Natural Equilibrium of Interest Rates

Contributor(s): Peter del Rio
Last Edited: 2017-08-17

October 5, 2016


Abstract

The behavioural model for the Natural Equilibrium of Interest Rates (NEIR) is the combination of the equilibrium of Interest Rates on Cash and the equilibrium of Excess Bank Reserves (EBR) and both are affected by the supply and demand for cash. Typically, there is more demand than supply for cash so EBR are flat in an equilibrium near 0 and reflecting the demand for cash interest rates are greater than 0%. Rarely, there is more supply than demand for cash so EBR are greater than 0, currently over $2 trillion, and reflecting the lack of demand for cash interest rates are flat in an equilibrium near 0%.

Prior to this working paper, the economics and behavioural models have not been formulated for when there is more cash than demand.


Contents

  1. Abstract
  2. Introduction
  3. Overview
  4. Behavioural Models
  5. Excess Bank Reserves
  6. References
∞ - includes Behavioural mathematics to graph on a X Y chart


Introduction

Not since the Great Depression has there been more cash than demand. The economics of World War II reignited the demand for cash which remained strong until the Financial Crisis of 2008. The over supply of cash comes from liquidation of loans and investments which shows up as Excess Bank Reserves (EBR), bank reserves above and beyond the reserves required by law. Rather than borrowing money to make investments and purchases, cash sits idle waiting for opportunities slowing down the velocity of money.

Typically, there has always been more demand than supply for cash so the economics and behavioural models have only been formulated for this situation. However, the economic behaviour for when there is more cash than demand is far different and many times opposite for when there is more demand than cash. The behavioural model for the Natural Equilibrium of Interest Rates (NEIR) takes into account whether there is more demand or supply for cash. The model for NEIR is the combination of the equilibrium of interest rates for cash and the equilibrium of EBR. Typically, demand is greater than supply for cash so EBR are flat in an equilibrium near 0 and interest rates are greater than 0%, reflecting the current demand for cash. Rarely, supply is greater than demand for cash so EBR are greater than 0, currently over $2 trillion and interest rates on cash are naturally flat in an equilibrium near 0%, reflecting the lack of demand for cash. When monetary policy is perceived through NEIR in a global economy with excess cash, central bank strategies are much different. In the past, economics have only been formulated for when there is more demand than supply for cash so the velocity of money was high. Many of these economic models were discovered in the 19th century written about isolated domestic economies with limited capacity. These isolated domestic economies were less productive with less technological advancements so demand for products and full employment put upward pressure on inflation. Beginning in the 1980's, the US worker began globally competing on wages to deliver products keeping payroll inflation low while globalization solved the capacity problem. In the 21st century, the old economic models overemphasize the risk of inflation from full employment.

Now, when there is more cash than demand as loans and investments are liquidated, the velocity of money slows down and the risk of inflation decreases. The oversupply of EBR causes the interest rate on cash to naturally trade near 0%. At near 0%, a near infinite amount capital is available to consumers and businesses to help increase demand to maximize the economy. So when there is more cash than demand, central banks should allow the equilibrium of interest rates on cash to naturally be 0%, use quantitative easing to protect against deflation; and once deflation is no longer a problem reduce the central bank's balance sheet with the yield curve anchored by short term interest rates naturally at 0%. This enables the central bank to normalize its balance sheet without causing a taper tantrum. The temporary cheap capital helps stimulate the economy until demand picks up again and interest rates naturally rise as EBR return to equilibrium at 0.

An equilibrium is a stable state of being where the value can be definitely determined. It is a force of nature where the balance and behaviour are determined by supply and demand. The advantage is the equilibrium value can be known real time rather than estimated by a mathematical model. The Natural Equilibrium of Interest Rates can be directly observed in the trading of EBR. The natural equilibrium is the combination of the equilibrium of excess bank reserves and the equilibrium of interest rates. Both equilibria are affected by the supply and demand for cash and only one can be at equilibrium either at 0 or 0%. Since commercial banks are required by law to maintain cash reserves; the amount of EBR can never go below 0, so the graph of the equilibrium of EBR is 0, when demand is greater than supply and then positive when supply is greater than demand. The inverse is true for the equilibrium of interest rates which is positive when demand is greater than supply and 0% when the supply is greater than demand. In short, when demand is greater than supply for cash, EBR are in an equilibrium at 0, interest rates reflect the demand for cash and inflation is a concern. When supply is greater than demand for cash, interest rates are in an equilibrium at 0%, EBR reflect the supply of cash and inflation is not a concern.

