Week 1: uncertainty and time
example: Covid-19, the war in Ukraine, the climate
change All these things influence the decision of
people, firms etc.
Incentives
Trade-offs
Interaction
Information
Time
1.1 Decisions involving uncertainty: Policy, investments, insurance, education (We don’t know what
will happen in the future, what the pay-offs will be. life is full of uncertainty
- Uncertainty: likelihood of outcomes (unknown)
- Risk: likelihood of outcomes (known)
What are risky outcomes?:
- Risk describes any economic activity in which there are uncertain outcomes
- Associated with any uncertain outcome are probabilities
- Probabilities are numbers between zero and one that indicate the likelihood that a particular
outcome will occur,
- In the absence of a known probability, economic agents have to estimate: they can estimate
based on frequency or based on subjective probability.
Decision problem involving risk:
Option 1: Job with a certain income of Y = 20 000
Option 2: Job with
- probability 0.5 that Y = y1 = 30 000
- probability 0.5 that Y = y2 = 10 000
Nature (not really a person of a firm) sudo player? chooses
probability is of an uncertain outcome
How to compaere 1&2 by calculating the expected values:
The expected value of Y: E(Y) = Pr1 x y1 + Pr2 x y2
With E(Y) = expected value of Y
Y1, y2 = payoffs
Pr1, Pr2 = probabilities of y1 and y2, respectively
The expected value of income from the second job in our example:
E(Y) = 0.5(30 000) + 0.5(10 000) = $20 000
This answer is the same as the certain option, how do we evaluate this?
We have to calculate the utility of the expected value U[E(Y)]:
The utility of the expected value is the utility an individual
has from receiving a certain amount of money equivalent to
the expected value of an uncertain outcome
,In our example (Option (2)): We know that the expected outcome which is uncertain, is 20 000. If we
just plug this 20 000 into the utility fucntion as if it is certain outcome than we calculte the utility of
the expected value. This means that the utility of the expected value doesn’t take into account the
expected risk.
E(Y) = 0.5(30 000) + 0.5(10 000) = €20 000
The utility of the expected value U[E(Y)] = U(€20 000)
U[E(Y) doesn’t consider the involved risk)
Than we have the expected utility E(U(Y)):
The expected utility is the sum of utilities of all possible uncertain outcomes, weighted with their
probability, that any particular outcome will be realized.
E(U(Y)) = 0.5U(€20 000) + 0.5U(€10 000)
E(U(Y)) does consider the involved risk
Approach of expected utility theory
This is also called Von Neumann – Morgenstern Expected Utility Theory: what does this precisely
contain?
Rational decision making with with risk/ uncertainty: a decision maker facing a decision
problem with a risky payoff (U(Y)) is rational if he chooses an action α that maximizes his
expected utility. That is α chosen if and only if
o E[U(Y)| α ] ≥ E[U(Y)| b ] for all b
Where:
o E[U(Y)| α ] = pa1U(y1) + pa2U(y2)
pa1 : probability of income Y1 if action α is chosen.
pa2 : probability of income Y2 if action α is chosen.
And E(U(Y)|b] equivalently.
Example (A gamble):
With a 60% chance, you will € 1000 and
with a 40% chance, you will win €2500
What is the expected value of this gamble?
E(Y) = 0.6 x 1000 + 0.4 x 2500 = €1600
The expected utility from the gamble is:
𝐸(𝑈(𝑌)) = 0.6 × 𝑈(€1000) + 0.4 × 𝑈(€2500)
Suppose a persons’ utility can be expressed as a function of money in the following way:
U(money) = √ money
Then the expected utility from the above gamble is: E(U(Y)) = 0.6 x √ 1000+ 0.4 x = √ 2500
E(U(Y)) = 0.6 x 31.62 + 0.4 x 50 ≈ 38.97
Utility of the Expected value of this gamble:
If instead this person were given the expected value of the gamble, €1600, for certain:
U(E(Y)) = = √ € 1600 = 40
In other words, the guaranteed amount of €1600 yields higher utility than the gamble that has an
expected value of €1600.
,Utility of expected value > expected utility
This shows that the individual does not like risk: (s)he is risk averse.
Risk attitudes:
The terms risk attitude, risk appetite, and risk tolerance are often used to describe an individual’s or
an organisation's attitude towards risk-taking. Usually three types:
- Risk averse:
o 𝑬(𝑼(𝒀)) < 𝑼[𝑬(𝒀)]
o Decision maker prefers the option with the certain income over the option with the
uncertain income, given the same expected value.
- Risk loving (seeking):
o 𝑬(𝑼)𝒀)) > 𝑼[𝑬(𝒀)]
o Decision maker prefers the option with the uncertain income over the option with
the certain income, given the same expected value.
- Risk neutral:
o 𝑬(𝑼(𝒀)) = 𝑼[𝑬(𝒀)]
o Decision maker is indifferent between the option with the certain income and the
option with the uncertain income, given the same expected value.
A risk averse person is willing to pay to avoid risk, but how much?
Risk averse person: E(U(Y)) < U[E(Y)] concave function
Risk loving person: convex function
Risk neutral: linear line
What is the certainty equivalent?
Certainty equivalent (CE):
The certainty equivalent is the certain payoff that
generates as much utility as the expected utility of
the gamble.
, ~
𝐶𝐸 → U( Y ) = E(U(Y))
It is determined by equating the utility function to the expected utility and solving for the
income (or wealth).
For our example gamble with utility function 𝑈(𝑌) =√ Y we calculated: 𝐸(𝑈(𝑌)) = 0.6 × 31.62 + 0.4 ×
50 ≈ 38.97 Hence:
~
U( Y ) = E(U(Y)) = 38.97
√~ Y = 38.97
~ 2
Y = 38.97 = 1518.66 = CE
Risk premium: the risk premium is the maximum willingness to
pay to eliminate risk. It is determined as the difference between
the expected value and the certainty equivalent:
E(Y) – CE
𝐸(𝑌) − 𝐶𝐸 = €1600 − €1518.66 = €81.34
Be careful: Risk premium is not always a horizontal line, it depends on what is on the axis.
Measures of risk aversion:
A consumer with a von Neumann-Morgenstern utility function can be one of the following:
1. Risk averse, with a concave utility function.
2. Risk-neutral, with a linear utility function,
3. Risk-loving, with a convex utility function.
The degree of risk-aversion a consumer displays is related to the curvature of their utility
function.
1. 2. 3.
Somehow, we have to find a way to
measure this curvature. This has be done by Arrow and Pratt. Arrow and Pratt define two measure of
risk aversion:
1. Arrow-Pratt measure of absolute risk aversion (ARA)
2. Arrow-Pratt measure of relative risk aversion (RRA)
ARA: How Absolute Risk-Aversion (amount of money, wealth of whatever at risk)
changes with wealth:
Percentage at risk
RRA: How Relative Risk-Aversion (percentage at risk) changes
= with wealth:
