Senator Tom Cotton’s (R-Ark.) recent NY Times op-ed on immigration—“Fix Immigration. It’s What Voters Want.”—gets a few things right, but the ultimate analysis is off the mark. Cotton’s thesis is that immigrants have flooded the labor market to such an extent that immigration is largely to blame for decades of wage stagnation. Immigration does sometimes have negative impacts on American workers—and we need to be clear about who the economic winners and losers are—but contrary to Cotton’s claims, there is scant evidence that immigration overall has kept wages low. Furthermore, Cotton ignores the real reasons wages have failed to rise for so many American workers.

But first, credit where credit is due.

Sen. Cotton deserves credit for calling out businesses that warn of looming labor shortages in low-skilled jobs despite any observable evidence that this is imminent (while research shows the opposite has been true during at least the past decade). Cotton claims, however, that these businesses mainly support immigration so that they can add additional workers to the labor market in order to lower wages. There’s a kernel of truth in that, but in reality, employers care more about hiring workers without power or a voice in the workplace; that’s what puts downward pressure on wages in low-skilled jobs.

Most Americans do want immigration fixed. They reject a system that leaves families terrified of separation because they fear deportation of undocumented moms, dads, brothers, and sisters, even if they’ve resided in the United States for decades and have jobs and (otherwise) clean criminal records. Cotton doesn’t mention any of this. He instead laments a “generation-long influx of low-skilled immigrants that undermines American workers.”

The real problem isn’t immigration, it’s a legal framework that leaves all low-wage workers and millions of migrant workers—both authorized and unauthorized—vulnerable to wage theft and exploitation. We have allowed employers to race to the bottom toward lower and lower labor standards.

Cotton is right to highlight the plight of American workers and to cite data from the Economic Policy Institute showing that since the late 1970s the least-educated workers in the United States have seen their wages decline. That’s a national emergency and should be treated as such by all members of Congress. Cotton also has a valid point when he alludes to the fact that the low-wage workers most likely to be hurt by new low-skilled immigration are immigrants who arrived earlier. But Cotton blames immigration and “automation and globalization” for wage stagnation without offering any solutions for raising wages other than shrinking the labor market.

Tight labor markets do tend to raise wages, but the decline in union representation, caused by weak labor laws and extremely hostile businesses—and the corresponding reduction in power workers have in the workplace—has been a far more important factor in keeping wages low. The declining real value of the federal minimum wage (worth 25 percent less than in 1968) is the other key factor.

So where does Sen. Cotton stand on collective bargaining? Does he support a $15 an hour national minimum wage? He doesn’t say. Even raising the minimum wage to $12 an hour would eliminate the largest estimated negative effects of immigrant participation in the low-wage labor market on the wages of U.S.-born workers.

But let’s wrestle with the impact of immigration on low-wage American workers. It is undeniable that certain aspects of the immigration system are undermining wages and working conditions, but we have to be honest and specific about what those are, and address them appropriately. When it comes to wages and labor standards, there are three main problems with the system and each has a corresponding policy solution:

(1) Since 1986, demand from American employers, coupled with the lack of an effective system to check whether workers are authorized to be employed in the United States, has resulted in an unauthorized immigrant population of about 11 million. The vast majority of unauthorized immigrants are employed, but because they lack work authorization and fear employer retaliation and deportation, they cannot effectively complain when they are paid below the minimum wage or when their employer subjects them to unsafe conditions at the workplace. That fear keeps unauthorized workers docile and diminishes the bargaining power of Americans who work alongside them. That’s partly why so many employers like to hire unauthorized workers. Unsurprisingly, data show that unauthorized immigrant workers suffer from wage and hour violations at higher rates than American workers.

Fixing the problem once and for all is simple but requires congressional action. Congress should pass a broad legalization for the current unauthorized immigrant population that simultaneously mandates the nationwide implementation of an electronic employer worksite verification system to deter future unauthorized migration—along the lines of the current E-Verify system, but with important amendments to remove its control from the hands of employers and to increase accuracy and due process. A system that uses a personal identification number (PIN) instead of a Social Security Number—like the one proposed by the Migration Policy Institute and discussed in former Secretary of Labor Ray Marshall’s book Immigration for Shared Prosperity—is one example of a system that could alleviate many concerns of worker and civil rights advocates. Employers should support this because it will make life easier for them.

