Monetary Policy Transmission, Central Bank Digital Currency, and Bank Market Power

2.1 Households

We consider an economy consisting of many identical and infinitely-lived households, with the measure normalized to one. The representative household values consumption, c _t and liquidity services, z _t + 1, according to a period utility function of the form

where v ∈ ( 0,1 ) is the relative weight of liquidity services in utility, ψ ∈ (0, 1) is the inverse intratemporal elasticity of substitution between consumption and liquidity, and σ > 0 is the inverse intertemporal elasticity of substitution between consumption-liquidity bundles across time. We assume that CBDC and deposits are imperfect substitutes: Liquidity services are derived from real holdings of CBDC, m _t + 1, and deposits, n _t + 1, according to a constant elasticity of substitution (CES) aggregator

where γ ∈ (0, 1) is the relative liquidity weight of bank deposits, and ϵ ∈ (0, 1) is the inverse elasticity of substitution between CBDC and deposits. The liquidity weight parameter, γ, captures how useful deposits are for the purpose of holding liquidity relative to the same quantity of CBDC. We follow Drechsler et al. (2017) in assuming that deposits are themselves a composite good issued by a set of N non-competitive banks. Each bank i has mass 1/N and produces deposits of a quantity n t + 1 i / N . The household values deposits at different banks such that

where η denotes the inverse elasticity of substitution between banks. The representative household can be thought of as an aggregation of many individual households who may have diverse preferences for holding deposits at different banks. Therefore, the representative household substitutes deposits imperfectly across banks, which implies that 0 < η < 1.

In our framework, in addition to deposits and CBDC, households can invest directly in capital. This is necessary for the model to feature realistic amounts of capital and liquid assets, as the aggregate amount of the former typically far exceeds the amount of the latter in modern economies.^[1] Since we found it to be not crucial for our results, we abstract from a labor supply choice to simplify the analysis and instead assume the representative agent to inelastically supply a constant amount of labor l ̄ . The household’s budget constraint is then given by

where k t + 1 h are direct holdings of capital, τ _t is the lump-sum tax net of government transfer, w _t is the wage rate, π _t is the dividends from firms and banks, R t k is the return on capital, R t + 1 m is the real gross interest rate on CBDC, and R t + 1 n , i is the real gross interest rate on deposits at bank i. We assume that the returns on CBDC and deposits are risk-free, i.e.  R t + 1 m and R t + 1 n , i are known at time t. The household, taking prices, profits, and taxes as given, solves

We now turn to the first-order optimality conditions of the household program. Detailed derivations are provided in the Appendix A.1. First, the household optimally allocates resources between deposits at individual banks according to

which closely resembles demand equations for differentiated consumption goods commonly derived in New Keynesian models. The relative share of deposits at bank i, n t + 1 i / n t + 1 , must relate negatively to its corresponding relative cost, χ t + 1 n , i / χ t + 1 n . Here, χ t + 1 n , i is the interest-rate differential between the risk-free rate, R t + 1 f , and the deposit rate offered by bank i

which represents the opportunity cost of holding deposits at bank i and which we hereafter refer to as deposit spread. The risk-free rate is defined in the standard way as the inverse of the expected value of the household’s stochastic discount factor, Λ _t+1,

where the stochastic discount factor is defined as follows:

Given that the household can freely invest in capital, the risk-free rate will equal the expected return on capital.

We further define χ t + 1 n to represent the index that can be shown to capture the deposit spread associated with one unit of the aggregate deposit bundle n _t + 1 given demand schedule (3):

This quantity can be interpreted as an aggregate price of deposits.

Next, optimization requires that the marginal rate of substitution between consumption and each of the liquid assets equals their respective costs. These conditions can be combined to derive an expression for the velocity of consumption

where χ t + 1 z is a weighted average of the spreads on deposits and CBDC

Here, the CBDC spread, χ t + 1 m , is defined similarly to the deposit spread, as the interest-rate differential between the risk-free rate and the CBDC rate,

and denotes the opportunity cost of holding CBDC. Thus, we interpret χ t + 1 z as being the household’s average cost of liquidity. Note that consumption velocity is increasing in this cost. As liquidity becomes more expensive, the household would want to economize on its liquidity holdings, and therefore, velocity increases. In the limiting case where the relative utility weight of liquidity goes to zero, i.e.  v → 0 , consumption velocity goes to infinity and the model economy converges to the standard “cashless limit” case. Moreover, the household’s demand for CBDC and deposits can be expressed as

We see that the relative demand for each liquid asset is increasing in their relative liquidity weights and decreasing in their costs relative to the average cost. In the special case where the liquidity weight of deposits goes to zero or the relative cost of deposits goes to infinity, CBDC becomes the household’s only source of liquidity, i.e. z _t + 1 = m _t + 1. The opposite occurs if the weight of deposits goes to one or the relative cost of CBDC goes to infinity.

