DeFi – Through the Economic Theory Lens

Loading

Decentralized Finance (DeFi) – Economic Theory Lens

Introduction

Crypto assets and crypto applications started with the release of Bitcoin in 2008. As the
first cryptocurrency, Bitcoin enabled secure storage and an exchange of digital value without
the use of a designated third party. However, from the start, the economic benefit of cryptocurrencies has been questioned: Within a stable, efficient monetary system, it is not clear
what the value proposition of a cryptocurrency really is.
In 2015, the Ethereum blockchain was introduced to develop the idea of running smart
contracts on a decentralized ledger. At first, the Ethereum blockchain was mainly a way
to issue new tokens. However, it became clear that the blockchain could serve as a host
for decentralized financial applications, based on the technology that runs smart contracts
on secure distributed ledgers. Using these smart contracts, individuals can then directly
engage in financial transactions without the use of third parties. This process, commonly
referred to as decentralized finance (DeFi), enables the elimination of costly, third-party-run
infrastructure when lending or trading.1
In this viewpoint article, we provide a primitive analysis of the DeFi value proposition, as
well as its limitations. We ask two main questions: What are the necessary conditions for
DeFi to provide value over traditional, intermediated lending relationships? And what are
the main limitations DeFi applications currently face? While DeFi is used for a variety of
applications, we focus on applications for lending, which compete directly with real world
intermediaries such as banks or financing companies.2
We begin with a simple setting where there is a need for borrowing and lending, but where
frictions, in the form of a double-sided commitment problem, make direct lending between
two parties expensive. Third-party intermediaries can alleviate these problems by guaranteeing the execution of the contract, but they require a fee to ensure that they themselves
1

Harvey et al. (2021) and Schär (2021) provide a non-technical, detailed discussion of DeFi architecture

and applications.
2
In particular, we do not look at decentralized exchanges, which are another promising application, but
which mainly facilitate the trading of crypto assets and thus do not necessarily compete with traditional
intermediaries in the mainstream.

1

have proper incentives.
This is where DeFi comes into play. Instead of relying on high fees, it can run a platform
that—based on a distributed ledger and smart contracts—can guarantee the execution of
a borrowing contract. Hence, DeFi can either substitute for traditional intermediation or
allow for better, bilateral loans between contracting parties. The value of DeFi lies therefore
in both disintermediation and financial inclusion.3
This value proposition of DeFi, however, runs into several key limitations. First, DeFi
requires stablecoins with low volatility and fairly stable collateral values to function properly.
Unfortunately, we are currently not quite there yet.4 However, the introduction of a wholesale
central bank digital currency (CBDC) and the tokenization of government securities provide
alternatives that could alleviate these shortcomings.
Second, DeFi applications may be too rigid in their execution of smart contracts. One role
intermediaries play is to adjust contract execution in the case of unforeseen contingencies.
In the future, however, technological advances may make it possible to reduce the incompleteness of smart contracts or automate possible renegotiation of such contracts.
Third, DeFi relies on external entities providing information—so-called oracles. We are
not aware of a solution to fully decentralize an oracle that provides real time information,
especially unquantifiable “soft” information, to a DeFi application in a tamper-proof way.
One alternative would be to have a designated party provide such infrastructure. In some
cases, financial markets might regard a central bank as a reliable neutral provider.
In the last few years, there has been a tremendous growth in DeFi in terms of both its
scale and scope. As shown in Figure 1.1, the total value locked (TVL) into DeFi increased
dramatically starting with the so-called “DeFi Summer” in 2020. After reaching its peak
at USD 250 billion in late 2021, the DeFi market saw a sharp decline in its TVL in the
3

Chiu and Koeppl (2019) provide an early analysis of such DeFi applications for settling securities based

on a proof-of-work blockchain compared with using costly intermediaries.
4
The TerraUSD crash has been a reminder that unbacked stablecoins have difficulties maintaining their
exchange rate pegs, while backed stablecoins (e.g., Tether, USD Coin, Dai) tend to be more stable. Notwithstanding their backing, all stablecoins became more volatile during the Terra crash in 2022.

2

second quarter of 2022, largely due to the general price crash in crypto assets during this
period. Nonetheless it remains at around USD 70 billion as of June 30, 2022. DeFi has also
been expanding its scope. Figure 1.2 shows the decompositions of the TVL across different
DeFi protocols. When combined, lending protocols and the closely related collateralized
debt positions (CDP)5 constitute the most important portion of the market, closely followed
by decentralized exchanges (Dexes). Aave is currently the largest lending protocol, and
MakerDAO is the largest CDP.
Figure 1.1: Total Value Locked (TVL) in DeFi (Source: DeFiLlama)
300

250

TVL USD Billion

200

150

100

50

0

The economics literature on DeFi is just emerging. Many existing studies focus on decentralized exchanges and explore how automated market makers function differently from
centralized exchanges; see Aoyagi and Ito (2021), Capponi and Jia (2021), Lehar and Parlour (2021), Park (2021). Another line of research investigates the economics of decentralized
stablecoins such as Dai issued by the MakerDAO protocol (d’Avernas, Bourany, and Vandeweyer (2021), Li and Mayer (2021), Kozhan and Viswanath-Natraj (2021)).
Economic papers on DeFi lending are more limited in number. Lehar and Parlour (2022) empirically study how decentralized lending platforms affect the prices of crypto assets through
liquidations of loans or collateral. Chiu et al. (2022) theoretically model a dynamic feedback
between price and liquidity in DeFi lending and study the implications for fragility.6 Our
5

The key difference between the two is that a lending protocol lends out tokens deposited by lenders,

while a CDP lends out tokens (typically stablecoins) minted by itself.
6
There is also a related literature studying applications of smart contracts and the economic trade-offs

3

Figure 1.2: Decomposition of DeFi TVL in 2022 Q1 (Source: DeFiLlama)
Others
12%
Dexes
29%

CDP
8%

Liquid Staking
9%

Yield
9%
Lending
22%

Bridge
11%

viewpoint article contributes to this literature by using a simple, stylized model of borrowing
to show what DeFi lending has to offer in terms of lower costs and better financial inclusion.
The article is organized as follows. In Section 2, we provide an overview of DeFi. Section
3 presents a simple model where lending involves either direct intertemporal trading or a
centralized intermediary. Section 4 studies DeFi as an alternative arrangement and discusses its value proposition and limitations. Section 5 concludes by suggesting how proper
infrastructure for DeFi may be built. Omitted proofs can be found in the Appendix.

