I. Consumer
Theory
Applications
Intermediate Microeconomics (22014)
I. Consumer Theory Applications
Instructor: Marc Teignier-Baqué
First Semester, 2011
I. Consumer
Outline Part I. Consumer Theory Applications
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Topic 2.
Intertemporal
Choice
Topic 3.
Uncertainty
1. Topic 0. Consumer Theory Review
1.1
1.2
1.3
1.4
1.5
Budget Constraints
Preferences
Utility Function
Choice
Slutsky Equation
2. Topic 1. Buying and Selling
3. Topic 2. Intertemporal Choice
4. Topic 3. Choice under Uncertainty
I. Consumer
Theory
Applications
Topic 0. Cons
Review
Budget
Constraints
Preferences
Utility Function
Choice
Slutsky Equation
Topic 1. Buying
and Selling
Topic 2.
Intertemporal
Choice
Topic 3.
TOPIC 0. CONSUMER THEORY REVIEW
Budget Constraints
Denitions
The consumer's
budget set
is the set of all aordable
bundles,
B (p1 , .., pn ; m) =
{(x1 , . . . , xn ) : x1 ≥ 0, . . . , xn ≥ 0 and p1 x1 + . . . + pn xn ≤ m} .
The
budget constraint
is the upper boundary of the budget
set.
Uncertainty
x2
m /p2
Budget constraint: p1x1 + p2x2 = m x2 
m
p
 1 x1
p2
p2
Budget set: the collection of all Budget
set: the collection of all
affordable bundles.
Budget
d
Set
m /p1
x1
I. Consumer
Theory
Applications
Topic 0. Cons
Review
Budget
Constraints
Preferences
Utility Function
Choice
Slutsky Equation
Topic 1. Buying
and Selling
Topic 2.
Intertemporal
Choice
Preferences
Denitions
The set of all bundles equally preferred to a bundle x' is the
indierence curve
containing x'. The slope of the
indierence curve at x' is the
marginal rate of substitution
(MRS) at x', which is the rate at which the consumer is only
just willing to exchange commodity 2 for commodity 1':
4x2
MRS (x ) = lim
4x →0 4x1
0
=
1
Topic 3.
x’’
x’  x”  x”’
z

x2
x’

Uncertainty
y
x’’
 x2
z
 x1
x”’
y
x1
dx2
dx1
I. Consumer
Theory
Applications
Topic 0. Cons
Review
Budget
Constraints
Preferences
Utility Function
Choice
Slutsky Equation
Topic 1. Buying
and Selling
Utility Function
Denition
A Utility function U (x ) represents a preference relation if
and only if
x ' x ↔ U (x ') > U (x )
x ' ≺ x ↔ U (x ') < U (x )
x ' ∼ x ↔ U ( x ') = U ( x )
Topic 2.
Intertemporal
Choice
Topic 3.
Uncertainty
I The general equation for an indierence curve is
U (x1 , x2 ) = k , where k
is a constant. Totally
dierentiating this identity we obtain that the MRS is
equal to the ratio of marginal utilities:
∂U
∂U
∂U
dx
dx1 +
dx2 = 0 ⇔ 2 = − ∂∂xU
∂ x1
∂ x2
dx1
∂x
1
2
I. Consumer
Theory
Applications
Topic 0. Cons
Review
Budget
Constraints
Preferences
Utility Function
Choice
Slutsky Equation
Topic 1. Buying
and Selling
Topic 2.
Intertemporal
Choice
Topic 3.
Uncertainty
Choice
Denition
A decisionmaker chooses the most preferred aordable
bundle, which is called the consumer's
ordinary demand
or
gross demand.
I The slope of the indierence curve at ordinary demand
(x ∗ , x ∗ ) equals the slope of the budget constraint:
1
2
MRS (x1∗ , x2∗ ) = −
p1
∂ U /∂ x1 p1
⇔
=
p2
∂ U /∂ x2 p2
x2
More preferred
bundles
x2 *
Affordable
bundles
x1 *
x1
I. Consumer
Slutsky Equation
Theory
Applications
Topic 0. Cons
price change are always the sum
pure substitution eect and an income eect .
Changes to demand from a
of a
Review
Budget
Constraints
Preferences
Utility Function
Choice
Slutsky Equation
I
Pure substitution eect :
change in demand due only
to the change in relative prices. What is the change in
Topic 1. Buying
and Selling
demand when the consumer's income is adjusted so
Topic 2.
that, at the new prices, she can only just buy the
Intertemporal
Choice
original bundle?
Topic 3.
Uncertainty
I
Income eect:
if, at the new prices, less income is
needed to buy the original bundle then real income is
increased; if more income is needed, then real income
is decreased.
I. Consumer
Slutsky Equation graphically
Theory
Applications
x2
Topic 0. Cons
Pure Substitution Effect Only
Review
Budget
Constraints
Preferences
Utility Function
Choice
Slutsky Equation
x2 ’
x2’’
Topic 1. Buying
and Selling
x1 ’
x1’’
x1
Topic 2.
Intertemporal
Choice
x2
Topic 3.
Uncertainty
Adding now the income effect
x2 ’
x2’’
x1 ’
x1’’
x1
I. Consumer
Theory
Applications
Topic 0. Cons
Review
Budget
Constraints
Preferences
Utility Function
Choice
Slutsky Equation
Slutsky Equation formally
(p1 , p2 ) be the initial price vector, m the
∗ ∗
level and (x1 , x2 ) the initial gross demand.
I Let
I Dene the Sltusky demand function
of good
i
adjusted to give the consumer just enough to consume
the initial bundle
(x1∗ , x2∗ ):
Choice
xis p10 , p20 , x1∗ , x2 ≡ xi p10 , p20 ; p10 x1∗ + p20 x2∗ 
∗
|
Topic 3.
Uncertainty


