House Republicans have introduced bills for a second phase of tax cuts, after the Trump administration announced plans to pursue "tax reform 2.0" as a follow-up to last year's legislation.

On Monday, House Republicans introduced three bills — the Protecting Family and Small Business Tax Cuts Act of 2018, the Family Savings Act of 2018 and the American Innovation Act of 2018 — to further slash taxes and make other changes to the nation's tax code. The Trump administration was criticized when last year's legislation, known as the Tax Cuts and Jobs Act, made the cut in the corporate tax rate permanent but allowed for cuts for individuals to expire after 2025. Republicans' new legislation would make cuts for individuals and small businesses permanent.

The package of bills comes less than two months ahead of the November midterm elections as Republicans hope to keep control of the chamber.

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"Last year we said goodbye to America's old, broken tax code," House Ways and Means Committee Chairman Rep. Kevin Brady said in a statement introducing the bills. "Under our new system, we're seeing incredible job growth, bigger paychecks, and a tax code that works on behalf of families and American businesses. Now it's the time to ensure we never let our tax code become so outdated again. We look forward to bringing these bills to the committee soon."

The bills also aim to ease rules around retirement savings and start-up companies, among other changes.

CBS News correspondent Nancy Cordes recently asked Speaker of the House Paul Ryan how he can convince Americans they need yet another tax bill.

"I think take a look at take-home pay," Ryan responded. "When you fill your taxes out this year take a look at your standard deduction is doubling. Look at the fact your rates are going down. Oh, and by the way, the tax cuts are doubling. And, oh, by the way, since February and March you've seen withholding tables adjusted so people are seeing higher take home pay and again, most utility companies in the country have actually lowered the utility bills of hardworking Americans who pay less for electricity than they did before as a result of the tax cut. So the results are there."

But not even all Republicans are hopeful.

New Jersey Rep. Leonard Lance, a moderate Republican, told CNBC last week Republicans' attempt to pass further tax cuts is an "exercise in futility."

"If we were to pass that here in the House it would be an exercise in futility because it could never pass in the Senate," he said.