# (solution) UNIVERSITY OF TORONTO Joseph L. Rotman School of

I want to answer to question The file with the question is attached.

UNIVERSITY OF TORONTO
Joseph L. Rotman School of Management
RSM332 PROBLEM SET #1 1. Suppose you own a farm that, if run efficiently, can produce corn according to the
following ?transformation? formula:
2 W1 = 90 × I03
where I0 is the number of bushels of corn planted in date 0, and W1 is the number of
bushels turned over to you in date 1, net after all payments to labor and other hired
inputs.
Your utility function of consumption at date 0 and consumption at date 1 is:
1 3 U (C0 , C1 ) = C04 C14
(a) If 64,000 bushels of corn are planted, what will be the net output of corn at date 1?
(b) If you set a target for output of 110,250 bushels, what is the minimum number of
bushels that must be planted?
(c) Suppose capital markets do not exist, and you can neither borrow, lend, or store
any corn at date 0. If you have 40,500 bushels of corn at date 0, what will your
production plan be? Your consumption plan? What will be the average rate of return
on investment of corn? What will be the rate of return on the marginal investment?
(d) If 64,000 bushels are planted, what will be the average rate of return on the investment? What will be the rate of return on the marginal investment?
(e) If a capital market exists, and the rate of interest is 50%, what will be your optimal
investment?
(f) If you have no corn at date 0, a capital market exists (50% rate of interest), and
you invest optimally, what is your equity in the venture? What will be your optimal
consumption plan? Outline your sources and uses of funds for date 0 and date 1.
(g) Will you loan your farm for a period for 36,000 bushels of corn? Why? If you have
no corn at date 0, and you decide to loan the farm, what will be your consumption
plan?
2. Your brother works as an engineer. Today is his 24th birthday. At his birthday party,
he asks for your advice on saving for his retirement. He plans to retire at 65 years old
and he expects to live for another 20 years afterwards. He wants an income of \$30,000
1 per year during his retirement years, to be paid annually on his birthday (starting from
his 65th birthday). He plans to save some amount at each birthday from the age 25 to
64. He thinks about saving a constant amount for the first 10 years and then increases
his saving at 3% each year until the last one before his retirement. The bank provides
two types of accounts. One account pays 6.9%/year compounded quarterly. The other
account pays 7%/year compounded annually?
(a) Which account would you recommend? Why?
(b) After choosing the proper account, how much should your brother save each year
for the first 10 years?
(c) What is the balance of your brother?s account right after he makes his deposit in
his saving account on his 50th birthday?
(d) In fact, your brother is not entirely sure how long he will live. Although he expects
to live until 85 years old, there is actually an equal probability that he will die at the
age of 75, 85, or 95. If this is the case, would you change your answer to part (b)?
3. Hoarding Tokens
You were told that the TTC is going to increase its fare from \$2.90 to \$3.00, effective
on January 1, 2017. As a result, you would like to buy some tokens (at \$2.90) and
save them for future use in 2017. However, there are rules on the purchase and use of
tokens. Assume that (1) tokens can only be purchased for personal consumption but
not for resale, and you use 2 tokens every day, (2) you can buy at most 10 tokens a
day, but since you need to use 2 tokens a day, the maximum number of tokens that you
can hoard is 8 per day. Suppose the annually compounded interest rate is 10%/year.
When should you start hoarding tokens?
4. Management Fee
You are about to invest some money in a bond fund. The management fee of the fund
is quite low, it only charges a fee of 1%/year on the assets managed. However, you do
not believe the bond fund manager has superior ability to beat the market and you
expect him to earn a return of 5%/year (before management fee) on the assets of the
fund. This is the same return that you (and everyone else) will be able to get but you
just do not want the hassle of managing your own money.
(a) Suppose you plan to leave the money in the bond fund for 20 years. For every
dollar that you invest in the bond fund today, how much are you effectively giving to
the fund manager for his service over the next 20 years? (Hint: How much are you
willing to give to the manager today if he is willing to waive the management fee in
the future.)
(b) How would your answer in part (a) change if you plan to leave the money in the
bond fund for a very long period of time (say for T years, where T is a very large
number). 2