If Brian chooses to consume c0 = 8=7 in period 0, explainwhat his intertemporal budget set B1 ((c0; 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 choosein period 1 from this budget set. Explain what property of his choicebehavior does this reáect.Now suppose that Brian is a naive (quasi-)hyperbolic discounted utility max-imizer characterized by an ìinstantaneousîutility function u (c) = 2c c2=2,a long-term discount factor = 1=2, and a short-term discount factor (orpresent bias) = 0:5
Question
If Brian chooses to consume c0 = 8=7 in period 0, explainwhat his intertemporal budget set B1 ((c0; 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 choosein period 1 from this budget set. Explain what property of his choicebehavior does this reáect.Now suppose that Brian is a naive (quasi-)hyperbolic discounted utility max-imizer characterized by an ìinstantaneousîutility function u (c) = 2c�c2=2,a long-term discount factor = 1=2, and a short-term discount factor (orpresent bias) = 0:5
Solution
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If Brian chooses to consume c0 = 8/7 in period 0, his remaining wealth for the next two periods will be 2 - c0 = 2 - 8/7 = 6/7.
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Given the interest rate of 100%, this wealth will double in the next period. So, in period 1, Brian's wealth will be 2 * 6/7 = 12/7.
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Therefore, his intertemporal budget set B1 in period 1 will be: B1 = {(c1, c2) : c1 + 1/2 c2 ≤ 12/7}.
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Now, let's check if the continuation of his original consumption plan (c1, c2) = (8/7, 8/7) is indeed optimal in period 1. Substituting these values into the budget constraint, we get: 8/7 + 1/2 * 8/7 = 12/7. So, the original consumption plan satisfies the budget constraint.
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To check if it's optimal, we need to compare the marginal utility of consumption in periods 1 and 2. The marginal utility is mu(c) = 2 - c. So, mu(c1) = mu(c2) = 2 - 8/7 = 6/7. Since the marginal utility is the same in both periods, the original consumption plan is indeed optimal.
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This reflects the property of Brian's choice behavior known as time consistency. Time consistency means that Brian's preferences over consumption in different periods remain the same over time. In other words, the plan that was optimal for him in period 0 continues to be optimal in period 1.
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Now, suppose that Brian is a naive (quasi-)hyperbolic discounted utility maximizer characterized by an instantaneous utility function u(c) = 2c - c^2/2, a long-term discount factor δ = 1/2, and a short-term discount factor (or present bias) β = 0.5. This means that Brian tends to overvalue present consumption relative to future consumption. As a result, he might choose a consumption plan in period 0 that is not time consistent. In other words, the plan that is optimal for him in period 0 might not be optimal in period 1.
Similar Questions
Now suppose that Brian is a naive (quasi-)hyperbolic discounted utility max-imizer characterized by an ìinstantaneousîutility function u (c) = 2c c2=2,a long-term discount factor = 1=2, and a short-term discount factor (orpresent bias) = 0:5.(e) (10 points) Show that his optimal consumption plan (c0; c1; c2) fromthe perspective of his period 0 self is equal to (1:4; 0:8; 0:8). Givenhe consumes c0 = 1:4 in period 0 (and hence saves 0:6) work out theamounts ^c1 and ^c2 that he will actually choose to consume in periods1 and 2.
(b) (10 points) Utilizing the fact that his marginal utility of consumptionmu (c) = 2 c, show that in period 0 his optimal consumption plan(measured in millions of dollars) is(c0; c1; c2) = 87 ; 87 ; 87.(c) (5 points) Explain qualitatively how this optimal plan would changeif the interest rate was greater than 1. Explain qualitatively how thisoptimal plan would change if the interest rate was less than 1.(d) (10 points) If Brian chooses to consume c0 = 8=7 in period 0, explainwhat his intertemporal budget set B1 ((c0; 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 choosein period 1 from this budget set. Explain what property of his choicebehavior does this reáect.
Utilizing the fact that his marginal utility of consumptionmu (c) = 2 c, show that in period 0 his optimal consumption plan(measured in millions of dollars) is(c0; c1; c2) = 87 ; 87 ; 87
Show that his optimal consumption plan (c0; c1; c2) fromthe perspective of his period 0 self is equal to (1:4; 0:8; 0:8). Givenhe consumes c0 = 1:4 in period 0 (and hence saves 0:6) work out theamounts ^c1 and ^c2 that he will actually choose to consume in periods1 and 2.For the last question, suppose that instead of being able to put money inthe bank to earn interest at the rate of 100% per period, Brian can insteadpurchase a quantity q 0 of an annuity at a per-unit price p. That is, eachunit of the annuity costs him p million dollars in period 0 and pays out 1million dollars in period 1 and 1 million dollars in period 2. For example, ifhe purchases the fraction 0:2 of a unit of the annuity in period 0 at a costof 0:2 p then the annuity will pay him 0:2 (of a million dollars) in period1 and 0:2 in period 2
Explain why his intertemporal budget set B can be de-scribed as follows:B =(c0; c1; c2) (0; 0; 0) : c0 + 12 c1 + 14 c2 2Initially suppose that Brianís choices over consumption streams are gov-erned by an exponential discounted utility family of preference relations,characterized by an ìinstantaneousîutility function u (c) = 2c c2=2 and adiscount factor = 1=2
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