A dealer in sports cards has a rare baseball card, and he’s trying to decide when to sell it. He knows its value will grow over time, but he could sell the card and invest the money in a bank account, and the value of the money would also grow over time due to interest. The question is: when should the dealer sell the card?
Experience suggests to the dealer that over time, the value of the card, like many collectibles, will grow in a way consistent with the following model:
where and are constants, is the value of the baseball card in dollars, and is the number of years after the present time.
1. Graph this function against when and .
2. What is the interpretation of ?
3. Plot the function for several different values of . What effect does
have on the value of the baseball card over time?
v(t)= Ce^(k(square root(t))
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.
6. If those values of the constants were accurate, then when should the dealer sell the card so as to maximize the amount in his retirement account when he turns 65? First estimate the answer using the graph, then use calculus to verify your answer.
7. 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?
8. Plot the function A(t) for several different values of r ,while holding k constant. What does a larger value of r imply about the best time to sell the card? Is that consistent with the meaning of r ?
9. Let t be the optimal time to sell the baseball card—i.e., the time that will maximize A(t). Use calculus to find t in the general model. Note that since the constants C,k,and r are part of the general model, they may be part of the solution as well.
10. Graph A(t) against t for different combinations of C,k,and, r and verify that
your expression for t does accurately predict when the best time will be to sell the baseball card.
11. Are the properties of t as it relates to k and r consistent with those you found in steps 7 and 8 above?
12. There is another way to decide when to sell the baseball card instead of thinking about putting the money from the sale into a retirement account. Suppose that today (time =0) the dealer puts an amount of money $M into a bank account that earns interest at an annual rate of r , compounded continuously, so that at time t the bank account will be worth $Me^(rt). If at time t the baseball card is sold for an amount
equal to V(t)=Ce^(k(sqrt(t)), how much money $M would the dealer have needed to invest initially in order for the bank account value and the baseball card value to be equal at the time of the sale? That amount $M is called the present value of the baseball card if it ends up being sold at time t.
Model the present value of the card as a function of the time when it is sold. Find the time when selling the card would maximize its present value. Is the answer consistent with the one you found earlier, in step 9 above?