Chukyo University Institute of Economics
Discussion Paper Series
September 2022

No. 2204
Effects of central bank digital currencies on economic growth:
A note on transaction costs

Akane Watanabe and Akira Yakita

12Sep/23Aug2022/11042022

Effects of central bank digital currencies on economic growth:
A note on transaction costs
Akane Watanabe1 and Akira Yakita2
1

Graduate School of Social Sciences, Nanzan University, Nagoya 466-8673, Japan
2
Faculty of Economics, Nanzan University, Nagoya 466-8673, Japan

Abstract
We study the growth effects of a central bank digital currency in a closed overlapping
generations economy with an AK production technology and cash-in-advance constraints.
Working generations can hold interest-bearing bank deposits as well as money balancesin their portfolios. Money can be central bank digital money or physical money. Holding
physical money involves transaction costs, whereas digital currency does not. Privatebanks intermediate transactions between workers and entrepreneurs. The results indicatethat replacing physical currency with central bank digital currency lowers transaction
costs, raises the balanced economic growth rate, and possibly reduces the long-term
inflation and nominal interest rates.

Keywords: transaction costs, central bank digital currency (CBDC), cash-in-advanceconstraint, balanced growth
JEL Classification: D15, E42, E44, E58

Correspondence: Email yakita@nanzan-u.ac.jp (A. Yakita)
11. Introduction
Central banks such as the Federal Reserve (Fed coin), the Europe Central Bank (digitalEuro), and People’s Bank of China (e-yuan), are exploring the benefits and drawbacks ofcentral bank digital currencies (CBDCs) in consideration of introducing them. 1 Bordo
and Levin (2017) characterize the key features of CBDC in their seminal paper. Pfister(2019) states that the main motive for replacing paper money with CBDC could be tolower transaction costs in the sense that digital currencies save cash balances needed intransactions. 2 KPMG (2016) reports that the transaction costs of using cash and checks
amount to 0.52 % of GDP in Singapore. In this paper, we examine the effect of replacing
physical currency with a CBDC on long-term economic growth through changes intransaction costs in a simple endogenous growth setting in this study.
Many recent studies have examined the macroeconomic and financial stability andwelfare effects brought about by the introduction of CBDCs (Agur et al., 2021; Barrdear
and Kumhof, 2021; Davoodalhosseini, 2021; Fernandez-Villaverde et al., 2021). Most ofthem assume the coexistence of CBDC and deposits in private financial intermediates(some also have physical currency) to analyze the differences in their effects on
macroeconomies. The initiation of a CBDC likely makes paper cash obsolete (Bordo andLevin, 2017). Brunnermeier and Niepelt (2019) establish a sufficient condition underwhich a swap of CBDC for bank deposits does not alter the general equilibrium of theeconomy. Andolfatto (2021) finds that the introduction of a CBDC has no detrimental
effect on bank lending activity and a properly designed CBDC does not threaten financial
stability. By contrast, Kim and Kwon (2022) show that, if a CBDC yields a positive
interest rate, its introduction might cause financial instability of the banking system (i.e.,bank panics). Keister and Sanches (2022) suggest that in designing a single universal
digital currency, policymakers must consider tradeoffs between enhancing efficiency inexchange and crowding out bank deposits, that is, decreasing investment. Nevertheless,the literature cited above does not consider economic growth effects of the CBDC supply.
Our focus in this study is the growth effects.
We consider that a CBDC is a national medium of exchange and a store of value; it isan account-based and interest-bearing claim on the central bank. It can be used in all
1

