Suppose that the dealer, who is 25 years old, decides to sell the card at time , sometime in the next 40 years: 0< or equal to t < or equal to 40. At that time , he’ll invest the money he gets for the sale of the card in a bank account that earns an interest rate of r , compounded continuously. (This means that after years, an initial investment of will be worth Ie^(rt).) When he turns 65, he’ll take the money that’s in his bank account and begin to draw on it for his retirement. Let A be the amount of money in his account when he turns 65.

Question

Suppose that the dealer, who is 25 years old, decides to sell the card at time , sometime in the next 40 years: 0< or equal to t < or equal to 40. At that time , he’ll invest the money he gets for the sale of the card in a bank account that earns an interest rate of r , compounded continuously. (This means that after years, an initial investment of will be worth Ie^(rt).) When he turns 65, he’ll take the money that’s in his bank account and begin to draw on it for his retirement. Let A be the amount of money in his account when he turns 65.

Plot the function A(t) for several different values of k, while holding r constant. What does a larger value of k imply about the value of the card over time?. And now, what does a larger value of k imply about the best time to sell the card? Do these two facts seem consistent with one another?

Answer 1

To plot the function A(t) for different values of k, while holding r constant, we need to understand the relationship between k and the value of the card over time.

In this scenario, k represents the fraction of the value of the card that the dealer receives when selling it. Since the dealer is 25 years old and plans to sell the card at some time between 0 and 40 years, the value of the card is expected to appreciate over time.

So, a larger value of k means that the dealer will receive a larger fraction of the increasing value of the card when selling it. This implies that the value of the card over time will have a greater impact on the amount of money in his bank account when he turns 65.

If the value of the card is expected to increase substantially over time, selling it earlier (in the earlier years) at a larger k value would yield a higher amount of money in his bank account when he turns 65. This is because the larger value of k allows him to capture a larger share of the increasing value of the card.

Therefore, a larger value of k implies that selling the card earlier (in the earlier years) would be more beneficial for accumulating higher retirement savings.

These two facts are consistent with each other - a larger value of k indicates a greater gain from selling the card earlier, which aligns with the idea that the card's value will appreciate over time and selling it sooner allows the dealer to capture more value.