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Non-constant parameters? BURN HIM AT THE STAKE!!!

I'm also not a trade economist, but it was one of my fields, so I might be able to shed a little light.
The balanced trade prediction is actually quite interesting from a history of economic thought standpoint. The conventional treatment of trade followed the kind of comparative advantage framework that I'm sure you've shown your classes a thousand times. The problem is that actual trade flows don't resemble the "naive" comparative advantage story. Trade is remarkably balanced, even as you dive down into particular sectors and industry. Countries appear to just be swapping the same kinds of stuff back and forth. What's up with that?
I think it was actually an engineer, or some other technical non-economist, who developed the Gravity Model of trade, based on his observations of empirical data. The empirical regularity of balanced bilateral trade is one of the strongest empirical regularities in economics (as you know, we don't generally do that well with quantitative predictions) and there was no theoretical basis for it for a long time after it had been generally accepted.
Eventually, models with plausible theoretical rationales were developed. Krugman's Nobel Prize was for developing the first such model, based on monopsonistic competition. That's not the accepted rationale anymore, but it did get the right answer.
Anyway, this was one of the few times when economics proceeded like a normal science, with observation of an empirical regularity coming long before a theoretically sound explanation.
As to why America's bilateral flows don't balance, everything you said is fine.
Oh, interesting, I didn't know that historical background. Thanks!
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I actually think what they’re doing isn’t a dumb place to begin the analysis, but they need to consider the role of American policies in creating those deficits.
I don’t really see how they can maintain the reserve currency without running trade deficits.
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Agreed. And the equation itself is actually correct, assuming you choose the right parameters, and only for small changes (partial equilibrium.)
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I’m not actually doing any math, but I think the formula fits with the Gravity Model, which has trade flows varying inversely with “distance”.
Distance is something like the combined obstacles to trade (no one really knows because it’s usually just estimated as a residual and they don’t give it much thought). So, you’d expect the same proportionality between directional trade flows and directional trade barriers.
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'I don’t really see how they can maintain the reserve currency without running trade deficits.'
It is rather easy. USD as the primary reserve currency means there is demand for USD for use in trade payments and as a SoV, often held by preference as USTs. While nations selling goods to the US gets them USDs that is not the only way they can access USD. US can issue USD in exchange for investments in other countries and as loans. It has always done this, but the demand for USTs was so strong it was easier to issue them than to make investments. The whole petrodollar paradigm was built upon giving the Saudis preferential access to USTs and then others wanted access as well. It became all too easy, rather than the trouble of managing investments you just issue debt obligations. Other nations and entities can access USD as a SoV in the form of USD denominated investments...but again, they preferred USTs and the US issued them. It was not essential, but a choice. A choice that has led to the current situation where USTs and other US government debt now impose a burden of debt servicing greater than the cost of maintaining the military. Reckless. The truth is that the demand for USTs has enabled the US government to easily fund seemingly endless fiscal deficits and this lack of fiscal discipline is now delivering the inconvenient reality of imminent insolvency. There was no functional need for the US to become hugely in debt in order to operate the primary global reserve currency- but rather it has turned out that way because there has been a chronic lack of discipline and an erroneous belief that the accumulation of debt could continue without any serious consequence.
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