There are $N$ consumers uniformly distributed along a linear city of unit length, served by two shops

located at opposite extremities of the city. The two shops sell an identical product, for which

consumers have unit demands. One shop has a constant marginal cost of $2,$ the other a constant

marginal cost of $4.$ Neither shop has fixed costs. The cost to consumers of travelling the length of

the city is $4.$ Calculate the equilibrium prices for the shops, and their profits in terms of $N.$