This post has been updated.
The Texas Republican would abolish the estate tax, tax on health-care plans under the Affordable Care Act - a law Cruz wants to scrap - and the alternative minimum tax. The payroll and corporate income tax would be replaced by a 16 percent business flat tax.
"Giant corporations will lose their loopholes and instead pay the exact same Business Flat Tax as small businesses. And billionaire hedge-fund managers will no longer pay a lower rate than working men and women," he wrote.
Cruz would create universal savings accounts that allow people to save up to $25,000 and defer taxes on that money. People would be allowed to tap into that money at any time.
“The virtue of a single tax rate is that the rate doesn’t rise as people work more and invest more. This means better incentives to increase output, and fewer distortions,” Cruz wrote.
In the debate Wednesday, Cruz said the 10 percent rate is the "lowest personal rate" that has been proposed. Sen. Rand Paul put forward a flat tax plan with a 14.5 percent rate.
Cruz has said that he wants to abolish the Internal Revenue Service and simplify the tax code. The plan, he said, would spur economic growth and "shrink Washington by getting rid of the rat’s nest of complex tax requirements, credits and loopholes."
"The plan should: spur robust economic growth and job creation, while raising after-tax income for all Americans; be dramatically simpler, to allow working people to file their taxes with a postcard or phone app; and shrink Washington by getting rid of the rat’s nest of complex tax requirements, credits and loopholes," he wrote.
The Texas Republican would eliminate corporate income tax and replace it with a flat business tax of 16 percent.
He would also do away with the estate tax, the alternative minimum tax, and tax on profits earned abroad. Cruz said it would end the payroll tax, but social security and medicare will remain fully funded.
Cruz said his tax plan was conceived with help from Arthur Laffer, President Ronald Reagan's tax adviser.
This post has been corrected to reflect the debate was Wednesday, not Thursday.