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Intermediate Microeconomics UA10

Homework 5 Fall 2025

Upload to Brightspace by midnight (11:59 pm) Thursday October 16

1. Consider a setting with two prizes Z= {21, 22} and let 2 be a preference order over lotteries over these prizes that satisfies axioms Al and A2: continuity and the substitution axiom. Suppose we are given that z> 2. Use the substitution axiom to show that the lottery 1 = (0.9,0.1) offering a 90% chance of prize z is strictly preferred to the lottery l'= (0.1,0.9) offering a 10% chance by breaking the lotteries into compound lotteries with a common lottery on the left branch and different pure prizes on the right branch.

2. Consider a setting with two prizes, Z={21, 22}, again with a preference order over lotteries over these prizes. Consider a compound lottery in which the prize is drawn as follows:

At the first stage, a fair coin is flipped.

- If the coin comes up heads, a fair six-sided die is rolled and the prize is z if the number on the second die is 4 or below while it is z2 if the number is 5 or 6.

- If the coin comes up tails, a fair coin is flipped again. In this case the prize is z if the second coin comes up heads, and z2 if it comes up tails.

(a) Use the tree diagram from class to illustrate the above compound lottery.

(b) What is the simple lottery to which the above compound lottery corresponds?

(c) Suppose that prize 21 is strictly preferred to 22 and that assumptions A1 (continuity) and A2 (substitution) both hold. What can you say about the preferences between the compound lottery in (a) and a single stage lottery that offers prize with probability 0.6, z2 with probability 0.4?

3. By hand, draw any strictly concave (i.e. strictly risk averse) utility function u: [0, 2000]→R. Answer each of the following questions using the graph you have just drawn.

(a) In each of the following comparisons, identify the gamble that is more highly preferred, and illustrate your reasoning geometrically.

i. $1,000 for sure vs. [(500;0.5), (2,000;0.5)].

ii. $1,000 for sure vs. (800;0.5), (2, 000; 0.5)].

(b) Identify the certainty equivalent to each of the following gambles, and in each case illustrate your reasoning geometrically.

i. p= [(1,000; 0.5), (2, 000; 0.5)].

ii. q = [(500; 0.8), (1, 000; 0.2)].