In the House, 45 GOP legislators voted in favor of the increase, while 40 voted to uphold the governor’s veto.

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The state is facing a $350 million budget shortfall. Brownback’s critics say the state’s persistent deficits are evidence that the economic benefits from reduced taxes are not always adequate to make up for reductions in revenue, as advocates of supply-side changes have sometimes claimed.

“I'm disappointed in the actions of our Senate today,” said state Rep. Melissa Rooker, a Republican from Fairway, Kansas, who supported the bill. “This was a balanced compromise that provided the revenue necessary to fund the basic needs of our budget and restore some semblance of solvency and sustainability.”

For both Brownback and his critics, the changes are a model for the policies that Republicans in Washington, D.C., might pursue on a national level now that they are in control of the federal government. One of President Trump’s advisers on economic policy during the campaign, Stephen Moore, also helped Brownback develop the changes he enacted beginning in 2012. Rep. Paul D. Ryan (R-Wis.), the speaker of the House, served as Brownback’s legislative director when Brownback was in Congress.

Ryan’s and Trump’s proposals for tax reform have important features in common with Brownback’s policies. Both reduce the number of income-tax brackets. Brownback’s policies and Ryan’s proposal treat income from legal entities typically used by small businesses more favorably than ordinary income. Likewise, the plan Trump advanced as a candidate appeared to reduce the tax rate on such earnings, known as “pass-through income,” but his proposal was ambiguous on this point.

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In the case of Brownback’s overhaul, pass-through income has been completely exempt from taxation. In 2012, the state had projected that about 200,000 pass-through entities would take advantage of the exemption. In fact, about 330,000 ostensible small businesses profited from the rule. That data suggests the reform encouraged tens of thousands of Kansans to claim their wages and salaries as income from a business rather than from employment.

That avoidance has contributed to repeated budget deficits, forcing state policymakers to take emergency measures, exhausting the state’s reserves and diverting money dedicated to maintaining highways to keep the state’s government operating.

The bill in Kansas would have eliminated the exemption for pass-through income and increased income taxes (although not to the rates that prevailed before Brownback took office), while eliminating tax reductions planned for the future. The state projected that the legislation would have increased revenue by $590 million in 2018.

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Moore, the former adviser to Brownback and Trump, acknowledged that avoidance by residents claiming business income was an issue, one that he said the Trump campaign also debated.

“That has been a problem, I agree,” he said. “One of the things we’re really struggling with is how to avoid that problem.”

Experts on taxation said that Kansas’s experience with the exemption for pass-through income should be a source of caution for GOP lawmakers in Washington considering a similar approach.

“It’s very expensive,” said Scott Drenkard, the director of state projects at the conservative-leaning Tax Foundation. “What we’ve seen in Kansas as a result of this is that the state has had a hard time making budget.”

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Yet Moore and other proponents of the policies say that reduced taxes, especially for small businesses, will help encourage economic growth. Kansas’s economic performance has been only middling over the past several years, but Moore argued that the state’s problems are a result of tepid growth nationally.

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“It’s certainly not a very Republican idea to be raising income taxes,” he said.

Identifying the precise effect of the tax overhaul on the economy is difficult. Overall, Kansas’s economy expanded by about 2.9 percent between 2011, when Brownback took office, and 2015, the latest year for which data are available. Over the same period, the gross domestic product increased 9.2 percent nationally, according to the federal Bureau of Labor Statistics.

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Rooker noted that the coalition voting to override Brownback’s veto in her GOP-dominated chamber included more Republicans than Democrats. Meanwhile, Republicans in Congress are also divided on how to approach the problem of tax reform, confronting similar disagreements on how drastically to reduce federal revenue.

Sen. Mitch McConnell (R-Ky.), the U.S. Senate majority leader, said in December that he would prefer tax reform that did not reduce federal revenue. Ryan's proposal would cost the government by about $2.5 trillion over a decade, according to an analysis from the nonpartisan Tax Policy Center.