hw1a - UNIVERSITY OF TORONTO Joseph L. Rotman School of...

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Unformatted text preview: UNIVERSITY OF TORONTO Joseph L. Rotman School of Management RSM332 PROBLEM SET #1 SOLUTIONS 1. (a) The optimal consumption plan is C = Y = 18 and C 1 = 0. Therefore, the utility would be U (18 , 0) = 0. (b) If we have access to the capital market but not the production opportunity, the consumption allocation problem can be set up as follows: max C ,C 1 C 2 5 C 3 5 1 s.t. C + C 1 1 + 0 . 25 = 18 . With the budget constraint we have C = 18- 4 5 C 1 , which can be substituted into the utility function so that we maximize the utility function with respect to C 1 , i.e., max C 1 18- 4 5 C 1 2 5 C 3 5 1 . Taking the first order derivative of U ( C ,C 1 ) with respect to C 1 and setting it to zero, we obtain dU dC 1 = 3 5 C- 2 5 1 18- 4 5 C 1 2 5- 4 5 2 5 18- 4 5 C 1 - 3 5 C 3 5 1 = 0 . Multiplying 5 C 2 5 1 18- 4 5 C 1 3 5 on both sides of the above equation, we have 3 18- 4 5 C 1 - 8 5 C 1 = 0 and C 1 = 13 . 5. Therefore, C = 18- 4 5 C 1 = 7 . 2. Today you need to lend Y- C = 10 . 8 units of goods to the capital market. The utility at the optimal consumption is U (7 . 2 , 13 . 5) = 10 . 50. 1 (c) When we have access to the production opportunity but not the capital market, the consumption allocation problem becomes max C ,C 1 C 2 5 C 3 5 1 s.t. C = 18- I C 1 = 10 q I Substituting the budget constraints into the utility function and we can maximize the utility with respect to I , i.e., max I (18- I ) 2 5 (10 I 1 2 ) 3 5 . Taking the first order derivative of U ( C ,C 1 ) with respect to I and setting it to zero, we obtain dU dI = 3 10 I- 7 10 (18- I ) 2 5- 2 5 (18- I )- 3 5 I 3 10 = 0 . Multiplying 5 I 7 10 (18- I ) 3 5 on both sides of the above equation, we have 3 2 (18- I )- 2 I = 0 and I = 54 / 7 = 7 . 71. Therefore, C = Y- I = 10 . 29 and C 1 = 10 I = 27 . 77. The utility at the optimal consumption is U (10 . 29 , 27 . 77) = 18 . 67. (d) Finally, when we have access to both the production opportunity and the capital market, we know that the investment problem and the consumption can be solved separately according to Fishers Separation Theorem. First, we solve the investment problem. Since the production displays a decreasing marginal return to investment, the optimal investment amount is achieved when the marginal return on investment is equal to the interest rate in the capital market, i.e., df ( I ) dI = 5 I = 1 + 0 . 25 . Therefore, I = 4 and I = 16. Once we determine the optimal investment amount, we can set up the consumption allocation problem by taking the optimal investment amount as given, i.e., max C ,C 1 C 2 5 C 3 5 1 s.t. C + C 1 1 + r = Y + f ( I ) 1 + r- I , 2 where the last two terms on the right hand side of the budget constraint represents the net present value of the investment in the production function. Therefore, the budget constraint becomes C + 4 5 C 1 = 18 + 16 = 34 ....
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hw1a - UNIVERSITY OF TORONTO Joseph L. Rotman School of...

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