The e-monetary theory

Abstract The author develops a dynamic model with two types of electronic money: reserves for transactions between bankers and zero-maturity deposits for transactions in the non-bank private sector. Using this model, he assesses the efficacy of unconventional monetary policy since the Great Recession. After quantitative easing, keeping the interest on reserves near zero too long might create deflation. The central bank can safely get out of the “low rate-cum-deflation” trap by “raising rate and raising money supply”.


Introduction
Nowadays, money mostly exists in the electronic form. According to data from the Federal Reserve Bank of St. Louis, the total stock of M1 in Dec 2019 is around USD 4000 billion, consisting of USD 1700 billion in currency and USD 2250 billion in checkable deposits. However, as the world currency, most US dollar bills are held outside US. Judson (2012) estimates that 60 percent of US dollar bills are in foreign countries. If we exclude that number from M1 and M2, currency only accounts for 17 percent of M1, 4.4 percent of M2 and 4.0 percent of MZM. 1 In this paper, we focus on a popular group of e-money issued by commercial banks, including checkable deposits, saving deposits and money market deposit accounts. Together they account for 80 percent of M2. For convenience, we call this group as zero-maturity deposits (ZMDs) thereafter.
ZMDs are different from currency in two salient features. First, in nature, currency is an IOU issued by the central bank while ZMDs are IOUs issued by commercial banks. In the language of economics, currency is outside money while ZMDs are inside money. Second, in the households' perspective, unlike currency, ZMDs can earn nominal interest. Banks pay interest for saving accounts and money market deposit accounts for a long time, but the tricky parts are checking accounts. In a perfectly competitive banking market, the interest rate on checkable deposits should be positive and follow the federal funds rate. 2 However, before 2012, under the Regulation Q, banks in US were prohibited from paying interest on checking accounts. During this period, banks still implicitly paid the demand deposit rate under the form of NOW (negotiable order of withdrawal) accounts, giving gifts or reducing the cost of additional services to their customers (Mitchell, 1979;Startz, 1979;Dotsey, 1983). Becker (1975) estimates that the implicit demand deposit rate in US during 1960-1968 was around 2.64 percent to 3.74 percent.
Since 2012, the Regulation Q has been no longer valid, and most banks are now paying interest rate on checkable deposits. Data in September 2016 of Federal Deposit Insurance Corporation (FDIC) show that the national average interest on checkable account is 0.04 percent, on saving account is 0.06 percent. These rates are low as the federal funds rate is near zero. If the federal funds rate is around 4 percent, these rates range likely from 1 percent to 2 percent. As a result of that, in the era of electronic money, it is more natural to model money as an interest-earning asset that provides liquidity service.
The main contribution of this paper is to build a dynamic general equilibrium model where there is only electronic money (currency does not exist-a cashless model). There are two forms of money in our model: ZMDs and reserves. ZMDs are inside money issued by commercial banks. They are used for settling transactions between every pair of agents in the private sector, except between bankers-bankers. In these interbank transactions, bankers have to use reservesanother type of e-money issued by the central bank. The amount of ZMDs that banks can issue is restricted by two constraints: the reserve requirement and the capital requirement. In our model, the central bank only controls the level of reserves while the level of the money supply (amount of ZMDs) depends on the interaction between the central bank, the commercial banks and the public (Mishkin, 2019, ch. 15). To the best of our knowledge, this is the first model investigating the dynamic interactions between interest on reserves (IOR), reserves, deposits, and output where payment is conducted electronically. We use our model to discuss unconventional monetary policy during and after the Great Recession. Here are some key results: i. After a shock making banks' capital constraint binding, an interest rate policy following a Taylor rule is not sufficient to recover the economy quickly. Both output and inflation are lower than their steady state levels for a long time.
ii. A central bank's large scale asset purchase (LSAP) program, with the aim of directly injecting money into the economy, is very efficient at dealing with the above situation in the short-run. Inflation will go up immediately after this program. The byproduct of LSAP is a huge amount of excess reserves in the banking system (Keister and McAndrews, 2009;Sheard, 2013); the reserve requirement is no longer binding; and interest on reserves (IOR) becomes the main tool to control the federal funds rate.
iii. After LSAP, the longer the federal funds rate is committed at the lower bound, the higher is inflation in the short run. As loans have the longer maturity than deposits, commitment to keep the short-term rate near zero for a long time pushes down the loan rate stronger and pushes up the inflationary expectation. However, in the long run, it might create a persistent deflation due to the Fisher Effect. The real short-term rate will slowly climb back to the long-run level. The endogenous money supply declines, and deflation realizes. This matches with the US data since the Great Recession ( Figure 1).
iv. It is not easy to safely escape from the "low rate-cum-deflation" trap. If the central bank raises rates (by raising IOR), the amount of banks' credits declines. The economy will suffer a short recession. Deflation is even more severe in the short run. Still, after a period of time, inflation will move back to the central bank's target in the long run. Therefore, the central bank falls into a dilemma between to raise or not to raise rates. Either way the outcome is not bright.
v. When raising IOR, if the central bank simultaneously commits to target the growth rate of the money supply in response to inflation, the inflation and output path will be stabilized. With the new tool IOR, the central bank somehow can manipulate both interest rate and money supply at the same time. These tools should be utilized simultaneously so that the central bank can hit the inflation target better.

