Question

Question 4
James Bennett also allocates wealth between youth and old age. He has no cash currently (in his youth), but will inherit $3000 in his old age. He can lend and borrow at the bank at 18% (that is, lending $1 in youth will give him $1.18 in old age). He has an investment opportunity that costs $12,000 now in his youth and has a payoff of $15,000 in his old age. This is the only investment opportunity available to him. What is the most he can consume in his youth?

*Make sure to input all currency answers without any currency symbols or commas, and use two decimal places of precision.

Answer 1

To determine the maximum amount that James Bennett can consume in his youth, we can analyze his financial situation and potential investment.

1. **Understanding the Investment**:
- Cost of the investment: $12,000 (today in youth).
- Payoff from the investment: $15,000 (in old age).

2. **Net Payoff Calculation**:
- If James makes this investment, he will need to pay $12,000 now but will receive $15,000 in the future.
- The net benefit of making the investment is:
\[
\text{Net payoff} = \text{Payoff in old age} - \text{Investment cost} = 15,000 - 12,000 = 3,000.
\]

3. **Wealth at Age**:
- In his old age, he will also inherit $3,000.
- Hence, if he invests the $12,000 today, his total wealth in old age will be:
\[
\text{Total wealth in old age} = \text{Payoff} + \text{Inheritance} = 15,000 + 3,000 = 18,000.
\]

4. **Future Value of Good Lending Scenario**:
- If James doesn't invest, he can lend money at this interest rate. For every $1 lent today, in his old age, he would have $1.18.
- The total inheritance of $3,000 would grow to:
\[
\text{Wealth at old age without investment} = 3,000 \times 1.18 = 3,540.
\]

5. **Investment Decision**:
- To maximize his consumption in youth, he needs to see if he can afford the investment. If he goes through with the investment, he will have $18,000 in old age.
- If he does not invest, he can only grow his inheritance, leading to a total of $3,540.

6. **Current Consumption in Youth**:
- If James desires to make the investment, to afford the $12,000 investment he can borrow:
- Since he will gain $18,000 in old age with the investment:
\[
18,000 = (C + \text{borrowing}) \times 1.18
\]
Here \( C \) is consumption in youth, and borrowing will be amount needed to cover investment.

7. **Let’s rearrange the equation with simplified borrowing**:
- To find maximum consumption in youth:
\[
C + 12,000 \times 1/(1.18) - 3000 = 0
\]
\[
C = 12,000 - 3000 \times 1/(1.18)
\]

- Calculate the future value of the investment:
- Without using tons of formulas:
- Calculate:
\[
3000/1.18 \approx 2542.37
\]

- This means $12,000 can be borrowed:
\[
C = 12,000 - 2542.37 = 9457.63
\]

Output the maximum amount James can consume in youth in two decimal precision:
\[
\boxed{9457.63}
\]