Intermediate Microeconomicsbvankamm/Files/340 Notes/ECON 301 N… · If there are 3 possible...

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Intermediate Microeconomics UNCERTAINTY AND RISK BEN VAN KAMMEN, PHD PURDUE UNIVERSITY

Transcript of Intermediate Microeconomicsbvankamm/Files/340 Notes/ECON 301 N… · If there are 3 possible...

Page 1: Intermediate Microeconomicsbvankamm/Files/340 Notes/ECON 301 N… · If there are 3 possible payoffs for a lottery ticket ($0, $5, and $50) and the payoffs have probabilities (0.5,

Intermediate Microeconomics

UNCERTAINTY AND RISKBEN VAN KAMMEN, PHDPURDUE UNIVERSITY

Page 2: Intermediate Microeconomicsbvankamm/Files/340 Notes/ECON 301 N… · If there are 3 possible payoffs for a lottery ticket ($0, $5, and $50) and the payoffs have probabilities (0.5,

Discrete probabilityProbability: The relative frequency with which an event occurs.◦ Usually expressed as a fraction of 1. Pr. ϵ [0,1].

Expected Value: An average of probable outcomes weighted by how probable each outcome is.

Fair Gamble: A gamble with an expected outcome of zero.

Risk: A gamble with a positive probability of loss.

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Expected value (formally)If there are 3 possible payoffs for a lottery ticket ($0, $5, and $50) and the payoffs have probabilities (0.5, 0.4, and 0.1) respectively, the expected value of the lottery ticket is:

𝐸𝐸 $ = 0.5 ∗ 0 + 0.4 ∗ 5 + 0.1 ∗ 50 = $7.

To generalize, if there are N outcomes, each with payoff 𝑉𝑉𝑛𝑛and probability 𝑝𝑝𝑛𝑛, then the expected value of the payoff is:

𝐸𝐸($) = �𝑛𝑛=1

𝑁𝑁

(𝑉𝑉𝑛𝑛 ∗ 𝑝𝑝𝑛𝑛) .

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Fair gamblesIf you are offered a lottery ticket with the previously mentioned probabilities and payoffs, it would be a fair gamble is the ticket costs $7.◦ 𝐸𝐸(𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔) = 𝐸𝐸(𝑝𝑝𝑔𝑔𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝) − 𝐶𝐶𝑝𝑝𝐶𝐶𝐶𝐶.◦ If the expected payoff and cost are the same, the expected value of

the gamble is zero.

Whether or not a consumer accepts a fair gamble says something about his risk preference.

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Risk aversion, neutrality, loveIf a person refuses a fair gamble as defined previously, he is averse to risk,◦prefers to have the status quo with probability 1.

If a person accepts any fair gamble, he is risk neutral.◦The expected outcome of the lottery and the status quo are equally

acceptable to him.

Willingness to accept gambles that are less than fair, merely for the “thrill” of the risk, is characterized as risk loving.

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Risk and utilityAs it turns out, we can model individuals’ risk preferences using utility theory.

Instead of examining the way individuals trade off consuming two goods, we consider a utility function with only one argument: income.

To demonstrate that an individual is risk averse, all we need is diminishing marginal utility of income.

Since income is a good, its marginal utility will always be non-negative, but the question is whether marginal utility diminishes as income increases or not.

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Diminishing marginal utility of income

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Risk aversionAny utility function that resembles the previous slide represents risk averse preferences.◦ A simple example will demonstrate this.

Begin from any arbitrary level of income. Mark it on the horizontal axis.

Trace vertically to the utility curve. That is the utility level the person has to begin.

Imagine that the person is offered a fair gamble with a 0.5 probability of winning/losing “x” dollars.

Move an equal distance in both directions from the initial income level.