2

What Is DeFi? An Overview

DeFi is an umbrella term for a variety of applications and projects in finance that attempt
to reduce the reliance on costly, third-party intermediaries. These trusted actors are often
replaced by smart contracts, which are immutable computer programs that guarantee execution of the contract. The concept of a smart contract was first introduced into computer
science by Nick Szabo, who describes them as “building blocks for digital markets” embedding contracts into software and making their breach expensive (Szabo, 1996). He likened
involved; see, for example, Bakos and Halaburda (2021), Cong and He (2019), Lee et al. (2022).

4

the idea to replacing a shopkeeper in a store with a pre-programmed vending machine.
The key development for DeFi was to link this idea to blockchain and distributed networks
with the introduction of the Ethereum project. Smart contracts are run on a blockchain,
which is a ledger simultaneously stored and updated across a distributed network of independent computers. As long as the blockchain is tamper-proof, smart contracts can be
guaranteed to execute within this network without the use of a third-party intermediary.

2.1

DeFi Architecture

DeFi is designed as a multi-layered architecture with three primary layers (see Figure 2.1).
The bottom one consists of the blockchain where the settlement of contracts occurs. The
middle one creates assets as tokens that can be stored and transferred on the blockchain.
The final one at the top contains the actual DeFi protocols that deploy the smart contracts.
Above all these layers, there is an additional interface where potential users can access the
application. Such applications can also integrate different protocols and offer wallets, which
are local programs to run applications in a user-friendly way.
Figure 2.1: Basic DeFi Layers and Examples

Protocols

Aave

Uniswap

Lido

Tokens

WETH

USDC

DAI

Blockchain

Ethereum

Most DeFi protocols are run as a permissionless environment where anyone can use the
protocol without third-party consent. Contracts can then freely interact with each other,
be built on top of other existing contracts and even function across different protocols. As
a result, DeFi protocols are composable. For example, one can write a smart contract that
builds on a lending protocol and an exchange protocol to create a margin trade protocol.
5

We now briefly discuss the different layers of the DeFi architecture in further detail.

2.1.1

Settlement Layer

Bitcoin was the first blockchain application with a financial focus. As its script is rather
simple, Bitcoin is mainly a system for recording ownership and transferring value. It is not
designed as a foundational layer for other protocols to build on. In contrast, Ethereum,
founded by the Ethereum Foundation, was specifically built to support the execution of
smart contracts.7
Currently, Ethereum is the main blockchain for DeFi protocols, with over 50% of value in
DeFi locked into it. While the majority of smart contracts are written on Ethereum, there
are many other blockchains such as Algorand, Avalanche, Binance Smart Chain, Cosmos,
Polkadot and Solana. These blockchains are designed to tackle issues such as scalability,
interoperability and the cost of achieving consensus when a blockchain is updated with new
information.

2.1.2

Token Layer

The process of tokenization allows users to create tokens building on the blockchain layer
at the bottom. On Ethereum, the most popular standards are ERC20 for creating fungible
(i.e., fully interchangeable) tokens and ERC721 for creating non-fungible tokens.
A token plays various roles in a protocol. For example, it can represent an IOU of a lending
pool issued to a lender, or governance rights issued to an equity holder in a protocol. Tokens
can also represent real-world assets. In particular, tokens can be designed as stablecoins
intended to represent a stable value with respect to a unit of account.
Stablecoins can be backed by an asset off the blockchain, such as the US dollar or securities
denominated in USD. Prominent examples are USD Coin (USDC) issued by Circle, Tether
(USDT) issued by Bitfinex and Binance USD (BUSD) issued by Binance. Such an arrange7

Ethereum has its own native cryptocurrency, Ether, which also serves to pay fees (“gas”) for running

the smart contracts.

6

ment typically requires a centralized, trusted third-party in the real world to hold reserves
of the underlying asset to back the token. Hence, strictly speaking, these efforts a re not
decentralized solutions.
Alternatively, one can issue a stablecoin that is backed by on-chain assets. A prime example
is the stablecoin Dai, created by MakerDAO. This coin is backed by Ether (ETH) and
other cryptocurrencies, for example USDC. The value of Dai is pegged to the US dollar
and is based on over-collateralization, specifically a 33% haircut. More generally, one can
create other synthetic tokens to replicate the income flows of real-world assets such as a
stock index or standardized derivatives.8

2.1.3

Protocol Layer

DeFi protocols are built on top of the settlement and token layers. As previously noted,
these protocols can serve a variety of purposes, the main ones being
• decentralized lending platforms
• decentralized exchanges for crypto assets
• customized derivatives
• asset management for crypto assets
with the first two being the most important examples.
Decentralized exchanges set up marketplaces to facilitate the spot trades of cryptocurrencies.
These protocols have mainly arisen to permit the direct conversion of different cryptocurrencies without the use of traditional currencies. Many of the exchanges replicate real-world
arrangements such as over-the-counter markets or limit-order books. A more novel type of
decentralized exchange is the so-called “automated market maker” (AMM) which automates
the price-finding mechanism based on the actual trades made on an exchange.
8

Other attempts to create stablecoins are based solely on the principle of a “currency board” that actively

intervenes to support a peg. However, such attempts have sometimes spectacularly failed (see, for example,
the crash associated with the Terra/LUNA protocol).

