This is a bit of a subtle point. In theory, free trade between countries A and B makes the pie larger (or, at least, no smaller depending on your precise model) for A and B compared to no trade. What is at issue is how the surplus pie is distributed. For instance, if you start off with each country having 10 slices, and trade creates an addition 5 slices, basic economic theory would allow for country A to end up with anywhere between 10 and 15 slices.

What protectionism does, from a theoretical standpoint, is provide some control over the distribution of the surplass.

Having said all of that, it should be remembered that we do not live in a spherical void. In practice, it is entirely possible for "free" trade to leave a country worse off then they would have been with no trade.

I tried to work through a proof of “free trade makes the pie no smaller,” and it seems like it only works if you’re considering one good.

I reasoned thusly:

1. Country A can produce good X at price p_a < price p_b the price at which domestic X is available in B.

2. No deadweight trade loss, so X is available at p_a in B

3. Thus for the same quantity n of X, B has a surplus of n(p_b - p_a).

4. But domestic producers of X have lost income, and if there isn’t something else for them to do, the total shrinkage in B’s economy may exceed the surplus (I haven’t worked out this term but it should be pretty easy), thus leaving B worse off.

Factor in deadweight losses and free trade definitely could leave B worse off, by forcing some of its economy to be unproductive.

> we do not live in a spherical void

This got me thinking how odd it is that economic theory only gets invoked in so far as it advocates free trade: https://en.wikipedia.org/wiki/Comparative_advantage#Criticis...

You can't really aggregate trade at the country level. Well, people do that all the time, but you can't expect it to follow the same rules as the real market.

People participating on the international trade are better off. There are plenty of cases of market failures (monopolies, etc), inertial problems (deflation, etc), or human capital destruction (due to worsened wealth distribution) predicted by theory already. Any of those can overwhelm the usual net positive contributions of trade.

>What protectionism does, from a theoretical standpoint, is provide some control over the distribution of the surplass.

Protectionism provides a mechanism to increase comparative advantage.

American economists drank a little too much of their own ricardian kool aid in the 90s and early 00s when they paternalistically tutted at China for enacting protectionist policies against her own good.

Now that China has stolen comparative advantage in electronics manufacturing, the CFO of Huawei is under house arrest and the tutting has all but dried up.