MANAGEMENT SCIENCE

Vol. 69, No. 11, November 2023, pp. 6684–6707

https://pubsonline.informs.org/journal/mnsc ISSN 0025-1909 (print), ISSN 1526-5501 (online)

A Model of Cryptocurrencies
Michael Sockin,a,* Wei Xiongb,c
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a
McCombs School of Business, University of Texas at Austin, Austin, Texas 78712; b Bendheim Center for Finance, Princeton University,
Princeton, New Jersey 08544; c School of Management and Economics, Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China
*Corresponding author
Contact: Michael.Sockin@mccombs.utexas.edu (MS); wxiong@princeton.edu, https://orcid.org/0000-0003-3592-9373 (WX)

Received: February 6, 2022 Abstract. We model cryptocurrencies as utility tokens used by a decentralized digital plat­
Revised: August 24, 2022; December 15, 2022 form to facilitate transactions between users of certain goods or services. The network effect
Accepted: March 11, 2023 governing user participation, in conjunction with the nonneutrality of the token price, can
Published Online in Articles in Advance: cause the token market to break down. We show that token retradeability mitigates this risk
April 11, 2023 of breakdown on younger platforms by harnessing user optimism but worsens this fragility
when sentiment trading by speculators crowds out users. Elastic token issuance mitigates
https://doi.org/10.1287/mnsc.2023.4756 this fragility, but strategic attacks by miners exacerbate it because users’ anticipation of
future losses depresses the token’s resale value.
Copyright: © 2023 INFORMS
History: Accepted by Agostino Capponi, Special Section of Management Science: Blockchains and Crypto
Economics.

Keywords: cryptocurrency • token price nonneutrality • optimism • platform fragility

1. Introduction other; Filecoin, which matches the demand and supply

The rapid growth of the cryptocurrency market in the for decentralized computational storage; and GameCre­
last few years promises a new funding model for inno­ dits, which finances the purchase, development, and
vative digital platforms. Rampant speculation and vola­ consumption of online games and gaming content. The
tility in the trading of many cryptocurrencies, however, development of these platforms is financed by the sale
have also raised substantial concerns that associate of tokens to investors and potential users through the
cryptocurrencies with potential bubbles. The failure of issuance of utility tokens.
the DAO only a few months after its initial coin offering We follow Sockin and Xiong (2023) to model a crypto­
(ICO) raised $150 million in 2016, together with a num­ currency as membership in a platform, which has been
ber of other similar episodes, particularly highlights the created by its developer to facilitate decentralized bilat­
risks and fragility of cryptocurrencies. Understanding eral transactions of certain goods or services among a
the risks and potential benefits of cryptocurrencies pool of users by using a blockchain technology. Users
requires a systematic framework that incorporates sev­ face difficulty in making such transactions outside the
eral integral characteristics of cryptocurrencies—their platform as a result of severe search frictions. The plat­
role in funding digital platforms and in serving as form fills the users’ transaction needs by pooling a large
investment assets for speculators and their integration number of users who need to transact with each other.
of blockchain technology with decentralized consensus A user’s transaction need is determined by its endow­
protocols to record transactions on the platforms. We ment in a consumption good and its preference of con­
develop such a model in this paper. suming its own good together with the goods of other
Our model analyzes the properties of cryptocurren­ users. As a result of this preference, users need to trade
cies on platforms that rely on network effects. Crypto­ goods with each other, and the platform serves to facili­
currencies cover a wide range of tokens and coins tate such trading. Specifically, when two users are ran­
facilitated by crypto technologies. For simplicity, we domly matched, they can trade their goods with each
anchor our analysis on utility tokens, but our model can other only if they both belong to the platform. Conse­
also be applied to coins and altcoins. Utility tokens are quently, there is a key network effect—each user’s
native currencies accepted on decentralized digital plat­ desire to join the platform grows with the number of
forms that often provide intrinsic benefit to partici­ other users on the platform and the size of their goods
pants.1 The benefits of utility tokens can range from endowments. If more users join the platform, each bene­
provision of secure and verifiable peer-to-peer transac­ fits more from joining the platform and is willing to pay
tion services to the maintenance of smart contracts. a higher token price. Sockin and Xiong (2023) highlight
Examples of such utility tokens include Ether, which that tokenization helps to decentralize the control of
enables participants to write smart contracts with each the platform and makes it possible for the platform to

6684
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6685

commit to not exploiting its users. This commitment, is specifically relevant to platforms with nonneutral
however, comes at the expense of not having an owner token prices but not platforms that adjust the number of
with an equity stake to subsidize user participation and tokens required for their services in response to token
maximize the platform’s network effect.2 price fluctuations.
Our analysis builds on two key features. First, a user’s Users’ optimism about token price appreciation can
benefit from using the token is increasing in the quan­ alleviate this instability by inducing users to join the
tity, rather than in value in fiat currency, of tokens that platform even when their transaction needs are low. In
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she holds. This assumption is motivated by the nonneu­ contrast, speculators’ sentiment exacerbates this fragil­
trality of money that underlies modern monetary the­ ity by raising the cost for users to participate and crowd­
ory. This can arise, for instance, because of stickiness on ing them out. Consequently, token retradeability is a
the platform in adjusting the number of tokens required powerful tool for improving platform performance when
for its services in response to token price fluctuation.3 it capitalizes on user optimism. In contrast, it harms per­
As a result of such stickiness, a shock to the token price formance when it incentivizes outsiders, like speculators,
can directly affect user participation. This, in turn, to hold tokens as well, as their enthusiasm acts as a tax
amplifies the price shock through the platform’s net­ on user participation and exacerbates the platform’s
work effect. Second, in our model, the supply of tokens instability. Furthermore, elastic token issuance mitigates
to users is decentralized in that token issuance follows a this fragility.
predetermined schedule and that token market partici­ Because the supply of tokens increases deterministi­
pants are atomistic and therefore, do not internalize cally over time, the platform exhibits life-cycle effects
how their trading impacts others. that are governed by the substitution of the token’s cur­
Our model features infinitely many periods, with rent convenience yield and expected capital gains, which
users and speculators holding different beliefs about the jointly determine the total token return to each user. The
capital gain from holding the token. In each period, a inflation of the token base over time lowers expected capi­
new generation of users chooses whether to join the tal gains by shifting out the token supply curve. As a
platform by purchasing tokens from both existing token result, the region of market breakdown and the relative
holders and from new token issuance by the platform. weight of the convenience yield in the total token return
In deciding whether to join the platform, a user trades increase over time. Both of these effects, in turn, raise the
off the cost of buying a token with the benefits from sensitivity of the user base to the current demand funda­
both transacting goods on the platform and expected mental and log token price volatility over time. We illus­
token price appreciation. Each user optimally adopts a trate that more mature platforms not only have lower
cutoff strategy to join the platform by purchasing the expected log token prices but also, higher log token price
token only if its goods endowment is higher than a volatility and that these life-cycle effects are more pro­
threshold. This threshold and the token price are jointly nounced for platforms whose fundamentals have weaker
determined by users’ token demand, which is based on growth rates. Consequently, the ability of retradeability
their common goods endowment and optimism about to harness the optimism of users to mitigate platform sta­
token price appreciation and the net supply of tokens by bility declines as the platform matures.
speculators; this is also determined by their sentiment To further illustrate how outsiders hamper platform
about token price appreciation. Despite the inherent performance, we extend the model to incorporate miners
nonlinearity induced by each user’s cutoff strategy, we who provide accounting and custodial services to record
derive the equilibrium in an analytical form and system­ transactions on the platform’s blockchain according to
atically characterize the platform’s performance. the proof of work (PoW) protocol. Each miner incurs a
Our analysis highlights the fragility of cryptocurrency computational cost in providing the service and is com­
platforms induced by the network effect of user partici­ pensated by the seigniorage from token inflation, which
pation and decentralized token trading. Because of the diminishes deterministically over time, and a transaction
network effect, users’ demand curve for tokens is hump fee, which is a fraction of the transaction surplus of the
shaped (rather than downward sloping), whereas the users on the platform. This trade-off determines the num­
net token supply faced by users is upward sloping. As a ber of miners on the platform. When the number of
result, even though a trivial equilibrium with zero user miners falls sufficiently low, some corrupt miners may
demand and zero token price always exists, the token choose to attack the cryptocurrency so that they can bene­
price may fail to simultaneously clear the supply and fit from creating fraudulent seigniorage and stealing
demand for tokens with positive user participation. In other miners’ transaction fees. Although such attacks do
this case, the token market breaks down, which occurs not directly lead the platform to fail, our analysis shows
when the platform’s demand fundamental is sufficiently that users’ anticipation of future losses from miner
weak. Such market breakdown represents a severe form attacks may exacerbate the fragility of the token market,
of distortion induced by token price on user participa­ especially when the mining cost is high. Consequently,
tion through the network effect. Note that this distortion having outsiders with whom there is a conflict of interest
 Sockin and Xiong: A Model of Cryptocurrencies
6686 Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS

with users exacerbates the instability of cryptocurrency Our paper is also related to the emerging literature on
platforms. cryptocurrencies. Our model shares a similar pricing
Our framework provides a rich set of empirical pre­ model but differs by deriving a strong network effect in
dictions for token price appreciation. As only part of the transaction benefits of the cryptocurrency as well as
users’ token return, the expected token price apprecia­ subtle interactions between strategic attacks by miners
tion is determined by the marginal user’s equilibrium and the cryptocurrency’s fragility. Cong et al. (2021)
condition—it equals the total cost of capital and partici­ also emphasize the strong network effect among plat­
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pation minus the convenience yield from transaction form users. They construct a dynamic model of crypto
surplus. Consistent with Liu and Tsyvinski (2021), our tokens to study the dynamic feedback between user
model predicts a role for both news and investor senti­ adoption and the responsiveness of the token price to
ment in explaining the time series of cryptocurrency expectations about future growth on the platform. In
price appreciation, not through risk premia but rather, contrast to the monetary neutrality assumed in their
by predicting the marginal user’s convenience yield. In model, which ensures that the token market is always
addition, our model can rationalize the momentum pat­ stable, our model assumes that the token price is non­
terns that they observe in token price appreciation neutral. This key assumption, together with the network
through the persistence of user participation costs and effect in user participation, underlies our mechanism
convenience yields, as well as the size effect that Liu et al. that induces platform fragility. In addition, we show
(2022) show in the cross-section of cryptocurrency price that miner attacks may exacerbate the platform fragility
appreciation. Nonfundamental shocks to token prices, through the users’ anticipation of losses from future
represented by user optimism and speculator sentiment attacks. Athey et al. (2016) model Bitcoin as a medium of
in our model, can also help explain reversals in crypto­ exchange of unknown quality that allows users to avoid
currency returns, consistent with the evidence of a bank fees when sending remittances, and they use the
“value” factor in Cong et al. (2022a). Importantly, our model to guide an empirical analysis of Bitcoin users.
asset pricing predictions are applicable only to tokens Schilling and Uhlig (2019) study the role of monetary
on platforms with nonneutral token prices and would policy in the presence of a cryptocurrency that acts as a
not apply, for instance, to (alt-)coins and tokens on plat­ private fiat currency. Mayer (2019) finds that specula­
forms with token neutrality, such as stablecoins and tors provide or take liquidity from adopters depending
nonfungible tokens (NFTs). on how volatile the platform fundamental is. In contrast
Our paper contributes to a literature that studies to these papers and as a key contribution of our analysis,
instability on cryptocurrency platforms. Cong et al. we examine token prices and platform performance
(2021b) show that network effects amplify utility token with a realistic information structure that allows us to
price but mitigate user base volatility. Biais et al. (2023) examine the role of optimism among users and senti­
develop a structural model of Bitcoin with transaction ment among speculators. Goldstein et al. (2019) show
benefits and costs from hacking and show that Bitcoin is that when there is token nonneutrality on an online plat­
subject to significant extrinsic volatility because of coor­ form, utility tokens that trade in secondary markets can
dination on sunspot equilibria. Pagnotta (2022) shows in act as a commitment device for an owner to price ser­
an equilibrium model of Bitcoin that the interaction vices competitively.
between the network of users and the investment of Our analysis also contributes to the literature on fric­
miners in network security amplifies Bitcoin price vola­ tions in consensus validation on cryptocurrency plat­
tility. Mei and Sockin (2022) illustrate how speculation forms. Easley et al. (2019) analyze the rise of transaction
as an outside option can slow down learning on token fees in Bitcoin through the strategic interaction of users
platforms with network effects and lead to participation and miners. Chiu and Koeppl (2022) consider the opti­
traps. In contrast to these papers, we highlight how the mal design of a cryptocurrency and emphasize the
network effect among users can lead to market break­ importance of scale in deterring double spending by
down when there is nonneutrality of the token price on buyers. Cong and He (2019) investigate the trade-off of
users’ participation decisions. Furthermore, this insta­ smart contracts in overcoming adverse selection while
bility is mitigated by user optimism and exacerbated by also facilitating oligopolistic collusion, whereas Biais
speculator sentiment because the latter shifts upward et al. (2019) consider the strategic interaction among
the supply curve of tokens. We further illustrate how miners. Pagnotta (2022) examines the strategic interac­
users’ anticipation of strategic attacks amplifies this fra­ tion among miners on the Bitcoin platform. Capponi
gility by reducing the token’s expected retrade value. et al. (2023) illustrate how the nature of mining may
Our mechanism also differs from the impact of specula­ lead to a concentration of mining power, whereas Abadi
tion on other asset classes, such as stocks and commodi­ and Brunnermeier (2018) examine disciplining writers
ties, in which speculation can increase price volatility to a blockchain technology with static incentives. Saleh
but not lead to a breakdown in which price and demand (2021) explores how decentralized consensus can be
both collapse to zero. achieved with the proof of stake protocol. Even without
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6687

