Seminar Economics and Psychology of Risk and Time Summary
Lecture 1: Introduction
- Attitudes toward risk depends on the framing of the options (gains vs. losses)
o We will look at several models that incorporate this reference dependence.
- People’s preferences are inconsistent over time. Whether people are patient depends
on the time horizon (near vs. distant future).
o We will look at several models that incorporate this “time inconsistency”.
- Other behavioural phenomena that will be discussed are uncertainty aversion, source
preferences and aversion towards losses.
Lecture 2: risk preferences 1
Normative (prescriptive) = how should people choose between risky alternatives?
Descriptive = how do people choose between risky alternatives?
Expected value maximization
- Prospect = a list of probabilities that things will happen, together with the monetary
payoff if they do happen.
- We say that someone faces a situation of risk if they know what could happen and how
likely it is. An example would be someone who bets $10 on the toss of a coin; they
know that there is a 50:50 chance it could be heads or tails, and, if it’s heads, they win
$10 and, if it’s tails, they lose $10.We say that someone faces a situation of
uncertainty if they do not know some of the possible outcomes or how likely they are.
An example would be someone booking a plane ticket, who is unlikely to know all the
possible delays or problems that could happen to change their experience of the flight.
- People used the maximization of expected values to decide what to choose when they
were facing risky prospects.
- However, Daniel Bernoulli showed that this can sometimes lead to very bad decisions
St. Petersburg Paradox. Therefore, EV maximization fails as a normative theory of
decision making under risk.
- Do people maximize according to EV? Van de Kuilen et al. (2013) showed that out of
a representative sample of 991 Dutch respondents, 83% preferred prospect B (sure
amount). Older and lower educated people choose the risky prospect more often.
- Risk averse = expected value of a prospect is preferred over prospect itself.
- Risk seeking = risky prospect is preferred over expected value of prospect.
- Risk neutral = indifferent between expected value of a prospect and the prospect
itself.
- people also do not maximize the expected value when choosing risky prospects
and thus, EV maximization is not an accurate descriptive theory of decision making
under risk as well.
Expected utility theory
- Bernoulli/von Neumann-Morgenstern proposed that risky prospects should be
evaluated by taking the expectation in terms of utility rather than in terms of expected
value.
,- To calculate expected utility we make use of a utility function u that translates money
into utility.
- The expected utility of a prospect is found by using the following formula:
o Where x is the total wealth + risky outcome(s)
o And u(x) is the utility function that transforms outcomes to utilities
o You can scale utility such that u(worst outcome) = 0 and u(best outcome) = 1
- People will then choose the prospect with the highest expected utility.
- The more concave the utility function, the more risk averse a person is and therefore,
he or she is more inclined to take on full insurance. Thus, the shape of the utility
function is important in determining what prospect gives the highest utility.
o Whether or not Alan is risk-averse or risk-loving will depend in a simple way
on the curvature of his utility function. If his utility function is concave, Alan
loses relatively more if his wealth goes down than he gains if his wealth goes
up, so he would rather not risk a loss for a gain, and he is risk-averse. If his
utility function is convex, then he loses relatively less if his wealth goes down
than he gains if his wealth goes up, so he would risk a loss for a gain and he is
risk-loving.The curvature of the utility function tells us a lot, therefore. risk
attitudes are determined by how much extra utility people derive from
additional outcomes.
o Expected utility theory is the reigning normative theory of decision making
under risk and its assumptions are deemed reasonable.
o A critical axiom/assumption is the independence axiom: if prospect X is
preferred to Y, then (p: X, 1-p:Z) is preferred to (p: Y, 1-p:Z) for all p and all
Z.
- The curvature of the utility function can be measured using 2 measures:
o Absolute risk aversion =
o Relative risk aversion =
o The second derivative of a function measures concavity
o The higher these numbers, the more risk averse.
o These 2 coefficients imply constant absolute risk aversion (CARA) or constant
relative risk aversion (CRRA)
o
o The beauty of the CARA and CRRA functions is that we need to know only
one number, the level of risk aversion, and then we can model choice with
risk, because we know what the utility function is and can use expected utility
to find what maximizes utility.
o Risk premium = EV - CE
- Do people maximize according to EU?
,Problems with EU theory
- Difference in certainty equivalent (= sure amount that makes a person indifferent
between receiving the prospect and the certain amount) becomes very small when
wealth levels increase. At normal levels, most people should not indicate strong
preferences between small scale gambles according to EU, but they do!