Related Literature
Our model is still in the general New Keynesian framework with the crucial sticky price feature. The important role of financial frictions in the New Keynesian has been emphasized for a long time (Bernanke et al., 1999;Christiano et al., 2004). Recently, many New Keynesian studies (Gertler and Kiyotaki, 2010;Curdia and Woodford, 2011;Gertler and Karadi, 2011) incorporate the banking sector to their models, aiming to answer what happened in the Great Recession and the role of the unconventional monetary policy. Several studies (Sargent and Wallace, 1985;Goodfriend, 2002;Ireland, 2014;Cochrane, 2014;Keister, 2016) discuss interest on reserves as an important unconventional monetary policy tool. Our paper differs mainly from this line of research in the existence of reserves, banking and money supply. We can characterize the micro-foundation link between bank reserves, banks' balance sheets, money supply, interest rate and output. We emphasize the importance of both money supply and interest rate in monetary policy when the central bank adjusts the interbank rate by IOR. On the money supply side, our approach is similar to Bianchi and Bigio (2014) and Afonso and Lagos (2015) when the central bank can increase the level of the money supply and cut down the federal funds rate by injecting more reserves into the banking system. These papers explicitly model the search and matching process of heterogeneous agents in the interbank market while our model is frictionless with identical bankers. Our model can connect the central bank policy to not only banks' balance sheet but also the production sector, which is missing in both Bianchi and Bigio (2014) and Afonso and Lagos (2015). The study of Brunnermeier and Sannikov (2016) also emphasizes the importance of inside money issued by banks. However, they emphasize the money function as a store of value in a risky environment while our paper focus on the common role of money -medium of exchange -in a deterministic setting.
On the money demand side, our model follows the cash-in-advance approach in Lucas and Stokey (1987). As our model does not include currency, the most liquid asset here should be ZMDs. In Ireland (2006, 2014), currency and deposits are bundled together and provide the liquidity service to households. We also extend the Clower constraint to investment (Stockman, 1981;Abel, 1985;Fuerst, 1992;Wang and Wen, 2006). Indeed, most empirical research, for example Friedman (1959) and Mulligan and Sala-i-Martin (1997), usually uses income, rather than consumption alone, to estimate the money demand function.

Notation
Let P t be the price of the final good. We use lowercase letters to represent the real balance of a variable or its relative price to the price of the final good. For example, the real reserves balance n t = N t /P t , or the relative price of the intermediate good to the final good is p m t = P m t /P t . The timing notation follows this rule: if a variable is determined or chosen at time t, it will have the subscript t. The gross inflation rate is π t = P t /P t−1 .

Time, Demographics and Preferences
Time is discrete, indexed by t and continues forever. The model is in the deterministic setting and has five types of agents: bankers, households, wholesale firms, retail firms, and the consolidated government.
There is a measure one of identical infinitely lived bankers in the economy. Bankers discount the future with the discount factor β . Each period, they gain utility from consuming the final consumption good c t . Their utility can be written as: There is also a measure one of identical infinitely lived households. Households discount the future with the discount factor β <β , so they will borrow from bankers in the steady state. 3 Each period, households gain utility from consuming the final consumption good c t and lose utility when providing labor l t to their own production. Household's utility can be written as: where ν is the inverse Frisch elasticity of labor supply. Wholesale firms and retail firms are owned by households. The consolidated government includes both the government and the central bank, so for convenience, we assume there is no independence between the government and the central bank.

Goods and Production Technology
There are three types of goods in the economy: final good y t produced by retailers, wholesale goods y t ( j) produced by wholesale firm j and intermediate good y m t produced by households. The final good y t could be used for consumption and investment.
Each period households self-employ their labor l t and use the capital stock k t−1 to produce the Banker Banker Loans: +S t Deposits: + q L t S t Loans: where α is the share of capital in the production function. Capital k depreciates with the rate δ k . Households also own a technology to convert one unit of final good y t to one unit of capital type k and vice versa. So each period they also make an investment i t = k t − δ k k t−1 . Households sell y m t to wholesale firms in the competitive market with price P m t . There is a continuum of wholesale firms indexed by j ∈ [0, 1]. Each wholesale firm purchases the homogeneous intermediate good y m t from households and differentiates it into a distinctive wholesale goods y t ( j) under the following technology: Then retail firms produce the final good y t by aggregating a variety of differentiated wholesale goods y t ( j):

