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Payments on Digital Platforms:
Resiliency, Interoperability and Welfare
WP 21-04
Jonathan Chiu
Bank of Canada
Tsz-Nga Wong
Federal Reserve Bank of Richmond
Payments on Digital Platforms: Resiliency,
Interoperability and Welfare
∗
Jonathan Chiu
Tsz-Nga Wong
Bank of Canada
Federal Reserve Bank of Richmond
February 17, 2021
Abstract
Digital platforms, such as Alibaba and Amazon, operate an online marketplace to facilitate
transactions. This paper studies a platform’s business model choice between accepting cash and
issuing tokens, as well as the implications for welfare, resiliency, and interoperability. A cash
platform free rides on the existing payment infrastructure and profits from collecting transaction
fees. A token platform earns seigniorage, albeit bearing the costs of setting up the system
and holding reserves to mitigate the cyber risk. Tokens earn consumers a return, insulating
transactions from the liquidity costs of using cash, but also expose them to the remaining cyber
risk. The platform issues tokens if the interest rate is high, the platform scope is large, and
the cyber risk is small. Unbacked floating tokens with zero transaction fees or interest-bearing
stablecoins can implement the equilibrium business model, which is not necessarily socially
optimal because the platform does not internalize its impacts on off-platform activities. The
model explains why Amazon does not issue tokens, but Alipay issues tokens circulatable outside
its Alibaba platforms. Regulations such as a minimum reserve requirement can reduce welfare.
Keywords: Digital tokens, payments, platforms, money, regulation
JEL Classification: E1, E5, G28
∗
We have benefited from the comments and suggestions of participants, especially Charlie Kahn, at conferences
and seminars. The views expressed here do not necessarily reflect those of the Bank of Canada, the Federal Reserve
Bank of Richmond, or the Federal Reserve System.
1
Introduction
Recent years have witnessed the emergence of private digital tokens issued by social and economic
platforms to facilitate transactions among users in a variety of contexts. Early digital tokens were
issued by social media networks to allow platform users to conduct peer-to-peer transfers.1 In the
past few years, many blockchain-based platforms have launched utility coins circulating in their
distributed networks as the only acceptable means of payment.2 In China, Alibaba’s and WeChat’s
digital wallets, which allow holders to accumulate redeemable, interest-bearing balances for online
and point-of-sale transactions, have dominated the retail payments system. In the United States,
retail giants Walmart and Amazon have also recently applied for patents related to cryptocurrencies.
Private tokens, especially those issued by major platforms with a large user base, could reshape
the retail payment landscape and influence the functioning of the monetary system. The influence
of private tokens has been an important theme of policy discussion.3 According to a central banks
survey by Auer, Cornelli, and Frost (2020), the announcement of Facebook’s Libra in 2019 was a
tipping point to urge the development of central bank digital currencies (CBDCs). The share of
central banks that are likely to issue a retail CBDC in one to six years doubled in 2019 to 20%.
Within this context, our paper studies the business model choice between running a cash platform and a token platform, as well as its welfare and policy implications. What trade-offs does a
platform face when choosing between accepting cash and issuing its own tokens? How does a token
platform design features such as transaction fees, token interests and reserves held for redemption?
How are the token prices determined? Is the platform’s choice socially optimal? What are the
implications for payment resiliency and interoperability? What is the scope for regulation?
The key trade-off between the two business models is as follows. A cash platform makes profits
by charging fees on platform transactions. However, the platform’s ability to extract consumer
surplus is constrained by consumers’ cash holdings. In particular, the platform cannot charge
profit-maximizing fees when consumers’ cash holdings are tight due to high inflation (or any other
costs associated with using cash). By issuing tokens, a platform can insulate platform transactions
from the costs associated with cash.4 Issuing tokens also provides the platform with an option
to use token sales, i.e., the seigniorage, as an alternative source of revenue. Unlike cash, tokens
can pay interest to the holders to boost their demand. However, tokens can lose transaction value
when the platform ceases to operate (e.g., due to cyber attacks), and hence tokens, unlike cash, are
subject to default risk. To support the transaction value of tokens, the platform may need to hold
reserves in liquid low-yield assets, such as cash, for consumers’ redemption.
1
Early examples are Tencent QQ, Second Life Linden and Facebook Credits. In June 2019, Facebook announced
its intention to introduce Libra, a blockchain-based stablecoin.
2
Examples are Ethereum tokens, used in the Ethereum system, as well as Filecoin and Sia, used in cloud storage
networks.
3
See Kahn, Rivadeneyra, and Wong (2020b) for a summary of the policy discussion about the private and public
provision of digital currencies.
4
This cash constraint was a driving force behind the development of some digital platforms. For example, Alibaba
launched its own payments product, Alipay, in 2004 partly due to the inefficiency of the “cash-on-delivery” payment
method on its Taobao platform.
2
We find that the optimal platform design features zero reserves. It is because a token platform can support the transaction value of tokens by simply engineering a sufficiently high yield
of tokens (e.g., fast-appreciating token prices). The platform also profits from selling tokens at
fast-appreciating prices. The trade-off is that the platform has to limit the quantity of tokens
sold or keep buying back tokens in order to sustain the appreciation. There are multiple ways to
implement the optimal yield of tokens. For example, the platform can charge a zero transaction
fee or pay interest on tokens.
Given the above trade-off, when would a platform choose to issue tokens instead of accepting
cash? When inflation is not too high, using cash is optimal, especially when setting up a token
system is costly. A platform chooses to issue tokens when inflation is sufficiently high (so that the
cash holding is tight), when the platform is sufficiently resilient to cyber attacks (so that tokens
are expected to circulate longer) and when its market share is sufficiently large (so that transaction
fees are not a good way to collect revenue). Interestingly, the platform’s choice of business model
is not necessarily socially optimal because it ignores the impact on consumers’ welfare. Depending
on parameter values, both cash and token platforms can be suboptimally adopted. For example,
the over-adoption of token platforms happens when the platform does not internalize the negative
impacts of rejecting cash on non-platform consumption.
Could policy intervention improve social welfare? When the platform’s business model choice is
suboptimal, a simple tax/subsidy scheme, if feasible, can induce the platform to make the optimal
decision. However, conditional on the adoption of a token system, its design can still be suboptimal
relative to the first-best allocation. Could imposing reserve requirements be welfare improving?5
We find that it is optimal for a token platform to minimize its reserve holdings, and hence a reserve
regulation is typically welfare reducing.
Our paper contributes to the young but fast-growing literature exploring the economics of
digital currencies. A large amount of research focuses on understanding the design and functioning
of cryptocurrencies, such as Bitcoin. This literature can be broadly divided into three lines of
research. The first line aims at understanding the pricing of cryptocurrencies, e.g., Biais, Bisiere,
Bouvard, Casamatta, and Menkveld (2020), Choi and Rocheteau (2020), and Schilling and Uhlig
(2019). The second line examines the functioning and robustness of the blockchain, e.g., Biais,
Bisiere, Bouvard, and Casamatta (2019), Eyal and Sirer (2014), Saleh (2020), Chiu and Koeppl
(2019), and Pagnotta (2020). The third line focuses on mining activities and the determination of
fees, e.g., Huberman, Leshno, and Moallemi (2017) and Easley, O’Hara, and Basu (2019).
There are important differences between general-purpose cryptocurrencies and platform-based
tokens. Unlike general-purpose cryptocurrencies, platform-based tokens are mainly used for activities on the platform, and the platform controls whether or not to circulate the tokens outside the
platform, subject to regulatory and technical restrictions. Also, platform-based tokens are often
5
For example, in China, the PBoC recently imposed 100 percent reserve requirements on third-party payment
groups. According to a report by Financial Times, Alipay and Tencent were poised to lose around $1 billion in
combined annual revenue (source: https://tinyurl.com/yax3hjjf).
3
redeemable in national currency or platform goods.6 Therefore, the value of platform-based tokens
depends on the functioning of the issuing platform as well as the issuer’s commitment to future
actions, such as redemption, investment and reserve holdings. Finally, the token prices and fees are
determined by a forward-looking profit-maximizing platform like a durable-goods monopoly. The
monopoly power of the platform comes from its network effect.
A more closely related research area focuses on the design of utility tokens issued by platforms
and the value of initial coin offerings as a mode of financing, e.g., Catalini and Gans (2018), Cong, Li,
and Wang (2020), Howell, Niessner, and Yermack (2020), Garratt and Van Oordt (2019), Sockin and
Xiong (2020), and You and Rogoff (2019). Gans and Halaburda (2015) study the design of private
digital currencies from a platform-management perspective and explain why a profit-maximizing
platform may choose to limit the functionality of digital currencies. This stream of research typically
focuses on partial-equilibrium analysis and thus does not study macro implications of platformbased tokens. Our paper complements this literature by highlighting the interaction between onplatform and off-platform activities in a general equilibrium setting.
Some recent studies develop general equilibrium models to investigate the macro implications of issuing central bank digital currencies, e.g., Andolfatto (2018), Barrdear and Kumhof
(2016), Chiu and Wong (2014), Chiu, Davoodalhosseini, Jiang, and Zhu (2019), Davoodalhosseini
(2018), Fernández-Villaverde, Sanches, Schilling, and Uhlig (2020), Keister and Sanches (2019) and
Williamson (2019). Zhu and Hendry (2019) study the competition between national currency and
general-purpose electronic money. To the best of our knowledge, this paper is the first one that
studies the welfare and policy implications of issuing platform-based digital tokens in a general
equilibrium setting.
Cyber risk and security issues have been investigated in the payment literature, for example,
Kahn and Roberds (2008), and Kahn, Rivadeneyra, and Wong (2020a) study the trade-off between
security and convenience. We explore how platform security influences payment design in a general
equilibrium environment.
Our model is based on Lagos and Wright (2005), which provides a tractable framework for
us to study the joint determination of the transaction values of cash and tokens and the general
equilibrium effects of a platform’s payment choice. See also the survey in Kahn and Roberds (2009).
Recent models of payment systems building on the Lagos-Wright framework include Li (2011) and
Monnet and Roberds (2008). Chiu and Wong (2015) adopt a mechanism design approach to
compare the essential payment features of digital currency with cash.
The rest of the paper is organized as follows. Section 2 describes the basic environment. Sections 3 and 4 examine, respectively, a cash platform and a token platform. Section 5 studies the
equilibrium business model and its welfare implications. Section 6 discusses the effects of token
platform regulation. Section 7 considers an extension with endogenous interoperability and pro6
In other words, government-issued fiat currencies and cryptocurrencies, like Bitcoin, are outside money, while
redeemable platform-based tokens are inside money. But unlike typical inside money, such as deposit, platform-based
tokens do not involve any maturity transformation.
4
vides examples calibrated to Amazon and Alibaba. Section 8 concludes. All proofs are provided in
the Appendix.
