Microeconomics 5.1: Intro to Uncertainty; Risk and Insurance Flashcards
- Uncertainty, expected wealth and expected utility; - Attitudes to risk; - Demand under uncertainty; - Insurance market: “fair” and “unfair” prices. (27 cards)
What is uncertainty?
It is a rule or function consisting of outcomes or states of nature (from a list of outcomes)
and probabilities linked to those outcomes
Describe what consumers’ choice is based on
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.
Describe variables in probability distributions which can take only two values, with
associated distribution of probabilities
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
Using wealth as an example how can you define probability distribution of a factor?
Describe what consumers’ utility will depend on in an uncertain environment
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)
Describe the concept of ‘expected utility’
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)
Describe & explain the parameter of ‘expected utility’
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.
Describe & explain ‘positive affine transformation’
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
How is utility defined in an uncertain environment?
Functional forms of utility and expected utility interpretation
What are the key assumptions and properties of expected utility?
✓ 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.
(ii) Expected utility is unique up to positive affine transformation.
Expected utility: parameters
What is the role of affine transformations in expected utility?
What are the different attitudes to risk in expected utility theory?
Risk averse consumer:
The expected value is preferred to the gamble:
𝑈 (𝐸 [𝑊]) > 𝐸 [𝑈(𝑊)]
Risk-loving consumer:
Random distribution is preferred to the expected value:
𝑈 (𝐸 [𝑊]) < 𝐸 [𝑈(𝑊)]
Risk neutral agent:
The expected utility of wealth is the utility of expected value:
𝑈 (𝐸 [𝑊]) = 𝐸 [𝑈(𝑊)]
Diagrammatic representation of risk attitudes
Briefly introduce how we model consumer demand under uncertainty?
✓ After introducing and analysing consumer behaviour under uncertainty, we can now proceed to analyse consumer decisions under uncertainty.
✓ We will need to build the budget constraint, which takes into account different possible outcomes (state-contingent);
✓ We will need to represent preferences with indifference curves;
✓ And finally, to find consumer’s demand, we will need to bring these building blocks together.
✓ This is easier done with an example: of the insurance market.
Using risk of car accident example, do we construct and interpret a state-contingent budget constraint?
What does the state-contingent budget constraint look like graphically?
Show & describe indifference curves under uncertainty
How do we derive the marginal rate of substitution (MRS) under uncertainty?
How do we determine optimal consumer demand under uncertainty?
✓ After examining separately state-contingent budget constraint and indifference curves, we can now bring them together.
✓ We can illustrate the affordable plans and find the most preferred state-contingent consumption plan.
Competitive Insurance: PCM
Competitive Insurance: fair price
Competitive Insurance: rational choice
