Chapter Five

1
Chapter Five

• Choice

2
Economic Rationality

• The principal behavioral postulate a
  decisionmaker chooses the most preferred
  alternative from those available.
• The available choices constitute the choice set.
  (Budget Set)
• Utility functions and resulting indifference
  curves determine preferences over goods.

3
Rational Constrained Choice

• Constrained Optimization Problem Consumer wants
  to get on the highest indifference curve subject
  to satisfying the budget constraint.
• Focus on the case where considering goods.

4
Rational Constrained Choice

• Objective
• Want to derive the ordinary demand for
  commodities given prices and income.
• Ordinary demands will be denoted byx1(p1,p2,m)
  and x2(p1,p2,m).
• This makes explicit that the amounts demanded may
  change when prices or income change.

5
Rational Constrained Choice

• Methodology
• Visual approach try to derive conditions that
  have to hold at the optimum choice of goods.

6
Rational Constrained Choice

• These conditions will depend on the types of
  Utility functions/Indifference curves that
  describe preferences.
• We will look at the following special cases of
  Utility Functions/Preferences
• Cobb-Douglas
• Perfect Substitutes
• Perfect Complements
• Quasi-linear
• Non-convex preferences

7
Rational Constrained Choice

• Illustration of how this methodology will work,
• Look at the case where we have strictly convex
  indifference curves that never cross either axis.

8
Rational Constrained Choice
Budget Constraint
x2
Higher indifference curves represent higher
levels of utility.
A
Is B optimal? Is A?
Affordablebundles
B
x1
9
Rational Constrained Choice
x2
Budget Constraint
More preferredbundles
Any point preferred to A is not affordable.
A
Affordablebundles
x1
10
Rational Constrained Choice
x2
More preferredbundles
A is optimal. The consumer cannot afford any
bundles preferred to A.
A
Affordablebundles
x1
11
Rational Constrained Choice
x2
(x1,x2) is the mostpreferred affordablebundle.
x2
x1
x1
12
Rational Constrained Choice

• If indifference curves have this shape (strictly
  convex and never cross either axis), we note that
  at the solution two conditions must hold.
• The budget is exhausted
• p1x1 p2x2 m
• The slope of the indifference curve equals the
  slope of the budget constraint.
• MRS-p1/p2

13
Rational Constrained Choice
x2
(a) (x1,x2) exhausts thebudget p1x1 p2x2
m.
x2
x1
x1
14
Rational Constrained Choice
x2
(b) The slope of the indiff.curve at (x1,x2)
equalsthe slope of the budgetconstraint. ie
MRS-p1/p2
x2
x1
x1
15
Rational Constrained Choice
Interpretation Notice at point A, MRSgt
p1/p2. This means that the consumer is willing
to give up more x2 then the market requires, in
order to get one more unit of x1.
At point B?
x2
A
x2
B
x1
x1
16
Rational Constrained Choice

• At point B, MRSlt p1/p2 or 1/MRS gt p2/p1 which
  implies that the consumer is willing to trade
  more x1 for another unit of x2 then the market
  requires.
• Therefore, the consumer increases utility by
  giving up x1 for x2.

17
Computing Ordinary Demands

• How can this information be used to locate
  (x1,x2) for given p1, p2 and m?
• In other words, how do we derive the ordinary
  demand functions?
• Use conditions (a) and (b) to solve for the
  demands if indifference curves are strictly
  convex and do not cross either axis.

18
Computing Ordinary Demands - a Cobb-Douglas
Example.

• Suppose that the consumer has Cobb-Douglas
  preferences.

19
Computing Ordinary Demands - a Cobb-Douglas
Example.

• We know the Cobb-Douglas indifference curves are
  strictly convex and never touch either axis.
• Therefore, we can solve for the demands using
  conditions (a) and (b).

20
Computing Ordinary Demands - a Cobb-Douglas
Example.

• Condition (a) require that (x1,x2) exhausts the
  budget so

(a)
21
Computing Ordinary Demands - a Cobb-Douglas
Example.