7

Decentralized lending platforms such as Aave, C.R.E.A.M. Finance, dYdX and Compound
function very much like traditional banking intermediaries, taking in deposits in stablecoins
and allowing borrowers to obtain funds in these stablecoins against crypto collateral. The
main attraction of these platforms is that crypto users can use their investments as collateral
to achieve additional functionality from their crypto holdings.
Other protocols generate value by introducing financial products that are hard to replicate
with traditional arrangements. A prime example is flash loans. Unsecured credit can be
provided under the condition that the loan must be repaid atomically within a single block
of the underlying blockchain. This means that the borrower receives the funds, uses and
repays them—all within the same blockchain transaction or in linked transactions within
the same block. Hence, either all transactions are carried out, or none are. Flash loans help,
for example, in arbitraging away price difference across different exchanges. For instance, if
two AMM pools price a token differently, an investor can obtain a flash loan from a lending
protocol, buy and sell in the two pools, and then repay the loan immediately, making a profit
from the price discrepancy. While flash lending improves market efficiency, it can also be
abused to launch so-called “flash attacks.” For example, an attacker can use a flash loan
to create an artificially large transaction on an illiquid exchange. The attacker can thereby
temporarily manipulate the price and potentially profit from it, if a lending protocol relies
on this price on the exchange to value collateral assets.

2.2

An Illustrative Example

Consider an example of arranging a secured loan using an intermediary such as a bank.
As illustrated in Figure 2.2, suppose Bob wants to borrow cash from Alice. He is willing to
pledge some assets as collateral. A financial intermediary—“the bank”—takes a cash deposit
from Alice and lends the cash to Bob against his collateral held in custody. The collateral
will be returned to Bob when he repays the loan. The bank will liquidate the collateral if
Bob defaults. Bob has an incentive to default whenever the value of the collateral drops
enough in value. A traditional arrangement that involves an intermediary is often referred
to as centralized finance (CeFi).
8

Figure 2.2: Secured Loan Example

How does a typical DeFi lending protocol replicate this loan arrangement? Assume that
the borrower takes out a loan in a stablecoin and secures the loan with a crypto asset.
Lenders deposit their tokens individually into a lending pool that is governed by a lending
protocol. To access funds in the lending pool, the borrower locks crypto collateral into a
smart contract. The borrowing terms are calculated based on a pre-programmed function
including the haircut on the collateral, the repayment schedule and the interest rate.
The smart contract is then executed according to its rules. Settlement is atomic in the
sense that the collateral is only returned when all obligations are met by the borrower.
Typically, the contract is over-collateralized to safeguard against default, but as the value of
the collateral can fluctuate, the need to liquidate becomes an issue.
To deal with liquidation, two elements are required. First, the crypto asset that serves as
collateral needs to be priced automatically using a price feed. This component of the arrangement, called an oracle, can be incorporated in the protocol or can come from a different
DeFi application. Second, a liquidation bot is needed to monitor open borrowing positions
and—using a smart contract—automatically liquidates under-collateralized positions.
This example makes it clear that decentralization is often a matter of degree, where different
steps in this lending relationship may or may not be decentralized. First, the borrower
and the lender need to be matched. This requires an effective matching engine or pool
to be set up where lenders contribute funds and borrowers can access funds within the
9

protocol. Second, the digital form of funds—stablecoins in our example, but possibly other
cryptocurrencies—are genuine representations of real-world cash balances. This requires a
reliable tokenization process and, in the case of stablecoins, a reliable way to manage them.
Third, the protocol needs to keep custody of the digital assets and allocate their ownership
reliably. This requires a well-designed smart contract and a tamper-proof blockchain on
which the contract is deployed. Fourth, there needs to be a mechanism to settle the contract
unambiguously. This may involve the liquidation of collateral when the price of the collateral
drops, which, as pointed out, necessitates an oracle and liquidation bots. Each of these
components can be implemented by either an intermediary or a smart contract. Most DeFi
applications decentralize the custody and settlement. Decentralizing other elements, such as
tokenization or the oracle, is much harder to achieve.

2.3

A Short Preview

In the next two sections, we present a formal discussion of the value and limitations of both
CeFi and DeFi. To do so, we focus on the example of decentralized lending just outlined and
compare three different lending arrangements where a borrower has collateral, but needs to
obtain liquidity against the collateral. One arrangement is a direct trading relationship with
a lender. The problem is that both sides to a collateralized loan contract cannot commit to
the terms of the contract. The borrower may default, while the lender may not return the
collateral.
A second lending arrangement solves the problem by having a financial intermediary take
custody of the collateral and execute the contract terms for a fee. The fee is necessary to give
incentives to this third party to safeguard the collateral and execute the contract properly.
An alternative way, the third arrangement, is to use a smart contract. DeFi can then be seen
simply as a different arrangement for custody and execution of the contract. We assume that
the contract has a lower cost than using the third party. However, the collateral being used
and, possibly, the stablecoin used for settlement as well, have higher volatility than their
real counterparts. This gives rise to a trade-off for the value offered by the smart contract.

10

Beyond the issue of volatility, such DeFi arrangements tend to suffer from several limitations
associated with the execution of the smart contract. Firstly, guaranteed execution comes at
the cost of too little flexibility ex post. Secondly, the contract needs an oracle that provides
price feeds that are necessary for executing the contract. These are potential areas where
public infrastructure can increase the value proposition of DeFi in the future.

3
3.1

CeFi—Intermediated Lending
Setup

We start off by looking at a simple borrowing environment that requires collateral to secure
a loan. There are two periods, t = 0, 1. Agents are risk neutral. We focus on two agents, a
lender and a borrower.
There is a single nonstorable consumption good, y, which serves as numeraire. The lender
has preferences given by
y0 + y1
and is endowed with eL units of numeraire goods in period t = 0. The borrower values the
numeraire good more than the lender in period t = 0,
(1 + v)y0 + y1 ,
where v > 0. The borrower is endowed with eB units of the numeraire good in t = 1, but has
no endowment of the good in period 0. We assume that eL and eB are sufficiently large for
them to conduct the financial arrangements discussed below. Since the borrower and lender
have different intertemporal marginal rates of substitution, there are gains from trade.
The borrower also has an endowment of one unit of a durable asset x in period 0. The
asset matures at the end of period 1, when it provides a payoff to its holder. For the typical
holder, the payoff of the asset is px1 (s) in units of the numeraire good, where s = ℓ, h denotes

11

the state of the world in period t = 1. We assume the two states are equally probable and
px1 (h) = (1 + δ)px
px1 (ℓ) = (1 − δ)px
so that the price of the asset is px in period 0. The state s is revealed at the beginning of
period 1, so that the price of the asset in period 1 is given by px1 (s).9
Since the asset is transferable, it also has the potential to serve as collateral. However, we
assume that the asset is not perfectly liquid. There are two components to this assumption.
First, when the borrower or lender sells the asset on the open market, there is a transaction
cost of L per unit initially invested.10 Second, we assume that for the lender, the asset yields
no payoff if held in period 1; instead, he or she must sell the asset to obtain any value,
thereby incurring the transaction cost. On the other hand, the borrower receives full benefit
from the asset in period 1 and therefore does not need to sell it. In addition, we assume that
px (ℓ) > L,

(1)

px (h) − px (ℓ) > L.

