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5 Appendix

A Proofs

Proof of Lemma 1. A CBDC can be designed in a manner that mimics cash: (θ, rcbdc) = (1, 0). From this, it directly follows that welfare in both ce and nce is higher than in an equilibrium without CBDC: in both ce and nce the central bank could attain the same welfare as in the equilibrium without CBDC, by setting θ = 1 and rcbdc = 0, but this policy combination is never optimal, as seen from (36) and (37) where θce < θnce < 1. Hence, W(1,0)<W(θnce,rcbdcnce)<W(θce,rcbdcce), where the last inequality follows from (38).

Proof of Lemma 2. Replacing from (36), (40) and (23) into (18)(20) gives the expressions for the shares of money, scce, sdce, and scbdcce, when all forms of money exist (ce), in terms of parameters only. We can then calculate the infima of scce, sdce, and scbdcce, respectively, over the parameter space defined by (17). This yields

inf scce=134inf sdce=217inf scbdcce=117

and therefore, given s¯117 in (17), it follows that ηd = ηcbdc = 0.35

Moreover, using (36) and (40), as well as (23), we can also verify that two necessary conditions for positive CBDC take up, which are subsumed by the CBDC design constraint (15), are also satisfied. These conditions are

rcbdc(1θ)ρ1(44)
θ>ρ(rdrcbdc)(45)

which respectively rule out the strict dominance of CBDC by cash and deposits (i.e., ensure that neither cash nor deposits offer all households a strictly better utility than CBDC) as per (5) and (7). First, since sup θce=1617<sup θnce=2324<1, while rcbdc = 0, condition (44) cannot be violated. Second, as inf (θceρrd)=117 over the parameter space in (17), (45) is never violated either (and this necessarily also holds for θnce, since θnce > θce).

Proof of Lemma 3. Wce (θ, rcbdc) can be determined by solving the following system of 11 equations in 11 unknowns, which gives the expression (30):

W=ρ(1+φk0+(A1φsd2)sd)sdEdscbdc1Ecbdc1scbdc2Ecbdc2scEcsd=θ+ρ(rdrcbdc)2scbdc=scbdc1+scbdc2scbdc1=θρ(rdrcbdc)2scbdc2=1+ρrcbdcθ2sc=1θρrcbdc2Ed=|012sd|=sd2Ecbdc1=|θ(sd+12scbdc1)|=(θsd12scbdc1)Ecbdc2=|θ(sd+scbdc1+12scbdc2)|=sd+scbdc1+12scbdc2θEc=|1(112sc)|=12scrd=2(A1φ)+ρrcbdcθ2+ρ

Similarly, the solution for Wnce (θ, rcbdc) is found by setting sc = 0 in the above, and solving. This yields the expression in (31).

Proof of Proposition 1. First, we note that Wce(θce,rcbdcce)>Wnce(θnce,rcbdcnce), per (38). Moreover, Wce(θce,rcbdcce)>Wnce(θ˜,r˜cbdc) per definition, as welfare under unconstrained optimal policies exceeds welfare under constrained optimal policies within a given equilibrium (namely, ce). Hence, as long as the unconstrained ce is feasible, it is optimal. Therefore, the relevant comparison centers on Wce(θce,rcbdcnce) versus Wce(θ˜,r˜cbdc) when the network effects constraint matters, that is, when θce+rcbdcce=θce>12s¯.

Second,

infA,φ,s¯[Wnce(θnce,rcbdcnce)Wce(θ˜,r˜cbdc)]>0ρ>1.431=ρ¯(46)

which means that for ρ<ρ¯, the policy combination (θ˜,r˜cbdc) always (i.e., for any values of other parameters) welfare dominates (θnce,rcbdcnce), and hence cash never vanishes under optimal policies. Instead, for ρ>ρ¯, there exist parameterizations, including the extremes of (Aφ)=52 and s¯=117, such that (θnce,rcbdcnce) welfare dominates (θ˜,r˜cbdc). That is, when ρ>ρ¯, cash can optimally be allowed to vanish, when network effects are strong enough (s) and the value of bank intermediation (Aϕ) is large enough.

