Question

Gerald is a CEO in Brainies Consulting, Inc. His income in the first year is m1 = $200 and in the second m2 = $200. Assume that the interest rate is r = 100%. His time horizon is limited to these two years.

(a) Find PV and FV of Gerald’s income

(b) Show on the graph (C1; C2) Gerald’s budget set. Mark PV, FV, and the slope of his budget line.

(c) Explain what borrowing/lending strategy gives Gerald each of the two “extreme” consumption points. How much does he borrow/lend in the first period, how much does he pay back/receive in the second period?

(d) Suppose his utility function is:

U(C1; C2) = ln(C1) + ln(C2)

Find Gerard’s optimal choice analytically and show it on the graph. Does the optimal consumption involve saving or borrowing?

Answer 1

Given:

(a) The Present Value and Future Value of Gerald's income is

(b) The graph showing PV, FV and the slope of the budget line is shown in the figure below, where the incomes of the two periods are shown on the 2 axes, with present consumption on the x-axis and future or next period consumption on the y-axis.

(c) For consumption in the current period equal the the extreme value of $300 (=PV), we must borrow a sum of $100, and consume nothing in the next period. In the next period, we get the income of $200, which is returned back along with interest and so we consume nothing in the next period. The calculations are as follows:

At time t=1, Period 1 Income + Borrowed Amount = $200 + $100 = $300 (Present Period Consumption)

At time t=2, Period 2 Income - Borrowed Amount - Interest on Borrowed Amount at 100% = $200 - $100 - $100 = $0 (Future Period Consumption)

For consumption in the future period equal the the extreme value of $600 (=FV), we must lend a sum of $200 in present period. In the next period, we get the income of $200, along with the lent amount and the interest earned on it. The calculations are as follows:

At time t=1, Period 1 Income - Lent Amount = $200 - $200 = $0 (Present Period Consumption)

At time t=2, Period 2 Income + Lent Amount + Interest on Lent Amount at 100% = $200 + $200 + $200 = $600 (Future Period Consumption)

(d)

The Optimum value of consumption occurs when the MRS equals the slope of the budget line.

Equating (i) and (ii), we get

Again, we know that the present value of wealth is equal to the present value of consumption. So, have the wealth equation as:

Using the value of current wealth (PV = $300) and plugging equation (iii), we get:

This optimal consumption can be obtained by Lending. It can be explained as:

At time t=1, Period 1 Income - Lent Amount = $200 - $50 = $150 (Present Period Consumption)

At time t=2, Period 2 Income + Lent Amount + Interest on Lent Amount at 100% = $200 + $50 + $50 = $300 (Future Period Consumption).

The following graph shows the optimal consumption, given the incomes of the two periods.