Extensions to Consumer theory Inter-temporal choice Uncertainty Revealed preferences

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 function. 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 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. Seems this is 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 future consumption (c 2 ) current consumption (c 1 ) The inter-temporal indifference curve 1.Strictly convex et decreasing 2.Corresponds to an inter-temporal utility function U(C1,C2)

Inter-temporal preferences A Imagine you are at A: Low current, high future consumption In order to increase your current consumption, you are willing to reduce your future consumption quite a lot future consumption (c 2 ) current consumption (c 1 )

Inter-temporal preferences A At B, you are willing to give up less future consumption than at A for the same amount of extra current consumption r a > r b You are more patient ! B future consumption (c 2 ) current consumption (c 1 )

Inter-temporal preferences A B 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 future consumption (c 2 ) current consumption (c 1 )

Inter-temporal budget constraint Let’s see what happens if your savings earn interest over time. A sum M t invested in t is worth M t+1 =M t × (1+ i ) after 1 year 01/08 : I invest 100 € 12/08 : I will receive 100 × (1+ i ) € If i = 0,05 (5%); M t+1 =100 × (1,05) =105 € A invested sum M t+1 was worth M t =M t+1  (1+ i ) a year earlier 12/08 : I receive 525 € 01/08 : 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 = p 2 = 1 Explicit interest rate: i Consumption : c Budget : b Two periods : 1 and 2

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 Current ValueFuture value B C 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 There are 2 ways of expressing this budget constraint

Inter-temporal budget constraint Current value budget constraint Future value budget constraint Generic budget constraint

Inter-temporal budget constraint future consumption (c 2 ) current consumption (c 1 )

Inter-temporal budget constraint A B C Maximum savings strategy Maximum borrowing strategy No borrowing / No lending future consumption (c 2 ) current consumption (c 1 )

Inter-temporal budget constraint A B C future consumption (c 2 ) current consumption (c 1 ) Effect of an increase of interest rates on the inter-temporal budget constraint

Inter-temporal choice A B E Inter-temporal choice of a lender future consumption (c 2 ) current consumption (c 1 ) C At E: MRS =slope of the budget constraint

Inter-temporal choice A B E Inter-temporal choice of a borrower C future consumption (c 2 ) current consumption (c 1 ) At E we still have r=i

Extensions to Consumer theory Inter-temporal choice 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 prizes 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.

Uncertainty Reminder 1: preferences are assumed convex Good 1 Good 2 y1y1 y2y2 Y x1x1 x2x2 X A combination z of extreme bundles x and y is preferred to x and y Z

Uncertainty Good Utility Reminder 2: Convex preferences imply a decreasing marginal utility : total utility is concave

Uncertainty Simple illustration of uncertainty : 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 Good Utility Diagram of the expected utility of the game : 3 U(3) 10 U(10) 17 U(17) 0.5*U(3) + 0.5*U(17)

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 Good Utility 3 U(3) 10 U(10) 17 U(17) The increase in utility following a win is smaller than the loss of utility following a loss This behavioural result is a central consequence of the hypothesis of convex preferences !! Risk aversion:

Uncertainty Now imagine that you do not have a choice, and you must play the game. This is a risky situation. Good Utility 3 U(3) 10 U(10) 17 U(17) X represents the insurance premium that you are willing to pay to avoid carrying the risk X

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. You can “sell” your risk to agents that are willing to carry it, against a payment. But beware of information problems !!

Extensions to Consumer theory Inter-temporal choice Uncertainty 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  A If it could be afforded, would bundle C be preferred to A ?  C Good 1 Good 2 Without further information, we don’t know... But imagine we know of a change in prices and incomes that makes C affordable

Revealed preferences Good 2 B is revealed preferred to C. Therefore B ≻ C  C A is revealed preferred to B. Therefore A ≻ B By transitivity, A is indirectly revealed preferred to C: A ≻ C  B Good 1  A

Revealed preferences Less desirable bundles Good 1 Good 2  C  B  A B is revealed preferred to C. Therefore B ≻ C A is revealed preferred to B. Therefore A ≻ B By transitivity, A is indirectly revealed preferred to C: A ≻ C

Revealed preferences Good 1 Good 2  C  A  B  Y  Z 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. Therefore Y,Z ≻ A Less desirable bundles

Revealed preferences Good 2  C  A  B  Y  Z Preferred bundle Good 1 Less desirable bundles

Revealed preferences Good 2  C  A  B  Y  Z Good 1 Preferred bundle Less desirable bundles

Revealed preferences Good 2  C  A  B  Y  Z Preferred bundle Approximation of the indfference curve Good 1 Less desirable bundles

Revealed preferences Good 2  C  A  B  Y  Z Approximation of the indfference curve Good 1

Revealed preferences Good 2  C  A  B  Y  Z Approximation of the indfference curve Good 1