Lastly, we derive the intertemporal Euler equation of the household, which has the form

with the standard interpretation of the representative agent equating the current-period marginal cost of savings, given by the marginal utility of consumption (left-hand side), with the next-period discounted expected return on savings (right-hand side). Notice that relative to a textbook real business cycle (RBC) model, the Euler equation (11) contains the term Ω t c

which summarizes the impact of liquidity services on the marginal utility of consumption.

2.2 Banks

There is a set of N non-competitive banks that produce differentiated deposit services and invest in capital and reserves. The balance sheet of a typical bank is

where k t + 1 i and r t + 1 i denote the bank’s capital and reserve holdings, respectively. We follow Niepelt (2024) and assume that maturity transformation requires bank resources. Banks incur a cost per unit of deposit funding, and a role for reserves is introduced by assuming that larger reserve holdings relative to deposits reduce these costs. Unlike in Niepelt (2024), we do not assume positive externalities of reserve holdings as it does not substantially affect our results. A bank’s per-unit cost of deposit, ν t i , is thus assumed to be a decreasing function of just its own reserves-to-deposits ratio, ζ t + 1 i = r t + 1 i / n t + 1 i ,

where ω, ϕ ≥ 0; φ > 1. We assume that all banks face the same cost function.

At time t, bank i decides on its reserve holdings and deposit rate, subject to its deposit demand schedule (3) and the balance sheet constraint (13). Returns on the bank’s assets, net of interest payments, are realized in the subsequent period. The bank retains no earnings and distributes its entire profit to the household every period. The date-t program of a typical bank is

where R t + 1 r is the gross interest rate on reserve balances.

We focus on a symmetric industry equilibrium: Since all banks are identical, they will choose identical deposit rates (and thus identical deposit spreads) and levels of reserve holdings, i.e.  χ t + 1 n , i = χ t + 1 n , j and r t + 1 i = r t + 1 j for all i and j. Identical deposit spread across banks, given banks’ demand schedule (3), implies that the household’s demand for each bank’s deposits is also identical, i.e.  n t + 1 i = n t + 1 j for all i and j. Using equations (1) and (5), we can establish that n t + 1 = n t + 1 i and χ t + 1 n = χ t + 1 n , i . Moreover, identical levels of reserve holdings mean that the aggregate reserve holdings of the whole banking sector are r t + 1 = ( 1 / N ) ∑ i = 1 N r t + 1 i = r t + 1 i . Then, the reserves-to-deposits ratios have to be equal across banks too, i.e.  ζ t + 1 i = ζ t + 1 . As all banks are identical, we hereafter drop the individual superscript i.

We now turn to the first-order conditions of a bank’s optimization problem. Detailed derivations are provided in the Appendix A.2. Firstly, the first-order condition with respect to reserves yields the bank’s desired reserves-to-deposits ratio

The ratio depends on the reserve spread, χ t + 1 r , which represents the opportunity cost of holding reserves

As the reserve spread increases and reserves become more expensive, the bank’s desired reserves-to-deposits ratio decreases, and its cost of deposit issuance increases. Equation (14) also shows that the bank’s choice of reserves equalizes their marginal (opportunity) cost of holding reserves, χ t + 1 r , to the marginal gain stemming from a lower cost of deposit issuance, − ( 1 − φ ) ϕ ζ t + 1 − φ .

Secondly, the first-order condition with respect to deposit rate yields the condition that determines the equilibrium deposit spread

The right-hand side of equation (16) shows the marginal cost of deposit issuance. The marginal unit of deposit not only implies an extra cost of ν t = ω + ϕ ζ t + 1 1 − φ , but also increases the cost for inframarginal units, given by − ∂ ν t ∂ ζ t + 1 ζ t + 1 . The two components add up to ω + φ ϕ ζ t + 1 1 − φ . The left-hand side of (16) shows the banks’ marginal benefit of raising deposit funding. The first term on the left-hand side, χ t + 1 n , entails (if positive) a return in excess of the reference risk-free rate. That is, deposits are a cheap source of funding for the bank and the spread denotes a marginal gain from deposit issuance. However, recall that the deposit spread represents an (opportunity) cost for the household. The second term on the left-hand side shows the decrease in the spread the bank must make in order to incentivize the household to provide more deposits

Expression (17) can be thought of as a markup over the marginal cost that the non-competitive bank imposes on the household. The deposit spread markup depends negatively on the elasticity of demand that the bank faces, given by

where s _t  ∈ [0, 1] is a relative weight

The elasticity of demand (18) shows that the changes in the demand for deposits are the sum of two effects. Firstly, suppose the bank decreases its deposit rate and thus widens its deposit spread. It raises the aggregate deposit spread index, χ t + 1 n , by the amount equal to its mass, 1/N. This makes deposits more costly overall for the household and induces a substitution away from deposits at a rate − 1 − s t ψ − s t ϵ < 0 . This aggregate effect is more pronounced in a more concentrated deposit market since the actions of each bank have a larger impact on the overall cost of deposits. Secondly, given a decrease in the deposit rate, its deposit spread increases by 1 − 1/N relative to the aggregate index. This induces an outflow of deposits from the bank at the rate of the elasticity of substitution between banks, 1/η. This interbank effect is larger when the elasticity of substitution between banks is large or when market concentration is low.