(2) The United States issues nonimmigrant visas that authorize employment to hundreds of thousands of workers from abroad every year in an alphabet soup of temporary foreign worker programs. The workers in these programs are among the most exploited laborers in the U.S. workforce because the employment relationship created by the visa programs leaves workers powerless to defend and uphold their rights due to fear of retaliation and deportation. Temporary foreign workers are usually tied to one employer and can’t change jobs if their boss is abusive or breaks the law, and the exorbitant fees charged to them by labor recruiters for employment opportunities in the United States leave workers indebted and indentured to both employers and recruiters.

That’s obviously bad for the foreign workers, but Americans suffer, too. There is abundant evidence that the laws and regulations governing nonimmigrant visa programs—such as H-2B and H-1B—permit employers to pay their temporary foreign workers much less than the local average wage for the jobs temporary foreign workers fill. Most visa programs like the J-1 and H-1B don’t even require that employers first recruit and offer jobs to unemployed U.S. workers before hiring a temporary visa holder. Most visa programs also have no minimum or prevailing wage rules at all and little or no enforcement. That’s why employers think they can get away with paying less than $2 an hour to skilled foreign workers employed in computer occupations, as one company in California attempted to do with their Indian workers on L-1 visas. We also know that less-skilled “legal” temporary foreign workers earn the same low wages as unauthorized immigrant workers in similar jobs; that means temporary foreign workers don’t have an incentive to work legally through visa programs—in fact, many authorized workers end up even worse off than unauthorized workers because of the debts they incur.

The solution to this is simple. Congress could get rid of temporary foreign worker programs altogether. Or it could update, simplify, and standardize the rules for all of them: first, requiring employers to recruit and offer jobs to any qualified U.S. workers, and second, in cases where employers can’t find qualified U.S. workers, ensuring that any temporary foreign workers hired are paid no less than the local average wage for the job. Congress should also appropriate more funding to the Labor Department to enforce the new rules and audit employers, and pass a law permanently banning any employer from hiring temporary foreign workers if it violates wage and hour or immigration laws.

These reforms are commonsense and obviously pro-immigrant, and they would raise wages for migrant and U.S. workers alike, but lobbying from business associations like the U.S. Chamber of Commerce and corporate-funded groups like Facebook founder Mark Zuckerberg’s FWD.us—groups that publicly sing the praises of immigrants and highlight their contributions to the American economy—have kept such reforms from being written into law because they would negatively impact the bottom lines of their members. During the negotiation of the comprehensive immigration reform bill that passed the Senate in 2013, the U.S. Chamber of Commerce’s price to support the bill was 200,000 more low-wage temporary foreign workers per year. And Facebook was part of a successful lobbying effort by the tech industry to keep wages low for foreign tech workers and block a provision that would have provided U.S. tech workers with a chance to apply for jobs before companies could hire from abroad.

(3) The third aspect of the U.S. immigration system that harms workers is its inflexibility. Specifically, there is no timely or effective way to adjust immigration levels to changing economic conditions in the United States. A study by economist Giovanni Peri shows that competition with lesser-skilled immigrants can be a problem for low-wage-earning native-born workers during a recession or a time of high unemployment. The effect generally equalizes over the long term, but to protect against harm in the short term, immigration levels should be lowered during economic downturns. Once the economy picks back up, levels could increase correspondingly.

The best way to do this would be to establish an independent commission on immigration and the labor market, staffed with technical experts who could advise Congress and recommend targets for immigration levels annually or semi-annually. There are models for such an expert advisory body—most notably the Migration Advisory Committee in the United Kingdom—and many experts and research institutes have called for one to be established. A commission could also work to improve data and transparency, recommend to Congress the best skill mix that new flows of immigrants should have, and create shortage occupation lists based on empirical evidence about the labor market, as many developed, net-immigration countries do. Employers could then have easier access to foreign workers in verifiable shortage occupations.

Unfortunately, Sen. Cotton is not calling for the immigration reforms America needs and he doesn’t have a track record of championing policies that would raise wages for American workers. If Sen. Cotton is serious about raising America’s pay, he should focus on policies that shift bargaining power from employers to workers instead of blaming stagnant wage growth on immigration. Deportations won’t raise wages—but legalizing unauthorized immigrants and fixing our immigration system, increasing the minimum wage, making new investments in infrastructure, reducing the trade deficit, and expanding collective bargaining rights and making it easier for workers to join and form unions—these things certainly will.