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Homework answers / question archive / 1)This problem uses calculus to compare two scenarios of consumer optimization
1)This problem uses calculus to compare two scenarios of consumer optimization.
a. Nina has the following utility function: U = ln(C1) + ln(C2) + 1
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?
b. David is just like Nina, except he always gets extra utility from present consumption. From the perspective of period 1, his utility function is U = 2ln(C1) + ln(C2) +ln(C3)
In period 1, how much does David decide to consume in each of the three periods? How much wealth does he have left after period 1?
c. When David enters period 2, his utility function is U = ln(C1) + 2ln(C2) + ln(C3)
How much does he consume in periods 2 and 3? How does your answer here compare to David’s decision in part (b)?
d. If, in period 1, David were able to constrain the choices he can make in period 2, what would he do?
Ans. A
The allocation of income made by consumers across time periods are such that the marginal rate of substitution between consumption in two periods is (1+r), here r is referred as the real interest rate.
The marginal rate of substitution between periods 1 and 2 of Mrs. N is expressed as follow:
MRS1,2 = C2/C1…… (1)
And the marginal rate of substitution between periods 2 and 3 of Mrs. N is expressed as follow:
MRS2,3 = C3/C2…… (2)
And the real interest rate is 0.
Now, Mrs. N’s optimal consumption bundle fulfills with budget constraint and the marginal conditions expressed as in Equation (1) and (2) which gives following results as follow:
C1 + C2 + C3 = $120,000 …… (3)
Hence, it leads to
C1 = C2 = C3 …… (4)
Putting Equation (4) in (3) gives following results as follows:
C1 = C2 = C3 = $40,000 …… (5)
Ans. B
Mr. D and Mrs. N has the same income and the real interest rate is still zero, but Mr. D gives priority of twice the weight on current consumption. This means that Mr. D’s marginal rate of substitution of period 1 and 2 is expressed as follows:
MRS1,2 = 2C2/C1…… (6)
And, Mr. D’s marginal rate of substitution of period 2 and 3 is expressed as follows:
MRS1,2 = C3/C2 …… (7)
Mr. D’s optimal bundle fulfills budget constraint and the marginal conditions of Equation (6) and (7) gives following results as follows:
C1 + C2 + C3 = $120,000 …… (8)
Since, C1 = 2C2 and C2 = C3, then Equation (8) gives following results as follows:
C1 + C2 + C3 = $120,000
2C2 + C2 + C2 =$120,000
4C2 = $120,000
C2 = $30,000
Thus, C1 = $60,000 and C2 = C3 = $30,000.
Ans. C
Mr. D present consumption twice of future consumption, which means that choices in period 2 satisfy following condition as follows:
C2 + C3 = $60,000 …… (9)
Since,C2 = 2C3, then this gives following results as follows:
C2 + C3 = $60,000
2C3 + C3 = $60,000
3C3 = $60,000
C3 = $20,000
Thus,C2 = $40,000 and C3 = $20,000.
Ans. D
During period 1, the optimal consumption of Mr. D in periods 2 and 3 are C2 = C3 = $30,000. However, in period 2, Mr., D the revises plan of optimal consumption is C2 = $40,000 and C3 = $20,000 . In order eliminate this situation, Mr. D required to constrained consumption in period 2 in stick to previous consumption plan. Mr. D’s preferences is an illustration of Laibson’s pull-of-instant-gratification model.
