Extensions to Consumer
theory
Inter-temporal choice
Uncertainty
Revealed preferences
Extensions to Consumer theory

Before we start, here are the dates of the
two class tests

Saturday 31st of October

Friday 27th of November, 16:15-18:15
Extensions to Consumer theory

We now know how a consumer chooses the
most satisfying bundle out of the ones it can
afford


From observing changes in choice that follow
changes in price, we can derive the demand
curve
But how useful (and realistic) is this theory?



We assumed that agents have no savings...
Does the theory still stand under uncertainty ?
We assume that stable preferences just “exist”.
Extensions to Consumer theory
Inter-temporal choice
The labour supply decision
Uncertainty
Revealed preferences
Inter-temporal choice


The typical agents spreads his consumption over
several periods of time: immediate consumption,
savings, borrowing, etc.
Consume today / consume tomorrow



Current preferences between goods are convex.
Is this also the case for inter-temporal choices ?
We need to define :


Inter-temporal preferences
Inter-temporal budget constraint
Inter-temporal preferences

Preference for current consumption

A unit of consumption today is “worth” more than a
unit of consumption tomorrow
 I’d rather receive 100 € today than 100 € next week !


If I give up some current consumption , I expect to
receive a return (r) in compensation.
There must exist a value of (r), a psychological
interest rate, for which I am indifferent between
current and future consumption

Would you rather receive 100 € today or 101 € next
week ? What about 120 € ?
Inter-temporal preferences
The inter-temporal indifference curve
1.
2.
Strictly convex et decreasing
Corresponds to an inter-temporal utility function
U(C1,C2)
future
consumption
(c2)
current
consumption
(c1)
Inter-temporal preferences
Imagine you are at A:
Low current, high future consumption
future
consumption
(c2)
A
In order to increase your current
consumption, you are willing to
reduce your future consumption quite
a lot
Slope =
c2
  1  r a 
c1
current
consumption
(c1)
Inter-temporal preferences
future
consumption
(c2)
c2
  1  r a 
c1
A
At B, you are willing to give up less
future consumption than at A for
the same amount of extra current
consumption
ra > r b
You are more patient !
B
c2
  1  r b 
c1
current
consumption
(c1)
Inter-temporal preferences
If c1 is low : (r) is high, you are
impatient
future
consumption
(c2)
If c1 is high : (r) is low, you are
patient
A
(1+r) : The MRS is a measure of
patience
B
current
consumption
(c1)
Inter-temporal budget constraint


The difference with the general case is that your
savings earn interest over time.
A sum Mt invested in t is worth Mt+1=Mt×(1+i) after
1 year




01/09 : I invest 100 €
12/09 : I will receive 100 × (1+i) €
If i = 0,05 (5%); Mt+1=100 × (1,05) =105 €
A invested sum Mt+1 was worth Mt=Mt+1(1+i) a
year earlier



12/09 : I receive 525 €
01/09 : I invested 525 (1+i) €
If i = 0,05 (5%); Mt = 525(1,05)= 500 €
Inter-temporal budget constraint

Simplification: invariable price p1 = p2 = 1
Explicit interest rate: i
Consumption : c
Budget : b
Two periods : 1 and 2

Consumption in period 2:






For a lender:
For a borrower:
Savings in period 1
c2 = b2 + (1+i)×(b1 – c1)
c2 = b2 + (1+i)×(c1 – b1)
Borrowing in period 1
Inter-temporal budget constraint

In general, one can write:
c2 = b2 + (1+i)×(b1 – c1)
If (b1 – c1) > 0  lender
If (b1 – c1) < 0  borrower
If (b1 – c1) = 0  neither
Inter-temporal budget constraint