Assets
There are three main types of financial assets (excluding reserves and deposits): bank loans B h t , share of wholesale firms x t and interbank loans B f t .
(a) Bank loans to households (B h t ) : We follow Leland and Toft (1996) and Bianchi and Bigio (2014) to model the loan structure between bankers and households. The market for bank loan is perfectly competitive and the price of loan is q L t . When a household wants to borrow 1 dollar at time t, bankers will create an account for her and deposit q L t dollars to her account. In the exchange for that, this household promises to pay .. dollars at time t + 1, t + 2,... t + n, t + n + 1... where n runs to infinity (Table 1). Loans are illiquid and bankers cannot sell loans. Let B h t be the nominal balance of loan stock in the period t, let S t be the nominal flow of new loan issuance, we have:  shares each period. Each share has a price υ t and pays a real dividend w t . The total number of wholesale firms' shares is 1.
In the LSAP, the central bank might purchase these shares to inject money into the market.
(c) Interbank loan (B f t ): Bankers can borrow reserves from other bankers in the federal funds market. The nominal gross interest rate in the federal funds market is the federal funds rate R f t .

Money
There are two types of electronic money in our economy: reserves n t and zero-maturity deposits m t .
(a) Reserve (n t ): is a type of e-money issued by the central bank. Only bankers have an account at the central bank, so only bankers have reserves. Each period, the central bank pays a gross interest rate R n t on these reserves. The rate R n t is decided solely by the central bank. Reserves are used for settling the transactions between bankers and bankers, bankers and central bank, bankers and government.
(b) Zero maturity deposit (m t ): is a type of e-money issued by bankers. Each period, banks pay the interest rate R m t for these ZMDs which is determined by the perfectly competitive market. Money m t is used for settling transactions in the non-bank private sector and the ones between households and bankers. These ZMDs are insured by the central bank, so they are totally safe. ZMDs and reserves have the same unit of account.
In the electronic payment system, there is a connection between the flows of reserves and deposits. For example, we assume that wholesale firm A (whose account at bank A) pays 1 dollar for household B (whose account at bank B). Then the flow of payment will follow Table 2.
For a transaction between the consolidated government and households, money still flows through banks, so we can think that this contains two sub-transactions. One is between the government and banks, which is settled by reserves. One is between banks and households, which is settled by ZMDs.
Under conventional monetary policy, the consolidated government targets the interbank rate by helicopter money or lump-sump tax on households. Each period, the central bank sends τ t dollars in checks to households. It can be thought as a shortcut of the open market operation process when the central bank purchases government bonds from the government (through banks). Then, the government transfers the payoffs to households (Table 3). In fractional reserve banking, the amount of τ t needed to change the federal funds rate is extremely small in comparison to the total money supply.
2.6 Timing within one period (i) Production takes place. Households sell intermediate goods to wholesalers, who, in turn, sell goods to retailers. All of the payments between households-wholesalers, wholesalers-retailer are delayed until the step (iv).
(ii) The loan market between bankers and households opens.
(iii) The final good market opens. Households need ZMD-in-advance to purchase the final good from retailers. Bankers create ZMD to purchase the final good from retailer.
(iv) Payments in the non-bank private sector are settled.
(v) The banking market opens. Banker can adjust the level of reserves by borrowing in the interbank market, receiving new deposits. The central bank might conduct LSAP here and transfer its profits to households at this stage.

Bankers
There is a measure one of identical bankers in the economy. These bankers have to maintain a good balance sheet under the regulation of the central bank. There are three types of assets on a banker's balance sheet: reserves (n t ), loans to households (b h t ), loan to other bankers (b f t ). His liability side contains the zero-maturity deposits that households deposit here (m t ).
Cost: We assume that the banker faces a cost of managing loan, which is θ b h t in terms of final goods.
On the timing of the market, it is worth noting that he can adjust the level of his deposits and The Fed Banks Public Reserves:+τ t Reserves:+τ t Deposits: +τ t Deposits:+τ t Net worth: +τ t Net worth:-τ t