strategic attacks, Capponi et al. (2021) demonstrate how (Kiyotaki and Wright 1993). If user i at t � 1 chooses to
miners can impose more subtle costs on users by leaking purchase the token, he purchases one unit at the equilib­
information about their transactions for front running. rium price Pt , denominated in the consumption numer­
aire. In the next period t + 1, each user from period t
2. The Model resells his token to future users and to speculators.
Consider a cryptocurrency that facilitates transactions We follow Sockin and Xiong (2023) to model the
on a decentralized digital platform. The platform serves users’ transactions on the platform. In each period, user
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to reduce search frictions among a pool of users who i is endowed with a certain good and is randomly paired
share a certain need to transact goods with each other. with a potential trading partner, user j, who is endowed
The benefits of participating on a utility token platform, with another good. Users i and j can transact with each
such as Ether or FileCoin, include securing transactions other only if both have the token. After their transaction,
and writing smart contracts to sharing gaming content user i has a Cobb–Douglas utility function over con­
and providing secure file storage. As the value of the sumption of his own good and the good of user j accord­
token may appreciate with the development of the plat­ ing to
form over time, the token also serves as an investable � � � �
Ci,t 1�ηc Cj,t ηc
asset for users and speculators to speculate about the Ui,t (Ci,t , Cj,t ; N t ) � , (1)
1 � ηc ηc
growth of the platform.
The model is discrete time with infinitely many peri­ where ηc ∈ (0, 1) represents the weight in the Cobb–
ods: t � 1, 2, : : : There are three types of agents on the Douglas utility function on his consumption of his trad­
platform: users, speculators, and validators. The suc­ ing partner’s good Cj,t and 1 � ηc is the weight on the
cess of the cryptocurrency is ultimately determined by consumption of his own good Ci,t : A higher ηc means a
whether the platform can gather a large number of stronger complementarity between the consumption of
users together. In each period, a new generation of the two goods. Both goods are needed for the user to
users purchases the cryptocurrency as the member­ derive utility from consumption. If one of them is not a
ship to the platform, and then, these users are ran­ member of the platform, there is no transaction, and
domly matched with each other to transact their goods consequently, each of them gets zero utility. This setting
endowments. The goods transactions are supported by implies that each user cares about the pool of users on
validators of the decentralized platform who act as ser­ the platform, which determines the probability of com­
vice providers and complete all user transactions. They pleting a transaction.
record these transactions in an indelible ledger called the The goods endowment of user i is eAi,t , where Ai,t is
blockchain. A key feature of the blockchain technology composed of a component At common to all users and
underpinning cryptocurrencies is that it is permission­ an idiosyncratic component εi,t :
less and verifies transactions through decentralized con­
Ai,t � At + τ�1=2
ε εi,t ,
sensus among an anonymous population of validators.
For now, we assume there are no issues of trust on the with εi,t ~ N (0, 1) being normally distributed and inde­
platform. We will extend the model in Section 4 to incor­ pendent withR each other, across time, and from At : We
porate decentralized miners who follow the PoW proto­ assume that εi,t dΦ(εi,t ) � 0 at each date by the strong
col to record transactions and may collude to strategically law of large numbers. The aggregate endowment At fol­
attack the platform. lows a random walk with a constant drift µ ∈ R:
�1=2 A
At � At�1 + µ + τA εt+1 ,
2.1. Users
There are overlapping generations of users who join the where εA t+1 ~ i:i:d N (0, 1): The aggregate endowment At
platform. In each period t, there is a pool of potential is a key characteristic of the platform. A cleverly designed
users, indexed by i ∈ [0, 1]. Each of these potential users platform serves to attract users with strong needs to trans­
is endowed with a different consumption good and act with each other. As we will show, a higher At leads to
needs to transact her good with another user so that more users on the platform, which in turn, implies a
each user can consume two goods. To complete such a higher probability of each user completing a transaction
transaction, both users need to participate on the plat­ with another user, and furthermore, each transaction
form by purchasing a unit of the cryptocurrency, which gives greater surpluses to both parties. One can, therefore,
we call a token of the platform. We can divide the unit view At as the demand fundamental for the cryptocur­
interval into the partition {N t , Ot } in each period, with rency and τε as a measure of dispersion among users in
N t ∩ Ot � ø and N t ∪ Ot � [0, 1]: Let Xi,t � 1 if user i the platform.4
purchases the token (that is, i ∈ N t ) and Xi,t � 0 if he We start with describing each user’s problem in
chooses not to purchase. An indivisible unit of currency period t, conditional on joining the platform and meet­
is commonly employed in search models of money ing a transaction partner, and then, we go backward to
 Sockin and Xiong: A Model of Cryptocurrencies
6688 Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS

describe his earlier decision on whether to join the plat­ platform’s matching services. As a result, token price
form. At t, when user i is paired with another user j on fluctuations directly affect user participation, which
the platform, we assume that they simply swap their may be amplified further by the network effect of user
goods, with user i using ηc eAi,t units of good i to participation.6
exchange for ηc eAj,t units of good j. Consequently, both An important aspect of our analysis is how the weights
users are able to consume both goods, with user i con­ of the token’s convenience yield and capital gain transi­
suming tion over the life of the platform. When the platform is
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young, there are few tokens in circulation, and users

Ci,t (i) � (1 � ηc )eAi,t , Cj,t (i) � ηc eAj,t
benefit more from the token price appreciation. When
and user j consuming the platform matures, there are many tokens in circula­
tion, and users benefit mostly from the convenience
Ci,t (j) � ηc eAi,t , Cj,t (j) � (1 � ηc )eAj,t :
yield from transactions on the platform. As we will ana­
As formally shown by Sockin and Xiong (2023), these lyze later, this transition underlies several interesting
consumption allocations between these two paired users life-cycle implications; more mature platforms might be
can be microfounded through a trading mechanism more vulnerable to market breakdown, younger plat­
between them. Furthermore, we can use Equation (1) to forms might have higher market capitalizations, and
compute the utility surplus Ui,t of each user from this token price volatility is increasing over time.
transaction. We now describe the information set, I i,t , of each
Before finding a transaction partner on the platform, user. In addition to observing the platform fundamen­
each user needs to decide whether to join the platform tal, At , each user knows the value of his own goods
by buying the token. In addition to the utility surplus, endowment, Ai,t : To facilitate our analysis of how users’
Ui,t , from the transaction, there is also a capital gain speculation of the token price may affect their participa­
from retrading the token, Pt+1 � RPt , with R ≥ 1 being tion in the platform, we also endow all users with a public
the interest rate for the holding period. We assume that signal about the next period’s innovation to aggregate
users have quasilinear expected utility and incur a linear endowment, εA t+1 , which by construction, is orthogonal to
utility gain equal to this capital gain net of a fixed partic­ At:
ipation cost κ > 0 if they choose to join the platform. The �1=2 Q
participation cost may be either pecuniary or mental Qt � εA
t+1 + τQ εt ,
and could represent, for instance, the cost of setting up a
where εQ t ~ i:i:d N (0, 1): This public signal is similar to a
wallet and installing the software necessary for participat­
“news” shock in the language of Beaudry and Portier
ing on the platform. Furthermore, we assume that each
(2006). Because the public signal only reveals informa­
user needs to give a fraction β of his utility surplus Ui,t
tion about next period’s At+1 , it only impacts users’
from the transaction as the service fee to the platform.
decisions through their beliefs about the next period’s
In summary, user i makes his purchase decision at t
token price, E[Pt+1 |I i,t ], and therefore, it represents a
according to
� � speculative shock to all the users. Even though we use
max E[(1 � β)Ui,t + Pt+1 | I i,t ] � RPt � κ Xi,t , (2) the term “user optimism” to denote the speculative
Xi,t
shock induced by the public signal Qt, the users are fully
where I i,t is the information set of user i at date t. Note rational in information processing in our model. Conse­
that the expectation of the user’s utility flow regards the quently, I i,t � σ({Ai,t , {Ps ,Qs }s≤t }) is user i’s full informa­
uncertainty associated with matching a transaction part­ tion set.
ner, whereas the expectation of the capital gain from It then follows that user i’s purchase decision is given
holding the token regards the uncertainty in the growth by
of the platform. By adopting a Cobb–Douglas utility �
1 if E[(1 � β)Ui,t + Pt+1 � RPt | I i,t ] ≥ κ
function with quasilinearity in wealth, users are risk Xi,t �
0 if E[(1 � β)Ui,t + Pt+1 � RPt | I i,t ] < κ:
neutral with respect to the token’s capital gain.5
By treating the token as a membership to the plat­ As the user’s expected utility is monotonically increas­
form, our model simplifies each user’s token demand to ing with his own endowment, regardless of other users’
a binary choice. In Appendix B, we consider a more gen­ strategies, it is optimal for each user to use a cutoff strat­
eral setting in which each user’s benefit from holding egy. This, in turn, leads to a cutoff equilibrium, in which
the token increases in the quantity of tokens that she only users with endowments above a critical level A∗t
holds. We show that our key results on platform fragility buy the token. This cutoff is eventually solved as a fixed
do not depend on this binary token demand assumption. point in the equilibrium to equate the token price, net of
Instead, what is key to our analysis is the nonneutrality of the expected resale value and participation cost, with
the token price—the platform does not adjust the num­ the expected transaction utility of the marginal user
ber of tokens required for each user to qualify for the from joining the platform. As each user’s participation
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6689

strategy also depends on his expected token resale value that they may trade overconfidently on noisy information
E[Pt+1 |I i,t ], the common optimism among users in­ or on spurious correlations that give rise to misspecified
duced by Qt helps to overcome their participation cost technical trading strategies. Hackethal et al. (2021), for
κ: Because a user receives all his transaction benefit instance, provide evidence that cryptocurrency investors
from holding one token, he will never buy a second and are prone to investment biases, to following technical
pay the participation cost for just the capital gain. This is analysis heuristics, and to investing in stocks with high
because the token price will equal the capital gain plus media sentiment. Fracassi and Kogan (2022) show that
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the transaction benefit of the marginal user. cryptocurrency investors trade based on pure technical
Given the cutoff strategy for each user who partici­ analysis.
pates if Ai,t ≥ A∗t , the total token demand of users Nt is Specifically, we assume the following token demand
given by curve for speculators:
Z ∞ � �
√ffiffiffiffi
Nt � Xi,t (I i,t )dΦ(εi,t ) � Φ( τε (At � A∗t )): (3) XS � Φ(yt ) � Φ yt + λ log(RPt ) � ζt ,
�∞
where ζt is the common sentiment shock of speculators
2.2. Token Supply and Speculators about the next-period token price. We separate specula­
Consistent with the common practice of decentralized tors’ sentiment from users’ optimism so that we can ana­
crypto platforms, we assume that the token supply, lyze their distinct effects on the token market equilibrium.
Φ(yt ), grows over time according to a predetermined This demand curve has the property that when specula­
schedule: tors, on average, are more optimistic (i.e., a higher ζt),
their demand is higher and tightens the supply of tokens
Φ(yt ) � Φ(yt�1 + ι), (4)
for users. In contrast, when the token price is higher, the
where Φ(·) is the normal cumulative distribution func­ usual downward-sloping demand effect leads to lower
tion. Token platforms may commit to a predetermined demand from speculators and a higher supply of tokens
inflation schedule, for instance, to mitigate incentives of for users. In Appendix C, we provide a parsimonious
validators to exploit users through excessive issuance microfoundation that derives this demand curve by agg­
but at the cost of suboptimal platform performance.7 It regating the dispersed token demand among a group of
also reflects that the inflation rules employed in practice atomistic active and passive speculators.
do not, for instance, condition on platform performance; Market clearing in the token market consequently
secondary market trading conditions; or the distribution imposes that
of tokens across users, speculators, and validators. This √ffiffiffiffi
Φ( τε (At � A∗t )) + Φ(yt ) � Φ(yt + λ log(RPt ) � ζt )
specification further captures, as in practice, that the
increase in supply from token inflation tapers over time. � Φ(yt ),
For PoW platforms, such as Bitcoin and Ethereum before
the Merge, the number of new coins and tokens created where we have substituted users’ token demand with
by inflation periodically halves over time, according to a (3). This condition implies a token price:
�√ffiffiffiffi �
predetermined schedule, so that the total supply asymp­ 1 τε 1 1
totes to a fixed limit.8 With our specification, at most a Pt � exp (At � A∗t ) � yt + ζt , (5)
R λ λ λ
unit measure of tokens exists. All of our key qualitative
results are unchanged, however, if instead we capped where the equilibrium token price Pt is a log-linear func­
token supply at some maximum y < ∞: tion of the platform’s demand fundamental At, the
In addition to the token inflation, we assume that there users’ participation threshold A∗t , the token supply yt ,
is a continuum of atomistic and myopic speculators who and speculator sentiment ζt.9
trade the token for investment and speculation purposes.
Speculators provide liquidity by buying tokens, including 2.3. Validators
those from the old generation of users, and then selling The platform requires record keeping of all transac­
them to the new generation of users. In our model, the tions. For the baseline model, we assume that the
trading of the token is fully decentralized in the sense that decentralized platform has a group of validators who
every participant is small and does not internalize the complete all user transactions each period without any
effects of her trading on others. frictions and record these transactions on the block­
We consider speculators to be outsiders to the platform chain.10 In a later section (Section 4), we expand the
who are distinct from the users who actually participate model to assume these validators record the transac­
on it. As such, they do not have private information about tions for a fee according to the PoW protocol and may
the platform’s fundamental or fully understand how to also attack the cryptocurrency. In the baseline setting,
interpret the implications of the same public information the payment to validators in period t is both the sei­
as the users. Instead, similar to Black (1986), we argue gnorage from the scheduled inflation of the token base,
 Sockin and Xiong: A Model of Cryptocurrencies
6690 Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS

Φ(yt�1 + ι) � Φ(yt�1 ), and the transaction fees from users, sets. After observing Qt , users share the same posterior
πt � (Φ(yt�1 + ι) � Φ(yt�1 ))Pt + βUt , belief about At+1 , which is normal with the following
conditional mean:
where Ut is the total transaction surplus on the platform. τQ
Validators have no use for tokens and potentially for  t+1 � At + µ + Qt :
τε + τQ
liquidity reasons, sell them immediately to speculators.
Assuming a cutoff strategy for users, we can integrate As we discussed earlier, the noise in Qt is a shock to the
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the expression for the expected utility of a user who users’ speculative optimism because it has no impact on
joins the platform, as derived in Sockin and Xiong their current surplus from transacting with other users
(2023), over Ai,t for Ai,t ≥ A∗t to arrive at the realized sur­ on the platform.
plus from user transactions: In each period, users sort into the platform according
! to a cutoff equilibrium determined by the net benefit of
At +12((1�ηc )2 +η2c )τ�1 �1=2 At � A∗t joining the platform, which trades off the opportunity of
Ut � e ε Φ (1 � ηc )τε + �1=2
τε transacting with other users on the platform and the
! expected token price appreciation with the cost of partic­
At � A∗ ipation. Despite the inherent nonlinearity of our frame­
· Φ ηc τ�1=2ε + �1=2 t :
τε work, we derive a tractable cutoff equilibrium that is
characterized by the solution to a fixed-point problem
In contrast to Sockin and Xiong (2023), we assume over the endogenous cutoff of the marginal user who
that validators can commit to these policies.11 As the purchases the token, A∗t , as summarized in the following
platform’s token base matures from inflation, the proposition.
compensation to the validators shifts from seignior­
age to transaction fees. Proposition 1. The rational expectations equilibrium ex­
hibits the following properties.
2.4. Rational Expectations Equilibrium 1. Regardless of other users’ strategies, it is optimal for each
Our model features a rational expectations cutoff equi­ user i to follow a cutoff strategy in purchasing the token:
librium, which requires the rational behavior of each �
1 if Ai,t ≥ A∗ (At , yt , Qt , ζt )
user and the clearing of the token market. Xi,t � :
0 if Ai,t < A∗ (At , yt , Qt , ζt )
• User optimization. Each user chooses Xi,t in each
period t to solve his maximization problem in (2) for 2. In the equilibrium, the cutoff A∗t solves the following
whether to purchase the token. fixed-point condition:
• In each period, the token market clears !
∗
Z ∞ (1�ηc )(A∗t �At )+At +12η2c τ�1 �1=2 A t � A t
(1 � β)e ε Φ ηc τε � �1=2 1{τ>t}
Xi,t (Ai,t , Pt )dΦ(εi,t ) � Φ(yt � ζt + λ log(RPt )), (6) τε
�∞ √ffiffiffi
τ
� λε (A∗t �At )�λ1 yt +λ1 ζt
+ E[Pt+1 |I t ] � κ � e , (7)
where each user’s demand Xi,t depends on its infor­ where τ is the stopping time for the breakdown of the plat­
mation set I i,t : The demand from users is integrated form because of the failure of the token market clearing
over the idiosyncratic component of their endow­
ments {εi,t }i∈[0,1] , which also serves as the noise in τ � {inf t : At < Ac (yt , Qt , ζt )},
their private information. with Ac (yt , Qt , ζt ) as a critical level for At, below which
Equation (7) has no root.
3. In each period t, there may be no or multiple equilibria
3. Equilibrium
with nontrivial user participation depending on the users’
We characterize the equilibrium in each period t when
expected token resale value.
At and ζt are publicly observable. In this case, the token
• If E[Pt+1 |I t ] � κ ≤ 0, Equation (7) has zero or
market is characterized by the following state variables:
two roots.
the users’ demand fundamental At , the token supply
• If E[Pt+1 |I t ] � κ > 0, Equation (7) has one or
yt , the users’ optimism driven by the public signal Qt ,
three roots.
and the speculators’ sentiment ζt. We use the notation
4. In the dynamic equilibrium, the token price P(At , yt ,
I t � {At , yt , Qt , ζt } to represent the state variables at t,
Qt , ζt ) is determined by Equation (5) according to the users’
which also represent the set of public information to all
equilibrium cutoff A∗t and how users coordinate on their
users. The public signal, Qt , contains information about
expectations of future equilibria.
At+1 , and thus, it is useful to users for forming their
expectations about the token price in period t + 1, Pt+1 : Proposition 1 characterizes the cutoff equilibrium in the
Given that all users have a common expectation about platform and confirms the optimality of a cutoff strategy
Pt+1 , we drop the i subscript from their information for users in their choice to purchase the token. Users in
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6691

each period sort into the platform based on their endow­ Figure 1. (Color online) An Illustration of the Left- and
ments, with those with higher endowments and thus, Right-Hand Sides of Equation (7)
more gains from trade entering the platform. In this cut­
2.5
off equilibrium, the token price is a correspondence of
the token market state variables (At , yt , Qt , ζt ), accord­
ing to Equation (5), with A∗t as an implicit function of 2
these state variables.
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Equation (7) provides a fixed-point condition to deter­

mine the optimal cutoff in each period. The left-hand 1.5
side (LHS) of Equation (7) reflects the expected benefit
to a marginal user with Ai,t � A∗t from acquiring a token 1
to join the platform; the first term is the expected utility
flow from transacting with another user on the plat­
form, whereas the other terms E[Pt+1 |I t ] � κ represent 0.5
the user’s expected next-period token price net of the
user’s participation cost κ: The right-hand side (RHS) of
0
Equation (7) reflects the cost of purchasing a token. 0 0.2 0.4 0.6 0.8 1
Figure 1 illustrates how the intersection of the two
sides, each of which is plotted against the number of √ffiffiffiffi
Note. The horizontal axis is the number of tokens Nt � Φ( τɛ (A∗t �
tokens Nt, determines the equilibrium cutoff A∗t . Note ∗
√ffiffiffiffi At )), which is an inverse and monotonic transformation of At � At :
that Nt � Φ( τɛ (A∗t � At )) is an inverse and monotonic
transformation of A∗t � At . The dashed hump-shaped
line depicts the left-hand side of Equation (7) in a bench­ trivial solution in which demand and supply are both
mark case when E[Pt+1 |I t ] � κ � 0. That is, it captures a zero at a zero token price. Whether there is also a nontri­
marginal user’s expected utility flow from transacting vial solution depends on whether the dashed hump-
with another user. This curve goes to zero when Nt goes shaped curve intersects the solid supply curve at some
to either zero or one. If Nt ↗ 1 (i.e., A∗t ↘ 0), the mar­ number of tokens greater than zero. As one can see, either
ginal user’s own endowment is so low that there is no this occurs twice or not at all if the solid supply curve lies
gain from transacting with the other user. On the other above the hump-shaped curve for Nt > 0. The latter case
hand, if Nt ↘ 0 (i.e., A∗t ↗ ∞), the equilibrium cutoff is is particularly important as it represents the breakdown
so high that there are no other users on the platform to of the token market and consequently, the failure of the
transact with the marginal user. This network effect platform. This happens when the expected utility from
makes her expected utility from transaction zero, despite transacting is strictly lower than the cost of acquiring the
her own high endowment. Once the two end points are token, either as a result of the small token supply yt or as
determined, it is intuitive that the marginal user’s exp­ a result of strong speculator sentiment ζt. Proposition 1
ected utility flow from transacting with another user on shows that these two curves do not intersect when At falls
the platform has a hump shape. Such a hump-shaped below a critical level Act (yt , Qt , ζt ), which is determined
demand curve is ubiquitous in the network effect litera­ by the other three state variables.
ture (e.g., Easley and Kleinberg 2010). In the absence of The terms E[Pt+1 |I t ] � κ may move the hump-shaped
the network effect, this demand curve is monotonically curve of the marginal user’s expected benefit from partic­
decreasing because the marginal user’s expected utility is ipating in the platform up or down relative to the bench­
simply increasing with her own endowment A∗t . mark case. If E[Pt+1 |I t ] � κ > 0, possibly as a result of
The right-hand side of Equation (7) is the supply the users’ optimism about future token price appreciation
curve of tokens and is represented by the solid upward- (i.e., a positive shock to Qt), the hump-shaped curve
sloping curve. It is an exponential function of the inverse moves up relative to the benchmark dashed curve in
of the normal Cumulative Distribution Function of Nt Figure 1. In this case, the bell curve may intersect with the
(i.e., Φ�1 (Nt )) because the number of users on the plat­ solid supply curve either once (as illustrated by the dot­
form is decreasing with the equilibrium cutoff A∗t and ted curve) or three times.
because the token price is an increasing function of the If E[Pt+1 |I t ] � κ < 0, either as a result of users’ pessi­
number of users as in Equation (5). In the absence of the mism or a high participation cost κ, the hump-shaped
network effect, this upward-sloping supply curve has a curve moves down relative to the benchmark dashed
unique intersection with a downward-sloping demand line in Figure 1, creating the possibility for the token
curve. In the presence of the network effect (i.e., the market to break down. That is, an increase in κ may lead
hump-shaped demand curve), however, there may be to the failure of the platform in which there is only a triv­
several possibilities, as is well known in the network ial solution. As each user does not account for his partici­
effect literature. Figure 1 illustrates that there is always a pation decision on other users through the network
 Sockin and Xiong: A Model of Cryptocurrencies
6692 Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS

effect, this externality exacerbates the effect of κ on the token market either by direct trading or by changing the
equilibrium user participation. Interestingly, users’ opti­ token issuance protocol.14
mism offsets the effect of their participation cost, which The following proposition characterizes the condi­
helps to overcome the network externality. tions for market breakdown to occur.
Finally, the dashed upward-sloping curve in Figure 1
Proposition 2. As a result of the network effect, only an
illustrates the impact of speculator sentiment on market
equilibrium with zero user participation exists (that is, the
breakdown. An increase in speculator sentiment (a higher
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token market breaks down) under the following conditions.

ζt) raises the solid black supply curve to the dashed black
1. The net speculative motive of users, E[Pt+1 |I ∗t ] � κ, is
supply curve. This higher supply curve no longer inter­
nonpositive.
sects with the dashed hump-shaped demand curve. In
2. The users’ demand fundamental is sufficiently low
this case, there is only a trivial equilibrium with zero user
(that is, At < Ac (yt , Qt , ζt )), or equivalently, speculator sen­
participation. Consequently, a higher speculator senti­
timent is sufficiently high (that is, ζt > ζc (At , yt , Qt )).
ment shifts up the token supply curve and makes it more
The critical level Ac (yt , Qt , ζt ) is decreasing in user opti­
difficult for there to be an equilibrium with nontrivial par­
mism Qt and increasing in speculator sentiment ζt and the
ticipation that clears the token market.
user participation cost κ:
3.1. Market Breakdown Proposition 2 characterizes the determinants of the fun­
When there is only the equilibrium with zero user par­ damental critical level Ac (yt , Qt , ζt ) for the token market
ticipation, the token market breaks down, and the plat­ breakdown to occur. On the demand side, the users’ spec­
form fails. Such market breakdown represents a severe ulative motive, driven by their optimism, helps to over­
form of market dysfunction stemming from the network come the participation externality. On the supply side,
effect in user demand for tokens. It is important to rec­ speculators’ sentiment has the opposite effect.
ognize that this breakdown is a result of two key fea­ To further illustrate the properties of the token market
tures of our model. First, token price fluctuations have a equilibrium, we provide a series of numerical examples
real effect on user participation because the platform based on the parameter values given in Table 1. We cau­
does not adjust the number of tokens required for users tion, however, that this exercise is not a calibration but
to participate.12 As discussed earlier, it is common for rather, an illustration of our model’s behavior. To disci­
crypto platforms not to adjust the number of tokens pline our numerical examples, we follow Cong et al.
required for their services in response to token price (2021b) and choose a growth rate for the platform fun­
fluctuations. One may still be concerned about the role damental of µ � 0:02 � 12 τ�1 1 �1
A (the term � 2 τA arises
played by each user’s binary token choice in driving the because At is equivalent to the log of the fundamental in
market breakdown. In Appendix B, we analyze a more their model), a risk-free rate of R � 1.05, and a degree of
general setting in which each user’s benefit from hold­ complementarity of ηc � 0:3. We also follow Cong et al.
ing tokens is monotonically increasing in the number of
(2022a) and choose a transaction fee rate of β � 0:001.
tokens she holds. This assumption maintains the non­
Finally, we choose a token supply inflation rate of 4%
neutrality of the token price but allows each user to
(ι � 0:04) based on Ethereum’s average inflation rate
choose a continuous number of tokens to hold. Interest­
before its conversion to proof of stake. We also choose
ingly, the token market may still break down because of
reasonable values for the remaining parameters.
the same mechanism illustrated by our main setting
Figure 2 depicts the fundamental critical level Ac
even in a more general setting, in which users choose
across speculator sentiment (the left panel), user opti­
tokens in a continuous quantity and face a general token
mism (the center panel), and token supply (the right
supply curve.
panel). When the platform fundamental A is below Ac,
The second key feature for market breakdown is that
the token market breaks down. The left panel shows
the token market is decentralized and no participant in
that as speculator sentiment increases, the crowding out
the market internalizes the effect of her trading on
effect of speculators holding more tokens lowers user
others. Such externalities are present in both users and
speculators. On the user side, no user accounts for the participation, shifting up the region of breakdown. In
network effect of her participation choice on other users. contrast, the center panel shows that an increase in user
On the speculator side, each speculator takes the token
price as given, which implies that when the token mar­ Table 1. Baseline Model Parameters
ket fails to find a market-clearing price, neither is a sin­
gle speculator present nor can a group of speculators Model parameters
coordinate with each other to offer a price to clear the Demand fundamental τA � 10, µ � 0:02 � 12 τ�1
A
users’ token demand.13 It is also important to note that Platform y0 � �:84, β � 0:001, ι � 0:04
on a decentralized crypto platform, the platform foun­ Sentiment τQ � 5, τζ � 2, λ � 1
Users τθ � 1, ηc � 0:3, κ � 0:04, R � 1:05
der by design is unavailable to support the secondary
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6693

Figure 2. An Illustration of the Market Breakdown Boundary for the Demand Fundamental Ac with Respect to Speculator Senti­
ment (Left Panel), User Optimism (Center Panel), and Token Supply (Right Panel)
-2 -2 -2
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-2.5 -2.5 -2.5

Ac

Ac

Ac
-3 -3 -3

-3.5 -3.5 -3.5

-4 -3.5 -3 -2.5 -2 -1.5 -1 -2 -1 0 1 2 0.6 0.7 0.8 0.9 1

t
Qt yt

Note. The model parameters are given in Table 1, and the baseline values for the current state are ζt � 0, Qt � 0, and yt � 0:92.