The aggregate elasticity of demand for deposits, as represented by the first term in equation (18), is a weighted average of their elasticity of substitution to consumption, 1/ψ, and CBDC, 1/ϵ. The weight s _t reflects the relative cost of CBDC m _t+1 within the liquidity bundle z _t + 1. When the interest rate on CBDC is high, the CBDC spread within the bundle is relatively low, leading to more outflow from deposits towards CBDC. This scenario makes the aggregate elasticity of demand close to 1/ϵ. Conversely, if CBDC becomes less attractive, consumption captures most of the substitution out of deposits.

We consider limit cases here. Suppose deposits at different banks are perfectly substitutable, i.e. η → 0. Then, the elasticity of demand goes to infinity. The bank becomes competitive and sets the deposit spread equal to its marginal cost of deposit issuance

Alternatively, suppose the deposit market is perfectly dispersed, i.e. 1/N → 0. The bank becomes monopsonistically competitive and charges a constant multiplicative markdown 1/(1 − η) over its marginal cost. Then, the deposit spread becomes

2.3 Firms

Competitive firms produce common consumption goods using capital and labor. The representative firm maximizes its profit by solving the following problem:

where α is the capital share of output, a _t is productivity, k _t and l _t are the firm’s demand for capital and labor, and δ is the capital depreciation rate. The first-order conditions of the firm pin down the capital return and the wage rate, respectively,

Since firms operate in a perfectly competitive environment and the production function exhibits constant returns to scale, equilibrium profits are equal to zero.

2.4 Consolidated Government

A consolidated government/central bank issues CBDC and reserves and invests in capital. The government incurs a per-unit cost, μ, when issuing (and managing) CBDC and a per-unit cost, ρ, when issuing (and managing) reserves. The budget constraint of the government reads

where k t + 1 g is the government’s capital holdings. We assume that the government sets the interest rates on CBDC and reserves, and the level of lump-sum tax. The specific way in which the government sets these interest rates will be discussed in detail in the next sections.

2.5 Market Clearing and Aggregate Resource Constraint

Market clearing in the labor market requires that firms’ labor demand equals the household’s inelastic labor supply, i.e.  l t = l ̄ . On the capital market, the firms’ capital demand has to equal the sum of the capital holdings of the household, banks, and the government

Lastly, total dividends distributed to the household must equal the sum of banks’ and firms’ profits

Aggregate resource constraint is derived by combining the budget constraints of the household and the government, market clearing conditions, and total dividends

where

The resource constraint has the standard interpretation that available output in the economy is split between consumption, c _t , and investment, k _t + 1 − k _t (1 − δ). However, there are resource costs associated with the provision of liquidity to the household, summarized by the term Q: μ per unit of CBDC and ν _t  + ζ _t + 1 ρ per unit of deposit. The resource cost of deposits has two terms because the banking sector incurs a cost of deposit issuance, ν _t , and the government incurs a cost of issuing reserves used by the banking sector, ζ _t + 1 ρ, to “back up” deposit issuance. As the household demands liquidity services in proportion to consumption, we can combine the terms c _t and Q _t , and rewrite the resource constraint as

where Ω t r c ≥ 1 is given by

2.6 Policy and Equilibrium

The consolidated government sets the interest rates on CBDC and reserves and elastically supplies these assets to households and banks to meet demand. A policy consists of R t + 1 m , R t + 1 r , τ t t ≥ 0 and an equilibrium conditional on policy consist of

1. a set of positive prices, w t , R t + 1 k , R t + 1 f , χ t + 1 m , χ t + 1 n , χ t + 1 z , χ t + 1 r t ≥ 0 ;

2. a positive allocation, { c t , k t + 1 } t ≥ 0 ;

3. and positive CBDC, deposits and reserves holdings, { m t + 1 , n t + 1 , z t + 1 , r t + 1 } t ≥ 0 ,

3.1 Real Effects of Monetary Policy

The two key conditions that characterize the equilibrium allocation, the Euler equation (11) and the resource constraint (25), all closely parallel the conditions of a textbook RBC model. The differences relative to an RBC model are the quantities Ω t + 1 c and Ω t + 1 r c . Importantly, the direct impact of liquidity on the household’s consumption/savings decision, captured by Ω t + 1 c , depends solely on the average cost of liquidity, χ t + 1 z . So we will mostly focus on the effects of policy on χ t + 1 z when studying transmission below. For this purpose, it is instructive to first lay down how the average cost of liquidity works through our model economy.

