Once again, I caveat that I don't study International Political Economy, but have opinions and such anyway, partly because I voyaged through some of that stuff as I studied for my PhD comprehensive exams and partly because I taught a smidge of it in my intro to IR classes.
Why do tariffs suck? My brother asked about this, so I thought I would give it a shot here.
First, they are a tax. They artificially raise the price of a good or service (seems mostly on goods since we care more about protest manufacturing jobs) so as to make domestically produced goods more competitive. Anyone who opposes taxes should oppose tariffs, but, well GOP hypocrisy is nothing new. The question with any tax, from my point of view, is whether a tax is progressive or regressive--who bears more of the burden: the relatively well off or the relatively less well off. For tariffs, it depends, I guess, on what the tariff is on and how the costs get passed on (more on that in a second). Anyhow, first, tariffs are taxes.
Second, as a tax, they make stuff more expensive without changing the value--so they are inflationary. We have gotten so used to low inflation that we have forgotten that inflation is bad for people with fixed incomes and people with low incomes. Given that we have not had much wage growth for what? Thirty years? Not good.
Third, more importantly, tariffs make stuff more expensive, so people have less money for other stuff. Here's the fun thing: it allows domestic producers to raise their prices to just below the level of the tariff, so they get to capture some profits. Woot? Depends on where those profits go. See aforementioned wage stagnation reference. The classic stat I remember from long ago was that various protections on steel jobs in the 1980s cost something like $200,000/job saved. Guess what? Steel workers were not getting $200k....
Fourth, as Milner reminds us, much of what we buy includes foreign parts, so raising the prices of inputs via tariffs increases the costs of other stuff, making them less competitive on world markets, such as .... cars made in the US these days. Seems like Trump wants to undo every Obama legacy, including saving the auto industry. Think about it: a Honda made somewhere else with European or Asian steel will cost less than a Honda made in the US with more expensive steel (either American or foreign with tariffs added). So, tariffs may help one industry but harm others.
Fifth, reciprocity is a core reality in International Relations for good or ill--countries respond to what one does. So, other countries put tariffs on American goods, making them less competitive in European and Asian markets. How are soybean farmers doing these days? Oh wait, the US government can pay off some of these folks. Sure, who gets these payments? Everyone who is harmed or those who are politically relevant to those in power? Markets may not be perfect (indeed, they need to be regulated to soften shocks), but political payoffs to the losers of tariffs is kind of a dumb way to deal with international trade. Instead, maybe invest money to train people in industries that are harmed by international trade, and, more importantly these days, automation?
Six, comparative advantage is a thing. Yes, some countries will out-compete the US because they have cheaper labor, weaker regulations, natural endowments, etc. We can't expect the US to have tariffs on those areas where the US is at a disadvantage and other countries not do the same. The comparative advantage logic, if allowed to play out, allows us all to get more stuff for lower prices and higher quality than if we all just relied on our own domestic producers. Yeah, it causes economic pain to some sectors. The reality is this: that pain is concentrated in some sectors while the rest of society benefits. Which is why it is natural politically to have high tariff (and non-tariff barriers): politicians pay more attention to the small groups with much pain than the larger groups where the pain is spread out. In a democracy, um, that ain't good. So, again, the question is how to deal with the pain experienced by the small groups. Tariffs are a crappy way to deal with such pain.
I have forgotten other dynamics and issues. It is the case that international trade can lead to races to the bottom--deregulation, low wages--but it probably also has something to do with the fact that there is less absolute poverty in the world now than ever before. Oh, and folks getting more income to work can ultimately lead to more pressure to improve labor regs and environmental policies. For instance, the legitimacy of the Chinese government now hinges more than ever on whether people can breathe.
There are tradeoffs involved, there is no perfect solution, but here's iron law of policy in the 21st century: if Trump is doing it, it is almost certainly dumb and self-destructive.