TOPIC4: INTER-TEMPORAL CONSUMPTION CHOICE

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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.

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• 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 consumers budget.
• The consumer derives utility from todays
  consumption(C0) and tomorrows consumption C1
  according to some utility function U(C0,C1)
  which generates an indifference map as shown in
  below.
•  

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Relate this diagram to the explanation in the
next slide
C1
o
C0
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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.
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• 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.

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• 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)

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OB ? M0(1r) M1
The negative sign represents the trade-off the
magnitude (1r) is the relative price of C0 in
terms of C1.
c1
B
The slope of the budget line (-OB/OA) -(1r) .
M1
c0
O
A
M0
OA ? M0 M1/(1r)
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Consumer choice The individual is a saver
c1
b
C1
M1
c0
O
A
M0
C0
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c1
A drop in the rate of interest Shifts AB down and
to the right
b
M1
c0
O
A
M0
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A fall in the rate of interest is a gain for the
borrower but a loss to the lender
B
M0 A A C0
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An increase in the interest rate reduces the
welfare of the borrower
c1
b
b
M1
1
2
c0
O
A A
M0
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An increase in the interest rate increases the
welfare of the lender
c1
b
2
b
In this case, as the interest rate goes up, C0
falls
1
M1
c0
O
A A
M0
The law of demand holds
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In this case, as the interest rate goes up, so
does C0.
c1
b
b
The law of demand does not hold
2
1
M1
c0
O
A A
M0
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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.

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• 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.

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• 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).

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C1
Individual has no access to an organized capital
market
Individual consumes OX units of C0.
B
Saves and invests AX units
Y
1
O X A
C0
Produces/consumes OY units of C1.
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C1
Saving CI
Individual has access to an organized capital
market
B
2
Investment IA
1
1
O C I A
C0
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Individual has access to an organized capital
market
C1
Saving -IC
1
B
2
1
Investment IA
O I A C
C0
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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.

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• 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 Dont buy the ticket.

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• 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.

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Utility
The utility from a gained is greater than the
disutility from a lost
O 1 2
Wealth
The utility function of a gambler
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Utility
The utility from a gained is less than the
disutility from a lost
O 1 2
Wealth
The utility function of a risk-averse person
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Utility
The utility from a gained is the same as the
disutility from a lost
O 1 2
Wealth
The utility function of a risk-neutral person
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• Consider this possibility. Individual says that  
• U(A) lt 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).

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• The value of U(A) is in between these extreme
  values.
• To capture this idea, we introduce the notion of
  EXPECTED UTILITY.
• .

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• 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) gt EU(A) gt U(0)

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• For a gambler, EU(A) gt U(B).
• Oppositely, for a risk-averse person EU(A) lt
  U(B)  
• and for a risk neutral person
• EU(A) U(B)

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Risk Preference and the market for insurance

• In this section we shall argue that
• there 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,

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• 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
•  
•  
•  

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• With a small probability ? the individual gets
  W0 unit s of money
• and with probability (1-?) he gets W1 gt W0.
• Let OB ? ?W0 (1-?)W1 .
• OB is then the EXPECTED VALUE of the monetary
  gain.

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Utility
U(OB)
EUA
O W0 C B W1
Wealth
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OB EVA pW0 (1-p)W1
EUA pU(W0) (1-p)U(W1)
U(EVA) gt EUA for a risk-averse person
W1-B is the minimum insurance premium
C is the certainty equivalent of lottery A
W1-C is the maximum insurance premium
BC is the consumer surplus
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• 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.

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• 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.

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• 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.

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• 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.

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• 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.

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• 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.

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• This is why you cannot purchase insurance against
  earthquakes or floods.
• A social risk is non-insurable.

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An agent has the utility functionU W2 defined
over wealth (W)
Utility
Examine his attitude to risk
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1
O 1 2
Wealth
The utility from a gained is greater than the
disutility from a lost
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U(W1)
Utility
U W2
EUA
U(EVA)
U(W0)
O
W0 EVA W1
Wealth
Relate this diagram to the explanation in the
next slide
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Let W0 100 and W1 200
Then U(W0) 10000 and U(W1) 40000
Suppose probability of getting W1 is 0.5
Then EVA 0.5200 0.5100 150
EUA 0.510000 0.540000 25000
25000 gt 22500
EU(A) gt U(EVA)
The agent is a gambler
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40000
Utility
U W2
Maximum Premium 41.89
25000
22500
10000
O
100 150 200
Wealth ()
158.11
No market for insurance
Minimum Premium 50
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Asymmetric Information and the Market for
Insurance
Moral Hazard Adverse Selection

•  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?

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• 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.

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• 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.  

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• 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.

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• 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.