The People’s Bank of China started demonstration experiments in several cities in
China in 2020.
2
Pfister (2019) mentions this as a motive for issuing a wholesale CBDC, which isdistinguished from general-purpose CBDC in BIS (2018). It has also been highlighted
that general purpose CBDCs make settlements and payments more efficient (Yanagawa
and Yamaoka, 2019). In the literature, terms of general-purpose CBDC, retail CBDC anduniversal CBDC seem to be used interchangeably.
2transactions, that is, it is a universal digital currency. 3 Therefore, in this study, a CBDC
is regarded as a perfect substitute for physical paper money in exchange. 4 We assume
that central bank determines the supply of money, both physical and digital, according toa certain price-level targeting regime in which inflation rate is kept constant over time. 5
In this study, the difference between CBDC and physical currency is that holding physicalcurrency needs transactions costs, whereas holding CBDC does not. We focus only on
this aspect of CBDC in this study. This means that individuals want to hold CBDC rather
than physical money. For our analytical purposes, we assume that initially, the ratio of the
CBDC in the total money holdings is constrained by the central bank to be less than one
so that individuals must hold both the CBDC and physical money in their portfolios. Our
research strategy is to study the effects of relaxing the constraint on the CBDC holding
ratio in individuals’ portfolios, and thereby analyzing the effects of introducing CBDC.
Time is discrete and runs infinitely. The lengths of the periods are normalized to unity.
Each generation lives for three periods: childhood (first), youth (second), and old age(third). Individuals are looked after by their parents in the first period, earn wage income,save for retirement in the second period, and consume their savings in the third period.
Retirement consumption requires that individuals hold money balances in advance.
Young individuals choose portfolio assets among physical and digital money and bankdeposits. Entrepreneurs invest in capital to be used in goods production in the next period.
Private banks intermediate transactions between depositors and entrepreneurs. The salientfeatures of our model setting are: (1) endogenous economic growth is driven by learningby-doing and knowledge spillovers among workers (i.e., an AK production technology);(2) money is introduced into the model by a cash-in-advance constraint approach à laHahn and Solow (1995) 6 ; (3) transaction costs is expressed as a linear function oftransacted amount of (physical) money, as introduced by Baumol (1952); and (4)3We are concerned only with CBDCs and not cryptocurrencies. Schilling and Uhlig(2019) analyze the situation wherein coexistence and competition between dollars andBitcoins (i.e., intrinsically worthless moneys). They study the behavior of the Bitcoin todollar exchange rate and derive conditions under which Bitcoin speculation cannot occur.
However, we can avoid such an issue in this study.
4
A general-purpose, account-based (indirectly held by the central bank), non-interestbearing CBDC is apparently considered in Japan (Bank of Japan, 2020).
5
Keister and Sanches (2022) assume a similar central bank monetary policy.
6
Alternatively, money has been introduced by the money-in-the-utility-function
approach. Abel (1987) and Yakita (1989) consider a money-in-the-utility approach inoverlapping generations settings. New monetarists such as Williamson and Wright (2010)
regard the cash-in-advance approach as at best of second-order importance in modelling
frictions in the exchange process.
3

transaction costs are real, that is, real resources are consumed.
The results are as follows. Relaxing the constraint on the CBDC ratio in individuals’
money holdings reduces transaction costs of money holdings by reducing physical money
holdings (i.e., improved efficiency in monetary transactions). The reduced transaction
costs free the disposable resources of young workers and thereby potentially increasebank deposits. If the increased bank deposits do not lower the interest rate much, then theincreased CBDC ratio induces young individuals to shift their portfolios toward bankdeposits. With private bank intermediation, this results in greater investment ofentrepreneurs and thereby raises the balanced growth rate. Under an AK production
technology with a constant interest rate, this outcome occurs because of a constant interest
rate. The reduction of transaction costs by the CBDC promotes economic growth.
The remainder of this paper is organized as follows. Section 2 introduces our modelwith individuals, government and central bank, private bank, and production sectors.
Section 3 considers balanced growth and presents an analysis of the effects of introducing
CBDC on the balanced growth rate. Finally, Section 4 concludes.