Typically, the demand for cash is greater than supply so EBR are in equilibrium at 0, less than $3 billion, and the interest rate paid reflects the demand for cash. The central bank can easily affect this equilibrium by either temporarily adding or removing cash in the "repo" market. Currently, there is over $2 trillion of EBR, this disequilibrium of EBR causes the equilibrium on interest rates to naturally be 0%, as commercial banks compete to lend out their EBR. However, the Federal Reserve pays 1.25% on EBR to keep interest rates on fed funds between 1 - 1.25% at a cost of $25 billion a year.


Overview

The monetary policy to stimulate demand to use up the excess supply of bank reserves has never been formulated. During the Great Depression the demand came from WWII.

Globally, the Federal Open Market Committee has been leading the way for monetary policy. The Board of Governors of the Federal Reserve System have some of the finest economist and mathematicians and Thomas Laubach, Director of Monetary Policy for the Federal Reserve Board has tackled with proficiency the mathematics for determining the "natural rate of interest". His words say it best:

Since Knut Wicksell introduced in 1898 the idea of the natural rate of interest - which we define to be the real short-term interest rate consistent with output equaling its natural rate and constant inflation - it has played a central role in macroeconomic and monetary theory (Wicksell, 1936). Estimates of the natural rate of interest, however, are very imprecise and subject to considerable real-time measurement error. This arises because the natural rate, like other latent variables, must be inferred from the data rather than directly observed.

The imprecision comes from economist using the Symmetric Properties of Equality to estimate the natural rate of interest rather than observing it realtime in the equilibrium of excess bank reserves. For example, though it is true short-term interest rates affect output, that does not necessarily mean through the Symmetric Properties of Equality, output affects short term interest rates - and certainly not in a global economy. Rather the natural rate of interest can be observed in the equilibrium of bank reserves. Now that computers can observe a million times per second, the central banks can create the technological transparency needed to quantify demand for cash in specific areas. When inflation is a concern, this transparency enables the central bank to determine where the inflationary pressures are coming from. This gives the central bank the precision to know where capital is demanded enabling either a micro adjustment through regulation to a specific area or a macro adjustment to interest rates by affecting the equilibrium of excess bank reserves in the "repo" market.

After the global financial crisis in 2008, the central banks are dealing with the excess bank reserves subjectively by creating a narrative rather than objectively dealing with them in their disequilibrium. The central banks chose different strategies to remove the excess supply of bank reserves to give the markets time to recover and generate demand. The Bank of Japan and European Central Bank chose to charge interest on excess bank reserves causing negative interest rates. Both the BOJ and ECB hoped to encourage banks to lend out excess cash and lower the sovereign debt burden. In the US, the FOMC in the name of normalization after the passage of the "Financial Services Regulatory Relief Act of 2006", Sec. 201 "Authorization for the Federal Reserve to Pay Interest on Reserves"; currently pays $20 billion a year in interest to depository institutions. The unintended consequences are it also slows down the US economy and encourages US banks to receive risk free interest from the Federal Reserve rather than lend out the cash to the private sector and buy US treasuries. Currently, by paying 1% on excess bank reserves causes the interest on the $20 trillion of national debt to be higher by $200 billion a year over time. Every 0.25% rise in interest rates represents $50 billion a year in interest on the US national debt.

The mathematics of the Impact of Monetary Policy shows that both a move to negative interest rates or positive interest rates when excess bank reserves are in disequilibrium is deflationary. When the supply of excess bank reserves are greater than demand, the central banks should let the interest rate paid on cash naturally be 0%. While demand is less than supply for cash inflationary pressures are subdued and the cheap capital encourages demand. Once demand picks up again, excess banks reserves will return back to 0 which will cause interest rates to naturally rise.

In a global economy, full employment is no longer inflationary, as there is always someone willing to do the job cheaper but rather it is the demand for cash that affects interest rates and inflationary expectations. The challenge for monetary policy is fiscal policy has been running historic deficits to maintain entitlement programs without the adequate tax revenue to finance them. Monetary policy must find the path for the greatest economic growth with 2% inflation and moderate long-term interest rates. By letting the natural rate of interest be 0% when excess bank reserves are in disequilibrium achieves these goals.


Behavioural Models

Capital Markets are a force of nature and as such have natural equilibriums that can be observed in real time. An equilibrium is a natural calm state and for our purposes is found in the balance of supply and demand for capital. Modeling the behaviour of cash, one-day capital is simple in that it can be graphed in 2D on a X Y chart using two sided limit functions, lim f(x){ to graph the behaviour of the equilibrium.