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Risk aversion

𝑈𝑈 𝑀𝑀0

𝑀𝑀0 𝑀𝑀0 + 𝑥𝑥𝑀𝑀0 − 𝑥𝑥

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Risk aversionTrace vertically from both of the potential outcomes of the gamble to the utility function.◦ These are the two payoffs of the gamble measured in utility terms.◦ The expected value is the midpoint of a line connecting these two

utility levels.The star on the following graph represents the expected utility resulting from the gamble.You can verify that it is less than the original utility level without the gamble.

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Risk aversion

𝑈𝑈 𝑀𝑀0

𝑀𝑀0 𝑀𝑀0 + 𝑥𝑥𝑀𝑀0 − 𝑥𝑥

𝑈𝑈 𝑀𝑀0 + 𝑥𝑥

𝑈𝑈 𝑀𝑀0 − 𝑥𝑥

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Certainty equivalenceWhat if—instead of voluntarily choosing the gamble—the individual was forced to partake. ◦ How much would he be willing to pay to avoid the risk of the

gamble?

Take the expected utility level from the gamble and trace horizontally to the left until you hit the utility curve.From there trace downward to the horizontal axis to find the level of income that will give the person the same utility with certainty that the gamble offers with risk.◦ This is called the individual’s certainty equivalence.

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Risk aversion

𝑈𝑈 𝑀𝑀0

𝑀𝑀0

𝑈𝑈 𝑀𝑀0 + 𝑥𝑥

𝑈𝑈 𝑀𝑀0 − 𝑥𝑥

C

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Willingness to pay for insuranceThe difference between 𝑀𝑀0 (initial income) and C (certainty equivalence) is the person’s willingness to pay for insurance.

The person would be willing to give up 𝑀𝑀0 − 𝐶𝐶 dollars with certainty to avoid the risk of losing x dollars from the gamble.◦ If someone is offering insurance for less than this amount, the person

in question will jump at the chance. ◦ If insurance costs him more than this, he will reluctantly bear the risk

instead of buying it.

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Uncertainty and informationInformation is the “cure” for uncertainty. If, for example, you had perfect information about whether the lottery ticket was a winner or not, there would be no risk:◦only buy the winning ticket, and don’t bother with the losers.◦You’ll be rich in no time if you can just know everything!

But realistically it is impossible to know many of the uncertain future events.◦And if you try to hold a lottery ticket up to an X-Ray machine to see if

it’s a winner, the clerk will probably kick you out of the store.

Sometimes, there are signals, that can be used to improve one’s information about the future, make better decisions, and partially reduce risk.

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ProbabilitiesPrior Probability: the ex ante (uninformed) probability that an event will occur.Likelihood: the probability of receiving a particular signal prior to an event’s occurrence.Joint Probability: the probability of a particular event and a particular event occurring.Bayesian Posterior Probability: the probability of an event occurring, given a particular signal.Bayes’s Rule: a formula for posterior probability as a function of likelihoods and prior probabilities.

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Bayes’s RuleIn this example there are two possible events, {𝐶𝐶1, 𝐶𝐶2} and two possible signals, {𝑧𝑧1, 𝑧𝑧2}.◦ 𝐶𝐶1 and 𝐶𝐶2 have prior probabilities, 𝑝𝑝 𝐶𝐶1 ,𝑝𝑝 𝐶𝐶2 .◦ 𝑧𝑧1 has likelihoods, 𝑝𝑝 𝑧𝑧1 𝐶𝐶1 ,𝑝𝑝 𝑧𝑧1 𝐶𝐶2 .◦ 𝑧𝑧2 has likelihoods, 𝑝𝑝 𝑧𝑧2 𝐶𝐶1 , 𝑝𝑝 𝑧𝑧2 𝐶𝐶2 .

The two signals are mutually exclusive, so exactly one signal is observed.◦ 𝑝𝑝 𝑧𝑧1 𝐶𝐶1 + 𝑝𝑝 𝑧𝑧2 𝐶𝐶1 = 1, and 𝑝𝑝 𝑧𝑧1 𝐶𝐶2 + 𝑝𝑝 𝑧𝑧2 𝐶𝐶2 = 1.