2
Environment
Our model is based on the alternating competitive markets of Lagos and Wright (2005). Time is
indexed by t ∈ N0 . Each period is divided into two stages. In stage 1, competitive markets for retail
goods open, and consumption opportunities arise. In stage 2, agents trade assets and numeraires.
The economy starts in stage 2 of period 0. There are three types of agents: merchants who can
produce retail goods in stage 1 and consume numeraires in stage 2, consumers who consume and
produce numeraires in stage 2 and consume retail goods if opportunities arise in stage 1, and a
platform that produces nothing but facilitates meetings between consumers and merchants in stage
1 as explained below. The population of consumers is normalized to one. When consumption
opportunities arise, there is a chance that the consumer can find a suitable merchant on or off
the platform. Consumers and the platform discount across periods with a factor β = 1/(1 + r),
where r > 0. Merchants live for a period with new merchants entering at the beginning of stage 1.
Starting from period 1, a cyber shock hits the platform with probability γ in stage 2, after which
the platform will cease to exist in the next period. A low γ captures that the platform is resilient
to constant cyber attacks in the background.
The preferences of merchants in period t are given by
−qt + ct ,
where qt is the retail goods produced in stage 1, and ct is the numeraire consumed in stage 2. The
preferences of consumers are given by
E
∞
X
β t {Iqt U (qt ) + Iyt U (yt ) + u (ct , ht )} ,
t=0
where U is the consumer’s utility in stage 1 such that U (k) = k 1−η / (1 − η) for k = q, y, where
λ
η ∈ (0, 1), and u is the consumer’s utility in stage 2 such that u (c, h) = ū[cλt + ϕ h̄ − ht ]1/λ ,
where ht ∈ 0, h̄ is the consumer’s production of numeraires in stage 2, λ ∈ [−∞, 1], ϕ > 0,
ū = [1 + ϕ1/(1−λ) ]−(1−λ)/λ and h̄ is sufficiently large. Iqt and Iyt are indicator variables capturing
the iid consumption opportunity. Conditional on no cyber shocks taking place, with probability
σ (1 − α), a consumption opportunity arises off the platform (i.e., Iq = 1, Iy = 0); with probability
σα, it arises on the platform (i.e., Iq = 0, Iy = 1); with probability 1 − σ it does not occur (i.e.,
Iq = 0, Iy = 0). After a cyber shock, with probability σ (1 − α), it arises off the platform (i.e.,
Iq = 1, Iy = 0); with probability 1 − σ (1 − α) it does not occur (i.e., Iq = 0, Iy = 0).7 Thus, a
higher α means the consumption opportunities are more likely to be met via the platform. The
7
The stochastic structure of the model is important for generating the endogenous linkage between platform and
non-platform activities in stage 1. This assumption also simplifies the analysis by making agents ex-ante homogeneous.
5
preferences of the platform are given by
x0 +
∞
X
(1 − γ)t−1 β t [(1 − γ) x1t + γx0t ] ,
t=1
where x0t (x1t ) is the platform’s consumption of numeraires in stage 2 of t if a cyber has (not) hit,
and x0 is the initial platform’s consumption.
Agents can only observe the actions and outcomes of their trades. Consumers are anonymous
and cannot commit. There is no enforcement or record-keeping technology. As a result, consumers
cannot borrow to finance their consumption because they would renege on their promise to repay
their debt. This generates the need for a medium of exchange. In this economy, there are perfectly
divisible storable objects called cash that can be used as a medium of exchange. The supply of
cash is maintained exogenously by a central bank’s lump-sum transfers to consumers in stage 2,
such that the real price of cash in period t is φt . We assume an exogenous monetary policy that
fixes the inflation rate at µ̄ = φt /φt+1 .
We interpret the platform as an online marketplace, such as Amazon and Alibaba, where the
platform does not act as a buyer/seller but intermediates goods and payments between consumers
and merchants who do not have direct contacts among themselves. This role allows the platform
to dictate the means of payments acceptable by the merchants on the platform.8 Also, note that
the amount paid by the consumer is not necessarily equal to the amount received by the merchant.
The difference is the transaction fee charged by the platform. In the next section, we will focus on
a cash platform that allows its users to use cash to transact on the platform. To trade on the cash
platform, in stage 1, consumers need to acquire cash balances in the previous stage 2 by selling their
production of numeraires. On the platform, consumers are subject to a CIA constraint. In addition,
the platform can charge a proportional transaction fee that is proportional to the transaction value
on the platform and is paid in cash.9 Later we will extend the model to study a token platform,
which issues an additional medium of exchange called tokens. We will then study the equilibrium
business model of the platform and its welfare implications.
We expect that, as long as the consumption shocks are not perfectly predictable, consumers will still hold both tokens
and cash and the basic results hold.
8
Specifically, in stage 1, (1) the platform monitors and controls only merchants’ flow of tokens; (2) the physical
changing hands of cash is only possible between an agent and platform but not among agents on the platform;
(3) consumers on the platform can find each other and trade among themselves privately, if they want to. In
this environment, due to (1) and (2), the platform can restrict the type of payment instruments acceptable by the
merchants and impose fees on their transactions. However, due to (3), the platform cannot monitor buyers or prohibit
side trades among themselves. So lump sum fees will be infeasible as consumers can avoid them by side-trades. These
are reasonable assumptions for an e-commerce platform built to connect anonymous buyers with online merchants.
Merchants cannot communicate with buyers easily outside the supervision of the platform. But there are many ways
for anonymous consumers to communicate among themselves outside the supervision of the platform.
9
Unlike lump-sum fees, proportional fees are not subject to side-trading; see Chiu and Wong (2015) for details.
It is now well-known in the IO literature that proportional fees are more profitable than lump-sum fees in general
environments with heterogeneity, see Shy and Wang (2011). Thus, we do not pursue the analysis of lump-sum fees
in this paper.
6
3
Cash platform
A cash platform allows consumers to use cash as the medium of exchange and makes profit solely
from charging the transaction fee τt on the (post-fee) transaction value on the platform. We solve
the merchant’s and consumer’s problems backward by first considering stage 1 of period t and then
stage 2 of period t − 1.
3.1
Merchant’s problem
In the competitive market of stage 1, merchants choose qt units of retail goods to be sold on the
platform and yt units of retail goods to be sold off the platform, taking as given the nominal price
Pt of retail goods, which is the same on or off the platform. The zero-profit condition implies
−qt + φt Pt qt = 0 = −yt + φt Pt yt , where −qt is the cost of producing retail goods, and φt Pt qt is the
real value of cash receipt. Hence, the equilibrium price of retail goods is Pt = 1/φt .
3.2
Consumer’s problem
At the beginning of stage 1 in period t, the value function of holding M units of cash is
Wt (M − Po
t qt )]
σ (1 −nα) [U (qt ) +
Pt yt
Vt (M ) = max
+σα U (yt ) + Wt M − 1−τt
qt ,yt
+ (1 − σ) W (M )
t
M
M
s.t.
(1)
≥ P t qt ,
Pt yt
,
≥
1 − τt
(2)
(3)
where qt is the quantity bought off the platform, yt is the quantity of retail goods bought on the
cash platform, Wt is the continuation value function in stage 2, and (2) and (3) are the associated
CIA constraints for non-platform and platform transactions respectively. With probability 1 − σ,
the consumer does not need anything in stage 1, and he brings M units of cash to stage 2. With
probability σ (1 − α), the consumer pays some merchants off the platform Pt qt units of cash for qt
units of retail goods, bringing the remaining M − Pt qt units of cash to stage 2. With probability
σα, the consumer pays on the cash platforms Pt yt / (1 − τt ) units of cash for yt units of retail goods,
of which Pt yt is paid to the merchants and τt Pt yt / (1 − τt ) to the platform as the transaction fees.10
At the beginning of stage 2 of period t − 1, the value function of a consumer holding M units
of cash is
Wt−1 (M ) =
max
{ct ,ht ,mt }∈R3+
{u (ct−1 , ht−1 ) + βVt (Mt )} , s.t.
(4)
10
Consistent with off-the-platform trades, we use Pt yt to denote the amount that on-platform merchants receive.
Denote Dt as the total payment the consumer pays, then the on-platform merchant receives (1 − τt ) Dt = Pt yt . Thus,
the transaction fee the platform receives is τt Dt = τt Pt yt / (1 − τt ).
7
ct−1 + φt−1 Mt = ht−1 + φt−1 M + Υt ,
where Υt is the real lump-sum transfer from the central bank needed to implement the price path
φt−1 = µ̄φt . Applying the envelope theorem to (4) and (1), we obtain
n
o
0 Mt − 1
1
+
σ
(1
−
α)
U
dWt (M )
dVt (M )
+
n
h Pt
i
o
= φt ,
= φt
0
t
dM
dM
+σα (1 − τt ) U (1 − τt ) M
−
1
Pt
,
(5)
+
where {X}+ ≡ max{X, 0}.
11
The first-order condition of (4) with respect to Mt implies that
i = σ (1 − α) U 0 (mt ) − 1
+
σα
(1 − τt ) U 0 [(1 − τt ) mt ] − 1
+
+
,
(6)
where i ≡ (1 + r) φt−1 /φt − 1 = (1 + r) µ̄ − 1 > 0 is the nominal interest rate and mt ≡ φt Mt is the
real balances. The rate i on the LHS of (6) captures the marginal (opportunity) cost of holding
cash, which the nominal interest forgone. The first and second terms on the RHS of (6) capture the
marginal returns of spending cash off the platform and on the platform, respectively. These terms
are positive whenever holding cash helps relax the binding CIA constraints, (2) and (3). Finally,
R
note that the aggregate stage-2 utility is a constant and equal to wt ≡ u [ct (A) , ht (A)] dFt (A) +
σαqt + σ (1 − α) yt + xt = h̄ for all t, where ct (A) and ht (A) are the solution to (11) given A and
Ft (A) is the distribution of A in the beginning of stage 2 at t.
We first derive the consumer’s choices (qt , yt , mt ) as a function of (τt , i), which is summarized
by Lemma 1.
Lemma 1 Define
−1/(1−η)
i
,
τC (i) ≡ 1 − 1 +
σ (1 − α)
i1/η
i −1/η h
m (τt ) ≡
1+
1 − α + α (1 − τt )1−η
.
σ
Under a cash platform, the consumer’s solution has two regimes.
(a). If τt ≥ τC (i), then qt = mt = [1 − τC (i)]1/η−1 , and yt = (1 − τt )1/η ≤ (1 − τt ) mt , where
equality holds when τt = τC (i);
(b). if τt ≤ τC (i), then qt = mt = m (τt ), and yt = (1 − τt ) mt .
The proof is provided in the Appendix. When τ is sufficiently high, the consumer’s on-platform
CIA constraint is not binding (case i). Inflation reduces off-platform transactions but has no effects
on on-platform transactions. Transaction fees lower on-platform transactions but have no effects
on off-platform transactions. When τ is sufficiently low, both CIA constraints are binding (case
ii). Inflation and transaction fees reduce consumption on both platforms. Note that case (ii) is
11
See Wong (2016) for the details of derivation.
