Choice under Uncertainty. Introduction Many choices made by consumers take place under conditions of uncertainty Therefore involves an element of risk.

Choice under Uncertainty

Introduction

• Many choices made by consumers take place under conditions of uncertainty

• Therefore involves an element of risk• We examine how the theory of consumer

choice can be used to describe such behaviour

Description of Risky Alternatives

• The consumer is concerned with the probability distribution of getting different consumption bundles.

• A probability distribution consists of a list of different outcomes or consumption and the probability associated with each outcome

• E.g. decision on how much vehicle insurance to buy or how much to invest on the stock market, is a decision on a pattern of probability distribution across different amounts of consumption.

• Each of these decisions involves a choice among risky alternatives.

Example: Lottery• Suppose a consumer has K100 & he is

contemplating buying a lotto ticket with a particular number.

• If that number is drawn in the lotto, the holder will be paid K200. This ticket costs K5.

• The two outcomes that are of interest are the event that the ticket is drawn and the event that it is not.

• The endowment amount the consumer would have if they did not buy the lotto ticket is:– K100 if the number of the ticket is drawn– K100 if it is not drawn.

• If the consumer did buy the ticket for K5 then the amount is:– K295 if the ticket is a winner– K95 if the ticket is not a winner

• The purchase of the lotto ticket changes the original endowment of probabilities of wealth in the two different circumstances.

Example: insurance• Suppose an individual has assets worth

K35,000 but there is a possibility that K10,000 worth of assets may be lost, e.g. car stolen.

• Suppose probability of this happening is P = 0.01.

• Then the probability distribution the person is facing is: – 1% probability of having K25,000 of assets – 99% probability of having K35,000.

• Insurance offers a way to change this probability distribution.

• Suppose an insurance contract will pay the person K100 if the loss occurs in exchange for a K1 premium.

• If the person decides to buy K10,000 worth of insurance, it will cost him K100.

• In this case he will have a 1% chance of having: K34,900 = (K35,000 assets - K10,000 loss + K10,000 insurance

payment - K100 insurance premium) if loss occurs.• and 99% chance of having:

K34,900 = (K35,000 assets -K100 insurance premium) if loss does not occur.

• The consumer ends up with the same wealth no matter what happens. He is fully insured against loss.

In general

• if this person purchases K kwacha of insurance and has to pay premium , then he will face the gamble:– probability 0.01 of getting K25,000 + K – and – probability 0.99 of getting K35,000

What amount of insurance will person choose?

• This depends on his preferences. • If conservative, he will purchase a lot of

insurance or take risks and not purchase any at all.

• Therefore, people have different preferences over probability distributions just like they have over the consumption of ordinary goods.

Contingent Consumption• The different outcome of a random event can be

thought of as being different states of nature. • In the insurance example, there are 2 states of nature:

the loss occurs or it does not. • A contingent consumption plan is a specification of what

will be consumed in each different state of nature. • Contingent means depending on something note yet

certain. • Therefore a contingent consumption plan depends on

the outcome of some event.

• People have preferences over different plans of contingent consumption just like they have over actual consumption.

• We can use the theory of choice developed so far to analyze the choices people make over consumption in different circumstances

• Using the example of insurance, we can describe the purchase of insurance in terms of indifference curves.

• Two states of nature: event that loss occurs and event that it does not.

• Contingent consumption plan: values of how much money you would have in each circumstance

K25,000

K35,000Endowment

ChoiceK35,000 - γK

K25,000 + K - γK Cb

Cg

• The endowment of contingent consumption is K25,000 in the “bad state” and K35,000 in the “good state”.

• Insurance offers a way to move away from this endowment point.

• By purchasing K Zmw of insurance, γK Zmw of consumption is given up in the good state in exchange of K – γK Zmw of consumption in the bad state.

• Thus consumption lost in the good state divided by the extra consumption you gain in the bad state is:

• This is the slope of the budget line. • We can also draw the indifference curve for contingent

consumption in the same manner as we did for choice under certainty.

