It is suggested that an extra degree of freedom is needed to construct a symmetric noncooperative price game in a market with n monopolists trading in n goods. This calls for the introduction of an n+1 good which can be interpreted as a commodity money. Where there are n monopolists using a commodity money in common a symmetric price or quantity noncooperative game can be constructed. The quantity game is examined. Necessary conditions are shown for the replicated game to have its noncooperative equilibria approach the competitive equilibria of the replicated market. It is demonstrated that unless there is “enough” commodity money convergence may not take place. There will be a “money shortage” and this will be reflected in a price for the commodity money higher than its utilitarian worth. This reflects the addition of a “shadow price” for the worth of relaxing the monetary capacity constraint.
Shubik, Martin, "A Theory of Money and Financial Institutions. Part III. The Missing Degree of Freedom: Commodity Money and Oligopoly in a General Equilibrium Model" (1971). Cowles Foundation Discussion Papers. 557.