(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.
Question
(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.
Solution
(b) To find the optimal consumption plan, we need to set the marginal utility of consumption equal to the marginal cost of consumption in each period. The marginal utility of consumption is given by mu(c) = 2 - c. The marginal cost of consumption in each period is the opportunity cost of not saving the money for future consumption, which is given by the interest rate r = 1.
Setting these equal to each other, we get:
2 - c = 1
Solving for c, we get:
c = 1
So, the optimal consumption in each period is $1 million. Therefore, the optimal consumption plan is (c0, c1, c2) = (1, 1, 1).
(c) If the interest rate was greater than 1, the opportunity cost of current consumption would be higher. This would incentivize Brian to consume less in the current period and save more for future consumption. So, the optimal consumption plan would involve lower consumption in period 0 and higher consumption in periods 1 and 2.
If the interest rate was less than 1, the opportunity cost of current consumption would be lower. This would incentivize Brian to consume more in the current period and save less for future consumption. So, the optimal consumption plan would involve higher consumption in period 0 and lower consumption in periods 1 and 2.
(d) If Brian chooses to consume c0 = 8/7 in period 0, his remaining wealth for periods 1 and 2 will be 2 - 8/7 = 6/7. This wealth can be saved to earn interest at the rate of r = 1, so his budget set in period 1 will be:
B1 = {(c1, c2) ≥ (0, 0) : c1 + 1/2 c2 ≤ 6/7}
To find the optimal consumption plan in period 1, we again set the marginal utility of consumption equal to the marginal cost of consumption. Solving this equation, we get:
c1 = c2 = 8/7
So, the continuation of his original consumption plan (c1, c2) = (8/7, 8/7) is indeed the optimal consumption plan in period 1.
This reflects the property of time consistency in Brian's choice behavior. Time consistency means that the optimal plan chosen at one point in time remains optimal at future points in time. In this case, the consumption plan chosen by Brian in period 0 remains optimal in period 1.
Similar Questions
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