**Author**: Eray Sabancilar

**Last updated**: 15.04.2020

# A Microeconomic Model for Parachains with Native Tokens

## Motivation

It is crucial for the prospective parachain networks that are planned to be deployed on Polkadot to achieve economic prosperity as well as to become decentralized, functional and competitive. In this note, we provide an agent based dynamic microeconomic model of a parachain with a native token in order to study the time evolution of its number of clients, revenues, expenses, token price and wealth distribution given the DOT price and the parachain cost and deposit.

## 1. Introduction

In this note, we first discuss various token designs that may be adapted by projects with different objectives. Then, we point out the importance of the initial token distribution on the economic success as well as decentralization. We introduce potential agents that may contribute to the token economy.

We then choose a simple token model with a few agents, namely, clients, collators and a treasury. We define the dynamic variables that we will use in the model.

We next introduce a simple money demand model based on Baumol–Tobin to calculate the token and DOT prices based on the interest & inflation rates, daily total payouts (incomes) and money supplies of the token and NPoS economies. In fact, we also use this pricing model to have an estimate of the parachain slot cost, $\mathcal{C}_p$, and deposit, $\mathcal{D}_p$. We stress that the parachain slot cost & deposit is determined via a candle auction as highlighted in the [Parachain Auction](https://research.web3.foundation/en/latest/polkadot/Parachain-Allocation.html) note. Therefore, the input parameters, $C_p$ and $D_p$ are to be replaced either with values obtained from a candle auction simulation or with realized values from the actual parachain auctions.

We finally set the initial conditions and design a utility function based demand model for the number of clients. We conclude with the results of our simulation.

## 2. Token Design

**Token Type**
There are many possibilities for a choice of a token type depending on the needs of the parachain network and its business model. A parachain project may desing its token with one or more of the following token features to name a few (see e.g., Ref. [1] for a comprehensive review):
- Utility
- Asset-Backed
- Spendable
- Tradable
- Fixed
- Schedule Based
- Inflationary
- Burnable
- Voting
- Payment

In what follows, we will consider a native utility token that is spendable, tradable and with schedule-based issuance to capture the generic features that are expected to be desired in most of the prospective parachain projects.

**Initial Token Distribution**
It is also important to setup the initial token distribution optimally so that the regulatory, strategic and financial needs of the parachain network are achieved. Tokens may be shared among different token holders as follows (see e.g., Refs. [2-5]):
- Private Investors
- Crowdfunding ICO Participants
- Reserved for Treasury
- Reserved for Initial Parachain Offering
- Founders & Employees
- Rewards to Service Providers/Collators and Investors
- Reserved for Ecosystem Funding
- Future Initial Exchange Offering
- Reserved for Future Fundraising

## 3. Agents

The agents that play a role in the parachain token economy may include (see also Refs. [6-8]):
- Service Providers
- Collators
- Liquidity Providers
- Staker/Lenders
- Experts/Consultants/Oracles (Data Provider)
- Users/clients
- Founders
- Developers
- Investors
- Treasury

In this version of the model, we only consider collators, clients, investors and a treasury.

## 4. Variables

We use various variables to keep track of the parachain token economy. In particular, we use the daily time series of the global NPoS variables as inputs:

- DOT Interest Rate: $r_{DOT}(t)$
- DOT Inflation Rate: $I_{DOT}(t)$
- DOT Money Supply: $M_{DOT}(t)$
- NPoS Payouts (Incomes): $Y_{DOT}(t)$
- DOT Price: $P_{DOT/USD}$
- Parachain Deposit and Cost: $\mathcal{D}_p(t), \mathcal{C}_p(t)$

in order to study the dynamics of the parachain token economy variables, where we denote the unit of a native token as TKN:

- Number of Clients: $N_{clients}(t)$
- Revenues
- Service Fees (TKN)
- Expenses
- Parachain Slot Cost (DOT)
- Collator Fees (TKN)
- Profits = Revenues - Expenses
- Profit Distribution Among the Agents
- Collators (TKN)
- Treasury (TKN)
- Token Accounts:
- Collator Wealth (TKN)
- Client Wealth (TKN)
- Rewards Reserve (TKN)
- Treasury reserves:
- Token Reserve (TKN)
- DOT Reserve (DOT)
- Token Price: $P_{TKN/USD}(t)$
- Opportunity Cost of Capital: $r_{opp}$

## 5. Initial Conditions

We set the initial conditions of the token economy model as follows.

- Initial Project Valuation: $10 M
- Initial Token Supply: 1 M
- Initial Token Price: $10
- Initial Token Distribution
- Private Investors: 10%
- Crowdfunding ICO Participants (30%)
- Founders & Employee (20%)
- Rewards to Collators (10%)
- Reserved for Ecosystem Funding (15%)
- Reserved for Future Fundraising (15%)

We assume that all the proceeds from the private investors and the crowdfunding ICO participants first get converted into DOTs and then goes to the treasury DOT reserve. This account is used as the reserve from which parachain deposit is withdrawn. We also assume that all the collators and active and potential clients come from the Crowdfunding ICO Participants accounts.

