How can he make Mexico pay? Are they legally bound to if the U.S passes a resolution saying so?
What Mr. Trump has to say on the matter: https://www.donaldjtrump.com/policies/immigration
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Donald Trump has posted this document on his website, which outlines how exactly the wall will be financed.
According to that document, Mexicans nationals (many working illegally) in America transfer $24 billion back to Mexico each year. Mexico does not provide a social safety net for its citizens, so this money is effectively supplanting any money the Mexican government would spend on welfare.
Also according to that document, the Patriot Act requires financial institutions to ask for proof of identity before opening accounts or initiating transactions. The executive branch is left to the details of that.
The wall proposal involves two immediate actions:
This proposal would mean that illegal aliens cannot wire transfer funds back to Mexico.
At this point, his policy requires negotiating with Mexico. Since Mexico effectively uses that $24 billion in earnings from Mexican nationals in America in place of providing social welfare programs, it will protest.
The deal? If Mexico agrees to pay for the wall, Donald Trump's administration won't implement the proposed change to the Patriot Act which would block the majority of the $24 billion in transfers back to Mexico.
And what if Mexico still refuses? Beyond blocking those transfers, he also proposes these as leverage in the negotiation. Some of them could also pay for the wall:
Okay, now to answer your questions.
How can he make Mexico pay?
See the above. Mexico will be incentivized to pay for the wall by the possibility of wire transfers from Mexican nationals working illegally in America back to Mexico. Other tools may be employed if this does not work.
Are they legally bound to if the U.S passes a resolution saying so?
No. The basis of our international order is sovereignty, which means the laws of one nation only apply to that nation. However, the proposal isn't that the US will pass a law requiring Mexico to pay for the wall. The proposal is that the United States will negotiate with Mexico; since the US has a lot of leverage on Mexico, they will almost have to agree.
I've written thousands of posts about the general topic since 2002, and Trump's plans are beyond unrealistic. I've detailed how his plan to pay for the Wall would fail here. It's certainly physically possible to build a giant wall, but politically it would be nearly impossible.
His attempts to build or pay for the wall would put him up against the richest people in the U.S. and the wealthy elites of Mexico. Those in the U.S. who'd oppose him include major banks and the Federal Reserve (who take or want to take a cut of remittances) and major businesses that rely in illegal labor either directly or indirectly. They'd fund politicians that opposed the Wall and withdraw donations from those who supported it. Almost all Democrats in Congress wouldn't need funding to oppose the Wall. It's highly likely that enough Democrats and Republicans in Congress would oppose it that it would be blocked in Congress.
However, even if it weren't, there are hundreds of left-leaning NGOs that would immediately file lawsuits to tie up the Wall in court. Some of those groups would get funding from the business interests mentioned above. They'd do to the Wall what they did to Arizona's SB1070.