Choice, Demand and
Uncertainty
Topic 4
1
Revealed Preferences
2
Revealed Preferences
• We get choices from preferences
• Preferences are not directly observable: we have to
discover preferences from choices.
• Consumption choices may reveals information about the
consumer’s preference.
• Suppose that we can observe the demands at different
budgets.
3
Revealed Preferences
• Assumptions:
– Preferences do not change while the choice data are
gathered.
– Preferences are strictly convex and monotonic (well
behaved)
• Convexity and monotonicity imply that the most
preferred affordable bundle is unique.
Revealed Preferences
• Suppose that the bundle x* is chosen when the
bundle y is affordable. Then x* is revealed directly
preferred to y (otherwise y would have been
chosen).
• That x is revealed directly as preferred to y can be
written as
p
x
D
y
Revealed Preferences
x2
The chosen bundle x* is
revealed directly as preferred
to the bundles y and z.
x*
y
z
x1
Revealed Preferences
• Suppose x is revealed directly preferred to y, and y is
revealed directly preferred to z. Then, by transitivity, x
is revealed indirectly preferred to z. Write this as:
p
so x
D
I
z
y and y
p
x
D
z =>
x
I
z.
Revealed Preferences
x2
z is not affordable when x* is chosen.
x*
z
x1
Revealed Preferences
x2
x* is not affordable when y* is chosen.
x*
y*
z
x1
Revealed Preferences
x2
z is not affordable when x* is chosen.
x* is not affordable when y* is chosen.
x*
y*
z
x1
Revealed Preferences
z is not affordable when x* is chosen.
x* is not affordable when y* is chosen.
So x* and z cannot be compared
directly.
x*
y*
z
But x*x*D
y*
and y*
z
so x*
x1
p
p
x2
D
I
z.
Axioms
• The Weak Axiom of Revealed Preference (WARP):
– If the bundle x is revealed directly as preferred to the
bundle y then it is never the case that y is revealed
directly as preferred to x.
• The Strong Axiom of Revealed Preference (SARP):
– If the bundle x is revealed (directly or indirectly) as
preferred to the bundle y and x is different from y,
then it is never the case that the y is revealed (directly
or indirectly) as preferred to x.
Demand
13
Demand
• The consumer’s demand functions give the optimal
amounts of each good as a function of prices and
income.
• The demand functions can be used to do comparative
statics:
– Understand how a choice responds to change in the
economics environment.
14
Own Price Changes
• 𝑝2 and 𝑚 are fixed.
𝑥2
𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚
𝑝1 = 𝑝1′
𝑥1
15
Own Price Changes
• 𝑝2 and 𝑚 are fixed.
𝑥2
𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚
𝑝1 = 𝑝1′
𝑝1 = 𝑝1′′
𝑥1
16
Own Price Changes
• 𝑝2 and 𝑚 are fixed.
𝑥2
𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚
𝑝1 = 𝑝1′
𝑝1 =
𝑝1′′′
𝑝1 = 𝑝1′′
𝑥1
17
Own Price Changes
• 𝑝2 and 𝑚 are fixed.
𝑥2
𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚
𝑝1 = 𝑝1′
𝑝1 =
𝑝1′′′
.
𝑝1 = 𝑝1′′
𝑥1
18
Own Price Changes
𝑝1
.
𝑥2
𝑝1 =
𝑝1 =
𝑝1′′′
(𝑥1′ 𝑝1′ )
𝑝1′
.
𝑥1
𝑝1 = 𝑝1′′
𝑥1′
𝑥1
19
Own Price Changes
𝑥2
. .
𝑝1 = 𝑝1′
𝑝1 =
𝑝1′′′
𝑝1 = 𝑝1′′
𝑥1′
𝑥1
20
Own Price Changes
𝑝1
𝑥2
..
(𝑥1′′ 𝑝1′′ )
. .
𝑝1 =
𝑝1 =
𝑝1′′′
𝑝1′
(𝑥1′ 𝑝1′ )
𝑥1
𝑝1 = 𝑝1′′
𝑥1′
𝑥1
21
Own Price Changes
𝑥2
.. .
𝑝1 = 𝑝1′
𝑝1 =
𝑝1′′′
𝑝1 = 𝑝1′′
𝑥1′
𝑥1
22
Own Price Changes
𝑝1
𝑥2
..
(𝑥1′′′ 𝑝1′′′ )
.
(𝑥1′′ 𝑝1′′ )
.. .