Net worth
Loans to other bankers: b f t reserves after households and firms pay for each other. When the different parties in the economy pay each other, he can witness that the deposits and reserves outflow from or inflow to his bank. Let e t be the net inflow of deposits and reserves go into his bank, he will treat e t as exogenous.
When the banking market opens, as the deposit market is perfectly competitive, he can choose any amount d t of deposit inflows or outflows to his bank. 4 LSAP: When the central bank purchases assets from the private sector, it enlarges the banker' balance sheet. Let ∆x t = x t −x t−1 be the additional number of shares that the central bank purchases at time t. If ∆x t < 0, the central bank conducts unwinding quantitative easing. This transaction could be described by Table 4. The banker treats ∆x t as an exogenous variable.
In each period, the banker treats all the prices as exogenous and choose { c t , n t , b h t , s t , m t , b f t , d t } to maximize his utility over a stream of consumptions: Reserve Flows (Equation 1): After receiving the interest on reserves, the previous reserve balance becomes R n t−1 n t−1 /π t . He also collects the payment from the interbank loan he lends out to other bankers in the previous period R f t−1 b f t−1 /π t . He can also increase his reserves by taking more deposits d t . When doing that, his reserves and his liability increase by the same amount d t (Table 5). That is the reason we see d t appear on both the equation (1) and (2). The similar effect can be found on τ t and υ t ∆x t when the central bank drops money or conduct LSAP. The banker treats τ t and ∆x t as given. Then, he can leave reserves n t at the central bank's account to earn IOR, or lend reserves to another bankers b f t with the rate R f t . Deposit Flows (Equation 2): He makes loans to households by issuing deposits or creating The Fed Bankers Households Securities: Reserves: Reserves: Deposits:   ZMDs (Table 1). So when he makes a loan (s t ), the balance sheet expands. When he collects the payoffs from loans to households (δ b b h t−1 /π t ), the balance sheet shrinks. 5 The banker also issues his own ZMDs to purchase the consumption good from retailers (c t ) and to pay for the cost (in terms of final goods) related to lending activities (θ b h t ) ( Table 6). It is noted that he cannot create infinite amount of money for himself to buy consumption goods as there exists the capital requirement and reserve requirement.
Reserve Requirement: At the end of each period, he has to hold enough reserves as a fraction of total deposits (Equation 4). 6 This constraint should be interpreted more broadly than the the real life reserve requirement in the US because the total ZMDs here include not only checkable deposits but also saving deposits and money market deposit account.
Capital Requirement: The second constraint plays the key role in our model -the capital requirement constraint. The left hand side of the equation (5) is the banker's net worth (capital), which is equal to total assets minus total liabilities. 7 The constraint requires the banker to hold capital greater than a fraction of total loans in his balance sheet. We assume that κ t is an exogenous variable reflecting the risk weight of the banker's asset. We later put the unexpected shock on this κ t to reflect the shock in the Great Recession. 8 Let γ t , µ r t and µ c t be respectively the Lagrangian multipliers attached to the reserves flows, reserves constraint and the capital constraint. The first order conditions of the banker's problem can be written as: And two complementary slackness conditions: Let B h t be the nominal debt stock that she borrows from bankers. Recalling from the section 2.4, each period she only pays a fraction δ b of the old debts. We impose an exogenous borrowing constraint for households with the debt limit b h t ≤ b h . After the loan market, she brings a t amount of ZMDs into the final good market. Basically, she faces the "ZMD-in-advance" constraint when the good market opens. So the amount of loans that she gets from banks will affect her demand for the final goods. In each period, she chooses Loan Market: ZMD-in-advance: Budget: Production: We assume that the household faces an exogenous borrowing constraint, rather than a collateral borrowing constraint like Kiyotaki and Moore (1997) and Iacoviello (2005). Our purpose is to emphasize that the mechanism of the shock transmission in our model is not related to the collateral constraint literature. Similar to the capital requirement, we impose the constraint on the face value of the loan.
Let η z And two complementary slackness conditions: As money plays the role of medium of exchange in our model, its value contains the liquidity part. In the steady state, the rate of return on money has to be less than 1/ β .
The equations (22) and (23) give us the marginal cost and the marginal benefit when the household borrows one more unit of loans from bankers and when she makes one more unit of investment. The equation (25) is a common asset pricing equation for the wholesalers' shares.

Retail Firms and Wholesale Firms
Follow Rotemberg pricing, we assume that each wholesale firm j faces a cost of adjusting prices, which is measured in terms of final good and given by: where ι determines the degree of nominal price rigidity. The wholesale firm j discounts the profit in the future with rate β i λ a t+i /λ a t . Her real marginal cost is p m t . In a symmetric equilibrium, all firms will choose the same price and produce the same quantity P t ( j) = P t and y t ( j) = y t = y m t . The optimal pricing rule then implies that:

The Central Bank and Government
The consolidated government Securities: x t Reserves: n t