- Rabin’s paradox = the utility curve must be curved quite a bit for small stakes already.
Risk aversion for small stakes implies an implausible degree of risk aversion for
larger stakes.
o Aversion to modest-stake risk has nothing to do with the diminishing marginal
utility of wealth.
- Allais paradox = people choose a certain amount in lottery A, but then choose the
riskier prospect in lottery B. This is caused by the overweighing of certainty (certainty
effect).
- Thus, EU fails as a descriptive theory of decision under risk:
o Rabin’s paradox = at normal wealth levels, most people should not indicate
strong preferences between small scale gambles implies a huge degree of
risk aversion.
o Common consequence version of the Allais paradox = adding or removing a
common outcome to/from 2 prospects should not affect preferences.
o Coexistence of insurance and gambling.
- EU works when preferences are transitive and independent.
We will look at two alternative descriptive models.
Disappointment theory
- The basic idea is to measure outcome utility relative to some prior expectation of what
utility will be. If outcome utility is below the expected level, then Alan experiences
disappointment. If the outcome utility is above the expected level, then he experiences
elation. Elation and disappointment are captured by some function, D. So, if the
outcome is x his utility will be u(x) + D(u(x) − prior), the outcome utility plus or
minus elation or disappointment. The utility of a prospect X is then written
- Disappointment provides a plausible explanation of the Allais paradox and fanning
out, therefore.
- Risk appears to have more to do with the probability domain instead of the outcome
domain.
Rank-dependent utility (RDU)
- A model of disappointment deviates from expected utility by changing how utility is
perceived. An alternative possibility is to change how probabilities are perceived.
This is the approach taken by rank-dependent expected utility.
, - One crucial component we need is a function π that weights probabilities. A
commonly used function is:
- Now that we know how to weight probabilities, we can find the utility of a prospect.
The first thing we need to do is rank the outcomes so that x1 is the worst outcome and
xn is the best. For now we shall assume that x1 ≥ 0, and so the worst outcome is not a
loss. The rank-dependent expected utility of a prospect X is then given by
for all i
- In the slides, pi and w are transformed:
-
-
- Follow the following procedure:
-
Lecture 1: Introduction
- Attitudes toward risk depends on the framing of the options (gains vs. losses)
o We will look at several models that incorporate this reference dependence.
- People’s preferences are inconsistent over time. Whether people are patient depends
on the time horizon (near vs. distant future).
o We will look at several models that incorporate this “time inconsistency”.
- Other behavioural phenomena that will be discussed are uncertainty aversion, source
preferences and aversion towards losses.
Lecture 2: risk preferences 1
Normative (prescriptive) = how should people choose between risky alternatives?
Descriptive = how do people choose between risky alternatives?
Expected value maximization
- Prospect = a list of probabilities that things will happen, together with the monetary
payoff if they do happen.
- We say that someone faces a situation of risk if they know what could happen and how
likely it is. An example would be someone who bets $10 on the toss of a coin; they
know that there is a 50:50 chance it could be heads or tails, and, if it’s heads, they win
$10 and, if it’s tails, they lose $10.We say that someone faces a situation of
uncertainty if they do not know some of the possible outcomes or how likely they are.
An example would be someone booking a plane ticket, who is unlikely to know all the
possible delays or problems that could happen to change their experience of the flight.
- People used the maximization of expected values to decide what to choose when they
were facing risky prospects.
- However, Daniel Bernoulli showed that this can sometimes lead to very bad decisions
St. Petersburg Paradox. Therefore, EV maximization fails as a normative theory of
decision making under risk.
- Do people maximize according to EV? Van de Kuilen et al. (2013) showed that out of
a representative sample of 991 Dutch respondents, 83% preferred prospect B (sure
amount). Older and lower educated people choose the risky prospect more often.
- Risk averse = expected value of a prospect is preferred over prospect itself.
- Risk seeking = risky prospect is preferred over expected value of prospect.
- Risk neutral = indifferent between expected value of a prospect and the prospect
itself.
- people also do not maximize the expected value when choosing risky prospects
and thus, EV maximization is not an accurate descriptive theory of decision making
under risk as well.
Expected utility theory
- Bernoulli/von Neumann-Morgenstern proposed that risky prospects should be
evaluated by taking the expectation in terms of utility rather than in terms of expected
value.
