Q. There are a 1000 costumers uniformly distributed on [0,3]. Each wants to buy 1 ice-cream. There are two firms which produce ice-cream costlessly and firm i charges p_i. Consumer's effective price is p_i+td where t is the transportation cost and d is the distance between the firm and the consumer. To show that there is a unique Nah equilibrium when firms are on 0th and 3th mark and no nash equilibrium when they are on 1th and 2th mark.
My attempt: I have shown that there is a unique Nash equilibrium in the first case. Each firm sells at 3t price and each faces a demand of 500.
For the second part I have shown that in equilibrium the prices will be necessarily equal. But then I am again getting that each firm will face a demand of 500. I thought I could show that there will exist a small epsilon such that when a firm cuts their prices by epsilon they will be able to increase their profits. It did not work. My reasoning was that between 1 and 2 there are around 333 consumers which cannot be equally divided and if some firm cuts their prices a little they will be able to attract the odd consumer. I am, however, not able to do this mathematically.
Working of " I thought I could show that there will exist a small epsilon such that when a firm cuts their prices by epsilon they will be able to increase their profits. It did not work."
I was able to show that in equilibrium the prices will be equal. Given that, I found that the consumer at the 1.5th mark is indifferent b/w buying from firm1 and firm2. Therefore, the demand that each firm faces is 500, and each makes a profit of 500p or 1500t (p will be 3t). Now suppose that firm1 cuts their prices by some epsilon. Now the agent at (1.5 +epsilon/2t)th mark is indifferent. Firm1 now faces a demand of (500 + 500 epsilon/3t). Profit of firm1 then is (p-epsilon)[demand]. I tried to find for which epsilon will this be > 500p (original profits). I am getting that this will hold for epsilon<p-3t = 0, which contradicts epsilon being strictly greater than 0.
I am not able to understand or see how the second case is differnt from the first one.
I am not looking for an answer, just a little nudge.