Macroeconmics at DKU Fall 2020

1. A Macroeconomic Model
for Liquity
Presented by William Zhao
Instructed by Dr. Luyao Zhang and Dr. Yulin Liu
Source: https://www.liquity.org/

2. Cryptocurrency
● Bitcoin, Ether etc.
● Pros: decentralized, liquid,
convenient…
● Cons: unstable, safety concerns,
environmental concerns…

3. Values of Cryptocurrencies are Unstable
Ether Price

4. Values of Cryptocurrencies are Unstable
Ether Price

5. Why are they unstable?
● What does stability mean?
● Exchange Rate with US Dollars/
Asset Price in US Dollars

6. Pricing Model

7. Supply
● Fixed
● No room for
monetary policy

8. Demand
● Where does the demand come from?
● All the illegal transactions (drugs, weapons, money laundering…);
● Investment in the currency and the financial assets in the currency;
● The Ether-based economic activity (smart contracts).
● Still, highly unstable demand

9. Why are they unstable?
● Compared to traditional sovereign-based currencies?
● Government enforcement ensures stable money demand.
● Monetary policies avoids deflation and controls inflation.

10. Stable Coins
● Stable coins are cryptocurrencies
designed to minimize the
volatility of the price of the stable
coin, relative to some "stable"
asset or basket of assets.
● e.g. USDT, AMPL
Price of AMPL

11. Introduction
● Liquity is a decentralized protocol
that allows Ether (ETH) holder to
obtain maximize liquidity against
their protocol without paying
interest.

12. Money Supply
● In real world, money
supply can be seen from
the size of the Central
Bank balance sheets.

13. Troves
● A borrower may obtain LUSD after depositing ETHs into a trove.
● A borrower can obtain in the form of LUSD coins at maximum
90.91% of the current dollar value of a trove, to maintain a collateral
ratio of 110%.
● An issuance fee is charged in the process.

14. Collateral Ratio
● Collateral Ratio is the ratio of value of Ether in the trove over the
quantity of LUSD tokens withdrawn.
● MCR = 110%, but it is recommended to maintain a CR > 150%.
𝐶𝑅 =
𝑃𝑒 𝑄 𝑒
𝑄 𝑑

15. Troves
● LUSDs are “printed” or created from the troves.
● Decentralized, permissionless, collateralized
● Total Supply = # of LUSDs withdrawn from all troves.

16. Money Demand
● Cash – Transactions – Liquidity Pool
● Savings – Stability Pool
● Total Demand = Liquidity Pool + Stability Pool

17. Stability Pool
● LUSD holders can deposit their LUSD tokens into the stability pool and
earn profits from doing so.

18. Trove Liquidation
● Troves will be liquidated if CR < 110% automatically and promptly.
● The LUSD tokens in the stability pool are used to pay back the debt
in the defaulted trove, and the ETH in the trove was transferred to
the stability pool and shared by all investors in the stability pool.

19. Trove Redemption
● Any LUSD holder
can redeem LUSD
at a price of $1.
● Creates a soft
price floor of (1-
redemption fee)

20. Trove Redemption
● Ex: LUSD currents trades at $0.95, current redemption fee is 1.9%.
How can an arbitrageur make risk-free profits?
● First, buy 150,000 tokens from the market with 150,000*$0.95 =
$142,500.
● Then, redeem the tokens and receive 150,000*(1-0.019) = $147117
worth of ETH.
● Finally, sell the ETH and earn $4617 of risk-free profits.

21. More Details
● Issuance fee and redemption fee as tools of monetary policy
● LQTY tokens backed by the revenues of fees
● “Recovery Mode”

22. Modeling and Simulation
● How can we verify the validity of the mechanism before launching the
project?
● Adjustments are not possible to make afterwards!

23. Structure
● Exogenous Factors: Ether price, natural rate, PE ratios…
● Model for money supply: simulation of liquidating, closing, opening, and
redeeming troves.
● Model for money demand: simulation of liquidity and stability pool.
● Supply = Demand, solve for LUSD price

24. Exogenous Factors
𝑃𝑡
𝑒
= 𝑃𝑡−1
𝑒
(1 + 𝜁𝑡
𝑒
)(1 + 𝜎𝑡
𝑒
)
𝜁𝑡
𝑒
∼ 𝑁 0, 0.01 , 𝜎𝑡
𝑒
= 0.0001
𝑃0
𝑒
= 200
𝐸(𝑃8760
𝑒
) = 𝐸(𝑃𝑡−1
𝑒
) 1 + 𝜎𝑡
𝑒
= 200 ∗ 1.0001 8760
= 480.23

25. Troves
● A data frame that
records information
of all existing troves

26. Troves
● At each period, undercollateralized troves are liquidated.
● Then, people might come to close, adjust, open, or redeem troves.
● A behavioral model for each step.

27. Opening Troves
● How many new troves are opened?
𝑁𝑡
𝑜
= Nt−1
o
Pt−1
l
Rt
o
− ft
i
𝛼
(1 + 𝜁𝑡
𝑜
)
𝜁𝑡
𝑒
∼ 𝑁 0, 0.01 , 𝑁0
𝑜
= 10

28. Opening Troves
● What’s the size of each trove?
𝑄𝑡
𝑑
𝑖 =
𝑃𝑡
𝑒
𝑄𝑡
𝑒
𝑖
𝐶𝑅∗(𝑖)
𝑄𝑡
𝑒
𝑖 ∼ 𝜒10
2
, 𝐸 𝑄𝑡
𝑒
𝑖 = 10, 𝑉𝑎𝑟 𝑄𝑡
𝑒
𝑖 = 20
𝐶𝑅∗
𝑖 ∼ 1.1 + 0.38𝜒5
2
, 𝐸 𝑄𝑡
𝑒
𝑖 = 300%, 𝑉𝑎𝑟 𝑄𝑡
𝑒
𝑖 = 120%

29. Money Supply

30. Stability Pool and Liquidity Pool
𝐷𝑡
𝑠
= 𝐷𝑡−1
𝑠
1 + 𝜁𝑡
𝑠
1 + 𝑅𝑡−1
𝑠
− 𝑅𝑡
𝑛 𝜃
● R^s is the return in stability
pool in the previous period,
which includes airdrop gain
and liquidation gain.
● R^n is natural rate.
𝐷𝑡
𝑙
= 𝐷𝑡−1
𝑙
1 + 𝜁𝑡
𝑙
𝑃𝑡
𝑙
𝑃𝑡−1
𝑙
𝛿
● P^l is the price of LUSD token
in the period.

31. Stability Pool and Liquidity Pool

32. LUSD Price
𝑆𝑡 = 𝐷𝑡
𝑙
+ 𝐷𝑡
𝑠
𝑆𝑡 − 𝐷𝑡
𝑠
= 𝐷𝑡
𝑙
= 𝐷𝑡−1
𝑙
1 + 𝜁𝑡
𝑙 𝑃𝑡
𝑙
𝑃𝑡−1
𝑙
𝛿
𝑃𝑡
𝑙
= 𝑃𝑡−1
𝑙
𝑆𝑡 − 𝐷𝑡
𝑠
𝐷𝑡−1
𝑙
1 + 𝜁𝑡
𝑙
1/𝛿

33. LUSD Price

34. Discussion
● The model survives 8760 repetitions.
● The model is consistent with the economic intuitions.
● The model considers the random factors in the real world.
● The model shows that LUSD price will remain stable.
● The model ignores the complexity of people’s behaviors. The behavioral
models used are not well verified.

35. Thank you!
● Questions?