Exercise3 A

Exercise Session 5, Nov 8th ; 2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua (Cissy) Chen, Natalia Guseva

Exercise 1 Let f : R2 ! R be ( f (x; y) =

xy(x2 y 2 ) x2 +y 2

0

if (x; y) 6= (0; 0); if (x; y) = (0; 0):

a) Show that all second-order partials exist, and check whether they are continuous. b) Does Schwartz’s Theorem apply at (x; y) = (0; 0)? Exercise 2 Let f (x) = x3 ; g(y) = y: a) Check whether f , g are strictly concave/strictly convex, concave/convex, pseudo-concave/pseudo-convex, quasi-concave/quasi-convex. b) Show that f + g is neither quasi-concave nor quasi-convex. c) Show that a pseudo-concave function can only achieve its global maximum at a zero gradient point. Exercise 3 Find the quadratic approximation around (0; 0) for f (x; y) = ex ln(1 + y): Exercise 4 A competitive …rm chooses the quantity of labor L to be hired in order to maximize pro…ts, taking as given the salary w and the value of a productivity parameter . That is, the …rm solves max( f (L) L

wL)

Assume that the production function f () is twice continuously di¤ erentiable, increasing, and strictly concave (i.e.,f 0 > 0; f 00 < 0) (a) Write the …rst order condition for the …rm’s problem, and verify that the second order su¢ cient condition for a maximum holds. (b) Interpret the …rst order condition as an equation that implicitly de…nes a labor demand function of the form L = L(w; ):Show, using the implicit function theorem, that @L @L < 0 and > 0: @w @

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