## References

Agarwal, R. and Kimball, M. (2015). Breaking Through the Zero Lower Bound.

, International Monetary Fund.*IMF Working Papers 15/224*Agarwal, R. and Kimball, M. (2019). Enabling Deep Negative Rates to Fight Recessions: A Guide.

, International Monetary Fund.*IMF Working Papers 19/84*Andolfatto, D. (2018). Assessing the Impact of Central Bank Digital Currency on Private Banks.

, Federal Reserve Bank of St. Louis.*Working Papers 2018-25*Assenmacher, K. and Krogstrup, S. (2018). Monetary Policy with Negative Interest Rates: Decoupling Cash from Electronic Money.

, International Monetary Fund.*IMF Working Papers 18/191*Athey, S., Catalini, C., and Tucker, C. (2017). The Digital Privacy Paradox: Small Money, Small Costs, Small Talk.

, National Bureau of Economic Research.*NBER Working Papers 23488*Bank for International Settlements (2018). Central Bank Digital Currencies.

.*Technical report, Basel Committee on Payments and Market Infrastructures*Barontini, C. and Holden, H. (2019). Proceeding with Caution - a Survey on Central Bank Digital Currency. BIS Papers 101, Bank for International Settlements.

Barrdear, J. and Kumhof, M. (2016). The Macroeconomics of Central Bank Issued Digital Currencies.

, Bank of England.*Bank of England working papers 605*Bech, M. L. and Garratt, R. (2017). Central Bank Cryptocurrencies.

*BIS Quarterly Review.*Bergara, M. and Ponce, J. (2018). Central Bank Digital Currencies: the Uruguayan e-Peso Case. In Masciandaro, D. and Gnan, E., editors,

.*Do We Need Central Bank Digital Currency? Economics, Technology and Institutions. SUERF Conference Volume*Biais, B., Bisière, C., Bouvard, M., and Casamatta, C. (2019). The Blockchain Folk Theorem.

32(5):1662–1715.*Review of Financial Studies,*Biais, B., Bisière, C., Bouvard, M., Casamatta, C., and Menkveld, A. J. (2018). Equilibrium Bit-coin Pricing.

, Toulouse School of Economics (TSE).*TSE Working Papers 18-973*Bjerg, O. (2017). Designing New Money - The Policy Trilemma of Central Bank Digital Currency.

, June 2017.*CBS Working Paper*Bolt, W. and Van Oordt, M. R. (2019). On the Value of Virtual Currencies.

.*Journal of Money, Credit and Banking, forthcoming*Bordo, M. D. and Levin, A. T. (2017). Central Bank Digital Currency and the Future of Monetary Policy.

, National Bureau of Economic Research.*NBER Working Papers 23711*Borgonovo, E., Caselli, S., Cillo, A., and Masciandaro, D. (2018). Between Cash, Deposit And Bitcoin: Would We Like A Central Bank Digital Currency? Money Demand And Experimental Economics.

.*BAFFI CAREFIN Working Papers 1875*Bounie, D., François, A., and Van Hove, L. (2017). Consumer Payment Preferences, Network Externalities, and Merchant Card Acceptance: An Empirical Investigation.

51(3):257–290.*Review of Industrial Organization,*Brunnermeier, M. K. and Niepelt, D. (2019). On the Equivalence of Private and Public Money.

, National Bureau of Economic Research.*NBER Working Papers 25877*Chakravorti, S. (2010). Externalities in Payment Card Networks: Theory and Evidence.

9(2):1–28.*Review of Network Economics,*Chiu, J., Davoodalhosseini, M., Jiang, J. H., and Zhu, Y. (2019). Central Bank Digital Currency and Banking.

, Bank of Canada.*Staff Working Papers 19-20*Chodorow-Reich, G., Gopinath, G., Mishra, P., and Narayanan, A. (2018). Cash and the Economy: Evidence from India’s Demonetization.

, National Bureau of Economic Research.*NBER Working Papers 25370*Davoodalhosseini, M. (2018). Central Bank Digital Currency and Monetary Policy.

, Bank of Canada.*Staff Working Papers 18-36*Diamond, D. W. and Rajan, R. G. (2001). Liquidity Risk, Liquidity Creation, and Financial Fragility: A Theory of Banking.

109(2):287–327.*Journal of Political Economy,*Donaldson, J. R., Piacentino, G., and Thakor, A. (2018). Warehouse Banking.

129(2):250 – 267.*Journal of Financial Economics,*Engert, W. and Fung, B. (2017). Central Bank Digital Currency: Motivations and Implications.

, Bank of Canada.*Discussion Papers 17-16*Engert, W., Fung, B., and Hendry, S. (2018). Is a Cashless Society Problematic?