The combinations of events and signals have joint probabilities:

𝑝𝑝 𝑧𝑧1, 𝐶𝐶1 = 𝑝𝑝 𝑧𝑧1 𝐶𝐶1)𝑝𝑝(𝐶𝐶1) and 𝑝𝑝(𝑧𝑧1, 𝐶𝐶2) = 𝑝𝑝 𝑧𝑧1 𝐶𝐶2)𝑝𝑝(𝐶𝐶2)𝑝𝑝(𝑧𝑧2, 𝐶𝐶1) = 𝑝𝑝 𝑧𝑧2 𝐶𝐶1)𝑝𝑝(𝐶𝐶1) and 𝑝𝑝(𝑧𝑧2, 𝐶𝐶2) = 𝑝𝑝(𝑧𝑧2|𝐶𝐶2)𝑝𝑝(𝐶𝐶2).

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Bayes’s Rule (continued)The overall probabilities of getting each signal are:

𝑝𝑝(𝑧𝑧1) = 𝑝𝑝 𝑧𝑧1 𝐶𝐶1)𝑝𝑝(𝐶𝐶1) + 𝑝𝑝(𝑧𝑧1 𝐶𝐶2 𝑝𝑝 𝐶𝐶2 , and𝑝𝑝(𝑧𝑧2) = 𝑝𝑝 𝑧𝑧2 𝐶𝐶1)𝑝𝑝(𝐶𝐶1) + 𝑝𝑝(𝑧𝑧2 𝐶𝐶2 𝑝𝑝 𝐶𝐶2 .

Bayes’s rule is a way of converting between conditional probabilities. It states that:

𝑝𝑝 𝐶𝐶1 𝑧𝑧1) =𝑝𝑝 𝑧𝑧1 𝐶𝐶1 𝑝𝑝 𝐶𝐶1

𝑝𝑝(𝑧𝑧1|𝐶𝐶1) + 𝑝𝑝 𝑧𝑧1|𝐶𝐶2,

𝑝𝑝 𝐶𝐶1 𝑧𝑧2) =𝑝𝑝 𝑧𝑧2 𝐶𝐶1 𝑝𝑝 𝐶𝐶1

𝑝𝑝(𝑧𝑧2|𝐶𝐶1) + 𝑝𝑝 𝑧𝑧2|𝐶𝐶2,

𝑝𝑝 𝐶𝐶2 𝑧𝑧1) =𝑝𝑝 𝑧𝑧1 𝐶𝐶2 𝑝𝑝 𝐶𝐶2

𝑝𝑝(𝑧𝑧1|𝐶𝐶1) + 𝑝𝑝 𝑧𝑧1|𝐶𝐶2, and

𝑝𝑝 𝐶𝐶2 𝑧𝑧2) =𝑝𝑝 𝑧𝑧2 𝐶𝐶2 𝑝𝑝 𝐶𝐶2

𝑝𝑝(𝑧𝑧2|𝐶𝐶1) + 𝑝𝑝 𝑧𝑧2|𝐶𝐶2.

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ExampleA local bar has a special on “Mystery Beers”:◦ a brown paper bag with a random bottle of beer in it for $1.

You inquire with the bartender about the kind of beer you are likely to get, and she tells you that:◦ 14

of them are Blatz, 14

are Miller High Life, 110

are Lakefront ESB, and 4

10are Sam Adams Boston Lager.

◦ Except for High Life’s clear bottle, they all come in a brown bottle.

As the barkeep is telling you this, you glimpse her opening the next Mystery Beer and can tell that it’s a brown bottle.◦ What is the probability of getting each brand, now that you

have this information?