and Selling
Intertemporal
as the demand
after the price change when income is
Topic 1. Buying
Topic 2.
x1s
income
I Take the derivative with respect to
∂ xs
1
∂ p1
∂ x1
=
∂ p1
=
p1
{z
m
}
on both sides,
∂ x1 ∂ x1 ∗
+
x
∂ p1 ∂ m 1
⇔
∂ x1s
∂ p1
|{z}
∂ x1 ∗
−
x
m 1}
| ∂{z
substitution eect income eect
I. Consumer
Slutsky Equation formally
Theory
Applications
Topic 0. Cons
Review
Budget
Constraints
Preferences
Utility Function
Choice
Slutsky Equation
Topic 1. Buying
and Selling
Topic 2.
Intertemporal
∂ x1s
I The sign of the susbstitution eect
∂ p1 is negative: the
cange in demand due to the susbstitution eect is the
opposite to the change in price
p1 ↓⇒ x1s ↑).
(p1 ↑⇒ x1s ↓ and
I If the good is normal, the sign of the income eect is
also negative: an increase in a price is like a decrease in
Choice
income, which leads to a decrease in demand; a price fall
Topic 3.
is like an income increase, which leads to an increase in
Uncertainty
demand.
∂ x s ∂ x1 ∗
∂ x1
= 1−
x1
∂ p1 ∂ p1 |∂ m
{z }
|{z}
(+)
(−) | {z
}
(−)
I. Consumer
Outline Part I. Consumer Theory Applications
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply
Topic 2.
Intertemporal
Choice
Topic 3.
Uncertainty
1. Topic 0. Consumer Theory Review
2. Topic 1. Buying and Selling
2.1
2.2
2.3
2.4
Endowments
Net Demand
Slutsky Equation
Labor Supply
3. Topic 2. Intertemporal Choice
4. Topic 3. Choice under Uncertainty
I. Consumer
TOPIC 1. BUYING AND SELLING
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply
I So far, consumers' income taken as exogenous and
independent of prices. In reality, consumers' income
coming from exchange by sellers and buyers.
Topic 2.
Intertemporal
Choice
Topic 3.
Uncertainty
I How are incomes generated? How does the value of
income depend upon commodity prices?
I How can we put all this together to explain better how
price changes aect demands?
I. Consumer
Endowments
Theory
Applications
I In this chapter, consumers get income from endowments.
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply
Topic 2.
Intertemporal
Choice
Topic 3.
Uncertainty
This makes the budget set denition change slightly.
Denition
The list of resource units with which a consumer starts is her
endowment ,
denoted by
ω = (ω1 , ω2 ).
Denitions
Given p1 and p2 , the budget constraint
with an endowment
ω = (ω1 , ω2 )
for a consumer
is
p1 x1 + p2 x2 = p1 ω1 + p2 ω2 .
and the
budget set
is formally dened as
{(x1 , x2 ) | x1 ≥ 0, x2 ≥ 0 and p1 x1 + p2 x2 ≤ p1 ω1 + p2 ω2 } .
I. Consumer
Theory
Applications
Endowments and Budget Sets
I Graphically, the endowment point is always on the
budget constraint.
Topic 0. Cons
Review
Topic 1. Buying
I Hence, price changes pivot the constraint around the
endowment point.
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply
x2
Topic 2.
Intertemporal
p 1 x 1  p 2 x 2  p 1 1  p 2  2
Choice
Topic 3.
Budget Set before price change
Uncertainty

