I am having great difficulty with this question... I don't know how to do from part c to part e, i managed to do a and b. Please can some one point me in the right direction.
1. Double Margins
Consider a market for a homogenous product with a demand function Q=a-P
The sole manufacturer of the product has a per unit cost of c and sells the product through a sole retailer who has no other costs apart from the per unit price that the manufacturer charges (d).
a) If the retailer takes d as given, what price maximizes his profit (PR)? What is the retailer’s profit (πR)?
b) Given that the manufacturer knows what price the retailer will set (as a function of d and a), what value of d (dM) maximizes the manufacturer’s profit? What is the corresponding profit (πM)?
c) Who, the retailer or the manufacturer, gets the largest profit?
d) If the manufacturer could bypass the retailer and sell directly to the consumers, show that its profits (πMR) would be larger that the sum of the profits that the retailer and manufacturer would earn if they acted independently (πR+πM)
e) Show that if the manufacturer charges the retailer d=c, plus a fixed fee of [(a-c)2]/4, it would earn the same as it would in b) above.
Part a: PR = (a+d)/2
ΠR = (a-d)2/4
Part b: PM = (a+c)/2
ΠM = (a-c)2/4