positive constants
I MRSx ,y = a = CONSTANT
b
I
always substitute at the same same rate (perfect subs).
Perfect Complements
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
I U (X , Y ) = aX + bY , where a, b are positive constants
I MRSx ,y = a = CONSTANT
b
I
always substitute at the same same rate (perfect subs).
Perfect Complements
I U (X , Y ) = minfaX , bY g
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
I U (X , Y ) = aX + bY , where a, b are positive constants
I MRSx ,y = a = CONSTANT
b
I
always substitute at the same same rate (perfect subs).
Perfect Complements
I U (X , Y ) = minfaX , bY g
I
How to get ICs:
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
I U (X , Y ) = aX + bY , where a, b are positive constants
I MRSx ,y = a = CONSTANT
b
I
always substitute at the same same rate (perfect subs).
Perfect Complements
I U (X , Y ) = minfaX , bY g
I
How to get ICs:
1
equate two arguments to …nd the cut-o¤ line
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
I U (X , Y ) = aX + bY , where a, b are positive constants
I MRSx ,y = a = CONSTANT
b
I
always substitute at the same same rate (perfect subs).
Perfect Complements
I U (X , Y ) = minfaX , bY g
I
How to get ICs:
1
2
equate two arguments to …nd the cut-o¤ line
start at a point on the line, then increase X while holding Y constant
and/or increase Y while holding X constant.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
I U (X , Y ) = aX + bY , where a, b are positive constants
I MRSx ,y = a = CONSTANT
b
I
always substitute at the same same rate (perfect subs).
Perfect Complements
I U (X , Y ) = minfaX , bY g
I
How to get ICs:
1
2
3
equate two arguments to …nd the cut-o¤ line
start at a point on the line, then increase X while holding Y constant
and/or increase Y while holding X constant.
bundles have the same level of utility as the initial bundle
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
I U (X , Y ) = aX + bY , where a, b are positive constants
I MRSx ,y = a = CONSTANT
b
I
always substitute at the same same rate (perfect subs).
Perfect Complements
I U (X , Y ) = minfaX , bY g
I
How to get ICs:
1
2
3
I
equate two arguments to …nd the cut-o¤ line
start at a point on the line, then increase X while holding Y constant
and/or increase Y while holding X constant.
bundles have the same level of utility as the initial bundle
see class notes for another way.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions
Cobb-Douglas utility function
I U (X , Y ) = AX a Y b , where A, a, b are positive constants
I MRSx ,y = aY and a measures relative importance of X and Y
bX
b
I (Constant returns to scale when a + b = 1)
Perfect Substitutes
I U (X , Y ) = aX + bY , where a, b are positive constants
I MRSx ,y = a = CONSTANT
b
I
always substitute at the same same rate (perfect subs).
Perfect Complements
I U (X , Y ) = minfaX , bY g
I
How to get ICs:
1
2
3
I
I
equate two arguments to …nd the cut-o¤ line
start at a point on the line, then increase X while holding Y constant
and/or increase Y while holding X constant.
bundles have the same level of utility as the initial bundle
see class notes for another way.
always consume in …xed ratio (perfect comp.)
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
8 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
(no income e¤ect)
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
(no income e¤ect)
One Good, One Bad
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
(no income e¤ect)
One Good, One Bad
I
U (X , Y ) = aX
Tianyi Wang (Queen’s Univerisity)
bY , where a, b are positive constants
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
(no income e¤ect)
One Good, One Bad
I
I
U (X , Y ) = aX bY , where a, b are positive constants
MRSx ,y = ba , ICs slope upward!
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
(no income e¤ect)
One Good, One Bad
I
I
I
U (X , Y ) = aX bY , where a, b are positive constants
MRSx ,y = ba , ICs slope upward!
e.g. return and risk; leisure and labor.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
(no income e¤ect)
One Good, One Bad
I
I
I
U (X , Y ) = aX bY , where a, b are positive constants
MRSx ,y = ba , ICs slope upward!
e.g. return and risk; leisure and labor.
Exercise: Draw IC for the following utility function
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
(no income e¤ect)
One Good, One Bad
I
I
I
U (X , Y ) = aX bY , where a, b are positive constants
MRSx ,y = ba , ICs slope upward!
e.g. return and risk; leisure and labor.
Exercise: Draw IC for the following utility function
I
U (X , Y ) = X 1 /2 Y 1 /2
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
(no income e¤ect)
One Good, One Bad
I
I
I
U (X , Y ) = aX bY , where a, b are positive constants
MRSx ,y = ba , ICs slope upward!
e.g. return and risk; leisure and labor.
