CHAPTER 12
UNCERTAINTY
Uncertainty ís a act of life. People face risks every time they take a shower,
walk across the street, or make an investment. But there are financial insti
tutions such as insurance markets and the stock market that can mitigaic
at least some of these risks. We will study the functioning of these mar
kets in the next chapter, but first we must study individual behavior with
Yespect to choices involving uncertainty.
12.1.ContingentConsumption
Since wie 0w know all about the standard theory of consumer choice, let's
try to uSe what we know to understand choice under uncertainty. The first
questi:n to ask is what is the basic thing" that is being chosen?
The corsumer is presumably concerned with the probability distri
bution of gettng different consumption bundles of goods. A probability
listribution consists of a list of different outcornes in this case, consUnip
tion budles and tie prsbabil1ty dss0ciated with each oitcoine. Vhen a
COnsurIeI decides }ow iih automobile insurance to buv or how CH t
, 216 UNCERTANTY (Ch, 12) (ONTINGENT CONSUMPTION 21:
invest in the stock market, he is in effect deciding on a pattern of probability and
distribution across different amounts of consumption.
For example, suppose that you have $100 now and that you are con probability .99 of getting $35, 000 - yK.
templating buying lottery ticket number 13. If number 13 is drawn in the
lottery, the holder will be paid $200. This ticket costs, say, $5. The two What kind of insurance will this person choose? Well, that depends on
outcomes that are of interest. are the event that the ticket is drawn and the his preferences. He might be very conservative and choose to purchase a lot
event that it isn't. of insurance, or he might like to take risks and not purchase any insurance
Your original endowment of wealththe amount that you would have if at all. People have different preferences over probability distributions in
you did not purchase the lottery ticket-is $100 if 13 is drawn, and $100 the same way that they have different preferences over the consumption of
if it isn't drawn. But if you buy the lottery ticket for $5, you will have ordinary goods.
a wealth distribution consisting of $295 if the ticket is a winner, and $95 In fact, one very fruitful way to look at decision making under uncertainty
if it is not a winner. The original endowment of probabilities of wealth is just to think of the money available under different circumstances as
in different circumstances has been changed by the purchase of the lottery different goods. A thousand dollars after a large loss has occurred may
ticket. Let us examine this point in more detail. mean a very different thing from a thousand dollars when it hasn't. Of
In this discussion we'll restrict ourselves to examining monetary gambles course, we don't have to apply this idea just to money: an ice cream cone
for convenience of exposition. Of course, it is not money alone that mat if it happens to be hot and sunny tomorrow is a very different good fTom
ters; it is the consumption that money can buy that is the ultimate "good" an ice cream cone if it is rainy and cold. In general., consumption goods will
being chosen. The same principles apply to gambles over goods, but re be of different value to a person depending upon the circumstances under
which they become available.
stricting ourselves to monetary outcomes makes things simpler. Second, Let us think of the different outcomnes of some random event as
we will restrict ourselves to very simple situations where there are only a being
few possible outcomes. Again, this is only for reasons of simplicity. different states of nature. In the insurance example given above there
were two states of nature: the loss occurs or it doesn't. But in general
Above we described the case of gambling in a lottery; here we'll consider
there could be many different states of nature. We can then think of
the case of insurance. Suppose that an individual initially has $35,000
worth of assets, but there is possibility that he may lose $10,000. For a contingent consumption plan as being a specification of what will
be consumed in each different state of natureeach different outcome of
example, his car may be stolen, or a storm may damage his house. Suppose the random process. Contingent means depending on something not yet
that the probability of this event happening is p =.01. Then the probability
distribution the person is facing isa l percent probability of having $25,000 certain, so a contingent consumption plan means a plan that depends on the
outcome of some event. In the case of insurance purchases, the contingent
of assets, and a 99 percent probability of having $35,000.
Insurance offers a way to change this probability distribution. Suppose consumption was described by the terms of the insurance contract: how
much money you would have if a loss occurred and how much you would
that there is an insurance contract that will pay the person $100 if the loss have if it didn't. In the case of the rainy and sunny days, the contingent
occurs in exchange for a $l premium. Of course the premium must be paid
whether or not the loss Occurs. If the person decides to purchase $10,000 consumption would just be the plan of what would be consumed given the
various outcomes of the weather.
dollars of insurance, it will cost him $100. In this case he will have a l
percent chance of having $34,900 ($35,000 of other assets$10,000 loss + People have preferences over different plans of consumption, just like
$10,000 payment from the insurance payment - $100 insurance premium) they have preferences over actual consumption. It certainly might make
and a 99 percent chance of having $34,900 ($35,000 of assets $100 in youfeel better now to know that you are fully insured. People make choices
that reflect their preferences over consumption in different circumstances,
surance prenium). Thus the consumer ends up with the same wealth no
matter what happens. He is now fully insured against loss. and we can use the theory of choice that we have developed to analyze
has to pay those choices.
In general, if this person purchases K dollars of insurance and If we think about a contingent consumnption plan as being just an ordi
a premium yk, then he will face the gamble:! nary consumption bundle, we are right back in the framework described in
probability .01 of getting $25, 000 +K - yK the previous chapters. We can think of preferences as being defined over
different consumption plans, with the "terms of trade" being given by the
budget constraint. We can then model the consumner as choosing the best
consumption plan he or she can afford, just as we have done all along.