𝑝1 =
𝑝1 =
𝑝1′′′
(𝑥1′ 𝑝1′ )
𝑝1′
𝑥1
𝑝1 = 𝑝1′′
𝑥1′
𝑥1
23
Own Price Changes
𝑝1
𝑥2
..
(𝑥1′′′ 𝑝1′′′ )
.
(𝑥1′′ 𝑝1′′ )
.. .
𝑝1 =
𝑝1 =
𝑝1′′′
(𝑥1′ 𝑝1′ )
𝑝1′
𝑥1
Demand curve
for good 𝑥1
𝑝1 = 𝑝1′′
𝑥1′
𝑥1
24
Own Price Changes
𝑥2
.. .
𝑝1 price offer curve
𝑥1
25
Own Price Changes
• Price Offer Curve: represents the optimal choice
as the price of good 1 changes.
• Demand Curve: plots the optimal choices of
good 1 as a function of its price.
– It typically have a downward slope (ordinary
goods).
– Exception: Giffen Goods.
26
Giffen Good
• 𝑝2 and 𝑚 are fixed.
𝑥2
Demand curve has
𝑝1
a positively
sloped part

px price offer
curve
Good x is
Giffen
𝑥1
𝑥1
27
Income Changes
• 𝑝1 and 𝑝2 are fixed.
𝑥2
𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚
𝑚 = 𝑚′
𝑥1
28
Income Changes
• 𝑝1 and 𝑝2 are fixed.
𝑥2
𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚
𝑚 = 𝑚′
𝑚 = 𝑚′′′
𝑚 = 𝑚′
𝑥1
29
Income Changes
𝑥2
𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚
.
..
𝑚 = 𝑚′
𝑚 = 𝑚′′′
𝑚 = 𝑚′
𝑥1
30
Income Changes – Normal Good
.
.
.
.
..
𝑚
𝑥2
𝑚 = 𝑚′
𝑚 = 𝑚′′′
𝑥1
Engel curve for
good 𝑥1
𝑚 = 𝑚′
𝑥1
31
Income Changes – Normal Good
𝑥2
.
..
Income offer
curve 𝑥1
𝑥1
32
..
.
Income Changes – Inferior Good
𝑚
𝑥2
..
.
𝑚 = 𝑚′
𝑚 = 𝑚′′′
𝑥1
Engel curve for
good 𝑥1
𝑚 = 𝑚′
𝑥1
33
Income Changes
• Income offer curve (expansion path): depicts the optimal
choice at different levels of income and constant prices.
• A good for which quantity demanded rises with income is
called normal.
• Therefore a normal good’s Engel curve is positively sloped.
• A good for which quantity demanded falls as income
increases is an inferior good.
• Therefore an inferior good’s Engel curve is negatively sloped.
34
Income and Substitution
Effects
35
Price Change
• The effect of a price change of either 𝑥1 or 𝑥2 on
the consumer is twofold:
– First, a change in the relative price ratio. After
the change of price, 𝑥1 is either cheaper or
dearer.
– Secondly, there is a change in the consumer
purchasing power.
36
Price Changes
• A price increase means less buying power, a
price decrease means more buying power.
• The reaction of the consumer to the change in
relative prices only is measured by the
substitution effect.
• how would the consumer change his
consumption if there were no accompanying
change in purchasing power?
37
Income and Substitution
• We decompose the total change into effects: income and
substitution.
• We hypothetically adjust the consumer’s income to restore
him to the level of real income he enjoyed before the price
change.
• Let’s assume a fall in 𝑝1 . It implies an increase in real income.
• We have to reduce the consumer’s income by drag the new
budget line back until it is just tangent to the original
indifference curve.
38
Fall in 𝑝1
𝑥2
𝑚/𝑝2
Price fall from 𝑝1′ to 𝑝1′′
E0
E1
0
𝑚/𝑝1′
𝑚/𝑝1′′
𝑥1
39
Fall in 𝑝1
𝑥2
𝑚/𝑝2
Price fall from 𝑝1′ to 𝑝1′′
E0
E1
E2
0
𝑚/𝑝1′
𝑚/𝑝1′′
𝑥1
40
Fall in 𝑝1
𝑥2
E1-E0: Total Effect (𝑥11 -𝑥10 )
E2-E0: Substitution Effect (𝑥12 -𝑥10 )
𝑚/𝑝2
E1-E2: Income Effect (𝑥11 -𝑥12 )
E0
E1
E2
0
𝑥10
𝑥12
𝑥11
𝑥1
41
Substitution Effect
• The substitution effect is always negative (price and
quantity consumed go in opposite directions).
