This problem requires the use of calculus to solve some consumer optimization pr

Question

This problem requires the use of calculus to solve some consumer optimization problems. Nina has the following utility function: U = ln C_1 + ln C_2 + ln C_3 She starts with wealth of $120,000, earns no additional income, and faces a zero interest rate. How much does she consume in each of the three periods? David is just like Nina, except he always gets extra utility from present consumption. From the perspective of period one, his utility function is U = 2 ln(C_1) + ln(C_2) + ln(C_3) In period one, how much does David decide to consume in each of the three periods? How much wealth does he have left after period one? When David enters period two, his utility function is U = ln(C_1) + 2 ln(C_2) + ln(C_3) How much does he consume in periods two and three? How does your answer here compare to David's decision in part (b)? If, in period one, David were able to constrain the choices he can make in period two, what would he do? Relate this example to one of the theories of consumption discussed in the chapter.

Explanation / Answer

U = lnC1 + lnC2 +lnC3

subject to: C1 + C2 + C3 = 12000

MRS = MU1/MU2 = P1/P2

C2/C1 = 1

Similarly, C2/C3 = 1

Therefore, C1 = C2 = C3

Putting this in budget constraint, C1 = C2 = C3 = 40,000

b) MU1/MU2 = 2C2/C1 = 1

MU2/MU3 = C3/C2 = 1

2C2 = C1

C3 = C2

Putting this in constraint,

2C2 + C2 + C2 = 120000

C2 = 30000 = C3

C1 = 60000

After first period he has 60000 left.

c) MU1/MU2 = C2/2C1 =1

C2 = 2C1

MU1/MU3 = C3/C1 = 1

Putting this in constraint,

C1 + 2C1 + C1 = 120000

C1 = 30000 = C3

C2 = 60000

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