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Change always comes from those that others brand and try to dismiss as "extreme," in this case all MMT did was explain a few things that conventional economic principles have been wrong about.
Best example we have from recent history is the years after the financial collapse, monetary and fiscal policy responses to our economic condition saw FoxNews run nonstop "inflation is coming" commentary with full time sellers of Gold filling up the advertising space. Not only did inflation not happen the way they claimed it would, too many people bought overpriced gold that will take generations to recoup those investments.
MMT started to explain things in terms of realized impacts in our fiat money system, that those who are stuck in the gold standard days could not explain.
So then the false information and misleading polls started to come out to do what conservative economics always does, invoke fear through "socialist" and "Venezuela" talk. All devoid of what MMT really says or why.
Contemporary economic theory doesn't fail to account for massive amounts of QE not materializing as inflation. I doubt Bernanke enacted QE because he paid attention to a small subset of unusual economists who claim they revolutionize monetary economics. The man worked with structural VAR, factor models, and DSGE models in his research and is an accomplished academic first and foremost. He sure didn't expect inflation would just rise on a massive scale, yet his framework was as close to contemporary mainstream theory as you could get it. Some of his work on financial frictions in DSGE models (1999) and his joint work with Boivin and Eliasz (2005) on the use of factor models in structural vector autoregressions still stimulate research to this day. His views are clearly similar to those of the bulk of the profession and he wasn't panicking about inflation. On the other hand, Peter Schiff was panicking on Fox News in front of an audience which wants to hear about government causing problems. Bernanke has a Ph.D. and conducts research, while Schiff only has a bachelor degree in economics and does not conduct research. It's not a stab taken at Schiff, as much as a statement about whose opinion more closely captures what up-to-date research in economics implies. This is further buttressed by the fact Krugman, Delong, and Summers didn't panic either about inflation. They all predicted beforehand that it would never rise above 3% during the crisis, some even suggesting it might even become negative during some months of the crisis depending on the measure you use for inflation. They're all prominent macroeconomists who hold mainstream views on economic theory.
Nowadays, what I would do if I had to accommodate QE without going into the complex and explicit modeling of its dynamics is to follow the advice of Wu and Zhang (2016) who proposed a nice way to summarize all monetary policy in a "shadow" interest rate that can become negative. This way, you can use a much more typical way to model monetary policy in an already very complicated model such as the one in Ascari, Phaneuf and Sims (2019) (pricing and wage setting Calvo-style frictions, financial borrowing by firms, firms produce intermediate goods in a network, firms face variable utilization rates of capital, adjustment costs to new investments, there is a whole continuum of goods and labor types, etc.). That model wouldn't imply massive inflation response under the ZLB to large QE, nor would it imply monetary policy is 100% impotent in such conditions... So, clearly, explaining these things is absolutely not beyond the scope of mainstream methods.
Moreover, it might be hard for you or anyone here to appreciate how good are our methods in economics since you likely never read a single paper on the subject. A good example is Ascari, Sims and Phaneuf (2019). I'll bring your attention to the quality of the match between theory and data. Their simulated economy matches evidence from the structural vector autoregression literature, as well as produce incredibly good match for contemporary and cross-correlations of variables like consumption, investment, output, real wages, etc. on top of correct volatility values for all of their cyclical component. Here is the paper in question:
APS March 2019.pdf
Page 36 onwards show statistics about how well the simulated economy (from the model) matches empirical data, as well as showing impulse response functions. Clear quantitative predictions in a transparent and flexible framework... That's why DSGE are so popular, period.