• Condition (b) requires MRS-p1/p2 at (x1,x2).
• Then

22
Computing Ordinary Demands - a Cobb-Douglas
Example.

• So the MRS is

23
Computing Ordinary Demands - a Cobb-Douglas
Example.

• Condition (b), that at (x1,x2), MRS -p1/p2
  requires

(b)
24
Computing Ordinary Demands - a Cobb-Douglas
Example.

• So now we know that

(a)
(b)
25
Computing Ordinary Demands - a Cobb-Douglas
Example.

• So now we know that

(b)
Substitute
(a)
26
Computing Ordinary Demands - a Cobb-Douglas
Example.

• So now we know that

(b)
Substitute
(a)
and get
This simplifies to .
27
Computing Ordinary Demands - a Cobb-Douglas
Example.
28
Computing Ordinary Demands - a Cobb-Douglas
Example.
Substituting for x1 in (a)
and solving for x2 then gives
29
Computing Ordinary Demands - a Cobb-Douglas
Example.
So we have discovered that the mostpreferred
affordable bundle for a consumerwith
Cobb-Douglas preferences
is
30
Computing Ordinary Demands - a Cobb-Douglas
Example.
x2
x1
31
EX 5.1 Cobb-Douglas Numerical Example

• Find the optimal demands for the two goods when
  the utility function is given by
• and p12, p21, and m100.

32
EX 5.1 Cobb-Douglas Numerical Example

• Two conditions must hold
• Budget Constraint (p1x1 p2x2 m)
• 2x1 x2 100
• MRS-p1/p2
• 2. -2x2 /3x 1-2 or x2 3x1

33
EX 5.1 Cobb-Douglas Numerical Example

• 1. 2x1 x2 100
• 2. x2 3x1
• Using these two equations to solve for the two
  unknowns (x1 and x2 ), we get the solutions
• x120 and x260.

34
Computing Ordinary Demands - a Cobb-Douglas
Example.
x2
100
At optimum, MRS-p1/p2-2
60
x1
50
20
35
Rational Constrained Choice Interior versus
Corner Solutions

• When x1 gt 0 and x2 gt 0 the demanded bundle is
  INTERIOR.
• Or we say that the constrained maximization
  problem has an INTERIOR SOLUTION.
• If one of the demands is exactly zero, then we
  say that the constrained maximization problem has
  a CORNER SOLUTION.
• The conditions for the optimal demand will depend
  on the shapes of indifference curves and whether
  or not it is possible to get a corner solution.

36
Examples of Corner Solutions -- the Perfect
Substitutes Case

• We now look at a case where corner solutions
  (when either x1 0 or x2 0) are possible
  Perfect Substitutes

37
Examples of Corner Solutions -- the Perfect
Substitutes Case

• Preferences that fall under the perfect
  substitutes case can be expressed by any utility
  function that takes the form
• 
  
  
  

38
Examples of Corner Solutions -- the Perfect
Substitutes Case
x2
Indifference Curves for this Utility Function
have linear indifference Curves with the slope
of -a/bMRS
x1
39
Examples of Corner Solutions -- the Perfect
Substitutes Case

• We consider 3 cases
• Case 1 MRS lt p1/p2
• Case 2 MRS gt p1/p2
• Case 3 MRS p1/p2

40
Examples of Corner Solutions -- the Perfect
Substitutes Case
Case 1 MRS lt p1/p2
x2
MRS -a/b
Slope of Budget Constraint -p1/p2
x1
Here the Consumer gets on the highest
Indifference curve by only consuming x2.
41
Examples of Corner Solutions -- the Perfect
Substitutes Case
Case 1 MRS lt p1/p2
x2
MRS -a/b
Slope -p1/p2
x1
42
Examples of Corner Solutions -- the Perfect
Substitutes Case

• Case 1 MRS lt p1/p2
• Optimal Demands
• x2m/p2 and x10

43
Examples of Corner Solutions -- the Perfect
Substitutes Case
Case 2 MRS gt p1/p2
x2
MRS -a/b
Slope -p1/p2
x1
44
Examples of Corner Solutions -- the Perfect
Substitutes Case