(2)

The sequence of events is as follows. In period 0, the lender can transfer some of the
endowment good to the borrower in exchange for the borrower’s asset. In period 1, the
borrower can in return transfer some endowment to the lender in exchange for the asset held
by the lender. Since the numeraire good serves the role of a means of payment, which is
used to settle trades, we will label it as cash.

Assumptions In period 0, in addition to spot transactions, the borrower and the lender
can establish two-period agreements. In effect, these arrangements have the borrower receiving cash in period 0 in return for temporarily handing over the collateral asset to the lender.
We assume that such trades are subject to these frictions:
9

We treat these prices as exogenous in the sense that they are not influenced by the use of the asset as

collateral in the lending relationship.
10
To keep comparisons consistent across arrangements, we assume that the transaction cost only applies
when dealing with the open market, not in direct interactions between the borrower and the lender.

12

1. State-contingent contracts whose terms depend on the realized value of the collateral
asset x are too costly to write.
2. Neither the lender nor the borrower can commit to period 1 exchanges that are not ex
post rational.

3.2

Direct Trading

We next look at the possibilities for the lender and borrower to directly trade with each
other.

Spot sale of collateral Suppose the borrower sells the collateral in period 0 to the lender
in order to finance consumption. Since the lender will resell it on the market, incurring the
cost L, the lender will pay px − L.11 Thus, the borrower’s payoff is
(1 + v)(px − L).

(3)

The borrower gains from a spot sale whenever this payoff is greater than the expected payoff
from retaining the asset or, in other words, if
px > L

1+v
,
v

(4)

which we assume throughout the analysis.

Direct Collateralized Lending Suppose now that the borrower asks for a cash loan C in
period 0 against a promised repayment R in period 1 and hands over the asset as collateral to
the lender to secure the loan. Our assumptions imply that both the lender and the borrower
need an incentive to settle the trade and that not settling the trade is costly due to the
cost L. The borrower defaults whenever the repayment is more costly than the value of the
collateral:
R ≥ px (s).
11

(5)

This is without loss of generality since the borrower could equivalently sell the asset on the market at

the same price.

13

The lender does not hand back the collateral if the repayment is too low, taking into account
the liquidation cost
R ≤ px (s) − L.

(6)

Given condition (2), the outcome is equivalent to a spot sale of the collateral unless one of
the following mutually exclusive conditions holds:
px (ℓ) ≥ R > px (ℓ) − L

(7)

px (h) ≥ R > px (h) − L.

(8)

or

When condition (7) is satisfied, the loan is repaid in state ℓ, but the lender refuses to return
the collateral for the state h, leading to liquidation. When condition (8) is satisfied, the loan
is repaid in state h, but the borrower defaults in state ℓ, leading to liquidation. Since payoffs
for the borrower are increasing in the cash advanced, we have the following result.12

Lemma 1. The borrower offers the contract (C, R) with
C = px −

L
2

(9)

and R ∈ {px (h), px (ℓ)}, which yields a payoff equal to
(1 + v)(px −

L
)
2

(10)

for the borrower.

Since the lender gets zero expected profits, the social welfare is equal to the borrower’s
payoff, so that direct lending always dominates the spot sale of collateral. We turn next to
the question of whether an intermediary can achieve an even better outcome.
12

See the Appendix for a formal proof.

14

3.3

CeFi Loans

We consider now a third agent, a banker who is hired to intermediate loans. The banker has
no personal funds; instead, the banker receives funds from the lender and lends them out to
the borrower in period 0. The terms of the contract are again denominated by (C, R) and
the borrower pledges asset x as collateral with the bank (see Figure 3.1).
Figure 3.1: CeFi Loan Arrangement

Like the lender, the banker has no direct use for the collateral. The banker, however, can
commit not to steal the collateral and sell it, but to execute the loan agreement. The banker
charges a fee ϕ > 0 (payable by the borrower in cash in period 1). The fee reflects the fact
that the bank is considered a trusted third party.13
13

The fee is exogenous in our framework. However, it can be interpreted as the flow return to the charter

value of the bank in a more general model. The outline of such a model is as follows. Denote the banker’s
discount factor by β. Assume that the bank incurs a one-time entry cost ϕ/(1 − β) to acquire the bank
charter. For simplicity, suppose the banker handles one loan each period. Then potential competition from
other entrants ensures that the banker is limited to a fee of ϕ per loan, and the bank’s profit is dissipated.

15

Since the borrower pledges the collateral with the bank, there is no incentive problem for the
lender anymore. Hence, the only incentive problem is the borrower defaulting on the loan,
which happens whenever
R + ϕ ≥ px (s).

3.4

(11)

Optimal CeFi Loan Contract

Suppose the contract maximizes the borrower’s payoff.14 When R + ϕ > px (h), the borrower
always defaults and the contract terms are similar to those of a spot sale, with an additional
cost ϕ. When px (h) ≥ R + ϕ > px (ℓ), the borrower defaults when the collateral value is low.
One can easily show that the contract terms are then similar to those of direct lending, again
with an additional cost ϕ. Since the bank charges a positive fee, there is no value offered by
the banker intermediating the loan in these cases.
Thus the banker offers a loan contract (C, R) that solves
max(1 + v)C + px − R − ϕ
C,R

(12)

subject to
C≤R

(13)

R + ϕ ≤ px (ℓ).

