Paul Daivson wrote that “New Keynesian theory was developed to replace Samuelson’s Keynesianism.” (Davidson. Interpreting Keynes ofr the 21st Century.Palgrave 2007, pp. 208-209)
In 1992, N. Gregroy Makiw and David Romer put out two volumes on “New Keynesians Economics published by MIT Press. Mankiw macroeconomics text also is based on New Keynesian ideas.
When Paul Krugman described new Keynesian economics, we noted that “Version of this story may be found in articles by several economists, notable Greg Mankiw; George Akerlof and Janet Yellen, Oliver Blanchard and Nobuhiro Kiyotaki, and Laurence Ball and David Romer.” (Krugman. 1994, Peddling Prosperity, WW. Norton and Company, P. 213.
Pikkety is steep in wealth distribution. To the extent that unequal distribution creates imperfection in the market and added to rigidity in wage and prices then his thought are in line with the new keynesians.
A key concern of new Keynesian economists is the aggregate effect of imperfect competition. An example is the macroeconomic effect of price rigidity in monopolistically competitive markets. Mankiw, Romer, and Krugman contributed to developing this branch, so they can be considered "new Keynesian economists." (Indeed, they appear in the list linked below.) But I do not think it is the case of Piketty, who is mainly concerned with institutions and inequality.
However, the contributions of these economists may go beyond imperfect competition. I would say, for example, that Krugman has been primarily concerned with economic geography and international economics.
Personally, I do not like to consider trends in economic thinking as isolated from each other, or as opposites.
I support Esteban Colla . Mankiw, Romer, and Krugman have clear common core of economic thinking. However, I believe those New Keynesians are wrong on why prices of almost all goods and services are rigid. They believe they are rigid because of imperfect competition. Perfect and imperfect competition is a wrong dichotomy, which is created by the wrong theory that prices are determined at the point of equality of demand and supply.
This idea (once called law of demand and supply) is very old. We can observe many variation among classical political economists. Neoclassical economists like Jevons, Walras, and Menger wanted to reformulate the law in a more precise (or formal) way but could not acknowledge where they were wrong. Marshall went a bit deeper into the question on how to define the demand and supply function. To define supply function he was obliged to assume decreasing returns to scale whereas in the real economy almost all industries are operating
under the condition of constant returns (constant marginal cost) or increasing returns to scale (if we includes fixed costs such as depreciation, R&F and management cost).
John Maynard Keynes was a great renovator of neoclassical economics, but could not base his new theory (principally the principle of effective demand) on a new price theory and remained halfway in his renovation. This is the main reason why anti-Keynesian neoclassical revolution (or counter-revolution) occurred in 1970's.
Mankiw, Romer, and Krugman do not recognize this fundamental fact and remain to support neoclassical economic (e.g., Arrow and Debreu theory of competitive economy) as microfoundations to their macroeconomics.
Parallel to New Keynesian economics, there is now another school called Post Keynesian. We counts three or four strands (fundamentalists, Sraffians, Kaleckian, and Minskians). They all refuse neoclassical economics
like Walras and Arrow and Debreu, but they have no alternative theory to replace neoclassical microeconomics. Consequently their theoretical basis is quite ambiguous and vulnerable. Many appeal to intuition or ontology but have produced no rigid alternative price theory. This is the reason why I asked the next question two years ago:
Post Keynesian economics had lacked its foundation, but this situation changed rapidly by the appearance of new theory of value and its associated theory on quantity adjustment theory. See
Book Microfoundations of Evolutionary Economics
You can read a simple introduction to the new theory of value:
The revival of the classical theory of values
Chapter The revival of the classical theory of values
You can easily read this by a draft version even though there are many minor revisions after this draft:
Conference Paper The Revival of Classical Theory of Values
Please note however that most important contribution of the above book lies in its quantity adjustment process (Chapters 3 to 2), which is comparable to Arrow and Debreu (1954). Chapter 2 explains when the price changes. Theorem
4.4 proves the existence of admissible value. It implies that price of a product does not change unless the production cost or the markup rate changes. Such a change of production cost occurs only when there is a change of production techniques (if markup rates remain constant). This is the deeper reason why most industrial products (goods or services) remain rigid. Mankiw and others are mistaken because they believe that prices exhibit rigidity by the existence of menu costs. The existence of admissible value means that prices do not change even if there is no menu costs (interpreted in the most wide sense).
In my previous post of February 23, 2020, I cited my book with Morioka and Taniguchi: Microfoundations of Evolutionary Economics. As it is claimed in the Preface of the book (you can read it at Book Microfoundations of Evolutionary Economics
) , we have claimed that our theory provides microfoundations for Post Keynesian economics too. Please read the book review by Marc Lavoie
then you will see how our book is related to Post Keynesian economics.
Whether a macroeconomics is New Keynesian or Post Keynesian is not easy to determine when you are looking at macroecoomic models. You must consider on what microtheory an economists draws. There is a clear difference between Walarassian or Arrow-Debreu theory and the new theory of value. New Keynesian economics draws on the former, Post Keynesian economics on the latter.