• Case 2 MRS gtp1/p2
• Optimal Demands
• x20 and x1m/p1

45
Examples of Corner Solutions -- the Perfect
Substitutes Case
So when U(x1,x2) ax1 bx2, the mostpreferred
affordable bundle is (x1,x2)where
if a/b gt p1/p2
and
if a/b lt p1/p2
46
Examples of Corner Solutions -- the Perfect
Substitutes Case
Case 3 MRS p1/p2
x2
MRS -a/b
Slope of BC -p1/p2
x1
47
Examples of Corner Solutions -- the Perfect
Substitutes Case
Case 3 MRS p1/p2 All the bundles in the
constraint are equally the most preferred
when the slope of the budget
constraint in is the same as the MRS.
x2
x1
48
EX 5.2 Perfect Substitutes Numerical Example

• Find the optimal demands for the two goods when
  the utility function is given by
• U(x1,x2) 2x1 x2
• and p11, p21, and m100.

49
EX 5.2 Perfect Substitutes Numerical Example

• MRS-2
• Slope of Budget Constraint-1
• So we know that the consumer is willing to give
  up more x2 then the market requires to get more
  x1, so the optimal demands are
• x1m/p1100
• x20

50
EX 5.2 Perfect Substitutes Numerical Example
Indifference curves are steeper than the budget
constraint. MRS2 gt p1/p2 1
x2
MRS -2
Slope -1
100
x1
100
51
The Perfect Complements Case

• We now look at how to solve for the optimal
  demands when consumers prefer to consume goods in
  perfect proportion to one another Perfect
  complements.
• Recall, the is the case when a utility function
  takes the form U(x1,x2)min(ax1,bx2)

52
The Perfect Complements Case

• U(x1,x2)min(ax1,bx2)
• Given this utility function, we know that
  indifference curves are right angled.
• We also know that if the consumer does not waste
  any money on goods that do not add additional
  utility it must be the case that ax1bx2 or x2
  ax1/b. This defines the line through the origin
  connecting the vertices of all the indifference
  curves.

53
The Perfect Complements Case
U(x1,x2) minax1,bx2
x2
x2 ax1/b
x1
54
The Perfect Complements Case
U(x1,x2) minax1,bx2
x2

MRS -
MRS is undefined
x2 ax1/b
MRS 0
x1
55
The Perfect Complements Case
U(x1,x2) minax1,bx2
x2
Budget Constraint
x2 ax1/b
x1
56
The Perfect Complements Case
U(x1,x2) minax1,bx2
x2
Which is the mostpreferred affordable bundle?
x2 ax1/b
x1
57
The Perfect Complements Case
U(x1,x2) minax1,bx2
x2
The most preferred affordable bundle
x2 ax1/b
x1
58
The Perfect Complements Case
U(x1,x2) minax1,bx2
x2
At the most preferred bundle (a) p1x1 p2x2
m(b) x2 ax1/b
x2 ax1/b
x2
x1
x1
59
The Perfect Complements Case
At the most preferred bundle (a) p1x1 p2x2
m(b) x2 ax1/b We use this information to
solve for x1 and x2.
60
The Perfect Complements Case
(a) p1x1 p2x2 m (b) x2 ax1.
Substitution from (b) for x2 in (a) gives p1x1
p2ax1/b m Solving for x1 .
61
The Perfect Complements Case
(a) p1x1 p2x2 m (b) x2 ax1/b.
Substitution from (b) for x2 in (a) gives p1x1
p2ax1/b mwhich gives Solving for x1 .
62
The Perfect Complements Case
After plugging in x1 in equation (b) and solving
for x2 , we have the final demands.
63
The Perfect Complements Case
U(x1,x2) minax1,bx2
x2
x2 ax1/b
x1
64
Ex 5.3 Perfect Complements Numerical Example

• Find the optimal demands for the two goods when
  the utility function is given by
• U(x1,x2) min2x1,x2
• and p11, p21, and m100.

