If you own a company and would take a salary of 1 million you get taxed on that 1 million. If instead the company pays 70k that year for rent on your condo, 35k on the lease of your vehicle, and another 20k on food and you draw no salary, so your taxable personal income is 125k. You keep the other 875k in the company until you can find more preferable tax treatment for that money.

Obviously this is a terribly contrived example, but it is somewhat descriptive of the type of strategy used to reduce taxable income. Things are obviously more complicated when you have numerous types of income from domestic and foreign sources.

I thought we were talking under the assumption that the corporate tax rate was the same as the personal tax rate (ie a flat tax).

Keeping it in the company would then be irrelevant (the more you keep in the company by not paying yourself a salary, the higher the corporate taxes).

Not if you have foreign holdings and don't repatriate all of the profits.

I also don't think you could easily manage raising the effective corporate tax rate very much without seeing further offshoring of profits, which is the present situation. We presently live in a world where capital movement has been liberalized and the movement of people(Schengen notwithstanding) has not. So even if you have some sort of profit/income agnostic flat tax, any business that can seek lower tax rates abroad will.