Net worth
The consolidated government uses the payoffs from tax or their assets to pay for the interest on reserves, then injects (drains)τ t amount of reserves and deposits by helicopter money (lump-sum tax) to target the interbank rate. All transactions are conducted in the electronic system.
The consolidated government have three tools for monetary policy: a) Helicopter money / lump-sum tax (τ t ): This is a conventional tool, and it appears in most macro models of monetary policy. When there is no excess reserves (the reserve requirement is binding), the consolidated government could target the federal funds rate by adjustingτ t . When there is a huge amount of excess reserves (the reserve requirement is not binding), a small deviation ofτ t does not affect the interbank rate.
b) Interest on reserves (R n t ): This is an unconventional tool, manipulating the interbank rate R f t = R n t when the banking system has a large amount of excess reserves.
c) LSAP / Unwinding QE (∆x t ): The central bank can manipulate the amount of money supply as well as the yield of wholesale firms' shares by adjusting ∆x t . Unlike the helicopter money, LSAP does not expand households' balance sheets but rather switches an illiquid asset x t (shares) by a liquid one m t (ZMDs) for households. Furthermore, LSAP also injects reserves into the banking system.
In the conventional monetary policy, we assume that the central bank follows a simple Taylor rule, fixing R n t at a constant level R n . We also assume that the central bank does not use the last tool LSAP (x t = 0). To connect with the common New Keynesian literature, we assume there is a lower bound for R f t that is greater than R n , so there are no excess reserves. 9 Later, we relax the assumption and examine the situation when the banking system is awash of excess reserves and the central bank controls the federal funds rate by adjusting R n t . We assume the inflation target in the long-term of the central bank is π. The conventional monetary policy could be characterized as: Interest on Reserves: Helicopter money targets interbank rate: LSAP: ii All markets clear: Net flows of reserves: d t + e t = 0 The interbank market: Good Market: The list of equations in equilibrium could be found in the Appendix B. If we consider a model without currency where all banks are identical in the equilibrium, the net flows of reserves to the representative banker will be zero. We also make the following assumption to ensure that in the steady state households will borrow from bankers.
Assumption 1. The discount factors of bankers and households satisfy: Intuitively, the above inequality guarantees that the real rate of long-term loan, which depends on bankers' discount rate, inflation target, and loan maturity, is smaller than households' discount rate. We also assume that in the long run, the inflation will be at the target level by restricting monetary policy in every regime to satisfy: The first equation (34) restricts the ratio of helicopter money to the total level of reserves in the long run. This ratio must be consistent with the inflation target of the central bank. The combination of (35) and (36) ensures that the central bank will not use the interest on reserves as its main tool to adjust the interbank rate in the steady state (however, IOR could be used in the short run). The last assumption guarantees that the central bank will not hold any wholesales firms' shares in the long run.
The relationship between the federal funds rate R f t , deposit rate R m t and interest on reserves R n t can be understood under the following theorem: Theorem 1. In equilibrium: i The lower bound of the federal funds rate and the deposit rate is the interest on reserves. In all cases, R n t ≤ R m t ≤ R f t ii When the constraint of reserve requirement is not binding, R f t = R m t = R n t .
Proof: Please see the Appendix A for all proofs.
There are two benefits of holding reserves for bankers. First, bankers can earn the interest on reserves that central bank pays them. Second, it helps bankers satisfy reserve requirement. The cost of holding reserves is the federal funds rate that they give up when they do not lend reserves in the interbank market. When the banking system has a large amount of excess reserves, the second benefit vanishes and the federal funds rate must be equal to the interest on reserves.
In reality, the deposit rate of ZMDs might be lower than the interest on reserves due to the bankers' cost of providing liquidity services and market power. We ignore these factors in this model to present the main mechanism cleaner. Theorem 2. In equilibrium, the total level of reserves is determined solely by the central bank: Bankers themselves cannot change the total level of reserves in the banking system. Lending or not lending to households will not change the total level of reserves. The appearance of the huge amount of reserves after the large scale asset purchase is just a byproduct of the central bank's policy. Later we will examine this kind of policy.

The Steady State
We use a to denote the steady state value of a variable a t . i. The banker's reserves constraint (4), the household's borrowing constraint (19) and the ZMD-in-advance constraint (14) are binding.
ii. The banker's capital constraint (5) is not binding.
Theorem 4. Under the assumption (1) and (2), the capital in every steady state (if exists) satisfies the following equation: where r m , α m , α l , α y are constants independent of k.
We make one more assumption to ensure that there exists a unique steady state. The uniqueness of the steady state will be very important as we mostly examine the global nonlinear dynamic of our model. Assumption 3. The parameters satisfy: where m is defined in (A.21), r m and α m are defined in (A.12) and (A.16).
The restriction on the parameter κ is to ensure that the capital constraint is not binding. The last two restrictions are to ensure that Equation (39) has a unique positive solution k * .
Theorem 5. Under the assumption (1)  For the bankers' parameters, we choose the discount factor β = 0.99 to match with the federal funds rate of 4% annually before the Great Recession. The reserve requirement is set as the ratio between reserves and the total ZMDs (including checking account, saving account and money market deposit account) before the financial crisis, which is around ϕ = 0.002. The monitoring cost θ is to calibrate to match with the real borrowing cost about 4.5% before the Great Recession. The loan amortization is set to match the average duration of loan (around two quarters). This value is set to 0 as an one-quarter loan in Bianchi and Bigio (2014). The risk weight κ is exogenously set so that 10 percent increase of κ from steady state will make the capital constraint binding. (Table  7) Most of the households' parameters are standard in the literature. The household's discount factor is set to match with the discount factors of impatient agents and entrepreneurs in Iacoviello (2005). The relative utility weight of labor is calibrated to fix the hours working at the steady state equal to 1. The Frisch elasticity of labor supply is equivalent to 2 (King and Rebelo, 1999). The only one that needs to be calibrated is the borrowing limit b h . We calibrate it to match with the ratio between total households' debts and households' income before the Great Recession -around 1.3 times.
For the firm parameters, the elasticity of substitution of wholesale goods is similar to the one in Gertler and Karadi (2011). The Rotemberg cost of changing price is round up to match the estimated value in Richter and Throckmorton (2016). For the central bank parameters, the interest on reserves at the steady state is set at 0.25% annually.

