Here's one broad way Mankiw and company are going wrong. I start with three preliminary points and then point to the going wrong:
(1) They have physics envy. In particular, they want to sit in a small, dark, closed room and with just pure thought, with little to no contact with any real economy, come up with some economics version of Newton's second law F = ma or Einstein's result in special relativity E = mc^2. Nonsense. Incompetent, intellectually bankrupt nonsense.
(2) With their physics envy, they want mathematical economics and to build mathematical models. E.g., for some decades after Dantzig's simplex algorithm, they fell in love with optimization. As optimization moved forward with nonlinear programming, the Kuhn-Tucker conditions, nonlinear duality, deterministic optimal control and the Pontryagin maximum principle, and stochastic optimal control, the mathematical economists continued to use such applied math to build their models. A shockingly large fraction of the Nobel prizes in economics are from just such nonsense. The math is rock solid; there are occasional good applications to particular, small problems, e.g., the climb, cruise, and descent of an airplane or what mix of refined products should come today from an oil refinery; an application to a real economy is nearly always just total BS.
(3) To keep their models relatively simple, they make a lot of assumptions. E.g., in finance, they assume perfect information and f'get about the guy on a motorcycle at an airport who asked the guy on the ramp where the plane was going (as in the movie Wall Street), naked shorting, etc. E.g., they make a lot of Brownian motion assumptions that then say that the LTCM disaster was wildly improbable, which it was not. They make assumptions about which workers are more productive and feed that into their optimization looking for some case of a non-inferior solution or Pareto optimality as if Chinese women had 20 fingers on each hand so could sew buttons much faster than women in South Carolina.
Going Wrong. They take their assumptions of their simplistic models and the corresponding conclusions and say that a real economy SHOULD be like that. So, they want to bend the real economy to their Procrustean bed of simplistic assumptions and conclusions. If physical science had done such a thing, then they would still be saying that the planets should move as Ptolemy said, that falling bodies should still move like Aristotle said, and that we should still be looking for flogiston.
This economic science is intellectually both incompetent and dishonest; it's contemptible, and dangerous.
Once I went through Samuelson's college text: I found NOTHING that made any sense at all except for his chapter on the Federal Reserve; in that chapter, he just described what the laws had established and was clear. All the rest of the book was total BS disconnected from any real economy. E.g., if we buy more transistors, then the price of each transistor has to go up. Right: Transistors used to cost several dollars each, and now, after buying many billions, maybe trillions, of transistors, we can buy a few hundred million for less than $100, retail. Wheat: For years 1800, 1850, 1900, 1950, and 2000, take the quantity in bushels of wheat produced in the US and the price per bushel of wheat, corrected for inflation, see that the quantity has gone way up and the price, way down. Similarly for iron, steel, chickens, and pork. We're talking total suckage. The very first things in academic econ, the supply and demand curves, just do not work in any meaningful way in a real economy; real economies mostly just don't work that way, guys. Total BS.
E.g., the econonuts argue for free trade based on simplistic nonsense. Their idea is that, for some global benefit or some Pareto optimality for the world, it is better if the production is where it is most efficient. So, yes, grow teak wood in Thailand, grow rubber in Viet Nam, mine tin in Indonesia, and pump oil in Saudi Arabia. Fine. Then, sew buttons in China? Sure, if the Chinese women had 20 fingers on each hand. But they don't. And a sewing factory in China has to struggle with bad situations for each of suppliers, legal system, transportation system, communications infrastructure, information technology infrastructure, etc. Still, the econonuts want to conclude that China is more efficient at sewing buttons. No they aren't: Instead, the Chinese government understands what the econ profs don't: China takes their young women and makes each of them "an offer they can't refuse", work for pennies a hour under whatever conditions, or else, period. It's not a matter of being efficient. So, businesses, careers, lives, and communities in South Carolina are ruined. So, we pay the former textile workers to do nothing or just let them die. I know: A few of the textile workers get to serve BBQ. From such destructive nonsense, instead of textile workers, I have a better suggestion for who deserves to die. They are doing it, and we should say so: The academic econ profs are KILLING Americans.
Then there's US competitiveness: In the 1950s the US had a great economy. Except for some points of information, biomedical, and materials technology, it's not clear that our standard of living is as high now. Our imports were meager, maybe some tin from Indonesia, etc. We didn't buy much from outside because the other industrialized economies were devastated and, thus, had little to sell us. But we did sell some products: Telephone systems, construction machinery, airplanes, etc. Then the idea is that since we could sell at what were astoundingly high prices, especially for the buyers, that is why our economy was doing well: NONSENSE. Total 100% nonsense: So, Joe went to work at, say, Caterpillar, and made great machines which we shipped to, say, France. France paid us in silly paper which we converted to gold. So, we accumulated a lot of gold. What good did this do Joe or the US? Next to none: In particular, Joe's labor got consumed in France instead of the US. Econonuts are confused. It is easy to see why, say, Japan, Saudi Arabia, or Jamaica needs foreign trade. But the US was doing just fine, thank you, as essentially a self-sufficient economy in the 1950s and, with foreign trade, is doing worse now. We are shipping our going businesses, market position, education, technology secret sauce, and intellectual property overseas, and the econonuts conclude that this is good. Total BS.
Now, sure, we want to import some oil: Saudi Arabia has oil and needs national defense, wheat, construction expertise, water and sewer systems, information technology, cars, etc., so we can swap. Fine.
So, why did we give away major parts of the US economy? Sure: Some econonuts had excuses for why this was optimal, and the Foggy Bottom types wanted to save the world by exporting the US economy. Meanwhile, back in the US, the citizens got it in the rear. US citizens were killed, and are still dying.
So, several large US industries got shipped to Taiwan, South Korea, Pakistan, and China, and whole states in the US had their economies devastated. There is a loss the econonuts don't count: The human capital that gets written off. Or, the econonuts assume that the textile workers and metal bending manufacturing workers can, of course, just move to Redmond and write software for Microsoft, which has so far likely never happened even once, to sell to China, which, of course, steals software and doesn't buy it.
The econ profs conveniently f'get about market manipulations we learned about in the US in the 1890s -- predatory marketing practices, etc. And they f'get about the assets of a going business, technology secret sauce, market position, etc. The simplistic econ models just don't count such things, so the econonuts assume that we shouldn't think about such things in the real economy and, thus, just give them away.
In school, I knew the applied math MUCH better than the econ profs, and their nonsense and its intellectual dishonesty were infuriating. After the first lecture, I asked the econ prof, nicely, what he was assuming -- continuity, differentiability, continuous differentiability, convexity, pseudo convexity, quasi convexity, or what. Then within an hour he had called my Ph.D. advisor and got me OUT of his class. That was not my intention but was GOOD.
Academic economics has NOTHING important to do with any real economy and is from irrelevant, incompetent, dishonest, and contemptible down to seriously dangerous.
We're talking ordinary crooks way down to the lawyers way, way down to bucket shop operators and from there way, way, way down to the politicians and from there, far, far down where just can't see at all, the econ profs. I know what we should export next: If all the econ profs were lined up on a cargo ship, it would be a good thing. And still better if the ship sank far out at sea.
The real economy is IMPORTANT, way, WAY too important ever to be touched by anything like econ profs.