Choice under uncertainty

Question 1

What are the 5 assumptions of rational consumer preferences?

Answer 1

Completeness, transitivity, continuous, monotonicity and independence

Question 2

Define completeness

Answer 2

Consumers can always compare/rank bundles. X≥Y or Y≥X or Y∼X – often violated

Question 3

Define transitivity

Answer 3

If X≥Y and Y≥Z then X≥Z (ceteris paribus)

Question 4

Define continuous (in context of consumers’ preferences)

Answer 4

If X≥Y and Z is very similar to Y (lies in a small radius of Y) then X≥Z (aka. tiny changes in bundles will not change preference ordering)

Question 5

Define monotonicity

Answer 5

Consider two lotteries, L and L’’. One lottery will be preferred to the other only if it assigns a higher probability of getting the higher prize (more is better)

Question 6

Define independence (in context of consumers’ preferences)

Answer 6

If we mix each of two lotteries with a 3rd one, the preference ordering over the two resulting lotteries does not depend on the particular third lottery being used. (introduction of a 3rd lottery does not change order of original 2 lotteries) (aka. I prefer X to Y if I prefer pX+(1-p)Z to pY+(1-p)Z - often violated - Allais paradox

Question 7

What does Allias Paradox show? Give the example

Answer 7

Shows a violation of the independence axiom. Experiment 1: 1U(£1M)>0.89U(£1M)+0.01U(£0M)+0.1U(£5M) Experiment 2: 0.89(£0M)+0.11U(£1M) <0.9U(£0M)+0.1U(£5M) Rearranging experiment 2 equations: 0.11U(£1M) <0.01U(£0M)+0.1U(£5M) 1U(£1M)-0.89(£1M) <0.01U(£0M)+0.1U(£5M) 1U(£1M<0.89U(£1M)+0.01U(£0M)+0.1U(£5M) Which contradicts the first experiment.

Question 8

What is a game of ultimatum

Answer 8

1 player gets a sum of money, and is tasked with splitting it between themselves and the other player. The second player can accept or reject the offer, if they accept, each player gets the proposed amount, if they reject, they get nothing  shows that people will reject things that they perceive to be unfair, even if it is not in their best interest to reject this - not rational

Question 9

Define different state of nature

Answer 9

Different outcomes of some random event

Question 10

Define contingent consumption plan

Answer 10

What will be consumer in different states of nature

Question 11

What is the expected utility theorem?

Answer 11

A utility function over any lottery can be written as the expected utility of the outcomes that make up this lottery

Question 12

What is expected utility?

Answer 12

Expected utility (U) = probability of state of nature X level of consumption in that state of nature. We assume states of nature are mutually exclusive

Question 13

What is St. Petersburg’s Paradox, and what does it show?

Answer 13

A casino offers a game of chance for a single player, in which a fair coin is tossed at each stage. The pot starts at £2, and is doubled every time a head appears. The game ends the first time a tail appears. Thus, the player wins £2 if a tail appears on the first toss, £4 if a head appears on the first toss and a tail on the second, £8 if a head appears on the first two tosses and a tail on the third, and so on. In short, the player wins £2k , where k is the number of tosses. Under expected utility theorem the value of this game is infinity, this is not what happens in real life. Shows that we can not simply define the utility of a lottery bu the expectation of the random variable associated with the lottery.

Question 14

What is a positive affine transformation, and what are its properties? (in context of transforming an expected utility indifference curve)

Answer 14

Any transformation of a function that can be written in the form: v(u)=au+b, where a>0 Transformation still represents the original preferences, and still has the expected utility function.

Question 15

What is insurance and how can it benefit consumers?

Answer 15

Someone has to give up γK in the good state to gain K-γK in the bad state. It can change the probability distribution of outcomes

Question 16

What is full / actuarially fair insurance?

Answer 16

Full insurance makes the expected consumption equal all the states of nature. γ=ρ

Question 17

Draw a graph of insurance - what does the slope of the budget constraint represent?

Answer 17

(∆Cg)/(∆Cb )=γK/(K-γK)=(-γ)/(1-γ) Slope of budget constraint = The rate at which we trade consumption in each state / price ratio

Question 18

Define risk aversion, and describe the utility function.

Answer 18

Risk averse = expected utility of L < utility of the expected wealth from L

Concave utility function - Eu(L)< u(E(L))

Question 19

Define risk netural, and describe the utility function.

Answer 19

Risk neutral = expected utility of L = utility of the expected wealth from L

Straight utility function - Eu(L) = u(E(L))

Question 20

Define risk loving preferences, and describe the utility function.

Answer 20

Risk loving = expected utility of L utility of the expected wealth from L

Convex utility function: Eu(L)>u(E(L))

Question 21

What is certainty equivalence?

Answer 21

Certainty equivalent = the amount that you are willing to accept instead of the lottery

u(C)=Eu(L)

You are indifferent between the gamble and the amount

Question 22

What is a risk premium?

Answer 22

Risk Premium = difference between gamble and certainty equivalent = E(L)-C

For a risk adverse individual, the certainty equivalent < expected amount of money from the gamble

Question 23

How can you measure risk aversion?

Answer 23

Either using the absolute risk aversion, or the relative risk aversion.

Question 24

What is the equation for absolute risk aversion, and what is the sign for risk adverse and risk lovers

Answer 24

R_a(W)= - u’‘(W) / u’(w)

+ve for risk adverse -ve for risk lovers

Question 25

What is the equiation for relative risk aversion, and what is the sign for risk averse and risk lovers?

Answer 25

R_a​(W)= - u’‘(W).w / u’(W)

+ve for risk adverse -ve for risk lovers