4-households

```Chapter 4 – Households
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Consumption
The objective in this chapter is to apply what we have learned
about feasible sets and preferences to more general settings.
Consider Marı́a, whose income is equal to 56. Everything is spent
on clothing and food.
I The price of food is 1.75
If she spends all on food she can afford 56/1.75 = 32units
I The price of clothing is 1.12
If she spends all on clothing she can afford 56/1.12 = 50units
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4.1. CONSUMPTION
food
I/pf = 32 •
B
pc /pf
•
C
A
•
I/pc = 50
clothing
FIGURE 4.1
Maria’s budget set and budget line
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Denote by I Marı́a’s income and
I pf = price of food, pc = price of clothing
I F the quantity of food Marı́a buys, C the quantity of clothing
Then, if (F , C ) is affordable we have
pf &times; F + pc &times; C ≤ I
(1)
If Marı́a spends all her income on food and clothing then we have
pf &times; F + pc &times; C = I
(2)
Marı́a’s feasible set is given by all the combinations (C , F ) that
satisfy Eq. (1).
The frontier of the feasible set is a straight line given by Eq. (2).
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When talking about a person or a household’s feasible set with
respect to consumption we use the term budget set (instead of
feasible set).
The boundary of the budget set —Eq. (2)— is the budget line.
We can rewrite the equation of the budget line as the equation of
a line in the (C , F ) space,
f =
pc
I
− c
pf
pf
(3)
The absolute value of the slope of this line is
MRT =
pc
pf
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In Chapter 3 we have seen that the optimal choice is such that
MRT = MRS
So, in the case of Marı́a, we have when her choice is optimal
MRS =
pc
pf
In English: Marı́a’s optimal consumption level is such that
How much food she’s willing to give
up for an extra unit of clothing
=
ration of price(pc /pf )
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Changes in income
If we increase an individual’s income the budget line will shift
upward.
But what about the optimal consumption?
I Consumption of both goods increases?
I Consumption of one goods increases, and decrease for the
other good?
That depends. . .
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I If the consumption of a good increases when income
increases, then the good is a normal good.
Examples: Uber, leisure, housing
I If the consumption of a good decreases when income
increases, then the good is an inferior good.
Examples: public transportation, junk food.
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4.1. CONSUMPTION
y
I2
py
b2
I1
py
b1
Case when x
is a normal good
U2
U1
y2
y1
•
C1
x1
y
U1
U2
•
C2
x
x2
I1
px
I2
px
Case when x
is an inferior good
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x
x1
y
U1
I2
py
y2
I1
py
x2
I1
px
I2
px
Case when x
is an inferior good
U2
b2
b1
y1
•
C2
•
C1
x2 x1
x
I1
px
I2
px
FIGURE 4.2
Effect of an increase in income
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Income and substitution effects of a price change
Consider a consumer with income I and choosing how much to
consume of good x and of good y .
I px = price of x
I py = price of y .
So the budget line intersects
I The x axis at
I
(she spends all of her income on x).
px
I The y axis at
I
py
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Suppose now the price of x changes from px to px0 (with px0 &gt; px ).
Hence, the budget line moves and now intersects
I The x axis at
I
(a lower point than before)
px0
I The y axis at
I
py
So the optimal choice of the consumer changes:
We see that the consumption of x drops (not surprising, x is now
more expensive).
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y
UC
I
py
U
A
bA
bC
yA
yC
•
C
•
A
x
xC
I
p0x
xA
I
px
Figure 1: Income eﬀect (arrow I) and substituti
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The change from A to C can be decomposed in two parts:
I A substitution effect;
I An income effect.
Intuition
I px increases. So the real income decreases:
less of x, so the real income decreases.
I If px increases, then the relative price of x (relative to y )
increases:
For 1 additional unit of x we need to sacrifice more units of y
than under px .
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I Substitution effect
The change of consumption due to x being relatively more
expensive than y .
A change in x cause by a change in relative prices, keeping
real income constant
I Income effect
The change due to the fact that the real income is lower.
A change in x cause by the change in real income (i.e.,
keeping relative prices constant).
15/35
y
I+ I
py
UC
U
A
bB
I
py
bA
bC
•
yB
yA
yC
•
C
B
I
•
A
S
x
xC
xB
I
p0x
xA
I
px
Figure 1: Income eﬀect (arrow I) and substitutio
16/35
Sign of income/substitution effect
I If x is a normal good, the income effect has the opposite sign
of price change:
Price increases, income effect is negative (consumption of x
decreases).