2. ModelWe assume a closed overlapping generations economy populated by individuals who live
for three periods. Each individual draws on parents in the first period (childhood), works
and saves for their retirement in the second period (young), and retires in the third period(old). The population grows at a constant rate, both generation by generation and period
by period. They encounter a cash-in-advance constraint: a worker needs the minimum
amount of money to consume at the beginning of the retirement period. They choose
portfolio assets among physical and digital money and deposits. Furthermore, transacting
between consumable resources and money during the young age incurs transaction costs.
Private banks intermediate between deposits of individuals and investment ofentrepreneurs. Aggregate output production technology is characterized by an AK
technology whose engine is doing-by-doing and knowledge spillovers among workers.
The government controls the money supply to keep the inflation rate constant anddistributes seigniorage revenue to the working generation in each period. For expositional
simplicity, we assume that there is no other government expenditure or taxes.
2.1 Money issue and transfer policy
The consolidated central bank issues both physical and (universal) digital currency. Whenthe economy grows, it is necessary to issue new money to control the price level. We
4assume that the central bank supplies money at a rate of µ and distributes theseigniorage to working individuals as a money transfer in each period. By enforcing thesame price level target for physical and digital money, the central bank should offer toconvert units of physical currency one-for-one into units of digital currency and vice versa.
Transferring physical money to workers does not imply paying interest on money
holdings in this case. Nevertheless, for digital currency, it can be regarded as the interest
payment. Then, the government’s budget constraint is

τ t Nt = µ ( M t / Pt ) or τ t = µ mt ,(1)

where τ t is a lump-sum transfer (including interest payments to CBDC) to a working
individual by government, which is measured in real terms; Nt is the number ofindividuals in their young working age; M t denotes the aggregate nominal money
balance, physical and digital; and Pt

represents the price level in period t .

mt = M t / ( Pt Nt ) represents real money balance per worker.
2.2 Individuals
Individuals are homogeneous in preference and labor productivity. Each individual
inelastically supplies a unit of labor in their second period of life. The budget constraintsof an individual working in period t in the working and retirement periods are writtenas
Pt wt + Ptτ t= Pt c1t + Pt st + (1 + µ )( M t / Nt ) + ε ( M / Nt ) + Pt zt , and(2)

Pt +1ct2+1 = Pt +1 (1 + rt +1 ) st + ( M t +1 / Nt ) ,(3)

where c1t and ct2+1 are consumption when young and old, respectively; st is realsavings (i.e., bank deposits) 7; wt is the real wage rate in period t ; and rt +1 is the realinterest rate on real savings in period t + 1 . An individual can carry forward the received
7

As is commonly assumed in the literature, we consider only the so-called indirect
financing of entrepreneurs in this study.
5money transfer to the old age period without incurring any transaction costs. By contrast,
we assume here that transactions between consumable resources and physical currency
require transaction costs, which is a linear function of the transacted amount of money as

ε m t + zt , where m t = M t / ( Pt Nt ) is the real physical money balance. Here, ε > 0 is aconstant cost per transaction, and zt ≥ 0 is a lump-sum part of the costs. The cost might
include a fee of opening a bank account. We assume that the transaction round trip ofresources-(physical) cash-resources occurs only once, as is usual in overlapping
generation settings. 8 Therefore, the transaction cost in (2) is the total cost of the “trip.”
Because of money transfers and the money supply rule of the central bank explained intheintroduction,itfollowsthat

M t +1 / Nt= (1 + µ ) M t / Nt,or

(1 + n)mt +1 =(1 + µ )mt / (1 + π t +1 ) in real terms, where 1 + π t +1 =Pt +1 / Pt is the (gross)inflation rate.
The budget constraints of the individual, (2) and (3), can be rewritten in real terms asfollows:
wt = c1t + st + mt + (ε m t + zt ) , and(1 + rt +1 ) st +ct2+1 =

(4)

1+ µ
mt ,
1 + π t +1

(5)

Nt / Nt −1 is a constant (gross) population growth rate.
where 1 + n =At this stage, we consider individuals’ portfolio choices by using (4) and (5). The return
rate of real savings, physical money holding, and digital money holding are 1 + rt +1 ,(1 + µ ) / [(1 + π t +1 )(1 + ε )] , and (1 + µ ) / (1 + π t +1 ) , respectively. It is readily known that

