1 Literature Review
1.1 Introduction
On the 29th January 2016, the Bank of Japan (BOJ) decided to implement the Negative Interest Rate (NIR) policy, which is to apply an interest rate of −0.1% to the excess balance of accounts that financial institutions hold at BOJ (Bank of Japan (2016)).
It remains ambiguous from a macroeconomic perspective, on whether the NIR could boost the economy. Keynes (1937)(pp.165-174, 194-209) and Hicks (1937) inducted that, a rate cut from a positive nominal interest rate to a less positive nominal interest rate can boost consumption and investment. The topic question arises — does the rate cut from a positive to a negative nominal interest rate share the same macroeconomic result as if the rate cut was within positive nominal interest rates? i.e. Does the NIR boost consumption and investment?
This section firstly discusses the motivating macroeconomic issues centring on Zero Lower Bound (ZLB), liquidity trap, and NIR, then reviews the existing microeconomic models for commercial banks and their credit issuing decisions, that can be used by Section 2 to address the said question.
This subsection firstly glimpses at the history of monetary policies in Japan, which leads to the discussion of ZLB and liquidity trap, followed by some theoretical attempts regarding NIR.
Figure 1: Plot of the short-term interest rate (uncollateralised overnight call rate, monthly average) in Japan from January 1986 to October 2017. Data Source: Bank of Japan (2017a) and Horiuchi & Otaki (2017)(pp.33,62-63).
Figure 1 plots the interest rate trend in Japan in the past three decades. In 1991, the bubble economy collapsed, and economic stagnation set in for an entire decade despite dramatic rate cuts (Iyoda (2010), pp.74-93). Svensson (2006) summarised the monetary policies in that decade as a combination of low interest rates, zero interest rates, and a quantitative easing. Despite a short recovery in the mid-2000s, the interest rate was cut back to near-zero again after the 2007-8 financial crisis (Rossi & Malavasi (2015), pp.3-17).
Svensson (2006)(pp.1,3,8) described that, Japan faced a ZLB in the late-1990s, which prevented BOJ from setting its interest rate at its optimal level. The concept of ZLB can be illustrated by using a basic inequality
1.2 ZLB, Liquidity Trap, and NIR
i≥0
where i% denotes the nominal interest rate set by the central bank.
(1.2.1)
[If the interest rate is lower than a certain level] … everyone prefers cash to holding a debt which yields so low interest rate. In this event the monetary authority would have lost effective control over the interest rate. (p.207)
The settings that Keynes (1937)(pp.199-209) used to claim the above statement are:
M = L(i, Y ) where M is money, Y is income.
(1.2.2)
(1.2.3)
Then, Hicks (1937) plotted the relationship between i and Y based on (1.2.3), which brings up the ZLB. A modern approach to describe the similar concept of liquidity trap and ZLB was provided by Schmitt-Groh ́e & Uribe (2009)(pp.89-92) as below:
(1.2.4)
∂u(c,M)
lim ∂M = 0 ∀c > 0
Although different literature may impose different mathematical assumptions on ZLB, most of their roots are originated from (1.2.1), which cannot describe the post-2016 Japanese monetary phenomenon that i = −0.1 < 0. This motivates the author to revise the classification of nominal interest rates, as shown in Subsection 2.1, in order to address issues on NIR.
Looking back to the 1930s, the settings on ZLB was derived based on the concept of liquidity trap which Keynes (1937) described as follows:
M→∞ ∂u(c,M) ∂c
where c is consumption,
u is the household utility function.
(1.2.4) quantified (1.2.2) by using the concept of marginal utility — as M becomes large enough, the marginal increase in utility for having one more unit of money is negligible compared to the marginal increase in utility for a unit increase in consumption. Schmitt-Groh ́e & Uribe (2009)(p.89) then claimed that (1.2.4) leads to “money demand approaches infinity as the nominal interest rate vanishes”. This corresponds to Keynes (1937), and implying (1.2.1), or perhaps i > 0 as infinity is not observed in reality.
As liquidity trap induces ZLB, and this is supported by major literature, e.g. Benhabib et al. (2001), Woodford (2003), and their proceedings; it is unreasonable to remove (1.2.1) in the macroeconomic models of this paper. So, this paper complies with the concept of (1.2.2) by inheriting the functional form of (1.2.3), whilst admitting the fact that “slightly negative interest rates”, e.g. i = −0.1 or i = −1.25 1 being accepted.
Coming back to one simple motivating question — why should the policy makers care about NIR?
