# 332mid08a - UNIVERSITY OF TORONTO Joseph L Rotman School of...

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Unformatted text preview: UNIVERSITY OF TORONTO Joseph L. Rotman School of Management Oct. 21, 2008 Ezer/Kan/Florence RSM332 MID-TERM EXAMINATION Pomorski/Zhou SOLUTIONS 1. (a) For Mr. Oh, his consumption at time 0 and time 1 are given by C = 1000- I and C 1 = 30 I 1 2 . Therefore, we can write his utility as U O = C 1 2 C 1 2 1 = (1000- I ) 1 2 (30 I 1 2 ) 1 2 = √ 30(1000- I ) 1 2 I 1 4 = √ 30(1000 I 1 2- I 3 2 ) 1 2 . Differentiating U O with respect to I , we obtain d U O d I = 1 2 √ 30(1000 I 1 2- I 3 2 )- 1 2 1000 2 I- 1 2- 3 2 I 1 2 . Setting the derivative equal to zero, we have 1000 2 I- 1 2- 3 2 I 1 2 = 0 ⇒ I = 333 . 33 . Therefore, the optimal investment by Mr. Oh is I * = 333 . 33. (b) If Mrs. Ner follows the investment decision of Mr. Oh, her consumption at time 0 and time 1 will be C = 1000- 333 . 33 = 666 . 67 and C 1 = 30 √ 333 . 33 = 547 . 72. It follows that her utility will be U N ( C ,C 1 ) = C 1 4 C 3 4 1 = (666 . 67) 1 4 (547 . 72) 3 4 = 575 . 30 . (c) The utility of Mrs. Ner is given by U N = C 1 4 C 3 4 1 = (1000- I ) 1 4 (30 I 1 2 ) 3 4 = 30 3 4 (1000- I ) 1 4 I 3 8 = 30 3 4 (1000 I 3 2- I 5 2 ) 1 4 . Differentiating U N with respect to I , we obtain d U N d I = 1 4 30 3 4 (1000 I 3 2- I 5 2 )- 3 4 3000 2 I 1 2- 5 2 I 3 2 . Setting the derivative equal to zero, we have 3000 2 I 1 2- 5 2 I 3 2 = 0 ⇒ I = 600 . 1 Therefore, the optimal investment by Mrs. Ner is I * = 600. At the optimal level of investment, the utility of Mrs. Neh is given by U N = (1000- I * ) 1 4 (30 I * 1 2 ) 3 4 = 631 . 19 . (d) When Mrs. Ner invests 333.33, her consumption at time 0 and 1 are given by C = 666 . 67 and C 1 = 547 . 72. In order for Mrs. Ner to have a utility of 631.19, she needs a consumption at time 0 to satisfy C 1 4 (547 . 72) 3 4 = 631 . 19 ⇒ C = 965 . 98 . This means that she needs an additional consumption of 965 . 98- 666 . 67 = 299 . 31 at time 0 for her to be indifferent....
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## This note was uploaded on 01/10/2011 for the course RSM RSM332 taught by Professor Booth during the Spring '10 term at University of Toronto.

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332mid08a - UNIVERSITY OF TORONTO Joseph L Rotman School of...

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