Problem 4. [20 points] Cost minimization Assume a continuum of consumers are uniformly distributed along the in terval [0, 1]. A social planner chooses the placements of three \"shops\". Con- sumers have linear transportation costs and visit the shop that is closest to their location. Each consumer incurs a cost of i if she or he travels a distance 1. Please note that to obtain full marks, you need to provide detailed solutions and show your work. (a) [10 points] Derive the locations (3*, b' and c' of the three shops that minimize aggregate transportation cost T(e, b, c). What is the corresponding minimum cost? (b) [10 points] Suppose now that two shops already exist: one at location 0 and the other at location 5 / 6. The social planner is choosing where to place another (the third} shop but can also relocate the existing two shops. The cost of relocating a shop depends on the distance required: if a shop needs to be \"mov \" distance d, then the relocation cost is %d2. What are the optimal locations :r\