Three firms employ two sets of workers each. Their production functions are a) f(x,y) = x + y xy b) f (x, y) = (x + y) a c) f (x, y) = xya where x and y are the number of workers of each groups and 0.5 < a < 1. Also, assume x > 0 and y 0. 1. (12 pts) At the point (x, y) = (1,1), are the Hessian of the production functions pos- itive definite, positive semidefinite, negative definite, negative semidefinite or indefinite? Note, for a Hessian matrix a C H = C b you can label the principal minors | Ha| = | a |, |Hb| = | b |, |H| = |H| 2. (24 pts) The profit functions of the firms are II (x, y) = pf (x, y) - wxx - Wyy where wx, Wy are their salary. a C = C b **Assume p, wx, W are constants, p > wx, p > wy, Wx # Wy. For each of the production functions listed above, - If you can, find the stationary points of the profit function where both types of workers are used (i.e. an interior solution). - Explain why or why not you can find a unique maximum for the profit function (maybe more than one justification would be valid). 3. (12 pts) For each of the production functions listed above, would the firm prefer choosing only one type of worker to maximize its profits? If so, would it prefer using only x or only y? What level of profit can the firm expect (assuming an infinite profit is possible)? 4. (6 bonus points) Consider the three descriptions below. In your opinion, which applies to which of the previous production functions? No need to justify, just state you answer. In firm 1 workers have complementary skills. In firm 2 both groups of workers perform the exact same task independently In firm 3 some workers like to work while listening to fusion jazz while the other like to listen to pop rock. Both group's music makes it hard for the other group to concentrate.