Do general equilibrium models include money? An economics professor told me that General Equilibrium models are a sophistication of barter. Does that imply that they don't include money? And if that is the case, are there any attempts to include it?

Moreover, does this have any implications on the way we understand macroeconomics?

Thanks.
While many general equilibrium models do not need to model money to approach the questions that they would like to answer, there are many models that do include money to address questions that need money to be a relevant feature of the model. These models do it in a variety of ways -- some might be more relevant than others. I will try and describe two of the approaches briefly.

Cash-in-advance

When I think of a cash-in-advance (CIA) model, the first models that come to mind are Lucas (1982) and Lucas Stokey (1987). Models that take this approach have agents saving in assets and money, but they must finance any consumption solely through their money holdings (basically assets are less liquid than money which isn't the worst approach). There are many critiques of the CIA of which I find Wallace (1998) to be one of the more thoroughly presented.

New Monetarist

More recently there has been a push to use models of money that are less "ad-hoc" about how money works. Some foundational papers of this field are the Kiyotaki Wright papers (1989, 1991, 1993). These papers build a very basic model of money and show what role it plays in economics -- There is a good discussion on Randall Wright's Wikipedia page under the sub-title "Research contribution." While these models confirmed some of the reasons for money to exist in a model, they were quite simplistic and lacked many other features of the real economy. A monetary course I took labeled these types of models as "Generation 1 models of money." There was a group of papers that were known as "Generation 2 models of money, but these shared many of the simplifications made in Generation 1 (so I won't discuss them here).

Since this time Lagos Wright (2004) introduced a more complete framework (we referred to these as "Generation 3 models of money." One of the issues faced by these new monetarist models is that in order to solve the model, one must keep track of the entire distribution of money -- This is computationally infeasible unless the authors introduce a simplification in a different area of the model (in the Lagos Wright model, they simplify the form of utility in order to be able to compute the distribution of money next period). There is on-going research in this area and, in my opinion, is an open area for improvement in the literature.

References


Nobuhiro Kiyotaki and Randall Wright. "On Money as a Medium of Exchange." Journal of Political Economics. 1989.
Nobuhiro Kiyotaki and Randall Wright. "A Contribution to the Pure Theory of Money." Journal of Economic Theory. 1991.
Nobuhiro Kiyotaki and Randall Wright. "A search-theoretic approach to monetary economics." American Economic Review. 1993.
Ricardo Lagos and Randall Wright. "A Unified Framework for Monetary Theory and Policy Analysis." Journal of Political Economy. 2005.
Robert E. Lucas Jr. "Interest Rates and Currency Prices in a Two Country Model." Journal of Monetary Economics. 1982.
Robert E. Lucas Jr and Nancy L. Stokey. "Money and Interest in a Cash-In-Advance Economy." Econometrica. 1987.
Neil Wallace. "A Dictum for Monetary Theory." Federal Reserve Bank of Minneapolis Quarterly Review. Winter 1998.