Budget Set after the price change
p '1 x 1  p '2 x 2  p 1'  1  p '2 2

x1
I. Consumer
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply
Net demand
Denition
The dierence between nal consumption and initial
endowment of a given good
of good i .
i , xi∗ − ωi , is called net demand
I The sum of the values of net demands is zero:
p1 x1 + p2 x2 = p1 ω1 + p2 ω2
⇔ p1 (x1 − ω1 ) + p2 (x2 − ω2 ) = 0.
Topic 2.
Intertemporal
Choice
Topic 3.
x2
x2
Uncertainty
At prices (p1’,p2’) the consumer
sells units of good 2 to acquire
more of good 1 . The net demand of good 1 is, therefore, positive and the net demand of good 2 is negative.
d
d f
d i
i
At prices (p1,p2) the consumer sells units of good 1 to acquire more units of good 2. The net demand of good 1 is, therefore, negative,and the net demand of good 2 is positive.
d2i
ii
x2 *


x2 *
p1 ( x1   1 )  p 2 ( x 2   2 )  0
p'1x1  p'2x 2  p1'  1  p'2 2

x1 *
x1
x1* 
x1
I. Consumer
Theory
Applications
Topic 0. Cons
Review
Price oer curve
Denition
Price-oer curve
contains all the utility-maximizing gross
demands for which the endowment is exchanged.
Topic 1. Buying
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply
x2
Sell good 1, buy good 2
Sell good 1, buy good 2
Topic 2.
Intertemporal
Choice
Topic 3.
Uncertainty

Buy good 1, sell good 2

x1
I. Consumer
Theory
Applications
Slutsky equation revisited
In an endowment economy, the overall change in demand
a pure substitution
eect , an (ordinary) income eect , and an endowment
income eect.
I Pure Substitution Eect : eect of relative prices change.
I Income Eect : eect of original bundle cost change.
I Endowment Income Eect: change in demand due
caused by a price change is the sum of
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply
Topic 2.
Intertemporal
only to the change in endowment value.
Choice
Price change from (p1’,p2’) to (p1”, p2’):
Topic 3.
x2
Uncertainty
Pure substitution effect
Ordinary income effect
Endowment income effect
x2 ’
2
x2 ”
x1 ’
1
x1 ”
x1
I. Consumer
Theory
Applications
Slutsky Equation revisited
good 1 and
Topic 0. Cons
Review
x1 (p1 , p2 ; m (p1 , p2 )) be the demand function of
m (p1 , p2 ) = p1 ω1 + p2 ω2 the money income.
Then, the total derivative of x1 with respect to p1 is
I Let
I
Topic 1. Buying
dx1 ∂ x1 ∂ x1
=
+
ω .
dp1 ∂ p1 ∂ m 1
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply
Topic 2.
Intertemporal
Choice
Topic 3.
Uncertainty
I Using that ∂ x1
∂ p1
dx1
=
dp1
∂ x1s
= ∂ p − ∂∂ xm x1∗ ,
1
1
∂ x1s
∂ p1
|{z}
we obtain
∂ x1 ∗
−
x
m 1}
| ∂{z
∂ x1
+
ω .
m }1
| ∂{z
substitution ord.-income end.-income
I Rearranging,
dx1 ∂ x1
∂ x1
=
+ (ω1 − x1 )
.
dp1 ∂ p1
∂m
|{z}
|{z}
(−)
(+)
I. Consumer
Slutsky equation revisited
Theory
Applications
Overall change in demand of normal good (demand increases
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply
Topic 2.
Intertemporal
Choice
Topic 3.
with income) caused by own price change:
I When income is exogenous, both the substitution and
(ordinary) income eects increase demand after an
own-price fall; hence, a normal good's ordinary demand
curve slopes down (thus, Law of Downward-Sloping
Demand always applies to normal goods when income is
exogenous).
Uncertainty
I When income is given by initial endowments,
endowment-income eect decreases demand if consumer
supplies that good (negative net demand); thus, if the
endowment income eect osets the substitution and
the (ordinary) income eects, the demand function
could be upward-sloping!
I. Consumer
An application: labor supply
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply
Topic 2.
Intertemporal
Choice
Environment description:
I A worker is endowed with
and
R
m
euros of nonlabor income
hours of time.
I Consumption good's price is
pc , and the wage rate is w .
I Worker decides amount of consumption good, denoted
by
C , and amount of leisure, denoted by R .
Topic 3.
Uncertainty
Budget constraint:
pc C = m + w R − R
=
m
pc C + wR
| +{zwR}
⇔
|
{z
}
Expenditures value
Endowment value
I. Consumer
Labor supply choice
Theory
Applications
C
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply

Budget constraint equation:

w
m  w R
m  w R
pc
C


p c
R

p c
C*
Topic 2.
Endowment point
Intertemporal
Choice
Topic 3.
Uncertainty
m

R
R*
leisure
demanded
labor
supplied
R
I. Consumer
Theory
Applications
Labor supply curve
Eect of a wage rate increase on amount labor supplied:
I Substitution eect: leisure relatively more expensive
Topic 0. Cons
decrease leisure demanded / increase labor supplied.
Review
Topic 1. Buying
I (Ordinary) income eect: cost original bundle
and Selling
Endowments
Net Demand
Slutsky Equation
Labor Supply
increases→ decrease leisure demanded / increase labor
supplied.
I Endowment-income eect: positive endowment income
Topic 2.
eect because worker supplies labor
Intertemporal
Choice
Topic 3.
Uncertainty
→
→
demanded / increase labor supplied.
⇒
Labor supply curve may bend backwards.
decrease leisure
I. Consumer
Outline Part I. Consumer Theory Applications
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
1. Topic 0. Consumer Theory Review
and Selling
Topic 2.
2. Topic 1. Buying and Selling
Intertemporal
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
Topic 3.
Uncertainty
3. Topic 2. Intertemporal Choice
3.1
3.2
3.3
3.4
3.5
Present and Future Values
Intertemporal Budget Constraint
Intertemporal Choice
Ination
Valuing Securities
4. Topic 3. Choice under Uncertainty
I. Consumer
TOPIC 2. INTERTEMPORAL CHOICE
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
I So far, only static problems considered, as if consumers
only alive one period or only static decisions.
and Selling
Topic 2.
Intertemporal
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
Topic 3.
Uncertainty
I However, in the real world people often make
intertemporal consumption decisions:
I
I
Current consumption nanced by borrowing now against
income to be received in the future.
Extra income received now spread over the following
month (saving now for consumption later).
I In this section, we study intertemporal choice problem
using a two-period version of our consumer's choice
model.
I. Consumer
Intertemporal Choice Problem
Theory
Applications
Topic 0. Cons
Review
I Notation:
Topic 1. Buying
and Selling
I
Topic 2.
I
Intertemporal
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
I
I
Let interest rate be denoted by r .
Let c1 and c2 be consumptions in periods 1 and 2.
Let m1 and m2 be incomes received in periods 1 and 2. .
Let consumption prices be denoted by p1 and p2 .
I Intertemporal choice problem:
m1 and m2 , and given consumption
p2 , what is the most preferred
intertemporal consumption bundle (c1 , c2 )?
I Given incomes
Topic 3.
prices
Uncertainty
I
p1
and
Need to know: the intertemporal budget constraint, and
intertemporal consumption preferences.
I. Consumer
Theory
Applications
Topic 0. Cons
Review
Present and Future Values
Denitions
Given an interest rate
r , the future value
of
M ¿ is the
value next period of that amount saved now:
Topic 1. Buying
FV = M (1 + r ) .
and Selling
Topic 2.
Intertemporal
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
The
of M ¿ is the amount saved in the
M ¿ at the start of the next period:
present value
present to obtain
PV =
Topic 3.
M
1+
r
.
Uncertainty
I Example:
I
I
Example: if r = 0.1 the future value of 100¿ is
100(1 + 0.1) = 110¿.
if r=0.1, the present value of 1¿ is the amount we have
to pay now to obtain 1¿ next period: 1+10.1 = 0.91.
I. Consumer
Theory
Applications
Intertemporal Budget Constraint
Case I: No ination , p1 = p2
I Consumption bundle when neither saving nor borrowing:
Topic 0. Cons
(c1 , c2 ) = (m1 , m2 )
Review
Topic 1. Buying
and Selling
I If all period 1 income saved for period 2:
(c1 , c2 ) = (0, m2 + (1 + r ) m1 )
Topic 2.
Intertemporal
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
I If all period 2 income borrowed in period 1:
m2
,0
(c1 , c2 ) = m1 +
1+r
Topic 3.
c2
Uncertainty
(c1, c2)  0, m2 (1 r)m1
c1 , c2   m1 , m2 
m2
m