optimism, which incentivizes more users to participate, based on the (dynamic) stability of the potential equilib­
has the opposite effect and shifts down the region of ria.15 Then, the following proposition derives several
breakdown. Taken together, these two panels illustrate comparative statistics of the equilibrium user participa­
the opposite effects generated by users’ optimism and tion and token price.
speculators’ sentiment on the fragility of the platform,
Proposition 3. The equilibrium has the following properties.
as formally established by Proposition 2.
1. Demand fundamental. The token price and the fraction
The right panel of Figure 2 shows that an increase in
of users who participate in the platform are increasing in the
token supply, by lowering the expected retrade value of
demand fundamental, At :
the token, increases the breakdown boundary; to the left
2. User optimism. The token price and the fraction of
of the line, there is always an equilibrium. When the
users who participate in the platform are increasing in user
token base is small, there are at least two advantages.
optimism, Qt :
First, it is easier to clear markets with a small pool of
3. Speculator sentiment. The fraction of users who partici­
users. Second, the expected growth of the token value is
pate in the platform is decreasing in speculator sentiment,
also higher. As the token supply inflates over time, the
ζt , whereas the token price is increasing (decreasing) in ζt
effects of token supply imply that the platform becomes
when A∗t � At is sufficiently negative (positive).
more fragile over time, as the token’s expected retrade
value falls and user participation is driven more by the Figure 3 illustrates the equilibrium token price across
flow of convenience yields from transactions on the the demand fundamental A for different values of spec­
platform. This pattern thus suggests that large market ulator sentiment (the left panel), user optimism (the cen­
capitalization tokens, such as Ethereum, might be more ter panel), and token supply (the right panel). The
fragile and thus, have more pronounced price volatility center panel shows that the token price is increasing
than small capitalization tokens. Interestingly, although with user optimism, as formally established by Proposi­
Cong et al. (2021b) emphasize the role of token resale in tion 3. The left panel shows that the token price is also
facilitating adoption, our model shows that it also helps increasing with speculator sentiment, which holds, as
to stave off the failure of the platform. established by Proposition 3, only when the demand fun­
damental is high. The difference across user optimism is
3.2. User Participation and Token Price more pronounced because user optimism increases user
For the simplicity of our analysis, we assume that all participation by raising users’ expectations of the token’s
users coordinate on the highest-price (i.e., the lowest- resale value, which in turn, raises the price today; specu­
cutoff) equilibrium in each period, regardless of how lator sentiment, in contrast, raises the token price but also
many equilibria exist. One can motivate this refinement crowds out user participation, which in turn, lowers the
 Sockin and Xiong: A Model of Cryptocurrencies
6694 Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS

Figure 3. An Illustration of the Token Price Across the Demand Fundamental for Different Values of Speculator Sentiment (Left
Panel), User Optimism (Center Panel), and Token Supply (Right Panel)
8 8 8
= -2.12 Q = -1 y = -0.28
=0 Q=0 y = 0.92
= 2.12 Q=1 y = 2.12
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6 6 6
Token Price

Token Price

Token Price
4 4 4

2 2 2

0 0 0
-5 -4 -3 -2 -1 0 1 -5 -4 -3 -2 -1 0 1 -5 -4 -3 -2 -1 0 1
At At At

Note. The model parameters are given in Table 1, and the baseline values for the current state are ζt � 0, Qt � 0, and yt � 0.92.

price, leading to a more muted overall effect on the token time. Because more of the token return for high µ plat­
price. Finally, the right panel shows that the token price forms is from the capital gains part of the token return,
is decreasing in token supply because it lowers the the user base is less sensitive to instantaneous fluctua­
expected retrade value of the token. tions in the demand fundamental, which drive the con­
venience yield. As such, we expect higher µ platforms to
have lower token price volatility. In contrast, as the
3.3. Life-Cycle Effects
token supply increases, both the region of market break­
Because our model is nonstationary with the token sup­
down and the importance of the convenience yield in
ply increasing deterministically over time, it has nuanced
token returns increase, leading to a more volatile token
implications for how platform performance varies over
price.
the platform’s life cycle. Central to understanding this
pattern is the tension between the contemporaneous con­
venience yield and the capital gains in each user’s total 3.4. Implications for Platform Design
return from holding the token. Because users are risk Our analysis raises a key issue that the network effect
neutral, the sum of the two pieces always equals the cost endemic to utility token platforms can lead to fragility
of carry plus the participation cost, R + κ=Pt , in equilib­ when a rigid token supply curve interacts with a de­
rium. Thus, when expected future token price appre­ mand curve that is subject to a network effect and non­
ciation is high, the current demand fundamental and neutrality of the token price. Consequently, policies that
convenience yield must be low. make the supply of tokens respond to speculative shocks,
The demand fundamental’s expected growth rate µ � such as a state-contingent token issuance schedule, or
1 �1
2 τA and the token supply yt are the two key model that subsidize users, such as a state-contingent transac­
parameters that determine the expected token price. We tion fee rate (β
in the model), can mitigate the risk of mar­
illustrate these effects in Figure 4 for two values of µ: A ket breakdown. We now discuss these two possibilities.
platform with a higher µ will, on average, see At trend By making the token supply curve more elastic and
upward over time, sustaining a high expected token leaning against speculator sentiment, a state-contingent
price, whereas a high yt depresses token prices across all token issuance policy can potentially help ensure a non­
values of At from supply saturation. The tension bet­ trivial participation equilibrium on the platform. Such
ween the convenience yield and the expected future an issuance policy would ideally condition not only on
token price also impacts the log token price volatility the token price but also, on the nonfundamental com­
over time. When the demand fundamental growth rate ponent of token supply (i.e., speculator sentiment).16
µ is high, the expected token price remains higher over To see the potential role of a state-contingent token
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6695

Figure 4. An Illustration of the Unconditional Expected Log Token Price (Left Panel) and Log Price Volatility (Right Panel) over
Time
4 18

2 16

0
14
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-2
Expected Log Token Price

Log Token Price Volatility

12

-4
10

-6

8
-8

6
-10

4
-12

-14 2

-16 0
0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
Time Time

Notes. The model parameters are given in Table 1. The solid lines indicate the case in which the growth rate of the fundamental is
µ � 0:02 � 12 τ�1 1 �1
A , and the dashed lines indicate the case in which µ � 0:10 � 2 τA .

issuance policy, suppose now that the token issuance ιt, Proposition 4 loads positively on speculator sentiment ζt
defined in (4), is time varying and contingent on the when the expected token retrade value E[Pt+1 |I t ] is
state of the platform (At , yt , Qt , ζt ). The following propo­ below κ. When E[Pt+1 |I t ] exceeds κ, there will be an
sition establishes a condition on ιt for an equilibrium equilibrium for any inflation rate. Suppose the platform
with nonzero user participation to exist. designer miscalibrates the token inflation schedule and
sets
Proposition 4. There exists a state-contingent issuance
schedule, ι∗t , as given by (A.6), such that an equilibrium ιt � �ζt � yt�1 � p∗t if E[Pt+1 |I t ] < κ,
with nonzero user participation exists at date t if ιt ≥ ι∗t ,
where p∗t is defined in (A.5) and depends on E[Pt+1 |I t ].
which ensures the minimal supply elasticity with respect
In this case, the token supply at date t is Φ(yt + λ log
to ζt.
(RPt ) � ζt ) � Φ(λ log(RPt ) � p∗t � 2ζt ), instead of Φ(λ log
We caution, however, that in practice, it may be diffi­ (RPt ) � p∗t ) under ι∗t . There are two effects of this miscali­
cult to implement a token issuance policy (ιt ≥ ι∗t ) that bration: one static and one dynamic. The static effect is
responds to conditions in the token’s secondary market. immediate. As a result of the miscalibration, the token
Realistically, nonfundamental shocks to token prices, supply schedule now loads more negatively on specula­
such as optimism and sentiment, are not directly ob­ tor sentiment ζt (by a factor of –2) rather than positively
servable, and conditioning on the token price and mar­ as implied by the minimal inflation schedule. As such,
ket outcomes, such as trading volume, may not be the token supply not only fails to buffer the speculator
enough to disentangle the sources of token price fluctua­ sentiment shock but also, doubles its impact. More subtle
tions. Such contingency, if miscalibrated, may make the is the dynamic effect. Because a more volatile inflation
supply of tokens excessively volatile, which would be at schedule affects future token prices, the expected retrade
variance with the proper functioning of the platform. It value E[Pt+1 |I t ] at time t is also impacted by this miscali­
may also introduce unnecessary uncertainty into the bration through p∗t . More generally, this dynamic effect
revenue of validators, making them reluctant to provide will depend on how the static miscalibration interacts
validation services. It may further buffet the platform with the token price and platform breakdown over time.
with nonfundamental fluctuations that impair perfor­ A state-contingent transaction fee rate may also poten­
mance and exacerbate the problem. Note that govern­ tially mitigate market breakdown by adjusting how
ments face similar issues in setting monetary policy: for much transaction surplus users need to give up to com­
instance, when deciding whether monetary policy should pensate the platform’s validators. A reduction of β repre­
respond to stock or housing market fluctuations. sents an effective subsidy for users to raise their demand
To illustrate how a miscalibrated token inflation sched­ curve. Similar to the case of a state-contingent token issu­
ule can harm platform performance, we recognize the ance policy, however, it may be difficult in practice to
necessary issuance schedule ι∗t to avoid breakdown from condition platform policy on unobservable speculator
 Sockin and Xiong: A Model of Cryptocurrencies
6696 Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS

sentiment in secondary markets, and the alternative of with many token platforms with PoW mining, miners
conditioning on the token price can even be destabilizing. also earn transaction fees because over time, the number
In addition, decentralized platforms lack an owner who of tokens created by inflation will diminish. It is thus nec­
has incentives to subsidize user participation, and conse­ essary to shift the compensation toward fees. Miners have
quently, the subsidy through a reduction of transaction no use for tokens and sell them to users and speculators.
fees is bounded from below by a transaction rate of β � 0, If NM,t miners join the platform at date t, each miner earns
limiting its effectiveness. Because transaction fees are βUt +(Φ(yt�1 +ι)�Φ(yt�1 ))Pt
�e�ξt in expected net gain.20
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N M, t
used to compensate validators, validators also have an
Suppose that when a strategic attack occurs, users
incentive to maintain high fees on the platform.
lose a fraction 1 � γ of their transaction surplus from
failed transactions in the current period as a result of
4. Mining and Strategic Attacks service delays and denials. The interruption of service
The risk of strategic attacks by validators is a central also reduces transaction fees by a fraction 1 � γ. Further­
concern for cryptocurrency platforms. Attacks on Bit­ more, we assume that a strategic attack occurs when­
coin Gold, ZenCash, Vertcoin, Monacoin, Ethereum ever
Classic, and Verge (twice) have already led to losses of
approximately $18.6 million, $550,000, $50,000, $90,000, (Φ(yt + ψι) � Φ(yt ))Pt
$1.1 million, and $2.7 million, respectively. Such attacks (Φ(yt�1 + ι) � Φ(yt�1 ))Pt + βγUt 2
include, for instance, 51% attacks under the proof of + ≥ αNM ,t , (8)
2
work protocol that lead to “double-spending” fraud
and transaction failures through denials of service.17 In where α, ψ > 0: On the left-hand side of this condition,
this section, we demonstrate that strategic attacks occur the first term has the interpretation of fraudulent sei­
when the platform fundamental is sufficiently weak. gnorage created by corrupt miners from double spend­
More importantly, the risk of such attacks in the future ing, and the second is a fraction γ of the mining fees, in
exacerbates the region of market breakdown by reduc­ the forms of legitimate seigniorage and transaction fees,
ing the token’s retrade value, which feeds back into the earned from mining the attack. The right-hand side is
likelihood of a strategic attack. This adverse feedback the cost of attack, which is a convex function of the num­
loop is novel to decentralized cryptocurrency platforms. ber of miners, reflecting that a larger pool of miners
To illustrate how consensus protocols can impact plat­ makes it increasingly costly for corrupt miners to acquire
form performance and stability, we consider a simple the necessary computing power for completing a 51%
extension of our setting in this section that incorporates attack. In Appendix D, we provide a microfoundation
proof of work mining. We focus on the most ubiquitous for this strategic attack condition, although all that we
type of attack on PoW blockchains, a 51%, attack, but require is that strategic attacks occur whenever the cost
our general insights will also be valid for other types of of mining is sufficiently high and the number of miners
attacks, such as a selfish mining attack, and other consen­ is sufficiently low.
sus protocols, such as proof of stake, provided that the Consider the incentives of miners to join the platform
interests of validators may conflict with those of users. at date t. With rational expectations, miners choose
We now assume that in each period, a new popula­ whether to join, fully anticipating the possibility of a
tion of potential miners mines the token by providing strategic attack. Miner j with the common mining effi­
accounting and custodial services using its underlying ciency ξt thus maximizes his expected gain:
blockchain technology.18 As in practice, there is free
entry of miners onto the platform. All miners provide Πj � max
Mj,t
computing power to facilitate transactions among users, �
(Φ(yt�1 + ι) � Φ(yt�1 ))Pt + (1 � (1 � γ)χt )βUt
subject to a cost of setting up the required hardware and
software to mine the token: e�ξt Mj,t , where Mj,t ∈ {0, 1} (1 + χt )NM,t
!
is the miner’s decision to mine and ξt measures the
miner’s mining efficiency by inversely parameterizing �e�ξt Mj,t , (9)
the miner’s cost of mining.19 This mining efficiency ξt is
common to all miners and follows an AutoRegressive where χt ∈ {0, 1} is the indicator for whether there is a
AR(1) process: strategic attack at date t. The (1 � (1 � γ)χt ) factor reflects
�1=2 ξ
ξt � ξt�1 + τξ εt , that the mining pool receives only γ of the total mining
revenue from completing less than half of the blocks
with εξt ~ i:i:d N (0, 1): Validators are now miners who when a strategic attack occurs.
are compensated with the transaction fee βUt , which is a Note that relative to the equilibrium characterized in
fraction of the transaction surplus, and the seigniorage Section 3, the miners’ common mining efficiency ξt
from token inflation, (Φ(yt�1 + ι) � Φ(yt�1 ))Pt : Consistent becomes an additional state variable. The following
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6697

proposition shows that strategic attacks occur when fundamental ξt, as formally derived in Proposition 5.
either At or ξt falls below a certain level. Although each strategic attack does not lead to the fail­
ure of the platform, the expected losses induced by
Proposition 5. The equilibrium has the following properties.
future attacks lead to a higher-threshold Ac for market
1. There exists a critical level ξa (At , yt , Qt , ζt ) such that
breakdown. As such, the possibility of strategic attacks
strategic attacks occur when ξt < ξa (At , yt , Qt , ζt ):
by miners also exacerbates platform fragility. This is
2. There exists a critical level Aa (yt , Qt , ζt , ξt ), which is
reflected by the raised dashed line in the center panel.
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decreasing in ξt , such that strategic attacks occur when