Third, whenever θce>θ¯=12s¯, it is necessarily true that

Wce(θ˜,r˜cbdc)Wce(θ¯,0)>0(47)

since (θ˜,r˜cbdc)=(θ¯,0) is within the possibility set of (θ˜,r˜cbdc) but is not optimally chosen, as seen from (41) and (42). Hence, the range of parameter values where Wce(θ˜,r˜cbdc)>Wnce(θnce,rcbdcnce) is broader than the range where Wce(θ¯,0)>Wnce(θnce,rcbdcnce). To put this in more concrete terms, consider ρ>ρ¯ and s¯=117. Then, the value of (Aϕ) that is large enough to induce a switch from ce to nce is higher when policies are set at (θ˜,r˜cbdc) than when they are set at (θ¯,0).

B Derivation of distributional effects

The foundations for Figure 4 are found by considering the impact of a CBDC on, respectively, deposit, cash and CBDC users. We use the term “after the introduction of a CBDC” to indicate the comparison between a world with cash and deposits only, and one where CBDC is available as an additional payments instrument.

B.1 Depositors

For a household that continues using deposits after the introduction of a CBDC, such as i = 0, nothing changes in terms of the payment preference aspect of utility through the introduction of a CBDC. Hence, her tradeoff centers on consumption, as represented by

Cd=1+rd+πT(48)
=1+rd(1sd)+φk0+(A1φsd2)sdrcbdcscbdc(49)

where T = rcbdcscbdc and π has been replaced using (10), (11), and (16). Further replacing for sd, scbdc, and rd with expressions as shown in the proof of Lemma 3, this gives a closed-form expression for Cd. From this expression, we obtain

Cdθ|rcbdc=0=ρ(Aφ2)+θ2(2+ρ)2<0(50)

which means that the introduction of a non interest-bearing CBDC always raises welfare for households that continue using deposits, because the introduction of a CBDC is equivalent to lowering θ from θ = 1 (cash equivalence) to a lower value.36 Put differently, the more intensely the CBDC competes with bank deposits (lower θ) the more it pushes up deposit rates, and the larger the welfare gains to depositors.

Moreover,

Cdrcbdc=4ρ(4(Aφ1)ρ+2rcdbc(8+ρ(5+ρ))+θ(4+ρ))2(2+ρ)2

where we find that at rcdbc = 0, this term is negative overall, given the parameter space in (17) and θ ≤ 1. Hence, a marginal CBDC interest rate cut from rcdbc = 0 to rcdbc < 0 always raises the welfare of depositors.

B.2 Cash holders

For a household that continues using cash after the introduction of a CBDC (provided cash remains in use), such as i = 1, welfare effects similarly center on consumption only, as her payments instrument preferences are unaffected. Contrary to depositors, however, the impact of a non interest-bearing CBDC on cash holders is straightforward: While depositors see gains from increased deposit rates that (more than) compensate for lost firm profit transfers, cash holders see only those lost profit transfers, and are therefore necessarily worse off: Ccθ|rcbdc=0>0. Those cash holders would be even worse off if network effects push cash out of use and they are forced to take solace in a CBDC that is more distant from their payment preferences.

The impact of negative CBDC rates is also straightforward for cash holders. As cash pays no interest, the only channels through which cash holders are affected are π, which rises as the CBDC rate declines (increased financial intermediation), and T, which is positive when CBDC interest rates are negative (CBDC holders are taxed, and the proceeds accrue to all households). That is, Ccrcbdc<0, as shown in Figure 4.

B.3 CBDC users

For households that switch to CBDC after it has been introduced, the key question is whether their gains in payment preferences outweigh lost consumption arising from bank disintermedia-tion. Former depositors switching to CBDC, always see a welfare improvement overall. If they did not, they would have remained depositors, since depositors see welfare gains from the introduction of a CBDC, as per (50). The i = θ household experiences the largest welfare gain from the availability of a CBDC, because the CBDC precisely meets her payments preferences.

