HW9_Sketch_Solutions

ECO 206Y1Y: Microeconomic Theory Prof. Freitas, University of Toronto Eco 206 HW9 Sketch Solutions These are sketch solutions only. If you still have questions, please stop by office hours. Q1 There are two possible states of the world C and D. With probability 0.1 state C occurs. Without insurance, the buyer has $6000 in state C and $6000 in state D. Her value of money in state C is u ( x ) = 10 x 0 . 5 and in state D it is u ( x ) = x 0 . 5 She is offered insurance at the rate of $0 . 2 per $1 of benefit paid in state D. (a) How many C state dollars is she willing to give up for an extra dollar in the state D without insurance? What is the intuition behind the number you found? (b) How many C state dollars does she have give up for an extra dollar in the state D with the given insurance? (c) How much insurance does she choose to buy? Explain the intuition behind your answer. (d) If the insurance contract changes to $0 . 1 per $1 of benefit paid out in state D. Will she fully insure? Explain the intuition behind your answer. Solution: (a) This question is asking for the MRS. Since we’re looking for one more $ in state D , I’m going to put state D consumption x D on the x-axis. This means | MRS | = 0 . 9 * 0 . 5 (6000) - 0 . 5 0 . 1 * 10 * 0 . 5 (6000) - 0 . 5 = 0 . 9 She is willing to give up $0 . 9 of state C insurance for $1 in state D . By asking you for the intuition I was trying to get you to examine the MRS formula more closely. Notice here that the endowment of money in both state is exactly the same. This means that her willingness is determined purely by the probability difference (0 . 9 vs 0 . 1 ) and the state dependent nature of her preferences ( u C ( x ) = 10 x 0 . 5 vs u D ( x ) = x 0 . 5 ) . Based on the probabilities along, state D is more likely, so that makes her more willing to give up money on state C , but 1$ is state C is more valuable from a state dependent perspective. (b) This question is asking for the OC. The safest way to do this is from the BC equation. We have x D = 6000 - p + p γ = 6000 + 1 - γ γ p x C = 6000 - p ⇒ p = 6000 - x C ⇒ x D = 6000 + 1 - γ γ (6000 - x C ) ⇒ x D = 6000 γ - 1 - γ γ x C Taking the derivative gives us dx C dx D = - γ 1 - γ = 0 . 2 0 . 8 = - 0 . 25 This means she has to give up $0.25 of state C consumption for an additional $1 of state D consumption. (c) We can look at the MRS vs OC at the endowment bundle to make a prediction where we will go intuitively. Since she is willing to pay more the state D consumption than she has to we would predict that she would use insurance to move money to state D. She would do that via a positive premium. Here even though 1$ in state C is more valuable the fact that the insurance pays out in the state with the 90% probability (which will never happen in real life because 202209 1 of 3 HW9

ECO 206Y1Y: Microeconomic Theory Prof. Freitas, University of Toronto insurance companies would go out of business) makes her move money to state D. L : 0 . 1 * 10 (6000 - p ) 0 . 5 + 0 . 9 6000 + 1 - γ γ p 0 . 5 FOC 0 . 5 √ 6000 - p = 0 . 9 * 0 . 5 * 1 - γ γ ( 6000+ 1 - γ γ p ) 6000 + 1 - γ γ p = (6000 - p ) 0 . 9 * 1 - γ γ 2 p 1 - γ γ 1 + 0 . 81 1 - γ γ = 6000 0 . 9 * 1 - γ γ 2 - 1 p = 6000 ( ( 0 . 9 * 1 - γ γ ) 2 - 1 ) 1 - γ γ ( 1+0 . 81 1 - γ γ ) γ = 0 . 2 ⇒ p = 4231 . 13 b = 21155 . 65 x C = 1768 . 87 x D = 22924 . 52 (d) If the insurance contract changes to $0 . 1 per $1 of benefit paid out in state D. Will she fully insure? Explain the intuition behind your answer. Here I just wanted to catch any memorization that relied on match insurance rate 0.1 to probably 0.1 to claim fair insurance. This is NOT fair insurance. Fair insurance is when the expected value with insurance is the same without insurance. Here without insurance we have an EV=6000.With insurance we have γ = 0 . 1 0 . 1 * (6000 - p ) + 0 . 9 6000 + 1 - γ γ p = 0 . 1 * (6000 - p ) + 0 . 9 (6000 + 9 p ) = 6000 + 8 p = 6000 Clearly this is unfair insurance in a way that is terrible for the insurance company. You can resolve earlier parts with γ = 0 . 1 to get γ = 0 . 1 ⇒ p = 5195 . 82 b = 25979 . 1 x C = 804 . 18 x D = 26783 . 28 She does not fully insure and infact because her OC now becomes - 1 9 it becomes even cheaper to move money into state D so she moves even more money into that state. Q2 There are two possible states of the world C and D. With probability 0.1 state C occurs. Without insurance, the buyer has $6000 in state C and $6000 in state D. His value of money in state C is u ( x ) = 10 x and in state D it is u ( x ) = x . He is offered insurance at the rate of $0 . 2 per $1 of benefit paid in state D. (a) How many C state dollars is he willing to give up for an extra dollar in the state D without insurance? What is the intuition behind the number you found? (b) How many C state dollars does he have give up for an extra dollar in the state D with the given insurance? (c) How much insurance does he choose to buy? Explain the intuition behind your answer. (d) If the insurance contract changes to $0 . 1 per $1 of benefit paid out in state D. Will her choice change? Explain the intuition behind your answer. Solution: 202209 2 of 3 HW9