(c1 , c2 )   m1  2 ,0 
1 r 

0
0
m1
c1
I. Consumer
Theory
Applications
Intertemporal Budget Constraint
I Given a period 1 consumption of
c1 , period 2
consumption is
Topic 0. Cons
c2 = m2 + (1 + r ) m1 − (1 + r ) c1
Review
Topic 1. Buying
|
and Selling
}| {z }
{z
intercept
slope
Topic 2.
Intertemporal
c2
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
m2  1  r m1
m2
Topic 3.
Uncertainty
0
0
m1
m1 
m2
1 r
c1
I Intertemporal budget constraint:
I
I
Future-valued form: (1 + r ) c1 + c2 = m2 + (1 + r ) m1
Present-valued form: c1 + 1c+r = m1 + 1m+r
2
2
I. Consumer
Intertemporal Choice
Theory
Applications
Topic 0. Cons
I Optimal intertemporal consumption bundle given by
Review
tangency point of intertemporal indierence curves and
Topic 1. Buying
intertemporal budget constraint:
and Selling
Topic 2.
Intertemporal
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
c2
c2
The consumer borrows.
The consumer saves
The consumer saves.
c2
*
Topic 3.
Uncertainty
m2
m2
c2
0
0
*
c1
m1
c1
*
0
0
m1
*
c1
c1
I. Consumer
Comparative Statics: Slutsky equation re-revisited
Theory
Applications
Topic 0. Cons
I The Slutsky equation for the change in
change in
p1
c1
due to a
is the same as the one seen in topic 1:
∂ c1
dc1 ∂ c1s
=
+ (m1 − c1 )
.
dp1 ∂ p1
∂m
|{z}
Review
Topic 1. Buying
and Selling
Topic 2.
|{z}
(−)
Intertemporal
(+)
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
I Since a change in
r
is equivalent to a change in
p1 , the
Slutsky equation is exactly the same.
I
Topic 3.
Uncertainty
I
If r ↑, the substitution eect (the rst term in the
equation above) is negative; if r ↓ the substitution eect
is positive.
The sign of the total income eect (the second term in
the equation above) depends on whether the consumer
is a saver or a borrower:
I If borrower (c > m ), total income eect is negative.
1
1
I If saver, (c < m ), total income eect is positive.
1
1
I Note: eects of
r ↓ are the opposite as eects of r ↑.
I. Consumer
Theory
Applications
Topic 0. Cons
Comparative Statics: Interest rate decrease
I Graphically, since slope budget constraint curve is
− (1 + r ),
r ↓ ⇒ attening budget constraint.
Review
Topic 1. Buying
and Selling
c2
Topic 2.
Intertemporal
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
m2
0
Topic 3.
Uncertainty
I Eects of
0
m1
c1
r ↓ on optimal intertemporal consumption
bundle:
I
I
Substitution eect: increase in cost future consumption
relative to present consumption.
Total income eect:
I
I
If saver, total income eect is negative.
If borrower, total income eect is positive.
I. Consumer
Comparative Statics: Interest rate decrease
Theory
Applications
I Total eect:
Topic 0. Cons
Review
c1 ?, c2 ↓
If borrower, c1 ↑, c2 ?
If saver,
Topic 1. Buying
and Selling
Topic 2.
Intertemporal
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
c2
c2
The consumer borrows.
The consumer saves.
Topic 3.
Uncertainty
c2
c2
*
**
m2
m2
0
*
c2
**
c2
0
*
c1
**
c1
m1
c1
0
0
*
m1 c1
**
c1
c1
I. Consumer
Ination
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
Denitions
The ination rate
is the rate at which the level of prices for
goods increases. It is equal to
and Selling
Topic 2.
π=
Intertemporal
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
Topic 3.
Uncertainty
The
p2
p2 − p1
⇔ 1+π =
p1
p1
real-interest rate , ρ ,
is an interest rate adjusted to
remove the eects of ination. It is equal to
1+ρ
=
1+
r
1+π
⇔ρ=
r −π
1+π
and, if the ination rate is small, it can be approximated by
the dierence between the interest rate and the ination rate:
ρ ≈ r − π.
I. Consumer
Theory
Applications
Intertemporal Budget Constraint
Case II: Ination , p2 = (1 + π) p1
I Intertemporal budget constraint with ination:
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Topic 2.
Intertemporal
p1 c1 +
p2 c2
(1 + r )
⇔ c1 +
= p1 m1 +
c2
1+ρ
= m1 +
p2 m2
(1 + r )
m2