8

Rotemberg (1984, p. 43) designates the lump-sum brokerage fee as “the individual’s
real cost of visiting the financial intermediary.” In this study, unlike Rotemberg (1984),
we assume that transaction costs are borne by individuals without paying to other agents.
Kimbrough (1986) alternatively assumes the transactions time per unit of consumption
depending on the ratio of money holdings to nominal consumption expenditure. By
contrast, Wang and Yip (1992) instead assume a shopping-time technology to feature thetransaction cost approach, where monetary transactions consume an individual’s time.
The last two studies consider real transaction costs in terms of time.
6

digital money is preferred to physical money by individuals because they are perfect
substitutes. Therefore, when the nominal interest rate on deposits is lower than the interest
rate on CBDC, that is, when 1 + rt +1 < (1 + µ ) / (1 + π t +1 ) , then individuals want tosubstitute real savings for digital money if possible, and vice versa. 9 If the central bankincreases the ratio of CBDC, then individuals willingly substitute CBDC for bankdeposits in their portfolios when 1 + rt +1 < (1 + µ ) / (1 + π t +1 ) . 10 However, recalling thatthe central bank constrains the ratio of individuals’ CBDC holdings to be less than one.
Designating the ratio as 0 ≤ φ < 1 , digital money balance per worker becomes φ mt , wherem t= (1 − φ )mt . 11
From (4) and (5), the lifetime budget constraint of the individual is written in periodt values as
wt = c1t + [1 −c2
1+ µ+ ε (1 − φ )]mt + zt + t +1 .
(1 + rt +1 )(1 + π t +1 )1 + rt +1(6)

The term (1 + rt +1 )(1 + π t +1 ) is the nominal (gross) interest rate in discrete time settings,
which is assumed to be greater than unity in this study.
Working individuals must hold money in advance to finance retirement consumption.
The cash-in-advance constraint is assumed to be of the type assumed in Hahn and Solow(1995):

θ Pt +1ct2+1 ≤ ( M t +1 / Nt ) or θ ct2+1 ≤ (1 + µ )mt / (1 + π t +1 ) ,(7)

where θ is an exogenously given constant ( 0 < θ < 1 ). The third-period consumption
goods are purchased with the receipt from real savings as θ → 0 , whereas they arepurchased with money as θ → 1 . Although this assumption imposes a kind of constraint
on individuals’ portfolios, we adopt this approach in this study for analytical simplicity.

We assume that (1 + ε ) ≥ (1 + µ ) / [(1 + rt +1 )(1 + π t +1 )] . Otherwise, an individual does
not hold money. This contradicts to our assumption of cash-in-advance constraints.
10
The logic is apparently consistent with the negative crowding-out effect of CBDC, asemphasized by Keister and Sanches (2022). By contrast, Williamson (2022) states thatthe disintermediation can be good for welfare in the absence of a bank monopoly.
11
Without constraints on the CBDC holding ratio, we have φ = 1 . For analyzing effects
of an increase in CBDC, we here rule out this case here.
97

We also assume that the cash-in-advance constraint is binding, as is commonly assumed
in the literature; that is, ct2+1 =(1 + µ )mt / [θ (1 + π t +1 )] or mt =ct2+1[θ (1 + π t +1 )] / (1 + µ ) .
Therefore, budget constraint (6) can be rewritten as
wt = c1t + {1 + θ [(1 + ε (1 − φ ))c2
(1 + π t +1 )(1 + rt +1 )− 1]} t +1 + zt .
1+ µ
1 + rt +1(8)

The utility maximization problem of a working individual is to maximize their lifetime
utility, subject to constraint (8). To obtain an explicit solution, we assume lifetime utility
as a log-linear function of =U t ln c1t + ρ ln ct2+1 , where ρ

is a discount factor( 0 < ρ < 1 ). From the first-order conditions for utility maximization, we obtain1( wt − zt ) , and1+ ρ(1 + rt +1 ) ρ=
ct2+1( wt − zt ) ,At +1 (1 + ρ )
=
c1t