• Proof:
– Let (𝑥1 , 𝑥2 ) be a bundle demanded at some prices
(𝑝1 , 𝑝2 ) and let (𝑦1 , 𝑦2 ) be a demanded bundle at
some other prices (𝑞1 , 𝑞2 ).
– Suppose the consume is indifferent between (𝑥1 , 𝑥2 )
and (𝑦1 , 𝑦2 )
42
Substitution Effect
– The following inequalities cannot be true
• 𝑝1 𝑥1 + 𝑝2 𝑥2 > 𝑝1 𝑦1 + 𝑝2 𝑦2
• 𝑞1 𝑦1 + 𝑞2 𝑦2 > 𝑞1 𝑥1 + 𝑞2 𝑥2
– So, these inequalities are true:
• 𝑝1 𝑥1 + 𝑝2 𝑥2 ≤ 𝑝1 𝑦1 + 𝑝2 𝑦2
• 𝑞1 𝑦1 + 𝑞2 𝑦2 ≤ 𝑞1 𝑥1 + 𝑞2 𝑥2
43
Substitution Effect
– Summing the true inequalities above and
rearranging:
• 𝑞1 − 𝑝1 𝑦1 − 𝑥1 + (𝑞2 − 𝑝2 )(𝑦2 − 𝑥2 ) ≤ 0
– We want to check the effect of one price (say the price
of good 1), keeping the other price constant (𝑞2 = 𝑝2 ):
• 𝑞1 − 𝑝1 𝑦1 − 𝑥1 ≤ 0
– Therefore, quantities and prices go in opposite
directions.
44
Income Effect
• The income effect however can be negative (normal
goods) or positive (inferior goods).
• A rise in the price of a normal good induces a negative
substitution effect and a negative income effect, both of which
act to reduce the demand for good.
• A rise in the price of an inferior good, however, induces a
negative substitution effect but a positive income effect, thus
the overall effect is ambiguous.
45
Inferior and Giffen Goods
• If, when the price of an inferior good rises, the positive
income effect dominates the negative substitution effect, we
have the case of a Giffen Good.
• That explains why demand for a Giffen Good rises (falls)
when price rises (falls).
• In other words, a Giffen good is a very inferior good.
46
Normal Good
A
C
E1
E0
E2
0
A
C
B
47
Inferior Good
A
E1
C
E0
E2
0
A
C
B
48
Giffen Good
A
E1
C
E0
E2
I0
0
A
C
B
49
Giffen Good
• Think of a product which you buy a lot of (a staple diet)
and which is of inferior quality.
• If the price of this product rises, your initial reaction is to
substitute this product with some other.
• But other products are much more expensive and your
staple diet is still the cheapest.
• So the substitution effect is minimal.
50
Giffen Good
• However, your purchasing power is less than before
because of the price increase.
• So you buy less of other goods and more of the product
which, although now more expensive, is still the
cheapest and a staple food!
• Such products are known as Giffen goods.
51
Hicks Substitution Effect
• When we decomposed the change in demand resulting
from a change in price into an income and substitution
effect, we did so by varying money income.
• When the price of good 𝑥1 fell, we ‘varied’ the
consumer’s money income to hold his real income
constant.
• Real income is defined as the consumer’s ability to enjoy
a particular level of utility.
52
Hicks Substitution Effect
• Varying money income to keep the utility constant is
known as Hicks Compensating Variation in money
income (HCV).
• HCV allows consumer to enjoy original level of utility at
the new relative price ratio.
• We ‘compensate’ the consumer for the change in price.
• Somehow odd for a price fall.
53
HSE – Price Fall
A
C
B
I1
I0
0
54
HSE – Price Rise
B
C
A
I1
I0
0
55
Slutsky Substitution Effect
• An alternative definition of real income is the ability to
consumer not a particular level of utility, but a particular
bundle of goods.
• We vary the consumer’s money income following a
change in price to permit him to consumer his original
bundle of goods at the new relative price ratio.
•
The is know as the Slutsky Compensating Variation
(SCV) in money income.
56
SSE – Price Fall
A
C
I0
B
I1
I2
0
57
SSE – Price Rise
B
C
I2
A
I0
I1
0
58
Slutsky Substitution Effect
• The Slutsky Substitution effect is also negative:
– From the graph in slide 60: all the points on the dotted line
where the amount of good 1 consumed is less than at
bundle A where affordable before the price change.
