Consider a situation of two farmers who grow different kinds of plants. Depending on the climate,...

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Consider a situation of two farmers who grow different kinds of plants. Depending on the climate, there are two states, $₁ and 52. Farmer 1 grows a plant that harvests much in state si but does not so in state s2. In contrast, Farmer 2's harvest will be better in state s2 but not in state s₁. To be precise, denote by (e₁,e₂) the income Farmer i can obtain from their harvest and suppose that (el,el) = (3,1) (e. e) = (1,3) Let 3 1 0 1 3 X₂ 45° be farmers' (common) utility function from the consumption plan (x,x) of farmer i for each state. In particular, the utility function is characterized with indifference curves convex toward the origin. Indifference curves. x₁ the direction of increase utility (a) Compute the utility levels for the farmers without any trade between them. (b) Now suppose that there is a financial market that issues two kinds of fi- nancial securities. Security 1 pays one unit of good in state s₁ and pays nothing in state s2. Security 2, in contrast, pays one unit of good in state $2 and pays nothing in state $₁. The prices of the two securities are set to be 1. The farmers can buy and sell the securities to manage the risks of their consumption plans. Let a be farmer i's holding of security t (note that a may be negative). Then the flow budget constraints for farmer i is given by • Initial period: a + a² = 0 • State s₁: x₁=₁ + a² • State S₂: x2= 2 + a²₂ Draw the budget line on (x,x)-space. (c) Derive the optimal consumption plan in the case with financial markets. (d) Compute the utility level in the case of financial markets. Consider a situation of two farmers who grow different kinds of plants. Depending on the climate, there are two states, $₁ and 52. Farmer 1 grows a plant that harvests much in state si but does not so in state s2. In contrast, Farmer 2's harvest will be better in state s2 but not in state s₁. To be precise, denote by (e₁,e₂) the income Farmer i can obtain from their harvest and suppose that (el,el) = (3,1) (e. e) = (1,3) Let 3 1 0 1 3 X₂ 45° be farmers' (common) utility function from the consumption plan (x,x) of farmer i for each state. In particular, the utility function is characterized with indifference curves convex toward the origin. Indifference curves. x₁ the direction of increase utility (a) Compute the utility levels for the farmers without any trade between them. (b) Now suppose that there is a financial market that issues two kinds of fi- nancial securities. Security 1 pays one unit of good in state s₁ and pays nothing in state s2. Security 2, in contrast, pays one unit of good in state $2 and pays nothing in state $₁. The prices of the two securities are set to be 1. The farmers can buy and sell the securities to manage the risks of their consumption plans. Let a be farmer i's holding of security t (note that a may be negative). Then the flow budget constraints for farmer i is given by • Initial period: a + a² = 0 • State s₁: x₁=₁ + a² • State S₂: x2= 2 + a²₂ Draw the budget line on (x,x)-space. (c) Derive the optimal consumption plan in the case with financial markets. (d) Compute the utility level in the case of financial markets.

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Without any trade between them each farmers income e e is dependent on the state of the climate Using the given utility functions the utility levels f... View the full answer