However, some of the i > θ CBDC holders would have been better off had CBDC not existed. After all, the household that is exactly indifferent between holding cash and holding CBDC experiences a welfare loss, since all cash holders lose welfare, and this household is indifferent between the welfare loss of continuing to hold cash, and the welfare loss from holding CBDC. CBDC holders with i marginally below this indifferent household would also certainly see an overall welfare loss. CBDC does not offer them enough of an attractive payment option to compensate for the loss in firm profit transfers. Finally, a negative CBDC rate acts as a tax on CBDC holders, and therefore reduces their welfare, as shown in Figure 4.

C Extensions

C.1 Constant returns to scale production function

The baseline model considers a decreasing returns to scale (quadratic) firm production function. Here, we show that central components of the optimal policy profiles we derived, as represented by equations (36), (37) and (40), are robust to the using a constant returns to scale production function. Instead of Y=(Ak2)k, we now replace (10) with

Y=Ak(51)

Following the same steps as in the main text, we obtain the following outcomes for optimal policies in ce

θce=1+ρ(Aφ1)2(52)
rcbdcce=0(53)

and in nce

θnce=2+ρ(Aφ1)3(54)
rcbdcnce=0(55)

Thus, the optimal unconstrained CBDC interest rate remains zero, in both ce and nce. Moreover, the CBDC is optimally made more similar to cash (i.e., to help preserve bank deposits) when the value of bank intermediation, (Aϕ), rises.37

C.2 Anonymity externalities

In this extension, we consider the possibility that anonymous means of payment, like cash, are associated with negative externalities, due to the potential for illicit activities. There can be legitimate reasons that households desire anonymous forms of money, but by providing for that demand, the illicit uses of anonymity are also bolstered.38 In particular, we now let the utility of household i be given by

Ui(j)=ρCj|xji|ηjβnixj(n)dn(56)

where β ∫ni xj(n)dn captures the notion of negative externalities from anonymous means of payment. Here, n ∈ [0,1] represents “all other households”.39 While every household with i > 0 likes anonymity in her own means of payments, every household also dislikes anonymity in other households’ transactions. The weight β ∈ [0,1] represents the extent to which the household dislikes others’ anonymity in payments transactions.

Following the same steps as before, we derive unconstrained optimal policies as

θce=2+ρ(2(Aφ)1)β(2+ρ)4+3ρβ(4+ρ)(57)
rcbdcce=2β4+ρ(Aφ)(4+ρ)2(3+ρ)4+3ρβ(4+ρ)(58)

which nest the solutions in (36) and (40) for β = 0.40 The most interesting aspect of these solutions is that, for any β > 0, rcbdc ≠ 0 is now optimal, even when network effects play no role. Depending on parameter values, rcbdcce can be either positive or negative. In particular, in relation to the value of bank intermediation, rcbdcce moves inversely with θce: A higher value of bank intermediation leads to a more cash-like optimal CBDC design and lower (including possibly negative) CBDC rates.

This inverse relation between optimal CBDC rates and optimal CBDC design parameter θ is intuitive, and derives from a ranking of forms of payment according to their anonymity externalities: cash is worst, deposits are best, and CBDC is somewhere in between, depending on its design. When CBDC design is optimally quite similar to cash, then it is also optimal to have negative CBDC rates, to push more households into deposits, and limit the anonymity externalities induced by the CBDC. Instead, when CBDC design is more similar to deposits, then a positive CBDC rate is optimal, to help attract more households away from cash.

C.3 Bank market power

We now consider banks that compete à la Cournot in the loans market, taking the actions of other banks as given. Each bank therefore internalizes that total loans and the interest rates on those loans depend on its individual lending as follows

L=l+(1v)LLl=1(59)
Rl=RL=1(60)

where RL=1 comes from equation (12). Here, ν represents the extent of bank market power, with the extremes of ν = 0 and ν = 1 representing, respectively, perfect competition (i.e., our baseline model) and a monopoly.

