Choice under uncertainty

Question 1

What is the basic consequence function, and what do the elements represent?

Answer 1

c(x,s)
x is the set of actions available to choose
s is the set of possible states of nature

Question 2

What are the 2 additional functions involved with decision-making under uncertainty?

Answer 2

• Probability function (π(s)): expresses the likelihood of each state of nature
• Utility function (v(c)): captures the individuals preferences for the different possible consequences

Question 3

What does the Neumann-Morgenstern Expected Utility Rule state?

Answer 3

It says that the utility of an action, x can be estimated as a mathematical expectation (weighted sum) of utilities of consumption, where the weights are the probabilities of states of nature

Question 4

What is the main implication of the Expected Utility Rule?

Answer 4

The independence assumption: the utility derived from each consequence is independent of the utility from every other consequence

Question 5

Explain the mathematical way of expressing the Independence Assumption

Answer 5

The marginal rate of substitution between any two pairs is independent of the quantity of a third outcome

Question 6

What is the concept of a fair bet?

Answer 6

A fair bet is one where the mathematical expectation of the outcomes is equal to the certain outcome

Question 7

Explain the shape of a risk averse person’s indifference curve for a fair bet

Answer 7

Concave. This implies a diminishing marginal utility as the individual’s utility rises by less from a gain than it falls due to a loss of equal magnitude

Question 8

Explain the shape of a risk loving person’s indifference curve for a fair bet

Answer 8

Convex. This implies an increasing marginal utility as the individual’s utility rises by more from a gain than it falls from a loss of equal magnitude

Question 9

Explain the shape of a risk neutral person’s indifference curve for a fair bet

Answer 9

Straight upwards sloping. Implies a constant marginal utility as the individual’s utility rises by a constant amount and the utility from a gain is equal to the utility from a loss of equal magnitude

Question 10

Where is the optimal choice for a decision under uncertainty?

Answer 10

The tangent between the indifference curve and the budget line

Question 11

How do you find the optimal choice mathematically?

Answer 11

• Find the slope of the budget line by rearranging its equation
• Find the first derivative from the utility function
• Equate the two

Question 12

What is the optimal choice for a risk averse person?

Answer 12

c1 = c2 = ĉ

The risk averse person will reject the fair bet in favour of the certain amount

Question 13

In general terms, how does the individual decide to insure?

Answer 13

The individual wants to maximise the utility from insuring by optimising consumption in the good and bad states

Question 14

How do you solve the insurance problem?

Answer 14

• Substitute the budget constraint into the objective function
• Differentiate it with respect to c_b
• Go back to the expression for c_g and plug in for γ and c_b
• Go back to the budget constraint to solve for K

Question 15

What does fair insurance mean in the insurance problem?

Answer 15

Fair insurance is when the company makes breaks even, therefore: γ = π

Question 16

What is the Certainty Equivalent?

Answer 16

The amount that the individual would accept for certain that would make them indifferent between taking the bet and not taking the bet