The budget constraint equalises the total intertemporal budget B with the total inter-temporal
consumption C :
B=C
Note: Again, all the
budget is spent !!
Just not in the
same period
There are 2 ways of
expressing this
budget constraint
Current Value
Future value
B
b1 
b2
(1  i )
b1  (1  i)  b2
C
c1 
c2
(1  i )
c1  (1  i)  c2
Inter-temporal budget constraint
BC
b2
c2
 b1 
 c1
1  i 
1  i 
b2  1  i   b1  c2  1  i   c1
Generic budget
constraint
Current value budget
constraint
Future value budget
constraint
Inter-temporal budget constraint
future
consumption
(c2)
B
 b2  1  i   b1
p2
Slope =
c2
  1  i 
c1
b
B
 2  b1
p1 1  i 
current
consumption
(c1)
Inter-temporal budget constraint
future
consumption
(c2)
b2  b1  1  i 
b2
B
Maximum savings strategy
C
No borrowing / No lending
Maximum borrowing strategy
A
b1
b2
 b1
1

i
 
current
consumption
(c1)
Inter-temporal budget constraint
future
consumption
(c2)
b2  b1  1  i 
Effect of an increase of interest rates on
the inter-temporal budget constraint
B
Slope =
b2
c2
  1  i 
c1
C
A
b1
b2
 b1
1

i
 
current
consumption
(c1)
Inter-temporal choice
future
consumption
(c2)
b2  b1  1  i 
Inter-temporal choice of a lender
At E:
B
MRS = slope of the
budget constraint
 1  r    1  i 
E
b2
r i
C
A
b1
b2
 b1
1  i 
current
consumption
(c1)
Inter-temporal choice
future
consumption
(c2)
b2  b1  1  i 
b2
Inter-temporal choice of a borrower
At E we still have r=i
B
C
E
A
b1
b2
 b1
1

i
 
current
consumption
(c1)
Extensions to Consumer theory
Inter-temporal choice
The labour supply decision
Uncertainty
Revealed preferences
The labour supply decision

Framework similar to the one used
previously:


Labour is treated like a regular commodity
Based on an agent deriving utility from
consumption and leisure


Indifference curve based on preference for
consumption (which requires income) and leisure
(free time)
Budget constraint based on working (which uses
up free time but provides and income)
The labour supply decision
The labour supply decision indifference curve
1.
2.
3.
Is strictly convex and decreasing
Corresponds to an utility function U(C,Λ) defined over
consumption (of an aggregate basket) and leisure
Leisure brings utility ⇔ labour brings disutility
Consumption
(C)
Leisure (Λ)
The labour supply decision

The labour supply decision budget
constraint


First of all: Workers can earn an income
independently of supplying labour (unearned
income).
Labour generates a disutility, but this is
compensated by a wage.
Cost of
consumption
C  P  I u  w max   
Unearned
income
Labour supplied
(defined as leisure
not taken)
The labour supply decision
The labour supply decision budget constraint
Consumption
(C)
I u  w max 
P
B
Maximum consumption strategy
C
I u  w max   
P
Iu
A
Λmax
Maximum leisure strategy
Leisure (Λ)
The labour supply decision
The labour supply decision
Consumption
(C)
The optimal point is
given by the tangency
between the budget
constraint and the
indifference curve
A
Leisure
mU C mU 

P
w
Labour
Λmax
Leisure (Λ)
The labour supply decision
Effect of an increase in unearned income
Consumption
(C)
An increase in unearned
income increases
consumption and leisure
(reduces labour supply)
B
A
Λmax
Leisure (Λ)
The labour supply decision
Effect of an increase in the wage rate
An increase in the wage
rate usually increases
consumption and
reduces leisure
BUT this depends on the
income/substitution
effects !!
An increase can reduce
labour supply
Consumption
(C)
B
A
Λmax
Leisure (Λ)
Extensions to Consumer theory
Inter-temporal choice
The labour supply decision
Uncertainty
Revealed preferences
Uncertainty


How do we calculate the utility of an agent
when there is uncertainty about which
bundle will be consumed?
Example: You’re trying to decide if you want
to buy a raffle ticket. What determines the
potential utility of buying this ticket?


The amount of prize and their value
The amount on tickets on sale
Uncertainty
Under uncertainty, the decision
process depends on expected utility
 Expected utility is simply the sum of
the utilities of the different outcomes
xi, weighted by the probability they will
occur πi.