(14)

The objective function captures that the borrower receives (1 + v)C in period 0 and earns
px − R − ϕ in period 1. The first constraint captures the lenders’ participation in the loan
arrangement, whereas the second one is necessary so that the borrower does not default.
If the banker absconds with the collateral, the bank loses its reputation and future business. In order to
induce honest behavior, the fee must thus satisfy
ϕ ≥ (1 − β)(px − L).
For a given ϕ > 0, this restriction is always satisfied for β sufficiently close to 1.
14
The borrower’s payoff is equivalent to social welfare for all lending arrangements since she receives all
surplus and banking has zero profits.

16

Since v > 0, we have that the objective function is increasing in R after substituting the
first constraint. Hence, we have the following result.

Lemma 2. The optimal CeFi loan arrangement is given by
C = R = px (ℓ) − ϕ

(15)

so that the payoff for the borrower is
(1 + v)px (ℓ) − (1 + v)ϕ + px .

(16)

The bank adds value since it can solve the two-sided commitment problem. Hence, there is
a trade-off between incurring the banker’s fee ϕ and the cost of inefficiently liquidating the
collateral L when there is default in the optimal direct lending arrangement. We have the
following result.
Proposition 3. CeFi lending is optimal if and only if


v
L ≥ L̄ =
(px (h) − px (ℓ)) + 2ϕ.
1+v

(17)

Discussion A bank loan is thus preferred, whenever the costs of default are large relative
to the costs of using a trusted third party. It is interesting to also look at the loan size relative
to the ex ante value of the collateral. This can be summarized by the haircut defined as
H =1−

C
.
px

(18)

A larger haircut means that the loan is more over-collateralized. The haircut on the bank
loan and direct lending are given by
1
H= x
p




1 x
x
(p (h) − p (ℓ)) + ϕ
2

(19)

and
H=

1 L
,
px 2

17

(20)

respectively. The haircut on the bank loan is thus always larger since
L < (px (h) − px (ℓ)) + 2ϕ.

(21)

Define next the quality of the collateral good by the volatility of its price
δ=

px (h) − px (ℓ)
.
2px

(22)

Holding the expected payoff px constant, the loan size for direct lending is unaffected by
changes in volatility, but the haircut associated with the bank loan increases. Consequently,
collateral that is more volatile decreases the attractiveness of a bank loan.

4

DeFi—Decentralized Lending

4.1

Setup

We now consider a DeFi platform that offers contracts without relying on a trusted thirdparty. Instead, the platform uses a blockchain to store and execute an atomic, smart contract.
The environment remains the same as in the previous section, except for the introduction of
two new assets.
First, there is now a second asset c, which is a crypto asset that can be stored on the
blockchain and kept safe within a smart contract. This solves the problem that the lender
can seize the collateral in period 1. Consequently, using a smart contract, the borrower and
lender can avoid the fee ϕ.15 The payoff in period 1 of this asset is given by

 (1 + ε)pc w.p. 0.5
pc1 =
 (1 − ε)pc w.p. 0.5.
In order to facilitate a clear comparison, we assume that pc = px and that both assets face
the same liquidation cost L. We can then allow the borrower to exchange one unit of asset
x against one unit of the crypto assets c at the start of period 0.
15

More generally, one could assume that there are costs associated with deploying the smart contract on

the blockchain. In what follows, we interpret ϕ as the cost saved by employing a smart contract in lieu of a
bank.

18

Second, there exists a stablecoin s which is used on the blockchain to settle the smart
contract.16 The stablecoin is necessary because cash cannot be tokenized on the blockchain.
The stablecoin is liquid, in that there are no costs to purchasing or selling it.
The value of stablecoins fluctuates according to

 (1 + η)ps
ps1 =
 (1 − η)ps

w.p. 0.5
w.p. 0.5

where η < ε. Hence, the crypto collateral is more volatile than the stablecoin and, consequently,
1+ε
1−ε
>
.
1+η
1−η

(23)

Stablecoins in period 0 trade at their expected value ps . Finally, we assume that the distributions of payoffs for the two assets and the stablecoin are independent.

4.2

DeFi Loans

The borrower now receives a loan (S, R) from the lender, where S is the size of the loan
in stablecoins and R is the promised repayment, also in stablecoins. The loan is executed
automatically by an atomic smart contract. Hence, the smart contract can avoid the fee ϕ
associated with a bank loan. If the borrower repays the loan, the smart contract returns
the collateral to the borrower. If the loan is not repaid, the smart contract automatically
liquidates the collateral, incurring the deadweight loss L (see Figure 4.1).17
It is always optimal for the borrower to convert the entirety of the loan into consumption
in period 0, then wait until period 1 to purchase the stablecoins to repay the loan. The
borrower defaults if and only if the obligation to repay the loan in stablecoins exceeds the
16

We assume that stablecoins, just like cash in CeFi, cannot be pledged as collateral in DeFi. In reality,

many major stablecoins (e.g., USDT, BUSD) are not accepted by DeFi lending platforms as collateral.
17
In principle, the liquidation costs for crypto assets can be different from L. It may be more costly to
liquidate crypto assets (for example, due to slippage when the asset is sold through an illiquid AMM) or less
costly (for example, due to the fungibility of some crypto assets).

19

value of the collateral, or
ps1 R > pc1 .

(24)

Figure 4.1: DeFi Loan Arrangement

Assume for the rest of this section that L is sufficiently large so that any default by the
borrower is suboptimal.18 Then the contract requires
R≤

1 − ε pc
1 + η ps

(25)

so that there is no default risk. Thus the optimal contract solves
max(1 + v)ps S + pc − ps R

(26)

S ≤ R.

(27)

S,R

subject to (25) and

18

The Appendix provides an analysis where it can be optimal for the DeFi loan to include some default

by the borrower, if L is sufficiently small. Interestingly, we also show there that DeFi optimally rules out
default whenever the stablecoin is not volatile (η = 0).

20

The objective function captures that the borrower swaps asset x for the crypto asset, receives
(1+v)Spc in consumption in period 0, and repays the obligations from the DeFi loan, pc1 −ps1 R,
in period 1. This yields
1 − ε pc
R=S=
1 + η ps
for the optimal DeFi loan, with the borrower’s payoff given by


1−ε
pc + pc .
v
1+η

(28)

(29)

Hence, we have the following result establishing that DeFi saves costs while preventing
default.

