A Little Model of Banks and the RBI's rupee strengthening campaign

As part of its latest effort to strengthen the struggling rupee, the Reserve Bank of India (RBI) announced it would increase its cap on bank deposit rates for the Indian diaspora. The RBI hopes that the rupee will strengthen as those living abroad convert their foreign currency into rupees to take advantage of these higher rates. The Monti-Klein model -- developed in the 1970s independently by Michael Klein and Mario Monti, who became Italian Prime Minister and is now Senator for Life -- shows why and when this policy will work, and helps estimate the policy's overall impact.

The Monti-Klein model assumes a monopolist bank and an interbank lending rate, $r$ -- similar to what central banks set to conduct everyday monetary policy. The monopolist bank sets deposit rates, $r_{d}$ and retail lending rates $r_{l}$ . Higher deposit rates attract more depositors, implying an upward sloping deposit supply curve. The bank profits as the gap between the interbank lending rate and the rate it pays on deposits widens. Lower lending rates attract more borrowers, implying a downward sloping lending curve. The bank's profit on loans increases with the gap between the interbank rate and what it charges on loans.

Putting this together, the bank's profit function is what it makes from lending plus what it makes from borrowing minus the operating costs, $C$ :

Π (r_{d}, r_{l}) = D (r_{d}) (r - r_{d}) + L (r_{l}) (r_{l} - r) - C

Where $D (r_{d})$ is the deposit supply curve and $L (r_{l})$ is the loan demand curve.

Our perfectly rational little monopolist then sets its deposit and lending rates, $r_{d}$ and $r_{l}$ , to maximize its profit, subject to $r_{d}$ staying below the cap, $k$ , which the RBI imposes and recently increased to 7.5 percent.

The RBI hopes to strengthen the rupee by getting banks to increase $r_{d}$ . In the model, the lending rate doesn't depend on $r_{d}$ , so we can ignore the $L (r_{l}) (r_{l} - r)$ term. Similarly, we can ignore the constraint because we know it will only bite if it is less than the optimal deposit rate. Finally, we assume the upward sloping supply of deposits has the following simple functional form:

D (r_{d}) = α + β r_{d}

The monopolist bank, then, maximizes $(α + β r_{d}) (r - r_{d}) - C$ . Solving this yields the optimal deposit rate $r_{d}^{*} = \frac{r β - α}{2 β}$ .

We can use this equation to assess the impacts of the RBI's policy change. First, if $r_{d}^{*}$ was already below the old ceiling, then the new policy will do nothing because the cap never bound. Second, plugging $r_{d}^{*}$ into the function for the deposit supply curve allows us to predict the total growth in deposits, assuming the old cap bound. UBS researchers likely did something similar when estimating this policy's impact.

Little model, big data #

We can estimate $α$ and $β$ by gathering data on bank deposit rates and the number of deposits and then running a simple linear regression. But if we think the relationship between the supply of deposits and the deposit rate is a non-linear combination of many more features, X, we can follow the above recipe, but estimate $D (r_{d}, X)$ with something like a neural network and then find the optimal $r_{d}$ with basic optimization techniques. The simple form tells us exactly what data we need.

The central problem, however, is whether the data we gathered allows estimating the causal impact of the deposit rate on the deposit supply. For example, what if the central bank raised interest rates, causing the stock market to crash? If investors sold their stocks and deposited their cash in banks, and banks passed the central bank's interest rate hike on to depositors, we would see both deposit rates and total deposits rise. But the rise in deposits isn't related to the deposit rate; rather, it's because the stock market crashed. With correlational data alone, an intervention that moves the deposit rate to $r_{d}$ may result in a deposit supply much different than past correlations would suggest. In other words, our estimates of $α$ and $β$ would be biased.

Since the core problem is with the data, a more complicated model, like a neural network, doesn't remove this bias. Nor would gathering even more biased data. Rather, we need something that changed the deposit rate, $r_{d}$ , and only impacted the deposit supply via its influence on $r_{d}$ . Something like a rumor of a competitor bank entering the market could fit this requirement. Only with these conditions met could we estimate the causal impact of the deposit rate on the supply of deposits.

Model Shortcomings #

A simple model like that above can guide the data we collect and help evaluate the likely policy impacts. But models are only useful if they point us in the correct direction. Although the Monti-Klein model provides insights into the RBI's policy, it has gaps.

The model shows that if the RBI increases the overnight rate $r$ , which is standard monetary policy, the optimal deposit rate would also increase. Since the supply of deposits is upward sloping, this monetary policy intervention should drive more deposits. But total deposits may fall when the overnight rate, $r$ , increases if banks don't pass much of the interest rate increase through to $r_{d}$ . Deposit beta captures the portion of the central bank's interest rate increase banks pass to their depositors. The Monti-Klein model implies a 0.5 deposit beta, but it doesn't consider competition for deposits from non-bank financial institutions.^[1] Low deposit betas may lead depositors to move their money out of banks and purchase assets like money market mutual funds or high-yield savings accounts, which can reduce deposits throughout the system. The New York Fed suggests that low deposit betas occur when banks have more deposits than they need, so they don't mind depositors moving their money elsewhere. In reality, parameters like deposit betas are dynamic and context dependent, not fixed at something like 0.5.

But for the RBI, the policy targets deposit rates, not interbank lending rates. If the current ceiling binds, then banks should increase their deposit interest rate to the new ceiling. This policy alone doesn't offer an obvious method for non-bank financial institutions to lure people out of the banking system.

The simple Monti-Klein model provides insight into how the RBI's policy may work to increase deposits. It also guides data gathering and model building, which can help estimate the likely impacts of the policy. Because it makes clear assumptions and predictions, we can dig deeper into things like deposit betas to understand when the model will fail.

Starting with $r_{d}^{*} = \frac{r β - α}{2 β} = \frac{r β}{2 β} - \frac{α}{2 β} = \frac{r}{2} - \frac{α}{2 β}$ implies that half of the overnight interest rate, $r$ , is passed to the optimal deposit rate. ↩︎

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