Question

In: Economics

Consider the following two-period Fisher model of consumption. Jamil earns $600 in the first period and...

Consider the following two-period Fisher model of consumption. Jamil earns $600 in the first period and $0 in the second period. The interest rate is 10 percent. His lifetime utility function is log(?1 ) + 0.5log(?2).

a) Find the optimal values of ?1 and ?2.

Answer: ?? = ???, ?? = ???

b) Suppose that the lifetime utility function changes to log(?1 ) + log(?2). Calculate the new optimal values of ?1 and ?2. How is the optimal value of ?1 in this question compared to the one in the pervious question? What is the intuition behind this change?

Answer: ?? = ???, ?? = ??? .The weight (or time discount factor) multiplied to the second-period utility function is larger, which means that Jamil is more patient. Hence he reduces the first period consumption and increases the second period consumption.

As you may have noticed, I have the answers, I just need to understand the step-by-step process of solving this question.

Solutions

Expert Solution

Income in period 1, Y1 = $600. Income in period 2, Y2 = $0. Interest rate is 10 percent.

Lifetime utility function U = log(?1 ) + 0.5log(?2).

a) Find the optimal values of ?1 and ?2.

Lifetime budget constraint is (1 + 10%)C1 + C2 = (1 + 10%)Y1 + Y2

1.1C1 + C2 = 1.1*600 + 0

1.1C1 + C2 = 660

From the utility function slope = MRS = -MUC1/MUC2 = -(1/C1) / (0.5/C2) = -2C2/C1

Slope of the budget constraint = -1.1/1 = -1.1

At the optimal choice, MRS = slope of the budget constraint

2C2/C1 = 1.1

C2 = 0.55C1

Use this in the budget constraint

1.1C1 + 0.55C1 = 660

C1 = 660/1.65 = 400

C2 = 0.55*400 = 220

Answer: ?? = ???, ?? = ???

b) New lifetime utility function changes to log(?1 ) + log(?2). MRS = -MUC1/MUC2 = -(1/C1) / (1/C2) = -C2/C1

Slope of the budget constraint = -1.1/1 = -1.1

At the optimal choice, MRS = slope of the budget constraint

C2/C1 = 1.1

C2 = 1.1C1

Use this in the budget constraint

1.1C1 + 1.1C1 = 660

C1 = 660/2.2 = 300

C2 = 1.1*300 = 330

Answer: ?? = ???, ?? = ??? .

Rahul Sunny answered 1 year ago

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