1. Suppose you own a farm that, if run efficiently, can produce corn according to the following transformation formula:

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W1 = 90 I0

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:

U(C0, C1) = C0 C1

1 4

3 4

(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 $30,000per 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 brothers 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).