Choice
Molly W. Dahl
Georgetown University
Econ 101 – Spring 2009
1
Economic Rationality

We’ve got budgets, we’ve got preferences
 What



are people going to choose?
Behavioral Assumption: a decisionmaker
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?
2
Rational Constrained Choice
x2
x1
3
Rational Constrained Choice
x2
x1
4
Rational Constrained Choice
x2
Affordable
bundles
x1
5
Rational Constrained Choice
x2
Affordable
bundles
x1
6
Rational Constrained Choice
x2
More preferred
bundles
Affordable
bundles
x1
7
Rational Constrained Choice
x2
More preferred
bundles
Affordable
bundles
x1
8
Rational Constrained Choice
x2
(x1*,x2*) is the most
preferred affordable
bundle.
x2*
x1*
x1
9
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).

10
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.

11
Rational Constrained Choice
x2
(x1*,x2*) is interior.
(a) (x1*,x2*) exhausts the
budget: p1x1* + p2x2* = m.
x2*
x1*
x1
12
Rational Constrained Choice
x2
(x1*,x2*) is interior .
(b) The slope of the indiff.
curve at (x1*,x2*) equals
the slope of the budget
constraint.
x2*
x1*
x1
13
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*).
14
Computing Ordinary Demands

How can this information be used to locate
(x1*,x2*) for given p1, p2 and m?
15
Computing Ordinary Demands - a
Cobb-Douglas Example.

Suppose that the consumer has CobbDouglas preferences.
U( x1 , x 2 )  x1axb2
16
Computing Ordinary Demands - a
Cobb-Douglas Example.


Suppose that the consumer has CobbDouglas preferences.
U( x1 , x 2 )  x1axb2
Then
U
a1 b
MU1 
 ax1 x 2
 x1
U
MU2 
 bx1axb2  1
 x2
17
Computing Ordinary Demands - a
Cobb-Douglas Example.

So the MRS is
a1 b
dx 2
 U/ x1
ax1 x 2
ax 2
MRS 



.
dx1
 U/ x 2
bx1
bx1axb2  1

At (x1*,x2*), MRS = -p1/p2 so
ax*2
p1
* bp1 *


 x2 
x1 .
*
p2
ap2
bx1
(A)
18
Computing Ordinary Demands - a
Cobb-Douglas Example.

(x1*,x2*) also exhausts the budget so
*
*
p1x1  p2x 2  m.
(B)
19
Computing Ordinary Demands - a
Cobb-Douglas Example.

So now we know that
* bp1 *
x2 
x1
ap 2
p1x*1  p2x*2  m.
(A)
(B)
20
Computing Ordinary Demands - a
Cobb-Douglas Example.

So now we know that
* bp1 *
x2 
x1
ap 2
Substitute
p1x*1  p2x*2  m.
and get
(A)
(B)
bp1 *
*
p1x1  p2
x1  m.
ap2
This simplifies to ….
21
Computing Ordinary Demands - a
Cobb-Douglas Example.
x*1 
am
.
( a  b)p1
22
Computing Ordinary Demands - a
Cobb-Douglas Example.
x*1 
am
.
( a  b)p1
Substituting for x1* in
p1x*1  p2x*2  m
then gives
*
x2 
bm
.
( a  b)p2
23
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( x1 , x 2 )  x1axb2
is
( x*1 , x*2 ) 
(
)
am
bm
,
.
( a  b)p1 ( a  b)p2
24
Computing Ordinary Demands - a
Cobb-Douglas Example.
U( x1 , x 2 )  x1axb2
x2
*
x2 
bm
( a  b)p 2
x*1 
am
( a  b)p1
x1
25
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* = m
 (b) the slopes of the budget constraint,
-p1/p2, and of the indifference curve
containing (x1*,x2*) are equal at (x1*,x2*).
26
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.

27
Examples of Corner Solutions -- the
Perfect Substitutes Case
x2
MRS = -1
x1
28
Examples of Corner Solutions -- the
Perfect Substitutes Case
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
x1
29
Examples of Corner Solutions -- the
Perfect Substitutes Case
x2
y
*
x2 
p2
MRS = -1
Slope = -p1/p2 with p1 > p2.
x*1  0
x1
30
Examples of Corner Solutions -- the
Perfect Substitutes Case
x2
MRS = -1
Slope = -p1/p2 with p1 < p2.
x*2  0
y
*
x1 
p1
x1
31
Examples of Corner Solutions -- the
Perfect Substitutes Case
So when U(x1,x2) = x1 + x2, the most
preferred affordable bundle is (x1*,x2*)
where
y 
* *
( x1 , x 2 )   ,0  if p1 < p2
 p1 
and

* *
( x1 , x 2 )   0,
y 

 p2 
if p1 > p2.
32
Examples of Corner Solutions -- the
Perfect Substitutes Case
x2
y
p2
MRS = -1
Slope = -p1/p2 with p1 = p2.
y
p1
x1
33
Examples of Corner Solutions -- the
Perfect Substitutes Case
x2
y
p2
All the bundles in the
constraint are equally the
most preferred affordable
when p1 = p2.
y
p1
x1
34
Examples of ‘Kinky’ Solutions -- the
Perfect Complements Case
x2
U(x1,x2) = min{ax1,x2}
x2 = ax1
x1
35
Examples of ‘Kinky’ Solutions -- the
Perfect Complements Case
x2
U(x1,x2) = min{ax1,x2}
MRS = - 
MRS is undefined
x2 = ax1
MRS = 0
x1
36
Examples of ‘Kinky’ Solutions -- the
Perfect Complements Case
x2
U(x1,x2) = min{ax1,x2}
Which is the most
preferred affordable bundle?
x2 = ax1
x1
37
Examples of ‘Kinky’ Solutions -- the
Perfect Complements Case
x2
U(x1,x2) = min{ax1,x2}
The most preferred
affordable bundle
x2 = ax1
x1
38
Examples of ‘Kinky’ Solutions -- the
Perfect Complements Case
x2
U(x1,x2) = min{ax1,x2}
(a) p1x1* + p2x2* = m
(b) x2* = ax1*
x2 = ax1
x2*
x1*
x1
39
Examples of ‘Kinky’ Solutions -- the
Perfect Complements Case
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
40
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
41
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
m
which gives *
x1 
p1  ap2
42
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
m
which gives *
*
am
x1 
; x2 
.
p1  ap2
p1  ap2
43
Examples of ‘Kinky’ Solutions -- the
Perfect Complements Case
x2
U(x1,x2) = min{ax1,x2}
*
x2 
x2 = ax1
am
p1  ap 2
x*1 
m
p1  ap 2
x1
44