I If x is an inferior good, the income effect has the same sign of
price change:
I The substitution effect has always the opposite sign of price
change: this is the law of demand.
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Changes in income and prices
We are now interested in two comparative statics problems:
I The impact of changes in (non-labor) income
I The impact of changes in the wage rate.
One could think that the optimal choice about income–leisure is
similar to the optimal choice about consumption (e.g., food v.
clothing).
It’s more complicated than that. . .
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Income–leisure is
I Similar to food–clothing:
The more of, say food, the less of clothing (and vice-versa).
I Different from food:
There are two sources of income:
I Labor income (so income–leisure is similar to food–clothing)
I Non-labor income: a change of income does not come from
less leisure.
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consumption
I + 24 w
24 w
•
I
•
c2
c1
leisure
24
Figure 1: Eﬀect of additional income
20/35
In the previous graph, an increase of (non-labor) income induced
an increase of leisure
⇒ leisure is a normal good
Things are different when the increase of income is from labor, i.e.,
the wage increases.
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The income effect is smaller than the substitution effect
consumption
24 w2
24 w1
•
c2
•
c1
leisure
24
Figure 1: Eﬀect of additional income
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The income effect is greater than the substitution effect:
The price of leisure (= the wage) increases, the chosen amount of
leisure increases.
consumption
24 w
•
c1
•
c2
leisure
24
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Labor supply
The demand for leisure can be reinterpreted as labor supply.
The supply of labor may react positively or negatively to a change
in the wage rate depending on the relative magnitudes of the
substitution and income effects.
24/35
Savings
Saving decisions can be thought as a choice between consumption
today and consumption in the future.
Like leisure–grade or food-clothing or leisure–labor we can analyze
the tradeoff between consumption today and consumption in the
figure.
The new concept we need is the interest rate, ra .
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Let us simplify the problem (a model!):
I There are two periods:
I First period: working
I Second period: retirement
I The income in while working is I
I No (labor) income while retired.
If x is the amount saved in period 1, then in period 2 that amount
is equal to x(1 + ra ).
What if the interest rate changes from ra to rb , with ra &gt; rb ?
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The income effect is smaller than the substitution effect
consumption during retirement (f )
I (1 + ra )
ba
fa
I (1 + rb )
•
A
bb
fb
•
B
savings
ca cb
consumption tod
I
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The income effect is greater than the substitution effect:
The price of today’s consumption (= the wage) increases, the
chosen amount of leisure increases.
consumption during retirement
I (1 + ra )
ba
fa
I (1 + rb )
fb
•
bb
A
B
•
savings
cb
ca
consumption to
I
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Insurance
Uncertainty is capture by assuming that the future is made up of
multiple possible states of the world.
For simplicity, let us assume that there are two states:
I h, when the consumer is healthy;
The consumer’s income is h1
I s, when the consumer is sick. The consumer’s income is s1
(= h1 − medical expenses).
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The framework we used can be applied to insurance.
Instead of choosing the amount of consumption (e.g., food &amp;
clothing) or labor supply (labor &amp; leisure) the consumer choose a
bundle consisting of
I income when not sick (state h)
I income when sick (state s)
This is done by choosing the level of insurance
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The idea of the insurance is very simple:
I The agent pays (today) a premium p
I If the agent is sick then she gets a payment q from the
insurance company.
If the initial situation is (h1 , s1 ) then when contracting the
insurance the agent’s income is
I h − p in stage h
I s + q in stage s
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When the indifference curves are about two states of the world
(the consumer does not know which will prevail) the shape of
indifference curves capture the agent’s risk aversion.
If the agent is risk neutral, the indifference curves are straight
lines.
All the points on the same indifference curve yield the same
expected utility
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Suppose that the consumer has a probability ρ to be sick.
Then the expected income is
ρh + (1 − ρ)s
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The insurance company proposes a set of contracts, which
amounts to a set of combinations of income for states h and s.
To each combination of incomes (one for each state) corresponds
implicitly to a premium and a coverage q.
This set of contracts acts like a budget constraints: the agent will
maximize her preferences given the constraint offered by the
insurance company.
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sick
45
E2
•
s2
s3
s4
• E4
•
E3
U2
U3
U1
•
s1
E1
healthy
h4
h2 h3
h1
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