Appendix 1: The theory of consumer's behavior

Appendix 1: The theory of
consumer’s behavior
preference, utility, indifference
curve, budget constraint,
optimal consumption plan,
demand curve
1

1. Preference ordering and
utility function
! Objects to be selected x, y, z
! Consumption set consumption goods bundle
x =(x 1 ,x 2 ,...,x n ) (Consumption vector)
! Preference relation (binary relation)
! Preference ordering
! Value judgment about the alternatives
! xRyx is strictly better than y or both are
indifferent (x is at least as good as y)
! xPyx is strictly better than y (prefer x to y)
! xIyx and y are indifferent
2

Nature of the preference
relation
! Reflectivity: xRx holds for any x.
! Completeness:
xRy or yRx holds for any x and y.
! Transitivity: If xRy and yRz, then xRz
holds for any x, y, and z.
! Preference ordering = reflectivity +
completeness + transitivity
rational preference (relation)
3

Utility function
! Utility function preference ordering
u=u(x),
u=u(x 1 ,x 2 ,...,x n )
! The function gives a large value to the
desirable alternatives
! Indifference curve utility function
! The curve shows the group of alternatives
to have the same desirableness
4

y
u=u(x 1 ,x 2 )
x 2
Indifference curve
x 1
5

x 2
u 0

2. The budget constraint
! Price (p 1 ,p 2 ,...,p n )>0
! Income M>0
! The quantity of purchase plan (The
quantity demanded) (x 1 ,x 2 ,...,x n )0
! The budget constraint budget set
! Expenditure Income
! p 1 x 1 +p 2 x 2 +…+p n x n M
! The budget line (budget constraint line)
! p 1 x 1 +p 2 x 2 +…+p n x n =M
7

The budget set
! The budget set budget constraint
! The set of the possible consumer goods
bundles which satisfy a budget constraint
(The opportunity set)
! p 1 x 1 +p 2 x 2 +…+p n x n M
! x 1 0, x 2 0,……,x n 0
! The inside and the boundary of the triangle
which was surrounded by the budget line,
the vertical axis, the horizontal axis
! The budget line : x 2 =-(p 1 /p 2 )x 1 +M/p 2
8

x 2
x 2 =-(p 1 /p 2 )x 1 +M/p 2
M/p 2
The budget set
M/p 1
0
x 1
9

M'/p 2
M/p 2
x 2
The parallel shift of
the budget line
M"

x 2
The turn of the budget line p" 1

3. The optimal consumption plan
! The indifference curve + budget constraint
The optimal consumption plan (x 1 *,x 2 *)
! The additional assumption about preference
Monotonicity: The more, the better.
Convexity (The law of diminishing
marginal rate of substitution)
The more, the better, and the less the worse.
People like a mean better than the extremes.
12

x 2
marginal rate of substitution
MRS=-dx 2 /dx 1 =u 1 /u 2
E
dx 2
u=u 0
0
dx 1
x 1
13

Explanation of MRS=u 1 /u 2
! Differentiate u=u(x 1 ,x 2 )
du=u 1 dx 1 +u 2 dx 2
! On the same indifference curvedu=0
-dx 2 /dx 1 =u 1 /u 2
! Since MRS=-dx 2 /dx 1
MRS=u 1 /u 2
14

x 2
A
The law of diminishing marginal
rate of substitution
MRS A >MRS B >MRS C
The indifference curve swelled
up for the original point
B
0
C
u=u 0
x 1
13

! The optimal consumption plan (x 1 *,x 2 *)
! Inner point C of the budget set not
optimal
Utility increases in moving to the boundary.
! Point A on the boundary not optimal
Utility increases in moving to the direction
of point E on the budget line.
! Point B on the boundary not optimal
Utility increases in moving to the direction
of point E on the budget line.
! Point E is optimal
the optimal consumption plan
14

x 2
To decide the optimal
consumption plan
A
C
E
monotonicity:
u 0

The optimal consumption plan
! The budget line and the indefference
curve touch.
! price ratio
= marginal rate of substitution
p 1 /p 2 =MRS=u 1 /u 2
! It satisfies the budget constraint
with the equal sign.
p 1 x 1 +x 2 p 2 =M
16

4. The derivation of the
demand curve
! The optimal consumption plan
change of the price
! The new optimal consumption plan
A wide range of prices
! The demand curve x 1 =d 1 (p 1 ,p 2 ,M)
! The curve (function) shows how the
demand plan changes when the price of
the good is changed, making the prices
of other goods and the income constant.
19

x 2
The derivation of
the demand curve
The price-consumption
curve
E
E 2
0
E 1
M/p 1
0
M/p 1
2
u=u 2
u=u 0
u=u 1
0
x 1
1
x 1
0
x 1
2
M/p 1
1
20
x 1

p 1
The demand curve of
the 1st good
p 1
1
p 1
0
x 1 =d 1 (p 1 ,p 2 ,M)
p 1
2
x 1
1
x 1
0
x 1
2 x 1
21

5. The income effect and the
substitution effect
! The effect of income change
The change of the quantity demanded
when the income is changed
The effect by the shift of the budget line
22

M'/p 2
x 2
The normal good
M/p 2
The income-consumption
curve
M"/p 2
M"/p 1 M/p 1
M'/p 1 x 1
23

The intermediate good
The inferior property
24

The effect of the price change
! The change of the price
i) The change of the real income (The
income effect)
ii) The change of the relative price (The
substitution effect)
! The compensation variation with
income (The compensation income)
How much income is needed after the price
change in order to maintain utility level
before the price change
25

x 2
x 2
2
x 2
1
x 2
0
E 1
E 2
E 0
The income effect and
the substitution effect
u=u 1
u=u 0
0 x
1
1 x
2
1 x
0
1
income effect substitution effect
x 1
26