Equilibrium of Excess Bank Reserves ∞

Around the globe, commercial banks must maintain a minimum amount of bank reserves with their central bank equivalent to a specified percentage of their liabilities. Since each bank by law must have a certain amount of bank reserves, excess bank reserves can never go below 0. If a bank needs reserves, it can borrow from a bank that has excess reserves and by the end of the day the objective is to have no excess reserves due to the opportunity cost of holding cash so all cash is put to work. While demand for cash is greater than supply, excess bank reserves are 0. When supply of cash is greater than demand, when excess bank reserves are greater than 0, the supply is simply positive. Typically, the excess supply is because of the demand destruction during a financial crisis, as was the case with the Great Depression and the Financial Crisis of 2008. The equilibrium of excess bank reserves is an equilibrium of the supply and demand for cash. When excess bank reserves are in an equilibrium at 0, demand is greater than supply and when the supply of excess bank reserves are greater than 0, supply is greater than demand. The model of this behaviour is graphed on the right and the mathematics to graph it are are below:


Equilibrium of Interest Rates for Cash ∞

When supply is greater than demand for excess bank reserves commercial banks compete to lend out their excess cash and cause interest rates on cash to naturally be 0%. Between 1942 - 2008, excess bank reserves have always been in a supply and demand equilibrium at 0 so the interest rate paid for cash reflected the demand for cash and inflationary expectations. The model of this behaviour is graphed on the right and the mathematics to graph it are are below:




The miracle of capitalism can be witnessed in the trading of excess bank reserves in the US federal funds market. Since commercial banks are required to maintain a minimum amount of reserves by law, reserves are traded between banks that need reserves and banks that have extra reserves. Up until 2008, excess bank reserves were in supply and demand equilibrium of 0 and the interest rate paid reflected the demand for cash and inflationary expectations. When excess bank reserves are in equilibrium the Federal Reserve is most effective in changing interest rates by affecting the equilibrium in the federal funds market with minimal costs. After the financial crisis in 2008, for the first time since the Great Depression, banks were forced to compete to lend out the oversupply of excess bank reserves causing federal funds to naturally trade at 0%.


Natural Equilibrium of Interest Rates ∞

The Natural Equilibrium of Interest Rates can be directly observed in the federal funds market, as the interest rate paid for excess bank reserves. Within this equilibrium of supply and demand, there are two outcomes which depend on the supply of excess bank reserves which are either 0 or greater than 0. Typically, excess bank reserves are 0, when the demand for cash is greater than supply and the interest rate paid reflects the demand for cash and inflation expectations. Currently and in the Great Depression, excess bank reserves are greater than 0, when the supply of cash is greater than demand and the interest rate paid is 0%, as commercial banks compete to lend out their excess reserves.


The mathematics for the Natural Equilibrium of Interest Rates is the combination of the limit function for interest rates and excess bank reserves. The limit functions are elegant in their symmetry and simplicity. However, I would argue the limit function for interest rates when the supply of excess bank reserves is less than demand, when x < 0 is not linear off of 0 but rather curved, possibly exponentially where f(x) is not the absolute value of x but rather x2 which takes into account the ever increasing pressures on interest rates as demand outstrips supply. But in the interest of elegance and symmetry we will keep the function linear to balance against a supply curve that is most certainly linear. That being said, let's go through the limit functions for the Natural Equilibrium of Interest Rates and prove their validity.

In the limit function for Interest Rates, x is the demand vs. supply of excess bank reserves and y is interest rates. The top half of the Interest Rates limit function, where supply is greater than demand for excess bank reserves is 0 because banks compete to lend out their excess reserves driving interest rates to 0% and banks would never pay interest to lend cash so interest rates would never go negative. This has been the experience since 2008, until the central banks artificially manipulated the interest rate on excess bank reserves where the Fed paid interest on reserves to raise rates and the BOJ and ECB charged interest on reserves to drive rates negative. In otherwords, without central bank manipulation interest rates on excess bank reserves would naturally be 0%. The bottom half of the Interest Rates limit function, experienced between 1941 and 2008, where demand is greater supply, so when x is negative f(x) multiplied by -x results in a positive y, where y is positive interest rates. Between 1941 and 2008, interest rates have always been positive when demand was greater than supply and excess bank reserves were 0.

The limit function for excess bank reserves is straightforward in that the outcome is a function of bank regulations where x represents demand vs. supply and y represents excess bank reserves. The top half of the limit function for excess bank reserves is when supply is greater than demand so when x > 0   f(x) = x, the quantity of excess supply of bank reserves. The bottom half of the function is by law kept at 0 because when demand is greater than supply, banks must borrow reserves to meet their minimum requirements causing excess bank reserves to never go negative. In fact, when you look at the actual graph of excess bank reserves since 1941, they have been at 0 until 2008.
Excess Bank Reserves

Commercial banks must maintain a minimum amount of bank reserves with their central bank equivalent to a specified percentage of their liabilities. Since each bank by law must have a certain amount of bank reserves, excess bank reserves can never go below 0. Between WWII and 2008, excess bank reserves were in equilibrium at 0, below $3 billion in the US, except during 9/11 when they reached $38 billion. Starting in September 2008, for the first time since the Great Depression, the supply of excess bank reserves were greater than demand. This disequilibrium is caused by capital moving from investments and loans into cash. By August 2014, excess bank reserves rose as high as $2.7 trillion in the US before leveling off as the US economy began to recover. The Federal Reserve Bank of St. Louis maintains transparent data as shown to the right.