Proposition 4. Suppose L ≥ L̄ and pc = px . DeFi without default dominates CeFi if and
only if

ϕ≥

v
1+v



1−ε
(1 − δ) −
pc .
1+η

(30)

The optimal DeFi contract has the advantage of saving the banker’s fee ϕ. DeFi is thus
optimal as long as the cost ϕ of relying on a trusted third party to execute the lending
arrangement is large enough. This captures the promise of DeFi to reduce the cost of
lending. To the contrary, DeFi relies on crypto collateral and stablecoins for settlement that
are both potentially more volatile. If the cash price of traditional collateral is more volatile
than the cash price of crypto collateral, then DeFi dominates even if the banker’s fee were
zero. Finally, when the lending arrangement or collateral assets become more valuable, DeFi
becomes less attractive. This points to a role for DeFi for less important lending markets.
Consider now a situation where ϕ is too high so that CeFi is dominated by direct trading.
DeFi can still be better than direct trading since it can rule out default for the contracting
parties without using the banker. This is summarized in the following result.19
19

For L sufficiently small and η > 0, it can be the case that a DeFi contract with some default is best.

See the Appendix for the analysis.

21

Proposition 5. Suppose L < L̄ and px = pc . DeFi without default dominates direct trading
if and only if

L≥2

v
1+v



ε+η
1+η



px .

(31)

Hence, DeFi also provides value in that it expands intermediation to lending contracts that
did not rely on intermediation before due to the high costs of using formal lending contracts.
In this sense, DeFi also fosters financial inclusion.

4.3

The Limitations of DeFi

Volatility of Stablecoins and Crypto Collateral It is instructive to consider more
carefully the effect of volatility on DeFi. To do so, we can compare haircuts for DeFi and
CeFi lending. The haircut for a CeFi loan can be written as
δ+

ϕ
,
px

(32)

while the haircut for a DeFi loan without default is given by
1−

1−ε
Sps
=1−
.
c
p
1+η

(33)

The lower the haircut on the DeFi loan, the more likely it is to be preferable to the CeFi
loan. If returns on ordinary and crypto collateral are the same and the DeFi haircut is
smaller than the CeFi haircut, then the DeFi loan is guaranteed to be superior.
The haircut formula makes it clear that volatility of stablecoins and crypto collateral are
important factors for the value proposition of DeFi. In general, crypto assets seem to be
more volatile than traditional, real assets used for collateral, such as Treasury bills. As the
volatility of the crypto collateral increases, the DeFi contract becomes less attractive relative
to the bank loan.
For example, ETH, being the native token on Ethereum, is often used as collateral in DeFi
lending. ETH is substantially more volatile than traditional collateral assets, with its value

22

sometimes fluctuating by 25% within a day. Such volatility can result in wide spreads for
loans, liquidations and losses from lending.20

Incomplete Contracts While a smart contract allows for guaranteed execution, it may
have to be incomplete. Consider a situation where with probability (1 − θ) the borrower
loses the endowment eB in period 1. In this event, even if the borrower has an incentive to
repay the loan, they cannot do so, being unable to acquire the settlement asset. Note that
this information is “soft” in the sense that one needs to verify the circumstances why the
borrower does not repay the loan.
The optimal DeFi loan is now given by
max(1 + v)ps S + θ(pc − ps R)
S,R

(34)

subject to
ps S ≤ θps R + (1 − θ)(pc − L)
R≤

1 − ε pc
1 + η ps

with the solution yielding the following payoff for the borrower:


1−ε
v θ
+ (1 − θ) pc − (1 + v)(1 − θ)L + pc .
1+η

(35)
(36)

(37)

To the contrary, a CeFi contract allows the bank to condition on the “soft” information
and forgive the loan repayment in that contingency, while returning the collateral to the
borrower. This avoids the liquidation cost. Assuming the fee ϕ also includes the expected
costs of verifying the borrower’s condition, we obtain
max(1 + v)C + px − θ(R + ϕ)
C,R

(38)

subject to

20

C ≤ θR

(39)

R + ϕ ≤ (1 − δ)px

(40)

On February 23, 2021, a record-high $115 million in DeFi lending positions were wiped out following a

large price decline of ETH.

23

for the CeFi lending arrangement. Hence, the borrower’s payoff is given by
vθ(1 − δ)px − (1 + v)θϕ + px .

(41)

This yields the following result.

Corollary 6. Assume px = pc . When smart contracts are incomplete, DeFi is better than
CeFi if and only if

ϕ≥

v
1+v






1−ε
1−θ
v x
x
1−δ−
p +
L−
p .
1+η
θ
1+v

(42)

Relative to the original case, DeFi becomes less attractive whenever the last term is positive.
As the DeFi smart contract cannot replicate the state contingency of the bank loan, there
needs to be an advantage for it to become optimal. When stablecoins are stable (η → 0),
such an advantage can arise if the crypto collateral is less volatile and its liquidation costs
are sufficiently small. In reality, however, crypto assets tend to be more volatile than most
real assets that serve as collateral. Hence, as more contract flexibility is required, DeFi tends
to lose its value when L is relatively large.

The Oracle Problem DeFi contracts rely on price feeds that correctly specify the value
of the collateral and the stablecoin. To see the problems that can arise from mispricing,
consider an example where pc = px = ps = 1 and, in particular, the borrower is aware that
pc = 1, but the oracle misspecifies the price of the collateral as21
ρ > 1.

(43)

1−ε
The borrower can then take out a DeFi loan at R̃ = ρ 1+η
, which results in a lower haircut

than with the correct price. Assume now for simplicity that



1+ε
1+η
ρ>
1−ε
1−η
21

More generally, ρ represents a distortion of the relative price

24

pc
ps .

