TOPIC4: INTER-TEMPORAL CONSUMPTION CHOICE

• Saving for tomorrow is a fact of life.

• Equally, we often spend more than what we currently earn or have by borrowing.

• We shall consider a two-period model of consumer choice.

• The consumer receives a given amount of money in period 0 (M0), and in period 1 (M1). The numbers M0 and M1 thus define the consumer’s budget.

• The consumer derives utility from today’s consumption(C0) and tomorrow’s consumption C1 according to some utility function U(C0,C1) which generates an indifference map as shown in below.

 

C0o

C1Relate this diagram to the explanation

in the next slide

An individual who borrows has a stronger

preference for C0 (steeper indifference curves).

•The slope of the indifference curve is the

MRS between C0 and C1.

The MRS is also called the discount factor.

• People with stronger preference for C0 have a smaller discount factor.

• They discount future happiness more.

• The discount factor is unique to an individual consumer, just like his/her indifference curves.

• The budget constraint of the consumer must recognise the fact that the consumer is free to borrow and/or lend at the market interest rate r.

• For a given value of r, the budget constraint is drawn below

• (We assume for simplicity that the unit price of both C0 and C1 is 1)

c0

c1

O A

B

M0

M1

OA M0 + M1/(1+r);

OB M0(1+r) + M1

The slope of the budget line = (-OB/OA)= -(1+r) .

The negative sign represents the trade-off; the magnitude (1+r) is the relative price of C0 in terms of C1.

c0

c1

O

A

b

M0

M1C*1

C*0

Consumer choice: The individual is a saver

Ac0

c1

O

b

M0

M1

A drop in the rate of interestShifts AB down and to the right

M0 A A’ C0

A fall in the rate of interest is a gainfor the borrower but a loss to the lender

B’

c0

c1

O

A’ A

b

M0

M1

b’

2 1

An increase in the interest rate reduces the welfare of the borrower

c0

c1

O

A’ A

b

M0

M1

b’

1

2

An increase in the interest rate increases the welfare of the lender In this case, as the

interest rate goes up, C0 falls

The law of demand holds

c0

c1

O

A’ A

b

M0

M1

b’

12

In this case, as the interest rate goes up, so does C0.

The law of demand does not hold

Inter-temporal Choice with Production

The analysis above assumes that the individual can reallocate consumption across time by borrowing/lending in a perfect capital market.

However, instead of having just the freedom to lend current resources, it may be more realistic to include the possibility of using the current resources to produce some goods that are consumable in the future, or INVESTMENT.

We shall therefore broaden our analysis by incorporating a production opportunity that allows current saving to be invested, leading to a greater level of output in the future.

• To start with a simple model, • suppose that the individual has no

access to a capital market. ,• that is, she is unable to

borrow/lend.• Also, for simplicity, we assume that

the individual is endowed with productive resources only for today (she does not receive any resources tomorrow).

C0

C1

O X A

BIndividual consumesOX units of C0.

1

Individual has no access to anorganized capital market

Saves and investsAX units

Produces/consumes OY units of C1.

Y

C0

C1

O C I A

B 2

Individual hasaccess to an organized capital market

11’

Investment= IA

Saving= CI

C0

C1

O I A C

B

2

Individual hasaccess to an organized capital market

1

1’

Investment= IA

Saving=-IC

TOPIC5: CONSUMER CHOICE UNDER RISK

  

• We have so far analysed consumer behaviour under certainty.

• The typical consumer has been assumed to have perfect information on every single economic variable.

• We shall now introduce the notion of RISK.

Consider the following choice problem.

Choice A: Buy lottery ticket for £1 that wins £2 with probability 0.5 or nothing with probability 0.5

Choice B: Don’t buy the ticket.

• Consumer preferences may be one of the following three types:

• 1. Risk Lovers or Gamblers would prefer choice A to B.

• 2. A Risk Neutral person would be indifferent between choice A and B. 

• 3. A Risk Averse person would prefer choice B over A.

Wealth

Utility

O £1 £2The utility function of a gambler

The utility from a £ gained is greater than the disutility from a £ lost

Wealth

Utility

O £1 £2The utility function of a risk-averse person

The utility from a £ gained is less than the disutility from a £ lost

Wealth

Utility

O £1 £2The utility function of a risk-neutral person

The utility from a £ gained is the same as the disutility from a £ lost

• Consider this possibility. Individual says that  

U(A) < U(B) = U (£1). That is, although both A and B offers the same

amount of money (£1) on average, the individual would rather have B.

But what is U(A)? It is certainly not as high as U(£2). Neither is it as low as U(£0).

The value of U(A) is in between these extreme values.

To capture this idea, we introduce the notion of EXPECTED UTILITY.

.

• We shall say that

• Expected utility of lottery A, or EU (A) 0.5 U(£2) + 0.5 U(£0) .