Numerical Method
We solve the perfect foresight equilibrium with the unexpected shock at t = 0 by assuming that after T = 300 quarters, the economy will converge back to the initial steady state. The initial position before the unexpected shocks is the steady state. Basically, we need to solve a large system of equations to determine the dynamic path of the economy. The transformation of the occasionally inequality constraints into equality constraints (Appendix C) ensures that every equation is continuous and differentiable.
For every application, we use homotopy continuation method (by gradually increasing the size of shocks) for solving this large system of equations, with the initial point starting from the steady state. Let x be the vector containing all endogenous variables in 300 periods and u be the vector containing all exogenous shocks. Let z be the scalar governing the homotopy process. We need to solve systems of equations: F(x, zu) = 0 When z = 0, the solution is the steady state itself. When z = 1, the solution is the dynamic equilibrium after shocks. We start from z = 0, then slowly increase to z = 1 by small steps. For each step, we use the solution in the previous step as an initial guess.
We use Ipopt in C++ written by Wachter and Biegler (2006) with the linear solver HSL 10 to conduct this homotopy continuation method. We then use Matlab to generate figures.

Federal Funds Rate Shock
We examine an interest rate shock in the Taylor rule and compare the mechanism of this model to the standard version in the New Keynesian literature.
Helicopter money targets: Interest on Reserves: R n t = R n LSAP: x t = 0 Shock: From the steady state, there is an unexpected shock at t = 0 with u f 0 = −2/400, then agents know that the shock will die slowly with ρ u = 0.6. Similar to the standard New Keynesian model: As P t is sticky, when the central bank cuts the i Banks play an important role in creating money. After the interest rate shock, the real money balance increases by 0.45 percent. Most of that new money is created by banks when they increase loans. The amount of money that the central bank actually "drops" to the economŷ τ to change the federal funds rate only accounts for 0.02 percent of this increase. So unlike the standard model in New Keynesian, our model focuses on the money creation process by commercial banks and the pass-through effect from the federal funds rate to the loan rate.
ii Without any adjustment cost functions, investment still well-behaves after the cut in the real interest rate. The constraint for a huge sudden jump of investment comes naturally from the ZMD-in-advance constraint and the borrowing constraint.

Financial Crisis -Taylor Rule Response
From the steady state, we illustrate a financial crisis by imposing an unexpected shock at κ t in the capital constraint. This is a simplified way to reflect a sudden increase in the "potential" bad loans in the bankers' balance sheets. This paper does not try answering the cause of the Great Recession, so this reduced form is neat to assess different monetary policy rules. In this section, the conventional monetary policy still follows the Taylor rule in (31) and (32).
Helicopter money targets: Interest on Reserves: R n t = R n LSAP: x t = 0 Shock: where ρ k = 0.95 is the persistence of the shock and κ 0 = 0.26, which is 18 percent higher than the one in the steady state level. The capital requirement switches to the binding mode in the short run. The response of the economy is illustrated in the Figure 3. The banking crisis is dangerous as it raises the spread between the prime rate and the federal funds rate. To satisfy the capital requirement (CR), bankers have to cut loans. Loan rate goes up even when the federal funds rate is cut down, as the shadow price of capital requirement µ c t is positive now.
Spread due to CR's binding Money supply eventually drops due to the consequence of the debt deleveraging process. Deflation will be persistent under the Taylor rule as the conventional monetary policy only focuses on the pass through of federal funds rate to the prime rate, which will not work in this case. Standard New Keynesian model emphasizes the importance of monetary policy in correcting the deviation of real rate from its natural level due the the price stickiness. Under the framework where the banking sector is modeled clearly, there are two other inefficiencies that monetary policy can intervene to improve the social welfare. The first inefficiency arises from the binding of the capital constraints, which freezes the credit market between bankers and households. The second inefficiency comes from the households' borrowing constraint itself. Unconventional monetary policy focuses on the money supply and asset price might be a good remedy for this situation. We only focus on the money supply (liquidity provision) in this paper.