• The choice of how much insurance to purchase given the two states of nature will be characterized by the tangency condition: MRS = price ratio.

Utility Functions and Probabilities

• How person values consumption in one state of nature compared to another will depend on the probability that the state in question will actually occur.

• In other words, the rate at which a consumer is willing to substitute consumption if loss occurs for consumption if it does not has something to do with how likely he thinks the loss will occur.

• Thus the utility function depends on the probabilities as well as the consumption levels.

• Let c1 and c2 represent consumption in states 1 and 2 and π1 and π2 be the probabilities that state 1 or 2 actually occurs.

• If the two states are mutually exclusive, then π2 = 1-π1.

• Thus the utility function can be denoted as:

• This is the function that represents the consumer’s preferences over consumption in each state.

Expected Utility

• One convenient form that the utility function might take is the following:

• Utility is a weighted sum of the some function of consumption in each state, , , where the weights are given by the probabilities , .

• If one of the states is certain, so that , then , is the utility of certain consumption in state 1. Similarly for .

• the expression: is the average utility, or expected utility of the consumption pattern.

• We refer to a utility function with this particular form as an expected utility function or a Von Neumann-Morgenstern (VNM) utility function.

• The expected utility function can be subjected to some kinds of monotonic transformation and still have the expected utility property.

• We say that a function v(u) is a positive affine transformation if it can be written in the form: v(u) = au + b where a > 0.

• A positive affine transformation simply means multiplying by a positive number and adding a constant.

• If you subject an expected utility function to a positive affine transformation, it not only represents the same preferences, but it will also still has the expected utility property.

• Thus we say that an expected utility function is unique up to an affine transformation.

• This means that you can apply an affine transformation to it and get another expected utility function that represents the same preferences.

• But any other kind of transformation will destroy the expected utility property.

Risk Aversion

• Suppose the consumer currently has K10 worth of wealth and he is contemplating a gamble that gives him a 50% probability of winning K5 and 50% probability of losing K5.

• His wealth will therefore be random: 50% of ending up with K5 and 50% of ending up with K15.

• The expected value of his wealth is K10 and the expected utility is:

Fig 1

Risk averse

12𝐾 5+

12𝐾 15 Wealth

Utility

K15K5

• The expected utility of wealth is the average of and labeled .

• The utility of the expected value of wealth is labeled

• In the diagram, the expected utility of wealth is less that the utility of the expected value wealth:

• In this case we say that the consumer is risk averse since he prefers to have the expected value of his wealth rather than face the gamble.

• In general, an individual is risk-averse if he prefers the utility of the expected value of a gamble than the expected utility of a gamble.

• Let g = gamble and u(g) = expected utility of a gamble, then:– Risk averse at g if – Risk neutral at g if – Risk loving at g if

• The risk-averse consumer has a concave utility function__the chord btw any 2 points of the graph of the utility function lie below the function.

• Thus concavity of the expected utility function is equivalent to risk aversion.

• Risk aversion implies that • The risk-loving consumer has a convex utility function. • The curvature of the utility function measures the

consumer’s attitude toward risk. • The intermediate case is a linear utility function for a

consumer who is risk neutral.

Fig 2

Risk loving

12𝐾 5+

12𝐾 15 Wealth

Utility

K15K5

• In Fig 1 the consumer prefers K10 with certainty to the gamble (lottery) itself

• But there will be some amount of wealth we could offer with certainty that would make the consumer indifferent between accepting that wealth and facing the gamble.

• We call this wealth the certainty equivalent of the gamble.

• When a person is risk averse and strictly prefers more money to less, the CE is less than the expected value of the gamble:

• A person is willing to pay some positive amount of wealth to avoid the gamble’s inherent risk

• This explains why people buy insurance• They are giving up a certain amount, (risk premium) to avoid

the risky outcome they are being insured against.• Willingness to pay to avoid risk is measured by the risk

premium

Certainty Equivalent & Risk Premium

Risk averse

12𝐾 5+

12𝐾 15 Wealth

Utility

K15K5 CE

P

Insurance Example 1

• A risk averse individual with initial wealth and VNM utility function must decide whether and for how much to insure his car.