1+ρ
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
I Intertemporal budget constraint curve:
c2 = (1 + ρ) m1 + m2 − (1 + ρ)c1
|
{z
intercept
}|
{z
slope
Topic 3.
Uncertainty
c2
  same effects as r 
1 r
m1  m2
1 
m2
p2
0
0
m1
p1
m1 
m2
1 r
c1
}
I. Consumer
Valuing Financial Securities
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Denition
A nancial security
is a nancial instrument that promises
to deliver an income stream.
Topic 2.
Intertemporal
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
I Example:
I
Topic 3.
Uncertainty
I
Consider a security that pays m1 at the end of period 1,
m2 at the end of period 2, and m3 at the end of period 3.
What is the most that should be paid now for this
security? The present value of this security!
PV = (1m+1r ) + m2 2 + m3 3 .
(1 + r )
(1 + r )
I. Consumer
Valuing Bonds
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Denition
A bond is a type of security that pays a xed amount x for
T years (its maturity date) and then pays its face value F .
Topic 2.
Intertemporal
Choice
Present and
Future Values
Intertemporal
Constraint
Intertemporal
Choice
Ination
Valuing Securities
Denition
1
Year
Income paid
x
Present Value
(1+r )
2
x
x
...
x
(1+r )2
...
...
T −1
x
T
F
(1+r )T −1
(1+r )T
x
F
Topic 3.
Uncertainty
I The value of the bond is its present value:
PV =
x
(1 + r )
+
x
2
(1 + r )
+ ... +
x
F
T −1 +
(1 + r )
(1 + r )T
I. Consumer
Outline Part I. Consumer Theory Applications
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Topic 2.
Intertemporal
1. Topic 0. Consumer Theory Review
2. Topic 1. Buying and Selling
Choice
Topic 3.
Uncertainty
Contingent BC
Preferences
Choice
Insurance
Diversication
3. Topic 2. Intertemporal Choice
4. Topic 3. Choice under Uncertainty
4.1
4.2
4.3
4.4
State-contingent budget constraints
Preferences under uncertainty
Insurance
Risk spreading
I. Consumer
TOPIC 3. CHOICE UNDER UNCERTAINTY
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
and Selling
I So far, dynamic problems considered had no uncertainty.
I However, in the real world people often make decisions
with uncertainty about future prices, future wealth, or
other agents' decisions.
Topic 2.
Intertemporal
Choice
Topic 3.
Uncertainty
Contingent BC
Preferences
Choice
Insurance
Diversication
I In this section, we study the choice problem under
uncertainty using a two-state version of our consumer's
choice model.
I Optimal responses: insurance purchase, risk
diversication.
I Example:
I
I
I
2 possible states of nature: car accident (loss of L¿),
no car accident.
Probabilities for each state: πa , πna .
Insurance: get K ¿ if accident by paying γ K ¿ as
insurance premium.
I. Consumer
State-contingent budget constraints
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
Denitions
A contract is
state contingent
if it is implemented only
and Selling
when a particular state of Nature occurs.
Topic 2.
A
Intertemporal
state-contingent consumption plan
species the
Choice
consumption to be implemented when each state of Nature
Topic 3.
occurs.
Uncertainty
Contingent BC
Preferences
Choice
Insurance
Diversication
I Example:
I
I
Consumption if no accident: cna = M − γ K
Consumption if accident: ca = M − L − γ K + K .
M +L and
Hence, K = ca −1−γ
cna = M − γ
ca − M + L
1−γ
=
M − γL −
1−γ
γ
1−γ
ca
I. Consumer
Theory
Applications
State-contingent budget constraints
Car Insurance example
I State-contingent budget constraint in car insurance
Topic 0. Cons
Review
example:
cna =
Topic 1. Buying
and Selling
Topic 2.
Intertemporal
Uncertainty
Contingent BC
Preferences
Choice
Insurance
Diversication
γ
ca
−
1−γ
1−γ
| {z } | {z }
intercept slope
Choice
Topic 3.
m − γL
Cna
Bundle with no consumption if accident: K 
M
LM
1 
Endowment bundle
Full insurance bundle (Ca =Cna )
Bundle with no consumption if no accident: K  M
M‐L
Ca

