PS8(08) - 2 MWG 6.C.8 Prove this in two ways(a as a...

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ECN 712 Fall 2009 Professor Schlee Problem Set VIII, Due Tuesday, 3 November 1. An investor has initial wealth of w and divides his wealth between a safe asset with a rate of return of 0, and a risky asset with a rate of return of x + t . The number t is a constant and the distribution of x is given by a c.d.f. F with bounded support satisfying R xdF ( x ) = 0 and 0 < F (0) < 1. The investor has a C 3 vN-M utility u with u 0 > 0 and u 00 < 0. Let α * ( t ) denote the optimal expenditure on the risky asset as a function of the constant t . (a) Confirm that α * (0) = 0–the simpler your argument the better–and α * ( t ) > 0 for t > 0. (b) Calculate α 0* (0) and confirm that it is positive. Show that its magnitude depends on the investor’s Arrow-Pratt risk aversion measure at the point w ; and the variance of the random return, x . Explain the relationship intuitively. (c) Evaluate : “The optimal investment α * ( · ) is nondecreasing in t on R ++ .” Prove it if true, provide a counterexample or explanation if it is false.
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Unformatted text preview: 2. MWG, 6.C.8. Prove this in two ways: (a) as a corollary of Pratt’s Theorem; and (b) directly. (Hint for (b): write R xu 00 dF ( x ) = R ( xu )( u 00 /u ) dF ( x ).) 3. MWG, 6.C.9. Add a part (e): For t = 0 , 1, let y t be a nondegenerate, zero-mean random variable with cdf F t . Let F 1 be strictly riskier than F : the expectation of any strictly concave function on the support of F is lower under F 1 than under F . If v 000 > 0, show that E [ v ( x + y 1 )] > E [ v ( x + y ]. Hence any increase in risk leads to more precautionary saving. 4. MWG, 6.E.2. (Suggestion: prove this two ways, using the first order conditions and using the SSCP. Either of course is fine on its own, but notice that the argument appealing to the SSCP is shorter and effortlessly deals with corner solutions.) 1...
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