Transcript Transparencies – 01/29/01 and 01/31/01 Classes

Choice
Q: What is the Optimal Choice?
Budget
constraint
x2
Z
Indifference curves
More
preferred bundles
X
Y
x1
A: Optimal Choice is X
Optimal choice:
indifference curve
tangent to budget line.
Does this tangency
condition necessarily
have to hold at an
optimal choice?
x2
x2
X
*
*
x1
x1
Perfect Substitutes
x2
x2
*
X*
*
x1
x1
Perfect Complements
x2
x2
X*
*
*
x1
x1
Q: Is Tangency Sufficient?
x2
Y
X
Z
x1
What is the General Rule?
If:
•
Preferences are well-behaved.
•
Indifference curves are “smooth” (no kinks).
•
Optima are interior.
Then:
Tangency between budget constraint and
indifference curve is necessary and sufficient
for an optimum.
Multiple Optima
x2
A way to avoid
multiplicity of optima,
is to assume strictly
convex preferences.
This assumption rules
out “flat spots” in
indifference curves.
optimal
bundles
x1
Economic Interpretation
At optimum: “Tangency between budget
line and indifference curve.”
p1
Slope of budget line:

p2
Slope of indifference curve: MRS ( x1, x 2 )
Tangency:
p1

 MRS ( x1, x 2)
p2
Interpretation
At Z:
x2 Z
p1
  MRS ( x1, x 2)
p2
X
At Y:
p1
  MRS ( x1, x 2)
p2
Y
x1
Tangency with Many Consumers
Consider many consumers with different
preferences and incomes, facing the same
prices for goods 1 and 2.
Q: Why is it the case that at their optimal
*
*
choice ( xi1 , xi 2 ) the MRS between 1
and 2 for different consumers is equalized?
Tangency with Many Consumers
A: Because if a consumer i makes an
optimal choice, then:
p1
*
*

 MRS i ( xi1 , xi 2 )
p2
Implication: everyone who is consuming the
two goods must agree on how much one is
worth in terms of the other.
Tangency with 2 Consumers
Indifference curves
of consumer 1:
x2
Indifference curves
of consumer 2:
X
Budget line:
Y
x1
Finding the Optimum in Practice:
a Cobb-Douglas Example
Preferences represented by:
u( x1, x 2)  c log x1  (1  c) log x 2
Budget line:
p1x1  p2 x2  m
Finding the Optimum in Practice:
a Cobb-Douglas Example
Mathematically, we would like to:
max c log x1  (1  c) log x 2
x1, x 2
such that
p1x1  p2 x2  m
Finding the Optimum in Practice:
a Cobb-Douglas Example
Replace budget constraint into objective
function:
m p1
x2 
p2

p2
x1
New problem:

 m p1 
max c log x1  (1  c) log   x1 
x1
 p 2 p 2 

Finding the Optimum in Practice:
a Cobb-Douglas Example
New problem:

 m p1 
max c log x1  (1  c) log   x1 
x1
 p 2 p 2 

First-Order Condition:
 p 2  p1
c
 1  c 
  0
x1
 m  p1 x1  p 2
Finding the Optimum in Practice:
a Cobb-Douglas Example
First-Order Condition:
 p 2  p1
c
 1  c 
  0
x1
 m  p1 x1  p 2
Rearranging:
p1
c  m  p1 x1 



  MRS ( x1, x 2)
1  c   p 2 x1 
p2
Finding the Optimum in Practice:
a Cobb-Douglas Example
First-Order Condition:
p1
c  m  p1 x1 




1  c   p 2 x1 
p2
m
*
Solve for x1 : x1  c
p1 *
p1 x 1
Expenditures share in 1:
c
m
Finding the Optimum in Practice:
a Cobb-Douglas Example
Q: How do I find
x2
*
?
A: Use the budget constraint:
m p1 * m p1 m
x2 

x1 
 c
p2 p2
p 2 p 2 p1
m
 1  c 
p2
*