65
Ex 5.3 Perfect Complements Numerical Example

• We know that two conditions must hold
• The optimum is on the line connecting the kink
  points of the indifference curves
• (1) x2 2x1
• The budget constraint holds
• (2) x1 x2 100

66
Ex 5.3 Perfect Complements Numerical Example

• (1) x2 2x1
• (2) x1 x2 100
• Solving these two equations yields
• x1 100/3
• x2200/3

67
Ex 5.3 Perfect Complements Numerical Example
U(x1,x2) min2x1,x2
x2
x2 2x1
100
200/3
100
x1
100/3
68
Quasi-linear Indifference Curves

• We now look at the case for Quasi-linear
  preferences.
• Recall that a Quasi-linear Utility Function is
  one that takes the form
• U(x1,x2)f(x1)x2
• Where f(x1) is a strictly concave function

69
Quasi-linear Indifference Curves
x2
Each indifference curve is a vertically shifted
copy of the others.
Each curve may intersectboth axes.
x1
70
Quasi-linear Indifference Curves

• Since indifference curves may intersect both
  axes, it may be possible to get either interior
  solutions or corner solutions.

71
Quasi-linear Indifference Curves
x2
An interior solution will attain, if the budget
constraint is tangent to the highest possible an
indifference curve somewhere in the first
quadrant.
x2
x1
x1
72
Quasi-linear Indifference Curves
A corner solution will attain, if the budget
constraint is tangent to an indifference curve
somewhere where one of the variables is zero or
negative. In this case, it
is optimal to spend
all income on the
other good.

• x2

Not feasible
x20
x1
x1
73
Quasi-linear Indifference Curves

• How to solve for optimal demands
• I. Assume an interior solution and use the
  following conditions to solve for the demands
• (1) MRS-p1/p2
• (2) p1x1p2x2m
• If both x1 and x2 are positive, you are done.
• II. If one of the demands is negative, set this
  demand to zero and spend all income on the other
  good.

74
Ex. 5.4 Quasi-linear Numerical Example

• Find the optimal demands for the two goods when
  the utility function is given by
• U(x1,x2) 5ln(x1) x2
• and p12, p24, and m100.

75
Ex. 5.4 Quasi-linear Numerical Example

• I. Assuming an interior solution
• (1) MRS-p1/p2
• 5/x1 .5 or x1 10
• (2) p1x1 p2x2 m
• 2x1 4x2 100
• Solving yields that x110 and x220. Since both
  are positive we have an interior solution.

76
Ex. 5.4 Quasi-linear Numerical Example
An interior solution will attain, if the budget
constraint is tangent to an indifference curve
somewhere in the first quadrant.

• x2

20
x1
10
77
Ex. 5.5 Quasi-linear Numerical Example CORNER
SOLUTION

• Find the optimal demands for the two goods when
  the utility function is given by
• U(x1,x2) 5ln(x1) x2
• and p12, p24, and m15.

78
Ex. 5.5 Quasi-linear Numerical Example CORNER
SOLUTION

• I. Assuming an interior solution
• (1) MRS-p1/p2
• 5/x1 .5 or x1 10
• (2) p1x1 p2x2 m
• 2x1 4x2 15
• Solving yields that x110 and x2-5/4.
• Since x2 is negative, then it is optimal to
  spend all income on x1.
• Solutions x1m/p115/27.5 and x20.

79
Ex. 5.5 Quasi-linear Numerical Example CORNER
SOLUTION
A corner solution will attain, if the budget
constraint is tangent to an indifference curve
somewhere where one of the variables is zero or
negative.

• x2

x2 0
10
x1
x1 7.5
-5/4
80
Examples of Corner Solutions -- the Non-Convex
Preferences Case
x2
Better
x1
81
Examples of Corner Solutions -- the Non-Convex
Preferences Case
x2
Budget Constraint
m/p2
x1
m/p1
82
Examples of Corner Solutions -- the Non-Convex
Preferences Case
x2
Which is the most preferredaffordable bundle?
m/p2
x1
m/p1
83
Examples of Corner Solutions -- the Non-Convex
Preferences Case
x2
The most preferredaffordable bundle
m/p2
x1
m/p1
84
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
85
Ex. 5.6 Non-Convex Preferences Numerical Example

• Find the optimal demands for the two goods when
  the utility function is given by
• U(x1,x2) (x1)2 (x2)2
• and p110, p210, and m100.

86
Ex. 5.6 Non-Convex Preferences Numerical Example

• Setting the utility level to a constant we can
  plot some indifference curves and see that they
  have the shape of quarter circles centered around
  the origin.

87
Ex. 5.6 Non-Convex Preferences Numerical Example

• x2

Here there are two solutions that yield the same
level of utility. Both are corner solutions
x2 10
x1
x1 10