Part A -

(A) Consider an individual who makes choices in two states of the world. The individual's preferences over lotteries that payoff x in state one and y in state two are described by the utility function:

U(x, y) = min_p∈[0,1] px + (1 - p)y + 0.5α(p-0.5)²    α > 0.

i. Does this individual exhibit ambiguity aversion?

ii. Does this individual satisfy certainty independence?

iii. Does this individual have linear or piecewise linear utility?

iv. Describe the individual's indifference curve through the prize x = y = 1.

(B) Consider a decision maker who prefers the gamble with prizes (x, y) to gamble with prizes (a, b) if

min_p∈[0,1] px + (1-p)y + γlog[p^p(1-p)^1-p] > min _p∈[0,1] pa + (1-p)b + γlog[p^p(1-p)^1-p]

where γ > 0. Show that this decision maker's behaviour does not satisfy the axiom of certainty independence. Do these preferences satisfy uncertainty aversion?

Part B -

Give examples of utility functions on compound lotteries or sets of compound lotteries that:

(a) exhibit a preference for commitment,

(b) exhibit a preference for early resolution of uncertainty,

(c) exhibit uncertainty aversion,

(d) exhibit none of the above.

Consider the utility function

U(x, y) = min_p∈[0,1] px + (1 - p)y + 0.5α(p - 0.5)²

Where x is the prize if a red ball is drawn from an urn and y is the prize if a black ball is drawn. In the urn there are 100 balls either red or black but in unknown proportions. An individual has the following preferences. They prefer the gamble that pays Ll if a red ball is drawn (and nothing if a black is drawn) to a gamble that pays £0.25 if a ball of either colour is drawn. What can you deduce about the value of a?

Consider the set of outcomes Y :- {1, 2, . . . , N}. For any subset A ⊆ Y let #A denote the number of elements of the set A. Define the function v(.) on subsets of Y as follows: v(A) := (#A)^α/N. What values can α ≥ 0 take if this function determines a convex set of probabilities on Y?

Give an example of a preference ordering on objective lotteries that does not satisfy the independence axiom.

Attachment:- Lecture Notes.rar