– A must be preferred to all bundles in the part of the dotted
line that lies inside the original budget set.
– The optimal choice of must be either A or some point of the
right of A.
– Price and consumption go in opposite directions.
59
Slutsky Substitution Effect
• It should be clear from the above analysis that
the Slutsky method over-compensates.
• The consumer would be on a higher indifference
curve than the original one were the
compensation made following the price increase.
60
Slutsky vs Hicks
• Hicks Substitution Effect: we need to know the exact
position of the indifference curves in order to learn
about the various effects of a price change.
– Conceptually correct, but not very applicable.
• Slutsky Substitution Effect: we do not need to know
anything about consumer preferences to implement it
and so Slutsky’s method can be implemented.
– Applicable, but over-compensates.
• For small change in prices the two substitution effects
are virtually identical.
61
Example 1
•
Susan’s preferences over pizza (x) and salad (y) are
given by the utility function 𝑈 𝑥, 𝑦 = 𝑥𝑦.
• Price of 𝑥, 𝑝𝑥 , is £4 per unit and that of 𝑦, 𝑝𝑦 , is £1 per
unit.
• Susan had a budget, m, of £120.
62
Example 1
• Compute:
i.
Susan’s optimal Choice.
ii. Calculate the income and substitution effect of a
decrease in 𝑝𝑥 to £3 per unit.
63
Example 1
i.
Susan’s optimal basket.
• We have worked with this function before. The
preferences are convex, and we can just check the
tangency condition:
• 𝑀𝑅𝑆 =
𝜕𝑈/𝜕𝑥
𝜕𝑈/𝜕𝑦
• 𝑀𝑅𝑆 =
𝑦
𝑥
=
=
𝑝𝑥
𝑝𝑦
4
1
64
Example 1
• From the MRS above:
• 𝑦 = 4𝑥
• Budget line is 4𝑥 + 𝑦 = 120
• Solving for 𝑦 and 𝑥:
• 𝑥 ∗ = 15 and 𝑦 ∗ = 60 (basket A)
• U(15,60) = 900
65
Example 1
y
Original Bundle A
60
O
A
15
x
66
Example 1
ii.
Income and substitution effect of a decrease in 𝑝𝑥 to £3 per
unit.
• The decomposition basket has to satisfy the new tangency
condition:
• 𝑀𝑅𝑆 =
𝑦
𝑥
=
3
1
• And give the same level of utility:
• 𝑈 = 900 => 𝑦𝑥 = 900
• Solving: 𝑥′ = 17.3 and 𝑦′ = 51.9
67
Example 1
Original bundle: A
y
Decomposition bundle: B
60
51.9
Substitution Effect: Movement A to B
X changes from 15 to 17.3
Y changes from 60 to 51.9
A
B
m’ = 17.3*3 + 51.9*1 = 103.8
The consumer is “compensated”
in 103.9 - 120= - 16.2
O
15 17.3
x
68
Example 1
• Calculate the final basket. Tangency Condition:
• 𝑀𝑅𝑆 =
𝑦
𝑥
=
3
1
• And budget constraint:
• 3𝑥 + 𝑦 = 120
• Solving the two equalities above to the get the final
basket: 𝑥′′ = 20 and 𝑦′′ = 60
69
Example 1
Original bundle: A
Decomposition bundle: B
Final bundle: C
y
60
51.9
O
C
A
B
15 17.3 20
Income Effect:
Movement B to C
X changes 17.3 to 20
Y changes from 51.9 to 60
x
70
Example 2
• Supply of Labour:
• Leisure is a good; that is, more of it is better than less.
• This means that work is bad, i.e., less work is better
than more.
• The consumer earns the wage rate w per hour of
work, so that the ‘price’ of an hour of leisure is w.
• The consumption C is ‘good’, that is, more is better.
71
Example 2
• Budget Constraint:
• Let’s assume the consumer has some income M
(non labour income).
• The consumer consumption:
• C = M + wL
• Where L is the amount of labour supplied.
72
Example 2
• The budget constraint can be written as follow:
– M = C - wL
• Negative price as we have seem before.
• Let’s assume that there is a maximum amount of labour
that can be supplied 𝐿ത .