Source: Board of Governors of the Federal Reserve System


Demand and Supply for Cash

Specifically, the demand and supply for cash affects the behaviour of excess bank reserves and interest rates. Generally, the equilibrium for cash affects the behaviour of micro and macro economics. This equilibrium is ruled by opportunity cost and as such is the binding force for the economy. Supply and demand is fundamental to the laws of economics and the equilibrium for cash is its foundation. The economy typically operates with excess demand for cash, except for those times after a financial crisis, where the economy suffers from demand destruction and there is an excess supply of cash.

Cash is a store of purchasing power. The behaviour of cash is ruled by the opportunity costs of being in cash, purchasing a product, making an investment or making a loan. These opportunity costs determine the equilibrium of supply and demand for cash. Typically, there is more demand for cash than supply, as all available capital is making purchases, making investments and making loans so the interest rate paid for cash reflects the demand for cash. However, when supply is greater than demand for cash, caused by lack of demand, causing the reduction in purchases, investments and loans, the interest rate paid for cash is 0%, as banks compete to lend out their excess cash.

There are many factors that affect the demand for cash beyond interest rates and quantitative monetary policy. There are the velocity of money, the elasticity of demand, regulations and taxes to name a few. But what is evident, is by observing the demand and supply for cash we can gain an insight into how much pricing pressure there is in the economy.


Inflation and Deflation

Inflation and deflation are ruled by the opportunity cost of purchasing now or later. The purchase price is ruled by the supply and demand for the product.

Prior to 1980, supply was a challenge as isolated domestic economies struggled to meet demand driving the price of products higher causing inflation. Currently in a global economy, the supply of a product is no longer a challenge, as nations compete for good jobs and technology adds to productivity. Rapid increases in technology will continue to put downward pressure on prices as productivity increases. This increase in productivity should be encouraged as workers finally become so productive they can be paid a good wage and the increased productivity keeps the cost of goods down. In sum, the supply of a product is abundant in a global economy.

In a global economy, it is demand that must meet supply. This demand can be directly observed in the supply of excess bank reserves. Typically, excess bank reserves are 0, so demand is greater than supply and this places upward pressure on inflation, as the pressure of opportunity cost is to purchase now. However, when supply is greater than demand and excess bank reserves are in a disequilibrium greater than 0 this places downward pressure on inflation, as the pressure of opportunity cost is to delay purchases.

In the decade that led to the Financial Crisis of 2008, much of the demand came from the purchasing power of wealth in real estate financed with Collateralized Debt Obligations (CDOs). Lack of regulation and oversight allowed "bad actors" to game the system and create an asset bubble.


Demand greater than Supply causes Inflation

Usually excess bank reserves are in equilibrium at 0 which means there is no extra cash. The demand for cash puts upward pressure on interest rates as consumers compete to make purchases, investments and loans. Prior to 1980, isolated domestic economies struggled to meet demand which drove inflation higher and created boom and bust cycles for the economy. After 1980, when globalization began to take hold, inflationary pressures lessened as global economies competed to meet demand.

The interest rate paid for cash reflects the demand for cash and inflationary expectations. This natural equilibrium keeps the economy in balance because as demand increases the interest rate to borrow cash increases hindering demand. Further demand destruction comes from monetary policy tightening, more stringent regulations and higher taxes.


Supply greater than Deflation causes Deflation

During those rare times when the supply of cash is greater than demand, due to demand destruction after a financial crisis. Monetary policy must allow the interest rates on cash to trade naturally at 0% and use quantitative easing to protect against further demand destruction. Restrictive regulations need to be removed and tax policy needs to be less burdensome.
References

  1. Kathryn Holston, Thomas Laubach, John C. Williams, December 15, 2016, "Measuring the Natural Rate of Interest: International Trends and Determinants" Federal Reserve Bank of San Francisco Working Paper 2016-11. Introduction

  2. Thomas Laubach and John Williams, November 2003, "Measuring the Natural Rate of Interest" Review of Economics and Statistics, Vol. 85, No. 4 , Pages: 1063-1070 (doi: 10.1162/003465303772815934) Abstract

  3. Ibid. Kathryn Holston, Thomas Laubach, John C. Williams, December 15, 2016

  4. "Inflation, consumer prices for the United States", FRED Economic Data, Federal Reserve Bank of St. Louis