Chapter Five
Choice
选择
Structure
?Rational constrained choice
?Computing ordinary demands
–Interior solution （ 内在解）
–Corner solution （ 角点解）
–“Kinky” solution
?Example,Choosing taxes
Economic Rationality
?The principal behavioral postulate is
that a decision-maker chooses its
most preferred alternative from those
available to it.
?The available choices constitute the
choice set.
?How is the most preferred bundle in
the choice set located?
Rational Constrained Choice
x1
x2
Affordable
bundles
More preferred
bundles
Rational Constrained Choice
Affordable
bundles
x1
x2
More preferred
bundles
Rational Constrained Choice
x1
x2
x1*
x2*
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is the most
preferred affordable
bundle.
Rational Constrained Choice
?The most preferred affordable bundle
is called the consumer?s ORDINARY
DEMAND （ 一般需求） at the given
prices and budget.
?Ordinary demands will be denoted by
x1*(p1,p2,m) and x2*(p1,p2,m).
Rational Constrained Choice
?When x1* > 0 and x2* > 0 the
demanded bundle is INTERIOR.
?If buying (x1*,x2*) costs $m then the
budget is exhausted,
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior.
(x1*,x2*) exhausts the
budget.
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior.
(a) (x1*,x2*) exhausts the
budget; p1x1* + p2x2* = m.
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior,
(b) The slope of the indiff.
curve at (x1*,x2*) equals
the slope of the budget
constraint.
Rational Constrained Choice
?(x1*,x2*) satisfies two conditions:
? (a) the budget is exhausted;
p1x1* + p2x2* = m
? (b) tangency,the slope of the budget
constraint,-p1/p2,and the slope of the
indifference curve containing (x1*,x2*)
are equal at (x1*,x2*).
Meaning of the Tangency Condition
?Consumer?s marginal willingness to pay
equals the market exchange rate.
? Suppose at a consumption bundle (x1,x2),
MRS=-2,-P1/P2=-1
– The consumer is willing to give up 2
unit of x2 to exchange for an additional
unit of x1
– The market allows her to give up only 1
unit of x2 to obtain an additional x1
? (x1,x2) is not optimal choice
? She can be better off increasing her
consumption of x1.
x1x1
x2
Computing Ordinary Demands
?Solve for 2 simultaneous equations.
–Tangency
–Budget constraint
?The conditions may be obtained by
using the Lagrangian multiplier
method,i.e.,constrained
optimization in calculus.
Computing Ordinary Demands
?How can this information be used to
locate (x1*,x2*) for given p1,p2 and
m?
Computing Ordinary Demands -
a Cobb-Douglas Example.
?Suppose that the consumer has
Cobb-Douglas preferences.
U x x x xa b(,)1 2 1 2?
Computing Ordinary Demands -
a Cobb-Douglas Example.
?Suppose that the consumer has
Cobb-Douglas preferences.
?The MRS is
U x x x xa b(,)1 2 1 2?
M R S dx
dx
U x
U x
ax x
bx x
ax
bx
a b
a b? ? ? ? ? ? ?
?
?
2
1
1
2
1
1
2
1 2
1
2
1
? ?
? ?
/
/
.
Computing Ordinary Demands -
a Cobb-Douglas Example.
?At (x1*,x2*),MRS = -p1/p2 so the
tangency condition (MRS = -p1/p2) is
? ? ? ? ?ax
bx
p
p
x bp
ap
x2
1
1
2
2
1
2
1
*
*
* *,
(A)
Computing Ordinary Demands -
a Cobb-Douglas Example.
?(x1*,x2*) also exhausts the budget so
p x p x m1 1 2 2* *,? ?(B)
Computing Ordinary Demands -
a Cobb-Douglas Example.
?So now we know that
x bpap x2 1
2
1
* *?
(A)
p x p x m1 1 2 2* *,? ?(B)
Computing Ordinary Demands -
a Cobb-Douglas Example.