The Euler equation (11) shows that the household’s consumption/savings choice depends on liquidity through the marginal utility of consumption, which changes with the average cost of liquidity according to

We see that the sign of the impact on the marginal utility of consumption depends on the relative magnitudes of ψ and σ. If the household’s intratemporal elasticity of substitution between consumption and liquidity is smaller than the intertemporal elasticity of substitution, i.e. ψ > σ, an increase in the cost of liquidity leads to a decrease in the marginal utility of consumption. This is driven by the fact that an increase in the cost of liquidity, according to (6), reduces the household’s demand for it. A decrease in the level of liquidity then decreases the marginal utility of consumption, and hence there is consumption–liquidity complementarity. On the other hand, when ψ < σ, an increase in the cost of liquidity leads to an increase in the marginal utility of consumption. In the case where ψ = σ, the household’s utility is separable in consumption and liquidity and the cost of liquidity has no direct impact on consumption/savings choices.

Moreover, the spreads on CBDC and deposits also show up in the aggregate resource constraint (25) through the term Ω t r c . This reflects the resource costs associated with liquidity provision, incurred by the government and the banking sector. In the special case where the household does not value liquidity services, i.e.  v → 0 , both Ω t + 1 c and Ω t + 1 r c converge to one. At this “cashless limit”, the cost of liquidity has no impact on the household’s consumption/savings since no liquid assets are held. Therefore, there are also no resource costs associated with liquidity provision. Then, the model collapses into a standard RBC model.

To conclude, we have seen that the household’s consumption/savings decision only depends on the average cost of liquidity, which in turn is a function of the spreads on CBDC and deposits. As these are also the sole endogenous determinants of the liquidity cost term Ω t + 1 r c , the government can affect the allocation only insofar as it affects these spreads. While the government controls the CBDC spread directly through the CBDC rate, the deposit spread is determined by the banking sector. But as we will see below, the government can influence its behavior through the interest rates on both reserves and CBDC.

3.2 Interest on CBDC

We now explain the channels through which the household’s average cost of liquidity can be influenced by the CBDC rate. Suppose the government lowers the CBDC rate so that the CBDC spread widens.^[2] Differentiating the average cost of liquidity, given by (7), with respect to the CBDC spread yields

This expression shows that the CBDC spread works through two channels: Firstly, it directly increases the cost of liquidity by the first term. The strength of this direct effect is increasing in the relative liquidity weight of CBDC, 1 − γ, and decreasing in how costly CBDC is relative to the average cost of liquidity, χ t + 1 m / χ t + 1 z . Comparing the direct effect with the household’s demand for CBDC (9), we see that it is just the share of CBDC in the total stock of liquidity, m _t + 1/z _t + 1. Intuitively, the more important CBDC is as a source of liquidity for the household, the larger the impact of its cost on liquidity’s average cost.

Secondly, the CBDC spread affects the cost of liquidity through the deposit side, given by the second term. The strength of this indirect effect is increasing in the relative liquidity weight of deposits, γ, and decreasing in how costly deposits are relative to the average, χ t + 1 n / χ t + 1 z . Comparing the indirect effect with the household’s demand for deposits (10), we see that it is equal to the product of the share of deposits in the total stock of liquid, n _t + 1/z _t + 1, and the change in the deposit spread caused by a change in the CBDC spread, ∂ χ t + 1 n / ∂ χ t + 1 m . Analogous to the direct effect, the more important deposits are as a source of liquidity the larger is this indirect effect. However, the sign and the magnitude of the second effect also depend on how the banking sector responds to an increasing CBDC spread, captured by ∂ χ t + 1 n / ∂ χ t + 1 m .

In this regard, the optimality condition (16) shows that the CBDC spread can influence the deposit spread through the bank’s marginal benefit of deposit issuance (left-hand side). Specifically, CBDC spread affects the elasticity of demand for deposits that the bank faces, given by (18). As we discussed previously, the demand elasticity depends on a weighted average of the household’s elasticities of substitution to consumption, 1/ψ, and CBDC, 1/ϵ. The CBDC spread determines this average through the relative weight s _t , given by (19). Taking the partial derivative of the demand elasticity (18) with respect to the CBDC spread, we get

The partial derivative shows that the marginal impact of CBDC spread is non-zero only if ψ ≠ ϵ. Intuitively, banks collectively face competition from CBDC and consumption for the household’s resources. Therefore, any outflow from deposits depends on the household’s elasticities of substitution to CBDC and consumption. The CBDC spread only influences the relative importance of these two sources of deposit outflow, indicated by s _t . If the household finds it as easy to substitute from deposits to consumption as it does to CBDC, i.e. ψ = ϵ, then the two sources of competition for the banks are equally important and the CBDC spread does not influence the banks’ deposit spread. In such a case, the equilibrium spread is set equal to the marginal cost of deposit issuance plus a constant markup, similar to the case where banks are monopsonistically competitive.

In general, it seems reasonable to expect that deposits will be more substitutable with CBDC than with consumption, i.e. ψ > ϵ. Then, an increase in the CBDC spread makes the demand elasticity for deposits (18) less negative in value and, in turn, decreases the marginal benefit of deposit issuance. The intuition is that when its spread widens, CBDC becomes a comparatively expensive source of liquidity and a larger fraction of potential substitution out of deposits will go to consumption (indicated by a decrease in s _t and more weight being put on 1/ψ). The elasticity of demand moves closer to 1/ψ, which is smaller than 1/ϵ, and thus decreases in absolute value. Therefore, an increase in the CBDC spread makes the household’s demand for deposits less elastic. For banks with market power, a less elastic demand means that in order to attract additional deposits from the household, the spread needs to be lowered by more than before. That is, the marginal benefit of deposit issuance decreases. Given a fixed marginal cost, this implies that the equilibrium deposit spread increases. In other words, an increase in the CBDC spread is akin to giving banks more market power. Banks take advantage of this and charge a higher spread on deposits in equilibrium.

