Exercise 5: Dividing the Market Suppose there are two producers (U1 and U2), each of them producing a different product, and two retailers (R1 and R2). R1 only sells product 1 and R2 only sells product 2. The market is divided in two territories (zones): t1 and 1). Both retailers operate in both zones. In zone It, market demand for product 1 is given by D] : ak[5 plk + dpzk] and for product 2 is Dz = wk [5 pzk + dplk], for k = a, b, with are + sq, = 1. For a retailer operating in zone k, cost function is given by the price it must pay for each unit sold (am for retailer 1 and mu for retailer 2), plus a transportation cost of tk for each unit sold. That is: C - = (w:~ + tk)q for retailer 1' selling in zone 1:. As for the producers, the rst one has a unitary cost of c1 > 0 while for the second one we have 2 > O Inwhat follows, assume: d 1, ta 1, ff, 2, an ab , c1 0.8 and C2 = 1.4. NIH a) (8 points) Set up and solve both retailer's problems at each zone. Find all four retail prices (PM, Pza, P15, P25) as functions of producers' prices (:01 and 02). 13) (6 points) Applying backward induction, set up and solve both producers' problem, to find each one's reaction functions c) (3 points) Find equilibrium producers' prices :01 and $02. Using these, find all four retail prices d) (3 points) Conclude this exercise by computing each firm's prots (the two producers and the two retailers)