A Simple Theory of Banking and the Relationship between Commercial Banks and the Central Bank

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1 A Simple Theory of Banking and the Relationship between Commercial Banks and the Central Bank Eric Kam 1 Ryerson University John Smithin 2 York University Abstract: This note provides an explanation of the relationship between the nominal and real lending rates of the commercial banks and central bank policy interest rates. It suggests that a real interest rate policy rule on the part of the central bank would influence the real lending rate perceived by commercial banks and their borrowers, and will affect the real economy via this route. There is a negative theoretical relationship between the ex post real lending rate and the rate of inflation. JEL Classifications: E50, E52, E58. Keywords: interest rate policy, banking, central banking, deposit rates, lending rates, monetary policy rules, risk, profit maximization. Introduction The purpose of this note is to present a simple profit maximizing theory of banking, and of the relationship between those commercial banks that have a "direct clearing" relationship with the central bank and the central bank itself. One of the starting points for the discussion is the critique of the "neoclassical" theory of banking due to Rymes (1998) and Rogers and Rymes (2000). Also relevant is the theory of "free banking" put forward by White (1984), a number of years ago. The issues involved have also been briefly discussed previously by MacKinnon and Smithin (1993, ), and, in somewhat more detail, by Tabassum (2007), in a doctoral thesis proposal. As has been pointed out elsewhere (Smithin 2009), in other areas of the macro model the application of a profit maximization or utility maximization methodology could reasonably 1

2 be criticized on the grounds that it is not, after all, a realistic description of the way in which decisions are actually made. For some types of firm behavior, however, and particularly that of financial and banking firms, there is an important caveat to this position. This is because, given the economic sociology of the system, such firms are constrained to behave in a certain way if they are to succeed on their own terms of reference in the existing environment. This remains true even if no comparable case could be made for (say) the behavior of consumers or workers. The Commercial Bank Balance Sheet Table I below is a stylized/simplified balance sheet of a commercial bank which makes loans L at a competitive interest rate i L, and accept deposits D, on which it pays a competitive deposit rate i D. It may choose to hold reserves, R, consisting of the non interest bearing liabilities of the central bank, but there are no formal reserve requirements. It is not a crucial requirement that the interest rate on reserves be equal to zero. However, it is crucial that the central bank determines this rate. In this case, we can say that they set the rate on reserves "at zero". Table 1: The commercial bank balance sheet Assets Liabilities Reserves R Deposits D Loans L Settlement Balances S R+L D+S Clearing with the other commercial banks is conducted via the central bank and if, at the end of the designated accounting period, a commercial bank finds itself with a negative settlement balance, S, at the central bank, they must pay a penalty interest rate, i o, determined by the 2

3 central bank itself. We can call this the "policy rate" of the central bank, and it is a proxy for such policy rates in the real world. These might historically have been "bank rates" or "discount rates" or the like. In the contemporary financial environment the policy rate would usually be an overnight intra bank lending rate of some kind (Atesoglu and Smithin 2008). The Optimization Problem for Bankers The profit maximization problem for the commercial bank is: (1) Max: Π = i L L i D D i O f ( x) dx (S R) - μl where the sum is the subjective cumulative probability distribution (in the view of the bankers themselves) of the likelihood of ending up with a negative settlement balance. For example, if (a big if) the bank was operating in an ergodic environment (Davidson 2009, 37 38) and the probability distribution was known to be normal, the figure would be 0.5. However, neither of these conditions is likely to hold in reality, so, in fact, the number simply "is what it is", so to speak. It is the bankers' own subjective estimate of what the risks are. Some of the empirical evidence on this, in fact, suggests a pass through coefficient that is less than one but larger than 0.5 (Atesolgu 2003/04). We can therefore simplify the notation by introducing the term σ, where: (2) σ =. 0 0 f ( x) dx f ( x) dx 0 Meanwhile, it is also tempting to interpret the term μ as the average cost per dollar of making bank loans, but a difficulty here is that it difficult to conceive of any precise analog to a textbook physical production function in the nebulous case of banking (Dow and Smithin 1999). It is probably safer to say that μ must be high enough to cover costs and earn a "normal" rate 3

4 of return for the banks given existing institutional arrangements, market structure, banking legislation and regulations. It is ultimately determined by the latter four sets of conditions. Now, substituting in from the bank balance sheet, the optimization problem becomes: (3) Max: Π = i L L - i D (L + R S) - i O σ(s R) - μl The choice variables for the bank are the volume of loans granted and the quantity of precautionary reserves they choose to hold. The first order conditions for the problem are therefore obtained by differentiating with respect to L and R, setting the results equal to zero: (4) i L i D = μ (5) i D = σi O Therefore, the mark up between commercial bank lending rates and deposit rates will be equal to μ, and the deposit rate will be a "mark down" from the central bank's setting of the policy rate. The degree of the mark down depends on the subject assessment of risk for a commercial bank of not "keeping in step" (Keynes 1971/1930, 23) with their rivals. In the past, a similar sort of result has been called the "two for one" rule (Rogers and Rymes 2000, 259), but this would depend on the assumption of a normal distribution which, we have argued, cannot be justified. Combining equations (4) and (5), we can see that there is linear relationship between policy rate and the bank lending rate, thus explaining how increases in the central bank policy rate may be transmitted to interest rates in general. This is: (6) i L = μ + σi 0 Now subtract the observed inflation rate, p, from both sides of equation (6). This gives: (7) i L p = μ + σr 0 (1 σ)p, 4