As we alluded to previously, market conditions in the deposit market also play a central role. If deposits at different banks are perfect substitutes or the deposit market is perfectly dispersed, the equilibrium deposit spread is determined without the influence of the CBDC spread. If the household does not differentiate between banks, each individual bank’s choice of how much deposits to issue does not matter for the equilibrium spread, which will equal the marginal cost of deposit issuance (20): the market is competitive. Similarly, if the deposit market is perfectly dispersed and the only source of market power is differentiation, the impact of each individual bank’s spread on the aggregate deposit spread goes to zero. The deposit market becomes monopsonistically competitive with a constant markup over marginal cost solely depending on the substitutability between banks, given by (21). In both cases, the government cannot use the CBDC spread to influence the banking sector.

To sum up, when the government decreases the CBDC rate and widens the CBDC spread, it directly increases the household’s average cost of liquidity and affects allocation. Moreover, a higher CBDC spread increases the spread on bank deposits, provided that banks have sufficient market power, which raises the household’s cost of liquidity further. The transmission of the CBDC rate through the banking sector is similar to the deposit channel of monetary policy proposed by Drechsler et al. (2017). The authors describe a situation in which the household holds cash issued by the government and deposits issued by banks with market power. Policy-makers can induce an increase in the deposit spread by increasing the household’s opportunity cost of holding cash, captured by the nominal interest rate on risk-free bonds. In our model, instead, the alternative to bank deposits is CBDC. The government can similarly affect banks’ deposit spread by changing the household’s opportunity cost of holding this alternative, i.e. its spread.

3.3 Interest on Reserves

While interest rates on central bank reserves have traditionally not been a particularly salient policy tool (for example, the Fed only started to remunerate reserves in 2008), we briefly analyse them as a stand-in for monetary policy instruments directed at the banking sector: Since the reserve rate shock effectively increases the banks’ cost of deposit provision, we expect that other (unmodeled) monetary policy shocks that affect consumers only through the banking system to be affected by deposit market power in a qualitatively similar way.

In particular, the interest on reserves affects the household’s average cost of liquidity only through its impact on the deposit spread. Suppose the government decreases the reserve rate so that the reserve spread increases.^[3] Taking the first derivative of the average cost of liquidity with respect to the reserve spread, we get

Notice that the marginal impact of the reserve spread is very similar to the indirect effect of the CBDC spread in (27). This is not surprising since both effects work through the banking sector. The impact of the reserve spread is the product of the share of deposits in the total stock of liquid, n _t + 1/z _t + 1, and the change in the deposit spread caused by the change in the reserve spread, ∂ χ t + 1 n / ∂ χ t + 1 r . Again, the more important deposits are as a source of liquidity, the larger is this effect. But, its sign and the magnitude also depend on how the banking sector responds to an increasing reserve spread, ∂ χ t + 1 n / ∂ χ t + 1 r . This key term is in turn determined by how banks react to the higher cost of deposit provision induced by χ ^r and depend to what extent potential outflows to CBDC are considered in banks’ deposit rate setting.

5.1 Policy Shocks

When analyzing the impact of a CBDC rate shock numerically below, we assume it to follow a log AR (1) process

where ρ ^m is the persistence parameter, R ^m is the steady state CBDC rate, and e t m is the exogenous shock. The exogenous shock is non-zero in the first period of the simulation and returns to zero afterward. In order to properly isolate the effect, when analyzing the CBDC rate, we assume that the reserve rate is set so that the reserve spread is constant at its steady state level, i.e. it fulfills

where R ^r is the steady state reserve rate. When we discuss the case of the reserve rates, equivalent assumptions are made, effectively interchanging the processes for R ^r and R ^m .

5.2 Impulse Responses

5.2.1 Response to a CBDC Rate Shock

Figure 1 shows the IRFs, as deviations from the non-stochastic steady state, to a negative 10 basis points shock to the quarterly CBDC rate. Naturally, the decrease in the CBDC rate immediately widens the CBDC spread by essentially the same magnitude as the aggregate capital stock and the risk-free rate changes little. The increasing CBDC spread raises the household’s average cost of liquidity in both the baseline N = 1 and the alternative N = 3 cases so that households choose to enjoy less liquidity services. This decreases the household’s demand for liquidity services but increases the household’s current marginal utility of consumption, reflected in a higher Ω t + 1 c . This is due to our calibration featuring ψ < σ. In other words, the household’s opportunity cost of saving, in utility terms, goes up. The household is incentivized to save less and increase current consumption.^[9] Overall, the effect of the cut is not overly strong in either case, reflecting its size and the assumed scenario of limited CBDC adoption.

