Sen.Ted Cruz (R-Tex.) and Carly Fiorina participate in the Republican presidential debate at the University of Colorado last month. (AP /Mark J. Terrill)

Many people would like to pay less in taxes, but every dollar that the government collects in taxes is a dollar it doesn't have to borrow. There's no way around that arithmetic unless the government spends less money at the same time, in which case it can borrow less, too.

Maria Bartiromo, a moderator in Tuesday night's primary debate on Fox Business Network, tried to draw out Sen. Ted Cruz (R-Tex.) on this math. Under his proposal to cut taxes, the federal government would give up about $3.6 trillion over the next decade, without accounting for effects on economic growth.

Cruz replied that the government wouldn't have to borrow that money, and that instead, he has a plan to reduce federal spending. That plan, he said, has "$500 billion in specific cuts." He also said that he would eliminate five federal agencies.

Cruz didn't list all five, though, instead naming the Commerce Department twice. And he his plan for how he'd achieve figure he cited -- $500 billion in savings -- is short on specifics.

"Five major agencies that I would eliminate: the IRS, the Department of Commerce, the Department of Energy, the Department of Commerce and HUD," Cruz said.

A document on Cruz's Web site lists five major agencies (the fifth is actually the Department of Education) along with 25 smaller programs that he'd abolish. Yet the document does notĀ itemize specifically how much the government would save on those programs under a President Cruz. The document lists only the top-line figure of $500 billion, without attributing that amount to particular reductions in spending.

The Congressional Budget Office has studied what it would mean to eliminate the departments of Commerce, Energy and Education. The budget for Education was about $55 billion in 2012 -- most of it in grants to local schools and in tuition aid for college students. Eliminating that department alone would save well over $500 billion over a 10-year period.

Yet under Cruz's plan, funding currently dedicated to education would instead be transferred to state officials to use as they see fit, so it isn't clear how much money this measure would save, if any.

"Eliminating a department while transferring its programs in essentially unchanged form to other departments or agencies would probably result in little or no budgetary savings, because most of the costs incurred by departments are the costs of the programs themselves," the analysts at the budget office noted.

Cruz also proposed eliminating the Internal Revenue Service. It is hard to know whether doing so would save any money. Without any agents on the beat, tax evasion could run rampant, with substantial costs for the government in uncollected taxes. The agency estimates that for every additional dollar it receives in funding, it collects an additional $7 in taxes owed by those who aren't paying their fair share.

Cruz might not have identified what he called "$500 billion in specific cuts" to the federal budget. He might well be able to achieve a reduction of $500 billion in spending over a decade, but it's worth noting that even under the rosiest assumptions, his tax plan would reduce federal revenues by more than that amount.

The independent Tax Foundation puts the cost of Cruz's plan at $768 billion over the next decade. Even if Cruz's plan would reduce spending by $500 billion over the same period, the federal debt would still increase.

The Tax Foundation analyzes tax proposals using a methodology that even many conservative economists say is too favorable to politicians who want to reduce taxes. If the group's assumptions are overly optimistic, Cruz's plan could cost even more.