5 where r 0 is the ex post real policy rate of interest, defined here as the nominal policy rate adjusted for the currently observed inflation rate, r 0 = i 0 p. This gives some insight into the discussion in Smithin (2007, 2009) that suggested a "real interest rate rule" for monetary policy. As first suggested in the original "Taylor rule" paper (Taylor 1993), as a practical matter this would have to involve a target for the ex post policy rate (as defined above) as the true expected inflation rate is not known. The question then is whether the similar ex post real bank lending rate in equation (7) can also be taken as a proxy for the real lending rate itself. If so, in the absence of any other reliable indicator on which borrowers can base their estimates, equation (7) can be written: (8) r = μ + σr 0 (1 σ)p Therefore if the term r in equation (8) does stand for the "real interest rate" involved in economic decision making (for example, the interest rate in an investment function, or in an IS curve in a macro model), equation (8) shows how central bank activities can have influence over this rate, and over the "real economy" in general. Notice also that there is a negative (rather than positive) relationship between inflation and real interest rates in this situation (Kam 2005, Kam and Mohsin 2006). Conclusion This note provides an explanation of the relationship between both nominal and real central bank policy interest rates, and the nominal and real lending rates of the commercial banking system. It suggests that a real interest policy rule on the part of the central bank would influence the real interest rate that is perceived by commercial banks and their borrowers as 5

6 relevant to their concerns, and will affect the real economy via this route 3. In this case, there is would also be a negative relationship between the ex post real lending rate and the rate of inflation. However, in a world where the central bank was following a Taylor rule, or similar, this "natural" relationship would likely be obscured because the central bank would then be deliberately raising or lowering the real policy rate whenever inflation rises or falls. Notes 1. Eric Kam is Associate Professor of Economics at Ryerson University, Toronto, Canada M5B 2K3; tel: +1 (416) ext. 6183, fax: +1 (416) , e mail: erickam@economics.ryerson.ca. 2. John Smithin is Professor of Economics in the Department of Economics and the Schulich School of Business, York University, Toronto, Canada M3J 1P3; tel: +1 (416) , ext.33623; fax: +1 (416) ; e mail: jsmithin@yorku.ca. 3. Smithin (2007, 2009) also demonstrates that this will ensure inflation stability. References Atesoglu, H. Sonmez 2003/2004. Monetary transmission federal funds rate and prime rate. Journal of Post Keynesian Economics 26: and John Smithin Un modelo macroeconomico simple. Economia Informa 346: Canadian monetary policy and the US federal funds rate. Applied Economics Letters 15: Davidson, Paul The Keynes Solution: The Path to Global Economic Prosperity. London: Palgrave Macmillan. Dow, Sheila C. and John Smithin The structure of financial markets and the "first principles" of monetary economics, Scottish Journal of Political Economy 46: Keynes, John Maynard The Collected Writings of John Maynard Keynes, V, A Treatise on Money, I, The Pure Theory of Money, ed. Donald Moggridge. Cambridge: Cambridge University Press. (First published 1930). 6

7 Kam, Eric A note on time preference and the Tobin effect. Economics Letters 89: and Mohammed Mohsin Monetary policy and endogenous time preference. Journal of Economic Studies 33: MacKinnon, Keith T. and John Smithin An interest rate peg, inflation and output. Journal of Macroeconomics 15: Rymes, Thomas K Keynes on anchorless banking. Journal of the History of Economic Thought 20: and Colin Rogers The disappearance of Keynes's nascent theory of banking between the Treatise and the General Theory. In What is Money?, ed. John Smithin, , London: Routledge. Smithin, John A real interest rate rule for monetary policy? Journal of Post Keynesian Economics 30: Money, Enterprise and Income Distribution: Towards a Macroeconomic Theory of Capitalism. London: Routledge. Tabassum, Aqeela Three Essays on the Impact of Financial Evolution on Monetary Policy. A dissertation proposal submitted to the Faculty of Graduate Studies in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Graduate Program in Economics, York University, Toronto. Taylor, John Discretion versus policy rules in practice. Carnegie Rochester Conference Series on Public Policy 39: White, Lawrence H Free Banking in Britain: Theory, Experience and Debate Cambridge: Cambridge University Press. 7