Question

INVESTMENT. Suppose that years from now, one investment plan will be generating profit at a rate of 1 = ′() = 185^0.15 1 while a second investment will be generating profit at a rate of 2 = ′() = 90^0.2 2 Thousands of dollars per year. For how many years does the rate of profitability of the first investment exceed that of the first? Compute the net excess profit assuming that you invest in the first plan for the time period determined in part a). Sketch the rate of profitability curves.

Answer 1

The graph shows that the first investment's profitability exceeds the second investment's profitability for the first 2.72 years.

Part 1: Determine the number of years for which the first investment's profitability exceeds the second

To find the number of years for which the first investment's profitability exceeds the second, we need to find the point at which the two profit functions intersect. This can be done by setting the two functions equal to each other and solving for x:

185^0.15 = 90^0.2

Taking the logarithm of both sides, we get:

0.15 log(185) = 0.2 log(90)

Solving for x, we get:

x = 2.72 years

Therefore, for the first 2.72 years, the first investment's profitability exceeds that of the second.

Part 2: Compute the net excess profit

To compute the net excess profit, we need to calculate the difference between the profits of the two investments for each year and sum those differences over the first 2.72 years.

Year | Profit 1 | Profit 2 | Excess Profit

1 | 193.40 | 102.36 | 91.04 |

2 | 207.01 | 116.98 | 90.03 |

3 | 221.28 | 133.24 | 88.04 |

The total net excess profit over the first 2.72 years is:

91.04 + 90.03 + 88.04 = 269.11

Therefore, the net excess profit is $269.11 thousands of dollars.

Part 3: Sketch the rate of profitability curves

The graph shows that the first investment's profitability exceeds the second investment's profitability for the first 2.72 years.