The price of bitcoin is determined by the market in which it trades. It is determined by how much someone is willing to pay for that bitcoin. The market sets the price of bitcoin as same as Gold, Oil, Sugar, Grains, etc. is determined. Bitcoin, like any other market, is subject to the rules of supply and demand. i.e.,
More Demand, Less Supply = Price Goes Up
More Supply, Less Demand = Price Goes Down
No one, in particular, sets the bitcoin’s price nor we can trade it in one place. Each market/exchange determines its price based on supply and demand.