The fate of the world economy may depend on what happens to a company most Americans have never heard of

On a tiny island off the coast of China, one company manufactures a product used across the globe for countless household products as varied as PCs and washing machines.

And as that island — Taiwan — worries about the threat of a standoff between the US and China, the world's economy holds its breath. That's because there could be trillions of dollars' worth of economic activity tied to that one company: Taiwan Semiconductor Manufacturing Company, the world's biggest chipmaker.

Industry watchers say an escalating dispute between the US and China...