"In practice, of course, the purists were unable to deliver, and the new tricks involve the 'modern macroeconomists' in ad hoc assumptions of their own that are at least as objectionable as the Keynesian macroeconomic generalizations that [Michael] Wickens objected to. We have already encountered one example, the 'Gorman preferences' needed to make the representative agent at least minimally plausible... Two others are equally incredible. The first is the 'no-bankruptcies' assumption in Walrasian models and the related 'No Ponzi' conditon that is imposed on D[ynamic] S[tochastic] G[eneral] E[quilibrium] models. This eliminates the possibility of default, and hence the fear of default (since these are agents with rational expectations, who know the correct model, and hence know that there is no possibility of default), and hence the need for money, since if your promise to pay is 'as good as gold', it would be pointless for me to demand gold (or any other form of money) from you. Money would be at most a unit of account, but never a store of value. The second is the unobtrusive postulate of 'complete financial markets', smuggled into Michael Woodford's Interest and Prices (Woodford 2003, p. 64), which means that all possible future states of the world are known, probabilistically, and can be insured against: this eliminates uncertainty, and hence the need for finance..." -- J. E. King, The Microfoundations Delusion: Metaphor and Dogma in the History of Macroeconomics (2012: p. 228)
Can General Equilibrium Theory find a role for money? Consider Frank Hahn's "On some problems of proving the existence of an equilibrium in a monetary economy" (1965). He considered the question posed above to be an unmet challenge at the time. Hahn did not think, for example, Don Patinkin's attempt to justify the use of money through the inconvenience of indirect transactions and through transactions cost fit comfortably in a General Equilibrium model. Hahn wanted to find a model in which the existence of money was essential, in some sense, in which an equilibrium with money differed from one without.
mainstream macroeconomists claim to base their approach on General Equilibrium Theory. Experts on GE (for example, Alan Kirman) have been saying for decades that this claim is dubious. The critiques that I am most aware of are based on price theory.
These modern macroeconomists claim to have available models incorporating money and finance. This availability does not mean that they are unwilling to deploy models without money in some contexts for some purposes. As far as I am aware, Woodford is widely cited and widely respected in current mainstream monetary economics.
I have wondered how and if mainstream macroeconomists address the problems highlighted by Hahn in incorporating money in General Equilibrium Theory. But I have not wondered enough to read much. If I take King as an authority, I can spare myself the trouble of establishing that mainstream macroeconomists are basically confused about their own monetary theory. Is this a sound conclusion?
(Colin Rogers is an expert on how the Cambridge Capital Controversies can be used to critique Wicksellian theories of money and of the natural rate of interest. Woodford and Rogers had an exchange of views in the Cambridge Journal of Economics a number of years ago.)