As our analysis highlights, the PoW protocol intro­
At < Aa (yt , Qt , ζt , ξt ):
duces several novel features to cryptocurrency plat­
3. Both an attack equilibrium and a no-attack equilibrium
forms. First, the anticipation of future attacks makes
can exist as a result of the positive relationship between the
such a strategic attack easier to execute. An attack lowers
benefits and costs of attacks.
the revenue each honest miner receives, which reduces
From Proposition 5, a strategic attack occurs when the the number of miners who join the platform and thus,
mining fundamental and/or the user demand funda­ lowers the cost of an attack. Interestingly, the decentra­
mental are sufficiently weak because in these situations, lized consensus protocol exacerbates the problem by
the number of miners is too small to deter a strategic dispersing the revenue from mining over the whole pop­
attack. Although the impact of each strategic attack is ulation of miners. As a result, an honest miner captures
transitory, the occurrence of strategic attacks is persis­ only a fraction of the revenue that is recovered by
tent because an attack will occur every period in which increasing its own mining power to preempt attacks.21
the platform is in the attack region. As attacks reduce the In this way, decentralized consensus averts the internali­
token price and thus, the incentives of miners to join the zation of incentives to ensure the platform’s security.
platform, it may be possible for both a no-attack equilib­ Second, the feedback effects from mining to the plat­
rium and an attack equilibrium to be self-fulfilling. form token’s intrinsic value through service delays and
Figure 5 depicts the strategic attack boundary (left denials are peculiar to the decentralized consensus pro­
panel) and the platform breakdown boundary with and tocol. Users are effectively also shareholders in the plat­
without mining (center panel) for τξ � 10, α � 0:8, and form through the retradeability of the token. As such,
ψ � 3: Miners choose to attack the cryptocurrency if the delays and expectations of future delays have an impor­
user fundamental At falls below the attack boundary Aa. tant impact on the token price because they reduce user
This attack boundary is decreasing with the mining participation and consequently, demand for the token.

Figure 5. An Illustration of the Strategic Attack Boundary (Left Panel), Market Breakdown Boundary (Center Panel), and Token
Price (Right Panel) with Respect to Mining Fundamental ξt
7 -2.3 8

6
7.5
-2.4
5

7
4
-2.5

3
6.5
Token Price
Aa

Ac

2 -2.6

6
1

-2.7
0
5.5

-1
-2.8
5
-2

-3 -2.9 4.5
0 2 4 6 8 0 2 4 6 8 10 0 2 4 6 8 10

t t t

Notes. The market breakdown boundary without mining (solid line) is for comparison. User optimism is turned off (τQ � 0) in this illustration,
and the lost surplus is set to γ � 12. In addition, platform participants always coordinate on the no-attack equilibrium when it exists. Baseline
values are At � 0.98, ζt � 0, and yt � 0.92.
 Sockin and Xiong: A Model of Cryptocurrencies
6698 Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS

These two features contribute to a rich dynamic NFTs. Cryptocurrency returns in our framework have
adverse feedback loop between strategic attacks and three components: a convenience yield of the marginal
market breakdown. The anticipation of future strategic user, which acts like a dividend; a capital gain from the
attacks lowers the expected retrade value of the token, token price appreciation; and an embedded discount in
which in turn, reduces users’ incentives to join the plat­ the token price to compensate users for their participa­
form and exacerbates both the problem of strategic tion cost. By the marginal user’s equilibrium condition
attacks and market breakdown. in (7), these three components satisfy the following rela­
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Finally, from Figure 5 (right panel), we see a nonlinear tionship:

relation between the mining fundamental and token (1 � β)Ut∗ E[Pt+1 | I t ] κ
price. When the mining fundamental is far away from R� + � :
Pt Pt Pt
the strategic attack boundary, an incremental change in
the efficiency of mining has a limited impact on the In contrast to fiat currencies, the expected capital gain
token price because the probability of an attack is small. can be quite positive, despite token inflation, and sub­
When the mining fundamental is close to the strategic stantial, which has attracted many speculators to the
attack boundary, however, a small change in the effi­ nascent asset class. In addition and novel to cryptocur­
ciency of mining can have a substantial impact on the rencies, the convenience yield is created by shareholders
token price, which in turn, leads to a substantial impact acting in their dual capacity as users of the platform,
on the platform’s stability. which gives rise to a feedback mechanism from the cryp­
Our insights about the adverse dynamic feedback tocurrency return to user participation. As the platform
loop associated with strategic attacks are also applicable matures and participation increases, the cryptocurrency
to other types of attacks beyond a 51% attack under the return transitions from being driven more by the capital
PoW consensus protocol. Another type of PoW attack is gain component to more by the convenience yield.22
a selfish mining attack, in which a miner secretly vali­ The empirical literature is mostly focused on the capital
dates blocks until she can broadcast it as the longest gain component of the cryptocurrency return, as it is
chain. Like a 51% attack, this attack is more difficult to directly measurable by the econometrician. In equilib­
execute when there are more miners, and each miner has rium, the expected excess capital gain can be expressed as
less probability of winning a block. Consequently, our E[Pt+1 | I t ] κ (1 � β)Ut∗
�R� � : (10)
analysis would also apply to this type of attack. Similarly, Pt Pt Pt
under the proof of stake protocol, the most prevalent type
Consistent with the empirical findings of Hu et al.
of strategic attack is a Sybil attack. Under a Sybil attack, a
(2019) and Liu and Tsyvinski (2021), the expected excess
rogue validator can acquire 51% of all staked tokens and
capital gain in our setting does not exhibit conventional
create false validator nodes to manipulate consensus on
risk premia. The capital gain may still exhibit predict­
the blockchain to engage in a distributed denial of service
ability through the underlying state variables that exp­
or “double-spending” attack. Like the 51% attack under
lain the convenience yield. These state variables are the
PoW, such a strategic attack is more difficult when the demand fundamental, user optimism, speculator senti­
revenue from validating transactions is higher and there ment, and token supply. Liu and Tsyvinski (2021), for
are a lot of tokens staked to compete for this revenue. Fur­ instance, show that investor attention, measured either
thermore, a Sybil attack is also harder when the token with Google searches or with Twitter post counts for
price is high because acquiring a 51% stake size is more “Bitcoin,” predicts future cryptocurrency returns, with
expensive. As in our analysis, an attack is more likely to positive (negative) attention, as measured by keywords,
occur when the platform is weak and user participation is positively (negatively) predicting future weekly ret­
low. By similar logic to our 51% attack analysis, the antici­ urns.23 Liu and Tsyvinski (2021) also find that investor
pation of a Sybil attack also reduces user incentives to join sentiment, measured as either the log ratio between the
the platform, which increases the region of market break­ number of positive and negative phrases of cryptocur­
down and strategic attacks. Consequently, our analysis of rencies in Google searches or the ratio of trading vol­
strategic attacks applies more generally to vulnerabilities ume to return volatility, predicts future cryptocurrency
of consensus protocols. returns. Such nonfundamental shocks to token prices,
represented by user optimism and speculator senti­
5. Empirical Implications ment in our model, can also explain reversals in crypto­
In this section, we discuss several empirical implications currency returns, consistent with the evidence of a
of our conceptual framework for cryptocurrency re­ “value” factor in Cong et al. (2022a).
turns. It is important to note that our model implications Our model also suggests that the participation cost
are specific to tokenized platforms that do not adjust the borne by users, which is not directly observed by the
number of tokens required for their services and would econometrician, is an additional channel of return pre­
not apply, for instance, to (alt-)coins, stablecoins, or dictability. As this cost effect is inversely related to the
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6699

token price and consequently, to market capitalization, goods or services. As a result of the strong network effect
our model predicts a size effect in the capital gain of among users to participate on the platform and the rigid­
cryptocurrencies. This prediction is consistent with Liu ity induced by market clearing with token speculators,
et al. (2022), who find a size factor in the cross-section of the market can break down so that there is only an equi­
cryptocurrency returns, with size measured as market librium with zero user participation. In such a setting,
capitalization, price, or maximum price. token retradeability plays an important role in harnes­
In addition, the persistence of the two return compo­ sing the optimism of users to mitigate this instability. In
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(1�β)U∗ contrast, it can exacerbate such fragility if it attracts spec­

nents Pκt and Pt t in (10) can lead to a positive autocor­
relation in the capital gain: ulators whose enthusiasm crowds out users. As a result
� ! of token inflation, this novel benefit of token retradeabil­
�
Pt+2 Pt+1 �� κ ∗
(1 � β)Ut+1 ity fades as the platform matures and the token price
Cov , �I t�1 � Cov � , becomes driven more by the current platform fundamen­
Pt+1 Pt � Pt+1 Pt+1
� ! tal. We further illustrate how consensus validation pro­
κ (1 � β)Ut∗ �� tocols can exacerbate the platform’s instability through
� �I t�1 > 0
Pt Pt � strategic attacks on the blockchain. The potential for
strategic attacks feeds back into the incentives both of
because the innovations Pt+1 �E[P Pt
t+1 | I t ]
and Pt+2 �E[P t+2 | I t+1 ]
Pt+1 miners to mine and of users to join the platform, which
are uncorrelated with rational expectations. This positive makes such attacks more likely. Our model also pro­
autocorrelation implies momentum, as empirically docu­ vides several implications for cryptocurrency price
mented by Liu and Tsyvinski (2021) in the prices of cryp­ changes that are broadly consistent with recent empiri­
tocurrencies. Furthermore, the momentum effect in our cal evidence.
model is independent of investor attention and senti­
ment, which is also consistent with Liu and Tsyvinski Acknowledgments
(2021), who find time series momentum over one- to The authors thank An Yan for a comment that led to this paper.
eight-week horizons that is not subsumed by their mea­ The authors also thank Will Cong, Haoxiang Zhu, and Aleh
sures of attention or sentiment. Tsyvinski as well as seminar participants at Instituto Tec­
Our model also highlights the importance of network nológico Autónomo de México, the National Bureau of Eco­
nomic Research Asset Pricing Meeting, the National Bureau of
effects in utility token pricing. Shams (2020) shows that
Economic Research Summer Institute, Tsinghua, University of
return comovement arising from overlapping exposures
British Columbia, University of North Carolina, and Yale for
to demand shocks is significantly stronger among “high helpful comments. The authors particularly thank Agostino
community-based” cryptocurrencies, whereas Schwenk­ Capponi (editor), an associate editor, and three referees for their
ler and Zheng (2021) find evidence of comovement highly constructive comments and suggestions.
among peer cryptocurrencies based on news reactions.
Cong et al. (2022a) also provide evidence of a network
Appendix A. Proofs of Propositions
factor that prices the cross-section of cryptocurrencies.
Proof of Proposition 1. We first examine the decision of
Finally, our extension with mining suggests that the
a user to purchase the token. We first recognize that each
capital gain from a cryptocurrency has a nonlinear rela­ user’s expectation about Pt+1 , E[Pt+1 |I t ], depends on each
tion with the marginal cost of mining. When the cost of user’s expectation of At+1 : By the Bayes rule, it is straight­
mining is low relative to the strategic attack threshold, forward to conclude that the conditional posterior of users
small changes in it have a muted impact on the capital about At+1 after observing At and Qt is Gaussian At+1 |I t ~
gain, as the potential loss from strategic attacks, which N (Â t+1 , τ̂ �1
A ), where the conditional estimate and precision
can be viewed as an extended form of the participation satisfy
cost in (10), is small. As the mining cost increases toward τQ
 t+1 � At + µ + Qt ,
the strategic attack boundary, however, incremental τε + τQ
changes become more relevant. Our model, therefore, τ̂ A � τε + τQ :
predicts that measures of mining costs should have
We define τ as the stopping time, at which the platform fails
more predictive power for the capital gain when there is
as a result of the breakdown of the token market. We shall
a nontrivial chance of strategic attacks, such as when the derive the conditions that determine τ later. Conditional on
hash rate or the number of miners is low. t < τ, the expected utility of user i, who chooses to purchase
the token at t, from transacting with another user is
6. Conclusion E[Ui,t |I t , τ > t, Ait , matching with user j]
This paper develops a model to analyze the price dynam­
ics and stability of cryptocurrencies. In our model, a cryp­ � e(1�ηc )Ai,t E[eηc Aj,t |I t ],
tocurrency comprises both an asset and a membership in which is monotonically increasing with the user’s own
a platform developed to facilitate transactions of certain endowment Ai,t . Note that E[eηc Aj,t |I t ] is independent of Ai,t
 Sockin and Xiong: A Model of Cryptocurrencies
6700 Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS

but dependent on the strategies used by other users. It then 3. If RHS < LHS(ż) or if RHS > LHS(z̈), then Equation
follows that user i will follow a cutoff strategy that is (A.3) has one root.
monotonic in its own type Ai,t : 4. If LHS(ż) < RHS < LHS(z̈), then Equation (A.3) has
Suppose that every user uses a cutoff strategy with a three roots.
threshold of A∗t . Then, the expected utility of user i is In the first scenario outlined, there is only an equilib­
! rium with trivial user participation, and the token market
∗
(1�ηc )Ai,t +ηc At +12η2c τ�1 �1=2 At � At
E[Ui,t |I t , τ > t] � e ε Φ ηc τε + �1=2 1{τ>t} breaks down. Note that At shifts up and down the left-
τε hand side of Equation (A.3). Thus, Equation (A.3) has no
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because losing a transaction is independent of the identities root when At is sufficiently small. For this situation to
of the two transacting parties. occur, the speculative motive, E[Pt+1 |I t ] � κ, must be non­
To determine the equilibrium threshold, consider a user positive; otherwise, Equation (A.3) has one or three roots.
with the critical endowment Ait � A∗t : As this marginal user This condition is also satisfied when At is sufficiently
must be indifferent to his purchase choice, it follows that small because E[Pt+1 |I t ] is increasing with At. Thus, the
token market breaks down when At falls below a certain
E[(1 � β)Ui,t + Pt+1 |I t , Ait � A∗t ] � RPt + κ, critical level, which we denote as Ac (yt , Qt , ζt ). Thus, the
which is equivalent to stopping time τ of the platform’s disbandment is
! τ � {inf t : At < Ac (yt , Qt , ζt )}:
(1�ηc )Ai,t +ηc At +12η2c τ�1 At � A∗t
(1 � β)e ε Φ ηc τ�1=2
ε + 1{τ>t}
τε
�1=2 Finally, note that because the only difference among users
is the value of their transaction benefit E[Ui,t |I t , τ > t],
+ E[Pt+1 |I t ] � RPt + κ, (A.1) which is monotonically increasing in Ai,t regardless of the
with Ai,t � A∗t : Fixing the critical value A∗t , the expected mass of users who join the platform, it follows that,
token price E[Pt+1 |I t ], and the price Pt , we see that the regardless of the strategies of other users, it is always
LHS of Equation (A.1) is monotonically increasing in Ai,t optimal for each user i to follow a cutoff strategy. w
because 1 � ηc > 0: This confirms the optimality of the cutoff Proof of Proposition 2. The first part of the proposition
strategy that users with Ai,t ≥ A∗t acquire the token to join follows from the derivation of Proposition 1 and the defi­
the platform and that users with Ai,t < A∗t do not. Because nition of Ac : This proof characterizes the determinants of
√ffiffiffiffi
Ai,t � At + εi,t it then follows that a fraction Φ(� τε (A∗t � the fundamental critical level Ac.
At )) of the users enters the platform and that a fraction With regard to speculator sentiment, notice from Equa­
√ffiffiffiffi
Φ( τε (A∗t � At )) chooses not to. As one can see, it is the tion (A.3) that, when E[Pt+1 |I t ] � κ
is nonpositive, there is
integral over the idiosyncratic endowment of users εi that a critical value of speculator sentiment ζc (At , yt , Qt ):
determines the fraction of potential users on the platform. ( ( � �
By substituting Pt from Equation (5) into Equation (A.1), �1=2 1 1 2 �1
ζt � λ log sup (1 � β)e (1�ηc )τε +λ zt +At +2ηc τε
c
we obtain an equation to determine the equilibrium cutoff zt
A∗t � A∗t (I t ): ))
! 1
λzt
A � A ∗ Φ(ηc τ�1=2
ε � zt ) + e (E[Pt+1 | I t ] � κ) + yt ,
At +(1�ηc )(A∗t �At )+12η2c τ�1 �1=2 t t
(1 � β)e ε Φ ηc τε + �1=2 1{τ>t}
τε
√ffiffiffi
τε
such that nontrivial equilibrium exists if ζt ≥ ζc (At , yt , Qt ),
∗ 1 1
+ E[Pt+1 | I t ] � e λ (At �At )�λyt +λζt + κ: (A.2) with the convention that ζct � �∞ if the argument in the log
√ffiffiffiffi ∗ is negative.
Define zt � τε (At � At ), which determines the population
It is straightforward to see that, in the high-price (low-
that buys the token. We can rewrite Equation (A.2) as
� � cutoff) equilibrium, the implicit function theorem implies
�1=2 1 dzt
(1 � β)e (1�ηc )τε +λ zt +At +2ηc τε Φ(ηc τ�1=2
1 2 �1
� zt )1{τ>t} that dζ > 0: Because the user participation is Φ(�zt ), it fol­
ε t
1 1 1 lows that an increase in ζt exacerbates the market break­
+eλzt (E[Pt+1 | I t ] � κ) � e�λyt +λζt : (A.3)
down region by lowering user participation. Because ζt is
Note the first term in the LHS of Equation (A.3) has a i.i.d., there is only this static impact of an increase in spec­
humped shape with respect to zt , and the second term is ulator sentiment on the equilibrium cutoff. As such, by
an exponential function of zt with a coefficient that may lowering user participation, it shifts up Ac (yt , Qt , ζt ) for
be either positive or negative. As the RHS of Equation any given pair of {yt , Qt }.
(A.3) is constant with respect to zt, this equation may We next consider how user optimism Qt impacts the
have zero, one, two, or three roots. market breakdown region. Because user optimism Qt
• If E[Pt+1 |I t ] � κ ≤ 0, the LHS has a humped shape with raises each user’s estimate of the resale value of the token
a maximum at z, and it may intersect with the RHS at zero or at date t + 1, it raises user participation and the token
two points. price at date t. Because Qt is i.i.d., this is the only impact
1. If LHS(z) < RHS, then Equation (A.3) has no root. of an increase in user optimism. As such, it shifts down
2. If LHS(z) > RHS, then Equation (A.3) has two roots. the market breakdown threshold, Ac (yt , Qt , ζt ), for any
• If E[Pt+1 |I t ] � κ > 0, the LHS is nonmonotonic with given pair of {yt , ζt }.
LHS(�∞) � 0, LHS(∞) � ∞, and one local maximum z̈ and Similarly, an increase in the user participation cost, κ,
one local minimum ż in (�∞, ∞), and it may intersect the deters user participation at all dates and therefore, exacer­
RHS at one or three points. bates the market breakdown by both increasing the cost
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6701

today and lowering the expected retrade value of the state space (At , yt , Qt , ζt ), the retrade value of the token is
token tomorrow through the reduced participation in the unaffected by changes in sentiment today. It is straightfor­
future. As such, it also shifts up Ac (yt , Qt , ζt ): w ward by the implicit function theorem to the equation that
Proof of Proposition 3. We first establish that the map ∂z̃ t dHt =dζt
�� :
from the demand fundamental At to the equilibrium user ∂ζt dHt =dz̃ t
cutoff for joining the platform is monotone when the
Because z̃ enters Ht symmetrically as z does in Equation
highest-price equilibrium is always played.24
(A.3), dHt =dz̃ t > 0 in the high-price equilibrium. In con­
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Suppose that the token price at date t + 1, Pt+1 , is increasing

trast, dHt =dζt is
in At for all (yt , Qt , ζt ) triples in the high-price equilibrium. � �
Then, because At follows a random walk, its cumulative distri­ �1=2
φ ηc τε � z̃ t � ζt
�1=2
bution function satisfies the Feller property, and the condi­ dHt =dζt ∝ (1 � ηc )τε � � �
�1=2
tional expectation operator preserves this relation: Φ ηc τε � z̃ t � ζt
� � � �
∂E[Pt+1 |I t ] ∂P(At + µ + εt+1 , yt+1 , Qt+1 , ζt+1 ) �1=2
φ ηc τε � zt
�E |I t > 0,
∂At ∂At � (1 � ηc )τ�1=2
ε � � �:
�1=2
Φ ηc τε � zt
where the expectation is take over εt+1 : Consequently,
E[Pt+1 |I t ] is increasing in At : Then, we can rewrite Equa­ Consequently, if zt is sufficiently small, then dHt =dζt > 0,
tion (A.3) as the function Gt: whereas if zt is sufficiently large, then dHt =dζt < 0: Because
� � ∂Pt ∂z̃ t ∂Pt
�1=2 1 1
∂ζt � � λ Pt ∂ζt , it follows that ∂ζt > 0 for zt sufficiently small
1 2 �1
Gt � (1 � β)e (1�ηc )τε +λ zt +At +2ηc τε Φ(ηc τ�1=2
ε � zt )1{τ>t}
1 1 1
+ eλzt (E[Pt+1 | I t ] � κ) � e�λyt +λζt ≡ 0: (A.4) and that ∂P ∂ζt < 0 for zt sufficiently large. Because zt �
t

∗
At � At , the result follows. w
Assuming the existence of an equilibrium with nontrivial
user participation, applying the implicit function theorem Proof of Proposition 4. From the proof of Proposition 1,
to Gt , one has that the left-hand side of Equation (7) is a hump-shaped curve
√ffiffiffiffi
in the number of tokens demanded Nt � Φ( τɛ (A∗t � At )),
∂zt ∂Gt =∂At
�� , whereas the right-hand side (the token price) can be
∂At ∂Gt =∂zt expressed as an exponential function of λ1 (ζt � Φ�1 (Nt ) �
where yt�1 � ιt ) for an inflation rate ιt. Notice that a higher ιt lowers the
� � � � exponential curve, whereas the fundamental cause of market
∂Gt �1=2 1 1 2 �1
� (1 � β)e (1�ηc )τε +λ zt +At +2ηc τε Φ ηc τ�1=2
ε � z t breakdown (only a trivial participation solution) is that the expo­
∂At nential curve is always above the hump-shaped curve for Nt > 0.
1 ∂E[P t+1 |I t ] Consequently, there exists a minimum i∗t such that the exponen­
+ eλzt > 0:
∂At tial curve just touches the peak of the log of the hump-shaped
In the high-price equilibrium, the RHS of Equation (A.3) curve. Equating the log of the hump-shaped demand curve with
intersects the hump-shaped curve of the LHS in zt on the the token price, we can define p∗ as
left side of the hump, and consequently, ∂G 25 8 h
∂zt ≥ 0: It then
t
√ffiffiffi Φ (n)+At +1η2 τ�1
1�ηc �1
∂zt >
> max λ log (1 � β)e τɛ 2 c ε
follows that, in the high-price equilibrium, ∂A < 0: There­ >
> n
t >
> � �
fore, user participation Φ(�zt ) is increasing in At : >
< �1=2
1 1 1
Furthermore, because Pt � e�λzt �λyt +λζt , it follows that Φ ηc τε � Φ�1 (n) 1{τ>t} if E[Pt+1 |I t ] ≤ κ
p∗ �
>
> i
∂Pt Pt ∂zt >
> +E[Pt+1 |I t ] � κ + Φ�1 (n)
�� > 0: >
>
∂At λ ∂At >
:
∞ if E[Pt+1 |I t ] > κ:
Consequently, Pt is increasing in At in the high-price equi­
librium. Because the choices of t and t + 1 are arbitrary, (A.5)
Pt is increasing in At generically if the high-price equilib­ This minimal state-contingent inflation rate then takes the form
rium is played at each date.
ι∗t � ζt � yt�1 � p∗t : (A.6)
Finally, because user optimism Qt enters into the user’s
problem by raising the expected resale token price, it raises Consequently, for ιt ≥ ι∗t , an equilibrium with nontrivial
user participation and the token price. In contrast, speculator participation exists. w
sentiment ζt lowers user participation by leading to nonfun­
Proof of Proposition 5. From the miner optimization
damental upward pressure on the token price. Because it also
Problem (9), it is straightforward to see that, with free
lowers user participation, the overall impact on the token
entry, miners must be indifferent to participating on the
price is ambiguous. To see this, we rewrite Equation (A.4) as
�1=2 1 2 �1
� � platform. Consequently, the number of potential miners
Ht ≡ (1 � β)e(1�ηc )τε (z̃ t +ζt )+At +2ηc τε Φ ηc τ�1=2
ε � z̃ t � ζ t 1{τ>t} who choose to mine is given by
1 1
� e�λz̃ t �λyt + E[Pt+1 | I t ] � κ � 0, (Φ(yt�1 + ι) � Φ(yt�1 ))Pt + β(1 � (1 � γ)χt )Ut ξt
NM,t � e :
1 + χt
where the change of variables z̃ now absorbs speculator sen­
1 1
timent, so that the price is Pt � e�λz̃ t �λyt : Because speculator Substituting the optimal number of miners, NM,t , from (9)
sentiment is i.i.d. and the equilibrium is Markovian in the into the attack condition given in (8) conjecturing an attack,
 Sockin and Xiong: A Model of Cryptocurrencies
6702 Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS

χt � 1, we can define Therefore, it must be the case that Aa (yt , Qt , ζt ; ξt ) � A a (yt ,