• Clearly, EU(A) is a weighted average of the two extreme values U(£2) and U(£0), using the probabilities as the ‘weights’.  

• It then follows that

•  U(2) > EU(A) > U(£0)

• For a gambler, EU(A) > U(B).

• Oppositely, for a risk-averse person EU(A) < U(B)  

• and for a risk neutral person

• EU(A) = U(B)

Risk Preference and the market for insurance

In this section we shall argue thatthere is no demand for insurance

from an individual who is a gambler or risk-neutral.

Risk-averse people will want to be insured against the risk,

but this alone does not guarantee the existence of a market as we shall shortly see.

We start with the case of a risk-averse person.

In order to generalise the argument above, let the individual face a risky situation A

described as follows:   

With a small probability the individual gets W0 unit s of money

and with probability (1-) he gets W1 > W0.

Let OB W0 + (1-)W1 . OB is then the EXPECTED

VALUE of the monetary gain.

Wealth

Utility

O W0 C B W1

EUA U(OB)

EUA = UW0) +U(W1)

W1-B is the minimum insurance premium W1-C is the maximum insurance premium

C is the certainty equivalent of lottery A

OB = EVA = W0 +W1

U(EVA) > EUA for a risk-averse person

BC is the consumer surplus

• It therefore seems that potentials of a market exist because the buyer could pay more than what the seller would charge.

• But what about administrative costs?• If these exceed distance BC in the

diagram above, there is no market! • As long as these costs are less than

BC, potentials of a market exist.•

• The nature of admin costs is such that they do not change proportionately with the size of the policy or that of the premium.

• On the other hand, notice that the consumer surplus BC is larger the greater the difference between W1 and W0.

• Hence it follows that the consumer will not bother to buy insurance if the extent of the loss is ‘small’.

• You should now be able to argue that a risk –neutral person will not buy insurance if there are positive administrative costs.

• What exactly are these ‘administrative’ costs?

• The insuring party typically ‘pools’ the risk faced by several of its customers.

• Suppose that 100 individuals each wants to purchase an insurance against unemployment.

• Assume that they each earn £40000 p.a. and nothing if s/he loses job. Let the job loss probability for each be 0.1.

• Here, the ‘fair bet’ premium is £4000.

How much extra the seller charges depends how the job losses are correlated.

Essentially, the seller hopes to pay out a claim (by a customer who lost his job) from the money paid in by those who do not lose their jobs.

This can be ideally achieved if the event that one customer loses his job is negatively correlated with the one in which some other customer loses his.

The worst scenario for the insurer is when these events are perfectly and positively correlated.

For example, if all the workers work in the same factory, there can be no single insurer who can sell unemployment insurance to all of them.

This is why you cannot purchase insurance against earthquakes or floods.

A social risk is non-insurable.

Wealth

Utility

O £1 £2The utility from a £ gained is greater than the disutility from a £ lost

1

4Examine his attitudeto risk

An agent has the utility functionU = W2 defined over wealth (W)

W0 EVA W1

Utility

O

EUA

U(W1)

U(W0)

U =W2

Wealth

U(EVA)

Relate this diagram to the explanation

in the next slide

25000 > 22500

Then EVA = 0.5*200 + 0.5*100= 150

Suppose probability of getting W1 is 0.5

Then U(W0) = 10000and U(W1) = 40000

EUA = 0.5*10000 + 0.5*40000 = 25000

Let W0 = 100and W1 = 200

EU(A) > U(EVA)

The agent is a gambler

100 150 200

Utility

O

25000

40000

10000

U =W2

Wealth (£)

22500

158.11

Maximum Premium =£41.89

Minimum Premium=£50

No market for insurance

Asymmetric Information and the Market for Insurance

 Are we more likely to leave the house/car door unlocked after purchasing insurance against burglary?

• Does the probability of a risky event occurring increase as we insure ourselves against it?

Moral Hazard Adverse Selection

• Some economists answer in the affirmative.

• If this is the case, then the profitability of the insurance seller may well be lower,

• thus reducing the willingness of the seller to sell insurance and the probability of market existence.

• This is the problem of ‘moral hazard’ due to asymmetric information.

• A second problem, and due to asymmetric information as well, is that of ‘adverse selection’.

• Suppose that individual A is a low-risk case who wishes to purchase insurance.

• Mr. A expects the premium to be low.  

• Indeed, the seller would offer him a low premium rates if he knew that A was low-risk.

• Unfortunately, he cannot distinguish Mr. A from Ms. B, a high-risk customer.

• The presence of the latter kind of customers drives the cost of insurance higher than what the likes of Mr. A ought to pay.

• On the other hand, such premiums are well below the rate that the high-risk customer ought to pay.

• Result- low-risk customers such as Mr. A pull out of the market; high-risk customers are the ones that remain.

• The seller has managed to attract the worse type of customers to his business.