# SJSU When to Sell the Basketball Signed by Kobe Bryant Questions

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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?