Definition of covered market

I do Industrial Organization course, and I've seen the phrase "market is covered" several times in my lecture slides, however, I'm struggling to find the formal definition of "covered market" and the condition when it's true. Can you please help me with that?

A market is covered if all consumers will choose to buy from at least one of the firms at the prevailing prices.

For example, consider a standard Hotelling model with two firms who are located at opposite ends of the unit interval. Suppose that each firm charges that same price $$p$$ and that consumers are uniformly located on $$[0, 1]$$. Finally, suppose that every consumer has unit demand and the same utility function $$u = v - tx - p$$ where $$v$$ is their valuation for the good, $$t$$ is the cost of transportation and $$x$$ is the distance that they travel to the firm from which they buy. In this example, every consumer will choose to buy a good if and only if the consumer located in the centre chooses to buy the good (since she has the furthest distance to travel). This is equivalent to asserting that

$$v - 0.5t - p \geq 0 \iff v \geq 0.5t + p .$$

That is, the market is covered when the valuations of the consumers are sufficiently high relative to the price and cost of transportation.

It is a market where firms compete for the marginal consumer.

It’s an opposite of having fully separated market where two firms do not compete for marginal customer.

An example of covered market would be market for bus rides where the departures are scheduled at the exactly same time - so there is competition for the marginal customer.

Example of separated marked would be when bus companies all schedule different times of departure so they don’t compete for the marginal customer.

• "Example of separated marked would be when bus companies all schedule different times of departure so they don’t compete for the marginal customer." So it is the 17:15 bus home or death? – Giskard Dec 13 '19 at 14:21
• @Giskard I don’t get it, I see the word departure often being applied to public transport, am I using in wrong way somehow? – 1muflon1 Dec 13 '19 at 14:38
• Separated market: there is no competition for the marginal customer. Thus for every customer there is exactly one acceptable departure time? If I don't catch my bus, the next one is not good enough? – Giskard Dec 13 '19 at 15:05
• @Giskard oh I see what you mean. Well I don’t think the models want to imply that but if you have some duopoly situation with two busses if they departure at same time then they will interact like duopoly on covered market and if they have different departure times more like a monopoly where they don’t compete for marginal customer and hence markets will be separated. If you think of the consumers utility as a function of desired departure time then you can say this, or at least I seen this terminology been used this way in IO papers – 1muflon1 Dec 13 '19 at 15:20
• "If you think of the consumers utility as a function of desired departure time" which of course, is the correct way to think about it, as anyone who has ridden the bus would know. There is a reason frequencies rise during peak hours, and it's not because consumers are indifferent to departure time. – heh Dec 13 '19 at 21:16