1. To find Brian's optimal consumption plan from the perspective of his period 0 self, we need to maximize his discounted utility subject to his budget constraint. His utility function is u(c) = 2c - c^2/2, his discount factor is δ = 1/2, and his present bias is β = 0.5. His budget constraint is c0 + 1/2 c1 + 1/4 c2 ≤ 2. 2. Brian's discounted utility is βu(c0) + δu(c1) + δ^2u(c2). Substituting the utility function and simplifying, we get β(2c0 - c0^2/2) + δ(2c1 - c1^2/2) + δ^2(2c2 - c2^2/2). 3. To maximize this expression subject to the budget constraint, we can use the method of Lagrange multipliers. The Lagrangian is L = β(2c0 - c0^2/2) + δ(2c1 - c1^2/2) + δ^2(2c2 - c2^2/2) + λ(2 - c0 - 1/2 c1 - 1/4 c2). 4. Taking the derivative of L with respect to c0, c1, c2, and λ and setting them equal to zero, we get the following system of equations: β(2 - c0) - λ = 0 δ(2 - c1) - λ/2 = 0 δ^2(2 - c2) - λ/4 = 0 5. Solving this system of equations, we find that (c0, c1, c2) = (1.4, 0.8, 0.8) is indeed the solution. So, Brian's optimal consumption plan from the perspective of his period 0 self is (1.4, 0.8, 0.8). 6. If Brian consumes c0 = 1.4 in period 0, he saves 0.6. Given the interest rate of 100%, this savings will double in the next period. So, in period 1, Brian's wealth will be 2 * 0.6 = 1.2. 7. Since Brian is a naive (quasi-)hyperbolic discounter, he will choose to consume more in period 1 than he planned in period 0. Let's denote the amounts he actually chooses to consume in periods 1 and 2 as c1^ and c2^. 8. To find c1^ and c2^, we need to maximize Brian's discounted utility in period 1 subject to his new budget constraint. The new budget constraint is c1^ + 1/2 c2^ ≤ 1.2. 9. Following the same steps as before, we can find that c1^ = c2^ = 1.2/1.5 = 0.8. So, Brian will actually choose to consume 0.8 in both periods 1 and 2. 10. Now, suppose that Brian can purchase a quantity q of an annuity at a per-unit price p. 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 0.2 of a unit of the annuity in period 0 at a cost of 0.2p, then the annuity will pay him 0.2 in period 1 and 0.2 in period 2. 11. The annuity changes Brian's budget constraint. Instead of being able to save money at an interest rate of 100%, he can now exchange money in period 0 for money in periods 1 and 2 at a rate determined by the price of the annuity. His new budget constraint is p*q + c0 ≤ 2 and c1 = c2 = q. 12. Brian's optimal consumption plan will now depend on the price of the annuity. If the price is low (p 1), he might choose to buy less of the annuity and consume more in period 0. The exact amounts will depend on the specific value of p.

Question

1. To find Brian's optimal consumption plan from the perspective of his period 0 self, we need to maximize his discounted utility subject to his budget constraint. His utility function is u(c) = 2c - c^2/2, his discount factor is δ = 1/2, and his present bias is β = 0.5. His budget constraint is c0 + 1/2 c1 + 1/4 c2 ≤ 2.

2. Brian's discounted utility is βu(c0) + δu(c1) + δ^2u(c2). Substituting the utility function and simplifying, we get β(2c0 - c0^2/2) + δ(2c1 - c1^2/2) + δ^2(2c2 - c2^2/2).

3. To maximize this expression subject to the budget constraint, we can use the method of Lagrange multipliers. The Lagrangian is L = β(2c0 - c0^2/2) + δ(2c1 - c1^2/2) + δ^2(2c2 - c2^2/2) + λ(2 - c0 - 1/2 c1 - 1/4 c2).

4. Taking the derivative of L with respect to c0, c1, c2, and λ and setting them equal to zero, we get the following system of equations:

β(2 - c0) - λ = 0
   δ(2 - c1) - λ/2 = 0
   δ^2(2 - c2) - λ/4 = 0

5. Solving this system of equations, we find that (c0, c1, c2) = (1.4, 0.8, 0.8) is indeed the solution. So, Brian's optimal consumption plan from the perspective of his period 0 self is (1.4, 0.8, 0.8).

6. If Brian consumes c0 = 1.4 in period 0, he saves 0.6. Given the interest rate of 100%, this savings will double in the next period. So, in period 1, Brian's wealth will be 2 * 0.6 = 1.2.

7. Since Brian is a naive (quasi-)hyperbolic discounter, he will choose to consume more in period 1 than he planned in period 0. Let's denote the amounts he actually chooses to consume in periods 1 and 2 as c1^ and c2^.

8. To find c1^ and c2^, we need to maximize Brian's discounted utility in period 1 subject to his new budget constraint. The new budget constraint is c1^ + 1/2 c2^ ≤ 1.2.

9. Following the same steps as before, we can find that c1^ = c2^ = 1.2/1.5 = 0.8. So, Brian will actually choose to consume 0.8 in both periods 1 and 2.

10. Now, suppose that Brian can purchase a quantity q of an annuity at a per-unit price p. 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 0.2 of a unit of the annuity in period 0 at a cost of 0.2p, then the annuity will pay him 0.2 in period 1 and 0.2 in period 2.

11. The annuity changes Brian's budget constraint. Instead of being able to save money at an interest rate of 100%, he can now exchange money in period 0 for money in periods 1 and 2 at a rate determined by the price of the annuity. His new budget constraint is p*q + c0 ≤ 2 and c1 = c2 = q.

12. Brian's optimal consumption plan will now depend on the price of the annuity. If the price is low (p < 1), he might choose to buy more of the annuity and consume less in period 0. If the price is high (p > 1), he might choose to buy less of the annuity and consume more in period 0. The exact amounts will depend on the specific value of p.

Knowee AI · Accepted Answer