Exercise: Draw IC for the following utility function
I
I
U (X , Y ) = X 1 /2 Y 1 /2
How to get ICs:
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
(no income e¤ect)
One Good, One Bad
I
I
I
U (X , Y ) = aX bY , where a, b are positive constants
MRSx ,y = ba , ICs slope upward!
e.g. return and risk; leisure and labor.
Exercise: Draw IC for the following utility function
I
I
U (X , Y ) = X 1 /2 Y 1 /2
How to get ICs:
1
…x level of utility, say U = 9
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
(no income e¤ect)
One Good, One Bad
I
I
I
U (X , Y ) = aX bY , where a, b are positive constants
MRSx ,y = ba , ICs slope upward!
e.g. return and risk; leisure and labor.
Exercise: Draw IC for the following utility function
I
I
U (X , Y ) = X 1 /2 Y 1 /2
How to get ICs:
1
2
…x level of utility, say U = 9
…nd 3 pairs of (X , Y ) on this IC.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Several Utility Functions (Cont’d)
Quasi-Linear
I
I
I
I
U (X , Y ) = v (X ) + bY , where b constant, v (.) any increasing function
v 0 (x )
MRSx ,y = b , independent of Y
have just enough curvature to get interior solution (more later)
(no income e¤ect)
One Good, One Bad
I
I
I
U (X , Y ) = aX bY , where a, b are positive constants
MRSx ,y = ba , ICs slope upward!
e.g. return and risk; leisure and labor.
Exercise: Draw IC for the following utility function
I
I
U (X , Y ) = X 1 /2 Y 1 /2
How to get ICs:
1
2
3
…x level of utility, say U = 9
…nd 3 pairs of (X , Y ) on this IC.
connect
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
9 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
I
A1: Two goods: (X1 and X2 ), example: bananas and apples
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
I
I
A1: Two goods: (X1 and X2 ), example: bananas and apples
A2: Your income: m
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
I
I
I
A1: Two goods: (X1 and X2 ), example: bananas and apples
A2: Your income: m
A3: P1 the price for good X1 and P2 is the price for good X2
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
I
I
I
A1: Two goods: (X1 and X2 ), example: bananas and apples
A2: Your income: m
A3: P1 the price for good X1 and P2 is the price for good X2
F
take as give for now, can be determined in equilibirum (Chapter2).
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
I
I
I
A1: Two goods: (X1 and X2 ), example: bananas and apples
A2: Your income: m
A3: P1 the price for good X1 and P2 is the price for good X2
F
I
take as give for now, can be determined in equilibirum (Chapter2).
A4: spend all your income on the two goods.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
I
I
I
A1: Two goods: (X1 and X2 ), example: bananas and apples
A2: Your income: m
A3: P1 the price for good X1 and P2 is the price for good X2
F
I
take as give for now, can be determined in equilibirum (Chapter2).
A4: spend all your income on the two goods.
BC expresses the combination of X1 and X2 the consumer can buy
given her/his income
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
I
I
I
A1: Two goods: (X1 and X2 ), example: bananas and apples
A2: Your income: m
A3: P1 the price for good X1 and P2 is the price for good X2
F
I
take as give for now, can be determined in equilibirum (Chapter2).
A4: spend all your income on the two goods.
BC expresses the combination of X1 and X2 the consumer can buy
given her/his income
written as P1 X1 + P2 X2 = m
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
I
I
I
A1: Two goods: (X1 and X2 ), example: bananas and apples
A2: Your income: m
A3: P1 the price for good X1 and P2 is the price for good X2
F
I
take as give for now, can be determined in equilibirum (Chapter2).
A4: spend all your income on the two goods.
BC expresses the combination of X1 and X2 the consumer can buy
given her/his income
written as P1 X1 + P2 X2 = m
I
What are the values of the intercepts
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
I
I
I
A1: Two goods: (X1 and X2 ), example: bananas and apples
A2: Your income: m
A3: P1 the price for good X1 and P2 is the price for good X2
F
I
take as give for now, can be determined in equilibirum (Chapter2).
A4: spend all your income on the two goods.
BC expresses the combination of X1 and X2 the consumer can buy
given her/his income
written as P1 X1 + P2 X2 = m
I
I
What are the values of the intercepts
What is their interpretation?
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
I
I
I
A1: Two goods: (X1 and X2 ), example: bananas and apples
A2: Your income: m
A3: P1 the price for good X1 and P2 is the price for good X2
F
I
take as give for now, can be determined in equilibirum (Chapter2).
A4: spend all your income on the two goods.
BC expresses the combination of X1 and X2 the consumer can buy
given her/his income
written as P1 X1 + P2 X2 = m
I
I
I
What are the values of the intercepts
What is their interpretation?
What does the shaded area represent?
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint
Cannot purchase in…nite goods due to budget constraint.