8
more likely when i is high because τC (i) is an increasing function. The idea is that the consumer
economizes on cash holdings when inflation is high, implying that the CIA constraint is more likely
to be binding on the platform.
3.3
Platform’s problem
The previous results suggest that, in setting the transaction fee, the platform has to take into
account not only the negative impacts on platform consumption yt given the money holding mt ,
but also the potential effects on the consumer’s choice of mt , which happens only in case (ii). The
platform maximizes the discounted sum of profit flow given the consumer’s solution:
ΠC ≡ max
τt
∞
X
(1 − γ)t−1 β t αστt
t=1
φt Pt yt
.
1 − τt
(7)
There is a measure ασ of consumers trading on the platform. Each consumer pays φt Pt yt / (1 − τt )
units of real balances of which the platform collects a fraction of τt . Notice that (7) is a repeated
static problem so its solution is time invariant. Using the two regimes of the consumer solution
characterized by Lemma 1, we can rewrite the repeated static problem as
ΠC
1
ασ
C
max ΠC
1 , Π2 , where
r+γ
≡ max G1 (τt ) s.t. τt ≥ τC (i) ;
ΠC
2
≡ max G2 (τt ) s.t. τt ≤ τC (i) ;
ΠC
=
τt
τt
where G1 (τ ) ≡ τ (1 − τ )1/η−1 and G2 (τ ) ≡ τ m (τ ). Here, the ΠC
1 function corresponds to case (i)
in Lemma 1, where τ is sufficiently high to make the CIA constraint non-binding on the platform.
Similarly, the ΠC
2 function corresponds to case (ii), where τ is sufficiently low to make the CIA
constraint binding. In setting the optimal fee, the platform compares the values in the unconstrained
C
and the constrained regions, ΠC
1 and Π2 , respectively.
Given these results, we can now derive the optimal fee τ chosen by the cash platform and the
equilibrium quantities (q, y) and platform profits ΠC , summarized in Proposition 1.
Proposition 1 Define
h
i
i∗ ≡ σ (1 − α) (1 − η)−(1−η) − 1 ,
h
i
ı̂ ≡ σ (1 − α) (1 − τ̄ )−(1−η) − 1 > i∗ ,
h
i1
η
1−η
τ̄ ≡ arg max τt α (1 − τt )
+ (1 − α)
∈ (η, 1) .
Under a cash platform, the platform’s solution has three regimes:
9
(a). i < i∗
(b). i ∈ [i∗ , ı̂]
(c). i > ı̂
τt
qt
yt
ΠC
η
[1 − τC (i)]1/η−1
(1 − η)1/η
τC (i)
[1 − τC (i)]1/η−1
[1 − τC (i)]1/η
τ̄
m (τ̄ )
(1 − τ̄ ) qt
ασ
r+γ G1 (η)
ασ
r+γ G1 [τC (i)]
ασ
r+γ G2 (τ̄ ; i)
The optimal fee is τ = η when the constraint (3) is non-binding, while τ̄ > η is the optimal
fee when the constraint is binding. When the CIA constraint is non-binding on the platform, it
is optimal to set a lower fee to maximize fee revenue. Setting τ = η is not feasible when the CIA
constraint is binding, and the platform needs to raise τ to τ̄ .
There are three cases. In case (a), as i < i∗ , consumers hold sufficiently high cash balances so
that the platform can set the fee at the unconstrained level τ = η. The on-platform consumption
y is not affected by inflation, while the off-platform consumption q and money holding m are not
affected by the fee rate.
As i rises and enters the range [i∗ , ı̂] in case (b), the CIA starts to bind on the platform. The
platform optimally raises the fee to τ = τC (i) ∈ [η, τ̄ ], which induces a level of platform spending
such that the CIA constraint is just binding. In this case, the marginal valuation of cash is not
determined by platform activities. The on-platform consumption y decreases with inflation, while
the off-platform consumption q and money holding m are still independent of the fee rate.
When i > ı̂ in case (c), the consumers hold so little cash that the platform chooses to set a
high τ = τ̄ , which implies the CIA is binding on the platform. In this case, the marginal valuation
of cash is determined by platform activities. The fee rate and the inflation rate affect the money
balances, the on-platform and the off-platform consumption.
To finish the study of the cash platform, we report some comparative statics in the following
proposition.
Proposition 2 Under a cash platform, the comparative statics are given by
∂τt
∂i
∂qt
∂i
∂yt
∂i
∂ΠC
∂i
∂τt
∂α
∂qt
∂α
∂yt
∂α
∂ΠC
∂α
∂τt ∂qt ∂yt
∂γ , ∂γ , ∂γ
∂ΠC
∂γ
(a). i < i∗
0
−
0
0
0
−
0
+
0
−
(b). i ∈ [i∗ , ı̂]
+
−
−
−
+
−
−
±
0
−
(c). i > ı̂
0
−
−
−
−
±
±
±
0
−
Inflation generally has a negative effect on consumption and platform profits, with the exception
of case (a), where on-platform activities are unconstrained by cash holdings. The effects of market
share α is more complicated. In case (a), α reduces the off-platform marginal value of cash, while
the on-platform marginal value of cash is not affected. This implies a lower q but a fixed y. In
case (b), the platform needs to raise τ to τC , which increases with α. Hence, the on-platform
consumption y declines with α, too. In case (c), the platform raises the fee to τ̄ . There are two
opposite forces with respect to α. First, the negative impact of the fee on cash holdings increases
10
with α. Second, the optimal fee τ̄ decreases with α. As a result, the overall effects on quantities
and profits are ambiguous. The effects of the resiliency parameter γ are also intuitive. It reduces
the (discounted) profits of the platform but has no effects on the fee rate or the quantities because
the platform has a repeated static problem.
4
Token platform
We now consider a different business model in which the platform issues private IOUs, called tokens,
in stage 2. In stage 1, consumers can then use these tokens to buy goods from merchants as the
only acceptable medium of exchange on the platform. A token platform can make profits from
charging transaction fees as well as selling new tokens. Issuing tokens incurs a setup cost κα, which
is increasing in the market share α of platform trades. This captures the expenses of arranging the
initial coin offering, building the new payment infrastructure and adhering to regulations. Define
pt as the real price of tokens (in numeraire) if the cyber shock does not hit the platform. The
nominal exchange rate of tokens is simply St ≡ pt /φt (the amount of cash traded for each unit of
tokens).
Notice that, unlike cash, the platform can fix the price of tokens pt . How? One simple implementation is that the platform announces a price pt and stands ready to trade (redeem and issue)
tokens in stage 2 at this targeted price. Tokens can keep circulating without redemption, which
could happen when the redemption price is set below the stage 2 market price. Obviously, in an
equilibrium with positive redemption, the market price is equal to the redemption price.12
Recall that the cyber shock hits with probability γ in stage 2. When this happens, the token
can no longer be used for future platform transactions. A merchant who has accepted the tokens
suffers the loss as the payment becomes worthless. Hence, the exchange value of tokens in stage 1
needs to price in the cyber risk in stage 2. To mitigate the token holder’s exposure to the cyber
risk, the platform can hold cash reserves to finance token redemption in case the platform collapses.
Specifically, bearers can redeem each token for Bt units of cash in stage 2 so that bt ≡ Bt /St is the
real redemption rate. If bt = 1, then the token is fully backed and hence risk-free. To deliver on
this promise, the platform issuing Kt units of tokens needs to hold Bt Kt units of cash reserves.
Before formally analyzing the token platform, we can briefly compare it to a cash platform
to highlight some differences. Under a cash platform, inflation reduces platform transactions and
profits whenever the CIA is binding on the platform. Using tokens to facilitate platform transactions
can avoid the inflation tax. There are, however, costs associated with the token platform. First,
issuing tokens incurs a setup cost. Second, token returns are risky due to the cyber risk, while cash
is always safe. Third, the platform is subject to the opportunity cost of holding cash reserves to
back the redemption price.
We now solve the equilibrium under a token platform. The merchant’s problem is the same as
that on the cash platform. To buy a unit of retail goods, the consumer needs to pay Pt = 1/φt units
12
An alternative way, which does not involve actively trading with consumers, is that the platform can pay token
holders interest rate in terms of tokens. See Section 4.3 for details.
11
of cash. Similarly, to buy retail goods using tokens, a consumer needs to pay Pt /[(1 − γ + γbt ) St ]
units of tokens, where the denominator captures the associated risk and the exchange rate. We can
then define PtT = Pt / (1 − γ + γbt ) as the cash price of goods for a token user.
4.1
Consumer’s problem
At the beginning of stage 1 in period t, the value function of a consumer holding M units of cash
and K units of tokens is
Vt (M, K) = max
qt ,yt
n
σα
U
(y
)
+
W
t
t M + St K −
PtT yt
1−τt
o
+σ (1 − α) [U (qt ) + Wt (M − Pt qt + St K)] s.t.
+ (1 − σ) W (M + S K)
t
t
(8)
PtT yt
,
1 − τt
≥ P t qt ,
St K ≥
M
(9)
(10)
where yt is the quantity of retail goods bought on the token platform. In stage 1, the nominal value
of the tokens held by the consumer is St K. The token-in-advance constraint is given by (9). With
probability σα, the consumer pays on the token platform PtT yt / (1 − τt ) units of tokens for yt units
of retail goods, of which PtT yt is paid to the merchant and τt PtT yt / (1 − τt ) to the platform as the
transaction fees. The expected nominal value of balances (cash and tokens) brought to stage 2 is
M + St K − PtT yt / (1 − τt ).
At the beginning of stage 2 in period t − 1, the value function of a consumer holding A units of
nominal balances (cash and tokens) is
Wt−1 (A) =
max
{ct−1 ,ht−1 ,Mt ,Kt }∈R4+
{u (ct−1 , ht−1 ) + βVt (Mt , Kt )} , s.t.
(11)
ct−1 + φt−1 Mt + pt−1 Kt = ht−1 + φt−1 A + Υt .
Applying the envelope theorem to (11) and (8), we obtain
∂Vt (M, K)
∂M
∂Vt (M, K)
∂K
h
i
= φt 1 + σ (1 − α) U 0 (mt ) − 1 + ,
h
= pt (1 − γ + γbt ) 1 + σα (1 − τt ) U 0 [(1 − τt ) kt ] − 1
i
+
,
where kt ≡ [1 − γ (1 − bt )] pt Kt denotes the real token balances. When bt = 1, the full real value of
tokens is always guaranteed. The first-order conditions of (11) with respect to Mt and Kt are
n
o
i = σ (1 − α) m−η
−
1
,
t
+
n
o
ρt = σα (1 − τt )1−η kt−η − 1 ,
+
12
(12)
(13)
where
ρt ≡
(1 + r) pt−1
−1
pt [1 − γ (1 − bt )]
is the token yield – the interest rate needed to compensate the token holder for bearing the cyber
risk, discounting, and changes in token prices. The RHS of (13) is the marginal value of holding
tokens, consisting of the liquidity value of using tokens on the platform. Under a token platform,
the consumer’s solution is given by
−1/η
i
,
= mt = 1 +
σ (1 − α)
ρt −1/η
1/η
= (1 − τt )
1+
,
σα
(1 − τt )1/η−1 1 + ρt −1/η , if ρt > 0,
σα
h
i
=
∈ (1 − τt )1/η−1 , ∞ , if ρt = 0.
qt
yt
kt
(14)
(15)
(16)
Intuitively, the on-platform consumption decreases with ρ, γ, τ and increases with b. Finally,
note that the aggregate stage-2 utility is a constant and equal to wt for all t, similar to the case of
the cash platform.
