Transcript Document 7186127

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.
Relate this diagram to the explanation
C1
in the next slide
o
C0
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)
OB  M0(1+r) + M1
c1
The negative sign represents
the trade-off; the magnitude
B
(1+r) is the relative price
of C0 in terms of C1.
M1
O
The slope of the budget
line = (-OB/OA)= -(1+r) .
M0
A
OA  M0 + M1/(1+r);
c0
c1
b
Consumer choice:
The individual is a saver
C*1
M1
O
C*0 M0
A
c0
c1
b
A drop in the rate of interest
Shifts AB down and to the
right
M1
O
M0
A
c0
A fall in the rate of interest is a gain
for the borrower but a loss to the
lender
B’
M0
A
A’ C0
c1
b’
An increase in the interest rate
reduces the welfare of the borrower
b
M1
2
O
M0
1
A’ A
c0
c1
b’
b
An increase in the interest rate
increases the welfare of the
2 lender
In this case, as the
interest rate goes up,
1
C0 falls
M1
O
M0
A’ A
The law of demand holds
c0
c1
b’
b
In this case, as the
interest rate goes up,
so does C0.
The law of demand
2
does not hold
1
M1
O
M0
A’ A
c0
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).
C1 Individual has no access to an
organized capital market
Individual consumes
B
OX units of C0.
Saves and invests
Y
1
AX units
O
X
A
C0
Produces/consumes OY units of C1.
C1 Saving
= CI
B
Individual has
access to an
organized capital
2
market
Investment
= IA
1
1’
O
C
I A
C0
C1
Saving
=-IC
1’
B
Individual has
access to an
organized capital
market
2
1
Investment
= IA
O
I
A
C
C0
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.
Utility
The utility from a £ gained is
greater than the disutility from a
£ lost
O
£1
£2
Wealth
The utility function of
a gambler
Utility
The utility from a £ gained is less
than the disutility from a £ lost
O
£1
£2
Wealth
The utility function of a riskaverse person
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
• 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 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,
but this alone does not guarantee
the existence of a market as we
shall shortly see.
We start with the case of a riskaverse 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.
Utility
U(OB)
EUA
O W0
C
B
W1
Wealth
OB = EVA = W0 + (1-)W1
EUA = U(W0) + (1-)U(W1)
U(EVA) > EUA for
a risk-averse person
W1-B is the minimum C is the certainty
insurance premium
equivalent of lottery
W1-C is the maximum A
insurance premium
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.
An
agent
has
the
utility
function
Utility
U = W2 defined over wealth (W)
4
Examine his attitude
to risk
1
O
£1
£2
Wealth
The utility from a £ gained is greater than
the disutility from a £ lost
Utility
U =W2
U(W1)
EUA
U(EVA)
U(W0)
O
W0
EVA
W1 Wealth
Relate this diagram to the explanation
in the next slide
Let W0 = 100 Then U(W0) = 10000
and W1 = 200 and U(W1) = 40000
Suppose probability
of getting W1 is 0.5
Then EVA = 0.5*200 +EUA = 0.5*10000 +
0.5*100= 150
0.5*40000 = 25000
25000 > 22500
EU(A) > U(EVA)
The agent is a
gambler
Utility
Maximum Premium =£41.89
U =W2
40000
25000
22500
10000
O
150158.11 200 Wealth
No market for Minimum Premium (£)
insurance
100
=£50
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?
• 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 highrisk customer ought to pay.
• Result- low-risk customers such as
Mr. A pull out of the market; highrisk customers are the ones that
remain.
• The seller has managed to attract the
worse type of customers to his
business.