Goodfriend (2002) reviewed the real business cycle (RBC) model, then Goodfriend (2016) used the result to derive
1 The central bank of Sweden has been implementing a −1.25% deposit rate since February 2016 (Sveriges Riksbank (2017)). 3
the followings:
rN =η1 +η2 +η3 −η4
where rN is the natural real interest rate,
η1 captures time preference,
η2 is the expected productivity growth,
η3 captures distortions adversely impacted by current income,
η4 captures distortions adversely impacted by expected future income.
(1.2.5)
So, if productivity is expected to decline, and future adverse factors are expected to incline, then η2 would decrease and η4 would increase. These lead to a decrease in rN . Despite the Fisher Equation allows some negative real interest rates as shown below (Fisher (1896), pp.348-374; and Carlin & Soskice (2015), pp.36-37):
r = i − E[π]
where r is the real interest rate, (1.2.6)
E[π] is the expected inflation.
The fact that E[π] can only be in a certain range plus i being restricted to ZLB would make r being restricted to a certain lower bound. In particular, the policy-induced real interest rate could be higher than the natural rate due to ZLB, i.e. r > rN , which leads to a deficiency of aggregate demand relative to potential output, as mentioned by Goodfriend (2016). This is consistent with the Japanese economic performance in the late 1990s and the late 2000s, where nominal interest rate hits ZLB whilst output deviates negatively from its potential (see Figure 1, and Iyoda (2010), pp.89-91).
Hence, if the nominal interest rate is allowed to be negative, then by (1.2.5) and Goodfriend (2016), the economic depression can be solved by a monetary policy which sets r = rN disregarding the ZLB, and Japan would be able to restore its economic potential by a sufficiently low NIR. Having this motivation, and to address the topic question, the author reviewed a macroeconomic paper by Schmitt-Groh ́e & Uribe (2009) which touched on NIR.
Schmitt-Groh ́e & Uribe (2009) inherited Benhabib et al. (2001) and Woodford (2003) on solving a dynamic macroeconomic model with Taylor Rule2. However, there were two ambiguous assumptions that Schmitt-Groh ́e & Uribe (2009) imposed:
Taylor Rule holds for all possible rates of inflation. (pp.92-93) (1.2.7)
For the household-side utility maximisation, the household-side budget constraint is the same function for any nominal interest rate. (pp.89-90, 95-97)
(1.2.8)
(1.2.7) means that central bank can set any NIR, which certainly conflicts with ZLB and liquidity trap, thus self-conflicts with its own settings for ZLB, e.g.(1.2.4). Also, for the case that central bank set i < 0, (1.2.8) effectively
2 Originated from Taylor (1993).
…assumes that households face a budget constraint with the exact i < 0, which is inconsistent with the reality in Japan. This is because, Japanese commercial banks did not impose NIR on their customers despite central bank implemented NIR to the commercial banks (Nakano (2016), pp.98-101; and Figure 2). Thus, individuals do not face a budget constraint with the negative interest rate that central bank imposes on commercial banks, i.e. the policy-induced interest rate is no longer a good proxy for the interest rate that households face in their budget constraint, so (1.2.8) is inconsistent with the empirical fact, especially when the nominal interest rate is negative.
Therefore, due to the self-conflicting (1.2.7) and inconsistent (1.2.8), policy makers shall not be convinced by Schmitt-Groh ́e & Uribe (2009)’s result regarding NIR.
Figure 2: After the implementation of NIR, various interest rates that commercial banks offer to customers fell, but still being positive, despite i = −0.1 and the short-term interest rate (uncollateralised overnight call rate) fell below zero. Data Source: Bank of Japan (2017a).
The lesson from Schmitt-Groh ́e & Uribe (2009) warns this paper not to conduct the macroeconomic analysis directly without any support from the microeconomic mechanism on NIR. In particular, because of the rationale behind liquidity trap, this paper should not simply delete (1.2.1) from the settings. Additionally, the fact that commercial banks did not set negative interest rates to households and firms (Figure 2) implies that, a closer look at commercial banks and credit mechanism is necessary, to address the topic question.
1.3 Bank and Credit
Motivated by Subsection 1.2, this subsection reviews papers regarding commercial banks and their credit issuing decisions.
Keynes (1937)(p.63) drew a one-to-one relationship between savings and investment from a macroeconomic perspective. However, it is up to banks which would decide whether the savings are being lent to households and firms (Carlin & Soskice (2015), pp.162-176).
Figure 3 plots the Loans-over-Deposits-ratio3 in Japan, which shows a decreasing trend in the past two decades, and the NIR did not seem to have a positive impact on it.
Figure 3: Plot of the Loans-over-Deposits-ratio over all domestically licensed banks in Japan (exclude Japan Post Bank Co.) from October 1993 to September 2017. Data Source: Bank of Japan (2017a) and Bank of Japan (2017b).