4.2
Platform’s problem
The initial profit flow of the platform in stage 2 is
x0 = p0 K1 − κα − µ̄b1 p1 K1 ,
which consists of the revenue from the initial issuance of K1 units of tokens each worth p0 units of
numeraire net of the setup cost and the amount of cash reserves the platform holds to back up its
redemption in the following period. Each token is redeemable for B1 = b1 p1 units of cash in period
1, and each unit of period-1 cash is worth µ̄ = φ0 /φ1 units of numeraire in period 0. The expected
profit flow in any subsequent period is
Kt+1
µ̄
φt PtT yt
x1t = pt bt Kt +
− Kt − bt+1 pt+1 Kt+1 + σατt
,
1+i
pt
1 − τt
φt PtT yt
x0t = σατt bt
.
1 − τt
If the cyber shock does not hit in stage 2, then the profit flow consists of the reserves brought
forward (pt bt Kt ), the sale of new tokens, [pt (Kt+1 − Kt )], the receipt of transaction fees in tokens
[σατt φt PtT yt / (1 − τt )], subtracting the expenses on new cash reserves (µ̄bt+1 pt+1 Kt+1 ). If the cyber
shock hits in stage 2, then token holders redeem the platform’s cash reserves, leaving an amount
σατt bt φt PtT yt / (1 − τt ) to the platform (which captures the reserves held to back the fee revenue
received by the platform).
13
o
n
P
t−1 t
β [(1 − γ) x1t + γx0t ] . By
The platform’s profit is given by ΠT = max x0 + ∞
t=1 (1 − γ)
collecting terms with the same t across periods, we can express the platform’s problem as
T
Π = −κα +
max
bt ,kt ,τt ≥0
(∞
X
(1 − γ)
t−1
t=1
β
t
τt yt
ibt
ρt kt + σα [1 − γ (1 − bt )]
−
kt
1 − τt 1 − γ (1 − bt )
)
.
The optimal design of a token platform is summarized by the following proposition.
Proposition 3 Under a token platform, the solution of the platform’s problem is given by
τt
∈ [0, η] , bt = 0, ρt =
σα
(η − τt ) ,
1−η
qt = [1 − τC (i)]1/η−1 ,
yt = (1 − η)1/η ,
σαη
ΠT = −κα +
(1 − η)1/η−1 .
r+γ
The platform monopoly designs the features {ρt , τt , bt } to extract the consumer’s surplus of the
platform trades. Rearranging terms, the expected profit flow in t is proportional to
σα yt−η − 1 yt −
ibt
kt
1 − γ (1 − bt )
In this economy, it is the purchasing power of tokens, yt =
(1 − τt )kt , that matters for the consumer
surplus, which is proportional to the first term above, yt−η − 1 σαyt , as the marginal utility of
retail consumption net of real price (which is unity) times the total retail consumption purchased
by the tokens. The platform can implement any yt by varying the reserve bt , the fee τ and the
token interest rate ρt , via the pricing kernel (13). Increasing reserves holding bt helps support
the purchasing power of token, albeit raising the opportunity costs ibt kt in terms of the interest
forgone. A less costly way to support the purchasing power of token without this opportunity cost
of the interest forgone is to reduce τt and ρt . Thus, it is optimal not to back tokens by any reserves.
Finally, the token is indifferent to a range of ρt and τt because charging a higher transaction fee
can always be replicated by selling token at a higher price without the transaction fee.
We first define the growth rate of the nominal token price as gt ≡ St /St−1 . The following
proposition reports some comparative statics results.
Proposition 4 Under a token platform, fixing τt ∈ [0, η), the comparative statics are given by
∂gt
∂qt
∂yt
∂ΠT
∂i
+
−
0
0
∂α
−
−
0
+
∂γ
+
0
0
−
14
Propositions 1 and 2 suggest that, under a cash platform, platform consumption y and profits
are both maximized when i < i∗ . Higher i tightens the CIA constraint on the platform
ΠC
and reduces platform transactions and profits. By issuing tokens, the platform can replicate the
maximum consumption and profits under a cash platform as the usage of tokens helps avoid inflation
tax. The platform can maintain a fixed consumption level y by adjusting the nominal price paths
of tokens (through gt ) in response to changes in i, α and γ. Interestingly, while a higher γ increases
the risk of holding tokens, the platform can simply compensate users by appreciating token prices
faster without the need to hold costly cash reserves.
4.3
Alternative implementation: interest-bearing stablecoins
An alternative way to implement the token solution is to pay interest in terms of tokens. One realworld example is the interesting-bearing Yu’ebao accounts offered by Alipay. Here, tokens become
a “digital asset” working like a money market mutual fund. As we will show, by administering the
interest rates, the platform can ensure that tokens are traded at par with cash, like stablecoins.
Suppose the platform pays an interest of νt (in terms of of tokens) for each unit of tokens held
in the beginning of stage 2. By modifying the definition to kt ≡ [1 − γ (1 − bt )] (1 + νt ) pt Kt , we
obtain the following first-order condition with respect to Kt :
n
o
(1 + r) pt−1
− 1 = σα (1 − τt )1−η kt−η − 1 .
pt (1 + υt ) [1 − γ (1 − bt )]
+
The optimal allocation of Proposition (4) can be implemented under pt = φt , by setting
υt =
1+i
− 1,
(1 + ρt ) (1 − γ)
where ρt = σαη/ (1 − η), bt = 0 and τt = 0. In this case, the platform issues stablecoins which pay
interests at a rate below the cyber-risk adjusted nominal interest rate.
4.4
Digression: acceptance of cash on token platforms
We have implicitly assumed that a token platform does not accept cash. We here consider the
general setting and show that accepting both cash and tokens are not optimal. Suppose the
platform has an additional option of running a dual payments business model where both cash
and tokens can be used on the platform. The consumer’s problem becomes
Pt qt )}
σ (1 −h α) {U (qt ) +Wt (M+ St K − M
P
y
t
+σα U ytM + ytK + Wt M − 1−τtM + St K −
Vt (M, K) =
max
t
qt ,ytM ≥0,yt ≥0
+ (1 − σ) W (M + S K)
t
t
15
PtT yt
1−τt
i
s.t.
(17)
≥ P t qt ,
Pt ytM
M ≥
1 − τtM
PtT ytK
St K ≥
,
1 − τtK
M
(18)
(19)
(20)
where qt denotes off-platform consumption paid with cash, ytM denotes on-platform consumption
paid with cash, and ytK denotes on-platform consumption paid with tokens. Note that all onplatform transactions are subject to transaction fees: τtM for cash and τtK for token.
We focus on the case where the consumer’s token holding is positive, otherwise, it is just a
de facto cash platform. This means the token-in-advance constraint (19) is binding, i.e., ytK =
1 − τtK kt . The shadow rate of tokens is given by the first order condition w.r.t. kt :
i
1 − τtK yt−η − 1 ,
n
o
n
o
i = σ (1 − α) m−η
+ σα 1 − τtM yt−η − 1
t −1
ρt = σα
h
+
(21)
(22)
+
where yt = ytM + ytK . We also focus on the case that cash is used on the platform (otherwise it is
just a de facto token platform). Since cash and tokens are perfect substitutes, cash usage on the
platform is “bang-bang”, i.e., either ytM = 1 − τtM mt or ytM = 0. The consumer uses all the cash
she holds only if
1 − τtM yt−η ≥ 1.
(23)
Here, the marginal benefit of using cash balances on platform consumption is 1 − τtM yt−η , and
the marginal cost is one (the forgone opportunity of buying numeraires with cash in stage 2). The
platform problem becomes
Π
D
= −κα + max
(∞
X
t−1
(1 − γ)
β ρt kt + σατtK kt + σατtM mt −
t
t=1
ibt
kt
1 − γ (1 − bt )
)
s.t. (21), (22) and (23).
,
(24)
Proposition 5 A dual platform is always dominated by a cash or a token platform in terms of
profits, i.e., ΠD ≤ max ΠC , ΠT .
This result helps us understand why blockchain-based platforms issue utility tokens as the only
means of payments. At the same time, there do exist dual systems where both tokens and cash are
accepted on the platform. The model can potentially be extended to generate this as an equilibrium
outcome. For example, if consumers are heterogeneous in terms of their costs of adopting tokens,
then the platform might choose to adopt a dual system where cash is used by high adoption cost
consumers (e.g. the elderlies) and tokens used by others. We leave this for future research.
16
5
Equilibrium business model and welfare
We now study the platform’s choice to accept cash or issue tokens and the welfare implications of
that choice.
5.1
Platform’s business model choice
Obviously, the platform chooses to issue tokens, instead of using cash, whenever ΠT > ΠC . The
three key parameters are i, γ, and α. Lemma 2 completely characterizes the parameter space such
that the platform issues tokens, instead of using cash.
Lemma 2 There exists a threshold interest rate ιΠ (α, γ) > i∗ such that ΠT > ΠC if and only if
i > ιΠ (α, γ). Moreover, ιΠ (α, γ) is a function decreasing in α but increasing in γ.
As discussed above, when i < i∗ , the tax collected on the cash platform is the same as the token
sale revenue collected from the token platform because the CIA constraint is not binding on the
cash platform. Hence, the token platform is dominated due to the additional cost. As i increases,
the CIA constraint on the cash platform starts to bind. The token platform dominates for high i
because the usage of tokens insulates the platform from the inflation tax.
The token platform dominates the cash platform when the share of platform trade (α) is sufficiently high. Two channels explain why this happens. First, a higher α increases the extensive
margin of platform trades, which raises the revenues of both the token and cash platform proportionally. Second, a higher α will have potential effects on the intensive margin of platform trades.
A higher α means there is less off-platform trade, implying a lower cash holding. When the CIA
constraint is not binding, the fee revenue per trade is unaffected. When the CIA constraint is binding, the fee revenue per trade falls in proportion to the cash holding. To counteract that effect, the
cash platform reduces τt . The overall effect of α is thus ambiguous on a cash platform. However,
a higher α does not change the intensive margin under a token platform because it does not alter
the consumer surplus per platform trade. Hence, the token platform does not want to change the
distortion. In sum, when α increases, the profit increases more under a token model than a cash
model. When the premium in profit justifies the fixed cost of issuing tokens, the token platform
dominates. Finally, as the platform becomes less resilient (higher γ), the platform is less willing to
issue tokens because γ reduces the expected returns to pay a setup cost to start a token platform.