• The probability that he will have an accident and incur a kwacha loss of L is

• How much insurance should he purchase?• Let us suppose insurance is available at an

actuarially fair price, i.e, one that yields insurance companies zero expected profits.

• If denotes the rate at which each kwacha of insurance can be purchased, the insurance co expected profits per kwacha insurance sold will be:

• With price per kwacha of insurance = • Our risk-averse individual will choose x to

maximize expected utility:• • Differentiating w.r.t and setting result to zero:

• = 0• =• Dividing both side by yields:• =• Because the individual is risk averse, so that the marginal

utility of wealth is strictly decreasing in wealth thus if the above is equal, then:

• Thus • In other words, when insurance is available at some fair price,

a risk averse individual will fully insure against risk.• Note that at the optimum, the individual’s wealth is constant

and equal to whether or not he has an accident.

Insurance Example

• Consider a person with current wealth K100,000 and faces the prospect of a 25% chance of losing her car worth K20,000 due to theft in 2015.

• Suppose this person’s VNM utility is given by • 1. Calculate the expected utility if the person• faces next year without insurance:• Expected utility =

• 2. Assuming that insurance is available at an actuarially fair price, calculate the expected utility if person completely insures the car.

• Fair insurance premium = K5,000 (25% of K20,000)• If car stolen: w = 80,000 + 20,000 – 5000 = 95,000• If car not stolen: w = 100,000-5,000 = 95,000• Expected utility =

• Person is made better off by purchasing fair insurance

• 3. What is the maximum premium would be paid to cover the expected value of the loss?

• Let maximum amount = • Expected utility

• Therefore in addition to K5,000, the person would be willing to pay up to K426 in admin costs to insurance co.

• Even when these costs paid, this person is well as he would be when facing the world uninsured.

Measuring risk aversion• We not only want to know whether someone is risk

averse but also how risk averse they are.• The most commonly measure of risk aversion is

defined as:

• The Arrow-Pratt measure of absolute risk aversion.• Because the distinguishing feature of risk averse

individuals is a diminishing marginal utility of wealth [ the Arrow-Pratt measure is +ve in such cases.

• Generally, is +ve, -ve or zero as the individual is risk averse, risk loving, or risk neutral, respectively.

Risk aversion and wealth

• Does risk aversion increases or decreases with wealth?

• Intuitively one might think that the willingness to pay to avoid a gamble decline as wealth increases bcos diminishing marginal utility would make potential loses less serious for high wealth individuals

• The intuitive answer is not correct bcos diminishing marginal utility makes gains from winning gambles less attractive

• Net result therefore indeterminate• It depends on the shape of the utility function• Arrow has proposed a simple classification of VNM

utility functions (or utility function segments) according to how varies with wealth

• A VNM utility function displays constant, decreasing or increasing absolute risk aversion over some domain of wealth if over that interval, remains constant, decreases, or increases with an increase in wealth respectively.

• Decreasing absolute risk aversion (DARA) is generally a plausible restriction to impose: the higher the level of wealth, the less averse to taking small gambles (risks)

• Under constant absolute risk aversion (CARA), there is no greater willingness to pay to avoid a gamble at higher levels of wealth

• Under increasing absolute risk aversion, the greater wealth, the more one is willing to pay to avoid a gamble, i.e. the more averse one becomes to accepting the same small gamble__a rather perverse behaviour.

Examples

• Calculate for the following utility functions and state what happens as wealth increases:

• 1.

• If utility is quadratic in wealth increases as wealth increases

• 2.

• If utility is logarithmic in wealth, decreases as wealth increases

• 3. Exponential utility function:

• Where s a +ve constant• Exhibits constant absolute risk aversion over

all ranges of wealth.

Relative Risk Aversion

• It is unlikely that the willingness to pay to avoid a given gamble is independent of a person’s wealth.

• A more appealing assumption may be that such willingness to pay is inversely proportional to wealth:

• Is a measure of relative risk aversion.