(9)
(10)

where

1 + ε (1 − φ )1+θ[(1 + π t +1 )(1 + rt +1 ) − 1] .
At +1 =
1+ µ

(11)

We assume that condition At +1 > 0 must hold for ct2+1 > 0 . From (4), (7), (9), (10), and(11), with some rearrangement of terms, we obtain the optimal real savings (i.e., bankdeposits) as follows:st =ρ

wt − zt.
1 + ρ 1 + θ 1 + ε (1 − φ ) (1 + π )(1 + r )
t +1
t +1
1−θ 1+ µ(12)Real savings decrease with the nominal interest rate (1 + π t +1 )(1 + rt +1 ) , whereas they
increase with wage income. The nominal interest rate is the opportunity cost of holding
money balances. An increase in the opportunity cost also increases the price of second
period consumption because it requires money in advance (see equation (8)). The
transaction costs, per transaction ε and lump-sum z , are likely to reduce the realsavings because of resource costs. Nevertheless, an increase in the ratio of digital money
holdings φ increases real savings and private bank deposits, ceteris paribus.
2.3 Production
8

Goods production uses labor and capital. Goods producers (i.e., entrepreneurs) borrowfunds from private banks and invest in capital used in the next period. They pay the grossinterest rate, which is equal to the marginal product, to the banks from which they borrow.
Aggregate production technology is assumed to be represented by a Cobb–Douglas
production function:
Yt = BKtα ( Et Nt )1−α

( B > 0 and 0 < α < 1 ),(13)

where Yt is the aggregate output in period t , Kt represents the aggregate capital stock
in period t , and labor productivity is assumed to be Et = ( Kt / Nt ) / a ( a > 0 ). This
formulation is developed by Grossman and Yanagawa (1993). Under the technology, theaggregate production function can be rewritten as Yt = Baα −1Kt , that is, an AK
technology. It is well known that the first-order conditions for profit maximization are
1 + rt =1 + r =α Baα −1 ,(14)

wt= (1 − α ) Baα −1kt ,(15)

where kt = Kt / Nt is the capital–labor ratio. The interest rate remains constant over time.
The wage rate increases proportionally with the capital-labor ratio. Capital fully
depreciates after it is used in production in a period.
2.4 Private banks
Private banks intermediate funds from depositors to entrepreneurs (i.e., goods producers).
For simplicity, we assume away transaction costs that are involved in theseintermediations in this study.
2.5 Market equilibrium
The equilibrium condition in the capital market is given as Kt +1 = st Nt , which is(1 + n)kt +1 =st in per worker terms. Using (12), (14), and (15), the equilibrium condition
can be rewritten as9(1 − α ) Baα −1kt − ztρ.
(1 + n)kt +1 =1 + ρ 1 + θ 1 + ε (1 − φ ) (1 + π )(1 + r )
t +1
t +1
1−θ 1+ µ(16)

The perfectly competitive labor market equilibrium ensures full employment. The
wage rate is equal to the marginal product of labor.
The money market equilibrium is represented by the condition M t= (1 + µ ) M t −1 . The
left-hand side is the nominal total money demand of the young working generation,
whereas the right-hand side is the total money supply of the old generation, which is
inflated by newly issued money in period t − 1 . The market-clearing condition isrewritten in per worker terms as(1 + n)mt =(1 + µ )mt −1 / (1 + π t ) .