We now consider the general-equilibrium effects of the platform’s decision to issue tokens. First,
to examine the effects on trading activities, we define the off-platform social surplus as U (q) − q
and the on-platform social surplus as U (y) − y. To study how the platform’s business model affects
the central bank’s seigniorage revenue, we define the equilibrium seigniorage from cash issuance as
(
S = φ0 M1 + µ̄b1 p1 K1 +
X
t=1
=
(1 − γ)t−1 β t
(1 − γ) [φt (Mt+1 − Mt ) − pt bt Kt + µ̄bt+1 pt+1 Kt+1 ]
γ −pt bt Kt + S̄
i+γ
i
b
γ
m+
k+
S̄,
r+γ
r + γ 1 − γ (1 − b)
r+γ
17
)
where S̄ is the continuation present value of the seigniorage revenue after the cyber shock hits. We
can use the above findings to obtain the following proposition.
Proposition 6 When the platform chooses to issue tokens instead of accepting cash,
1. on-platform consumption and social surplus go up,
2. off-platform consumption and social surplus go down,
3. seigniorage revenue goes down.
When the platform issues tokens instead of accepting cash, it insulates the platform transactions
from the inflation tax. This will not only increase profits but also increase the total trade surplus
on the platform. However, this has a spillover effect on off-platform transactions. The switch to
a token platform lowers the demand for cash, and hence the off-platform transactions and surplus
go down as the CIA constraint becomes more binding. Finally, since the token platform does not
hold cash reserves, the central bank’s seigniorage revenue will decline too.
5.2
Welfare implications
We now investigate whether the business model chosen by the platform also maximizes social
welfare. The welfare of this economy is defined as the discounted sum of utilities of consumers,
merchants and the platform, which are given by
WC
WT
1
γβ
σα U y C − y C + σ (1 − α) U q C − q C + wt +
W̄;
r+γ
r+γ
1
γβ
= −κα +
σα U y T − y T + σ (1 − α) U q T − q T + wt +
W̄,
r+γ
r+γ
=
where W C is the welfare under a cash platform, W T is the welfare under a token platform, and
W̄ is the continuation welfare after the cyber shock. The following lemma describes when issuing
tokens maximizes social welfare.
Lemma 3 There exists a threshold rate ιW (α, γ) > i∗ such that W T < W C if and only if i <
ιW (α, γ). Moreover, ιW (α, γ) is a function decreasing in α if i ≤ ı̂, but increasing in γ.
The effects of i and γ on welfare are similar to those on the platform’s profits. When i <
i∗ ,
a token platform and a cash platform support the same consumption allocations and profits.
Therefore, the token platform is dominated due to the additional setup cost. As i increases,
both platform profits and consumption drop when a cash platform is chosen. Hence, a token
platform is preferred when the inflation rate is sufficiently high because tokens insulate the platform
consumption from inflation tax. Furthermore, an increase in γ reduces the expected benefits of
setting up a token platform.
18
Figure 1: Profits vs welfare (γ = 0.8)
The effects of α on welfare can be different from those on the platform’s profits when i > ı̂. In
particular, for i > ı̂, it could well be that W C overtakes W T when α becomes higher. This is due
to the spillover effects on off-platform surplus discussed in the previous proposition. On the other
hand, ΠC never overtakes ΠT . This is because the cash demand decreases when α becomes higher.
Since the cash platform relies on cash trades for revenue, it responds by cutting the transaction fee
to mitigate the reduction in cash holding. Doing so reduces the distortion on the platform trades,
thus the welfare under the cash platform can be higher than it is under the token platform.
The equilibrium choice of the platform is not socially optimal when the welfare could increased
by making the alternative choice. This happens when ΠT > ΠC but W T < W C or when ΠT < ΠC
but W T > W C . In the former, a token platform is chosen in equilibrium but a cash platform
yields higher welfare. By using Lemmas 2 and 3, the following corollary summarizes the welfare
properties of the equilibrium business model.
Corollary 1 A cash platform is a socially optimal business model if and only if i ≤ min ιΠ (α, γ) , ιW (α, γ) .
The measure of this parameter subspace decreases with α if i ≤ ı̂ but increases with γ. A token
platform is a socially optimal business model if and only if i ≥ max ιΠ (α, γ) , ιW (α, γ) . The
measure of this parameter subspace increases with α if i ≤ ı̂ but decreases with γ.
The equilibrium choice of business model can be suboptimal because the platform only considers
its profits and does not internalize the effects on the consumers’ surplus (merchants have zero
surplus). In particular, a token platform is suboptimally adopted when i ∈ (ιΠ (α, γ) , ιW (α, γ))
and a cash platform is suboptimally adopted when i ∈ (ιW (α, γ) , ιΠ (α, γ)). A simple way to
correct these suboptimal outcomes is to impose a tax T on token platform adoption (subsidy when
negative). The government can induce the platform to choose the socially optimal business model
by imposing on the token platform a tax T = (W C − ΠC ) − (W T − ΠT ).
19
Table 1: Welfare properties of platforms
5.3
Cash platform
Token platform
off-platform
q
0.3674
0.3607
Cash platform
Token platform
0.3674
0.0258
Example 1:
on-platform
y
0.3368
0.3527
α = 0.96, i = 0.34, γ = 0.8
welfare profits
buyers’surplus
W
Π
Ū q
Ū y
Ū
0.0755 0.0354 0.0031 0.0369 0.0401
0.0761 0.0345 0.0031 0.0385 0.0416
Example 2: α = 0.99, i = 0.34, γ = 0.8
0.3377
0.0755 0.0354 0.0008 0.0393
0.3527
0.0754 0.0355 0.0001 0.0397
0.0401
0.0399
Numerical examples
We study a numerical example with the following parameter values: β = 0.97, σ = 1, η = 0.08,
κ = 0.001. Figure 1 compares the relative ranking of a cash platform and a token platform in
terms of social welfare and the platform’s profits for different levels of interest rate i and platform
fragility α. As implied by Corollary 1, the platform’s choice of its business model (i.e., cash vs.
token platforms) can be socially optimal or not. In particular, there are altogether four regions in
the (i, α) parameter space, depending on whether the platform’s choice and the social planner’s
choice are aligned or not. Note that the region where cash is optimal is hump-shaped for higher
values of α due to the trade-off between the positive effect on on-platform surplus and the negative
effect on off-platform social surplus.
To illustrate that the equilibrium business can be suboptimal, in Tables 1, we decompose welfare
into profits and consumers’ surplus (denoted by Ū ) under the cash and token platforms. In the
first example, the platform suboptimally chooses not to issue tokens. As shown in the upper panel
of Table 1, switching from a cash platform to a token platform increases consumers’ surplus, which
is big enough to offset the platform’s profit loss. In the second example, the platform suboptimally
chooses to issue tokens. As shown in the lower panel of Table 1, switching from a token platform
to a cash platform can significantly increase off-platform consumption and hence increase welfare.
To understand these findings, note that in choosing its business model, the platform considers
only the effects on its profits:
σα
Π − Π = −κα +
r+γ
T
C
η
τ
yT −
yC
1−η
1−τ
.
However, the platform does not fully internalize the effects on the consumers’ surplus, which is
20
given by
C
T
σ
y
y
T
C
T
C
T
C
+ (1 − α) U q − U (q ) − q + q
+
α U y − U (y ) −
.
r+γ
1−η 1−τ
|
{z
}
{z
}
|
change in off-platform surplus Ū q
change in on-platform surplus Ū y
This is a weighed sum of the effects on consumers’ on-platform surplus Ū y and off-platform surplus
Ū q with the weight given by α. As α goes up, the on-platform surplus goes up, while the on-platform
surplus goes down. In the first (second) example where α is lower (higher), the on-platform (offplatform) surplus change dominates. Since the platform fails to fully internalize the social benefits
(costs) of issuing tokens in the first (second) example, the platform tends to under-issue (over-issue)
tokens.
6
Token platform regulation
The above discussion concerns the optimality of adopting a token platform. Conditional on its
adoption, the allocation under a token platform can still be suboptimal. Relative to the first-best,
there is under-consumption of platform goods when the purchasing power of tokens is so low that
yt ≤ 1. Recall that the purchasing power of tokens is yt = (1 − τt ) kt . From (15), we know that
the demand for tokens is low when the transaction fee is high, or the shadow rate is high (tokens
depreciate quickly), or the cyber risk is high, or the reserve backing is low. Since the token platform
charges the minimal transaction fee, τt = 0, the other three are the potential sources of welfare loss.
A natural question is whether the government can impose certain policies to reduce welfare loss.
In reality, widely circulated priavte currencies are often subject to a minimum reserve requirement
(e.g., PBoC’s regulation on Alipay and WeChat). We consider below whether regulating the reserve
holding of the platform is welfare-improving. In the Appendix, we also study whether providing
deposit insurance can reduce the impacts of the cyber risk.
Suppose the central bank introduces a minimum reserve requirement on the token platform such
that at least a fraction b̄ of the token value is backed by cash reserves, i.e., bt ≥ b̄. The platform
problem becomes
ΠR = −κα + max
kt ,τt ≥0
(∞
X
)
t−1
(1 − γ)
β t [ρt − i + (1 − γ) σατt ] kt
t=1
The equilibrium quantities and profits are given by the following proposition.
21
.
Proposition 7 Under a reserve requirement, the solution of a token platform is bt = b̄, τt = 0,
ρt =
"
#
1
ib̄
,
σαη −
1−η
1 − γ 1 − b̄
qt = [1 − τC (i)]1/η−1 ,
1/η
1
−
η
,
yt =
i
b̄
1 + σα
1−γ (1−b̄)
σαη
ΠR = −κα +
y 1−η .
r+γ t
The token platform under a higher reserve requirement features lower trading activities yt .
This is not due to a higher transaction fee as the platform sets τt at 0. Recall that from (15),
the demand for tokens is lower if the reserve backing is lower or the shadow rate is higher. The
reserve requirement increases cash reserves and actually increases the demand for tokens. However,
the platform takes advantage of the higher demand by charging a far higher shadow rate ρ, which
ultimately pushes down the token demand. But this does not mean the platform makes more profit
because holding reserves is costly and the higher shadow rate is to compensate that. Overall, both
the consumer surplus and platform profit fall after a higher reserve requirement. Hence, conditional
on the usage of a token platform, a higher reserve requirement only reduces welfare. Nevertheless,
a sufficiently high reserve requirement could improve welfare when the platform’s choice of token
issuance is suboptimal (see Corollary 1) because it induces the platform to use cash instead of
issuing tokens. The following corollary summarizes these findings.
Corollary 2 The token platform under a higher reserve requirement features a lower gt , a lower
platform trade volume and a lower platform profit. Reserve requirement potentially results in a
higher welfare only if the platform ceases to issue tokens.
