1. When developing a framework moving from analyzing financial decisions with certainty to those with uncertainty, we make certain assumptions about the market conditions that must hold, as well as the properties of individual preferences (utility). Explain some of these assumptions and why they are important.

2. You have a log utility function ?(?) = ln (?), and your current level of wealth is $5,000. a. Suppose you face a 50/50 chance of winning or losing $1,000. If you can buy insurance that completely removes the risk for a fee of $125, will you buy it, or take the gamble? b. Suppose you accept the gamble and lose, so you now have $4,000. What will you do if you are given the same choice to buy the insurance for $125 or take the gamble?

3. Suppose you have a utility function ?(?) = ?????, where ? > 0. a. Graph the function b. Does the function exhibit positive marginal utility of wealth and risk aversion? c. Does the function have increasing or decreasing absolute risk aversion? d. Does the function have increasing or decreasing relative risk aversion?

Please write down the answer without AI. Thank you!