I ran into an interesting question online: what is neoclassical macroeconomics? I use neoclassical economics as a synonym for “modern mainstream academic macroeconomic theory,” which may not coincide with other definitions. That said, I think it captures what people are interested in, as opposed to any other narrower definition. The terms in the above definition are vague, and so I will spell the definition out in more detail below.
What I find interesting about this question is that I think the usual postKeynesian approach to this question is misleading. The usual charge is that neoclassical economics has real world political connotations. I think that the real connotations are for academic politics. The real world political content of neoclassical economics just reflects that it is being developed by individuals and groups that have a conservative bent. (Note that I deliberately wrote conservative and not free market, as the focus is more on serving the interests of the governing elites, and developed countries are mixed economies.)
“Formal” Definition
The defining characteristic of neoclassical macroeconomics is that it is generally put forth in the form of mathematical models that follow a particular pattern. (Although neoclassical economists might do empirical work, I do not consider econometric work to be “neoclassical economics” — a statistical tool is a statistical tool.) I do not believe that the “pattern” can be formalised as a mathematical statement, as mathematical statements are given in terms of precisely defined sets. What model fits the “pattern” is a question of psychology, and not real analysis. That is, any attempt to create a formal set covering “neoclassical models” will be invalidated by somebody coming up with what is arguably a “neoclassical model” that is not a member of that set.
I summarise the pattern as follows. The core of the model structures involve intertemporal optimisation by “household agents,” along with optimisation problems with other actors (firms, the central bank, etc.). Actual economic outcomes are determined by an “equilibrium” resulting from agents setting exchange ratios between goods and money.
Note that there is an infinite number of classes of models that can fit within this structure, and since these classes typically have free parameter values that are real numbers, there is also an infinite number of models within each class. If we have two different models within the same class, they can still have very different implications due to the behavioural differences. (E.g., if we look at the simplest different equations, models of the same order can be either stable or unstable based on the coefficients in the equations.)
As an example of the slipperiness of the definition, keep in mind that there are no little people in households inhabiting mathematical sets. A “household optimisation problem” is just an optimisation problem that is allegedly embedded in a larger mathematical system. But somebody else can choose a different objective function (“utility function”) that has the result that the model no longer fits the original set that was supposed to contain “neoclassical models,” but psychologically, it is still a “household optimisation problem” and thus would be considered in practice to be a “neoclassical model.”
The models can either be discrete time or continuous time, and may have sensible time axes (monthly, quarterly) or insane ones (“generations” in overlapping generation (OLG) models). In the sensible time axis category, the socalled Real Business Cycle (RBC) model would be a canonical model.
One could interpret all the sensible classes of models that fall under this definition as being RBC models plus “refinements.” I am unable to give a precise date when “neoclassical economics” starts (i.e., what qualified as “modern mainstream macroeconomics”), but the release of the RBC model would be the latest cut off date.
Real World Political Content
Although the models can be used as cheerleading for capitalism (“market outcomes are optimal!”), one is free to add “imperfections” that result in “capitalist outcomes — boo!”
However, I would argue that the conservative edge of the theory comes from the people producing the theory. Central banks are key to funding and validating the theory. One of the advantages of being a Canadian Prairie Populist is that my thinking is not clouded by tribal right/left politics: the Bank of Canada and the private banks is one of the clubs for Canada’s governing elites, and you are either a member of that club or not. A club that includes private banks and a central bank that enacts policy for the central government does not fit into a simplistic “free market versus socialism” political spectrum.
Is Neoclassical Economists Coherent?
If we move beyond the dubious Welfare Theorems of Economics, we can then ask: is the neoclassical project theoretically coherent? Well, look at the definition I gave. You have an extremely wide of classes of models that can be looked at, and each class includes an infinite number of models. Why would you not expect to find a model that can fit almost any specified behaviour? Unfortunately, neoclassical economics education is designed to weed out people who would ask nihilistic questions like that. Instead, the attitude is that the objective of an economics education is to train people to pick the “best” model for a question at hand. This is obviously disastrous if the models themselves are internally inconsistent.
The way real applied sciences deal with this issue is that they have successful models, which can then create a standard to judge other models. Macroeconomics is notable for its complete lack of successful mathematical models, so this is not possible.
This is why arguing that there is a political bias to neoclassical economics is straining at a gnat but swallowing a camel. In order to believe that neoclassical has a political agenda, you have to believe it is an internally consistent theory in the first place.
Brilliant Academic Politics
Neoclassical economics is where it is because it is an amazing evolutionary development in response to the “publish or perish” environment.

The infinite number of model classes — with inconsistent lessons to be learned — guarantees that articles can be published forever.

Results “proving” that central banks ought to be given free rein to do whatever they want is extremely convenient from the perspective of where funding and recognition comes from. Although one could probably come up with models that suggest that the central bank should be under the control of elected officials, attempting to publish such a model would be career limiting.

By eliminating other approaches — particularly literary approaches favoured by postKeynesians — largely eliminates any challenges the neoclassical paradigm.

Any theoretical failing of favoured models just means that researchers need to look at more complicated models.
Taken together, they make the neoclassical paradigm extremely robust to change. Even if researchers have doubts about the current batch of models, they have been trained to believe that a new generation of models that rely on the same underlying assumptions is just around the corner. All we need is just a bit more mathematics, and all the problems with predicting the future will go away.
Concluding Remarks
The defining characteristic of neoclassical economic education is that it produces people who are clearly unwilling to question core theoretical assumptions of neoclassical economics. The belief that “science advances one funeral at a time” relies on the assumption that there exists young scientists that wants to impose a new paradigm. Instead of reading the philosophy of science, one needs to study anthropology, and in particular how magical practice stuck around in complex societies.