Poole (1968) analysed the Loans-Deposits relationship by using a short-term model to explain the optimal cash holding for a commercial bank, which depends on multiple variables including the interest rate offered by the central bank4, discount rate, subjective probability distribution on net deposit accretion, and etc.. However, if interest rates were allowed to be negative, then the model needs to be redefined.
Goodfriend & McCallum (2007) drew an outline for the commercial bank’s balance sheet, then found solutions in the macroeconomic model set by Woodford (2003)(pp.299-311). However, ZLB (1.2.1) was assumed, thus the result did not address NIR. This paper takes the initiative of Goodfriend & McCallum (2007) and considers a view from the
3 Loans-over-Deposits-ratio is L where L is the aggregated Loans that banks lent, and D is the aggregated Deposits that banks held. D
4 Federal Funds Rate in the literature.
balance sheet.
Adrian & Shin (2010)(pp.8-12) modelled a balance sheet of a leveraged investor (Figure 4a), then solved the optimal holding of securities. This paper makes some adjustments towards Adrian & Shin (2010) as shown in Figure 4b. The purpose of such adjustments is to reframe the balance sheet from a leveraged investor to a more “defensive”5 Japanese commercial bank.
(a) Investor’s Balance Sheet as per Adrian & Shin (2010)
(b) Commercial Bank’s Balance Sheet for this paper
Figure 4: The modification on Balance Sheet that this paper makes on Adrian & Shin (2010)
Stiglitz & Weiss (1981)(pp.393-398,401-402) made an important microeconomic conjecture on credit rationing. His view of commercial bank — the bank shall maximise its expected profit, is taken by this paper. Additionally, this paper adjusts the profit function as plotted in Figure 5. The purpose is not only to simplify the calculation, but also to make the generalisation of borrowing-lending market feasible, as the probability distribution of ρ would be binary after the adjustment.6
5 As Nakano (2016)(p.177) described.
6 See Figure 5 for notation meanings. See the remark part of Example 5 for an explanation on probability distribution.
Figure 5: The modification on net return to bank that this paper makes on Stiglitz & Weiss (1981)
There is, however, a lack of justification on an assumption by Stiglitz & Weiss (1981). That is, the non-linear relationship between the supply of loan and the bank’s profits. Adrian & Shin (2013) used Value-at-Risk (VAR) model to explain the non-linear relationship between bank’s asset and lending7. But the VAR model was derived based on the balance sheet of a leveraged investor from Adrian & Shin (2010) (Figure 4a), so it would be inconsistent to adopt the VAR model to a commercial bank, where balance sheet structure largely differs from a leveraged investor. Therefore, this paper takes the idea from Adrian & Shin (2013) and use the microeconomic concept of utility theory8 to analyse the non-linear relationship between bank’s asset and lending.
In summary, the review of existing papers on Bank and Credit brings useful microeconomic ideas to the model building in Section 2.
7 Traded securities in the literature.
8 Pantsulaia (2016), Kubrusly (2015), Cohn (2013), Hogg et al. (2012), Barbera et al. (2004a), Barbera et al. (2004b), Pitman (1999), and Mas-Colell et al. (1995)(pp.5-39, 167-205) are useful foundations to this paper’s settings on utility functions and their maximisations.
2 Project Outline
2.1 Aim and Setting
The aim of this paper is to assess whether NIR boosts consumption and investment in Japan, by considering the commercial bank’s credit issuing decisions. Subsection 2.2 builds up a microeconomic model on bank and credit. Furthermore, Subsection 2.3 applies the microeconomic result to the macroeconomic context thus addresses the topic question.
Firstly, as the central bank is no longer restricted by the ZLB that characterised by (1.2.1), this paper re-classifies the central bank interest rate (i%) into the following four intervals9:
Interest Rate Interval (−∞, iE ) [iE , 0) [0, iP ] (iP , ∞)
Name Impossible NIR Effective NIR Near zero positive interest rate Strictly positive interest rate
Someone may ask, what are actually the iE and iP ? Empirical observations in the last two decades in Japan (Figure 1) has shown that 0.1% is almost a zero interest rate — the nominal interest rate stabilised around 0.1% due to the ZLB, thus iP ≈ 0.1. Also, the fact that central bank of Sweden has been implementing a −1.25% interest rate means that iE < −1.25 (Sveriges Riksbank (2017)). However, due to insufficient literature identifying what is the actual boundary between Impossible NIR and Effective NIR, this paper does not assign value to iE. For simplicity and clarity, this paper sets the timing as a two-period economy, with periods t = 1 and t = 2. This paper also supposes the rate cut happens at t = 1, which cuts interest rate from i(0) ∈ [0, iP ] to i(1) ∈ [iE , 0), and interest rate does not change thereafter.
