Price is up because it takes 18-24 months to produce lithium on the margin. The primary source of lithium is solar concentrated brine, so today’s supply is peak Covid planning decisions.
The people that had the guts to invest in lithium production in the middle of Covid when the price was in the tank, are currently harvesting that lithium and getting a massive payday.
Cost is literally another word for price - I'm not sure what you're trying to convey by trying to distinguish the two. When a battery manufacturer goes to buy a ton to lithium, the cost they have to pay has indeed increased 10x.
The price of extracting a ton of lithium from the ground may be the same, but it's not enough to keep up with demand. Which is why the cost of lithium on the market is skyrocketing.