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Example (continued)Prior probabilities: 𝑝𝑝(𝐵𝐵𝑔𝑔𝑔𝑔𝐶𝐶𝑧𝑧) = 0.25, 𝑝𝑝(𝑀𝑀𝑀𝑀𝑔𝑔𝑔𝑔𝑔𝑔𝑀𝑀) = 0.25, 𝑝𝑝(𝐿𝐿𝑔𝑔𝐿𝐿𝑔𝑔𝑝𝑝𝑀𝑀𝑝𝑝𝐿𝐿𝐶𝐶) = 0.1, and 𝑝𝑝(𝑆𝑆𝑆𝑆) = 0.4.

Probability of brown bottle, given brand: 𝑝𝑝(𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿|𝑆𝑆𝑆𝑆) = 𝑝𝑝(𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿|𝐿𝐿𝑔𝑔𝐿𝐿𝑔𝑔𝑝𝑝𝑀𝑀𝑝𝑝𝐿𝐿𝐶𝐶) = 𝑝𝑝(𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿|𝐵𝐵𝑔𝑔𝑔𝑔𝐶𝐶𝑧𝑧)= 1; 𝑝𝑝(𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿|𝑀𝑀𝑀𝑀𝑔𝑔𝑔𝑔𝑔𝑔𝑀𝑀) = 0.

Probability of brown bottle and brand:𝑝𝑝(𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿,𝐵𝐵𝑔𝑔𝑔𝑔𝐶𝐶𝑧𝑧) = 0.25,𝑝𝑝(𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿, 𝑆𝑆𝑆𝑆) = 0.4,

𝑝𝑝(𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿, 𝐿𝐿𝑔𝑔𝐿𝐿𝑔𝑔𝑝𝑝𝑀𝑀𝑝𝑝𝐿𝐿𝐶𝐶) = 0.1,𝑝𝑝(𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿,𝑀𝑀𝑀𝑀𝑔𝑔𝑔𝑔𝑔𝑔𝑀𝑀) = 0.

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Example (continued)Probability of signal (brown bottle)

𝑝𝑝 𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿 = .25 + .4 + .1 = 0.75.Probability of brand, given brown bottle:

𝑝𝑝 𝐵𝐵𝑔𝑔𝑔𝑔𝐶𝐶𝑧𝑧 𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿 =.25.75

=13

, 𝑝𝑝 𝑆𝑆𝑆𝑆 𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿 =.4

.75=

815

,

𝑝𝑝 𝐿𝐿𝑔𝑔𝐿𝐿𝑔𝑔𝑝𝑝𝑀𝑀𝑝𝑝𝐿𝐿𝐶𝐶 𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿 =.1

.75=

215

,𝑝𝑝(𝑀𝑀𝑀𝑀𝑔𝑔𝑔𝑔𝑔𝑔𝑀𝑀|𝑔𝑔𝑀𝑀𝑝𝑝𝑏𝑏𝐿𝐿) = 0.

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Value of informationThe information communicated by a signal can have value to the decision-maker if it leads him to better decisions.

To illustrate this, consider another example.◦ There are two possible events that can occur, {run, pass} with prior

probabilities {0.47,0.53}.◦ There are two possible signals that can occur, {Full Back, No Full Back} with

likelihoods, {𝑝𝑝 𝐹𝐹𝐵𝐵 𝑀𝑀𝑟𝑟𝐿𝐿 = 0.6, 𝑝𝑝 𝐹𝐹𝐵𝐵 𝑝𝑝𝑔𝑔𝐶𝐶𝐶𝐶 = 0.3}.◦ There are two possible actions for the defense confronted with this decision,

{man, zone}.◦ The payoffs for the defense that plays “man” are:

{5 if run, -2 if pass}.◦ The payoffs for the defense that plays “zone” are:

{0 if run, 3 if pass}.

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Value of the decision without informationIf the defense has no signal to go on, it chooses the action with the highest expected value:

𝐸𝐸(𝑝𝑝𝑔𝑔𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝|𝑔𝑔𝑔𝑔𝐿𝐿) = 5 0.47 − 2 0.53 = 1.29𝐸𝐸 𝑝𝑝𝑔𝑔𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑧𝑧𝑝𝑝𝐿𝐿𝑔𝑔 = 0 0.47 + 3 0.53 = 1.59.