The bank’s profit maximization problem is given by

maxl{(R(l)rd)l}(61)

where the bank recognizes the dependence of loan rates on an individual bank’s lending decision: R depends on l. This yields the first order condition

R(l)+R(l)ll=rd(62)

Moreover, deposit market equilibrium is derived from D = L, where D is from sd in (19):

L=ρ(rdrcbdc)+θ2(63)

Together, (12), (62), and (63) provide three equations in three unknowns, L, R and rd. Replacing Rl=1 from (60), and l = νL, we can solve this to attain

L=ρ(A1rcbdcφ)+θ2+ρ(1+v)(64)
R=Aφ1ρ(A1rcbdcφ)+θ2+ρ(1+v)(65)
rd=2(A1φ)+(1+v)(ρrcbdcθ)2+ρ(1+v)(66)

Following the same steps as before, we again derive welfare and, from there, optimal policies

θce=8+2ρ(2v+(Aφ)(4+ρ+2ρv)1)ρ2(1+(2v)v)16+3ρ2(1+v)2+8ρ(2+v)(67)
rcbdcce=2v1+3ρ(A1φ)16+3ρ2(1+v)2+8ρ(2+v)(68)

where for ν = 0 we retrieve our earlier solutions for optimal policies in (36) and (40). Indeed, by comparing the above expressions to (36) and (40), we can see the direction in which ν > 0 pulls optimal policies. That is, using the expressions for θce and rcbdcce in (67) and (68), we numerically obtain that, over the parameter ranges in (17):

inf θceθce|v=0=2795372,sup θceθce|v=0=0inf rcbdccercbdcce|v=0=35163,sup rcbdccercbdcce|v=0=0

and therefore ν > 0 means that both θce and rcbdcce are lower than with ν = 0. This emanates from the fact that greater market power in lending helps insulate banks from the negative impact of a CBDC. Although increased competition for retail funding still drives up banks’ deposit rates, banks with market power partly compensate by also raising loan rates. In view of banks’ increased ability to withstand the impact of a CBDC, the optimal CBDC design moves closer to deposits (lower θ), although the policy maker partly insulates the impact of this move by also cutting CBDC rates into negative territory.

C.4 Alternate equilibria under suboptimal policies

Table 1 listed three equilibria that do not occur under optimal policies. However, these equilibria can come about if policies are set suboptimally.

CBDC and cash Per Lemma 2, deposits never vanish under optimal policies. This is intuitive, since without deposits, our model yields zero intermediation, and the production of consumption goods shuts down. Nevertheless, it is easy to show that suboptimal policies could yield this equilibrium. For instance, for θ = 0, if the CBDC rate is set such that

rcbdc>Aφ1(69)

then this ensures that rcbdc > rd (by equation (23)), while the payments profile (θ = 0) is equivalent to deposits. Hence, the CBDC strictly dominates deposits in this case: no household would choose to hold deposits.

CBDC only Any arbitrarily high rcbdc would kill off both deposits and cash. Households would be paying for these CBDC interest payments through the lump-sum tax T, and therefore this scenario brings only disadvantages to households, who lose payment instrument variety and the productive benefits of bank intermediation, without gaining anything in return.

Cash and deposits There are three ways that a suboptimally designed CBDC could lead a situation where the design constraint (15) is violated such that there is no uptake of CBDC, and only cash and deposits are in use. First, CBDC could be designed in such a way that it is strictly dominated by cash, and violates (44). Second, CBDC design could imply that bank deposits are a strictly preferred form of payment, which occurs when (45) is violated. Third, even if the CBDC is not strictly dominated by cash or deposits, its design could be such that network effects prevent the buildup of a critical mass of CBDC users (15).

