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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.

1 answer

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: \[ \boxed{9457.63} \]