Introduction to consumer choice and demand decisions

1
Lecture 4

• Introduction to consumer choice and demand
  decisions

2
Four key elements in consumer choice

• Consumers income
• Prices of goods
• Consumer preferences
• The assumption that consumers maximise utility

3
Consumption Choice Sets

• A consumption choice set is the collection of all
  consumption choices available to the consumer.
• What constrains consumption choice?
• Budgetary, time and other resource limitations.

4
Budget Constraints

• A consumption bundle containing x1 units of
  commodity 1, x2 units of commodity 2 and so on up
  to xn units of commodity n is denoted by the
  vector (x1, x2, , xn).
• Commodity prices are p1, p2, , pn.

5
Budget Constraints

• Q When is a bundle (x1, , xn) affordable at
  prices p1, , pn?
• A When p1x1 pnxn mwhere m is
  the consumers (disposable) income.

6
Budget Constraints

• The bundles that are only just affordable form
  the consumers budget constraint. This is the
  set (x1,,xn) x1 ³ 0, , xn ³ 0 and
  p1x1 pnxn m .

7
Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
x1
m /p1
8
Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
x1
m /p1
9
Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
Just affordable
x1
m /p1
10
Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
Not affordable
Just affordable
x1
m /p1
11
Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
Not affordable
Just affordable
Affordable
x1
m /p1
12
Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
the collection of all affordable bundles.
Budget Set
x1
m /p1
13
Budget Set and Constraint for Two Commodities
x2
p1x1 p2x2 m is x2 -(p1/p2)x1 m/p2
so slope is -p1/p2.
m /p2
Budget Set
x1
m /p1
14
Budget Constraints
x2
Opp. cost of an extra unit of commodity 1 is
p1/p2 units foregone of commodity 2.
-p1/p2
1
x1
15
Budget Constraints
x2
Opp. cost of an extra unit of commodity 1 is
p1/p2 units foregone of commodity 2.
And the opp. cost of an extra
unit of commodity 2 is
p2/p1 units foregone
of commodity 1.
1
-p2/p1
x1
16
For example.
x2
5x1 10x2 50 is x2 -(5/10)x1 50/10
so slope is -1/2.
50 /105
Budget Set
x1
50 /510
17
Budget Sets Constraints Income and Price
Changes

• The budget constraint and budget set depend upon
  prices and income. What happens as prices or
  income change?

18
How do the budget set and budget constraint
change as income m increases?
x2
Original budget set
x1
19
Higher income gives more choice
x2
New affordable consumptionchoices
Original and new budget constraints are parallel
(same slope).
Original budget set
x1
20
Higher income gives more choice
x2
5x1 10x2 60
6
5
5x1 10x2 50
Original budget set
x1
10
12
21
How do the budget set and budget constraint
change as income m decreases?
x2
Consumption bundles that are no longer affordable.
Old and new constraints are parallel.
New, smaller budget set
x1
22
For example.
x2
5
5x1 10x2 50
4
5x1 10x2 40
New, smaller budget set
x1
10
8
23
Budget Constraints - Income Changes

• Increases in income m shift the constraint
  outward in a parallel manner, thereby enlarging
  the budget set and improving choice.
• Decreases in income m shift the constraint
  inward in a parallel manner, thereby shrinking
  the budget set and reducing choice.

24
Budget Constraints - Price Changes

• What happens if just one price decreases?
• Suppose p1 decreases.

25
How do the budget set and budget constraint
change as p1 decreases from p1 to p1?
x2
m/p2
-p1/p2
Original budget set
x1
m/p1
m/p1
26
How do the budget set and budget constraint
change as p1 decreases from p1 to p1?
x2
m/p2
New affordable choices
-p1/p2
Original budget set
x1
m/p1
m/p1
27
Example
x2
m/p2
New affordable choices
-1/2
-1/5
Original budget set
x1
50/510
50/225
28
Budget Constraints - Price Changes

• Reducing the price of one commodity pivots the
  constraint outward. No old choice is lost and
  new choices are added, so reducing one price
  cannot make the consumer worse off.

29
Budget Constraints - Price Changes

• Similarly, increasing one price pivots the
  constraint inwards, reduces choice and may
  (typically will) make the consumer worse off.

30
Rationality in Economics

• Behavioral PostulateA decision maker always
  chooses its most preferred alternative from its
  set of available alternatives.
• So to model choice we must model decision makers
  preferences.

31
Preference Relations

• Comparing two different consumption bundles, x
  and y
• strict preference x is more preferred than is y.
• weak preference x is as at least as preferred as
  is y.
• indifference x is exactly as preferred as is y.

32
Preference Relations

• Strict preference, weak preference and
  indifference are all preference relations.
• Particularly, they are ordinal relations i.e.
  they state only the order in which bundles are
  preferred.