To give a concrete example, we replace rd from (23) into (45). This yields

(1+ρρ)θ+rcbdc>Aφ1(70)

which means that when the policy combination (θ, rcbdc) is set such that the condition above is violated, as for example for a sufficiently negative rcbdc, deposits strictly dominate CBDC.

D Deriving a linear city of payments preferences

This appendix provides a stylized model highlighting how a linear-city model of payments preferences can be derived from microfoundations. The model is based on the notion that payments privacy can have value for households, when their digital transactions data can be used by private companies with monopoly power. We concentrate on a simple setup with cash and deposits only, and show how a "line" between these can arise endogenously, including a cutoff that determines household sorting. Once a spectrum of this sort is derived, formulating the intermediate case of a CBDC is a relatively straightforward extension.41

In this model, deposit-based payments are processed by a fintech provider (or a bank that has a similar business model), which is capable of tracking all transactions and is legally unencumbered to use this data to its own benefit. The fintech company is also the sole provider of credit in the economy, and provides loans to households. Moreover, the only means that the fintech company has to assess the creditworthiness of its customers is by parsing their transactions data. For simplicity, we abstract from explicitly modeling deposit and lending markets and interest rates here, and instead focus purely on household choice based on the characteristics of deposits versus cash.

There are two types of products for households to purchase in this economy: G (Good) and B (Bad), where B can be considered a type of sin product, such as alcoholic beverages or cigarettes. Credit quality is inferred from the share of its income that a households spends on G. We assume identical incomes across households, and each household i determines what fraction γ (i) to spend on good G. Each household has a preferred share of its income that it would like spend on each type of product: we denote by p (i) the ideal fraction of household i’s income spent on good G. Households are heterogeneous in their ideal consumption patterns. In particular, households are uniformly distributed on p (i) ∈ [0,1]. Moreover, any distance between a household’s ideal and actual consumption allocation, comes at a quadratic disutility cost to the household: (γ (i) − p (i))2.

The key distinction between cash and deposits here, is that deposit transactions are monitored, while cash transactions are not. Monitoring matters because of the credit scores being assigned to households by the fintech company. For households using cash, the company cannot assign individualized credit scores, but rather uses an aggregate credit score, based on the consumption pattern of the average cash user. That is, all cash users are pooled together, in this respect. Instead, deposit using households are differentiated by the fintech company according to their own purchase behavior.

Importantly, once households use deposits for any fraction of their payments, they are unable to hide their overall purchase pattern from the fintech company. Endogenously, the model contains full revelation, because households have known, identical incomes.42 If the fintech company observes a depositor using only a fraction γ (i) of income, and fully using it on G, then the company infers that the household used the rest of its income to purchase B using cash. It is in this sense that deposits and cash cannot be effectively mixed: while the household is technically capable of mixing, the choice for using deposits at all, immediately implies full revelation: payments privacy is undiversifiable.

The aim of this appendix is purely qualitative, and as such we choose simple functional forms to highlight the relevant tradeoff. In particular, we let credit scores be a linear function of γ (i) (for depositors) and assume that the utility derived from a higher credit score also enters linearly in the household’s utility function. Household utility is given by

U(i)=λE[γ(i)j(i)](γ(i)p(i))2(71)

where j (i) is household i’s chosen form of money, namely either d (deposits) or c (cash), λ is a parameter that weighs the utility value of the welfare score as compared to approximating the household’s ideal consumption shares, and

E[γ(i)|j(i)]={γ(i) if j(i)=dγ^ if j(i)=c(72)

where γ^ equals the average share of G purchased by cash holders. Since households are atomistic, a given cash holder will always consume exactly the same as her bliss point: γ (i) = p (i) when j (i) = c.

Instead, a depositor will solve the following optimization problem

maxγ(i){λγ(i)(γ(i)p(i))2}(73)

leading to optimal consumption share of G

γ(i)=λ2+p(i)(74)

where λ2 parameterizes the extent of overconsumption of G induced by monitored transactions.