• Adding w𝐿ത in each side of the budget constraint:
– M + w𝐿ത = C + w(𝐿ത − 𝐿)
73
Example 2
• Let R denote leisure:
• R = 𝐿ത − 𝐿
• The total amount of time available for leisure: 𝑅ത = 𝐿ത
• The amount of consumption the consumer can enjoy
ҧ
without working: 𝐶=M
• The budget constraint becomes:
• 𝐶ҧ + w𝑅ത = C + wR
74
Example 2
• We can now write The consumption as a function of the
hours of leisure:
• C = 𝐶ҧ + w𝑅ത - wR
• If the consume decides not work:
• C = 𝐶ҧ
• If he decides not have any leisure:
• C = 𝐶ҧ + w𝑅ത
75
Example 2
C
The slope of the budget line
is –w.
It represents the
opportunity cost (the price
of one extra hour of leisure)
𝐶ҧ
P
O
𝑅ത
R
76
Example 2
C
As the wage rate increases,
the Budget line pivots
upwards around P.
𝐶ҧ
P
O
𝑅ത
R
77
Example 2
C
As the wage rate increases,
the supply of labour
ത
ത
increases from (𝑅-R’)
to (𝑅-R’’)
P
O
R’’ R’
𝑅ത
R
78
Example 2
• In the figure above we can see the effect of a
wage increase on the supply of labour.
• We can decompose the change in price in
substitution and income effects.
• The analysis assumes the plausible, that leisure
is a normal good.
79
Example 2
C
As the wage rate increases,
the supply of labour
ത
ത
increases from (𝑅-R’)
to (𝑅-R’’)
P
O
R’’’ R’’ R’
𝑅ത
R
80
Example 2
• The income effect and the substitution effect
work in opposite directions, as long as it is
assumed that leisure is a normal good.
• In the above case, the income effect partly
neutralises the substitution effect, but not fully,
so that as W goes up, leisure decreases.
81
Example 2
• The above analysis suggests that as wages go up,
labour supply increases; the supply curve of
labour is upward sloping.
• But what if the income effect outweighed the
substitution effect?
82
C
Example 2
P
O
R’ R’’
𝑅ത
R
83
Example 2
• Empirical evidence corroborates with the theory.
• At a lower wage rate, the substitution effect is
stronger; if the rate increases, supply of labour
increases as well.
• At a higher level of wages, however, a further
wage increase may well outweigh the
substitution effect and the labour supply fall.
84
Example 2
Wage
Rate
O
The backwardbending supply
of labour
Labour Supply
85
Example 2
• We can introduce the discussion of over-time
payments here.
• In the next fig., the initial equilibrium is at point
1 where the wage line is flatter.
• If the individual is offered a higher over-time
pay, she moves to a new equilibrium (point 2) as
the wage line gets steeper and on a higher
indifference curve.
86
Example 2
C
2
A possible effect of over-time
on choice of leisure
1
P
R
87
Intertemporal Choice
88
Intertemporal Choice
• We extend consumer analysis.
• We consider now the possibility of saving and
borrowing.
• Saving for tomorrow is a fact of life.
• Equally, we often spend more than what we
currently earn or have by borrowing.
89
Model
• We shall consider a two-period model of consumer
choice.
• The amount of consumption in each period is
denoted by (𝑐1 , 𝑐2 ).
• Prices of consumption in each period are constant at
1.
• In each period the consumer receives an amount of
money denoted by (𝑚1 , 𝑚2 ).
90
No Borrowing
𝑐2
Budget Constraint
when interest rate is
zero and no borrowing
is allowed.
𝑚2
𝑚1
𝑐1
91
Borrowing
• Consumer can lend money at some interest rate, r.
• Suppose the consumer is a saver.
• She will earn interest on 𝑚1 − 𝑐1
• The amount she can consume next period:
• 𝑐2 = 𝑚2 + (1 + 𝑟)(𝑚1 − 𝑐1 )
92
Borrowing
• Rearranging:
• (1 + 𝑟)𝑐1 + 𝑐2 = (1 + 𝑟)𝑚1 + 𝑚2
• Future value expression
• Alternatively:
• 𝑐1 +
𝑐2
1+𝑟
= 𝑚1 +
𝑚2
1+𝑟
• Present value expression
93
Budget Constraint
𝐶2
(1 + 𝑟)𝑚1 +𝑚2
The slope of the budget
line =-(1+r).