� � Qt , ζt ; ξt ), and therefore, the strategic attack region can be
1 1
f (yt , Pt , E[Ut | I t ]) � Φ(yt + ψι) � Φ(yt ) � Φ(yt � ι) Pt characterized as
2 2 �
1 αe2ξt 1 ξt < ξa (At , yt , Qt , ζt )
+ βγE[Ut | I t ] � χt �
0 ξt ≥ ξa (At , yt , Qt , ζt )
2 4
� �2
� � β or alternatively,
Φ(yt ) � Φ(yt � ι) Pt + Ut : �
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2 1 At < Aa (yt , Qt , ζt ; ξt )
χt �
There is an attack whenever f (yt , Pt , Ut ) > 0: It is clear 26
0 At ≥ Aa (yt , Qt , ζt ; ξt ):
because ξ enters only through the quadratic term that
In addition, we recognize from (A.7) that because a higher
there exists a threshold ξc (At , Qt , ζt ) such that β
ξt lowers the critical 2 Ut , all else equal, it follows that
{χt � 1 : ξt < ξa (At , Qt , ζt )}, Aa (yt , Qt , ζt ; ξt ) is decreasing in ξt :
One may be concerned that no mining equilibrium may
where
exist if, conditional on no attack, miners want to attack the
ξc (At , yt , Qt , ζt ) � blockchain, whereas conditional on an attack, no miner ex
� �
1 Φ(yt + ψι) � 12Φ(yt ) � 12Φ(yt � ι) Pt + 12βγUt post wants to attack the blockchain. This does not occur
log β
:
2 a
((Φ(yt ) � Φ(yt � ι))Pt + E[Ut |I t ])2 because the (convex) cost of attacks from fewer miners falls
4 2
faster than the benefit from the attack from lower revenue.
Assume now that E[Ut |I t ] and Pt are (weakly) increasing To see this, notice that the only endogenous object deter­
in At whenever Pt is positive, and we define Pt � 0 when­ mined by users is A∗t , and a strategic attack raises A∗t , lower­
ever a market equilibrium does not exist. Define ing prices and transaction fees, by reducing the benefit of
β
(Φ(yt ) � Φ(yt � ι))Pt + 2Ut joining the platform for all users. This is equivalent to a fall
, xt � in At to some à t : Because if an attack that would occur at At
2
and rewrite f (yt , Pt , E[Ut |I t ]) as would also occur at A′t < At , by these arguments, it follows
� � that if a strategic attack would occur when users and miners
f (yt , Pt , xt ) � Φ(yt + ψι) � Φ(yt ) Pt + xt � αe2ξt x2t :
do not anticipate an attack, it would also occur if it is antici­
Notice that f (yt , Pt , xt ) is concave in xt , increasing for xt < pated. Consequently, such a strategic attack inconsistency
1
from zero to 4αe12ξt , and then, decreasing to �∞ for issue does not arise.
2αe2ξt n o Furthermore, although there cannot be an inconsistency
xt > 2αe12ξt : It has two roots at xt ∈ 0, αe12ξt : in the attack decision on the platform, there can be self-
It then follows that a strategic attack occurs whenever fulfilling prophecies, in which both the no-attack and
xt ≤ αe12ξt , or when At is sufficiently small. This occurs attack equilibria can be sustained. This arises because
because Ut and Pt are (weakly) increasing in At and both the benefit (Φ(yt + ψι) � Φ(yt ))Pt and the cost xt �
because Ut and Pt converge to zero as At → �∞, as there αe2ξt x2t of an attack are positively correlated.
is no benefit to any (positive measure of) users joining the Finally, we verify that the token price and transaction fees
platform. Consequently, because Pt and Ut are (weakly) are indeed (weakly) increasing in At : Let us conjecture that
increasing in At , it follows there is a connected set A t � {At : the token price, Pt , and transaction fees are (weakly)
At < Aa (yt , Qt , ζt ; ξt )}, where Aa (yt , Qt , ζt ; ξt ) � infAt {f (yt , Pt , increasing in At : We further define Pt � 0 whenever there is
xt ) � 0}, such that χt � 1 when At < A t : market breakdown. Under this assumption, strategic attacks
In contrast, when At is sufficiently large, it must be the occur when At is sufficiently small by these arguments. It
case that limAt →∞ f (yt , Pt , xt ) < 0 because the highest-order then follows that strategic attacks preserve the monotonicity
terms in Pt and Ut are quadratic through �x2t : Consequently, of Pt in At from Proposition 3, confirming the conjecture.
a Similarly, because a higher token price is associated with a
there is a connected set A t � {At : At > A (yt , Qt , ζt ; ξt )}, where
higher user population and consequently, higher transaction
A a (yt , Qt , ζt ; ξt ) � supAt {f (yt , Pt , xt ) � 0}, such that χt � 0 when
fees, this confirms our second conjecture. Further, because
At > A t : the strategic attacks occur when the mining fundamental,
Consequently, it follows that there is a strategic attack ξt , is sufficiently small and mining has no direct impact on
when At ∈ A t and no attack when At ∈ A t : What remains is platform performance when there is no strategic attack, it
to determine if A t ∪ A t � R or if there are more strategic follows that the token price and user participation are
attack regions for some At > A t : Notice now that f (yt , Pt , xt ) (weakly) increasing in ξt : w
is a quadratic function of xt and by Descartes’ rule of signs,
has at most one positive root, which we know must exist by Appendix B. A More General Setting
these arguments. Consequently, f (yt , Pt , xt ) has one zero
In this appendix, we illustrate the robustness of our key
when, substituting for xt ,
sffi�
ffiffiffiffiffiffiffiffiffiffiffiffiffi�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi insight about the fragility of token platforms when there
β 1 1 2 Φ(yt�1 + ψι) � Φ(yt ) are network effects and token nonneutrality. To do this,
E[Ut |I t ] � 2ξ + +4 Pt we first consider a static version of our model and then,
2 αe t αe2ξt αe2ξt
discuss the role of token retrading and a more general
� (Φ(yt ) � Φ(yt � ι))Pt : (A.7) endowment distribution.
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6703

Suppose, as in the main model, that there is a continuum The marginal user with endowment Ai,t � A∗t is indifferent to
of users who choose whether to join the platform to joining the platform, which imposes dropping t subscripts
exchange goods with each other. If a user joins the platform α ∗ 1 2 �1

and matches with a trading partner, she derives utility over α1�α (1 � α)(1 � β)e(1�ηc )(At �At )+At +2ηc τε
!
her own good Ci and that of her trading partner Cj: A∗ � A α

� �1�ηc � �ηc Φ ηc τ�1=2

ε � �1=2
� κP1�α : (B.1)
Ci Cj τε
Ui (Ci , Cj ; N ) � ,
1 � ηc ηc As in the main model, the token price reflects the mar­
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where N is the set of users on the platform. User i receives ginal user’s convenience yield.
an endowment eAi and pays a fraction β of his trade surplus Let the supply of tokens be x(P, y, ξ), where y is the
in transaction fees. To join the platform, the user has to pay amount of outstanding tokens and ξ is a negative supply
a participation cost κ: In contrast to the main model, how­ shock, such as a shock to speculator sentiment. We assume
ever, a user also receives utility from his token holdings Xi that x(·, y, ξ) is a strictly increasing function for all (y, ξ) and
such that she receives a Cobb–Douglas convenience yield that x(0, y, ξ) � 0. This more general token supply curve can
over his holdings and her expected trading benefit: reflect an arbitrary predetermined token issuance schedule.
Market clearing in the token market then implies that
ui � Xαi ((1 � β)E[Ui,t (Ci ,Cj ; N ) | Ai ])1�α , Z ∞ �α�1�α
1

Xi (Ai )dΦ(log Ai ) � (1 � β)U � x(P, y, ξ), (B.2)

with share weights α ∈ (0, 1) and 1 � α, respectively. This �∞ P
preference for token holding could reflect an unmodeled
where U is the total transaction surplus:
convenience from holding tokens. In contrast to Cong et al. � � !
(2021b, 2022a), we do not impose token neutrality. As such, A+12 (1�ηc )2 +η2c τ�1
ε A � A∗
we assume users have a preference for the token balance U�e Φ (1 � ηc )τ�1=2
ε + �1=2
τε
rather than its nominal value in the numeraire good. We !
could also allow for the convenience yield to be increasing A � A∗
Φ ηc τ�1=2
ε + �1=2
:
in the token price, P, provided this benefit does not impose τε
token neutrality and is not sufficiently convex that it leads
Substituting (B.1) into (B.2), we arrive at the following
to a trivial corner solution in which all users participate
condition:
regardless of the price. !
If the tokens are sold at a uniform price P and users α �(1�ηc )(A∗ �A)+12(1�ηc )2 τ�1 �1=2 A ∗
� A
have quasilinear preferences, then user i solves the follow­ κe ε Φ (1 � ηc )τε � �1=2
1�α τε
ing optimization program:
� X(P, y, ξ)P: (B.3)
max (Xαi,t ((1 � β)E[Ui,t (Ci,t ,Cj,t ; N t ) | Ai,t ])1�α
X i, t The left-hand side of (B.3) is strictly decreasing in A∗ � A ∈
�Pt Xi,t � κ)1{Xi,t >0} : (�∞, ∞) from ∞ to zero, whereas the right-hand side is a
horizontal line with value X(P, y)P for all values of A∗ � A.
If a user does not join the platform, she receives an out­
Consequently, there always exists a solution for the token
side option normalized to zero.
price P from (B.3). Because x(P, y, ξ)P is a strictly increasing
Furthermore, from our main analysis, recall that if users fol­
function of P, we can invert it to express the price as
low a cutoff strategy and join the platform if Ai,t ≥ A∗t , then ! !
∗
1 2 �1 α A�A∗ +12(1�ηc )2 τ�1 A � A
E[Ui (Ci,t , Cj,t ; N t ) | Ai,t ] � (1 � β)e(1�ηc )(Ai,t �At )+At +2ηc τε P�f �1
κe ε Φ �1=2
(1 � ηc )τε + �1=2 ; y, ξ
! 1�α τε
A∗t � At
�1=2
Φ ηc τε � �1=2 : � p(A � A∗ ; y, ξ),
τε
It is then immediate that if a user intends to join the plat­ where f �1 (·, y, ξ) is the inverse of x(P, y, ξ)P for a given (y,
form, her optimal choice of tokens is ξ) pair. Because the left-hand side of (B.3) is strictly decreas­
� �1�α1 ing in A∗ � A, p(A � A∗ ; y, ξ) is a strictly decreasing function
α 1 2 �1
of A∗ : If all users join, then X(P, y)P � ∞, which suggests a
Xi,t � (1 � β)e(1�ηc )(Ai,t �At )+At +2ηc τε
Pt price of p(∞; y, ξ) � ∞:
!
A∗t � At Although we can always find a unique price for a given
�1=2
Φ ηc τε � �1=2 : participation cutoff A∗ , we must now find A∗ from (B.1)
τε
by rewriting the condition as
As such, the maximized utility of user i is α
!
( α1�α (1 � α)(1 � β) A∗ +1η2c τ�1 A∗ � A
�α�1�α
α e 2 ε Φ ηc τ�1=2
ε �
max
1 2 �1
(1 � α)(1 � β)e(1�ηc )(Ai,t �At )+At +2ηc τε κ �1=2
τε
P α
! ) � p(A � A∗ ; y,ξ)1�α : (B.4)
�1=2 A∗t � At
Φ ηc τε � �1=2 � κ, 0 : Notice that the left-hand side of (B.4) is hump shaped in
τε A∗ , tending to 0 at A∗ ∈ {�∞, ∞}, whereas the right-hand
 Sockin and Xiong: A Model of Cryptocurrencies
6704 Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS

side is a strictly decreasing function of A∗ from ∞ at A∗ � matching with another user, E[1{Aj ∈N } ], is zero. Similarly, if all
�∞ to 0 for A∗ � ∞: Consequently, we have a situation users join the platform, then A∗ � �∞, and the left-hand side is
similar to that in the main model, in which the inverse again zero. Consequently, the left-hand side of (B.6) is hump
demand curve p(A � A∗ ; y, ξ) may remain above the hump- shaped in A∗ , as in Figure 1.
shaped curve for A∗ > �∞. In this case, only a trivial solu­ It is immediate then that the assumption of a normally
tion may exist. If p(A � A∗ ; y, ξ) shifts upward, for instance, distributed endowment process is not essential for our
because of a more negative supply shock, ξ, then the set market breakdown analysis.
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of A for which there is a nontrivial solution shrinks. Con­

sequently, the token market fragility illustrated in our Appendix C. Microfoundation of
main model is still present in this more general model. Speculator Demand
In this appendix, we provide a parsimonious model of
B.1. Token Retradeability speculators to aggregate their trading. We assume that
Until now, we have abstracted from token retradeability that there are overlapping generations of speculators. At each
comes with a dynamic setting. We can incorporate this easily date, two types of speculators participate in the token
into our analysis by assuming that tokens can be resold after market. The first is a group of passive speculators who
users transact for a final value δ=y per token, where the division enter at the beginning of each date and acquire all of the
by y reflects the dilution of final value based on the amount of Φ(yt ) tokens from the previous generation of users, valida­
outstanding token supply. In this situation, the user’s token tors, and speculators. These passive speculators provide
demand is liquidity to exiting token holders and do not engage in
� �1�α1 any additional token trading.
α ∗ 1 2 �1
The second is a group of active speculators of unit mass
Xi � (1 � β)e(1�ηc )(A �Ai )+At +2ηc τε
P � δ=y who choose whether to short sell tokens based on their
!
�1=2 A∗ � A expectations of the next-period token price. We assume
Φ ηc τε � �1=2 , that active speculator k has a noisy expectation of the
τε
next-period token price:
and (B.1) becomes
! ES,k [Pt+1 | I t ] � eζk,t RPt , (C.1)
∗
α ∗ 1 2 �1 A �A where I t is the public information set, RPt is the required risk-
α1�α (1 � α)(1 � β)eA +2ηc τε Φ ηc τ�1=2
ε � �1=2
τε neutral return for holding the token to the next period, and
� κ(P � δ=y)1�α :
α
(B.5) ζk,t ~ i:i:d N (ζt � yt , 1). ζk,t represents speculator k’s sentiment,
and ζt ~ i:i:d N (0, σ2ζ ) represents speculators’ common senti­
Note that the issues remain the same as in the static model, ment shock in period t. That speculator sentiment is decreasing
except now the term δ=y shifts down the right-hand side of in the token supply, yt, represents a time trend that speculators’
(B.5). Consequently, having a high retrade value increases the enthusiasm for a new platform declines as the platform
region of existence of a nontrivial solution, and this effect is matures. Each active speculator can short either zero or one
dampened by more tokens that have been issued, y. These token. In deciding whether to short sell a token, speculator k
again echo the results of our main model. faces an opportunity cost for her position of (RPt )1+λ for
λ > 0. As such, she chooses XSk,t ∈ {�1, 0} to maximize
B.2. More General Endowment Distribution
We now relax the assumption that the endowments of users at USk,t � max[ES,k [Pt+1 |I t ] � (RPt )1+λ ]XSk,t :
XSk,t
each date are normally distributed. Instead of assuming a nor­
mal distribution, we let the endowment of agent i follow a gen­ Choosing XSk,t � �1 indicates that the speculator is short
eral distribution: Ai,t ~ G(Ai,t |At ) with support A ∈ [�∞, A]. selling a token. Substituting with Equation (C.1), it is
Assuming users follow a cutoff strategy, straightforward to see that speculator k follows a cutoff
� �1�α1
policy of short selling a token with a cutoff at the senti­
α ment level λ log(RPt ):
Xi,t � (1 � β)e(1�ηc )Ai,t E[eηc Aj,t 1{A∈N t } ]:
Pt �
0 if ζk,t ≥ λ log(RPt )
As such, the maximized utility of user i is XSk,t �
�1 if ζk,t < λ log(RPt ):
�� � α �
α 1�α As a result, the aggregate demand of the speculators XS is
max (1 � α)(1 � β)e(1�ηc )Ai,t E[eηc Aj,t 1{Aj,t ∈N t } ] � κ, 0 :
P the sum of their passive and active positions
Z λ log(RPt )
It is immediate that it is again optimal for each user to follow a XS � Φ(yt ) � dΦ(ζk,t )
cutoff strategy and join the platform if Ai,t ≥ A∗t . The marginal �∞
� �
user with endowment Ai,t � A∗t is indifferent to joining the plat­
� Φ(yt ) � Φ yt + λ log(RPt ) � ζt :
form, which imposes (after dropping t subscripts)
α ∗ α
α1�α (1 � α)(1 � β)e(1�ηc )A E[eηc Aj 1{Aj ∈N t } ] � κP1�α : (B.6) It should be clear that we use these two types of speculators to
separately capture their collective buying and selling activities.
As in the main model, the token price still reflects the marginal Although this structure is somewhat mechanical, the aggregate
user’s convenience yield. If the marginal user has A∗ � A, then demand curve for speculators has sensible economic properties;
the left-hand side of (B.6) is zero because the probability of it increases with speculator sentiment ζt and decreases with the
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6705