(17)

2.6 Lump-sum transaction costs
We now specify the lump-sum part of the transaction costs involved in holding physicalmoney. The lump-sum part is assumed to be proportional to the per worker output; thatis,zt = b(Yt / Nt ) ( b ≥ 0 ),(18)

where b is a constant. In an endogenous growth economy, it is natural to assume thatlump-sum costs vary with per worker output. If the lump-sum costs are constant, then
their size relative to the wage rate becomes negligible as time goes infinite. This seems
to be implausible. 12 We assume that b is sufficiently small, that is, (1 − α ) > b .
Then, from (15) and (18), (16) can be rewritten in aggregate terms as
Kt +1 =[(1 − α ) − b]Baα −1Kt .
1 + ρ 1 + θ 1 + ε (1 − φ ) (1 + π )(1 + r )
t +1
t +1
1−θ 1+ µρ

(19)

3. Balanced growth
Before considering the dynamics of the system, we first consider the initial period, thatis, t = 0 . The initial levels of K 0 , N −1 , and M 0 are given. Money balances are
12

All transaction costs would disappear totally with full digitalization, although we ruleout this possibility in this study.
10

sharedequally

by

the

old

generation

N −1 .

From

(3),

we

have

c02 =
(1 + r )( K 0 / N −1 ) + ( M 0 / N −1P0 ) , where the price level P0 has been determined inperiod t − 1 . The price level in period t = 1 or, equivalently, the inflation rate=
π1 ( P1 / P0 ) − 1 , is determined in period t = 0 so that consumption plan c22 satisfies
the cash-in-advance constraint with equality. 13
We define a balanced growth path as a path on which the aggregate real endogenous
Y +1 Kt +1 M t +1 / Pt +1 (1 + n)mt +1==() ≡ 1 + γ , where
variables grow at the same rate, t=YtKt
M t / Pt
mt

γ designates the balanced growth rate. 14 The dynamics of the system are represented by

the difference equation (19), which has two endogenous variables, Kt and π t .
Completing the dynamic analysis requires one more equation governing the movement in

π t . 15 Nevertheless, it is difficult to solve the dynamics explicitly in a dynamic system of
the two equations. In this study, we are concerned with the effect of the changes in thetransaction costs on the long-term growth rate. Therefore, we assume the existence andstability of such a balanced growth path. If the central bank keeps issuing growth moneyon the balanced growth path, that is, controlling µ to satisfy condition 1 + µ =1 + γ ,then we have π = 0 on the balanced growth path from (17). This is considered as theprice-level target discussed in the introduction. In this case, the targeted inflation rate iszero. Although this assumption seems to be restrictive, it is useful for isolating the effectsof transaction costs on balanced growth, thereby neutralizing the effects of inflation.
Recall that our purpose is to present an analysis of the effects of replacement of physicalcurrency with digital currency on balanced growth rates.
Under the monetary policy of 1 + µ =1 + γ , that is, when π = 0 , the economy
achieves a balanced growth rate. 16 The balanced growth rate is obtained from (19) as

13

The resource constraint is explained in Appendix A.
We assume that γ > n as is commonly assumed in the literature. Otherwise, perworker income approaches zero as time approaches infinity.
15
The other difference equation of Kt and π t is explained in Appendix B. If thesystem is not stable, then examining the comparative static effects of parameter changes
makes no sense.
16
The case in which the money supply rate (and the CBDC interest rate) is kept constant14

11

[(1 − α ) − b]Baα −1
ρ
,1+ γ =
1 + ρ 1 + θ 1 + ε (1 − φ ) (1 + r )1−θ 1+ µ(20)

