The debate over reducing the federal deficit is focusing primarily on spending cuts, but also on taxes. Most serious, and honest, appraisals of the massive budget deficit suggest both will be needed. But before taxes are increased, the government should be doing a better job of collecting taxes.

It’s been a week since federal income taxes were due, and it’s worth remembering that the so-called tax gap, or the difference between what is owed and what is collected by the Internal Revenue Service, is estimated to be about $350 billion a year.

Not only is that a significant amount of money, but the idea that so many taxpayers avoid paying their fair share threatens the credibility of the tax-collection system. The $350 billion tax gap suggests the system is unfair — that honest taxpayers are unfairly burdened. Higher compliance and tougher penalties for tax cheats are necessary.

Yet, as part of the budget agreement reached earlier this month for the rest of the fiscal year, Congress decided that the IRS would get no additional funds to hire more agents. This IRS budget freeze was imposed despite the fact that studies have shown that each dollar spent on increased IRS enforcement produces about $10 of additional revenue.

A significant amount of the tax gap comes from high-income people who use high-priced accountants and lawyers to avoid paying their fair share of taxes. Some of that activity is legal, some is not.

Closing the tax gap would do much more to help close this year’s budget gap than the paltry $38 billion in spending cuts approved in the recent budget deal that threatened to shut down the federal government.

But even without additional funds for more auditors and investigators, the IRS is directing more attention to millionaires, increasing audits of the super-rich. That’s a smart move, demonstrating that the IRS expects everyone to pay their fair share of taxes. Too often, the super-wealthy seem to operate by a different set of rules — with the apparent approval of officials in Washington, D.C.

More audits of the wealthiest taxpayers will improve the perception of fairness, especially if audits produce higher tax revenues and penalties.

According to the IRS, taxpayers making between $5 million and $10 million a year now have about a one-in-10 chance of being audited. This is a 50 percent increase in audit rates for the top income levels.

Increased IRS auditing, at all income levels, produced about $57 billion in extra tax revenue in 2010, up about 18 percent from the year before. And more impressive returns might still be coming, since audits of high-income taxpayers are often so complex that final resolution, in terms of additional tax collections, might take years.

In another promising move to close the tax gap, in 2009, federal officials reached an agreement with the Swiss government to release information on about 4,500 wealthy Americans with Swiss bank accounts — and suspected of using those accounts to avoid U.S. taxes.

Overall, the IRS audits only about 1.1 percent of all returns. But the latest effort to put more focus on the top income earners is logical — and should produce more bang for the buck.

If the budget deficit results in taxes being increased on the wealthy — or on nearly everyone — it’s important to first ensure that compliance is as high as possible and, therefore, the tax gap is as low as possible. While increased taxes might be necessary, along with major spending cuts, to help trim the federal budget deficit, it’s not fair to increase taxes on honest taxpayers while allowing dishonest taxpayers — at any income level — to avoid paying their fair share.

In the coming months, there will be vigorous debates about tax rates, tax loopholes and simplifying the tax code. That debate is healthy, but there can be no debate over the idea that all taxpayers — especially the wealthiest — should pay their fair share.