1 The Greek letter Y, gamma, is pronounced "gam-ma."
UNCERTAINTY
Uncertainty ís a act of life. People face risks every time they take a shower,
walk across the street, or make an investment. But there are financial insti
tutions such as insurance markets and the stock market that can mitigaic
at least some of these risks. We will study the functioning of these mar
kets in the next chapter, but first we must study individual behavior with
Yespect to choices involving uncertainty.
12.1.ContingentConsumption
Since wie 0w know all about the standard theory of consumer choice, let's
try to uSe what we know to understand choice under uncertainty. The first
questi:n to ask is what is the basic thing" that is being chosen?
The corsumer is presumably concerned with the probability distri
bution of gettng different consumption bundles of goods. A probability
listribution consists of a list of different outcornes in this case, consUnip
tion budles and tie prsbabil1ty dss0ciated with each oitcoine. Vhen a
COnsurIeI decides }ow iih automobile insurance to buv or how CH t
, 216 UNCERTANTY (Ch, 12) (ONTINGENT CONSUMPTION 21:
invest in the stock market, he is in effect deciding on a pattern of probability and
distribution across different amounts of consumption.
For example, suppose that you have $100 now and that you are con probability .99 of getting $35, 000 - yK.
templating buying lottery ticket number 13. If number 13 is drawn in the
lottery, the holder will be paid $200. This ticket costs, say, $5. The two What kind of insurance will this person choose? Well, that depends on
outcomes that are of interest. are the event that the ticket is drawn and the his preferences. He might be very conservative and choose to purchase a lot
event that it isn't. of insurance, or he might like to take risks and not purchase any insurance
Your original endowment of wealththe amount that you would have if at all. People have different preferences over probability distributions in
you did not purchase the lottery ticket-is $100 if 13 is drawn, and $100 the same way that they have different preferences over the consumption of
if it isn't drawn. But if you buy the lottery ticket for $5, you will have ordinary goods.
a wealth distribution consisting of $295 if the ticket is a winner, and $95 In fact, one very fruitful way to look at decision making under uncertainty
if it is not a winner. The original endowment of probabilities of wealth is just to think of the money available under different circumstances as
in different circumstances has been changed by the purchase of the lottery different goods. A thousand dollars after a large loss has occurred may
ticket. Let us examine this point in more detail. mean a very different thing from a thousand dollars when it hasn't. Of
In this discussion we'll restrict ourselves to examining monetary gambles course, we don't have to apply this idea just to money: an ice cream cone
for convenience of exposition. Of course, it is not money alone that mat if it happens to be hot and sunny tomorrow is a very different good fTom
ters; it is the consumption that money can buy that is the ultimate "good" an ice cream cone if it is rainy and cold. In general., consumption goods will
being chosen. The same principles apply to gambles over goods, but re be of different value to a person depending upon the circumstances under
which they become available.
stricting ourselves to monetary outcomes makes things simpler. Second, Let us think of the different outcomnes of some random event as
we will restrict ourselves to very simple situations where there are only a being
few possible outcomes. Again, this is only for reasons of simplicity. different states of nature. In the insurance example given above there
were two states of nature: the loss occurs or it doesn't. But in general
Above we described the case of gambling in a lottery; here we'll consider
there could be many different states of nature. We can then think of
the case of insurance. Suppose that an individual initially has $35,000
worth of assets, but there is possibility that he may lose $10,000. For a contingent consumption plan as being a specification of what will
be consumed in each different state of natureeach different outcome of
example, his car may be stolen, or a storm may damage his house. Suppose the random process. Contingent means depending on something not yet
that the probability of this event happening is p =.01. Then the probability
distribution the person is facing isa l percent probability of having $25,000 certain, so a contingent consumption plan means a plan that depends on the
outcome of some event. In the case of insurance purchases, the contingent
of assets, and a 99 percent probability of having $35,000.
Insurance offers a way to change this probability distribution. Suppose consumption was described by the terms of the insurance contract: how
much money you would have if a loss occurred and how much you would
that there is an insurance contract that will pay the person $100 if the loss have if it didn't. In the case of the rainy and sunny days, the contingent
occurs in exchange for a $l premium. Of course the premium must be paid
whether or not the loss Occurs. If the person decides to purchase $10,000 consumption would just be the plan of what would be consumed given the
various outcomes of the weather.
dollars of insurance, it will cost him $100. In this case he will have a l
percent chance of having $34,900 ($35,000 of other assets$10,000 loss + People have preferences over different plans of consumption, just like
$10,000 payment from the insurance payment - $100 insurance premium) they have preferences over actual consumption. It certainly might make
and a 99 percent chance of having $34,900 ($35,000 of assets $100 in youfeel better now to know that you are fully insured. People make choices
that reflect their preferences over consumption in different circumstances,
surance prenium). Thus the consumer ends up with the same wealth no
matter what happens. He is now fully insured against loss. and we can use the theory of choice that we have developed to analyze
has to pay those choices.
In general, if this person purchases K dollars of insurance and If we think about a contingent consumnption plan as being just an ordi
a premium yk, then he will face the gamble:! nary consumption bundle, we are right back in the framework described in
probability .01 of getting $25, 000 +K - yK the previous chapters. We can think of preferences as being defined over
different consumption plans, with the "terms of trade" being given by the
budget constraint. We can then model the consumner as choosing the best
consumption plan he or she can afford, just as we have done all along.
1 The Greek letter Y, gamma, is pronounced "gam-ma."