(44)

so that the DeFi loan defaults for sure. The borrower’s payoff is then given by


1−ε
(1 + v)ρ
,
1+η
but the surplus from lending is only given by


1−ε

− L,
1+η

(45)

(46)

the difference being the lender’s payoff.
Recall that, under the optimal CeFi arrangement, the borrower’s payoff is
v(1 − δ) − (1 + v)ϕ,

(47)

which is equivalent to total surplus. This implies that the borrower would prefer the DeFi
loan, even though this is not socially optimal whenever




1−ε
1−ε
(1 + v)ρ
> v(1 − δ) − (1 + v)ϕ > vρ
− L.
1+η
1+η

(48)

Since the first inequality always holds by assumption (44), we have the following result.
Corollary 7. Suppose L > (1 + v)ϕ. If the oracle overprices the crypto collateral, there is
inefficient adoption of DeFi.

When the crypto collateral is mispriced, the borrower would like to take advantage by using
a DeFi loan and then default on the loan. Since the lender bears the liquidation cost from
the unanticipated default, there is now a wedge between the borrower’s payoff and social
welfare. In the extreme case, where lending does not add much value (v → 0), the borrower
would still prefer DeFi lending, even though there would be no surplus generated at all from
lending per se. This is akin to the borrower arbitraging against the oracle.22
22

Such a situation occurred for example during the TerraUSD collapse in May 2022. As a result of the

extreme volatility in the price of LUNA tokens, the price feed used for DeFi smart contracts denominated
in the LUNA token was significantly higher than the actual market value of the token. Attackers exploited
the price discrepancy to obtain loans collateralized by an inflated LUNA token from the underlying Venus
protocol. This led to a loss of about $11.2 million for the protocol until the protocol increased the haircut
of LUNA from 45% to 100%, essentially stopping lending against LUNA collateral altogether.

25

Further Limitations—Externalities and Anonymity When there are many borrowers
and lenders paired on the DeFi platform, there can be an interdependency between individual
smart contracts via price externalities. Suppose the value of the crypto collateral depends on
the selling pressure in the market according to α(n)pc (s), where n is the fraction of collateral
being liquidated in the market.23
For α(0) = 1 and α′ (n) < 0, there is a fire sale externality. Given the period 0 belief
that n = 0, the individually optimal DeFi contract between a borrower and a lender sets
R =

1−ε
.
1+η

In period 1, however, if some borrowers switch their belief to n > 0, then all

borrowers are induced to default. This leads to a self-fulfilling fire sale equilibrium, which is
socially inefficient due to the liquidation cost L. Such a scenario is less likely to occur with
CeFi arrangements, as the banker may have an incentive to avoid liquidation of collateral,
if this were to compromise the fee.
Another limitation of DeFi is that—in principle—borrowers are anonymous on the platform.
This anonymity tends to arise from the fact that individuals can assume several identities
on DeFi platforms. Hence, rationing of credit is limited because individual borrowers cannot
be uniquely identified.
Interestingly, this also precludes the use of more complicated, nonlinear contracts. The
reason is that these would give rise to arbitrage where borrowers can achieve better outcomes
by taking out smaller, but more individual, loans. Hence, DeFi contracts tend to be linear
in their pricing.

5

Building a Proper Infrastructure for DeFi

Some of the limitations of DeFi could be alleviated by building a proper third-party-provided
infrastructure. An institution—possibly public—that took on certain tasks could improve
the functionality of DeFi applications. This is ironic, since DeFi is supposed to be built on
a fully decentralized infrastructure. The first of such tasks is to improve the quality of the
assets used in DeFi, both stablecoins and collateral assets. For stablecoins, the obvious issue
23

This price externality is particularly likely to occur with an illiquid pool on a decentralized exchange.

26

is to ensure their stability. One possibility is for a central bank to issue a CBDC that can be
tokenized and transacted on public blockchains. DeFi applications would then have access
to a standardized, riskless settlement asset.
Alternatively, central banks could simply ensure that the issuers of stablecoins have access to
their balance sheets. As a consequence, stablecoins could operate as a narrow bank where the
issued coins are fully backed by deposits at the central bank. This would clearly remove any
ambiguity with respect to the backing of a coin. But it would also foster private innovation to
create tokenizable and programmable stablecoins. Prudential regulation would be necessary,
since DeFi could otherwise be used as a means for regulatory arbitrage without adding any
value per se.
There are also opportunities to improve the quality of the collateral that can be used by
DeFi applications. One way to foster the adoption of DeFi is to tokenize standard collateral
such as government-issued securities. Developing this capacity is a public good, which in
turn could foster incentives to tokenize private assets within the same infrastructure.
Finally, the provision of oracles, like the provision of any other information, is a classic
public good. For some oracles, a public institution with little incentive to manipulate the
information could prove a more trustworthy guarantor of quality; for other oracles, a private
institution might be the best provider.
In short, DeFi holds the promise to make financial arrangements more efficient and more
inclusive. A natural way to lever these promises into concrete success is by building a proper
infrastructure to support innovative, private applications. Such infrastructure, however, is
likely to rely on some designated third party, making the notion of a fully decentralized
system somewhat of an illusion (Aramonte et al., 2021).

References
Aoyagi, J. and Y. Ito (2021) “Coexisting Exchange Platforms: Limit Order Books and
Automated Market Makers”, SSRN: https://ssrn.com/abstract=3808755