Figure 1:

Impulse responses to 10 basis points decrease in CBDC rate. Figure shows the impulse responses of key economic variables to a 10 basis points decrease in the CBDC rate under two different market structures (monopsony N = 1 and oligopsony N = 3).

Nevertheless, there is a substantial difference in the relative magnitudes of the responses, with e.g. the overall impact of the shock on aggregate consumption being almost twice as large in the baseline case with N = 1. As outlined above, in our model, the real effects of a CBDC rate are transmitted via its effect on the liquidity cost χ ^z and the impact of the CBDC spread on this term can be decomposed into a direct effect and an indirect effect, shown in (27). Now, Figure 2 displays this decomposition of the responses of the liquidity cost: The green dashed lines show the direct effects and the red solid lines show the indirect effects. The sum of the lines equals the original impulse responses of the cost of liquidity in Figure 1.

Figure 2:

Decomposition of response of cost of liquidity. Figure shows the decomposition of the response of the average cost of liquidity into direct and indirect effects following a 10 basis points decrease in the CBDC rate. The responses are shown for two market structures (monopsony N = 1 and oligopsony N = 3).

We immediately notice that in the case with N = 1, the indirect effect dominates the direct one and is responsible for the bulk of the χ ^z response. In contrast, with N = 3, the indirect effect is much smaller and dominated by the direct one, resulting in an overall weaker response from the cost of liquidity and consumption.

Recall that the direct effect is equivalent to the ratio of CBDC to liquidity services, m _t + 1/z _t + 1. Since CBDC constitutes only a small fraction of the household’s portfolio, the direct effect is small. However, the indirect effect is the product of the ratio of deposits to liquidity services, n _t + 1/z _t + 1, and the marginal change in the deposit spread induced by the CBDC spread, ∂ χ t + 1 n / ∂ χ t + 1 m . If deposit market concentration is large and the key determinant of bank market power, the impact of each bank’s action on the aggregate is larger. In effect, bank(s) compete less with each other and more with CBDC: Changes in the CBDC spread enter into the banks’ competitive considerations to a much larger extent and the deposit spread is much more sensitive to the CBDC spread. However, in the alternative case where deposit market concentration is low, the “threat” from CBDC matters little for the banks. The equilibrium deposit spread is to a larger extent determined by the differentiation margin, captured by 1/η.

Interestingly, the different assumptions on deposit market power even lead to qualitatively different predictions on how the banking sector is affected by the changing CBDC rates: In the monopsonist N = 1 setting, its decrease induces the bank to raise the deposit spread by so much that their resulting lower return induces the households to not only hold less CBDC but also less deposits. This is reminiscent of Chiu et al. (2023), whose theoretical model predicts decreased financial intermediation for lower CBDC rates in the presence of bank market power.^[10] With N = 3, though, the relatively smaller reaction of the deposit spread instead results in the representative agent choosing to hold more deposits after the shock increased the opportunity cost of CBDC. This suggests that the sources of bank market power is key for the equilibrium impact of CBDC rates: Without knowledge of whether they are driven by concentration or differentiation, measures of bank competition such as deposit markdowns are not informative about the effect of a given CBDC rate on aggregate deposit holdings and bank intermediation.^[11]

To further illustrate how deposit market concentration affects the model economy’s response to the CBDC rate, Figure 3 displays the indirect effect’s relative contribution to the change of χ ^z upon impact of the same CBDC rate shock for different N’s: As in the cases analyzed more extensively above, all these model versions are calibrated to be consistent with the same steady state moments, including the deposit markdown. As we already know from Figure 2, the indirect effect dominates for the case of a banking monopoly (N = 1) but declines relatively quickly when allowing for more deposit providers. Nevertheless, even with a larger number of banks, such as N = 10, the indirect effect’s contribution remains firmly positive and of significant relative size.

Figure 3:

Relative sizes of the indirect effect by bank concentration. Figure shows the relative contribution of the indirect effect to the change in the average cost of liquidity following a 10 basis points decrease in the CBDC rate with different number of banks.

Overall, the above exercises thus support our assertion that the transmission of CBDC rate changes should depend importantly on indirect effects shaped by the degree and type of deposit market power.

5.2.2 Response to a Reserve Rate Shock

To now also check how deposit market concentration affects the transmission of monetary policy tools directed at the banking sector, we briefly analyze the aggregate effects of a reserve rate shock: Figure 4 shows the IRFs of the economy to a 10 basis points decrease in the reserve rate. This effectively increases banks’ costs of deposit provision, causing them to also increase their spreads which in turn raises the cost of liquidity in both specifications. The higher cost of liquidity then affects the allocation through the same mechanisms as described in the CBDC case above.

Figure 4:

Impulse responses to 10 basis points decrease in reserve rate. Figure shows the impulse responses of key economic variables to a 10 basis points decrease in the reserve rate under two different market structures (monopsony N = 1 and oligopsony N = 3).