In this case, playing “zone” is the wiser move in the absence of information.

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Bayesian posterior probabilitiesHow does the signal of a Full Back in the backfield change the defense’s expectations?Joint probabilities:

𝑝𝑝 𝑀𝑀𝑟𝑟𝐿𝐿,𝐹𝐹𝐵𝐵 = 0.6 0.47 = 0.282𝑝𝑝 𝑀𝑀𝑟𝑟𝐿𝐿,𝑁𝑁𝐹𝐹𝐵𝐵 = 0.4 0.47 = 0.188𝑝𝑝 𝑝𝑝𝑔𝑔𝐶𝐶𝐶𝐶,𝐹𝐹𝐵𝐵 = 0.3 0.53 = 0.159𝑝𝑝 𝑝𝑝𝑔𝑔𝐶𝐶𝐶𝐶,𝑁𝑁𝐹𝐹𝐵𝐵 = 0.7 0.53 = 0.371

Signal Probabilities:𝑝𝑝 𝐹𝐹𝐵𝐵 = 0.282 + 0.159 = 0.441

𝑝𝑝 𝑁𝑁𝐹𝐹𝐵𝐵 = 0.188 + 0.371 = 0.559

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Bayesian posterior probabilitiesPosterior Probabilities: joint divided by signal probability.

𝑝𝑝(𝑀𝑀𝑟𝑟𝐿𝐿,𝐹𝐹𝐵𝐵) =0.2820.441

= 0.639

𝑝𝑝(𝑀𝑀𝑟𝑟𝐿𝐿,𝑁𝑁𝐹𝐹𝐵𝐵) =0.1880.559

= 0.336

𝑝𝑝(𝑝𝑝𝑔𝑔𝐶𝐶𝐶𝐶,𝐹𝐹𝐵𝐵) =0.1590.441

= 0.361

𝑝𝑝 𝑝𝑝𝑔𝑔𝐶𝐶𝐶𝐶,𝑁𝑁𝐹𝐹𝐵𝐵 =0.3710.559

= 0.664

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Conditional optimal actionsIf the defense observes the Full Back in the backfield, what is its best decision?◦ Compare the expected values of the two decisions using Bayesian

posterior probabilities:𝐸𝐸(𝑝𝑝𝑔𝑔𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 | 𝐹𝐹𝐵𝐵,𝑔𝑔𝑔𝑔𝐿𝐿) = (0.639)(5) − (0.361)(2) = 2.473𝐸𝐸(𝑝𝑝𝑔𝑔𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 | 𝐹𝐹𝐵𝐵, 𝑧𝑧𝑝𝑝𝐿𝐿𝑔𝑔) = (0.639)(0) + (0.361)(3) = 1.083

◦ Man is the better decision when you observe the signal, “FB”.

If no Full Back is in, what is best?𝐸𝐸(𝑝𝑝𝑔𝑔𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 | 𝑁𝑁𝐹𝐹𝐵𝐵,𝑔𝑔𝑔𝑔𝐿𝐿) = (0.336)(5) − (0.664)(2) = 0.352𝐸𝐸(𝑝𝑝𝑔𝑔𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 | 𝑁𝑁𝐹𝐹𝐵𝐵, 𝑧𝑧𝑝𝑝𝐿𝐿𝑔𝑔) = (0.336)(0) + (0.664)(3) = 1.992

◦ Zone is the better decision when you observe the signal, “NFB”.

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Value of informationThe expected value of the decision over the possible signals is:𝐸𝐸 𝑝𝑝𝑔𝑔𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑔𝑔𝐿𝐿𝑝𝑝 𝐶𝐶𝑀𝑀𝑔𝑔𝐿𝐿𝑔𝑔𝑔𝑔 = 0.441 2.473 + 0.559 1.992= 2.204.

◦ The defense observes signal “FB” with probability 0.441, and when it does, its optimal action (man) has an expected value of 2.473.

