Microeconomics 5.1: Intro to Uncertainty; Risk and Insurance

Question 1

What is uncertainty?

Answer 1

It is a rule or function consisting of outcomes or states of nature (from a list of outcomes)
and probabilities linked to those outcomes

Question 2

Describe what consumers’ choice is based on

Answer 2

Consumer’s choice is based on probability distribution:
 For each of the states of nature, the rational representative consumer will have a contingency
plan or a contingent consumption plan.
 The representative consumer will have preferences over different contingent
consumption plans.
 We will model the choice of the representative consumer in the same way: the best
consumption plan he/she can afford.

Question 3

Describe variables in probability distributions which can take only two values, with
associated distribution of probabilities

Answer 3

We need to think in terms of probabilities:
 Suppose there are only two possible outcomes. We can then note:
1 > Pr 𝑋 , Pr 𝑌 > 0 and Pr 𝑋 + Pr 𝑌 = 1
Pr (𝑉 = 𝑣) = {Pr(𝑣bottom right1) , 𝑖𝑓 𝑉 = 𝑣1
{1 − Pr(𝑣bottom right1) , 𝑖𝑓 𝑉 = 𝑣2

Question 4

Using wealth as an example how can you define probability distribution of a factor?

Answer 4

We can define wealth in an uncertain (risky) situation: we can define the
expected value of wealth, if we know the probabilities distribution:
E[W] = Uppercase Epsilon with “n” on top and “𝑖=1” underneath, 𝑝bottom right𝑖 𝑤bottom right𝑖
 Where 𝑝bottom right𝑖 is the probability associated to wealth 𝑤bottom right𝑖 obtained in situation 𝑖.

Question 5

Describe what consumers’ utility will depend on in an uncertain environment

Answer 5

 In an uncertain environment, consumer’s utility will depend not only on the
consumption, but also on the probability distribution.
 Suppose two states of nature, first one with consumption 𝑐1 which occurs
with probability 𝜋1 and second one with consumption 𝑐2 which occurs with
probability 𝜋2.
 Consumer’s utility will be:
𝑈(𝑐bottom right1, 𝑐bottom right2 , 𝜋bottom right1, 𝜋bottom right2)

Question 6

Describe the concept of ‘expected utility’

Answer 6

 Expected utility or Von Neumann-Morgenstern utility function:
𝑈(𝑐1, 𝑐2 , 𝜋1, 𝜋2) = 𝜋1𝑢(𝑐1) + 𝜋2𝑢(𝑐2)
 This utility function has nice additive properties.
 Expected utility is defined within Von Neumann-Morgenstern axiomization (list
of constraints imposed on preferences over considered lotteries).
We can say that agents participate in lotteries and define a lottery 𝐿 in terms
of an outcome set 𝑊 = {𝑤1, 𝑤2, … 𝑤𝑖} and a probability distribution 𝜋 = {𝜋1, 𝜋2, … 𝜋𝑖}.
 We then can define the expected utility of the lottery 𝐿 before participation
in the lottery:
E[U(L)] = Capital Epsilon with ‘n’ on top and ‘𝑖=1’ underneath’ 𝜋bottom right𝑖 𝑢(𝑤bottom right𝑖)

Question 7

Describe & explain the parameter of ‘expected utility’

Answer 7

 We are making the Independence assumption:
 Each state of nature is considered separately; the probability of state 1 occurring
doesn’t depend on the probability of state 2 ; there is no interaction between
outcomes.
 This means that there is no additional utility for a very large gain just because it is so
rare.
 Note: this assumption is made additionally to the three assumptions of
completeness, reflexivity, and transitivity.
 The expected utility has two excellent properties: (i) we can rank complex lotteries with multiple
outcomes by calculating the expected utilities of simple two-outcome lotteries and (ii) unique up to
positive affine transformation.

Question 8

Describe & explain ‘positive affine transformation’

Answer 8

 A positive monotonic transformation will ensure that the new utility function
represents the same preferences;
 A positive affine transformation ensures not only that the transformed
utility function represents the same preferences, but also has the expected
utility property.
𝑉(𝑈) = 𝑎𝑈 + 𝑏, 𝑎 > 0
 Consider previous examples:
𝑐^𝜋bottom right1𝑐^1−𝜋bottom right2 and 𝜋bottom right1𝑙𝑛(𝑐1) + 𝜋bottom right2𝑙𝑛(𝑐2)
 Same preferences, different expected utility.
 Expected utility function is unique up to an affine transformation

Question 9

Describe different consumers with different attitudes to risk

Answer 9

1. Risk averse consumer:
    The expected value is preferred to the gamble;
    Risk averse consumer has a concave utility function.
   - *For any example, this is how their graph would look: graph with whatever context on…
2. Risk-loving consumer:
    Random distribution is preferred to the expected value;
    Risk-loving consumer has a convex utility function.
3. Risk neutral agent:
    The expected utility of wealth is the utility of expected value;
    Consumer doesn’t care about the risk, only expected value.
   - The curvature of the utility function measures consumer’s attitude towards
   risk
   …