provided the balanced growth path is stable. From (20) we obtain the following
comparative static results: 17
dγ
dγ
dγ
(21)
< 0 and
< 0.
> 0,
dε
db
dφ
Summarizing the above arguments, we obtain the following propositions.
Proposition 1 Assume a stable balanced growth path on which growth money is supplied.
Then, an increase in the ratio of the CBDC ( φ ) raises the growth rate.
Proposition 2 Assume a stable balanced growth path on which growth money is supplied.
Then, a decrease in the cost per unit of transactions ( ε ) and a decrease in lump-sumtransaction costs ( b ) raise the growth rate.
We now discuss the underlying intuition behind the propositions. The increased CBDC
ratio induces individuals to hold more digital money in their portfolios, instead of physicalmoney, which incurs transaction costs when held. In other words, individuals can reducetransaction costs of holding total money by increasing the ratio of digital money holdingsin their portfolios in response to the policy change. The decreased transaction costs raise
their disposable resources by freeing real resources, ceteris paribus. The increased
disposable resources induce young individuals to accelerate their asset accumulation andhence, in this case, bank deposits. Thus, capital accumulation is accelerated through bankintermediation. Because capital accumulation is the engine of this model, the decreasedtransaction costs boost economic growth. It is noteworthy here that an increased CBDC
does not crowd out bank deposits through changes in the interest rate in this model (seeSubsection 2.2). In our model with AK production technology, the interest rate isindependent of the capital–labor ratio and remains constant over time.
A decrease in the lump-sum cost factor also lowers the cost burden and frees disposable
resources. Therefore, lump-sum cost reduction boosts economic growth. This result is not
at 1 + µ ≠ 1 + γ is provided in Appendix C.
17
It is noteworthy that 1 + µ =1 + γ on the balanced growth paths with growth-money
supply policies. Therefore, (20) is an implicit equation of the balanced growth rate. Under
the
money
supplypolicy,(20)
becomes
1 + γ = ρ (1 − α ) Baα −1 / (1 + ρ ) − θ [1 + ε (1 − φ )](1 + r ) / (1 + θ ) .
12

contrary to the arguments of Rotemberg (1984), who assumes a neoclassical infinitely
lived representative agent model. However, in this study, the costs are not paid to financial
brokers; in other words, they are real resource costs. If CBDCs lower monetarytransaction costs, then the introduction or replacement of CBDCs promotes economic
growth.
Under Propositions 1 and 2, the growth money supply policy maintains the long-term
inflation rate at zero. Therefore, the money supply policy has no substitution effects
between physical capital and real balances; that is, there is no Tobin effect (Tobin,1956). 18

4. Concluding remarks
Our results show that if a balanced growth path exists, increases in the ratio of CBDC
raise economic growth. Individuals’ portfolio adjustments reduce transaction costs ofholding (total) money. This is true regardless of the government’s money supply rule. In
other words, if CBDCs reduce transaction costs, the replacement of physical money witha CBDC exerts positive effects on economic growth by freeing the disposable resources
to investment if the real interest rate does not change much. The replacement with CBDC
potentially lowers the inflation rate and hence the nominal interest rate.
Because of a simple model, our study has drawbacks. A possible analytical extension
would be to consider a different endogenous growth setting, such as that with an R&D
investment engine. When the interest rate is endogenized, the interest rate on the CBDC
might affect the real interest rate (Keister and Sanches, 2022). We have not addressed on
the importance of privacy in transactions, which is for example emphasized in Williamson(2022). Another extension is to consider the effects of other policies in addition to money
supply and lump-sum transfer policies. The optimum can be attained with multiple policy
measures in overlapping generations settings (Crettez et al., 2002; Gahvari, 2007).

Appendix A: Walras’ law and the resource constraint
We

have

Yt =+
(1 r ) Kt + wt Nt

from

wt Nt + τ t Nt= c1t Nt + st Nt + [(1 + µ )mt + ε m t ]Nt + zt Nt
18

(13),

(14),
from

and

(15);
(4);

A constant real interest rate implies that the nominal interest rate also remains constantin the long term.
13

Nt ; τ t Nt = µ mt Nt from (1);(1 + r ) Kt + mt Nt =ct2 Nt −1 from (5) and (1 + n) Nt −1 =and st Nt = Kt +1 from the capital market-clearing condition. From these equations, we
obtain the resource constraint:
12
 t + zt ) Nt , orY=t ct Nt + (1 − θ )ct Nt −1 + Kt +1 + ( mt + ε m

Yt = c1t Nt + ct2 Nt −1 + Kt +1 + (ε m t + zt ) Nt ,(A1)

where θ ct2 Nt −1 = mt Nt is from the cash-in-advance constraint. The left-hand side of(A1) is the aggregate output, which can be consumable and investable. The right-handside is the sum of the goods consumption of the two generations, physical capital
accumulation, and real resource costs of money transactions. The resource costs of moneytransactions are borne by individuals, such as the sunk costs of monetary transactions,
without payment to any party in this study.
Appendix B: Balanced growth path
From budget constraint (3) and the cash-in-advance constraint, we obtainct2+1 =
(1 + r ) st / (1 − θ ) .