?The solution to the simultaneous
equations is:x am
a b p1 1
*
( ),? ?
x bma b p2
2
*
( ),? ?
Computing Ordinary Demands -
a Cobb-Douglas Example.
So we have discovered that the most
preferred affordable bundle for a consumer
with Cobb-Douglas preferences
U x x x xa b(,)1 2 1 2?
is (,)
( ),( ),
* * ( )x x a m
a b p
b m
a b p1 2 1 2? ? ?
Computing Ordinary Demands -
a Cobb-Douglas Example.
x1
x2
x a ma b p1
1
*
( )? ?
x
b m
a b p
2
2
*
( )
?
?
U x x x xa b(,)1 2 1 2?
Rational Constrained Choice,
Summary
? When x1* > 0 and x2* > 0
and (x1*,x2*) exhausts the budget,
and indifference curves have no
?kinks?,the ordinary demands are
obtained by solving:
? (a) p1x1* + p2x2* = y
? (b) the slopes of the budget constraint,-
p1/p2,and of the indifference curve
containing (x1*,x2*) are equal at (x1*,x2*).
Rational Constrained Choice
?But what if x1* = 0?
?Or if x2* = 0?
?If either x1* = 0 or x2* = 0 then the
ordinary demand (x1*,x2*) is at a
corner solution (角点解 ) to the
problem of maximizing utility subject
to a budget constraint.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
x yp2
2
* ?
x1 0* ?
MRS = -1
Slope = -p1/p2 with p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
x yp1
1
* ?x2 0
* ?
MRS = -1
Slope = -p1/p2 with p1 < p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
So when U(x1,x2) = x1 + x2,the most
preferred affordable bundle is (x1*,x2*)
where ?
?
??
?
?? 0,
p
y)x,x(
1
*
2
*
1
and ?
?
??
?
??
2
*
2
*
1 p
y,0)x,x(
if p1 < p2
if p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 = p2.
y
p1
y
p2
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
All the bundles in the
constraint are equally the
most preferred affordable
when p1 = p2.
y
p2
y
p1
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Which is the most preferred
affordable bundle?
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
The most preferred
affordable bundle
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
The most preferred
affordable bundle
Notice that the,tangency solution”
is not the most preferred affordable
bundle.
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = 0
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = - ?
MRS = 0
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = - ?
MRS = 0
MRS is undefined
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Which is the most
preferred affordable bundle?
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
The most preferred
affordable bundle
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
(a) p1x1* + p2x2* = m
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
(a) p1x1* + p2x2* = m
(b) x2* = ax1*
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
Substitution from (b) for x2* in
(a) gives p1x1* + p2ax1* = m
which gives
.
app
amx;
app
mx
21
*
2
21
*
1 ????
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x mp ap1
1 2
* ?
?
x
am
p ap
2
1 2
* ?
?
Choosing Taxes,Various Taxes
?Quantity tax,on x,(p+t)x
?Value tax,on px,(1+t)px
–Also called ad valorem tax
?Lump sum tax,T
?Income tax,
–Can be proportional or lump sum
Income Tax vs,Quantity Tax
?Original budget,p1x1 + p2x2 = m
?After quantity tax,
(p1+t)x1 + p2x2 = m
?At optimal choice (x1*,x2*)
–(p1+t)x1* + p2x2* = m (5.2)
–Tax revenue,R*=tx1*
?With an income tax,budget is:
p1x1 + p2x2 = m- tx1*
Income vs,Quantity Tax
? Proposition,(x1*,x2*) is affordable under
income tax
? Equivalent to,prove that (x1*,x2*) satisfies
budget constraint under income tax,
? Or,budget constraint holds at point (x1*,
x2*).
p1x1* + p2x2*= m- tx1*
? Which is true according to (5.2).
? It is not an optimal choice because prices
are different.