, Bank of Canada.*Discussion Papers 18-12*Fung, B. and Halaburda, H. (2016). Central Bank Digital Currencies: A Framework for Assessing Why and How.

, Bank of Canada.*Discussion Papers 16-22*Garratt, R. and van Oordt, M. (2019). Privacy as a Public Good: A Case for Electronic Cash.

, Bank of Canada.*Staff Working Papers 19-24*Goodfriend, M. (2016). The Case for Unencumbering Interest Rate Policy at the Zero Lower Bound.

.*Paper presented at the August 2016, Jackson Hole Conference*Gopinath, G. and Stein, J. C. (2018). Banking, Trade, and the Making of a Dominant Currency.

, National Bureau of Economic Research.*Working Paper 24485*He, D., Leckow, R. B., Haksar, V., Mancini-Griffoli, T., Jenkinson, N., Kashima, M., Khiaonarong, T., Rochon, C., and Tourpe, H. (2017). Fintech and Financial Services; Initial Considerations.

, International Monetary Fund.*IMF Staff Discussion Notes 17/05*Kahn, C. M., Rivadeneyra, F., and Wong, T.-N. (2019). Should the Central Bank Issue E-Money?

, Federal Reserve Bank of St. Louis.*Working Papers 2019-3*Katz, M. L. and Shapiro, C. (1985). Network Externalities, Competition, and Compatibility.

75(3):424–440.*The American Economic Review,*Keister, T. and Sanches, D. R. (2019). Should Central Banks Issue Digital Currency?

, Federal Reserve Bank of Philadelphia.*Working Papers 19-26*Kim, Y. S. and Kwon, O. (2019). Central Bank Digital Currency and Financial Stability.

, Economic Research Institute, Bank of Korea.*Working Papers 2019-6*Krugman, P. (1979). Increasing Returns, Monopolistic Competition, and International Trade.

9(4):469–479.*Journal of International Economics,*Lagarde, C. (2018). Winds of Change: The Case for a New Digital Currency.

.*Remarks at the Singapore Fintech Festival*Mancini-Griffoli, T., Martinez Peria, M. S., Agur, I., Ari, A., Kiff, J., Popescu, A., and Rochon, C. (2018). Casting Light on Central Bank Digital Currencies.

, International Monetary Fund.*IMF Staff Discussion Notes 18/08*Masciandaro, D. (2018). The Demand for a Central Bank Digital Currency: Liquidity, Return and Anonymity. In Masciandaro, D. and Gnan, E., editors,

.McAndrews, J. (2017). The Case for Cash.

, Asian Development Bank Institute.*ADBI Working Papers 679*Meaning, J., Dyson, B., Barker, J., and Clayton, E. (2018). Broadening Narrow Money: Monetary Policy with a Central Bank Digital Currency.

, Bank of England.*Bank of England working papers 724*Merrouche, O. and Nier, E. (2012). Payment Systems, Inside Money and Financial Intermediation.

21(3):359–382.*Journal of Financial Intermediation,*Miccoli, M. (2019).

, Bank Runs and Welfare. mimeo.*Central Bank Digital Currencies*Niepelt, D. (2019). Reserves for All? Central Bank Digital Currency, Deposits, and their (Non)-Equivalence.

*International Journal of Central Banking, forthcoming.*Norges Bank (2018). Central Bank Digital Currencies.

*Norges Bank Papers No 1/2018.*Prasad, E. (2018). Central Banking in the Digital Age: Stock-Taking and Preliminary Thoughts.

, Hutchins Center on Fiscal and Monetary Policy at Brookings.*Discussion paper*Rochet, J.-C. and Tirole, J. (2006). Externalities and Regulation in Card Payment Systems.

5(1):1–14.*Review of Network Economics,*Rogoff, K. (2016).

Princeton University Press, 1st edition.*The Curse of Cash.*Stein, J. C. (2012). Monetary Policy as Financial Stability Regulation.

127(1):57.*The Quarterly Journal of Economics,*Sveriges Riksbank (2017). Central Bank Digital Currencies.

.*Technical report, 1st interim report on the e-krona project*Sveriges Riksbank (2018a). Special Issue on the e-Krona. Technical report, Sveriges Riksbank Economic Review.

Sveriges Riksbank (2018b). The Riksbank’s E-Krona Project. Technical report, Report 2.

Wakamori, N. and Welte, A. (2017). Why Do Shoppers Use Cash? Evidence from Shopping Diary Data.

49(1):115–169.*Journal of Money, Credit and Banking,*Wright, R., Tekin, E., Topalli, V., McClellan, C., Dickinson, T., and Rosenfeld, R. (2017). Less Cash, Less Crime: Evidence from the Electronic Benefit Transfer Program.

, 60(2):361–383.*Journal of Law and Economics*Yao, Q. (2018). A Systematic Framework to Understand Central Bank Digital Currency.