To deal with the default risk of a deposit-taking institution, a commonly used regulation is
deposit insurance. In the appendix, we study and show that the provision of deposit insurance
does not affect welfare because its effect is offset one-to-one by the corresponding decrease in gt ,
while the profits and allocation stay unchanged.
7
Endogenous interoperability
One concern about private digital currencies is that the lack of payment interoperability across
platforms can increase the fragmentation of the payment system (Brunnermeier, James, & Landau,
2019). We have already shown that a token platform has no incentives to accept cash. We will now
examine whether the platform wants to promote the circulation of tokens outside the platform.
Suppose the platform has an option to circulate tokens and provide payment services outside
the platform. We call these “circulatable tokens”. In reality, private currencies that are widely
22
accessible tend to be subject to regulation due to concerns about financial risk and consumer
protection. For example, China’s central bank requires payment firms such as Alipay and WeChat to
deposit reserves of client funds with a centralized custodian. To capture this real-world regulation,
we assume that the token platform has to satisfy a minimum reserve ratio b ∈ [0, 1]. The platform
cannot charge fees on transactions outside the platform. To simplify the analysis, we also assume
τt = 0 on the platform. The consumer problem becomes
Wt M + St K − PtT yt
σα U (yt ) +
Vt (M, K) = max
+σ (1 − α) U qtM + qtK + Wt M − Pt qt + St K − PtT qtK
qt ,yt
+ (1 − σ) Wt (M + St K)
s.t.
(25)
St K ≥ PtT yt , M ≥ Pt qtM , St K ≥ PtT qtK .
(26)
The first order conditions are given by
i ≥ σ (1 − α) (mt + kt )−η − 1 + , equal when mt > 0,
o
n
i
−η
−η
ρt = σα kt − 1
+ σ (1 − α) min kt , 1 +
−1
.
σ (1 − α)
+
+
(27)
(28)
The consumer’s solution is given by the following lemma.
Lemma 4 Under a circulatable token platform, the consumer’s solution has two regimes:
(a). i ≤ (1 − α) ρt
h
1+
qtM = mt
i−1/η
i
σ(1−α)
(b). i > (1 − α) ρt
− kt
0
qtK = yt = kt
−1/η
t −i
1 + ρσα
−1/η
1 + ρσt
Circulatable tokens and cash are in direct competition off-platform. When i is sufficiently high
(case ii), tokens can drive out cash and become the only payment instrument in the retail sector
(even though there is still demand for cash as a reserve asset). When i is not too high (case i),
cash is still used off-platform.
The platform problem is
ΠCT = −κα + max
bt ≥b,kt
(∞
X
t−1 t
(1 − γ) β ρt −
t=1
)
ib
kt
1 − γ (1 − b)
Rearranging term, the platform problem becomes
ΠCT
1
1
CT
max ΠCT
, where
1 , Π2
r+γ
≡ max GCT
1 (ρt ) s.t. i ≤ (1 − α) ρt ;
ΠCT
2
≡ max GCT
2 (ρt ) s.t. i > (1 − α) ρt ,
ΠCT
= −κα +
ρt
ρt
23
i=2.5%
b=0.05
0.35
0.36
k
k
0.34
0.3
0.32
0.3
0.25
0
0.02
0.036
0.06
0
0.035
0.034
0.034
0.032
0.033
0.03
0.032
0.028
0.031
0.5
1
b
i=2.5%
0.036
CT
CT
0.04
i
b=0.05
0.026
0
0.02
0.04
0.06
0
0.5
1
b
i
Figure 2: Supply of Circulatable Tokens
h
where GCT
(ρ)
≡
ρ−
1
ib
1−γ(1−b)
i
1+
ρ−i
σα
−1/η
h
and GCT
(ρ)
≡
ρ−
2
ib
1−γ(1−b)
i
1+
ρ −1/η
.
σ
We use a numerical example to illustrate the incentives of a platform to make tokens circulatable.
Figure 2 plots the supply of circulatable tokens k 0 and profits ΠCT as a function of i and b. Whenever
b > 0, the platform needs to keep cash reserves which involve a marginal cost i. As a result, both
token issuance and profits go down as the reserve ratio b increases. The interest rate has non-linear
effects. When i is low, increasing i makes cash a worse payment alternative outside the platform,
allowing the platform to issue more tokens and make a higher profit. When i is too high, further
increasing i will lower token issuance as platform’s profits decline with i. Figure 3 shows the
platform’s business model choice between issuing non-circulatable tokens, circulatable tokens and
using cash. Under this parameterization, we find that a cash platform is profit-maximizing only
for sufficiently low i and sufficiently high b. The platform issues non-circulatable tokens when i or
b are sufficiently high. In other cases issuing circulatable tokens maximizes profits. In particular,
in the absence of the minimum reserve requirement, a platform either accepts cash (for low i) or
issues circulatable tokens (for high i).
7.1
Calibrated examples
We use the model to study two calibrated examples.
Example 1. Cash platform: Amazon
We consider an annual model and set β = 0.97. We pick µ to generate i = 1.5% which is
the 3-month T-bill rate in 2020 Q1. We set σ = 1 to minimize matching frictions. With Amazon
exemplifying the modeled cash platform in our model, we calibrate the trading parameter α and
preference parameter η to match the market share and trading fees of this e-commerce platform.
First, for each item sold on their platform, Amazon charges the seller a referral fee which is a
24
Figure 3: Platform Choice
flat percentage typically about 15%.13 Second, according to the Department of Commerce, the
e-commerce share of retail in 2020 Q1 is 12%. Since Amazon accounts for about 50% of the US
e-commerce market, its market share is about 6%.14 We set α = 0.12 and η = 0.13 to minimize
the sum of squared percentage differences (see Table 2).
Under our parameterization, i∗ = 11%. Since i < i∗ , we are in case (i) considered in Proposition
3. This implies that it is not profitable for the platform to issue tokens for any κ and γ. Suppose
the setup cost of the token system is κ = 0.03 (about 1% of y T ) and the probability of collapse is
1%. Our model then suggests that it is socially optimal not to issue tokens for any γ. However,
when i goes above 12% or when α goes above 90%, issuing tokens maximizes both profits and social
welfare. As shown in Section 5, this is because high inflation constrains a cash platform’s ability
to charge fees especially when its market share is large. When there is no reserve requirement, the
platform will find it profitable to issue a circulatable tokens for when κ < 1.8. Hence the model
suggests that the regulatory costs in the U.S. (captured by κ or b) must be sufficiently high to
prevent the platform from issuing circulatable tokens.
Example 2. Circulatable tokens: Alibaba
We now use the model to study the case of Alibaba in China. As shown in Table 3, we maintain
the same preference parameter η = 0.13, and set a small γ = 1% to capture the reliability of the
platform. We pick µ to generate i = 3.5% which is the average 3-month T-bill rate in China in
2017. We set σ = 1 to minimize the matching frictions, and set α = 0.11 to match the fact that
Alibaba accounts for about 11% of China’s total retail market. With a higher interest rate in
China, the platform has a higher incentive to issue tokens, but issuing a non-circulatable token is
still unprofitable. Issuing a circulatable token is profitable when κ and b are sufficiently low. For
13
14
Source: https://www.junglescout.com/blog/amazon-fba-fees/
Source: https://tinyurl.com/y4xsyz6b
25
Table 2: Parameter Values for Example 1
Parameters
Discount factor
Nominal interest rate
Prob. of trading
Prob. of platform trades
Preference parameter
Notation
β
i
σ
α
η
Value
0.97
1.5%
1
0.12
0.13
Notes
Standard in literature
3-month T-bill rate in 2020
Minimal search fractions
Amazon share of retail share
Amazon fee 15%
example, when b = 0, the threshold fixed cost is κ = 5.2; when b = 0.5, the threshold fixed cost is
κ = 2.3; when b = 0.99, the threshold fixed cost becomes κ = 0.02. Hence the model suggests that
the regulatory costs in China must be sufficiently low to induce Alibaba to issue circulable tokens
in 2004. Indeed, some researchers have argued that the success of Alipay in China was due to the
very tolerant financial regulation: Alipay was introduced in 2004, but it did not obtain the official
payment license until 2011 (Huang, 2020). Our numerical exercise suggests that, if there had been
a 100% reserve requirement from the start, the platform could have chosen not to issue tokens.
Another reason for the Alipay’s success is the fact that China’s electronic payment infrastructure
was under-developed before the introduction of Alipay. Compared to the U.S. and other developed
economies, credit cards, debit cards and bank cards were much less used in China. As a result,
the introduction of Alipay helped facilitate transactions that were previously infeasible.15 Here, we
slightly modify the model to illustrate this idea. Suppose, due to payment frictions, cash payments
can only be used for fraction δ < 1 of transactions on the platform. In contrast, tokens can be
used to facilitate all transactions on the platform. Hence the introduction of tokens can expand the
extensive margin of platform transactions. Suppose δ = 0.86, then the platform has an incentive
to issue non-circulatable tokens whenever κ < 0.18.16 Hence, introducing tokens is more profitable
when it helps overcome payment frictions relative to existing payment instruments.
Overall, this numerical exercise identifies three potential explanations for the differences between
Amazon and Alibaba. First, the opportunity cost of holding cash is higher in China (captured by
i). Second, the regulatory costs are lower in China (captured by b, κ). Third, the payment
infrastructure was under-developed in China before the introduction of Alipay (captured by δ).
8
Conclusion
We develop a general equilibrium model to study a platform’s business model choice between
running a cash platform and a token platform, as well as the welfare and policy implications
of this choice. We show that the equilibrium business model choice is not necessarily socially
15
For example, Alibaba provided its first payment service when a Chinese student had difficulty paying a seller in
Japan to buy a second-hand camera in 2003 (Huang, 2020).
16
This parameter choice is motivated by the fact that about 80% of Chinese adults have a bank account in 2017.
The corresponding number in U.S. is 93%. We normalize the value of σ for the U.S. to 1.
26
Table 3: Parameter Values for Example 2
Parameters
Discount factor
Nominal interest rate
Prob. of trading
Prob. of platform trades
Preference parameter
Notation
β
i
σ
α
η
Value
0.97
3.5%
1
0.11
0.13
Notes
Standard in literature
3-month Treasury bond rate in 2017
Minimal search fractions
Alibaba accounts for 11% of total retail in China
Same as that in Example 1
optimal because the private platform does not internalize its impacts on the economy, especially
the spillover effects on non-platform transactions. We also extend the model to endogenize payment
interoperability. The model suggests that the need to impose a reserve requirement depends on
the nature of the risk associated with the platform, as well as the platform’s market share and the
central bank monetary policy.