? Conclusion,The optimal choice must be
more preferred to (x1*,x2*)
Choice
选择
Structure
?Rational constrained choice
?Computing ordinary demands
–Interior solution （ 内在解）
–Corner solution （ 角点解）
–“Kinky” solution
?Example,Choosing taxes
Economic Rationality
?The principal behavioral postulate is
that a decision-maker chooses its
most preferred alternative from those
available to it.
?The available choices constitute the
choice set.
?How is the most preferred bundle in
the choice set located?
Rational Constrained Choice
x1
x2
Affordable
bundles
More preferred
bundles
Rational Constrained Choice
Affordable
bundles
x1
x2
More preferred
bundles
Rational Constrained Choice
x1
x2
x1*
x2*
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is the most
preferred affordable
bundle.
Rational Constrained Choice
?The most preferred affordable bundle
is called the consumer?s ORDINARY
DEMAND （ 一般需求） at the given
prices and budget.
?Ordinary demands will be denoted by
x1*(p1,p2,m) and x2*(p1,p2,m).
Rational Constrained Choice
?When x1* > 0 and x2* > 0 the
demanded bundle is INTERIOR.
?If buying (x1*,x2*) costs $m then the
budget is exhausted,
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior.
(x1*,x2*) exhausts the
budget.
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior.
(a) (x1*,x2*) exhausts the
budget; p1x1* + p2x2* = m.
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior,
(b) The slope of the indiff.
curve at (x1*,x2*) equals
the slope of the budget
constraint.
Rational Constrained Choice
?(x1*,x2*) satisfies two conditions:
? (a) the budget is exhausted;
p1x1* + p2x2* = m
? (b) tangency,the slope of the budget
constraint,-p1/p2,and the slope of the
indifference curve containing (x1*,x2*)
are equal at (x1*,x2*).
Meaning of the Tangency Condition
?Consumer?s marginal willingness to pay
equals the market exchange rate.
? Suppose at a consumption bundle (x1,x2),
MRS=-2,-P1/P2=-1
– The consumer is willing to give up 2
unit of x2 to exchange for an additional
unit of x1
– The market allows her to give up only 1
unit of x2 to obtain an additional x1
? (x1,x2) is not optimal choice
? She can be better off increasing her
consumption of x1.
x1x1
x2
Computing Ordinary Demands
?Solve for 2 simultaneous equations.
–Tangency
–Budget constraint
?The conditions may be obtained by
using the Lagrangian multiplier
method,i.e.,constrained
optimization in calculus.
Computing Ordinary Demands
?How can this information be used to
locate (x1*,x2*) for given p1,p2 and
m?
Computing Ordinary Demands -
a Cobb-Douglas Example.
?Suppose that the consumer has
Cobb-Douglas preferences.
U x x x xa b(,)1 2 1 2?
Computing Ordinary Demands -
a Cobb-Douglas Example.
?Suppose that the consumer has
Cobb-Douglas preferences.
?The MRS is
U x x x xa b(,)1 2 1 2?
M R S dx
dx
U x
U x
ax x
bx x
ax
bx
a b
a b? ? ? ? ? ? ?
?
?
2
1
1
2
1
1
2
1 2
1
2
1
? ?
? ?
/
/
.
Computing Ordinary Demands -
a Cobb-Douglas Example.
?At (x1*,x2*),MRS = -p1/p2 so the
tangency condition (MRS = -p1/p2) is
? ? ? ? ?ax
bx
p
p
x bp
ap
x2
1
1
2
2
1
2
1
*
*
* *,
(A)
Computing Ordinary Demands -
a Cobb-Douglas Example.
?(x1*,x2*) also exhausts the budget so
p x p x m1 1 2 2* *,? ?(B)
Computing Ordinary Demands -
a Cobb-Douglas Example.
?So now we know that
x bpap x2 1
2
1
* *?
(A)
p x p x m1 1 2 2* *,? ?(B)
Computing Ordinary Demands -
a Cobb-Douglas Example.