I. Consumer
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Topic 2.
Intertemporal
Choice
Topic 3.
Uncertainty
Contingent BC
Preferences
Choice
Insurance
Diversication
Preferences under Uncertainty
I To know what is the agents' choice, we need to know
their preferences about the dierent state-contingent
consumption plans.
I Utility across state-contingent consumption plans is a
function of the consumption levels and probabilities at
each state,
U (c1 , c2 , π1 , π2 ).
Denition
A utility function
utility
or
U (c1 , c2 , π1 , π2 ) satises the expected
von NeumannMorgenstern
property if it can be
written as the weighted sum of the utility at each state,
where the weights are the probabilities of each state:
U (c1 , c2 , π1 , π2 ) = π1 v (c1 ) + π2 v (c2 )
It satises the independence property, which means that the
utility in a given state is independent of the utility in other
states.
I. Consumer
Theory
Applications
Topic 0. Cons
Risk aversion
Denition
We say an agent is
risk averse
if the expected utility of
Review
wealth is lower than the utility of expected wealth,
Topic 1. Buying
if it is higher, and
and Selling
risk neutral
risk lover
if it is equal.
Topic 2.
Intertemporal
I Example:
Choice
I
Topic 3.
Uncertainty
I
Contingent BC
Preferences
Choice
Insurance
Diversication
I
I
Lottery: 90¿ with probability 1/2, 0¿ with prob 1/2.
Utility levels: U($90) = 12, U($0) = 2.
Expected utility: EU=1/2*12+1/2*2=7.
Expected money value: EM=1/2*90+1/2*0=45.
Risk lover consumer
Risk lover consumer
Risk averse consumer
Risk neutral consumer
Risk neutral consumer
12
12
EU 7
EU=7
U(45) EU 7
U(45)=EU=7
12
U(45)
EU=7
U(45)
2
2
0
45
90
W lh
Wealth
2
0
45
90
Wealth
0
45
90
Wea
I. Consumer
Indierence Curves
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
and Selling
Topic 2.
Intertemporal
Choice
I State-contingent consumption plans that give equal
expected utility are equally preferred and on the same
indierence curve.
I Slope of indierence curves:
EU = π1 U (c1 ) + π2 U (c2 ) ⇒ dEU = π1
Topic 3.
Uncertainty
Contingent BC
Preferences
Choice
Insurance
Diversication
π1
∂ U (c 2 )
∂ U (c 1 )
dc1 + π1
dc2
∂ c1
∂ c2
∂ U (c2 )
∂ U (c 1 )
dc1 + π2
dc2 = 0 ⇒
∂ c1
∂ c2
dc2
π1 ∂ U (c1 ) /∂ c1
=−
dc1
π2 ∂ U (c2 ) /∂ c2
C2
Indifference curves
EU1 < EU2 < EU3
EU3
EU2
EU1
C1
I. Consumer
Theory
Applications
Topic 0. Cons
Review
Topic 1. Buying
Choice under uncertainty
I The optimal choice under uncertainty is the most
preferred aordable state-contingent consumption plan.
I In the car insurance example, the optimal consumption
and Selling
plan is where the slope of indierence curves is tangent
Topic 2.
to the budget constraint:
Intertemporal
γ
πa ∂ U (ca ) /∂ ca
=
πna ∂ U (cna ) /∂ cna 1 − γ
Choice
Topic 3.
Uncertainty
Contingent BC
Preferences
Choice
Insurance
Diversication
Cna
Most preferred affordable plan
m
Affordable plans
m L
m  L