While our paper has provided novel economic and policy insights, we have abstracted from
some interesting aspects. When the platform can exert private efforts to reduce the cyber risk,
there is a moral hazard in the choice of platform security. Reserve holdings can potentially increase
the platform’s “skin in the game,” implying that imposing a minimum reserve requirement could
sometimes be welfare enhancing. One could also endogenize the platform’s monopoly power by
endogenizing the network effect via the entry of merchants or studying the competition between
multiple private platforms. It would also be interesting to explore other policy tools, such as the
monetary policy of the central bank. Also, offering payment services can have other benefits to the
platform. For example, “Big Tech” platforms can profit from the acquisition and monetization of
user data (Chiu & Koeppl, 2020). We leave these extensions for future research.
27
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Appendix
A
Proof of Lemma 1
Proof. Note first that, given τt ≤ 1 and η < 1, constraint (2) is always binding, implying
that we must have qt = mt . If (2) were not binding then we have m−η
< 1, which implies
t
(1 − τt )1−η m−η
<
1
and
hence
the
RHS
of
(6)
is
zero,
contradicting
the
LHS
which is strictly
t
positive.
n
o
= 0,
We then consider the two possible cases for equation (6): (i) (1 − τt )1−η m−η
−
1
t
+
n
o
−η
>
0
.
In
case
(i),
equation
(6)
becomes
i
=
σ
(1
−
α)
m
−
1
,
and (ii) (1 − τt )1−η m−η
−
1
t
t
+
so we have mt = [1 + i/ [σ (1 − α)]]−1/η = [1 − τC (i)]1/η−1 . Since constraint (3) is not binding
in case (i), using equation (5) the first order condition of (4) with respect to yt implies that
yt = (1 − τt )1/η ≤ (1 − τt ) mt .
h
i
−η
1−η
In case (ii), equation (6) becomes i = σ (1 − α) m−η
−
1
+
σα
(1
−
τ
)
m
−
1
, so we
t
t
t
have mt = m (τt ). Since constraint (3) is binding in case (ii), we have yt = (1 − τt ) qt .
B
Proof of Proposition 1
We first establish the following lemma, which derives some properties of the objective functions in
these two regions.
Lemma 5 We have
(i) G1 (τ ) ≤ G2 (τ ) if and only if τ ≥ τC (i) ;
(ii) G1 (τ ) is single peaked at τ = η; G2 (τ ) is single peaked at τ = τ̄ , where τ̄ ∈ (η, 1).
Proof. Part (i) is obvious after arranging terms of G1 (τ ) and G2 (τ ). To show part (ii), differentiating log G1 (τ ) we have
d
1
1
1
log G1 (τ ) = −
−1
,
dτ
τ
η
1−τ
which is strictly decreasing from infinity to negative infinity when τ goes from 0 to 1, attaining
zero at τ = η. So G1 (τ ) is single peaked at τ = η. Differentiating log G2 (τ ; i) we have
d
1 1−η
α (1 − τ )−η
log G2 (τ ) = −
,
dτ
τ
η 1 − α + α (1 − τ )1−η
which is strictly decreasing from infinity to negative infinity when τ goes from 0 to 1, attaining
zero at τ = τ̄ . So G2 (τ ) is single peaked at τ = τ̄ . Finally, note that τ̄ solves the FOC of G2 (τ ),
which is 1 = (1 − τ̄ )−η (τ̄ /η − 1) / (1/α − 1). Evaluating the FOC at τ = η and τ = 1 we have 0
and ∞. So τ̄ must lie within (η, 1).
Given the results of Lemma 5, now we are ready to prove Proposition 1.
Proof. Consider the following three cases (the possible ranking of the τ threshold and the two
peaks): (a) τC (i) < η < τ̄ ; (b) τC (i) ∈ [η, τ̄ ]; (c) τC (i) > τ̄ > η.
Consider case (a). Since G2 (τ ; i) is single peaked at τ = τ̄ but τ̄ > τC (i), we must have
maxτt ≤τC (i) G2 (τt ; i) = G2 [τC (i)t ; i]. Notice that
C
ΠC
2 = max G2 (τt ; i) = G2 [τC (i) ; i] = G1 [τC (i)] < max G1 (τt ) = Π1 .
τt ≤τC (i)
τt ≥τC (i)
31
Thus we have ΠC =
ασ
r+γ G1 (η)
and τ = η. This corresponds to case i of Lemma 1, and hence
1/η−1
qt = [1 − τC (i)]
and yt = (1 − η)1/η . Rearranging terms, we have τC (i) < η is equivalent to
i < i∗ , corresponding to part (a) of Proposition 1.
Consider case (b). Since G1 (τ ) is single peaked at τ = η but η ≤ τC (i), we must have
maxτt ≥τC (i) G1 (τt ) = G1 [τC (i)]. Notice that
C
ΠC
2 = max G2 (τt ; i) = G2 [τC (i) ; i] = G1 [τC (i)] = max G1 (τt ) = Π1 .
τt ≤τC (i)
τt ≥τC (i)
Thus we have τ = τC (i) and ΠC =
ασ
r+γ G1 [τC
(i)]. This corresponds to case i of Lemma 1, and
1/η−1
hence qt = [1 − τC (i)]
and yt = [1 − τC (i)]1/η . Rearranging terms, we have τC (i) ∈ [η, τ̄ ] is
equivalent to i ∈ [i∗ , ı̂], corresponding to part (b) of Proposition 1.
Consider case (c). The premise of case (c) implies that
C
ΠC
1 = max G1 (τt ) = G1 [τC (i)] = G2 [τC (i) ; i] < max G2 (τt ; i) = Π2 .
τt ≥τC (i)
τt ≤τC (i)
ασ
G2 (τ̄ ; i). This corresponds to case i of Lemma 1, and hence
Thus we have τ = τ̄ and ΠC = r+γ
qt = m [τC (i)] and yt = (1 − τ̄ ) qt . Rearranging terms, we have τC (i) > τ̄ is equivalent to i > ı̂,
corresponding to part (c) of Proposition 1.
C
Proof of Proposition 2
Proof. We omit the proof of the comparative statics of i and γ as they are tedious but straightforward. Given the definition of τC (i), we have
∂τC (i)
> 0.
∂α
(C.1)
Notice that
∂ 2 G2 (τ ; i)
∂α∂τ
∂ 2 G2 (τ ; i)
∂2τ
= G2 (τ̄ ; i)
τ =τ̄
τ =τ̄
∂2
log G2 (τ ; i)
∂α∂τ
∂2
= G2 (τ̄ ; i)
log G2 (τ ; i)
∂τ 2
=−
τ =τ̄
=−
τ =τ̄
1 (1 − α) (1 − η)
G2 (τ̄ ; i) ,
α
τ̄ − η
(1 − α) (1 − τ̄ ) + η [τ̄ − η]
G2 (τ̄ ; i) .
τ̄ 2 (1 − η) (1 − α) (1 − τ̄ )
Since τ̄ > η given by Lemma 5, we have
∂ τ̄
∂ 2 G2 (τ ; i)
=−
∂α
∂α∂τ
τ =τ̄
∂ 2 G2 (τ ; i)
∂2τ
< 0.
τ =τ̄
In the case of i < i∗ , τt and yt are independent to α so their comparative statics is zero.
t
Given qt = [1 − τC (i)]1/η−1 in this case, we have ∂q
∂α < 0. Since G1 (η) is independent to α, given
C
ασ
ΠC = r+γ
G1 (η) in this case we have ∂Π
∂α > 0.
∗
In the case of i ∈ [i , ı̂], note that the signs of the comparative statics of qt and yt are the same,
which follows the comparative statics of qt in the case of i < i∗ . Since we have η ≤ τC (i) in the case
of i ∈ [i∗ , ı̂] (see the proof of Proposition 1), we have G01 [τC (i)] < 0 as τ = τC (i) is on the RHS of
C
ασ
the single peak τ = η of G1 (τ ). Applying (C.1) on ΠC = r+γ
G1 [τC (i)], we have ∂Π
∂α ambiguous.
32
In the case of i > ı̂, since τ̄ solves G02 (τ̄ ; i) = 0, we have
−1
1−η τ̄ 1 − η
α = 1 + (1 − τ̄ )
−1
.
η 1 − τ̄
Substituting the above to qt , we have
qt
yt
i −1/η τ̄ − η + η (1 − τ̄ )η −1/η
,
=
1+
σ
τ̄ (1 − η)
i −1/η
τ̄ − η + η (1 − τ̄ )η −1/η
=
1+
,
(1 − τ̄ )
σ
τ̄ (1 − η)
where are non-monotone in τ̄ , thus we have
D
∂qt ∂yt
∂α , ∂α
and
∂ΠC
∂α
ambiguous.
Proof of Proposition 3
Proof. After substituting ρt , the platform problem is a repetition of the following static problem:
σα
Π = −κα +
r+γ
T
max
[(1 − τt ) kt ]1−η + (1 − τt ) kt −
bt ,kt ,τt ≥0
ibt
kt .
1 − γ (1 − bt )
The solution is simply bt = 0 and (1 − τt ) kt = (1 − η)1/η . Substituting the above solution, we
have proved Proposition 3.
E
Proof of Proposition 4
Proof. We omit the proof as it is straightforward from Proposition 3.
F
Proof of Proposition 5
Proof. Combining (21) and (23) cash is used on the platform if
yt−η − 1 ≥
33
τtM
,
1 − τtM
(F.1)
Substituting ρt of (21) into (24), the platform problem is
ΠD
ibt
σα y −η − 1 ytK −
k
t
t
1−γ(1−bt )
, s.t.(22) and (F.1)
= −κα + max
(1 − γ)t−1 β t
M
+σα τt M yt − ytK
t=1
1−τ
t
ibt
∞
σα y −η − 1 yt −
X
k
t
t
h
1−γ(1−b
it)
(1 − γ)t−1 β t
, s.t.(22) and (F.1)
≤ −κα + max
M
−σα y −η − 1 − τt M yt − ytK
t
t=1
1−τt
(∞
)
X
ib
t
≤ −κα + max
(1 − γ)t−1 β t σα yt−η − 1 yt −
, s.t. (F.1)
kt
1 − γ (1 − bt )
∞
X
t=1
T
≤ Π ,
where the second inequality follows (F.1). If (F.1) is not satisfied, then the dual platform is simply
a token platform thus we have ΠD = ΠT . If (21) holds with strict inequality, then the dual platform
is simply a cash platform thus we have ΠD = ΠC .