◦ The defense observes signal “NFB” with probability 0.559, and when it does, its optimal action (zone) has an expected value of 1.992.

If we subtract from this average, the value of the uninformed optimal decision (zone every time), we get the value of the information.

Value of Information = 2.204 − 1.59 = 0.614.

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InformationInformation has value to any individual regardless of his risk preference.Perfect Information (no uncertainty) gives an upper bound for the value of a decision◦ In this case perfect information would be worth

0.47 5 + 0.53 3 = 3.94.◦ The signal that we have used as information is “imperfect” in the

sense that it doesn’t fully reveal what is going to happen.

Economists (and particularly econometricians) spend a lot of time trying to find reliable signals that will help forecast future events so they can give better advice to policy makers.

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SummaryExpected Value is an average of outcomes weighted by their probabilities.

A fair gamble has an expected value of zero.

Risk averse people will refuse fair gambles because they don’t like the risk of loss.

Risk aversion comes from a utility function that has diminishing marginal utility of income.

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SummaryInsurance is one way to deal with risk.Gaining information is another way of dealing with risk if there exists a signal that can be used as information.Incorporation of new information into expectations is done by Bayes’s Rule and the calculation of Bayesian Posterior probabilities.Information has value when it enables the decision maker to make optimal decisions more often◦ better decisions than he would have without the information.

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ConclusionUncertainty is not the exception in Economics; it is the norm.

Uncertainty in the present can have even more interesting consequences when an economic agent considers a longer time horizon.◦ Many Macroeconomics classes consider dynamic problems (over

time), so maybe it is fitting that we conclude this class with some of the tools that will help you think about intertemporal choices.

The next lesson concerns intertemporal choice directly.

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Risk preference in utility functionsTwo popular functional forms for representing different risk preferences are:◦ Constant absolute risk aversion (CARA) and◦ Constant relative risk aversion (CRRA).

Where the absolute and relative risk aversion concepts are defined by “how concave” the function is:

Coefficient of absolute risk aversion ≡ 𝜌𝜌 =−𝑈𝑈′′ 𝐶𝐶𝑈𝑈′ 𝐶𝐶

and

Coefficient of relative risk aversion ≡ 𝜃𝜃 =−𝐶𝐶 ∗ 𝑈𝑈′′ 𝐶𝐶

𝑈𝑈′ 𝐶𝐶.

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CARAConstant absolute risk aversion (CARA) functions have the form:

𝑈𝑈 𝐶𝐶 = − exp −𝜌𝜌𝐶𝐶 ,𝜌𝜌 > 0;𝑈𝑈′ 𝐶𝐶 = 𝜌𝜌 ∗ exp −𝜌𝜌𝐶𝐶 ;𝑈𝑈′′ 𝐶𝐶 = −𝜌𝜌2 exp −𝜌𝜌𝐶𝐶 .

And you can see that the coefficient of absolute risk aversion,−𝑈𝑈′′ 𝐶𝐶𝑈𝑈′ 𝐶𝐶

= 𝜌𝜌,

is constant.

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CRRAConstant relative risk aversion (CRRA) functions have the form:

𝑈𝑈 𝐶𝐶 =𝐶𝐶1−𝜃𝜃 − 1

1 − 𝜃𝜃;𝑈𝑈′ 𝐶𝐶 = 𝐶𝐶−𝜃𝜃;𝑈𝑈′′ 𝐶𝐶 = −𝜃𝜃𝐶𝐶− 𝜃𝜃+1 .

And you can see that the coefficient of relative risk aversion,−𝐶𝐶 ∗ 𝑈𝑈′′ 𝐶𝐶

𝑈𝑈′ 𝐶𝐶= 𝜃𝜃,

is constant. The larger 𝜃𝜃 is, the more risk averse the individual is.◦ I.e., the faster his marginal utility diminishes and the more concave his utility

function is.◦ If 𝜃𝜃 = 0, the utility function is linear and marginal utility is constant. So the

person would be risk neutral.