Ca
I. Consumer
Theory
Applications
Topic 0. Cons
Insurance
Fair Insurance
Denition
fair or competitive
Review
We say an insurance is
Topic 1. Buying
economic prot of the insurer is zero, or, equivalently, if the
and Selling
Topic 2.
if the expected
¿ insurance is the probability of the insured state.
price of a 1
Intertemporal
Choice
Topic 3.
I Car insurance example:
Uncertainty
Contingent BC
Preferences
Choice
Insurance
Diversication
γK
|{z}
−πa K − (1 − πa ) 0
|
{z
}
= 0 ⇒ γ = πa
revenues expected expenditures
I If the insurance is fair, the optimal choice of risk-averse
consumers is full insurance:
πa ∂ U (ca ) /∂ ca
πa
∂ U (ca ) ∂ U (cna )
=
⇒
=
πna ∂ U (cna ) /∂ cna 1 − πa
∂ ca
∂ cna
Hence, for risk averse consumers,
ca = cna .
I. Consumer
Theory
Applications
Topic 0. Cons
Insurance
Unfair Insurance
Denition
unfair
Review
We say an insurance is
Topic 1. Buying
expected economic prots.
and Selling
if the insurer makes positive
Topic 2.
Intertemporal
Choice
Topic 3.
I If the insurance is unfair, the optimal choice of
risk-averse consumers is less than full insurance:
Uncertainty
Contingent BC
Preferences
Choice
Insurance
Diversication
γK
|{z}
−π K − (1 − πa ) 0
| a
{z
}
>0⇒
revenues expected expenditures
Hence,
πa ∂ U (ca )/∂ ca
πna ∂ U (cna )/∂ cna
γ
= 1−γ
implies that
∂ U (ca ) ∂ U (cna )
>
∂ ca
∂ cna
ca < cna .
so, for risk averse consumers,
γ
πa
>
1−γ
πna
I. Consumer
Diversication
Theory
Applications
I Asset diversication typically lowers (or keeps) expected
Topic 0. Cons
Review
earnings in exchange for lowered risk. This is going to
Topic 1. Buying
be the case as long as the asset prices are not perfectly
and Selling
Topic 2.
correlated across states.
Intertemporal
Choice
Topic 3.
Uncertainty
Contingent BC
Preferences
Choice
Insurance
Diversication
I Example: two rms, two states (prob. 1/2), agent with
100
¿ to spend in rms' share.
I
I
Firm A: shares' cost 10¿, prots per share in state 1
100¿, in state 2 20¿.
Firm B: shares' cost 10¿, prots per share in state 1
20¿, in state 2 100¿.
Prots in 1
Prots in 2
Expected prots
10 shares of A
10 shares of B
5 of A, 5 of B
200¿
1000¿
600¿
1000¿
600¿
200¿
600¿
600¿
600¿