As we have shown earlier, the impact of the reserve spread on the cost of liquidity is the product of the ratio of deposits to liquidity services, n _t + 1/z _t + 1, and the marginal change in the deposit spread induced by shock, ∂ χ t + 1 n / ∂ χ t + 1 r . We see in Figure 4 that the increasing reserve spread pushes up the deposit spread, and in contrast to the CBDC shock, the aggregate effect is actually slightly more pronounced with lower deposit market concentration. In that case, the banks’ marginal benefit of deposit issuance (left-hand side of (16)) is less sensitive to changes in the deposit spread. When a shock increases the marginal cost of deposit provision (right-hand side of (16)), the banks then increase their deposit spread by more than in a case with higher market concentration. Intuitively, when deposit provision is less concentrated, the “price” banks charge on deposits is to a greater extent dictated by their marginal costs.^[12] The deposit spread is then more sensitive to changes in the reserve spread or alternative shocks with similar effect. Thus, the results above suggest that deposit market concentration has quite different implications for the transmission of traditional monetary policy tools compared to CBDC.

6.1 Optimal Policy

To start, we briefly discuss the Ramsey optimal policy in the context of our model. As many of the results are similar to those in Niepelt (2024), we just summarize the key takeaways here and relegate a more detailed discussion and derivations to Appendix B.

In our framework, for a given CBDC design, the long-run (steady state) socially optimal policy is mainly concerned with inducing an efficient liquidity mix, trading off the utility benefits of bank deposits and CBDC with the resource costs of providing them. Assuming that the Ramsey planner has access to a sufficiently rich set of tools, it sets policy rates so that the resulting spreads equal the unit costs of providing CBDC and reserves, i.e. χ ^m  = μ and χ ^r  = ρ, respectively. At the same time, deposit market power is addressed using a deposit subsidy. Thus, the optimal CBDC rate is independent of deposit market concentration if the government has another tool to influence the latter. The deposit subsidy, however, will generically depend on the number of banks N and the parameter η governing how easily households can substitute between banks. This implies the planner still needs to consider deposit market power and its different sources.

What if the planner does not have access to the deposit subsidy, a theoretical policy tool without obvious real world equivalent? If deposits and CBDC are imperfect substitutes, the planner will in general not be able to achieve the first best allocation, as Niepelt (2024) demonstrates for the special case with N = 1. While our model does not provide for a closed-form solution to the restricted planner problem, we searched numerically for steady state welfare-maximizing policies: In the baseline model with N = 1, our optimization routine suggests χ ^m *^,1 ≈ 0.0032 and χ ^r *^,1 ≈ 0.71 × 10⁻⁴ as optimal policy. In contrast, if N = 3 and bank market power is also due to differentiation, we obtain χ ^m *^,3 ≈ 0.0033 and χ ^r *^,3 ≈ 0.56 × 10⁻⁴.^[13]

The result that χ ^r * < ρ in either case seems intuitive: the planner can partly compensate the missing deposit subsidy by subsidizing the banks indirectly using the return on reserves, which she chooses to remunerate more generously in the restricted solution. Furthermore, we notice that this indirect subsidy is smaller in the high concentration scenario, i.e. χ ^r *^,1 > χ ^r *^,3. At the same time, the planner pays a higher return on money in that scenario, χ ^m *^,1 < χ ^m *^,3. Again, this makes sense: if concentration is low and differentiation is an important determinant of the deposit markdown, a higher CBDC return has a smaller influence on the deposit rates set by the banks and the consolidated government has to rely more on the indirect subsidy through reserves. Hence, both spreads deviate more from the unconstrained optimum. In the high concentration N = 1 case, we just have the reverse.

In conclusion, the above analysis indicates that the source of bank market power is also a pivotal determinant of how a constrained planner would use CBDC- and reserve policy rates if she cannot subsidize deposit provision directly. Naturally, similar as for direct deposit subsidies, an indirect bank subsidy through generous reserve rates may not be feasible due to political economy reasons (or only to some extent): Either increases bank profits, an outcome one could imagine to be unpopular with the public.^[14] Thus, it remains interesting to check to what extent interest payments on CBDC can improve long-run welfare and through which ways.

6.2 Long-Run Effects of CBDC Rates

We now turn to investigating how paying (higher) interest on CBDC affects steady state efficiency and welfare. For the purpose of making the related analysis more practically relevant, we divorce it from the optimal policy analysis in the subsection above: There may be various reasons why policymakers may not be able or willing to subsidize banks directly or indirectly, e.g. equity considerations. Instead, we just consider the long-run effects of a higher CBDC return compared to our baseline steady state(s). Recall that we calibrated the latter to be in line with what currently appears to be a plausible scenario for CBDC introduction, in that the digital currency does not pay a nominal return and is designed so that households choose to hold a CBDC amount comparable to physical currency in circulation. Thus, we effectively answer how steady state welfare would be affected if the government were to actually pay a higher real return on the digital currency in that scenario and how they come about.

