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Unformatted text preview: UNIVERSITY OF TORONTO Joseph L. Rotman School of Management RSM332 PROBLEM SET #1 SOLUTIONS 1. (a) The net output of corn at date 1 = 400 50,000 = 89,442.72. (b) We know from the transformation formula that W 1 = 400 I . Thus to harvest W 1 at date 1, we need to plant I = W 1 400 2 . In order to harvest 100,000 bushels at date 1, we must plant 100 , 000 400 2 = 62,500 bushels at date 0. (c) In this case, we maximize our utility by maximizing the minimum of consumption at date 0 and date 1. In order to do this, we want to have the same consumption at both date 0 and date 1. Consumption at date 0 is 22,500 I and consumption at date 1 is 400 I . To maximize utility, we set: 22,500 I = 400 q I I + 400 q I 22,500 = 0 . Solving the quadratic equation, we have: q I = 400 + q 400 2 4( 22,500) 2 = 50 . Therefore, I = (50) 2 = 2,500, and the optimal consumption at dates 0 and 1 are C = 22,500 2,500 = 20,000 and C 1 = 400 2,500 = 20,000, respectively. Average rate of return on investment = W 1 I 1 = 400 2 , 500 2 , 500 1 = 700% . Rate of return on the marginal investment = dW 1 dI 1 = 200 I 1 = 200 50 1 = 300% . (d) Average rate of return on investment = 400 22 , 500 22 , 500 1 = 166 2 3 % . Rate of return on the marginal investment = 200 I 1 = 200 150 1 = 33 1 3 % . (e) The optimal investment will be such that the marginal rate of return will equal the interest rate. So from (d), it can be seen that the optimal investment is 22,500. 1 (f) Optimal consumption plan: again you want the consumption in both periods to be the same. So we have: C = C 1 . But this consumption plan must fulfill the budget constraint: C + C 1 1 + r = I + W 1 1 + r C + C 1 1 1 3 = 22,500 + 60,000 1 1 3 C + C 1 1 1 3 = 22,500 . Solving the two equations, we have C = C 1 = 12,857.14. The value of your equity in the farm is simply its net present value and it is equal to 22,500 bushels of corn. The sources and uses of the funds at date 0 and date 1 are: Sources Uses Date 0 Borrowing 35,357.14 Plant 22,500 Consume 12,857.14 Date 1 Receive corn 60,000 Repayment of Loan 47,142.86 Consume 12,857.14 (g) Yes, loan the farm for a period at 23,100. Equity in the farm was shown to be 22,500 in (f). Again, you wish to consume equal amounts in both periods. Similar to (e), we solve C = C 1 and C + C 1 1 + r = 23,100 . Solving the equations, we get C = C 1 = 13,200. Now the residual amount in date 0 is lent at 33 1 3 %. 2. (a) Let y = 1 / (1 + r ) n , we have: 200 y + 100 y 2 = 200 y 2 + 2 y 2 = 0 y = 1 3 . 2 Since y has to be positive, we drop the negative root and therefore y = 3 1 (1 + r ) n = 1 3 1 ! 3 + 1 3 + 1 !...
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 Fall '08
 RAYMONDKAN

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