The negative sign
represents the trade-off
𝑚2
𝑚1
𝑚2
𝑚1 +
1+𝑟
𝐶1
94
Preferences
• The consumer has preferences over today and
tomorrow consumption.
• We consider the case of well behaved
preferences.
• Convexity is a natural assumption.
95
Preferences
• Given the preferences we can examine the
optimal choice (𝑐1 , 𝑐2 ).
• If the consumer chooses a point where 𝑐1 < 𝑚1 ,
he is a lender.
• If 𝑐1 > 𝑚1 the consumer is a borrower.
96
Lending and Borrowing
𝐶2
“Neither a borrower, nor a
lender be; For loan oft loses
both itself and friend, And
borrowing dulls the edge of
husbandry.”
(1 + 𝑟)𝑚1 +𝑚2
𝑚2
𝑚1
𝑚2
𝑚1 +
1+𝑟
𝐶1
97
Lender
𝐶2
(1 + 𝑟)𝑚1 +𝑚2
𝑐2
𝑚2
𝑐1 𝑚1
𝑚2
𝑚1 +
1+𝑟
𝐶1
98
Borrower
𝐶2
(1 + 𝑟)𝑚1 +𝑚2
𝑚2
𝑐2
𝑚1 𝑐1
𝑚2
𝑚1 +
1+𝑟
𝐶1
99
Comparative Statics
• We consider changes in the interest rate r.
• An increase in the interest rate makes the curve
steeper.
• The opposite happens with decrease in the
interest rate.
• The effects on the consumer will be different if
he is a lender or a borrower.
100
𝐶2
Comparative Statics
(1 + 𝑟)𝑚1
+𝑚2
A fall in the rate of interest is
a gain for the borrower
but a loss to the lender
(1 + 𝑟′)𝑚1
+𝑚2
𝑚2
𝑚1
𝑚1 +
𝑚2
1+𝑟
𝑚2
𝑚1 +
1 + 𝑟′
𝐶1
101
Increase in the Interest Rate
An increase in the
interest rate reduces
the welfare of the
borrower
2
1
102
Increase in the Interest Rate
An increase in the interest rate
increases the welfare of the lender
2
1
In this case, as the
interest rate goes up,
C1 falls
103
Increase in the Interest Rate
2
1
In this case, as the
interest rate goes up,
so does C1.
104
Summarizing
• Increase in the interest rate:
• A lender will remain a lender (he will be better off for
sure).
• A borrower may continue a borrower (he will be worse off
for sure) or may become a lender (the welfare result is
unknown).
• Decrease in the interest rate:
• A lender may continue a lender (he will be worse off for
sure) or may become a borrower (the welfare result is
unknown).
• A borrower will remain a borrower (he will be better off
for sure).
105
Example
• Susan lives for two periods, 1 and 2. She receives a
cash amount of £200 in period 1 and £300 in period
2.
• Her preferences over consuming today (c1) and
consuming tomorrow (c2) are given by the utility
function U(c1,c2 ) = 2c1c2.
• Assume that price of c1 and that of c2 is £1 per unit
and the interest rate is fixed at r =0.25
106
Example
i.
If Susan consumes c1 = 80, her saving is?
• Answer: Saving = 200 -80*1 = 120
ii. If Susan consumes c1 in general, her saving is?
• Answer: Saving = (200 - c1 )
iii. If Susan lends (200 - c1 ) at r = 0.25 her money in period
2 is?
• Answer: Period 2 money= (200 - c1)*(1+0.25) + 300
= 550 – 1.25c1
107
Example
iv. How many units of C2 can Susan buy with this
money?
• Answer: c2 = 550 – 1.25c1
v. So the equation of Susan’s budget constraint is?
• Answer: c2 + 1.25c1 = 550
vi. The ‘price’ of c1 per unit is?
• Answer: 1.25
108
Example
• Problem:
• max 𝑈(𝑐1 , 𝑐2 ) = 2𝑐1 𝑐2
(𝑐1 ,𝑐2 )
• Using a maximization technique (Lagrange
multiplier or tangency condition):
• c1* = 550/(2*1.25) = 220
• c2* = 550/(2*1) = 275
109
Inflation
• Now, price of c1 is different from the price of c2
• For convenience let’s assume price of c1 is £1 and price of
c2 is p2 (price of consumption tomorrow).