?The solution to the simultaneous
equations is:x am
a b p1 1
*
( ),? ?
x bma b p2
2
*
( ),? ?
Computing Ordinary Demands -
a Cobb-Douglas Example.
So we have discovered that the most
preferred affordable bundle for a consumer
with Cobb-Douglas preferences
U x x x xa b(,)1 2 1 2?
is (,)
( ),( ),
* * ( )x x a m
a b p
b m
a b p1 2 1 2? ? ?
Computing Ordinary Demands -
a Cobb-Douglas Example.
x1
x2
x a ma b p1
1
*
( )? ?
x
b m
a b p
2
2
*
( )
?
?
U x x x xa b(,)1 2 1 2?
Rational Constrained Choice,
Summary
? When x1* > 0 and x2* > 0
and (x1*,x2*) exhausts the budget,
and indifference curves have no
?kinks?,the ordinary demands are
obtained by solving:
? (a) p1x1* + p2x2* = y
? (b) the slopes of the budget constraint,-
p1/p2,and of the indifference curve
containing (x1*,x2*) are equal at (x1*,x2*).
Rational Constrained Choice
?But what if x1* = 0?
?Or if x2* = 0?
?If either x1* = 0 or x2* = 0 then the
ordinary demand (x1*,x2*) is at a
corner solution (角点解 ) to the
problem of maximizing utility subject
to a budget constraint.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
x yp2
2
* ?
x1 0* ?
MRS = -1
Slope = -p1/p2 with p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
x yp1
1
* ?x2 0
* ?
MRS = -1
Slope = -p1/p2 with p1 < p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
So when U(x1,x2) = x1 + x2,the most
preferred affordable bundle is (x1*,x2*)
where ?
?
??
?
?? 0,
p
y)x,x(
1
*
2
*
1
and ?
?
??
?
??
2
*
2
*
1 p
y,0)x,x(
if p1 < p2
if p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 = p2.
y
p1
y
p2
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
All the bundles in the
constraint are equally the
most preferred affordable
when p1 = p2.
y
p2
y
p1
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Which is the most preferred
affordable bundle?
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
The most preferred
affordable bundle
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
The most preferred
affordable bundle
Notice that the,tangency solution”
is not the most preferred affordable
bundle.
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = 0
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = - ?
MRS = 0
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = - ?
MRS = 0
MRS is undefined
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Which is the most
preferred affordable bundle?
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
The most preferred
affordable bundle
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
(a) p1x1* + p2x2* = m
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
(a) p1x1* + p2x2* = m
(b) x2* = ax1*
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
Substitution from (b) for x2* in
(a) gives p1x1* + p2ax1* = m
which gives
.
app
amx;
app
mx
21
*
2
21
*
1 ????
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x mp ap1
1 2
* ?
?
x
am
p ap
2
1 2
* ?
?
Choosing Taxes,Various Taxes
?Quantity tax,on x,(p+t)x
?Value tax,on px,(1+t)px
–Also called ad valorem tax
?Lump sum tax,T
?Income tax,
–Can be proportional or lump sum
Income Tax vs,Quantity Tax
?Original budget,p1x1 + p2x2 = m
?After quantity tax,
(p1+t)x1 + p2x2 = m
?At optimal choice (x1*,x2*)
–(p1+t)x1* + p2x2* = m (5.2)
–Tax revenue,R*=tx1*
?With an income tax,budget is:
p1x1 + p2x2 = m- tx1*
Income vs,Quantity Tax
? Proposition,(x1*,x2*) is affordable under
income tax
? Equivalent to,prove that (x1*,x2*) satisfies
budget constraint under income tax,
? Or,budget constraint holds at point (x1*,
x2*).
p1x1* + p2x2*= m- tx1*
? Which is true according to (5.2).
? It is not an optimal choice because prices
are different.
? Conclusion,The optimal choice must be
more preferred to (x1*,x2*)