Assumptions:
I
I
I
A1: Two goods: (X1 and X2 ), example: bananas and apples
A2: Your income: m
A3: P1 the price for good X1 and P2 is the price for good X2
F
I
take as give for now, can be determined in equilibirum (Chapter2).
A4: spend all your income on the two goods.
BC expresses the combination of X1 and X2 the consumer can buy
given her/his income
written as P1 X1 + P2 X2 = m
I
I
I
I
What
What
What
What
are the values of the intercepts
is their interpretation?
does the shaded area represent?
is the slope of the line?
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
10 / 32
Budget Constraint (Cont’d)
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
11 / 32
Budget Constraint (Income Increase)
1
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
12 / 32
Budget Constraint (Price Change)
1
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
13 / 32
Budget Constraint (summary)
Changes in income cause BC shift.
Changes in price cause BC rotate about intercepts.
Application 4.4
I
I
I
I
I
Suppose consumer chooses between two phone plans.
Plan 1: $40 per month for call up to 450min, additional minute costs
$0.40
Plan 2: $60 per month for call up to 900min, additional minute costs
$0.40
Income $500, price of other goods $1.
draw BC.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
14 / 32
Optimal Choices: Interior Solution
Consumer maximizes utility while facing his/her budget constraint.
Let consumer choices over x and y .
Let U (x, y ) denotes her utility function and I denotes her income.
Consumer’s problem can be formally expressed as:
I
max U (x, y )
(x ,y )
subject to : Px x + Py y
I
Note in our case budget constraint holds at equality, since consumer
does not derive utility from holding income.
Interior: At optimum, consume both x and y .
How to choose?
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
15 / 32
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
16 / 32
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
17 / 32
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
18 / 32
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
19 / 32
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
20 / 32
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
21 / 32
Graph Summary
1
Start at arbitrary bundle A, know there is an IC goes through A.
2
Ask if can increase utility by shifting IC to the right.
3
Stop if IC detaches from BC (non-a¤ordable).
4
Utility maximized when IC "just touches" BC.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
22 / 32
Optimality Condition
Notice at optimum slope of BC equals slope of IC (tangency).
Slope of IC =
∆Y
∆X
and MRSx ,y =
Therefore, Slope of IC =
Slope of BC =
∆Y
∆X
=
MU x
MU y
MU x
MU y
Px
Py
Then we have the following Optimality Condition:
I
MU x
MU y
=
Px
Py
To understand, rewrite is as:
I
MU x
Px
=
MU y
Py
Marginal Utility derived from last dollar equals.
"bang for the buck"
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
23 / 32
Optimality Condition (Con’t)
Why bundles like A and B are not optimal?
For instance, at A slope of IC is steeper than slope of BC:
MUx
MUy
MUx
Px
Px
, or
Py
MUy
Py
Last dollar on X gives more utility than on Y .
How to solve Consumer’s Problem? In 3 steps:
1
2
3
Calculate MUx and MUy .
Use Optimality Condition to express X in terms of Y .
Sub into BC to solve for Y . Then get X from step 2.
See class notes for example.
See class notes for opportunity cost argument.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
24 / 32
Optimal Choices: Corner Solution
At optimum, some goods are not consumed (when chooses among
multiple goods).
In our case, only one good is consumed.
Graphically, for any interior points on BC, slope of IC is
steeper/‡atter than slope of BC.
See class notes for example (short sales).
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
25 / 32
Example: Savings Decision
A simple two-period model: present (1) and future (2).
Consumer cosumes in both period
I
U (C )
U (C1 , C2 ) = U (C1 ) + 1 +ρ2
ρ rate of time preference, usually a small positive number (0.05).
Income endowment in each period I1 and I2 .
Can borrow or save at given interest rate r .
Write down BC, draw Budget Line.
Present consumer’s problem and solve for optimum.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
26 / 32
Consumer’s problem:
I
U (C )
max U (C1 , C2 ) = U (C1 ) + 1 +ρ2
(C 1 ,C 2 )
subject to : C1 + 1C+2r
MRSC1 ,C2 =
MU C 1
MU C 2
Slope of BC =
I1 + 1I+2 r
0
= (1 + ρ) UU 0 ((CC12 ))
(1 + r )
U 0 (C )
Then at optimum (1 + ρ) U 0 (C12 ) = 1 + r
When will she save/borrow?
I
0
U (I 1 )
(1 + ρ ) U
0 (I ) Q 1 + r
2
See class notes for example.
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
27 / 32
Changing Interest Rate
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
28 / 32
Changing Interest Rate
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
29 / 32
Changing Interest Rate
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
30 / 32
Changing Interest Rate
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
31 / 32
Di¤erent Interest Rate for Borrowing and Saving
Tianyi Wang (Queen’s Univerisity)
Lecture 5
Winter 2013
32 / 32