Chapter 5 Choice

Report
Course: Microeconomics
Text: Varian’s Intermediate
Microeconomics
1
The principal behavioral postulate is
that a decision-maker chooses its most
preferred alternative from those
available to it.
 The available choices constitute the
budget set.
 How is the most preferred bundle in the
budget set located?

2
x2
x1
3
Utility
x2
x1
4
Utility
Affordable, but not
the most preferred
affordable bundle.
x2
x1
5
Utility
The most preferred
of the affordable
bundles.
Affordable, but not
the most preferred
affordable bundle.
x2
x1
6
x2
One will choose the
bundle on the outermost
indifference curve.
Therefore, it is the one
that just touch the
budget constraint.
Affordable
bundles
x1
7
x2
(x1*,x2*) is the most
preferred affordable
bundle.
x2*
x1*
x1
8


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).
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

When x1* > 0 and x2* > 0 the demanded
bundle is INTERIOR.
If buying (x1*,x2*) costs $m then the
budget is exhausted.
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


Assuming monotonicity and smooth
indifference curve, for an interior solution
(x1*,x2*), it 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*).
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
Condition (b) can be written as:

dx2
dx1


 MRS 
MU 1
MU 2

p1
p2
So, at the optimal choice, the marginal
willingness to pay for an extra unit of good
1 in terms of good 2 is the same as the
price you actually need to pay.
If the price is lower than your willingness
to pay, you would buy more, otherwise buy
less. Adjust until they are equal.
12
The consumer problem can be
formulated mathematically as:
One way to solve is to use
Lagrangian method:
13
Assuming the utility function is
differentiable, and there exists interior
solution
 The first order conditions are:

14

So, we have the same condition as before
that MRS equals price ratio.
15

So, for interior solution, we can solve for
the optimal bundle using the two
conditions:
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
Suppose that the consumer has CobbDouglas preferences.
a b
U( x1 , x 2 )  x1 x 2

Then
MU1 
MU2 
U
 x1
U
 x2
a1 b
x2
 ax1
a b 1
 bx1 x 2
17

So the MRS is
MRS 

 U / x1
 U / x2
a 1 b
1
2
a b 1
1
2
ax

x

bx x
ax2
.
bx1
At (x1*,x2*), MRS = p1/p2 so
*
2
*
1
ax
bx

p1
p2
x 
*
2
bp1
ap2
*
1
x .
18

So now we know that
bp1 *
*
x2 
x1
ap 2
*
*
p1x1  p 2x 2  m.
(A)
(B)
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
So now we know that
Substitute
bp1 *
*
x2 
x1
ap 2
*
*
p1x1  p 2x 2  m.
(A)
(B)
and get
bp1 *
*
p1x1  p 2
x1  m.
ap 2
This simplifies to ….
20
*
x1 
am
( a  b )p1
.
21
*
x1 
am
( a  b )p1
.
Substituting for x1* in
*
*
p1x1  p 2x 2  m
then gives
*
x2 
bm
( a  b )p 2
.
22
So we have discovered that the most
preferred affordable bundle for a consumer
with Cobb-Douglas preferences
a b
U( x1 , x 2 )  x1 x 2
is
* *
( x1 , x 2 ) 
(
am
,
bm
( a  b )p1 ( a  b )p 2
)
.
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x2
a b
U( x1 , x 2 )  x1 x 2
*
x2 
bm
( a  b )p 2
*
x1 
am
( a  b )p1
x1
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
With the optimal bundle:
* *
( x1 , x 2 ) 

(
am
,
bm
( a  b )p1 ( a  b )p 2
)
.
Note the expenditures on each good:


a
b
( p1 x , p2 x )  
 ( a  b) m, (a  b) m 
.


*
1

*
2
It is a property of Cobb-Douglas Utility
function that expenditure share on a
particular good is a constant.
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1. Tangency condition
is only necessary but
not sufficient.
2. There can be
more than one
optimum.
3. Strict
Convexity
implies
unique
solution.
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



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.
E.g. it happens when the indifference
curves are too flat or steep relative to the
budget line.
28
The indifference
curves are too steep in
this case to touch the
budget line.
The willingness to pay
for good 2 is still lower
than the price of good
2 even when x2 is zero.
29



The mathematical treatment for corner
solution is more complicated.
We have to consider also the inequality
constraints. (x1 ≥0 and x2 ≥0)
We skip the details here.
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x2
MRS = 1
x1
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x2
MRS = 1
Slope = -p1/p2 with p1 > p2.
x1
32
x2
MRS = 1
Slope = -p1/p2 with p1 > p2.
x1
33
x2
MRS = 1
x2 
*
m
p2
Slope = -p1/p2 with p1 > p2.
*
x1  0
x1
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x2
MRS = 1
Slope = -p1/p2 with p1 < p2.
*
x2  0
x 
*
1
m
p1
x1
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So when U(x1,x2) = x1 + x2, the most
preferred affordable bundle is (x1*,x2*)
where
m 

(x , x )  
 p ,0 
 1 
*
1
*
2
if p1 < p2
and

m 

(x , x )  
0
,


p
2 

*
1
*
2
if p1 > p2.
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x2
m
p2
MRS = 1
Slope = -p1/p2 with p1 = p2.
x1
m
p1
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x2
m
p2
All the bundles in the
constraint are equally the
most preferred affordable
when p1 = p2.
x1
m
p1
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x2
x1
39
x2
x1
40
x2
Which is the most preferred
affordable bundle?
x1
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x2
The most preferred
affordable bundle
x1
42
x2
Notice that the “tangency solution”
is not the most preferred affordable
bundle.
The most preferred
affordable bundle
x1
43
x2
U(x1,x2) = min{ax1,x2}
x2 = ax1
x1
44
x2
U(x1,x2) = min{ax1,x2}
MRS = -

x2 = ax1
MRS = 0
x1
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x2
U(x1,x2) = min{ax1,x2}
MRS = -

MRS is undefined
x2 = ax1
MRS = 0
x1
46
x2
U(x1,x2) = min{ax1,x2}
x2 = ax1
x1
47
x2
U(x1,x2) = min{ax1,x2}
Which is the most
preferred affordable bundle?
x2 = ax1
x1
48
x2
U(x1,x2) = min{ax1,x2}
The most preferred
affordable bundle
x2 = ax1
x1
49
x2
U(x1,x2) = min{ax1,x2}
(a) p1x1* + p2x2* = m
(b) x2* = ax1*
x2 = ax1
x2*
x1*
x1
50
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
-Intuitively, it is clear to have (b) because we
don’t want to spend money on something that
cannot raise my utility.
-Substitution from (b) for x2* in (a)
gives p1x1* + p2ax1* = m
which gives
*
x1

m
p1  ap2
;
*
x2

am
p1  ap2
.
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x2
U(x1,x2) = min{ax1,x2}
*
x2 
x2 = ax1
am
p1  ap 2
*
x1 
m
x1
p1  ap 2
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
Recall
MRS 
p1
p2
As price is the same for all individuals,
every individual has the same MRS at
the optimal consumption bundle, even if
they have very different preferences.
 Thus, it can be interpreted as a social
willingness to pay for good 1 in terms of
good 2.

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In this chapter we put together budget
constraint and preference together to
analyze consumption decision.
 Typically, if we have interior solution,
then the optimal condition is
MRS=p1 / p2.
 It can also have corner solution if one
good is not consumed at all.

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

Next, we will look at comparative statics.
In particular, how will prices and
income changes affect the optimal
consumption bundle?
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