Financial Crisis -Large Scale Asset Purchase (LSAP)
Now, assume that central bank injects money directly into the market by purchasing a large sacle of wholesale firms' shares in the crisis. The central bank is assumed to keepτ t and interest on reserves R n t constant at the steady state level. Before time 0, x t = 0. At time 0, there is an unexpected shock of purchasing assets from the central bank in response instantly to the unexpected shock on κ t . Then from t = 1, the central bank will slowly sell these assets back to the market. The speed of unwinding LSAP is governed by parameter ρ x . For the dividends earned from holding securities, we assume that the consolidated government (after paying IOR) will give the remaining back to households under the form of lumpsum transfers. The unconventional monetary policy and the shock process could be described as: Helicopter money:τ t =τ, ∀t ≥ 0 Interest on reserves: R n t = R n , ∀t ≥ 0 LSAP: x t = ρ x x t−1 , x 0 is given where ρ x = 0.98 be the persistence of the asset purchasing shock and x 0 = 0.0008. We assume that the central bank does not follow the Taylor rule anymore. They still fix the interest on reserves at the constant level R n and only use that asset purchase/sale program to adjust the money supply. Figure 4 shows the reaction of the economy to this monetary policy.
Here are some important remarks for LSAP's effect: i The excess reserves skyrockets and the long duration of the federal funds rate at the lower bound: When the central bank purchases assets from the private sector, they inject simultaneously money supply into the market and banking reserves into the banking system. When the level of reserves increases by 700 percent, the reserve constraint is no longer binding, µ r t = 0. Otherwise, money supply will increase by 700 percent, which is impossible as there is an exogenous borrowing constraint (for households) and capital constraint (for banks). As we assume that the central bank fixes IOR at a constant level, it is synonymous that the federal funds rate will be at the lower bound for a long time, around 25 years (100 quarters) in our model. After a long unwinding quantitative easing process, the reserve requirement will be binding again. The federal funds rate climbs back to its long run level. The whole transition process can take around 80 years in our model.
ii Positive effect in the short-run: The combination of the new money injection and the long duration of the federal funds rate at the lower bound steers the economy out of recession quickly, unlike the case with the Taylor rule. Due to the effect of forward guidance on the inflationary expectation, if the central bank commits to let the federal funds rate at the low level for a long time, the real lending rate will decline sharply. It combines with the relaxation of the liquidity constraint, stimulating the household's demand and pushing up inflation. This point in our research is identical to the New Keynesian literature.
iii Negative effect in the long-run: After inflation jumps up in the short run, it starts declining, below the central bank's target in the medium run. This phenomenon can be explained by the Fisher Effect. In the long run, the real short-term rate will be back to the long-term level. As R f t = R n , the deflation must realize to increase R f t /π t+1 . This negative effect in the long-run is robust to the different package size of LSAP ( Figure 5).
iv Intuitive Explanation: The explanation could be seen through the real interest rate. In the short run, monetary policy can lower the real interest rate due to sticky price. However, in the long run, the real interest rate (real rate return on capital) has tendency to move to the long run level. When the central bank keeps the interbank rate at 25 basis points, the rate of saving account R m will be at 25 basis points as deposits and interbank loans have the same short-term maturity. However, the real return on capital in the long-run recovers to the pre-crisis level. In equilibrium, the real return on deposits (plus the liquidity premium) must follow the real return on capital. The endogenous money supply declines gradually. Deflation must realize to ensure the condition in Fisher Effect.

Interest on Reserves (IOR) as Monetary Policy Tool
IOR: To raise or not to raise?
In Section 6.5, we know that after the LSAP program without adjusting R n t , the inflation -the central bank's main target -is high in the short run but below the target in the long-run. How long should the central bank keep the federal funds rate at the zero lower bound? And if the central bank decides to raise rate, what is a good strategy for the central bank?
In this section, we still conduct the experiment similar to the previous section with one twist. We assume that after T u periods, the central bank will raise IOR. Then IOR will be brought back to the steady state level after T d period. We choose the different level for T u at 20, 40 and 80 quarters to see the effect of the prolonged low interest rate environment on output and inflation in the short Helicopter money:τ t =τ, ∀t ≥ 0 Interest on reserves: Here are some remarks from our experiment: (Figure 6) i. The longer is the duration of the federal funds rate at the lower bound, the higher is inflation in the short run. This forward guidance effect is well-documented in the New Keynesian literature when the central bank commits to set the short-term at the zero lower bound for a long time (Eggertsson and Woodford, 2003). However, the hyperinflation never happens in our model even with 20 years that rate is set at the lower bound. Due to the household's borrowing constraint and banker's capital constraint, the amount of the money supply is restricted even with the huge amount of excess reserves in the banking system.
ii. The longer is the duration of the federal funds rate at the lower bound, the bigger is the negative effect on output and deflation in the long run. It emphasizes that our model is Keynesian in the short run, but Fisherian in the long run.
iii. The endogenous money supply drops sharply when the central bank raises rates. As price is sticky, the real fed funds rate and real lending rate must go up after this rate hike. Hence, the total of amount of bank credits declines, also implying a huge fall in the money supply. However, after some periods, the neo-Fisherian effect dominates the Keynesian effect, stabilizing inflation at the target level. After all, the central bank still needs to pay a big cost for a rate hike in the short run.
The last point implies an important hint for monetary policy when the central bank decides to raise rate. The central bank can still stabilize inflation and the aggregate demand if it commits to a rule of targeting the money supply at the time of raising rates. The appearances of interest on reserves and electronic payment system allow the central bank to manipulate both the money supply and interest rate at the short run, which is very different from New Keynesian focusing only on the short-term rate. In this sense, our research is near to Monetarism in the long run. The growth rate of the money supply always decides the inflation path in the long run.

