Brian is about to take early retirement at the age of 45. His current wealth is...

    

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Brian is about to take early retirement at the age of 45. His current wealth is $2 million and he plans to use this wealth to fund his consumption over the remaining three periods of his life.¹ Any money that he does not spend on consumption in period 0 can be put in the bank where it will earn interest at the rate of 100% per period. That is, r = 1, so if chooses not to consume an amounts of his $2 million in period 0 he will have s (1+r) 2s available at the beginning of period 1. Similarly, any money he does not spend on consumption in period 1 can be put in the bank where it will also earn interest at the rate of r = 1. = (a) (5 points) Explain why his intertemporal budget set B can be de- scribed as follows: B = = {(co, C₁, C₂) ≥ (0,0,0) : co + 120₁ + 1/ 02 ≤ 2} (b) (10 points) Utilizing the fact that his marginal utility of consumption mu (c) = 2c, show that in period 0 his optimal consumption plan (measured in millions of dollars) is 888 (6, ci, G;) = (§¾, 7, 7 ) . (c) (5 points) Explain qualitatively how this optimal plan would change if the interest rate was greater than 1. Explain qualitatively how this optimal plan would change if the interest rate was less than 1. (d) (10 points) If Brian chooses to consume c = 8/7 in period 0, explain what his intertemporal budget set B¹ ((co, c₁, c₂)) will be in period 1. Show that the continuation of his original consumption plan (c₁, c₂) = (8/7,8/7) is indeed the optimal consumption plan for him to choose in period 1 from this budget set. Explain what property of his choice behavior does this reflect. Now suppose that Brian is a naive (quasi-)hyperbolic discounted utility max- imizer characterized by an "instantaneous" utility function u (c) = 2c-c²/2, a long-term discount factor d = 1/2, and a short-term discount factor (or present bias) B = 0.5. (e) (10 points) Show that his optimal consumption plan (co, c₁, c₂) from the perspective of his period 0 self is equal to (1.4, 0.8, 0.8). Given he consumes c = 1.4 in period 0 (and hence saves 0.6) work out the amounts ĉ₁ and ĉ2 that he will actually choose to consume in periods 1 and 2. For the last question, suppose that instead of being able to put money in the bank to earn interest at the rate of 100% per period, Brian can instead purchase a quantity q ≥ 0 of an annuity at a per-unit price p. That is, each unit of the annuity costs him p million dollars in period 0 and pays out 1 million dollars in period 1 and 1 million dollars in period 2. For example, if he purchases the fraction 0.2 of a unit of the annuity in period 0 at a cost of 0.2 x p then the annuity will pay him 0.2 (of a million dollars) in period 1 and 0.2 in period 2. (f) (10 points) If the unit-price p of the annuity is equal to + 1+r (+)² = 1+1, what type of Brian considered above would be strictly better off purchasing a quantity of the annuity instead of putting his savings in the bank. How many units will each of these types demand? Brian is about to take early retirement at the age of 45. His current wealth is $2 million and he plans to use this wealth to fund his consumption over the remaining three periods of his life.¹ Any money that he does not spend on consumption in period 0 can be put in the bank where it will earn interest at the rate of 100% per period. That is, r = 1, so if chooses not to consume an amounts of his $2 million in period 0 he will have s (1+r) 2s available at the beginning of period 1. Similarly, any money he does not spend on consumption in period 1 can be put in the bank where it will also earn interest at the rate of r = 1. = (a) (5 points) Explain why his intertemporal budget set B can be de- scribed as follows: B = = {(co, C₁, C₂) ≥ (0,0,0) : co + 120₁ + 1/ 02 ≤ 2} (b) (10 points) Utilizing the fact that his marginal utility of consumption mu (c) = 2c, show that in period 0 his optimal consumption plan (measured in millions of dollars) is 888 (6, ci, G;) = (§¾, 7, 7 ) . (c) (5 points) Explain qualitatively how this optimal plan would change if the interest rate was greater than 1. Explain qualitatively how this optimal plan would change if the interest rate was less than 1. (d) (10 points) If Brian chooses to consume c = 8/7 in period 0, explain what his intertemporal budget set B¹ ((co, c₁, c₂)) will be in period 1. Show that the continuation of his original consumption plan (c₁, c₂) = (8/7,8/7) is indeed the optimal consumption plan for him to choose in period 1 from this budget set. Explain what property of his choice behavior does this reflect. Now suppose that Brian is a naive (quasi-)hyperbolic discounted utility max- imizer characterized by an "instantaneous" utility function u (c) = 2c-c²/2, a long-term discount factor d = 1/2, and a short-term discount factor (or present bias) B = 0.5. (e) (10 points) Show that his optimal consumption plan (co, c₁, c₂) from the perspective of his period 0 self is equal to (1.4, 0.8, 0.8). Given he consumes c = 1.4 in period 0 (and hence saves 0.6) work out the amounts ĉ₁ and ĉ2 that he will actually choose to consume in periods 1 and 2. For the last question, suppose that instead of being able to put money in the bank to earn interest at the rate of 100% per period, Brian can instead purchase a quantity q ≥ 0 of an annuity at a per-unit price p. That is, each unit of the annuity costs him p million dollars in period 0 and pays out 1 million dollars in period 1 and 1 million dollars in period 2. For example, if he purchases the fraction 0.2 of a unit of the annuity in period 0 at a cost of 0.2 x p then the annuity will pay him 0.2 (of a million dollars) in period 1 and 0.2 in period 2. (f) (10 points) If the unit-price p of the annuity is equal to + 1+r (+)² = 1+1, what type of Brian considered above would be strictly better off purchasing a quantity of the annuity instead of putting his savings in the bank. How many units will each of these types demand?

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Brians intertemporal budget set B can be described as follows B Co C1 C2 0 0 0 Co C11r C21r2 2 This ... View the full answer