Auction Strategies and Utility Functions: Homework 1 for AGEC695, Assignments of Agricultural engineering

Download Auction Strategies and Utility Functions: Homework 1 for AGEC695 and more Assignments Agricultural engineering in PDF only on Docsity! AGEC695 Homework 1 Due: September 11, 2007 1. The dominant strategy in Dutch and first-price, sealed-bid auction is given by σ (v) = v − ∫ v v FV (u) (N−1) du FV (v) (N−1) . (a) Show that σ (v) < v for v > v. (b) Show dσ (v) /dv > 0. (c) Show ∂σ (v;N ) /∂N > 0. (hint: you may need to use the Leibniz’ formula: Let F (t) = ∫ b(t) a(t) f (t, x) dx for all t, where a, b and f are differentiable functions of t. F ′ (t) = f (t, b (t)) b′ (t)− f (t, a (t)) a′ (t) + ∫ b(t) a(t) f ′t (t, x) dx.) 2. When there is a binding reserve price r, the dominant strategy in Dutch and first-price, sealed-bid auction is σ (v) = v − ∫ v r FV (u) (N−1) du FV (v) (N−1) , v ≥ r. (a) Show that ∂σ (v; r) /∂r > 0 for v > r. (b) Does the result in part (a) suggest that the seller can always in- crease expected payoff by raising the reserve price? Why? 3. Let U (Y ) be the utility function of an individual who is strictly risk averse. Suppose U (0) = 0. Show that for Y > 0, U (Y ) U ′ (Y ) > Y. 1