The value of money in a model economy. Has this been addressed?
What you are asking about is "General Equilibrium with Money", and there is literature on the matter.  

To get you started, check out   

Duffie, D. (1990). Money in general equilibrium theory. in Friedman and Hahn (eds), Handbook of monetary economics, vol. 1, 81-100.

The chapter starts with the sentence


  This chapter provides conditions for the existence of general monetary equilibrium in a simple setting, and discusses the role of
  money in general equilibrium theory.