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

(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.

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

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) = 2cc2=2,a long-term discount factor  = 1=2, and a short-term discount factor (orpresent bias) = 0:5

The marginal propensity to consume is A. Options A, B and C B. ΔC/ΔY C. The slope of the consumption function D. Coefficient c in the equation C = C + cYd

Now suppose that Brian is a naive (quasi-)hyperbolic discounted utility max-imizer characterized by an ìinstantaneousîutility function u (c) = 2cc2=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.