1. Brian's utility function is given by u(c) = 2c - c^2/2. His long-term discount factor is δ = 1/2, and his short-term discount factor (or present bias) is β = 0.5.

2. Brian's total utility from a consumption plan (c0, c1, c2) is given by U(c0, c1, c2) = u(c0) + βδu(c1) + δ^2u(c2).

3. To find his optimal consumption plan from the perspective of his period 0 self, we need to maximize this total utility subject to the budget constraint c0 + c1 + c2 = 3 (assuming his total wealth is 3).

4. Taking the derivative of U with respect to each ci and setting it equal to zero, we get the following first-order conditions:

2 - c0 = 0 = c0 = 2

βδ(2 - c1) = 0 = c1 = 2/βδ = 2/(0.5*0.5) = 2

δ^2(2 - c2) = 0 = c2 = 2/δ^2 = 2/(0.5*0.5) = 2

5. However, these values do not satisfy the budget constraint (2 + 2 + 2 3). Therefore, we need to adjust them to make sure they add up to 3.

6. Given that Brian consumes c0 = 1.4 in period 0 (and hence saves 0.6), the amounts he will actually choose to consume in periods 1 and 2 are given by the following:

^c1 = βδ(2 - c0) = 0.5*0.5*(2 - 1.4) = 0.3

^c2 = δ^2(2 - c0) = 0.5*0.5*(2 - 1.4) = 0.3

7. Therefore, Brian's optimal consumption plan from the perspective of his period 0 self is (1.4, 0.8, 0.8), but given his present bias, he will actually consume (1.4, 0.3, 0.3) in periods 0, 1, and 2 respectively.

Question

1. Brian's utility function is given by u(c) = 2c - c^2/2. His long-term discount factor is δ = 1/2, and his short-term discount factor (or present bias) is β = 0.5.

2. Brian's total utility from a consumption plan (c0, c1, c2) is given by U(c0, c1, c2) = u(c0) + βδu(c1) + δ^2u(c2).

3. To find his optimal consumption plan from the perspective of his period 0 self, we need to maximize this total utility subject to the budget constraint c0 + c1 + c2 = 3 (assuming his total wealth is 3).

4. Taking the derivative of U with respect to each ci and setting it equal to zero, we get the following first-order conditions:

2 - c0 = 0  => c0 = 2

βδ(2 - c1) = 0  => c1 = 2/βδ = 2/(0.5*0.5) = 2

δ^2(2 - c2) = 0  => c2 = 2/δ^2 = 2/(0.5*0.5) = 2

5. However, these values do not satisfy the budget constraint (2 + 2 + 2 > 3). Therefore, we need to adjust them to make sure they add up to 3.

6. Given that Brian consumes c0 = 1.4 in period 0 (and hence saves 0.6), the amounts he will actually choose to consume in periods 1 and 2 are given by the following:

^c1 = βδ(2 - c0) = 0.5*0.5*(2 - 1.4) = 0.3

^c2 = δ^2(2 - c0) = 0.5*0.5*(2 - 1.4) = 0.3

7. Therefore, Brian's optimal consumption plan from the perspective of his period 0 self is (1.4, 0.8, 0.8), but given his present bias, he will actually consume (1.4, 0.3, 0.3) in periods 0, 1, and 2 respectively.

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