The choice between cash and deposits then boils down to a comparison of utility under household optimal consumption. A household chooses deposits over cash if and only if utility as a depositor (setting γ(i)=λ2+p(i)) is greater than utility as a cash holder (which equals λγ^). This becomes the following condition for choosing deposits:

λ4+p(i)>γ^(75)

which can also be written as

p(i)>γ^λ4=p¯(76)

This implies a sorting of households, such that households with p(i)>p¯ choose deposits, while households with p(i)<p¯ choose cash. That is, those households whose preferences favor a relatively large share of G consumption, are more eager to engage in a full revelation relationship with the fintech provider, in order to reap the benefits of an improved credit score. Instead, households with a relatively larger preference for consuming B, choose cash, opting out of a depositor relationship with the fintech provider that effectively "forces" them to overconsume G in order to appear more creditworthy. Overall, then, this model shows that heterogeneity in consumption preferences can translate into heterogeneous payment instruments choice.

1

We would like to thank Todd Keister, Morten Bech, Maria Soledad Martinez Peria, Tommaso Mancini-Griffoli, Marcello Miccoli, Beat Weber, Baozhong Yang, Jacky So, Garth Baughman, and audiences at the IMF, the Federal Reserve, the Bank of England, the ECB, the Bank of Israel, Cambridge University, the 12th Paul Woolley Centre Conference, the 12th Swiss Winter Conference on Financial Intermediation, the 19th FDIC/JFSR Conference, the ONB/BIS/CEBRA Conference on Digital Currencies, the Atlanta Fed Conference on the Financial System of the Future, the ADBI Conference on Fintech, the IMF's 2nd Annual Macro-Finance Conference, and the RESMF-FRBIF Workshop on Financial Cycles and Central Banking for helpful comments.

1

For an overview of ongoing CBDC initiatives, see Mancini-Griffoli et al. (2018), Bank for International Settlements (2018) and Prasad (2018). In a survey of 63 central banks, a third of central banks perceived CBDC as a possibility in the medium term (Barontini and Holden, 2019). Notably, the central banks of China, Norway, Sweden, and Uruguay are actively investigating the possibility of introducing a CBDC. The Sveriges Riksbank is expected to decide on the introduction of an eKrona in 2019, while Uruguay’s central bank has run a successful pilot (Bergara and Ponce, 2018; Norges Bank, 2018; Sveriges Riksbank, 2018a).

2

See Mancini-Griffoli et al. (2018) for other design aspects of CBDCs, which are mostly of an operational nature, such as the means to disseminate, secure and clear CBDCs.

3

We parameterize and vary the degree to which bank financing of firms provides efficiency gains. On the special role of depository institutions in intermediation, see Diamond and Rajan (2001) and Donaldson et al. (2018), as well as Merrouche and Nier (2012) for supporting empirical evidence.

4

Empirical research on payment instruments choice attributes a central role to heterogeneous preferences (Wakamori and Welte, 2017). For empirical work measuring preferences for anonymity and the potential demand for CBDC, see Athey et al. (2017), Borgonovo et al. (2018) and Masciandaro (2018).

5

This possibility is increasingly enabled by technological developments, as for instance discussed by Yao (2018) in the Chinese context, and forms the basis for the microfoundations that we develop in Appendix D.

6

Nevertheless, a CBDC is certain to raise aggregate welfare in our framework, but only if it is optimally designed. Moreover, even when aggregate welfare rises, there are distributional effects, and some households are worse off due to CBDC availability. We analyze these distributional effects in Section 3.3.

7

A central bank could attempt to mitigate the decline in bank lending by providing banks with cheap liquidity to replace lost deposits. However, this may not be feasible for two reasons. First, banks’ ability to intermediate funds may depend on their reliance on deposits (see e.g., Diamond and Rajan, 2001; Donaldson et al., 2018). Second, this policy would permanently expose the central bank to credit risk.