EU  x1 , x2 ,..., xn ;  1 ,  2 ,...,  n  
 1  U  x1    2  U  x2   ...   n  U  xn 
Uncertainty
Reminder 1: preferences are assumed convex
Good 1
x1
X
A combination z of
extreme bundles x
and y is preferred to
x and y
Z
y1
Y
x2
y
2
Good 2
Uncertainty
Reminder 2: Convex preferences imply a
decreasing marginal utility : total utility is concave
Utility
Good
Uncertainty

Simple illustration


You have 10 units of a good and you are invited
to play the following game.
A throw of heads or tail:
 Probability of success or failure is 0.5

The stake of the game is 7 units:
 Outcome if you win: 17 units
 Outcome if you loose: 3 units

Are you willing to play ?
Uncertainty
Diagram of the expected utility of the game :
Utility
U(17)
U(10)
0.5*U(3) + 0.5*U(17)
U(3)
3
10
17
Good
Uncertainty



In this example, the expected result does not
change the expected endowment of the agent.
 The player starts with 10 units and the net
expected gain is 0.
Even though the expected outcome is the same
as the initial situation, the mere existence of
the game reduces the utility of the agent.
Why is that ?
Uncertainty
Utility
Risk aversion:
U(17)
The increase in
utility following a
win is smaller than
the loss of utility
following a loss
U(10)
U(3)
3
10
17
Good
This behavioural result is a central consequence of
the hypothesis of convex preferences !!
Uncertainty
Now imagine that you do not have a choice, and you
must play the game. This is a risky situation.
Utility
U(17)
U(10)
U(3)
X represents the
insurance premium
that you are willing to
pay to avoid carrying
the risk
X
3
10
17
Good
Adapting to Uncertainty/risk

Insurance:



Diversification behaviour:



Agents are willing to accept a smaller endowment to mitigate the
presence of risk
A risky outcome, however, does not impact all agents. Insurance
spreads this risk over all the agents: This is known as the
mutualisation of risk.
Imagine you sell umbrellas: your income depends on the weather,
so your future income is uncertain.
How can you make your income more certain? Sell some icecreams on the side !!
Financial markets:


Spread the risk over many assets instead of concentrating it on a
few.
They also allow you to “sell” your risk to agents that are willing to
carry it, against a payment.
Extensions to Consumer theory
Inter-temporal choice
The labour supply decision
Uncertainty
Revealed preferences
Revealed preferences

Up until now we have assumed that preferences
and indifference curves are given, and are stable




This assumption was required for the purpose of
developing a theory of choice !
But we’ve never directly observed them. How do we
know we’re right?
We can reverse the theory: we work backwards
from the optimal bundle and the budget constraint
to get to the indifference curve.
Past choices/decisions reveal your preferences
Revealed preferences




If we have information on the bundles chosen by
consumers in the past,
If we have information on the changes in prices and
incomes for the duration of the period,
Then we can determine the indifference curves of
the agent and verify if preferences are stable
through time.
The process of revealed preferences:


Gives us information on the indifference curves
Allows us to check the realism of the assumptions
behind consumer theory and the test the coherence
of consumers when they make choices.
Revealed preferences
If it could be afforded, would bundle
C be preferred to A ?
Good 1
Without further information, we
don’t know...
A

But imagine we know of a change in
prices and incomes that makes C
affordable
C

Good 2
Revealed preferences
B is revealed preferred to C.
Therefore B ≻ C
Good 1
A is revealed preferred to B.
Therefore A ≻ B
By transitivity, A is indirectly revealed
preferred to C: A ≻ C
A

B

C

Good 2
Revealed preferences
B is revealed preferred to C.
Therefore B ≻ C
Good 1
A is revealed preferred to B.
Therefore A ≻ B
By transitivity, A is indirectly revealed
preferred to C: A ≻ C
A

B

Less
desirable
bundles
C

Good 2
Revealed preferences
Similarly, if I see that the consumer
chooses Y then Z as his income
increases, I can conclude that these
are revealed preferred to A.
Good 1
Y
Therefore Y,Z ≻ A

A

B

Less
desirable
bundles
Z

C

Good 2
Revealed preferences
Good 1
Y

Preferred bundle
A

B

Less
desirable
bundles
Z

C

Good 2
Revealed preferences
Good 1
Y

Preferred bundle
A

B

Less
desirable
bundles
Z

C

Good 2
Revealed preferences
Good 1
Y

Preferred bundle
A

B

Less
desirable
bundles
Z

C

Approximation of the
indfference curve
Good 2
Revealed preferences
Good 1
Y

A

B

Z

C

Approximation of the
indfference curve
Good 2
Revealed preferences
Good 1
Y

A

B

Z

C

Approximation of the
indfference curve
Good 2