27

Aramonte, S., W. Huang and A. Schrimpf (2021) “DeFi Risks and the Decentralisation
Illusion”, BIS Quarterly Review, December.
Bakos, Y. and H. Halaburda (2021) “Blockchains, Smart Contracts and Connected Sensors:
Substitutes or Complements?”, NYU Stern School of Business, August 1.
Capponi, A. and R. Jia (2021) “The Adoption of Blockchain-based Decentralized Exchanges”,
arXiv preprint arXiv:2103.08842
Chiu, J. and T. V. Koeppl (2019) “Blockchain-based Settlement for Asset Trading”, Review
of Economic Studies, 32, pp. 1716–1753
Chiu, J., E. Ozdenoren, K. Yuan and S. Zhang (2022) “On the Inherent Fragility of DeFi
Lending”, Manuscript
Cong, L. W. and Z. He (2019) “Blockchain Disruption and Smart Contracts”, The Review
of Financial Studies, 32, pp. 1754–1797
d’Avernas, A., T. Bourany and Q. Vandeweyer (2021) “Are Stablecoins Stable?”, Manuscript
Harvey, C. R., A. Ramachandran and J. Santoro (2021) DeFi and the Future of Finance,
New York: John Wiley & Sons
Kozhan, R. and G. F. Viswanath-Natraj (2021) “Decentralized Stablecoins and Collateral
Risk”, WBS Finance Group Research Paper
Lee, M., Martin, A. and R. M. Townsend (2021) “Optimal Design of Tokenized Markets”,
SSRN: http://dx.doi.org/10.2139/ssrn.3820973.
Lehar, A. and C. A. Parlour (2021) “Decentralized Exchanges”, SSRN: http://dx.doi.org/
10.2139/ssrn.3905316
Lehar, A. and C. A. Parlour (2022) “Systemic Fragility in Decentralized Markets”, SSRN:
https://ssrn.com/abstract=4164833
Li, Y. and S. Mayer (2021) “Money Creation in Decentralized Finance: A Dynamic Model
of Stablecoin and Crypto Shadow Banking”, SSRN: https://ssrn.com/abstract=3757083
Park, A. (2021) “The Conceptual Flaws of Constant Product Automated Market Making”,
28

SSRN: https://ssrn.com/abstract=3805750
Schär, F. (2021) “Decentralized Finance: On Blockchain-and Smart Contract-based Financial Markets”, Federal Reserve Bank of St. Louis Review vol. 103(2), pages 153–174, April.
Szabo, N. (1996) “Smart Contracts: Building Blocks for Digital Markets” EXTROPY: The
Journal of Transhumanist Thought, 16 18.2:28

A

Proof of Lemma 1

Note first that, by assumption, px (h) − L > px (ℓ).
Consider R ∈ [px (ℓ) − L, px (ℓ)]. The borrower then solves
1
max (1 + v)C + (px (ℓ) − R)
C,R
2

(49)

subject to
1
1
C ≤ R + (px (h) − L)
2
2

(50)

since the lender keeps the collateral in the high state. After substituting the constraint
with equality, the objective function is increasing in R. Hence, we have that R = px (ℓ) and
C = px − L/2.
Next, consider R ∈ [px (h) − L, px (h)]. The borrower’s problem is now given by
1
max (1 + v)C + (px (h) − R)
C,R
2

(51)

subject to
1
1
C ≤ R + (px (ℓ) − L)
2
2

(52)

since the borrower defaults in state ℓ. The solution is R = px (h), and again we have
C = px − L/2.
For all other cases, the collateral is always liquidated. Hence, the outcome is identical to a
spot sale, and thus dominated by this arrangement.

29

B

Optimal DeFi Contracts with Default

For clarity, we focus on the case where pc = ps = 1. The results are unaffected by this
assumption. When R >

1+ε
,
1−η

the borrower always defaults. This is equivalent to a spot

trade.
1+ε 1+ε
, 1−η ]. Then, the borrower defaults unless the crypto collateral has a
Consider now R ∈ ( 1+η

high value and the stablecoin has a low value. The borrower’s problem is then given by
1
max(1 + v)S + (1 + ε − (1 − η)R)
S,R
4

(53)

subject to
1
1
1
S ≤ (1 − η)R + ((1 − ε) − L) + ((1 + ε − L).
4
2
4

(54)

The borrower’s payoff from the optimal contract is then
3
v − (1 + v) L
4

(55)

which is worse than direct lending.
1−ε 1+ε
Consider next R ∈ ( 1−η
, 1+η ]. Then, there is default whenever the crypto collateral has a

low value. Thus, the borrower solves
1
max(1 + v)S + ((1 + ε) − R)
S,R
2

(56)

subject to
1
1
S ≤ R + ((1 − ε) − L)
2
2
so that the payoff from the contract is given by


L
L
v 1+ε
+ (1 − ε) − (1 + v) < v − (1 + v) .
2 1+η
2
2

(57)

(58)

Hence, the contract is again dominated by direct lending.
1−ε 1−ε
Consider then a repayment R ∈ ( 1+η
, 1−η ]. Since there is default only when the crypto

30

collateral has a low value, but the stablecoin has a high value, the optimal contract solves
1
1
max(1 + v)S + ((1 + ε) − R) + ((1 − ε) − (1 − η)R)
S,R
2
4

(59)

subject to
1
1
1
S ≤ R + (1 − η)R + ((1 − ε) − L).
2
4
4
Hence, the borrower’s payoff from such a contract is given by



η 1 − ε
L
v 1−
− (1 + v) .
2
1−η
4

(60)

(61)

The borrower can thus reduce the default probability and potentially obtain a cheaper loan
when the stablecoin has a lower value in period 1.
Finally, when R ≤

1−ε
,
1+η

there is no default. This is the case analyzed in the main body of

the paper. We can summarize our result as follows.

Lemma 8. The optimal DeFi contract is given by no liquidation and
R=S=

1−ε
1+η

(62)

if and only if
L
v

(1 − ε)
2
1+v



η(2 − η)
1 − η2


.

(63)

Otherwise, the optimal contract includes some liquidation and satisfies
1−ε
1−η



L
η 1 − ε
S = 1−
− .
2
1−η
4

R=

(64)
(65)

When liquidation costs are low, the optimal DeFi contract includes default in the worst
possible state, i.e., when the value of the crypto collateral is low and the stablecoin’s value is
high. If there is no difference in the volatility of the two collateral assets, the DeFi contract
with default always dominates direct lending. The reason is simply that the haircut is smaller
due to the lower likelihood of default.
31

If the liquidation costs are large, the optimal DeFi contract does not include default. More
interestingly, however, whenever the stablecoin has a small enough volatility (η → 0), the
optimal DeFi contract never allows for default.
These results are somewhat an artifact of our model. First, contracts are not state contingent.
However, a DeFi contract with default can increase the state contingency relative to direct
lending. Second, we assume risk neutrality and iid shocks to crypto collateral as well as
stablecoins. The advantage of DeFi with some default may disappear if we relax these
assumptions.