Introduction
The money multiplier framework models aggregate broad money supply ($M$) as the product of reserve or high-powered money ($H$) and the money multiplier ($m$). In a fractional reserve banking system, the multiplier is governed by public portfolio preferences—encapsulated by the currency-to-deposit ratio ($c$)—and banking liquidity management represented by the reserve-to-deposit ratio ($r$).
Mathematical Derivation of the Multiplier Effect
Under standard central banking theory, broad money supply is expressed as:
$$M = m \cdot H = \left(\frac{1+c}{c+r}\right) H$$
where $c = C/D$ represents the currency-deposit ratio held by the non-bank public, and $r = R/D$ denotes the aggregate reserve-deposit ratio maintained by commercial banks ($r < 1$).
To assess the sensitivity of the money multiplier to changes in the currency-deposit ratio, take the partial derivative of $m$ with respect to $c$:
$$\frac{\partial m}{\partial c} = \frac{(c+r)(1) - (1+c)(1)}{(c+r)^2} = \frac{r-1}{(c+r)^2}$$
Since commercial banks operate under fractional reserve banking, aggregate reserves are strictly a fraction of deposits ($r < 1$). Consequently, the numerator $(r - 1)$ is strictly negative, yielding:
$$\frac{\partial m}{\partial c} < 0$$
Thus, a reduction in the currency-deposit ratio ($c$) mathematically causes an increase in the money multiplier ($m$).
Theoretical vs. Realised Impact on Money Supply
- Theoretical Potential Expansion: By invalidating high-denomination bank notes, demonetisation compulsorily channelled currency into commercial bank deposits, forcing an abrupt fall in $c$. A smaller currency leakage implies that fewer reserves exit the banking system as physical currency, theoretically enabling a larger round of credit creation and expanding the potential money supply.
- Contraction of Reserve Money ($H$): In the immediate aftermath, the demonetisation of specified bank notes caused high-powered money ($H$) to fall sharply because currency in circulation collapsed faster than deposits rose, depressing base money.
- Spike in the Effective Reserve Ratio ($r$): Due to transactional friction and muted consumer and industrial credit demand, banks were unable to intermediate the massive liquidity influx immediately. Instead, commercial banks accumulated substantial excess reserves ($ER$), causing the effective reserve ratio ($r$) to spike and dampening the multiplier.
- Regulatory Sterilisation: The Reserve Bank of India stepped in to mop up surplus banking liquidity through an extraordinary 100% Incremental Cash Reserve Ratio (ICRR) and the issuance of Market Stabilisation Scheme (MSS) bonds, which artificially raised $r$ and restricted immediate monetary expansion.
- Short-Run Dichotomy: While the broad money multiplier ($M_3/M_0$) jumped arithmetically from approximately 5.6 to over 8.5 due to the denominator effect of plunging $H$, realized narrow money ($M_1$) contracted sharply and broad money ($M_3$) growth decelerated significantly.
Endogenous Money Creation and Structural Shifts
From a Post-Keynesian endogenous money perspective, banks do not passively multiply pre-existing reserves into loans. Lending is constrained by creditworthy demand and collateral value rather than reserve availability. The expansionary impact of a lower $c$ materialized only over the medium to long term as remonetisation stabilized base money, while structural shifts toward digital payments and formal banking channels sustained a lower structural currency-deposit ratio.
Conclusion
While standard money multiplier theory predicts that a fall in the currency-deposit ratio expands money supply via an enlarged multiplier, the immediate outcome of demonetisation was constrained by a declining monetary base and administrative liquidity sterilisation. Over the long run, however, the permanent shift away from physical cash into banking and digital payment ecosystems enhances financial intermediation and increases the systemic credit multiplier.