But gold doesn't make itself easier to find if too few mine it. Bitcoin does.

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to think mining has no effect on price and only demand moves price is to dent the fundamentals of btc as compare to gold or silver or any precious metal.
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The point is that the offer of new btc stays pretty stable as long as one miner keeps crunching numbers. In other words, with difficulty adjustment a single miner can keep ALL the mineable btc in existence. This isn't true for gold: it does continue to exist even if no one mines it, but has little financial or industrial use when it's buried deep down.