Raise rate and raise money supply -Money Supply Rule
We do an experiment similar to (P4) but at the time of raising IOR, the central bank also commits to a money supply rule (massive helicopter money if necessary) to target the inflation rate. The π 80 (f) Inflation -long run money supply rule simply responds to the deviation of the inflation rate from its target: where ρ m = 0.5 is the coefficient showing how much the central bank will change the growth rate of the money supply in response to inflation. To create the same interbank rate path as the one in the previous section, we assume this money supply rule only applies since the time the central bank decides to raise rates. The complete list of exogenous shocks and unconventional monetary policy for this experiment can be written as follows: ii. Money supply targeting is extremely efficient in stabilizing inflation and output. The inflation is anchored at the target since the time the central bank targets the growth rate of the money supply in our model.
iii. At the time of raising rate (after 20 quarters), to stabilize the inflation and avoid a severe short recession, money supply targeting implies that the central bank should conduct a massive helicopter money. With this commitment, the central bank can increase the household's expectation about inflation path when raising rates. As a result of that, the real interest rate does not change. Do we really need two types of electronic money (reserves and deposits) in a macroeconomic model to understand the effect of monetary policy? The answer is that it depends on the research question. If we want to investigate the effect of monetary policy before the Great Recession -when there were no excess reserves, a monetary policy with an interest rule (on bond yield) alone could capture the whole effect of monetary policy. As a result of that, adding reserves and deposits makes our model more complicated without giving us new insights. However, if we want to understand the unconventional monetary policy after 2008, reserves and deposits are "must-have" elements in models. When the main tool of the central bank is IOR, the effect of monetary policy depends on both money supply and interest rate. Therefore, we need reserves to understand IOR and we need deposits (created by banks) to understand money supply.

Determinacy
As our model is non-linear in nature with inequality constraints, it requires the global determinacy rather than local determinacy. The method of log-linearization is not helpful in this case. Our model guarantees the uniqueness of the steady state under some conditions. By numerical methods, we also show that there exists a transition path converging to the unique steady state that shares many salient features with reality. However, the limitation of our model is that it does not have a theoretical proof to show the uniqueness of this transition path. Hopefully, the future research could address this issue or coming up with criteria of equilibrium selection (homotopy continuation method might be used for equilibrium selection).

Monetary Policy Stance
Our model also gives new insight to the stance of monetary policy. Economists often refer to a loose (tight) monetary policy when the central bank cuts (raises) interest rate. We show that it is not complete true in the unconventional monetary policy. A raise on nominal interest rate (by raising IOR) is not a tight monetary policy if the central bank couples it with an increase in the money supply (by LSAP or helicopter money). A cut on nominal interest rate is not a loose monetary policy if the central bank simultaneously sells a huge amount of agency MBS or government bonds back to market.

Bank -Intermediaries or Money Creation Vehicles?
In our model, banks do not play a role of intermediaries, but rather special financial institutions that can create inside money under regulations. Banks also directly affect the economy (even though their consumption only accounts 0.3% of total output in my model). A model nested both functions of banks will be very interesting, but it is a very challenging task to achieve. The beginning point to achieve that should be the family structure of Gertler and Karadi (2011), which is a promising approach but beyond the scope of this paper.

Conclusion
Our research shows that, when the central bank controls the federal funds rate by adjusting interest on reserves, the path of the interbank rate alone does not provide full information on the stance of monetary policy. The endogenous money supply can complete go down when the federal funds rate is near zero for a long time. However, if the central bank simply raises rates, the economy will fall into a short recession. Deflation will be worse in the short run after the interest rate is lifted. Basically, the central bank falls into a dilemma to raise or not to raise rate, where outcome is not bright in either way. One feasible solution for the central bank is to target the growth rate of the money supply in response to inflation when they raise rates. The key insight of our paper is that the central bank can adjust simultaneously the interbank rate and the money supply in this era. With that, they can completely avoid the negative short term effect of raising rates and do a better job at hitting the inflation target.
(A.4) and (A.5) → π = π. Money supply rule ensures that inflation reaches to its target in the steady state.
From (32), we have: Under the assumption (2): R n + ε f < π/β , we get R f = π/β . The equation (A.1) can be rewritten in the steady state as: When R f = π/β , we get µ c = 0, the capital constraint is not binding (if steady state exists). As R f > R n , from the Theorem 1, µ r > 0, or the reserve requirement is binding.

C Numerical Method
There are 5 occasionally binding inequality constraints in our model: the reserve requirement, the capital requirement, the ZMD-in-advance, the household's borrowing constraint and the Taylor rule of the central bank.