Chapter Five
Choice
消费者最优选择
Where Are We Doing in This
Chapter?
?After modeling a consumer?s choice set
and his preference (represented by utility
functions),we now put them together
and model how he/she makes optimal
choice.
?In mathematical terms,this is a
constrained maximization problem;
?In economics,this is a rational choice
problem.
Rational Constrained Choice
Affordable
bundles
x1
x2
More preferred
bundles
Rational Constrained Choice
?The most preferred affordable bundle
is called the consumer?s ORDINARY
DEMAND 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.
(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) the slope of the budget constraint,
-p1/p2,and the slope of the
indifference curve containing (x1*,x2*)
are equal at (x1*,x2*).
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.
?Then
U x x x xa b(,)1 2 1 2?
MU Ux ax xa b1
1
1 1 2? ? ?
?
?
MU Ux bx xa b2
2
1 2 1? ? ?
?
?
Computing Ordinary Demands -
a Cobb-Douglas Example.
?So the MRS is
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.
?So the MRS is
?At (x1*,x2*),MRS = -p1/p2 so
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
? ?
? ?
/
/
.
*
22
*
11
xp
xp
b
a ?
(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 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
?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 to the problem of
maximizing utility subject to a budget
constraint.
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 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
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*
(a) p1x1* + p2x2* = m
(b) x2* = ax1*
Summary,
Three Steps to Find the Optimal
Choice of the Consumer
?Step 1,Draw the budget set;
?Step 2,Draw the indifference curves;
?Step 3,Locate the point of optimal
choice and calculate the
solution.