61(3):033101.*Science China Information Sciences,*

## 5 Appendix

### A Proofs

**Proof of Lemma 1.** A CBDC can be designed in a manner that mimics cash: (*θ*, *r _{cbdc}*) = (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

*r*= 0, but this policy combination is never optimal, as seen from (36) and (37) where

_{cbdc}*θ*<

^{ce}*θ*< 1. Hence,

^{nce}**Proof of Lemma 2.** Replacing from (36), (40) and (23) into (18)–(20) gives the expressions for the shares of money, *ce*), in terms of parameters only. We can then calculate the infima of

and therefore, given *η _{d}* =

*η*= 0.

_{cbdc}^{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

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 *r _{cbdc}* = 0, condition (44) cannot be violated. Second, as inf

*θ*, since

^{nce}*θ*>

^{nce}*θ*).

^{ce}**Proof of Lemma 3.** *W ^{ce}* (

*θ*,

*r*) can be determined by solving the following system of 11 equations in 11 unknowns, which gives the expression (30):

_{cbdc}Similarly, the solution for *W ^{nce}* (

*θ*,

*r*) is found by setting

_{cbdc}*s*= 0 in the above, and solving. This yields the expression in (31).

_{c}**Proof of Proposition 1.** First, we note that *ce*). Hence, as long as the unconstrained *ce* is feasible, it is optimal. Therefore, the relevant comparison centers on

Second,

which means that for __ s__) and the value of bank intermediation (

*A*–

*ϕ*) is large enough.

Third, whenever

since *A* — *ϕ*) that is large enough to induce a switch from *ce* to *nce* is higher when policies are set at

### 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

where *T* = *r _{cbdc}s_{cbdc}* and

*π*has been replaced using (10), (11), and (16). Further replacing for

*s*,

_{d}*s*, and

_{cbdc}*r*with expressions as shown in the proof of Lemma 3, this gives a closed-form expression for

_{d}*C*. From this expression, we obtain

_{d}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,

where we find that at *r _{cdbc}* = 0, this term is negative overall, given the parameter space in (17) and

*θ*≤ 1. Hence, a marginal CBDC interest rate cut from

*r*= 0 to

_{cdbc}*r*< 0 always raises the welfare of depositors.

_{cdbc}#### 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:

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,

#### 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

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

and in *nce*

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, (*A* – *ϕ*), 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

where *β ∫*_{n≠i} *x*_{j(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

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, *r _{cbdc}* ≠ 0 is now optimal, even when network effects play no role. Depending on parameter values,

*θ*

^{ce}: 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

where *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

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

Moreover, deposit market equilibrium is derived from *D* = *L*, where *D* is from *s _{d}* in (19):

Together, (12), (62), and (63) provide three equations in three unknowns, *L*, *R* and *r _{d}*. Replacing

*l*=

*νL*, we can solve this to attain

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

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 *θ ^{ce}* and

and therefore *ν* > 0 means that both *θ ^{ce}* and

*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

then this ensures that *r _{cbdc}* >

*r*(by equation (23)), while the payments profile (

_{d}*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 *r _{cbdc}* 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 *r _{d}* from (23) into (45). This yields

which means that when the policy combination (*θ*, *r _{cbdc}*) is set such that the condition above is violated, as for example for a sufficiently negative

*r*, deposits strictly dominate CBDC.

_{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

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

where *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

leading to optimal consumption share of *G*

where *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

which can also be written as

This implies a sorting of households, such that households with *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.

^{}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 *k*_{0} > 1 to ensure that lending frictions always bind such that *k* < *k*_{0}.

^{}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

^{}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, *r _{cbdc}* ≥ – (1 –

*θ*)

*ρ*

^{−1}and

*θ*>

*ρ*(

*r*–

_{d}*r*), 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 (

_{cbdc}*θ*= 1) that pays negative rates (

*r*< 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.

_{cbdc}^{}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

^{}26

The restriction (*A* – *ϕ*) > 1 ensures that aggregate output (and hence consumption) increases in financial intermediation in equilibrium. This follows directly from the derivative *k* ≤ 1, is always positive for (*A* − *ϕ*) > 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 __ 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* + *ρr _{cbdc}* > 1 – 2

__–__

*s**g*(0) without any qualitative impact on our analysis.

^{}30

Given (17), these optimal policies can range between

^{}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, *θ ^{nce}* >

*θ*; second,

^{ce}*θ*is flatter.

^{nce}^{}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

^{}36

Formally, we can verify that *A* – *ϕ* → 1 the expression becomes *θ* ≤ 1. Hence,

^{}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, *r _{d}* =

*A*–

*ϕ*– 1 and therefore the deposit rate is irresponsive to

*θ*and

*r*

_{cbdc}^{}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.