Suppose you and I consider the following game: We both put $y on the table, then flip a coin. If it comes up heads, I get everything on the table, and if it comes up tails, you get everything on the table.
A. Suppose we both have an amount z > y available for consumption this week and both of us are risk averse.
(a) Draw my (weekly) consumption utility relationship given that I am risk averse. On your graph indicate the expected value of the gamble and the expected utility of the gamble assuming that we are playing with a fair coin — i.e. a coin that comes up heads half the time and tails the other half.
(b) I agree to participate in the gamble if I think the coin is fair?
(c) Now suppose that I exchanged the game coin for a weighted coin that comes up heads more often than tails. Illustrate in your graph how, if the coin is sufficiently unfair, I will now agree to participate in the gamble.
(d) Now consider both of us in the context of an Edgeworth Box, and suppose again that the coin is fair Draw an Edgeworth box with consumption xH under “heads” on the horizontal and consumption xT under “tails” on the vertical axis. Illustrate our “endowment” bundle E before the gamble and the outcome bundle A if we do gamble.
(e) Illustrate the indifference curves through E and A. Will we gamble? Is it efficient not to gamble?
(f) Suppose next that I have an unfair coin that is weighted toward coming up heads with probability δ > 0.5. How do my indifference curves change as a result?
(g) You do not know about the unfair coin, but you are delighted to hear that I have just sweetened the gamble for you: If the coin comes up heads, I agree to give you a fraction k of my winnings. Draw a new Edgeworth box with the endowment bundle E and the outcome bundle B implied by the change I have made to the gamble.
(h) Can you illustrate how both of us engaging in the gamble might now be an efficient equilibrium?
(i) True or False: If individuals have different beliefs about the underlying probabilities of different states occurring, then there may be gains from state-contingent consumption trades that would not arise if individuals agreed on the underlying probabilities.
B. Suppose that the function u(x) = ln x allows us to represent both of our preferences over gambles using the expected utility function. Suppose further that z and y (as defined in part A) take on the values z = 150 and y = 50.
(a) Calculate the expected utility of entering this gamble (assuming a fair coin) and compare it to the utility of not entering. Will either us agree to play the game?
(b) Suppose that I paid you a fraction k of my winnings in the event that “heads” comes up. What is the minimum that k has to be for you to agree to enter the game (assuming you think we are playing with a fair coin)?
(c) If I agreed to set k to the minimum required to get you to enter the game, determine the lowest possible δ that an unbalanced coin must imply in order for me to want to enter the game.
(d) Suppose my unbalanced coin comes up heads 75% of the time. Define the expected utility function for me and you as a function of xT and xH given that I know that the coin is unbalanced and you do not.
(e) Define p as the price for $1 worth of xH consumption in terms of xT consumption. Suppose you wanted to construct a linear budget (with price p for xH and price of 1 for xT ) that contains our “endowment” bundle as well as the outcome bundle from the gamble (in which I return k of my winnings if the coin comes up heads). Derive p as a function of k.
(f) Using our expected utility functions and the budget constraints (as a function of k), derive our demands for xH and xT as a function of k
(g) Determine the level of k that results in an equilibrium price and then verify that the resulting equilibrium output bundle is the one associated with the gamble we have been analyzing. Call this k∗ and illustrate what you have done in an Edgeworth Box.
(h) Is the allocation chosen through the gamble efficient when k = k∗?
(i) Suppose I had offered the lowest possible k that would induce you to enter the game instead —i.e. the one you derived in (b). Would the allocation chosen through the gamble be efficient in that case? Could it be supported as an equilibrium outcome with some equilibrium price?
(j) Illustrate what’s different in an Edgeworth Box in part (i) than in part (g).