Consumer Equilibrium
• Now we have all the tools we need to analyze
consumer choice
• Suppose a consumer faces the constraints
Consumer Choice
– p1x1 + p2x2 ≤ m and x1 ≥ 0, x2 ≥ 0.
• The consumer’s problem is to choose and to
maximize utility (i.e. to attain highest possible
indifference curve) s.t. the above constraints,
ECON 370: Microeconomic Theory
Summer 2004 – Rice University
• That is, the consumer is choosing the most
preferred point in the budget set.
Stanley Gilbert
Econ 370 - Consumer Choice
Optimal Choice
2
Rational Constrained Choice
• Provided that this is an “interior” equilibrium,
x2
– That is, that it involves strictly positive amounts of all goods
• And assuming all our assumptions (A1 – A4) hold:
• Then we have equality of
Optimal Choice
– Slope of budget constraint (-p1 / p2)
– Slope of indifference curve (MRS)
– Ratio of marginal utilities (MU1/MU2)
• And our budget is exhausted: p1x1* + p2x2* = m
• Interpretation
Preferences
– Willingness to trade (-slope of IC) = p1 / p2
– Marginal utility per dollar equalized across goods
Budget
Econ 370 - Consumer Choice
x1
3
Econ 370 - Consumer Choice
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1
x2
•
Interior solution
•
Slope of indifference curve = Slope of
budget constraint
•
(that is) MRS = -p1 / p2
•
budget is exhausted: p1x1* + p2x2* = m
What about savings?
Saving for Retirement
Characteristics of Optimal Choice
We said that generally all money is
spent
•
That is p1x1* + p2x2* = m
•
How do we deal with savings?
– By assuming our consumption choices
include spending today v. spending in
the future.
Spending Today
x1
Econ 370 - Consumer Choice
•
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Econ 370 - Consumer Choice
Finding the Optimum
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Example
• We will demonstrate two methods for finding the
optimal choice
• Assume for this example that
– Preferences are Cobb-Douglass
– U = xαy1 – α
– Budget constraint is pxx + pyy ≤ m
– MRS = p1 / p2
– Maximizing the Utility function
• (At least) one other that I will not cover:
MU1 MU 2
=
p1
p2
– Mathematically, it is indistinguishable from the first
Econ 370 - Consumer Choice
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Econ 370 - Consumer Choice
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2
MRS Solution
MRS Solution (cont)
We will use x* and y* to represent our solution
In order to calculate MRS, we first calculate Marginal Utilities
MU x =
∂U
= αxα −1 y1−α
∂x
MU y =
α y* p x
=
1 − α x* p y
∂U
= (1 − α )xα y −α
∂y
Solving for y*:
Then we calculate MRS as:
MRS =
y* =
αxα −1 y1−α
dy ∂u ∂x MU x
α y
=
=
=
=
dx ∂u ∂y MU y (1 − α )xα y −α 1 − α x
Since MRS equals the slope of the budget line, we get:
So:
Econ 370 - Consumer Choice
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p x x* + p y
• Our problem is:
=m
Gives y * =
*
x =
Then
y* =
Substituting this into
– max{u(x, y)} = max{xαy1 – α}
αm
– Subject to the requirement that pxx + pyy ≤
px
1 − α px *
x
α py
m.
• Remembering that all money will be spent, we have:
– pxx + pyy = m
(from before)
• From here, the simplest way to proceed is to substitute the
modified constraint into problem to be solved:
(1 − α )m
– y = (m – pxx) / py
py
*
So, our solution is: x =
Econ 370 - Consumer Choice
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Maximization Solution
Finally:
α
1−α
1 − α px *
p x*
p x x* = x = m
x = p x x* +
α
α py
α
Econ 370 - Consumer Choice
MRS Solution (cont 2)
If
1 − α px *
x
α py
Since all money is spent, we also have: pxx* + pyy* = m
α y px
=
1−α x py
p x x*
(from previous page)
αm
px
*
y =
• Substituting into the original problem gives:
(1 − α )m
{
}
{
max xα y1−α = max xα (m − p x x )1−α pαy −1
py
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Econ 370 - Consumer Choice
}
12
3
Maximization (cont)
Maximization (cont 2)
Remembering our calculus, maximizing a function requires:
df ( x )
=0
dx
[
d 2 f (x )
≤0
dx 2
and
{
Since all this mess equals zero, we can eliminate common terms:
⇒ α ⎜⎛
⎝
}
So
max xα (m − p x x )1−α pαy −1
⇒
d α
x (m − p x x )1−α pαy −1 = 0
dx
⇒
pαy −1 αxα −1 (m − p x x )1−α − p x (1 − α )xα (m − p x x )−α = 0
[
]
pαy −1 αxα −1 (m − p x x )1−α − p x (1 − α )xα (m − p x x )−α = 0
m − px x ⎞
⎟ − (1 − α ) p x = 0
x
⎠
⇒ α (m − p x x ) − (1 − α ) p x x = αm − αp x x − p x x + αp x x = αm − p x x = 0
]
Econ 370 - Consumer Choice
⇒
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x=
αm
Exactly as before
px
Econ 370 - Consumer Choice
Maximization (cont 3)
Cobb-Douglas Observation
• Note that for Cobb-Douglas utility function
Verifying the Second Order Condition:
[
d α −1 α −1
p y αx (m − p x x )1−α − p x (1 − α )xα (m − p x x )−α
dx
]
– Demands are linear in income
– Expenditure shares are constant
– Expenditure shares sum to one
(reorganizing it a bit)
1−α
−α ⎤
⎡ d m
d m
= pαy −1 ⎢α ⎛⎜ − p x ⎞⎟
− p x (1 − α ) ⎛⎜ − p x ⎞⎟ ⎥
dx ⎝ x
⎠
⎠ ⎦
⎣ dx ⎝ x
−
α
−α −1 ⎤
⎡
m m
m m
= pαy −1 ⎢ − α (1 − α ) 2 ⎛⎜ − p x ⎞⎟ − p xα (1 − α ) 2 ⎛⎜ − p x ⎞⎟
⎥ <0
⎠
⎠
x ⎝x
x ⎝x
⎣
⎦
Econ 370 - Consumer Choice
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x* =
Econ 370 - Consumer Choice
αm
px
y* =
(1 − α )m
py
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4
“Corner” Solutions
Corner Example
• An “interior” solution was earlier described as:
x2
– involving strictly positive amounts of all goods
– And all our assumptions (A1 – A4) hold
Optimal Choice
• If a solution does not meet these requirements, it is
called a “Corner” Solution
• Other methods have to be used to solve it
Preferences
Budget
Econ 370 - Consumer Choice
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Econ 370 - Consumer Choice
x1
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5