(A2)

Using the cash-in-advance constraint, we havePt +1 M t +1 / (ct2+1Nt ) M t +1 st −1 1
1 + µ st −1
.
=
= =
2
Pt
M t st 1 + n 1 + n st
M t / (ct Nt −1 )(A3)

From (12) and (A3), we obtainθ 1 + ε (1 − φ )(1 + π t )(1 + r )][(1 − α ) − b]Baα −1kt −1
1+ µ1−θ 1+ µ
1 + π t +1 =,1 + n [1 + θ 1 + ε (1 − φ ) (1 + π )(1 + r )][(1 − α ) − b]Baα −1kt +1t
1−θ 1+ µ[1 +(A4)
where we use (18) and

Yt = Baα −1Kt . In periodt , the variables

πt ,kt −1 (= Kt −1 / Nt −1 ) , and kt (= Kt / Nt ) have already been determined. Therefore, (A4)
determines π t +1 for these predetermined variables. (A4) may yield multiple solutions.
14

The dynamic stability must be determined using (16) and (A4). However, it is alsodifficult to study the stability and uniqueness of dynamic systems.
It is noteworthy that (A3) shows Pt +1 / Pt = 1 on the balanced growth path if thegrowth money is provided (i.e., 1 + µ =1 + γ ), because (ct2+1Nt ) / (ct2 Nt −1 ) = 1 + γ . Whenπ t = 0 , the dynamics of the system are given by (19).
Kt +1
=

ρ

1
[(1 − α ) − b]Baα −1Kt .
1 + ρ 1 + θ 1 + ε (1 − φ ) (1 + r )1−θ 1+ µ(A5)

Because capital stock grows on a balanced growth path, we obtain a balanced growth rate
as presented by (20).
Appendix C: Constant money supply rate
In Appendix C, we assume that the money supply rate is constant. If the money supply
policy is not for growth money supply (i.e., if 1 + µ ≠ 1 + γ ), then (A4) provides the
dynamics of the inflation rate. On the balanced growth path on which

1 + γ [= (1 + n)mt +1 / mt ] = (1 + µ ) / (1 + π ) for a given constant money supply rate µ , wehave 1 + π = (1 + µ ) / (1 + γ ) . Therefore, it follows that π > (<)0 when µ > (<)γ . Whenthe money supply rate is higher (lower) than the balanced growth rate, then the inflation
rate is positive (negative).
It is noteworthy that the balanced growth and inflation rates are determined
simultaneously by the two dynamic equations, (16) and (A4). Although it is also difficult
to examine stability even if we assume the existence of balanced growth paths, we obtainthe comparative static results as
dπ
dγ
(A6)
>0.
= 0 and
dµ
dµ
dπ
dγ
< 0.
> 0 and
dφ
dφ

(A7)

Therefore, under a constant ratio of CBDC with a constant money supply rate policy, an
increase in the money supply rate (i.e., CBDC interest rate) does not affect the balanced
growth rate and increases the inflation rate. An increase in the CBDC ratio increases thebalance growth rate and reduces the inflation rate. The growth effects of the transaction
costs ε and b are qualitatively the same as those in (21). The superneutrality result in15

(A6) comes from the assumed monetary policy in this case, in which the increased money
supply rate does not affect the discounted return rate to real money holding (see Crettez
et al., 1999).

Declaration of Conflicting Interests
The authors declare no potential conflict of interest with respect to the study, authorship,or publication of this article.
Data Availability Statement
Data sharing is inapplicable to this paper because no new data were analyzed.

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