# Micro 091021 by yurtgc548

VIEWS: 39 PAGES: 30

• pg 1
```									   Micro 1015

Risk and Uncertainty
Introducing risk and uncertainty

• Many economic decisions have to be taken without
being certain of the consequences
• When we buy certain goods, we cannot tell for sure in
advance what their quality is, so we won’t know how
much we will appreciate them
• When we decide to train for a particular job, we do not
know for sure what the future wage levels in that job are
going to be
• When we buy some type of capital asset we cannot tell
for sure what return it will earn, or what it’s future price is
going to be
• so in these cases, how do consumers decide?
decisions under uncertainty

• Even when there is uncertainty of the above types,
consumers have some idea of the possible outcomes
that can occur
• It is also reasonable to assume that, as in the case when
there is no uncertainty, consumers’ preferences over
outcomes can be represented by a utility function
• So if we represent an outcome by a vector x of relevant
variables, we know the consumer’s utility u(x). The
problem, however, is that we are not sure that the
outcome will be x; it may also be y, say, where y is a
vector with different elements. We also know u(y)
– so here x and y may stand for one of the goods we buy being of
high or low quality, the wages of our future job being high or low,
the return on the asset we buy being high or low …
decisions under uncertainty

• Next, it is reasonable to believe that we also have some
idea about the likelihood of the different outcomes. To
capture this we make the strong assumption that in the
case where the outcome may be x or y, the consumer is
willing to assign a probability p that the outcome will be x
• Note that the probability that the outcome will be x must
be less than or equal to 1. Moreover, if the outcomes can
only be x or y, then the probability that it will be y is just
(1-p)
• Now suppose the outcome if the consumer does not
take some action will be z with certainty. If he does take
the action, the outcome will be either x (with probability
p) or y
decisions under uncertainty

• So if he does not take the action, his utility will be u(z),
while if he takes the action his utility will be either u(x) or
u(y)
• The expected utility hypothesis states that in this kind of
situation with uncertainty but specified outcomes and
probabilities, the consumer will choose the action that
maximizes his expected utility
• Expected utility in this case is defined as the weighted
average of the utilities of each possible outcome, with
the weights being the probabilities of each outcome
• Thus in the above example, the consumer will choose
the action that may result in either x or y if
pu(x)+(1-p)u(y)>u(z)
expected utility

• Note that this way of making the decision doesn’t
guarantee that the consumer will make the right
decision. For example, suppose the inequality holds and
the consumer chooses to take the action, but that
u(y)<u(z). Then if the consumer is unlucky and outcome
y occurs, he will be worse off than if he hadn’t taken the
action
• However, on average a consumer who consistently
follows the expected utility rule will be better off than if he
does not
• So the theory of decision under uncertainty is similar to
the theory of decisions under certainty expect that now
we maximize expected utility, not just “actual” utility
restrictions on the utility function

• In the no-uncertainty case, we found that a given
preference ordering could be represented by many
different utility functions
– for example, if u(x) could represent the preference ordering, it
could also be represented by f(u(x)) as long as f was an
increasing function
• in the uncertainty case, a given preference ordering can
also be represented by many different utility functions
• however, if u(x) represents the consumer’s preference
ordering, then any other utility function representing the
same preference ordering (say, v(x)) must be of the form
v(x) = A + Bu(x), B greater than or equal to zero
A, B constants
restriction on the utility function

• As an example, suppose that u(x) represents the utility
function of the consumer in the previous example, and
suppose that u(x)=16, u(y)=49, and u(z)=31.36. Suppose
also that p=0.5. Since 0.5(16+49)=32.5>31.36, we
conclude that the consumer should take the action
• Now suppose we instead represent the utility function by
the square root of u(x). The square root certainly is an
increasing function. But the square root of 31.36 is 5.6,
which is larger than 0.5(4+7), so with this utility function
we conclude that the consumer should not take the
action
• So the square root of u(x) does not represent the same
preference ordering under uncertainty as u(x) does
restrictions on the utility function

• To see that a transformation of the form v(x)=A+Bu(x) will
properly represent the preference ordering, suppose we
represent all possible outcomes under decision 1 as
vectors x(i) which occur with probabilities p(i), while the
outcomes under decision 2 are vectors y(j) with
probabilities p(j). Then the inequality

 p [ A  Bu( x )]  
i   i            i          j
p j [ A  Bu( y j )]
will hold for any A and any B>0 if it holds for A=0, B=1.
• In the next few slides, we consider a consumer whose
utility depends on one variable only, namely his income
Y, but whose income is uncertain (may take on any of
many possible values)
Risk

• Notice that in this case, if Y can take on many values Y(i)
and we know the probabilities p(i), we can compute the
mean, variance, and standard deviation of Y.
• The for a given mean value of of Y, the variance (or
equivalently, the standard deviation) can be thought of
as a measure of the risk that the consumer faces
because the outcome of the income variable is uncertain
• The next question is: if we think of consumers who face
different probability distributions of income, with different
means and variances, which ones will they prefer?
