Answers to Chapter 3 Exercises - Luiscabral.net

Substituting this value into our inverse demand function, we obtain the optimal bid ofb = 2m .6 ⇥ 1m = 1.4m3.19. Competitive pressure. Suppose that a firm’s profits are given by ⇡ = ↵ + (e)+✏,where ↵ denotes the intensity of product market competition, e e↵ort by the manager, and✏ a random shock. The function (e) is increasing and concave, that is, d /de > 0 andd 2 /de 2 < 0.In order for the firm to survive, profits must be greater than ⇡. The manager’s payo↵is >0 if the firm survives and zero if it is liquidated, that is, if profits fall short of theminimum target. The idea is that if the firm is liquidated, then the manager loses his orher job and the rents associated with it.Suppose that ✏ is normally distributed with mean µ and variance2 , and that µ>⇡.Show that increased product market competition (lower ↵) induces greater e↵ort by themanager, that is, de/d↵ < 0.Answer:The manager’s payo↵ is given by⇣⌘P = P ↵ + (e)+✏>⇡ e✓ ⇣(e)⌘ ◆= 1 F ⇡ ↵ ewhere P(x >y) is the probability that x>yand F (✏) is the probability that ✏ is less thanx (cumulative distribution function). Taking the derivative with respect to e, the manager’schoice of e↵ort level, we getdPde =⇣⌘ d (e)f ⇡ ↵ (e)dewhere f(✏) is the density function of ✏. Sinceµ>⇡,wehaveµ>⇡ ↵ (e). Thereforef ⇡ ↵ (e) is in the increasing portion of f. It follows that an increase in ↵ leads toa decrease in f ⇡ ↵ (e) ; and this, in turn, implies a lower dP /de. Finally, a lowerdP /de implies a lower value of e. In words, a decrease in the degree of competition (higher↵) decreases the marginal benefit from managerial e↵ort (dP /de), and ultimately leads toa lower e↵ort of managerial e↵ort (e).113