• The amount of money the consumer can spend in period
2:
• 𝑝2 𝑐2 = 𝑝2 𝑚2 + (1 + 𝑟)(𝑚1 − 𝑐1 )
• The amount of consumption available in period 2:
• 𝑐2 =
1+𝑟
𝑚2 +
(𝑚1
𝑝2
− 𝑐1 )
110
Inflation
• The inflation rate, π, is rate at which prices grow.
• Since 𝑝1 = 1:
• 𝑝2 = 1 + π
• The amount of consumption available in period 2
becomes:
• 𝑐2 =
1+𝑟
𝑚2 +
(𝑚1
1+π
− 𝑐1 )
111
Real Interest Rate
• Let’s create a new variable ρ, the real interest rate:
• 1+ρ=
1+𝑟
1+π
• The budget constraint becomes:
• 𝑐2 = 𝑚2 + (1 + ρ)(𝑚1 − 𝑐1 )
• One plus the real interest rate measures how much extra
consumption the consumer can get in period 2 by given
up one unit of consumption in period 1.
112
Real Interest Rate
• The interest rate on dollars is called nominal rate of
interest.
• The relationship between the real and the nominal
interest rates are given by:
• 1+ρ=
1+𝑟
1+π
• We can rearrange to get an explicitly expression for
ρ:
• ρ=
1+𝑟
1+π
−1=
1+𝑟
1+π
1+π
−
1+π
=
𝑟−π
1+π
113
Real Interest Rate
• If the inflation is small, we can use the following
approximation:
• ρ≈𝑟−π
• The real interest rate is nominal rate minus the
inflation.
• In practise, we know the nominal interest rate
for the next period but not the inflation.
114
Several Periods
• Three period model.
• The consumer can borrow or lend at an interest rate r.
• The price of consumption in period 2 in terms of
consumption in period 1 is 1/(1+r).
• For period 3:
• If the consumer invests £1 in period 1, it will grow to
(1+r) in period 2 and to (1+r)2 in period 3.
• If he has 1/ (1+r)2 in period 1, it will be £1 in period 3.
115
Several Periods
• The budge constraint is given by:
• 𝑐1 +
𝑐2
1+𝑟
+
𝑐3
(1+𝑟)2
= 𝑚1 +
𝑚2
1+𝑟
𝑚3
+
(1+𝑟)2
• The price of consumption in period t in terms of
today’s consumption is:
• 𝑝𝑡 =
1
(1+𝑟)𝑡−1
116
Several Periods
• The interest rate may change over time:
• The model before can be generalized:
•
𝑐2
𝑐3
𝑐1 +
+
1+𝑟
(1+𝑟1 )(1+𝑟2 )
𝑚3 1
(1+𝑟1 )(1+𝑟2 )
=
𝑚2
𝑚1 +
1+𝑟1
+
• We will examine only situations with constant
interest rate.
117
Several Periods
• How the present value changes over time.
• The present value of £1 T years in the future:
Rate
1
2
5
10
15
20
25
30
.05
.95
.91
.78
.61
.48
.37
.30
.23
.10
.91
.83
.62
.39
.24
.15
.09
.06
.15
.87
.76
.50
.25
.12
.06
.03
.02
.20
.83
.69
.40
.16
.06
.03
.01
.00
118
Bonds
• Bonds are used by governments and
corporations to borrow money.
• The agent who issues the bond, promises to pay
a fixed amount of dollars x (the coupon) each
period until a certain date T (the maturity date),
at which the which the borrower will pay the
amount F (the face value).
119
Bonds
• The stream of payments is (x,x,x,…,F)
• If the interest rate is constant, the present value of the
bond is:
• 𝑃𝑉 =
𝑥
𝑥
+
(1+𝑟)
(1+𝑟)2
+ ⋯+
𝐹
(1+𝑟)𝑇
• The present value goes down if the interest rate
increases:
• When the interest rate goes up the price now for £1
delivered in the future goes down.
120
Bonds
• Perpetuity is a type of bond that makes
payments forever.
• Consider a perpetuity that promises to pay £x a
year forever.
• The present value is:
• 𝑃𝑉 =
𝑥
(1+𝑟)
𝑥
+
(1+𝑟)2
+⋯
121
Bonds
• Factor 1/(1+r) out:
• 𝑃𝑉 =
1
1+𝑟
𝑥+
𝑥
(1+𝑟)
+
𝑥
(1+𝑟)2
+⋯
• This is x plus the present value:
• 𝑃𝑉 =
• 𝑃𝑉=
𝑥
𝑟
1
1+𝑟
𝑥 + 𝑃𝑉
122