Uncertainty

Question 1

What is a risk environment

Answer 1

If the probability distribution is known,

E.g forcecast showing the P of rain tomorrow.

Question 2

What is consumers choice based on?

Answer 2

Probability distribution

E.g going out based on probability of it raining

Question 3

How does the consumer account for the possible states of nature e.g raining or not?

Answer 3

The consumer will have a contingent consumption plan.

Assigns a bundle to a given state of nature

Question 4

How to express the probabilities

Answer 4

Pr(𝑋) + Pr(𝑌)=1
Sum of probability = 1

Question 5

Before utility, we have to find wealth.

In risk environments, we need to look at expected wealth E[W].

How to calculate the expected value of wealth?

Answer 5

E[W] = Σ PiWi

Pi is probability associated to wealth in situation i.

E.g (Wealth from rain x probability if it rains) + (wealth of no rain x probability of no rain)

Question 6

Utility function under uncertainty - what influences utility? (2)

Answer 6

Consumption (obviously)
But probability distribution too

Question 7

How to express expected utility? (Von Neumann function)

2. Positive monotonic transformation utility function

Answer 7

U (c₁,c₂,π₁,π₂) = π₁u(c₁) + π₂u(c₂)

Utility of consumption level 1 u(c₁) occurs at probability of π₁

2.
U (c₁,c₂,π₁,π₂) = π₁ln(c₁) + π₂ln(c₂)

Question 8

General expected utility function

Answer 8

E[U(L)] = Σ πi u(wi)

Treat as a lottery, where we know the probability but not the actual outcome. Expected utility of the lottery is the sum of the weighted sums of utility of situations.

E.g Probability of outcome i x utility of wealth at i

Question 9

What assumptions do we make for expected utility (4)

Answer 9

Independence assumption - 2 states of nature are considered separately, the probability of state 1 occurring does not depend on probability of state 2. (No interaction)
It means there is no additional utility if it was unlikely, or less utility if probability was large.

E.g Probability of rain tomorrow 80%. If it doesn’t rain, the probability was still 80%

Also make completeness (Compare), reflexivity (bundle is at least as good as itself) and transitivity assumptions.

Question 10

Lottery example

Question 11

Positive monotonic transformation vs positive affine transformation

Answer 11

Positive monotonic transformation - new utility function represents the same preferences.

Positive affine transformation ensures the new utility function represents same preferences, and also same expected utility.

Question 12

Expected utility vs expected wealth formula

Answer 12

E[U(w)] = Σ μi u(wi)

E(W) = Σ piwi