8

Beyond satisfying household preferences, the disappearance of cash may reduce economic activity when a portion of the population is unable or unwilling to transact with digital payment methods because of digital illiteracy or informality. See Chodorow-Reich et al. (2018) for an empirical assessment of such costs.

9

There is also a sizeable policy literature discussing the financial stability effects of CBDC (see, e.g., Bech and Garratt, 2017; Fung and Halaburda, 2016; He et al., 2017; Kahn et al., 2019).

10

In our framework, CBDC interest rates embody any type of subsidy or cost associated with holding CBDC. For example, the pilot conducted by the central bank of Uruguay offered subsidies to CBDC holders (Bergara and Ponce, 2018). Moreover, we focus on the steady state effects of CBDC rates on financial intermediation and cash use, rather than their implications for monetary policy over the business cycle. On the relationship between CBDC and monetary transmission, see Agarwal and Kimball (2015, 2019), Assenmacher and Krogstrup (2018), Barrdear and Kumhof (2016), Bordo and Levin (2017), Bjerg (2017), Davoodalhosseini (2018), Goodfriend (2016), Meaning et al. (2018), and Niepelt (2019).

11

The role of strategic coordination and adoption equilibria has also been considered in the literature on cryptocurrencies (Biais et al., 2019, 2018; Bolt and Van Oordt, 2019).

12

We abstract from default risk on bank deposits, which is negligible in normal times due to deposit insurance and implicit bailout guarantees.

13

While some legal jurisdictions allow for deposit accounts that offer a degree of anonymity, these accounts are typically incompatible with payments services. Moreover, providing anonymity in deposits may undermine their complementarity with relationship lending (see e.g., Donaldson et al., 2018).

14

We adopt a uniform distribution for the sake of tractability. Our qualitative results generalize to any single peaked distribution with continuous support and sufficient weight in the tails to ensure that, absent a CBDC, both deposits and cash are sustained as payment instruments.

15

We assume that all forms of money are traded on par.

16

This notion is further explored in Appendix D, which provides an example of how a Hotelling linear-city setup of payments preferences can be microfounded.

17

This can be interpreted as a zero-capital central bank: any revenue that the central bank makes is immediately paid out to households, and any capital shortfall arising from CBDC costs directly leads to a recapitalization through a lump-sum tax.

18

The manner in which we combine consumption with payment preferences bears similarity to the utility function adopted in Gopinath and Stein (2018).

19

See Appendix C.3 for an extension where we allow for market power in the bank loans market.

20

We impose the restriction k0 > 1 to ensure that lending frictions always bind such that k < k0.

21

We adopt a quadratic functional form in the interest of tractability. Appendix C.1 considers a constant returns to scale technology as an alternative. In a derivation available upon request, we also generalize the quadratic technology to the form Y=(AΓk2)k and show that results are robust to varying Γ.

22

The liquidation value is also in terms of consumption goods. The liquidation of projects can be microfounded in a framework similar to Stein (2012) where projects are sold to outside buyers with a lower marginal valuation. While we do not explicitly incorporate outside buyers into our model, doing so would have no impact on welfare provided these buyers are non-resident and/or projects are priced at their opportunity cost to outside buyers. In the interest of tractability, we also assume that funds from liquidated projects cannot be used towards financing other projects. This could be due to a combination of information asymmetries and timing. For example, the time required for outside buyers to verify and pay for a project may exhaust the time for implementation by firms.

23

An implicit assumption in our model is that the central bank does not allow any agent to take a short position in CBDC (i.e., the central bank does not grant CBDC credit to other parties). This precludes arbitrage opportunities by entities without payment preferences, such as banks, which might prefer funding themselves with CBDC rather than deposits. Based on CBDC studies currently underway at central banks, we consider this a realistic assumption.

24

The design constraint subsumes two conditions, rcbdc ≥ – (1 – θ) ρ−1 and θ > ρ(rdrcbdc), which respectively rule out the strict dominance of CBDC by cash and deposits (i.e., ensure that neither cash nor deposits offer all households a strictly better utility than CBDC) as per (5) and (7). For example, a completely cash-like CBDC (θ = 1) that pays negative rates (rcbdc < 0) would violate the first condition, such that all households have a strict preference for cash over CBDC. Because of network externalities, these conditions are necessary, but not sufficient, for positive CBDC take-up.

