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Mathematical Methods for Economics Project

Problem: When to Sell the Basketball Signed by Kobe Bryant?

A seller in ebay has a rare basketball, which was used in 2012 London Olympics and was signed by

the basketball giant, Kobe Bryant. The seller is trying to decide when to sell it, and he is asking for

980 USD at this stage. He knows its value will grow over time, but he could sell it 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 seller sell the basketball? Experience suggests to the seller that

over time, the value of the basketball, like many other collectibles, will grow in a way consistent

with the following model:

V (t) = Ae?

p

t

where A and ? are constants, and V (t) is the value of the basketball in dollars at the time t years

after the present time.

1. Plot this function against t when A = 980 and ? = 0.5.

2. What is the interpretation of A?

3. Plot the function V (t) for several di?erent values of ?. What e?ect does ? have on the

value of the basketball over time?

1

Suppose that the seller, who is 35 years old, decides to sell this basketball at time t, sometime

in the next 30 years: 0 ? t ? 30. At that time t, he will invest the money he gets from the sale in a

bank account that earns an interest rate of r, compounded continuously, which means that after t

years, an initial investment of B USD will be worth Bert USD. When he turns 65, he will take the

money in his bank account for his retirement. Let M (t) be the amount of money in his account

when he turns 65, where t is the time at which he sells his basketball.

4. Write down the closed-formed expression of M (t).

5. Plot your function M (t) against t when A = 980, ? = 0.5, and r = 0.05.

6. If those values of the constants were accurate, then when should the seller sell the basketball to maximize the amount in his retirement account when he turns 65?

7. Plot the function M (t) for several di?erent values of ?, while holding r constant. What

does a larger value of ? imply about the value of the basketball over time? (Refer back to

question 3.) And now, what does a larger value of ? imply about the best time to sell the

basketball? Do these two facts seem consistent with one another?

8. Plot the function M (t) for several di?erent values of r, while holding ? constant. What

does a larger value of r imply about the best time to sell the basketball? Is that consistent

with the meaning of r?

9. Let to be the optimal time to sell the basketball, i.e., the time that will maximize M (t).

Try to find to in the general model. Note that your solution should be a function of the

constant variables A, ? and r.

10. Plot M (t) against t for di?erent combinations of A, ? and r, and verify that your expression for to does accurately predict when the best time will be to sell the basketball.

11. Are the properties of to as it relates to ? and r consistent with what you found in step 7

and step 8?

12. There is another way to decide when to sell the basketball instead of thinking about

putting the money from the sale into a retirement account. Suppose that today (time=0) the

seller puts an amount of money W 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 W ert . If at

p

?

t

time t the basketball is sold for an amount equal to V (t) = Ae , how much money W would

the seller 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 W is called the present value of

the basketball if it ends up being sold at time t. Model the present value of the basketball as

a function of the time t 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?

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Explanation & Answer:

12 Question

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Excel software

Mathematical Methods for Economics

Basketball Signed by Kobe Bryant

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