Chapter 5Choice

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Chapter 5 Choice

• Course Microeconomics
• Text Varians Intermediate Microeconomics

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Economic Rationality

• 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?

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Rational Constrained Choice
x2
x1
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Rational Constrained Choice
Utility
x2
x1
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Rational Constrained Choice
Utility
Affordable, but not the most preferred affordable
bundle.
x2
x1
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Rational Constrained Choice
The most preferredof the affordable bundles.
Utility
Affordable, but not the most preferred affordable
bundle.
x2
x1
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Rational Constrained Choice
x2
One will choose the bundle on the outermost
indifference curve. Therefore, it is the one that
just touch the budget constraint.
Affordablebundles
x1
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Rational Constrained Choice
x2
(x1,x2) is the mostpreferred affordablebundle.
x2
x1
x1
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Rational Constrained Choice

• The most preferred affordable bundle is called
  the consumers ORDINARY DEMAND at the given
  prices and budget.
• Ordinary demands will be denoted byx1(p1,p2,m)
  and x2(p1,p2,m).

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Rational Constrained Choice

• When x1 gt 0 and x2 gt 0 the demanded bundle is
  INTERIOR.
• If buying (x1,x2) costs m then the budget is
  exhausted.

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Rational Constrained Choice

• 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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Rational Constrained Choice

• Condition (b) can be written as
• 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.

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Mathematical Formulation
The consumer problem can be formulated
mathematically as
One way to solve is to use Lagrangian method
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Mathematical Treatment

• Assuming the utility function is differentiable,
  and there exists interior solution
• The first order conditions are

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Mathematical Treatment

• So, we have the same condition as before that MRS
  equals price ratio.

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Mathematical Treatment

• So, for interior solution, we can solve for the
  optimal bundle using the two conditions

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Computing Ordinary Demands - a Cobb-Douglas
Example.

• Suppose that the consumer has Cobb-Douglas
  preferences.
• Then

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Computing Ordinary Demands - a Cobb-Douglas
Example.

• So the MRS is
• At (x1,x2), MRS p1/p2 so

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Computing Ordinary Demands - a Cobb-Douglas
Example.

• So now we know that

(A)
(B)
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Computing Ordinary Demands - a Cobb-Douglas
Example.

• So now we know that

(A)
Substitute
(B)
and get
This simplifies to .
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Computing Ordinary Demands - a Cobb-Douglas
Example.
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Computing Ordinary Demands - a Cobb-Douglas
Example.
Substituting for x1 in
then gives
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Computing Ordinary Demands - a Cobb-Douglas
Example.
So we have discovered that the mostpreferred
affordable bundle for a consumerwith
Cobb-Douglas preferences
is
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Computing Ordinary Demands - a Cobb-Douglas
Example.
x2
x1
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Demand for Cobb-Douglas Utility Function

• With the optimal bundle
• Note the expenditures on each good
• It is a property of Cobb-Douglas Utility function
  that expenditure share on a particular good is a
  constant.

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Special Example
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Special Examples
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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Corner Solution

• 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.

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Special Example
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.
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Corner Solution

• 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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Examples of Corner Solutions -- the Perfect
Substitutes Case
x2
MRS 1
x1
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Examples of Corner Solutions -- the Perfect
Substitutes Case
x2
MRS 1
Slope -p1/p2 with p1 gt p2.
x1
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Examples of Corner Solutions -- the Perfect
Substitutes Case
x2
MRS 1
Slope -p1/p2 with p1 gt p2.
x1
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Examples of Corner Solutions -- the Perfect
Substitutes Case
x2
MRS 1
Slope -p1/p2 with p1 gt p2.
x1
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Examples of Corner Solutions -- the Perfect
Substitutes Case
x2
MRS 1
Slope -p1/p2 with p1 lt p2.
x1
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Examples of Corner Solutions -- the Perfect
Substitutes Case
So when U(x1,x2) x1 x2, the mostpreferred
affordable bundle is (x1,x2)where
if p1 lt p2
and
if p1 gt p2.
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Examples of Corner Solutions -- the Perfect
Substitutes Case
x2
MRS 1
Slope -p1/p2 with p1 p2.
x1
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Examples of Corner Solutions -- the Perfect
Substitutes Case
x2
All the bundles in the constraint are equally
the most preferred affordable when
p1 p2.
x1
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Examples of Corner Solutions -- the Non-Convex
Preferences Case
x2
Better
x1
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Examples of Corner Solutions -- the Non-Convex
Preferences Case
x2
x1
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Examples of Corner Solutions -- the Non-Convex
Preferences Case
x2
Which is the most preferredaffordable bundle?
x1
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Examples of Corner Solutions -- the Non-Convex
Preferences Case
x2
The most preferredaffordable bundle
x1
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Examples of Corner Solutions -- the Non-Convex
Preferences Case
Notice that the tangency solutionis not the
most preferred affordablebundle.
x2
The most preferredaffordable bundle
x1
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Examples of Kinky Solutions -- the Perfect
Complements Case
U(x1,x2) minax1,x2
x2
x2 ax1
x1
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Examples of Kinky Solutions -- the Perfect
Complements Case
U(x1,x2) minax1,x2
x2

MRS -
x2 ax1
MRS 0
x1
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Examples of Kinky Solutions -- the Perfect
Complements Case
U(x1,x2) minax1,x2
x2

MRS -
MRS is undefined
x2 ax1
MRS 0
x1
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Examples of Kinky Solutions -- the Perfect
Complements Case
U(x1,x2) minax1,x2
x2
x2 ax1
x1
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Examples of Kinky Solutions -- the Perfect
Complements Case
U(x1,x2) minax1,x2
x2
Which is the mostpreferred affordable bundle?
x2 ax1
x1
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Examples of Kinky Solutions -- the Perfect
Complements Case
U(x1,x2) minax1,x2
x2
The most preferred affordable bundle
x2 ax1
x1
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Examples of Kinky Solutions -- the Perfect
Complements Case
U(x1,x2) minax1,x2
x2
(a) p1x1 p2x2 m(b) x2 ax1
x2 ax1
x2
x1
x1
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Examples of Kinky Solutions -- the Perfect
Complements Case
(a) p1x1 p2x2 m (b) x2 ax1.
-Intuitively, it is clear to have (b) because we
dont want to spend money on something that
cannot raise my utility.
-Substitution from (b) for x2 in (a) gives
p1x1 p2ax1 mwhich gives
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Examples of Kinky Solutions -- the Perfect
Complements Case
U(x1,x2) minax1,x2
x2
x2 ax1
x1
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Implications of MRS condition

• Recall
• 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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Summary

• 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 MRSp1 / p2.
• It can also have corner solution if one good is
  not consumed at all.

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Whats next?

• Next, we will look at comparative statics.
• In particular, how will prices and income changes
  affect the optimal consumption bundle?