33
Preference Relations

• denotes strict preference x y means
  that bundle x is preferred strictly to bundle y.

p
p
34
Preference Relations

• denotes strict preference x y means
  bundle x is preferred strictly to bundle y.
• denotes indifference x y means x and y are
  equally preferred.

p
p
35
Preference Relations

• denotes strict preference so x y
  means that bundle x is preferred strictly to
  bundle y.
• denotes indifference x y means x and y are
  equally preferred.
• denotes weak preferencex y means x is
  preferred at least as much as is y.

p
p
36
Preference Relations

• x y and y x imply x y.
• x y and (not y x) imply x y.

p
37
Assumptions about Preference Relations

• Completeness For any two bundles x and y it is
  always possible to make the statement that either
  x y or
  y x.

38
Assumptions about Preference Relations

• Reflexivity Any bundle x is always at least as
  preferred as itself i.e.
  x x.

39
Assumptions about Preference Relations

• Transitivity Ifx is at least as preferred as
  y, andy is at least as preferred as z, thenx is
  at least as preferred as z i.e. x y and
  y z x z.

40
Indifference Curves

• Take a reference bundle x. The set of all
  bundles equally preferred to x is the
  indifference curve containing x the set of all
  bundles y x.
• Since an indifference curve is not always a
  curve a better name might be an indifference
  set.

41
Indifference Curves
x2
x x x
x
x
x
x1
42
Indifference Curves
x2
z x y
p
p
x
z
y
x1
43
Indifference Curves
I1
All bundles in I1 are strictly preferred to all
in I2.
x2
x
z
I2
All bundles in I2 are strictly preferred to
all in I3.
y
I3
x1
44
Indifference Curves
x2
SP(x), the set of bundles strictly preferred
to x, does not include
I(x).
x
I(x)
x1
45
Indifference Curves Cannot Intersect
I2
x2
From I1, x y. From I2, x z. Therefore y z.
But from I1 and I2 we see y z, a
contradiction.
I1
p
x
y
z
x1
46
Slopes of Indifference Curves

• When more of a commodity is always preferred, the
  commodity is a good.
• If every commodity is a good then indifference
  curves are negatively sloped.

47
Slopes of Indifference Curves
Good 2
Two goodsa negatively sloped indifference curve.
Better
Worse
Good 1
48
Slopes of Indifference Curves

• The slope of an indifference curve is its
  marginal rate-of-substitution (MRS).
• How can a MRS be calculated?

49
Marginal Rate of Substitution
x2
dx2 MRS dx1 so, at x, MRS is the rate at
which the consumer is only just willing to
exchange commodity 2 for a small amount of
commodity 1.
x
dx2
dx1
x1
50
Utility Functions

• A preference relation that is complete,
  reflexive, transitive and continuous can be
  represented by a continuous utility function.
• Continuity means that small changes to a
  consumption bundle cause only small changes to
  the preference level.

51
Utility Functions

• A utility function U(x) represents a preference
  relation if and only if x x
  U(x) gt U(x) x x
  U(x) lt U(x) x x
  U(x) U(x).

p
p
52
Utility Functions

• Utility is an ordinal (i.e. ordering) concept.
• E.g. if U(x) 6 and U(y) 2 then bundle x is
  strictly preferred to bundle y. But x is not
  preferred three times as much as is y.

53
Utility Functions Indiff. Curves

• Consider the bundles (4,1), (2,3) and (2,2).
• Suppose (2,3) (4,1) (2,2).
• Assign to these bundles any numbers that preserve
  the preference orderinge.g. U(2,3) 6 gt
  U(4,1) U(2,2) 4.
• Call these numbers utility levels.

p
54
Utility Functions Indiff. Curves

• An indifference curve contains equally preferred
  bundles.
• Equal preference ? same utility level.
• Therefore, all bundles in an indifference curve
  have the same utility level.

55
Utility Functions Indiff. Curves

• So the bundles (4,1) and (2,2) are in the indiff.
  curve with utility level U º 4
• But the bundle (2,3) is in the indiff. curve with
  utility level U º 6.
• On an indifference curve diagram, this preference
  information looks as follows

56
Utility Functions Indiff. Curves
x2
(2,3) (2,2) (4,1)
p p
U º 6
U º 4
x1
57
Marginal Utilities

• Marginal means incremental.
• The marginal utility of commodity i is the
  rate-of-change of total utility as the quantity
  of commodity i consumed changes i.e.
  

58
Marginal Utilities and Marginal
Rates-of-Substitution

• The general equation for an indifference curve
  is U(x1,x2) º k, a constant.

59
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 choice set.
• How is the most preferred bundle in the choice
  set located?

60
Rational Constrained Choice
x2
More preferredbundles
Affordablebundles
x1
61
Rational Constrained Choice
x2
(x1,x2) is the mostpreferred affordablebundle.
x2
x1
x1
62
Rational Constrained Choice
x2
(x1,x2) is interior .The slope of the
indiff.curve at (x1,x2) equals the slope of
the budget constraint, or MRS-p1/p2 i.e.
MU1/MU2 -p1/p2
x2
x1
x1