token price Pt. This particular functional form facilitates tracta­ number of miners NM,t reflects that it is increasingly diffi­
bility of our model without necessarily driving any of our key cult to acquire more mining power because of additional
results. This microfoundation also makes it clear that each spec­ hardware and electricity costs.29 To join the strategic
ulator is atomistic and therefore, cannot internalize the impact attack, a potential attacker has to pay a participation cost,
of his trading on others. which can be viewed as the cost or disutility of coordinat­
ing with the other attackers. We normalize this cost to
Appendix D. Microfoundation of Strategic Attack one in the numeraire good.
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In this appendix, we provide a microfoundation for the Suppose that NM,t miners provide mining services at
strategic attack condition in the main paper. Specifically, date t and that a fraction pt of miners attacks and splits
we examine whether rogue miners wish to collude to the proceeds from the attack equally. They then need to
engage in a 51% “double-spending” attack. This requires acquire half of the total mining power, and consequently,
that a group of miners amasses enough computational they must acquire NM,t in additional mining power. An
power, compared with the rest of the mining community, attack will occur when the benefit—the fraudulent sei­
to be able to verify, on average, the majority of transac­ gnorage and additional half of the seignorage and transac­
tions on the blockchain. Conceptually, by winning enough tion fees—is greater than the cost of doubling the existing
blocks to add to the blockchain, these corrupt miners will be computing power of the mining community:
able to eventually validate their own blocks on the longest � �
1
chain or to mine secretly a second chain longer than the current Φ(yt + ψι) � Φ(yt ) + (Φ(yt ) � Φ(yt � ι)) Pt
2
blockchain and broadcast it to the mining community as the 1 2
legitimate chain. When this occurs, these miners can reverse + βγUt � αNM ,t ≥ 0:
2
their own transactions to undo their expenditures, returning
their spent tokens to their wallet to be spent again. This is the When this happens, a strategic attack occurs. When this
so-called “double-spending” problem. By creating duplicate condition is satisfied, however, all miners will want to
tokens, the strategic attack temporarily increases the token sup­ attack the platform, which will dilute the mining power and
ply through fraudulent inflation.27 undermine a strategic attack. As this cannot be an equilib­
The benefits and costs of a 51% attack are linked to par­ rium, the miners must play a mixed strategy when a strate­
ticipation by both users and miners. As more miners join gic attack is possible. The probability of a miner attacking,
the mining pool, the probability of completing any trans­ pt , is the date t probability then ensures that every miner is
action and adding it to the blockchain falls, increasing the indifferent to attacking based on the outcome of an i.i.d.
effective computational cost of attacking the currency. In draw of a Bernoulli random variable with Pr(Attack) � pt : By
addition, user and miner participation also increases the the weak law of large numbers, exactly a fraction pt of the
computational cost of an attack through the difficulty of existing mining pool will attack. This probability satisfies
mining each transaction or the hash rate. Many PoW pro­ that the fraction p1t of the revenue from attacking is offset by
tocols, such as those of Bitcoin and Ethereum, set the hash the disutility of participation
� �
rate to maintain a fixed average time for new blocks to be Φ(yt + ψι) � 12Φ(yt ) � 12Φ(yt � ι) Pt + 12βγUt � αNM
2
,t
added to the blockchain, and the hash rate increases in � 1 � 0,
pt NM,t
the number of users and miners to prevent blocks from
being added too quickly. As a consequence, having more from which follows, when pt > 0, that
subscribers and a more diverse mining pool can make the � �
platform more secure. Φ(yt + ψι) � 12Φ(yt ) � 12Φ(yt � ι) Pt + 12βγUt � αNM
2
,t
pt � ;
We assume that miners lack commitment, which is con­ NM,t
sistent with the static incentives miners face because of
otherwise, there is no attack. Consequently, we can inter­
free entry (e.g., Abadi and Brunnermeier 2018). Any miner
pret the strategic attack condition (8) as arising from a 51%
can attack the blockchain by engaging in a 51% attack to
attack on the currency, and the possibility of attack leads to
“double spend” the coins received from seigniorage. If
a stability boundary in the state space of the platform.
corrupt miners attack the blockchain, the strategic attack
artificially inflates the token base by Φ(yt + ψι) � Φ(yt ), for
ψ > 0, and the miner sells these additional tokens to earn Endnotes
(Φ(yt + ψι) � Φ(yt ))Pt in additional revenue. These addi­ 1
In contrast, coins (and altcoins), such as Bitcoin and Litecoin, are
tional tokens have to be absorbed by users and specula­ fiat currencies that are maintained on a public blockchain ledger by
tors by increasing the effective token supply to Φ(yt + ψι): a decentralized population of record keepers, whereas security
In addition, because the corrupt miners add over half the tokens are financial assets that trade in secondary markets on
blocks to the blockchain, they earn 50% of the transaction exchanges and whose initial sale is recorded on the blockchain of
the currency that the issuer accepts as payment. Coins are typically
fees from users and seigniorage. As a result of increased
created through “forks” from existing currencies, such as Bitcoin
waiting times and service denials, users also experience a
Gold from Bitcoin, and by airdrops, in which the developer sends
loss in expectation a fraction 1 � γ of their trade surplus.28 coins to wallets in an existing currency to profit from the price
To acquire 51% of the computing power, corrupt miners appreciation of its retained stake if the new currency becomes
must replicate the mining power of the existing NM,t widely adopted. Security tokens are typically sold through ICOs
2
miners by expending a convex technological cost αNM ,t , structured as “smart contracts” on existing blockchains, such as that
where α > 0: That the cost is convexly increasing in the of Ethereum.
 Sockin and Xiong: A Model of Cryptocurrencies
6706 Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS

2
We depart from Sockin and Xiong (2023) along several substantive its inflation rate as a function of the proportion of DOT tokens that
dimensions. First, in terms of emphasis, their focus is on platform gov­ are staked but tries to maintain about 10% per year (https://wiki.
ernance, whereas our focus is on token price dynamics and platform polkadot.network/docs/learn-staking).
stability. Second, in terms of information structure, in their setting the 9
We implicitly assume a frictionless secondary market for tokens.
fundamental underpinning the aggregate transaction surplus of users See, for instance, Capponi and Jia (2021) for liquidity issues associ­
is the only fundamental; in this paper, we extend their analysis to ated with cryptocurrency exchanges.
include not only a time-varying token supply but also, nonfundamental 10
In contrast to traditional multisided platforms, such as in Evans
fluctuations in the token price from the optimism of users and the senti­
(2003) and Rochet and Tirole (2003), the owner issues a native token
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ment of speculators. Third, although the token price is endogenous in

to users that has a floating exchange rate with other tokens and cur­
both settings, in ours it is determined by market clearing in a secondary
rencies instead of collecting discriminating participation fees. This
token market; however, in Sockin and Xiong (2023), it is set by the
potentially buffers the pricing of the platform’s services from exter­
developer that controls the supply of tokens. Finally, in terms of empiri­
nal shocks, such as monetary policy shocks to fiat currencies, by
cal implications, Sockin and Xiong (2023) use their model to explain
denominating them in the native token and disciplines their valua­
cross-sectional patterns in ICOs; this paper instead focuses on time
tion through price discovery in financial markets.
series patterns (e.g., momentum, reversal, life-cycle effects, relation to
11
investor attention) and cross-sectional patterns (e.g., size effect) in cryp­ We assume the owner completes all transactions without censor­
tocurrency returns. ship or charging monopoly markups. See Huberman et al. (2021)
3 for how proof of work-decentralized consensus can overcome these
Although transaction fees paid on decentralized crypto platforms
issues at the cost of transaction delays. We also assume that the
can adjust to token price fluctuations, such as gas fees under the
owner can commit to a token inflation schedule. See Cong et al.
proof of stake protocol, the services on these platforms are more
(2022a) for a setting in which the owner cannot commit.
rigid in the number of tokens required for their services. To claim a
12
username on Decentraland, for instance, requires 100 MANA This feature also contrasts the neutrality of the token price
tokens regardless of their value in US Dollars. Such costs are neces­ adopted by Cong et al. (2021b). In their model, each user’s benefit
sary to engage with other users on the platform and consequently, from holding a token is determined by the market value of her
contribute to a user’s decision to participate. token holdings in the numeraire rather than the number of tokens.
4 13
In an earlier version of the paper, we considered an extended setting Even though the market breakdown is a severe form of market
in which the platform fundamental is unobservable. In that setting, dysfunction, it may not present an arbitrage opportunity to specula­
users use their endowments as a private signal about this fundamen­ tors for several reasons. First, a platform’s tokens derive all their
tal; the token price and the transaction history on the blockchain act as value from the convenience yield that users receive from transact­
public signals that aggregate their dispersed information. This second ing on the platform. Thus, whether a platform’s token price can
public signal reflects that the blockchain technology supporting cryp­ recover from zero is ultimately determined by users rather than
tocurrencies acts as an indelible and verifiable ledger that records the speculators. Second, it is difficult for a platform that relies on net­
decentralized transactions that take place on the platform. In this work effects to recover once users have lost interest in it. The case
extended setting, we show that informational frictions attenuate the of Myspace after the rise of Facebook is a particularly salient exam­
risk of breakdown by dampening price volatility and platform ple of the fickleness of network effects.
14
performance. Although one may argue that validators can alter the token sup­
5
As Liu and Tsyvinski (2021) find little evidence that cryptocurren­ ply schedule to mitigate market breakdown, achieving consensus
cies load on conventional sources of systematic risk, such as market among stakeholders to alter token inflation on decentralized plat­
or style factors, such an assumption for the token market is realistic. forms is extremely difficult in practice. For instance, as EthHub
6 describes of the Ethereum platform, “[a]s Ethereum is a decentra­
The nonneutrality of the token price is highly realistic on many
lized network, the Monetary Policy cannot be successfully modified
crypto platforms. On Axie Infinity, for example, users can breed an
unless there is overwhelming consensus from the aforementioned
axie up to seven times using Small Love Potion (SLP) tokens accord­
stakeholders” (https://cryptobriefing.com/ethereum-sound-money-
ing to a rising scale (i.e., currently 900 SLP for the first breed up to
like-bitcoin/).
15,300 for the seventh). On Socios, users buy fan tokens associated
15
with specific sports teams that convey certain benefits and voting The second (high-cutoff) and third (highest-cutoff) equilibria
rights on team decisions. These features are independent of the may or may not exist at any given date depending on the expected
token price. On Friends with Benefits (FWB), users can purchase a retrade value of the token. As such, they are dynamically unstable,
local membership with 5 FWB tokens and a global membership and we can eliminate them as predictions for the equilibrium out­
with 75 FWB tokens. In addition, users currently pay one FWB come. In addition, the second (high-cutoff) equilibrium is unstable,
token to get access to the newsletter and five for access to the global even when fixing the token’s expected retrade value. Introducing a
network of token-gated parties. small amount of noise into users’ participation decisions, for
7 instance, and letting this noise become arbitrarily small would
Online platforms often face severe commitment issues. Facebook, for
ensure convergence away from this second equilibrium to the
instance, changed its data policies over time (for example, Beacon in
highest-price equilibrium.
2007 and the 2008 Terms of Service update) and settled with the Federal 16
Trade Commission in 2011 for violating privacy promises. Amazon Although such an analysis of optimal platform policy is beyond
engages in “copycat” practices on two-sided platforms that harm sell­ the scope of this paper, Mei and Sockin (2022) show that a platform
ers. A rigid token issuance schedule consequently represents one safe­ owner may find it optimal to inflate the token base to ensure nontri­
guard that decentralization token platforms have in place to protect vial user participation when incentives to speculate are particularly
users. See Sockin and Xiong (2023) for an analysis of trade-offs associ­ severe.
17
ated with decentralization. This issue has also received significant attention in the literature.
8
Such rigid, predetermined inflation schedules are ubiquitous in See, for instance, Budish (2018), Pagnotta (2022), and Chiu and
practice. Solana, for instance, currently has an annual inflation rate Koeppl (2023).
18
of 8% that is scheduled to decrease by 15% per year to a long-term In practice, several miners are randomly drawn from a queue to
inflation rate of 1.5% (https://blockdaemon.com/products/white- compete to complete each transaction, and miners often pool their
label-validator/how-solana-staking-works/). Polkadot instead sets revenue to insure each other against the risk of not being selected.
 Sockin and Xiong: A Model of Cryptocurrencies
Management Science, 2023, vol. 69, no. 11, pp. 6684–6707, © 2023 INFORMS 6707

See Cong et al. (2021a) for an extensive analysis of this issue. Our Budish E (2018) The economic limits of Bitcoin and the blockchain.
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19
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Management Sci. 69(11):6455–6481.

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