mining doesn't influences the highest price. but influences the lowest, because under an x price the mining costs more than the mined coins.
Please explain how it puts a lower limit on the price. If the cost of mining is higher than the value of the mined coins, what exactly causes the price to rise? How does the cost of mining prevent people from selling for a lower price or encourage people to pay more?
... if the cost of mining increases sufficiently ..., they will simply stop producing. ...
... Well, then producers will have to offer their bitcoin at increasingly higher prices in the market just to break even. ...
You have contradicted yourself. First you wrote that if the cost rises, then miners stop mining. Then you wrote that if the cost rises, they continue to mine but they hold the coins, waiting for a higher price.
I agree with your first statement, that if costs exceed value, miners will stop mining.
But, your second statement assumes that miners are operating as an informal cartel in which they agree to restrict supply of bitcoins in the market. I hardly think this is a realistic assumption. First, mined bitcoins are only a small part of the market. A mining cartel would have very little influence on the price. Second, miners have to pay their bills regardless of what happens to the bitcoin price. Many miners don't have the luxury of sitting on their coins waiting for a higher price.
Miners don't control the market, they are controlled by it.
It is irrelevant if miners are only a small percentage of the stock of bitcoins, because it is only producers who consistently offer them to the market at all times. oil producers are only a small part of the oil market. wheat farmers are only a small part of the wheat market. And yet marginal cost = marginal product = selling price, always and everywhere. Read an economics text book if this still confuses you (it is hard to wrap your mind around, I know!). I can give you some good recommendations.
Your second point point about not sitting on them is precisely why they offer them for sale in the market as they are produced. They are not speculating, they are in the business of producing them and immediately selling them. I never assert that miners hold on to coins waiting for a better price.
So let me be more clear as to why I think you see a contradiction. There are two forces at work here, one manifests itself in scenario 1 and the other scenario 2.
[1] in a world where there are heterogeneous miners, some with different efficiency or energy costs etc., then as difficulty (cost) increases, it will weed out all but the most efficient miners. Why? Let's say there is a point at which Miner A has a break-even of $250 and Miner B has a break-even of $300. Miner A & B will both offer $300, but for B this will be his absolute minimum. Nobody pays $300, so Miner A will offer $299 but miner B will still offer $300. Miner A will offer $298 but miner B will still offer $300. And so on until miner A's offer is $250 and miner B still offers $300. Assuming there are many like Miner A they will all be in competition with each other driving the offer quickly down to their $250 limit. This is individualistic competition not a cartel.
[2] in a world where the majority of miners are subject to the same efficiency (largely homogeneous), say now most miners are all like miner A, and nobody is able to increase their efficiency or lower their electricity cost any further. Their break even now is $250 and so they offer $250 in the market. Another miner adds his hashpower to the network so now all of their break even price rises to $251. Nobody offers $250 anymore, they all offer $251. Another dozen miners join the network, increasing the difficulty even more. Nobody offers $251 they all offer $260. This is not a cartel, this is each individual looking out for their own self-interest. The invisible hand somebody once called it.
Going back to scenario 1. say the market offer is $250 and nobody is willing to pay $250. The best bid is $240. Will a miner sell $240? probably not unless his breakeven is $240 or lower. But a speculator, trader, market maker etc. just might sell $240s. However, once those have been sold @ $240, the best offer will soon return to $250 after those lower offers have been cleared.
Next, if bitcoin is indeed currency, most people enter the market to buy bitcoin. They then "sell" their bitcoin by spending them on goods or services and do not ever offer them in exchange for dollars, so offers will be lifted. If my job is to produce and sell bitcoin I do not speculate on their price (just as oil companies do not speculate on the price of oil, they make it and sell it). So if my cost is $249 and I can sell $250, I will do that all day long. But why not raise my offer to $251? Because if I am competing with many other producers to sell my bitcoins and they also have a cost of $249 they can offer $250 and nobody will take my $251s. So I will have to at least match - or even improve if I can - that best offer. If the world price for oil is $40 a barrel and I am offering $41 nobody will buy my barrel of oil. If it cost me $41, well I am just out of luck - there must be a lower cost producer out there it turns out.
Finally, this mechanism happens over time. There will be periods of time, days, weeks maybe many weeks, where spikes in demand may drive up the price or spikes in supply may depress the price at levels well above or below the cost of production price. Extrinsic value drivers will tend to keep the market price, in fact, above the avg. cost of production.