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Unformatted text preview: UNIVERSITY OF TORONTO Joseph L. Rotman School of Management Oct. 19, 2010 Florence/Kan RSM332 MID-TERM EXAMINATION Simutin/Tolias/Yang SOLUTIONS 1. (a) When we have access to the production opportunity but not the capital market, the consumption allocation problem is max C ,C 1 C C 1 s.t. C = 12- I , C 1 = 40 q I . Substituting the budget constraints into the utility function and we can maximize the utility with respect to I , i.e., max I (12- I ) × 40 I 1 2 . Differentiating U ( C ,C 1 ) with respect to I and setting it to be zero, we obtain d U d I =- 40 I 1 2 + (12- I ) × 20 I- 1 2 = 0 . Multiplying I 1 2 / 20 on both sides of the above equation, we have (12- I )- 2 I = 0 and I * = 4. Therefore C * = Y- I * = 8 and C * 1 = 40 √ I * = 80. (b) When we have access to both the production opportunity and the capital market, we know that the investment problem and the consumption problem can be solved separately according to the Fisher’s Separation Theorem. We first 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., d f ( I ) d I = 20 √ I = 1 + 1 3 . 1 Therefore √ I * = 15 and I * = 225. 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 C 1 s.t. C + C 1 1 + r = Y + f ( I * )- I * (1 + r ) 1 + r , where the last term on the right hand side of the budget constraint represents the net present value of the production and it is NPV = 40 × 15- 225 × 4 3 4 3 = 225 . Therefore the budget constraint becomes C + 3 4 C 1 = 237 . There are different ways to solve the optimal consumption problem. The first method is to substitute C = 237- (3 / 4) C 1 in the utility function U ( C ,C 1 ) = C C 1 = [237- (3 / 4) C 1 ] C 1 . Then differentiating the utility function with respect to C 1 , we obtain d U d C 1 = 237- 3 2 C 1 . Setting it equal to zero, we obtain C * 1 = 158 and C * = 118 . 5. Alternatively, we can solve consumption allocation problem by setting MRS =- (1 + r ): MRS =- ∂U/∂C ∂U/∂C 1 =- C 1 C =- (1 + r ) =- 4 3 . Together with C + 3 4 C 1 = 237, we obtain C * = 118 . 5 and C * 1 = 158. We need to borrow C * + I *- Y = 331 . 5 units of goods today to finance part of the investments in the production and the consumption. (c) According to the Fisher’s Separation Theorem, when investors have access to a per- fect capital market, the investment problem and the consumption allocation problem can be done separately. Furthermore, the investment problem is to maximize the net present value (NPV) of the production and has nothing to do with individual investor’s utility. Therefore, both you and Larry would choose the same optimal investment level and the NPV of the production is the fair price of the production. From part (b), weand the NPV of the production is the fair price of the production....
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This note was uploaded on 11/29/2011 for the course RSM 332 taught by Professor Raymondkan during the Winter '08 term at University of Toronto.
- Winter '08