• The answer depends on the nature of their utility
functions
Risk aversion

• In the picture on the next slide, we measure income on
the horizontal axis and utility on the vertical axis. A
consumer with a utility function like solid curve A
displays risk aversion, whereas a consumer with a utility
function like dashed curve B is a risk lover
• For simplicity, we assume that income can take on only
two values, Y(1) and Y(2), and that the probability of
each is ½. As the curves have been drawn, two
consumers with utility functions depicted by A and B get
the same expected utility in this situation
• However, consider now a consumer who is offered the
choice between this distribution on the one hand, and
the mean income with certainty, on the other hand
Utility                          B

A

EA

E

EB

Y(1)   mean Y   Y(2)       Y

• In this diagram, the expected utility of both consumer A
and consumer B is E when income is either Y(1) or Y(2)
with probability 0.5.
• The utility the consumer would get from receiving mean
Y with certainty is different, however. Consumer A would
receive utility EA (at the level where the vertical through
mean Y intersects curve A) while consumer B would
• Thus consumer A is risk averse: he gets a higher level of
(expected) utility from receiving the mean level of income
with certainty than he gets from the lottery with Y(1) and
Y(2)

• Consumer B, on the other hand, is a risk lover. He
receives less (expected) utility from getting mean income
with certainty than he gets from taking the gamble
– note: actual utility and expected utility is the same for the case
where the consumer gets a given level of utility with certainty
• Note also the thick horizontal line segment to the right of
E. It measures the risk premium for (risk averse)
consumer A
• It can be interpreted as the amount of money that person
A would be willing to give up in order to eliminate the risk
when he will either get Y(1) or Y(2). That is, his expected
is the same as his expected utility in the risky situation
Measuring risk aversion

• Can we measure the degree of a person’s risk
aversion? One possible answer is by looking at the
second derivative of the utility function relating utility to
income
• The first derivative is the slope of utility function (which
is arbitrary, since units of utility are arbitrary). The
second derivative tells us how fast the slope changes
(the “curvature” of the utility function)
• The Arrow-Pratt measure of (absolute) risk aversion,
denoted by r, is given by:

 d 2u (Y )  u ' ' (Y )
r       2

dY         u ' (Y )
Risk aversion
• Note that this measure is positive for a person who is
risk-averse (that is, whose utility function displays
diminishing marginal utility of income). It is zero for a
risk-neutral person, and negative for a risk lover (a
person such as person B in the earlier example
• It can be proved that the Arrow-Pratt measure of risk
aversion is related to the size of a person’s risk
premium. Specifically, the larger is r, the larger is the
risk premium associated with any given level of risk
• An example of a utility function with a constant level of
absolute risk aversion is

u (Y )  e  rY , r  0
Risk aversion

• One can verify that the absolute risk aversion for this
utility function is r.
• One can also define a measure of Relative Risk
Aversion (RAV):
 u ' ' (Y )Y
RAV  rY 
u ' (Y )
• An example of a utility function that displays constant
relative risk aversion (treating utility as a function of
consumption), is

C 1
u (C ) 
1
Risk aversion

• This utility function is frequently used in
macroeconomics and other models. It has a relative
coefficient of risk aversion given by σ, and draws
attention to the fact that the coefficient of risk aversion
is the same as the elasticity of the marginal utility of
consumption or income
• Note also that if the consumer behaves in accordance
with expected utility theory, the parameter σ of the utility
function can, in principle, be estimated on the basis of
observable data (by looking at how much the consumer
is willing to pay in order to reduce risk)
Risk and insurance

• If most people are risk averse, they must be paid a risk
premium in order to take on risk. But conversely, this
also means that they are willing to pay in order to reduce
risk.
• Specifically, in the earlier example, suppose that the
difference between Y(2) and Y(1) is interpreted as a
random loss that happens to half the population
– that is, if there is no loss, a person’s annual income is Y(2). If
there is a loss, it is Y(1). In the example, half the population
suffers this loss in a given year
– This is consistent with the idea that in this population, mean
income is Y(mean), but in a given year the actual income is
either Y(2) or Y(1)
Risk and insurance

• In this situation, there should be a market for risk pooling
(insurance)
• Specifically, suppose insurers could offer plans under
of time. The insurance plan would then promise that for
those who were unlucky and suffered a loss (that is,
received an income of Y(1)), the plan would pay out a
lump-sum amount X. Those who did not suffer a loss
(that is, received an income of Y(2) would not get a
payout).
• Suppose we consider a very generous plan: it would
actually pay the whole difference [Y(2)-Y(1)] to those
who otherwise would have an income of Y(1)!
Risk and insurance, cont’d

• What would be the expected payout from such an
insurance plan for each insured person? Answ: simply
0.5(Y(2)-Y(1)). Since half the population would receive
an income of Y(1), the plan would have to pay out this
amount to half the population
• How large would the cost of the plan have to be? Answ:
more than 0.5(Y(2)-Y(1)) per insured person, since this
is the average amount that the plan would have to pay
out, per insured person
– in reality, it would have to charge a bit more than that, in order to
also earn enough revenue to pay the costs of plan
administration, profits on invested funds, etc.