As a starting point for analyzing how the CBDC rate can affect the long-run aggregate allocation and efficiency, Figure 5 briefly affirms the (by now perhaps obvious) point that indirect effects also matter in the long run and more so with higher deposit market concentration: It simply plots the long-run equilibrium deposit rate for different (gross) CBDC rates, starting from the initial steady states calibrated for Section 4. In these, deposit spreads are equal by construction for N = 3 and N = 1. The downward curves are consistent with what we have discussed so far: a higher R ^m makes CBDC more attractive, exerting downward pressure on the deposit spread which is substantially more pronounced in the high concentration (N = 1) case. Again, this is because the large bank competes only with CBDC without interbank pressure, causing it to adjust its deposit spread more in case the central bank increases the CBDC rate.

Figure 5:

Deposit spread against different CBDC rate changes. Figure shows how the deposit spread responds to different CBDC rates. The responses are shown for two market structures (monopsony N = 1 and oligopsony N = 3).

Now, to understand how this matters for efficiency, we calculate corresponding consumption equivalent (CE) welfare gains achieved by the household, which measure how much percent of consumption the representative household would be willing to sacrifice to live in the steady states with higher R ^m compared to the original one. To further assess what fraction of the gains is due to indirect effects through the banking sector, we also calculate the CE gain under the counterfactual assumption that banks’ deposit spreads are kept fixed at the level calibrated in the baseline steady state with R ^m  = 0.98: The difference between the true and counterfactual CE gains then isolates the contribution of the indirect effects to welfare.^[15]

Figure 6 illustrates these CE welfare effects for the by now familiar cases of a monopsonistic bank (N = 1) and the oligopsonistic environment with three banks (N = 3). Clearly, the model suggests that paying a higher real return on CBDC has welfare benefits, which reflects the results by Niepelt (2024) who finds CBDC to be a very efficient means of liquidity provision: As the higher rate on CBDC induces the representative agent to hold relatively more of it, the aggregate costs of liquidity provision are reduced. Overall, the contribution of the indirect effect to aggregate welfare is substantial and can account for almost a third of the overall CE gains with N = 1. With N = 3, the share is smaller but still noticeable. Of course, this is again because with differentiation being the more important source of deposit market power, the smaller banks react less to the decreasing CBDC spreads.

Figure 6:

CE welfare gain versus Counterfactual welfare gain. Figure compares the consumption-equivalent (CE) welfare gains and counterfactual welfare gains given different CBDC rates. The comparisons are shown for two market structures (monopsony N = 1 and oligopsony N = 3).

Considering the results above, it seems that indirect effects through the banking sector can be quantitatively important for the welfare gains a central bank can achieve by paying positive interest on CBDC. This is particularly the case if deposit market power is determined by concentration and reiterates the point that the source of bank market power is an important determinant of the model effects of CBDC policy. Interestingly, given that welfare gains are higher for N = 1 than for N = 3, the exercise above also suggests that for given policy rates, moving from low to high banking concentration can potentially increase aggregate efficiency as the differing deposit rates change the composition of the aggregate liquidity mix.

7.1 Short-Run Analysis

Figures A1 and A2 in Appendix C show the impulse responses to a 10 basis points decrease in the CBDC rate with the alternative specifications described above. We see that the main takeaways from Section 5 still stand: An increase in the CBDC spread increases consumption and lowers capital investment. The extent to which deposit market power is shaped by concentration still has a noticeable impact: Higher market concentration noticeably amplifies the impulse responses, although both the aggregate impact of the shock as well as the relative contribution of concentration decrease (increase) for higher (lower) ϵ. Naturally, if the representative agent finds it harder to substitute deposits with CBDC, banks need to be less concerned with the impact of its return R ^m on deposit demand, which dampens the indirect effect and in turn the strength of the overall response. If there is thus less scope overall for indirect effects due to lower substitutability, the impact of deposit market concentration on the effects of the shock is lower.

We also consider how the response to the reserve shock is shaped by η, for which Appendix Figures A3 and A4 display the IRFs for again the cases ϵ = 1/4 and ϵ = 1/10, respectively. In contrast to CBDC, lower (higher) substitutability now increases (decreases) the aggregate effects of the shock but also dampens (strengthens) the impact of concentration N. With less scope for substitution between CBDC and deposits, banks have more scope to react to their increases in costs due to the lower reserve rates, amplifying their aggregate effects.

Overall, the above results imply that the attention central banks will need to pay to deposit market power and its sources is going to depend on how they design their potential CBDC: In case it becomes rather easy for consumers to switch between deposits and the digital currency, it is going to matter more for the effects of monetary policy.

7.2 Long-Run Analysis

Similar considerations as for the short-run analysis above apply to the long-run effects of the CBDC rate: Figures A5 and A6 illustrate the CE welfare gains for different levels of the elasticity of substitution between CBDC and bank deposits for the cases of a monopsonistic bank (N = 1) and the oligopsonistic setting (N = 3), respectively. In either case, a higher elasticity of substitution (lower value of ϵ) corresponds to overall stronger welfare effects of the CBDC rate in the long run, as it ends up affecting deposit spreads and hence the aggregate liquidity mix more.