Transcript Extensions to Consumer theory
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 31 st of October Friday 27 th 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. Strictly convex et decreasing 2. Corresponds to an inter-temporal utility function U(C1,C2) future consumption (c 2 ) current consumption (c 1 )
future consumption (c 2 ) Inter-temporal preferences Imagine you are at A : Low current, high future consumption A In order to increase your current consumption, you are willing to reduce your future consumption quite a lot Slope =
c
2
c
1
r a
current consumption (c 1 )
future consumption (c 2 ) Inter-temporal preferences
c
2
c
1 A
r a
At B , you are willing to give up less future consumption than at consumption A for the same amount of extra current r a > r b You are more patient !
B
c
2
c
1
r b
current consumption (c 1 )
future consumption (c 2 ) Inter-temporal preferences A If
c 1
is low : (r) is high, you are impatient If
c 1
is high : (r) is low, you are patient (1+r) : The MRS is a measure of patience B current consumption (c 1 )
Inter-temporal budget constraint The difference with the general case is that your savings earn interest over time .
A sum M t 1 year invested in t is worth 01/09 : I invest 100 € M t+1 =M t × (1+
i
) after 12/09 : I will receive 100 × If
i
= 0,05 (5%); M t+1 =100 × (1+
i
) € (1,05) =105 € A invested sum year earlier M t+1 was worth 12/09 : I receive 525 € M t =M t+1 (1+
i
) a 01/09 : I invested 525 (1+
i
) € If
i
= 0,05 (5%); M t = 525 (1,05)= 500 €
Inter-temporal budget constraint Simplification: invariable price
p 1
Explicit interest rate:
i
Consumption :
c
Budget :
b
Two periods : 1 and 2
= p 2
= 1 Consumption in period 2: For a lender: For a borrower:
c
2
c
2 =
b
2 =
b
2 Savings in period 1 + (1+
i
) × (
b
1 + (1+
i
) × (
c
1 – –
c
1 )
b
1 ) Borrowing in period 1
Inter-temporal budget constraint In general, one can write:
c
2 =
b
2 + (1+
i
) × (
b
1 –
c
1 ) If (
b
1 –
c
1 ) > 0 lender If (
b
1 –
c
1 ) < 0 borrower If (
b
1 –
c
1 ) = 0 neither
Inter-temporal budget constraint The budget constraint equalises the total inter temporal budget B with the total inter-temporal consumption C : B = C Note: Again, all the budget is spent !! Just not in the same period
Current Value Future value
There are 2 ways of expressing this budget constraint B C
b
1 (1
b
2
i
)
c
1 (1
c
2
i
)
b
1
c
1
i
)
i
)
b
2
c
2
Inter-temporal budget constraint
B
C b
2 1
b
2
i
1 1
c
2
i
c
1
i
b
1
c
2 1
i
c
1 Generic budget constraint Current value budget constraint Future value budget constraint
Inter-temporal budget constraint future consumption (c 2 )
B p
2
b
2
i
b
1 Slope =
c
2
c
1
i
B p
1
b
2
b
1 current consumption (c 1 )
Inter-temporal budget constraint future consumption (c 2 )
b
2
b
1 B Maximum savings strategy
b
2
b
1 C No borrowing / No lending 1
b
2
i
A Maximum borrowing strategy
b
1 current consumption (c 1 )
Inter-temporal budget constraint future consumption (c 2 )
b
2
b
1 B Effect of an increase of interest rates on the inter-temporal budget constraint
b
2
b
1 Slope =
c
2
c
1 C
i
A 1
b
2
i
b
1 current consumption (c 1 )
Inter-temporal choice future consumption (c 2 )
b
2
b
1 B E Inter-temporal choice of a lender At E : MRS = slope of the budget constraint
r
r
i
i
C
b
2
b
1 A 1
b
2
i
b
1 current consumption (c 1 )
Inter-temporal choice future consumption (c 2 )
b
2
b
1 B Inter-temporal choice of a borrower At E we still have r=i
b
2 C
b
1 E A 1
b
2
i
b
1 current consumption (c 1 )
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. Is strictly convex consumption and decreasing 2. Corresponds to an utility function U(C,Λ) defined over (of an aggregate basket) and leisure 3. 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 Unearned income
C
P
I u
w
max 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
I u A Λ max Maximum leisure strategy Leisure (Λ)
The labour supply decision Consumption (C) The labour supply decision The optimal point is given by the tangency between the budget constraint and the indifference curve A
mU C P
mU
w
Leisure Labour Λ max Leisure (Λ)
The labour supply decision Effect of an increase in unearned income Consumption (C) B An increase in unearned income increases consumption and leisure (reduces labour supply) A Leisure (Λ) Λ max
The labour supply decision Consumption (C) Effect of an increase in the wage rate A B 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 Leisure (Λ) Λ max
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
x i
, weighted by the probability they will occur
π i .
1 1 , 2 ,...,
x n
; 1 , 2 ,...,
n
2
n
Uncertainty Reminder 1: preferences are assumed convex Good 1 x 1 X Z A combination z of extreme bundles x and y x and is preferred to y y 1 x 2 y 2 Y 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 U(17) U(10) U(3) Risk aversion: The increase in utility following a win is smaller than the loss of utility following a loss 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 X represents the insurance premium that you are willing to pay to avoid carrying the risk 3 10 17 Good
Adapting to Uncertainty/risk Insurance: 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 .
Diversification behaviour: 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 ice creams 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 Good 1 A If it could be afforded, would bundle C be preferred to A ?
Without further information, we don’t know...
But imagine we know of a change in prices and incomes that makes C affordable C Good 2
Revealed preferences Good 1 B is revealed preferred Therefore B ≻ C to C . A is revealed preferred Therefore A ≻ B to B . By transitivity, preferred A to C: A is indirectly revealed ≻ C A B C Good 2
Revealed preferences Good 1 A B Less desirable bundles B is revealed preferred Therefore B ≻ C to C . A is revealed preferred Therefore A ≻ B to B . By transitivity, preferred A to C: A is indirectly revealed ≻ C C Good 2
Revealed preferences Good 1 Y A B Less desirable bundles Similarly, if I see that the consumer chooses are Y then Z as his income increases, I can conclude that these revealed preferred to A .
Therefore Y,Z ≻ A C Z Good 2
Revealed preferences Good 1 Y A B Preferred bundle Less desirable bundles C Z Good 2
Revealed preferences Good 1 Y A B Preferred bundle Less desirable bundles C Z Good 2
Revealed preferences Good 1 Y A B Preferred bundle Less desirable bundles C Z Approximation of the indfference curve Good 2
Revealed preferences Good 1 Y A B C Z Approximation of the indfference curve Good 2
Revealed preferences Good 1 Y A B C Z Approximation of the indfference curve Good 2