2010-Problem-Session-4

2010-Problem-Session-4 - minimizing the cost of producing a...

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Stanford University Department of Management Science and Engineering Optional Problem Session 4 Winter 2010 Friday, February 5, 2010 Practice Problem 4.1 (Second Order Stochastic Dominance) Consider lotteries ˜ x = ± 1 , 1 2 ; 3 , 1 2 ² , and ˜ y = ± ±, 1 8 ; 2 , 3 4 ; 4 - ±, 1 8 ² where ± (0 , 1) (i) Compute and compare the mean and variance of each lottery. (ii) For a suﬃciently small ± , can you ﬁnd an increasing concave utility function u ( z ) such that Eu x ) > Eu y )? Does ˜ y second order stochastically dominate ˜ x ? (iii) Should the results of (i) and (ii) be interpreted as a contradiction? Provide an intuitive explanation for part (ii). (iv) Can you ﬁnd an increasing concave utility function ˆ u ( z ) such that E ˆ u y ) > E ˆ u x )? Does ˜ x second order stochastically dominate ˜ y ? Practice Problem 4.2 (Characteristics of the Conditional Factor Demand and Cost Func- tion) Consider a production function f : R M R L and a competitive ﬁrm facing the problem of
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Unformatted text preview: minimizing the cost of producing a vector q when input prices are w , i.e. min w z f ( z ) q where w R M ++ , z R M + , and q R L + . Assume that there is a twice continuously dierentiable solution function z * ( w,q ) to the minimization problem. z * ( w,q ) corresponds to the input vector that minimizes the cost of producing at least q when input prices are w , and is referred to as conditional factor demand . (i) Show that the associated cost function C ( w,q ) = w z * ( w,q ) is homogeneous of degree one in w . (ii) Show that if q q (componentwise) then C ( w, q ) C ( w,q ). (iii) Show that C ( w,q ) is concave in w . (iv) Show that 5 w C ( w,q ) = z * ( w,q ), and use this to show that D w z * ( w,q ) is symmetric and negative semidenite. Can D w z * ( w,q ) be negative denite? 1...
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