battery Li content = 80g / (kW hour) [1]
Gigafactory total projected capacity 2020 = 85 GW hr/year [2]
Gigafactory lithium metal demand = (80 g / (kW hour)) (85 GW hr/year) = 6800 t/year
Gigafactory lithium hydroxide demand (anhydrous) = (6800 t/year) (23.95 g/mol / (6.941 g/mol)) = 23500 t/year
Gigafactory lithium hydroxide demand (monohydrate) = (6800 t/year) (41.96 g/mol / (6.941 g/mol)) = 41000 t/year
(FT article was evidently talking about anhydrous lithium hydroxide)
price lithium hydroxide = USD 14000/t [3]
price lithium carbonate = USD 10000/t [3]
price lithium (from hydroxide) = USD 14000/t (6.941 g/mol / (23.95 g/mol)) = USD 4060/t
price lithium (from carbonate) = USD 10000/t (2 6.941 g/mol / (73.89 g/mol)) = USD 1880/t
(N.B. just cost of raw materials, not including cost of lithium extraction from salt by electrolysis)
Gigafactory demanded lithium cost (from hydroxide) = (USD 4060/t) (6800 t/year) = USD 27 million/year
Gigafactory demanded lithium cost (from carbonate) = (USD 1880/t) (6800 t/year) = USD 13 million/year
Estimated world lithium reserves = 14 million t [4]
Gigafactory proportional use of lithium reserves = (6800 t/year) / (14 million t) = 0.05%/year
World production of lithium (2015) = 32500 t/year [4]
Gigafactory proportional use of lithium production = (6800 t/year) / (32500 t/year) = 21%
The FT article's estimate of 24000 t/year checks out for anhydrous lithium hydroxide. I do not know if 50000 t/year is an accurate number for worldwide lithium hydroxide production in 2015, but lithium hydroxide is not the only source for lithium and this would imply a worldwide supply of lithium of just 14500 t/year, which is contradicted by the figure of 32500 t/year from the USGS [4].
The estimated lithium demand of the Gigafactory (which matches the FT article's figures) are for the Gigafactory's total demand (both Powerwalls and cars) with the factory at full operating capacity in 2020. There is a number of years yet for lithium production to scale up.
Even taking the Gigafactory's projected demands in 2020 against lithium production in 2015, it requires 'only' 21% of the market. Tesla would be a very large customer, perhaps the single largest in the wourld, but Tesla wouldn't exactly be required to corner the entire market.
Supply and demand dynamics dictate that with increased demand, price will increase. The cost of the raw lithium salts would be on the order of USD 20 million/year for Tesla. Even with a price vastly inflated from here, it would be a small cost against Tesla's operating income.
A single Gigafactory operating at full capacity would take over 2000 years to exhaust the USGS's estimate of world lithium reserves of 14 million t.
[1]: http://electronics.stackexchange.com/a/34504
[2]: http://www.zdnet.com/article/panasonic-to-pour-billions-of-y...
[3]: http://www.abc.net.au/news/2016-01-13/lithium-graphite-the-n...
[4]: http://minerals.usgs.gov/minerals/pubs/commodity/lithium/mcs...