2.2 Assumptions, Definitions, and the Bank-Credit Model
The aim of this subsection is to show that, NIR damages bank profits,10 which further leads the bank to a dilemma between choosing to lend to riskier borrowers or to retain cash.11 Eventually, the bank may have to decrease the total lendings.
Assumption 1 There are only one central bank and one commercial bank in the economy. The commercial bank (hereafter called “the bank”) has the balance sheet as shown in Figure 4b.
The above assumption is set not only for mathematical simplicity, but also based on the analysis done by Masi et al. (2011), which suggested that many Japanese firms have connections with only one or two banks, so assuming one commercial bank would partially model the actual scenario. As a result, we can have an identity based on the balance sheet as follows:
A0 +A1 +A2 +A3 =L0 +L1 +L2 (2.2.1) Definition 1 A differentiable function ic : [iE , +∞) → [0, +∞) is said to be a customer-side interest rate function if it satisfies
the following conditions:
ic(x) < x whenever x ≥ iP (2.2.2)
i′c(x) ≥ 0 ∀x ∈ [iE,+∞) (2.2.3)
The above definition models the behaviour of the bank on setting interest rates to its customers. This is to make the model consistent with the empirical observations on the interest rates that Japanese commercial banks offer to the customers who make deposits (Nakano (2016), pp.98-101; and Figure 2). An example of the above function is illustrated in Example 4. The following definition sets the notations for analytic purposes, and relates the interest rates to the bank’s balance sheet by modelling interest payments to its customers and interest receipts from (or payments to) the central bank.
Definition 2 a) ∀X ∈ {L0, L1, L2, A0, A1, A2, A3}, we write the value of X in period 1 before the rate cut as X(0) and after
the rate cut as X(1). We write the value of X in period 2 as X(2).
b) Ceteris paribus, given (x, y, z) where (x, y, z) ∈ {(A(0), L(0), i(0)), (A(1), L(1), i(1))}, then A(2) = x − 0.01 i (z) y. 00000c
c) Ceteris paribus, given (x, y) where (x, y) ∈ {(A(0), i(0)), (A(1), i(1))}, then A(2) = (1 + 0.01y)x. 111
d) Given L0,L1,L2, a strategy S is a decision to allocate the Assets-side of the balance sheet, subject to (2.2.1).
The following assumption marks the foundation of this paper’s analysis. Notice that, Appendix A may be useful to refer to, if
the mathematical expressions stated below and further are not clear.
Assumption 2 The bank chooses the strategy which maximises its expected utility. That is, mathematically:
given the bank’s utility function U, given a finite set of strategies {S1,…,Sn} with their associated Probability Functions {P1, …, Pn} and random variables {X1, …Xn}. The bank chooses the strategy S∗ where
S∗ = argmax EXj [U(Sj)] (2.2.4) j ∈{1,…,n}
9 This is motivated by Ness ́en (2016).
10 This is suggested empirically by Nakano (2016) (pp.98-101, 137, 178), and Barwell (2016)(pp.43-46).
11 The mechanism that central bank lowers interest rate leads to commercial banks lending to riskier borrowers has been tested to be
true, econometrically by Jim ́enez et al. (2008). Also see the remark part of Example 5 for an explanation on the dilemma. 9
(0) {P1 , …, Pn }, random variables {X1
(0) (0) (1)
23012
Assumption4∃λ,λ ∈(0,1)suchthat∀t,A(t)≥λ(L(t)+L(t)+L(t))and A(t)≥λ(L(t)+L(t)+L(t)) 01 00012 11012
Assumption 5 For a given individual in A2 or A3 market, the probability for that individual being able to pay back is independent of the time. L0 is exogenous over time. L1 and L2 are only dependent on the strategy.
Assumptions 3 and 4 are set to make the model more consistent with the reality. 3 means there is a threshold that the bank can lend at maximum, subject to its total liabilities. Similarly, 4 means that the bank needs some cash for daily operations.13Assumption 5 is set to reduce the potential modelling difficulty as if those variables are endogenous over time
or affected by uncontrollable factors, the magnitude of change would be ambiguous. Now, the main theorems are as follows: Theorem 1 For any utility function U, any finite set of strategies {S1,…,Sn} with their associated probability functions
, …, Xn }, and the chosen strategy S . After the rate cut from i (1)
to i , the strategies {S1,…,Sn} correspond to {X1 ,…,Xn }; then the new chosen strategy S has the property card(ΩS) ≥ card(ΩS ). 14
Theorem 2 With the same given definitions as Theorem 1, there exists cases where card(ΩS) > card(ΩS ).
Example 5 shows a case where the Theorem 1 is satisfied, with economic interpretations. To rigorously see the economic meaning of the above theorems, a lemma is needed to support.
Originally published 15.10.2019