25

While our model is not quantitative in nature, empirical evidence suggests that network effects only begin to play a significant role when the use of a payments instrument becomes very small, as respresented by s¯117. For instance, in Canada, cash is widely accepted although only about 10 percent of transactions in value terms are conducted with cash (Engert et al., 2018). In contrast, in Sweden, where network effects on cash are becoming a source of concern, cash use stands near 1 percent of transactions value (Sveriges Riksbank, 2017). We discuss the outcome when cash demand is too low to sustain cash, even absent the introduction of CBDC, at the end of Section 3.1.

26

The restriction (Aϕ) > 1 ensures that aggregate output (and hence consumption) increases in financial intermediation in equilibrium. This follows directly from the derivative dYdk, which, given k ≤ 1, is always positive for (Aϕ) > 1.

27

The three equilibria referred to as never occurring under optimal policy are further discussed in Appendix C.4, which considers outcomes under suboptimal CBDC design. The equilibria referred to as “impossible under any policy” are ruled out by the parameter restrictions which imply that, when there is no CBDC, the lowest possible shares of deposits and cash, respectively, are inf sd=722 and inf sc=322, both of which are above s. The derivations for these results are available upon request.

28

Resolving multiplicity in favor of the cashless equilibrium shifts the boundary condition to θ + ρrcbdc > 1 – 2sg (0) without any qualitative impact on our analysis.

29

The design constraint (15) is slack under optimal policies as per Lemma 2.

30

Given (17), these optimal policies can range between θce[717,1617] and θnce[712,2324].

31

This is formally derived in Proposition 1 below.

32

In addition to optimal policy derived in the Proof of Proposition 1, the exact shape of Figure 2 relies on two more properties from (36) and (37): first, θnce > θce; second, θce(Aφ)<θnce(Aφ)<0 and therefore the slope of θnce is flatter.

33

Appendix C investigates the robustness of this key result. We find that the optimality of zero CBDC rates (absent network effects) is robust to the specification of the production function. However, when banks have market power (Appendix C.3), or when anonymous payments instruments create negative social externalities (Appendix C.2), the optimal CBDC rate can deviate from zero.

34

See Appendix B for the underlying derivations.

35

This also remains valid in nce where inf sdce=16 and inf scbdcce=112.

36

Formally, we can verify that Cdθ|rcbdc=0<0 by noting that in (50) when Aϕ → 1 the expression becomes θρ2(2+ρ)2 and when Aφ52 it becomes 12ρ+θ2(2+ρ)2, both of which are smaller than 0 given ρ32 and θ ≤ 1. Hence, Cdθ|rcbdc=0<0 always holds over the parameter space in (17).

37

Decreasing and constant returns to scale production functions do lead to a different bank response to CBDC competition. Under decreasing returns to scale, banks push back against the competition through higher deposit rates (and also lending rates in Appendix C.3). Instead, in the constant returns to scale setup, rd = Aϕ – 1 and therefore the deposit rate is irresponsive to θ and rcbdc

38

The magnitude of negative externalities from cash is a topic of intense debate (Engert et al., 2018; McAndrews, 2017; Rogoff, 2016; Wright et al., 2017).

39

Given that each individual agent is atomistic, the space of all agents excluding one agent remains defined on [0, 1].

40

The same holds for the nce solutions. These are not shown here in the interest of brevity, but are available on request.

41

See also Garratt and van Oordt (2019), who develop a payments model with privacy as a public good, where each consumer fails to internalize that her payments data is used to price discriminate among future consumers, and privacy in government issued electronic cash can create social value.

42

More generally, the underlying assumption can be seen as a requirement on deposit-opening households to reveal their income to the fintech provider.