Formally, we find:
Proposition 6 A regime in which neither country imposes a CIA is not a subgame perfect Nash equilibrium.
Proof. Consider the configuration in which neither country imposes a CIA requirement and then each chooses their inflation rate. The payoff from this cannot exceed that obtained under monetary union, Vми, since the outcome under monetary union selected the inflation rate that was jointly optimal. Further, from Proposition 5, we know that if neither country imposes a local CIA requirement and each sets the rate of money creation to zero, this is the same outcome as that obtained under monetary union. this
Suppose then that the home country defects from this proposed equilibrium and establishes a CIA. In this case, we consider a steady state equilibrium associated with the (CIA, no CIA) cell of the regime choice game. By following an equilibrium path resulting from the defection of the home government, we axe able to argue that the conjectured equilibrium in which neither country imposes a CIA is not subgame perfect.