,- To calculate expected utility we make use of a utility function u that translates money
into utility.
- The expected utility of a prospect is found by using the following formula:
o Where x is the total wealth + risky outcome(s)
o And u(x) is the utility function that transforms outcomes to utilities
o You can scale utility such that u(worst outcome) = 0 and u(best outcome) = 1
- People will then choose the prospect with the highest expected utility.
- The more concave the utility function, the more risk averse a person is and therefore,
he or she is more inclined to take on full insurance. Thus, the shape of the utility
function is important in determining what prospect gives the highest utility.
o Whether or not Alan is risk-averse or risk-loving will depend in a simple way
on the curvature of his utility function. If his utility function is concave, Alan
loses relatively more if his wealth goes down than he gains if his wealth goes
up, so he would rather not risk a loss for a gain, and he is risk-averse. If his
utility function is convex, then he loses relatively less if his wealth goes down
than he gains if his wealth goes up, so he would risk a loss for a gain and he is
risk-loving.The curvature of the utility function tells us a lot, therefore. risk
attitudes are determined by how much extra utility people derive from
additional outcomes.
o Expected utility theory is the reigning normative theory of decision making
under risk and its assumptions are deemed reasonable.
o A critical axiom/assumption is the independence axiom: if prospect X is
preferred to Y, then (p: X, 1-p:Z) is preferred to (p: Y, 1-p:Z) for all p and all
Z.
- The curvature of the utility function can be measured using 2 measures:
o Absolute risk aversion =
o Relative risk aversion =
o The second derivative of a function measures concavity
o The higher these numbers, the more risk averse.
o These 2 coefficients imply constant absolute risk aversion (CARA) or constant
relative risk aversion (CRRA)
o
o The beauty of the CARA and CRRA functions is that we need to know only
one number, the level of risk aversion, and then we can model choice with
risk, because we know what the utility function is and can use expected utility
to find what maximizes utility.
o Risk premium = EV - CE
- Do people maximize according to EU?
,Problems with EU theory
- Difference in certainty equivalent (= sure amount that makes a person indifferent
between receiving the prospect and the certain amount) becomes very small when
wealth levels increase. At normal levels, most people should not indicate strong
preferences between small scale gambles according to EU, but they do!
- Rabin’s paradox = the utility curve must be curved quite a bit for small stakes already.
Risk aversion for small stakes implies an implausible degree of risk aversion for
larger stakes.
o Aversion to modest-stake risk has nothing to do with the diminishing marginal
utility of wealth.
- Allais paradox = people choose a certain amount in lottery A, but then choose the
riskier prospect in lottery B. This is caused by the overweighing of certainty (certainty
effect).
- Thus, EU fails as a descriptive theory of decision under risk:
o Rabin’s paradox = at normal wealth levels, most people should not indicate
strong preferences between small scale gambles implies a huge degree of
risk aversion.
o Common consequence version of the Allais paradox = adding or removing a
common outcome to/from 2 prospects should not affect preferences.
o Coexistence of insurance and gambling.
- EU works when preferences are transitive and independent.
We will look at two alternative descriptive models.
Disappointment theory
- The basic idea is to measure outcome utility relative to some prior expectation of what
utility will be. If outcome utility is below the expected level, then Alan experiences
disappointment. If the outcome utility is above the expected level, then he experiences
elation. Elation and disappointment are captured by some function, D. So, if the
outcome is x his utility will be u(x) + D(u(x) − prior), the outcome utility plus or
minus elation or disappointment. The utility of a prospect X is then written
- Disappointment provides a plausible explanation of the Allais paradox and fanning
out, therefore.
- Risk appears to have more to do with the probability domain instead of the outcome
domain.
Rank-dependent utility (RDU)
- A model of disappointment deviates from expected utility by changing how utility is
perceived. An alternative possibility is to change how probabilities are perceived.
This is the approach taken by rank-dependent expected utility.
, - One crucial component we need is a function π that weights probabilities. A
commonly used function is:
- Now that we know how to weight probabilities, we can find the utility of a prospect.
The first thing we need to do is rank the outcomes so that x1 is the worst outcome and
xn is the best. For now we shall assume that x1 ≥ 0, and so the worst outcome is not a
loss. The rank-dependent expected utility of a prospect X is then given by
for all i
- In the slides, pi and w are transformed:
-
-
- Follow the following procedure:
-