G
Proof of Lemma 2
Proof. Denote ∆Π (i) ≡ ΠC − ΠT /α. Using Proposition 1 and Proposition 3, we have
∗
κ if i < i
"
−1/(1−η) #
−1/η
i
i
1/η
σ
1+
− η (1 − η)
1− 1+
if i ∈ [i∗ , ı̂]
κ + r+γ
{z
}
|
σ (1 − α)
σ (1 − α)
G1 (η)
{z
}
|
∆Π (i) =
G
[τ
(i)]
1
C
i1/η
i −1/η h
1/η
1−η
σ
− η (1 − η) if i > ı̂
κ + r+γ 1 +
τ̄ 1 − α + α (1 − τ̄ )
{z
}
|
σ
{z
}
|
G1 (η)
G2 (τ̄ ;i)
Notice that ∆Π (0) > 0 and ∆Π (∞) < 0. Using Proposition 2, we have
∂∆Π (i) /∂i ≤ 0 (strict
∗
Π
Π
inequality when i > i ), so there exist a unique ι such that ∆Π ι = 0 and ΠC < ΠT if and
only if i > ιΠ . Notice that
0 if i < i∗
∂∆Π (i)
∂ΠC α
∗
=
∂i 1−α < 0 if i ∈ [i , ı̂]
h
i1/η−1 h
i
∂α
στ̄ (1+ σi )−1/η
1−η
1−η
− η(r+γ)
1 − α + α (1 − τ̄ )
1 − (1 − τ̄ )
< 0 if i > ı̂
Using
∂ιΠ
∂α
= −
∂∆Π(ιΠ )
∂α
∂∆Π(ιΠ )
∂i
≤ 0 (strict inequality when i > i∗ ), we have proved that ι is
decreasing in α. Finally, notice that G1 (η) > G1 [τC (i)]
> G2 (τ̄ ; i), we have ∂∆Π (i) /∂γ ≥ 0
Π
Π
∂∆Π
ι
∂∆Π(ιΠ )
(
)
(strict inequality when i > i∗ ) and hence ∂ι
=
−
≥ 0.
∂γ
∂γ
∂i
34
H
Proof of Proposition 6
Proof. We consider the three cases in proposition 1. In case (a), a token platform is not adopted.
In case (b), y C < y T , according to propositions 1 to 3. Also, q C = q T = [1 − τC (i)]1/η−1 . In case
(c), y C < y T , according to lemma 1 to 3. Also, q C = m (τ̄ ) and q T = [1 − τC (i)]1/η−1 in this case.
At i = ı̂, we have m (τ̄ ) = [1 − τC (ı̂)]1/η−1 . For i > ı̂, we have m (τ̄ ) fixed and [1 − τC (i)]1/η−1
declining in i. Therefore, we have q C ≥ q T in this case. The third statement is obvious given the
defination of S and the result that reserve is b = 0.
I
Proof of Lemma 3
Proof. Define ∆W (i) ≡ W C − W T /α. Recall the allocations under the cash and token platforms are given by
τ
qtC
qtT
ytC
(a). i < i∗
(b). i ∈ [i∗ , ı̂]
(c). i > ı̂
η
[1 − τC (i)]1/η−1
[1 − τC (i)]1/η−1
(1 − η)1/η
τC (i)
[1 − τC (i)]1/η−1
[1 − τC (i)]1/η−1
[1 − τC (i)]1/η
τ̄
m (τ̄ )
[1 − τC (i)]1/η−1
(1 − τ̄ ) qt
ytT
ΠC
ΠT
(a). i < i∗
(1 − η)1/η
(b). i ∈ [i∗ , ı̂]
(1 − η)1/η
ασ
r+γ G1 (η)
ασ
r+γ G1 [τC (i)]
ασ
r+γ G2 (τ̄ ; i)
κα
− r+γ
+
κα
− r+γ
+
κα
− r+γ
+
(c). i > ı̂
(1 − η)1/η
Using Proposition 1 and Proposition 3, we have
(1 − η)1/η−1
σαη
r+γ
σαη
r+γ
σαη
r+γ
(1 − η)1/η−1
(1 − η)1/η−1
κ if i < i∗ ;
κ + σ u y C − y C − u y T + y T if i ∈ [i∗ , ı̂] ;
r+γ
∆W (i) =
σ
C − yC − u yT + yT
κ
+
u
y
r+γ
if i > ı̂.
σ
(1/α − 1) u q C − qtC − u q T + q T
+ r+γ
In the case of i ∈ [i∗ , ı̂], it is straightforward to see that ∂∆W(i)
< 0. In the case of i > ı̂,
∂i
C
C
differentiating the above and using the fact that yt = (1 − τ̄ ) qt from Proposition 1, we have
∂∆W (i)
∂i
(
≤ 0 if i ≤h ı̂
i
= α(r+γ)
1+
σατ̄
i
∂qC
∂i
−
∂q T
∂i
i
if i > ı̂
For the derivative of i, we have
1+
σατ̄ ∂q C
∂q T
1
−
=
i
∂i
∂i
ησ (1 − α)
1+
where
σατ̄
H (i) = 1 +
− (1 − α)−1/η
i
α
1−
1+
35
i
σ (1 − α)
!1/η+1
i
σ
h
−1/η−1
H (i) ,
1 − α + α (1 − τ̄ )1−η
i1/η
.
The sign of
∂∆Wt (i)
∂i
in the case of i > ı̂ depends on H (i), which is decreasing in i. Notice that
#1/η
"
α (1 − τ̄ )1−η
σατ̄
> 0, H (∞) = 1 − 1 +
< 0.
H (ı̂) ≡ α +
ı̂
1−α
Also,
∂∆W(i)
∂i
∂∆W(i)
∂i
= β −1 κα > 0 and
∝ −ΠT < 0, so there exists a unique ιW such that
∆W(i)
∂i
i=ιW
= 0 and W T < W C if and only if i < ιW . For the derivative of α, using Proposition 1
and Proposition 3 we have
∂∆W (i)
≤ 0 if i ≤ ı̂,
∂α
∂∆W (ιW )
∂∆W (ιW )
∂ιW
Thus we have ∂α ≡ −
< 0 if i ≤ ı̂. For the derivative of γ, notice that
∂α
∂i
∂∆W
∂γ
Thus we have
J
ιW
∂ιW
∂γ
= −
≡−
σ
u yC − yC − u
(r+γ)2
+ σ(1/α−1)
u yC − qC −
(r+γ)2
∂∆W (ιW )
∂γ
∂∆W (ιW )
∂i
yT + yT
u qT + qT
=
κ
>0
r+γ
i=ιW
> 0.
Proof of Proposition 7
Proof. The repeated static problem is reduced to
σα
ΠT = −κα +
r+γ
"
(
max
bt ≥b̄,yt ,τt ≥0
# )
i
γτ
+
b
t
t
σα
yt1−η − 1 +
yt ,
1 − τt 1 − γ (1 − bt )
where the solution is simply bt = b̄ and τt = 0. The first order condition with respect to yt implies
yt =
1−η
1+
i
b̄
σα 1−γ (1−b̄)
1/η
.
Substituting the above solution, we have proved Proposition 7.
K
Proof of Lemma 4
Proof. Consider two cases of (27): in the first case (mt > 0), we have i = σ (1 − α) (mt + kt )−η − 1
and in the second case (mt = 0) we have i > σ (1 − α) (mt + kt )−η − 1 + . In the first case, we
have mt + kt = {1 + i/ [σ (1 − α)]}−1/η and kt−η ≥ (mt + kt )−η = 1 + i/ [σ (1 − α)] > 1. Thus, we
36
+
,
h
i
have ρt = σ α kt−η − 1 + σi , and hence kt = 1 +
mt + kt = 1 +
i
σ (1 − α)
−1/η
ρt −i
σα
> kt =
1+
−1/η
ρt − i
σα
. The premise mt > 0 is satisfied if
−1/η
⇔ i ≤ (1 − α) ρt .
In the second case, rearranging terms we have kt−η < 1 + i/ [σ (1 − α)]. Thus, we have kt =
−1/η
1 + ρσt
.
37
K.1
Online Appendix: Deposit insurance
To deal with the collapse risk, a commonly used regulation is deposit insurance. One can treat the
platform as a bank and tokens as deposits. Suppose the central bank requires a token platform
to pay a premium at the rate ϕ if the cyber shock does not hit. When the shock realizes, the
insurance will compensate token holders for the loss of the token value. Since the token holders are
fully insured against the cyber risk, the nominal price of retail goods in terms of tokens is equal to
that in terms of cash, i.e., PtT = Pt = 1/φt . In this case, it is obvious that the platform never holds
reserves (bt = 0), because the loss is fully covered by deposit insurance. The shadow interest rate
of tokens becomes
n
o
ρt = σα (1 − τt )1−η kt−η − 1 .
+
The initial profit flow of the platform in stage 2 is
x0 = p0 K1 − κα.
The expected profit flow in the latter period is
τt φt PtT yt
.
1 − τt
n
o
P
t−1 t
(1
−
γ)
β
x̄
The platform’s profit is given by ΠD = max x0 + ∞
t . By collecting terms of
t=1
the same t across periods, the platform’s profit can be expressed as
x̄t = (1 − γ)pt [Kt+1 − Kt − ϕKt ] + σα
ΠD
1
p0 K
P∞
t−1 t
pt
pt+1 Kt+1
β (1 − γ) pt+1
+
= −κα + max
t=1 (1 − γ)
h
i
kt ,τt ≥0
P
+ ∞ (1 − γ)t−1 β t −(1 − γ) (1 + ϕ) pt Kt + σα τt φt PtT yt
t=1
1−τt
(∞
)
X
= −κα + max
(1 − γ)t−1 β t [1 + ρt − (1 − γ) (1 + ϕ) + σατt ] kt ,
kt ,τt ≥0
(K.1)
t=1
where the platform takes as given the premium rate ϕ. In equilibrium, the premium rate of the
deposit insurance is actuarially fair, such that
(1 − γ)ϕ = γ.
(K.2)
We assume that the insurance premium (and payout after the cyber shock) are rebated to consumers
as lump sum transfers.
By substituting yt = (1 − τt ) kt , the repeated static problem is reduced to
max
yt ,τt ≥0
n
o
yt1−η − yt ,
which is independent to τt . Thus, τt is indeterminate, as long as ρt ≥ 0, which implies τt ∈ [0, η].
The first order condition with respect to yt implies
yt = (1 − η)1/η .
Substituting the above solution, we can derive the following proposition.
38
Proposition 8 The platform is indifferent to any τt ∈ [0, η]. The equilibrium allocation of a token
platform is the same, i.e., the deposit insurance does not affect the equilibrium allocation and profits
of the token platform, i.e.,
ρt = σα
η − τt
,
1−η
qt = [1 − τC (i)]1/η−1 ,
yt = (1 − η)1/η ,
σαη
ΠD = −κα +
(1 − η)1/η−1 .
r+γ
Corollary 3 The deposit insurance does not affect welfare but slows down the growth rate of the
nominal token price g.
Here is why deposit insurance is neutral in this economy. Recall that the platform can always
self-insure against the cyber risk by holding reserves b. Deposit insurance is cheaper than holding
reserves because buying deposit insurance does not incur the opportunity cost of nominal interest i.
However, this cheaper insurance option is not useful for the platform because the platform chooses
to hold zero reserves in the absence of deposit insurance. Instead, the platform compensates the
consumers’ expected loss from the cyber risk by engineering a higher token price growth gt . Thus,
the rise of insurance premium is offset one-to-one by the corresponding decrease in gt , while the
profits and allocation stay unchanged.
39
@FedFRASER