Actuarially fair insurance

nothing to run the insurance plans, so that its costs
actually equal 0.5(Y(2)-Y(1)), and that this is the
– a hypothetical insurance plan for which the premium is equal to
the expected benefit payouts (with no administration costs, etc)
is called an actuarially fair insurance plan
• Note also that implicitly, we are making use here of the
law of large numbers: even though we don’t know how
much we have to pay out as benefits for an individual
insured person, we know that on average, we have to
pay out 0.5(Y(2)-Y(1)) for each person
– the law of large numbers says that if we have a large number of
insured persons, the average payment will be very close to that
amount
Actuarially fair insurance

• But for a person who is insured by such a plan, what is
his expected utility?
• Answer: it is the same as the expected utility of a person
who receives the mean income for certain
– if the person receives an income of Y(2), his net income (given
that he has paid the premium of 0.5(Y(2)-Y(1)) ahead of time) is
just Y(2)- 0.5(Y(2)-Y(1))=0.5(Y(1)+Y(2)), the mean income
– if the person receives an income of Y(1), he receives a payment
0.5(Y(2)-Y(1)). Taking this into account, his net income will also
be 0.5(Y(1)+Y(2)), the mean income
• So his expected utility is EA in the diagram: the same as
a person who gets the mean income, with no risk!
Risk, return, and portfolio choices

• A large part of the literature regarding choice under risk
deals with the problem of how consumers should
allocate their private savings (“wealth”, or “assets”)
• A whole range of assets are available in the market.
They may differ in terms of their expected (mean) yield,
but also in their risk (variability of yield)
• So buying a given asset is like buying a lottery, but to
some extent one can choose the odds
• Intuitively, if most people in the market are risk averse
(so that, in order to buy a more risky asset, they must be
compensated by a higher expected return), one would
expect that riskier assets (with higher variance, or
standard deviation) would have a higher yield on
average
Diversification and risk pooling

• However, tendency for riskier assets to have higher
yields can, to a considerable extent, be offset by the
technique of diversification
– diversification consists in pooling a large number of assets with
uncorrelated risks; because of the law of large numbers, a
portfolio of such assets will have a much lower risk than any
individual asset would have
• Note also that risk pooling can be privately profitable: if
risky assets earn returns that are much higher than less
risky ones, financial professionals can earn a profit by
buying such assets and pooling them, earning a high
rate of return with low risk (after pooling)
• They can then resell the pooled assets to risk-averse
investors
Risk pooling and diversification

• Nevertheless, risk pooling is not costless, so it will
remain true that, in equilibrium, assets with higher risk
will earn a higher average return. Moreover, “market risk”
cannot be eliminated through risk pooling for a given
asset class. Thus, for example, funds invested in the
stock market (even in pooled “mutual funds”) will tend to
carry higher risk than funds kept in savings accounts
• Therefore, an investor with a certain amount of funds to
invest faces the problem of what portion of his assets to
invest in safe (low-risk) assets with a low expected
return, and what portion in higher-risk assets with higher
expected return
• The text gives an example, pp 179-183 in my copy
Asset allocation example

that has an expected return Rf with zero risk (variance),
and one that has an expected return of Rm, with an
actual return of rm, and standard deviation σm. Suppose
also that the investor has some degree of risk aversion,
so that her expected utility can be written U=U(Rp,σp),
where the subscript p refers to her entire portfolio
– “portfolio”: weighted sum of all assets
• Let b be the fraction of her portfolio that she allocates to
the risky asset (with subscript m). We thus have
Rp=(1-b)Rf+bRm
• We also have σp=b σm
Asset allocation, concluded

• These two equations can be used to create a curve that
gives us the consumer’s tradeoff between risk and return
– the consumer can trade return for risk be changing b
– the text refers to this as a “budget line”
• Also note that one can think of U(Rp, σp) as a utility
function, and construct a map of indifference curves,
each one corresponding to a given level of utility
• Note, though, that these indifference curves will slope
upwards, not downwards (why? because risk is not a
• So what is the consumer’s optimum choice? Answ:
where she reaches the highest possible utility; at that
point an indifference curve must be tangent to the
budget line (as usual)
Rp

Risk-return
Rm

Rp*

Rf

σp*   σm       σp
The “price of risk”

• The test refers to the ratio (Rm-Rf)/ σm as “the price of
risk”. It tells the consumer how much extra return she
can get by assuming more risk
• Note that the curve showing the risk-return tradeoff in the
diagram has been extended beyond the point where
σp=σm; that is beyond the point where b = 1. Does that
make sense?
– yes in principle; the concept of “leveraging”
• Note that the equilibrium price of risk in financial markets
depends not only the degree of consumer risk aversion,
but also on the efficiency of the financial system in
eliminating risk by diversification and risk pooling

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