For instance, in a standard AS-AD New Keynesian model, the two policies would be perfectly linearly dependent, as they both end up influencing the only control available to the policymaker, the interest rate, either through open market operations or via the macroprudential instrument.
In one possible modelling of the two channels, suggested by Angelini, Neri and Panetta in the above-mentioned Bank of Italy paper, monetary policy would set the level of the nominal interest rate, as in standard models, while countercyclical macroprudential policy would influence the differential between the interest rate on bank loans and the rate on deposits by setting a capital ratio.