Bertrand competition with homogenous good and Hotelling's spatial model

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.

• Unfortunately "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." is very poor reasoning. Assume that there are $1000/3$ customers in the interval $[1,2]$, most of these models are continuous. Oct 4 '21 at 17:55
• Can you please edit your question so that it includes the calculations for this part: "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." Oct 4 '21 at 17:56
• I understand that most of these models are continuous, but in my understanding, this particular one is not. Since, there are a discrete number of consumers. Am I interpreting it incorrectly?
– Pc1
Oct 4 '21 at 18:24
• There is no way for me to know that. Nothing you wrote in the Q suggests that the number of consumers is discrete. Oct 4 '21 at 18:37
• There are exactly 1000 consumers distributed uniformly b/w [0,3].
– Pc1
Oct 4 '21 at 18:57