## 6. NPoS Economics
We use global variables that we obtain from the NPoS economics model as an input for calculating the DOT price process, the parachain slot cost and the deposit amount. In particular, we use the nominal interest rate, $r_{DOT}(t)$, inflation rate, $I_{DOT}(t)$, daily DOT payouts, $Y_{DOT}(t)$ and the DOT money supply, $M_{DOT}(t)$ as inputs.

### 6.1. DOT Price

We model the DOT price using the DOT money supply, $M_{DOT}(t)$, and a liquidity demand $L_{DOT}(Y_{DOT}(t), r_{DOT}(t))$ that depends on the the total DOT payouts, $Y_{DOT}(t)$, and the nominal interest rate, $r_{DOT}(t)$.

**Baumol-Tobin Money Demand Model**

The Baumol-Tobin money demand model is based on the tradeoff between the opportunity cost of not incurring an interest and holding money in a liquid form for spending. The optimal spending is achived when the cost of spending is minimized (see e.g., Refs. [9,10]).

Consider an agent with income $Y_{DOT}$ kept in an account yielding interest $r_{DOT}$ (this is analogous to nominators staking in Polkadot to earn interest on their DOTs). If this agent wants to split his DOT spending in $\mathcal{N}$ transactions, there are two costs associated with spending them: i) Transaction cost, $C_{DOT}\mathcal{N}$, where $C_{DOT}$ is the transaction fee per transfer of illiquid (staked) funds into a liquid account, ii) Opportunity cost of capital due to not collecting interest is $r_{DOT}~m_{DOT}$, where $m_{DOT} = Y/\mathcal{N}$ is the amount of average liquid DOTs. We can define the Lagrangian cost function, $\mathcal{L}(\mathcal{N})$, as
\begin{align}
\mathcal{L}(\mathcal{N}) = C_{DOT}\mathcal{N} + \frac{r_{DOT}~Y_{DOT}}{\mathcal{N}}.
\end{align}
Assuming that the agent minimizes its total cost, we have the following Lagrange equation giving us the optimal number of spending, $\mathcal{N}$:
\begin{align}
\frac{\partial \mathcal{L}}{\partial \mathcal{N}} = C_{DOT} - \frac{r_{DOT}~Y_{DOT}}{2\mathcal{N}^2} = 0 \iff \mathcal{N} = \sqrt{\frac{r_{DOT}~Y_{DOT}}{2 C_{DOT}}}
\end{align}
Thus, the optimum amount of money held in a liquid account, or the liquidity demand is
\begin{align}
L_{DOT}(Y_{DOT}, r_{DOT}) \equiv m_{DOT} = \frac{Y_{DOT}}{\mathcal{N}} = \sqrt{\frac{C_{DOT}~Y_{DOT}}{2~r_{DOT}}}
\end{align}

**DOT Price**
Assuming that the money supply and demand are in equilibrium, the DOT price in USD is given by
\begin{align}
P_{DOT}(t) = \frac{N_{DOT/USD}(t)~M_{DOT}(t)}{L_{DOT}(Y_{DOT}(t), r_{DOT}(t))},
\end{align}
where $N_{DOT/USD}(t)$ is the normalization of the price from the token exchange market, $L_{DOT}(Y_{DOT}(t), r_{DOT}(t)) = \sqrt{\frac{C_{DOT}~Y_{DOT}}{2~r_{DOT}}}$ is the money demand that we calculated using the Baumol-Tobin model. In what follows, we assume for simplicity that $N_{DOT/USD}(t)$ is constant, which we will relax in the upcoming versions of our DOT price model with a stochastic process.

### 6.2. Parachain Slot Deposit and Cost

Ideally we want to have actual values of the parachain deposits from a candle auction or at least a simulation of it since the parachain slot deposits in Polkadot are determined via a candle auction (for a detailed description, see [Parachain Auction](https://research.web3.foundation/en/latest/polkadot/Parachain-Allocation.html)). Nevertheless, we can use the demand based simple pricing model we discussed above in order to get an idea about possible values of parachain deposits and costs for a given period of time.

In Polkadot, a parachain slot cost is paid indirectly via inflation by locking a deposit $D_{\tau}$ for a period of $\tau$ days and receiving the nominal amount back at the end of that period. The cost of a parachain slot for this period is
\begin{align}
\mathcal{C}_\tau =\left[\prod_{t=1}^{\tau}(I_{DOT}(t)+1) - 1 \right] D_\tau.
\end{align}

Thus, the required deposit amount is simply
\begin{align}
D_\tau = \frac{\mathcal{C}_\tau}{\left[\prod_{t=1}^{\tau}(I_{DOT}(t)+1) - 1 \right]}.
\end{align}

We can estimate the cost $\mathcal{C}_\tau$ assuming that the validator payouts are partially (or completely) financed by parachain slots, i.e.,
\begin{align}
\mathcal{Y}_{DOT}^{\tau} = f_p \sum_{t=1}^{\tau} \frac{Y_{DOT}(t)}{\prod_{i=1}^{t}(I_{DOT}(i)+1)} = N_\tau~\mathcal{C}_\tau,
\end{align}

where $f_p \in (0,1]$ is the fraction of the amount of payouts to be funded by parachain slot payments, $N_\tau$ is the number of parachain slots to be auctioned in a period $\tau$. Thus, we can estimate the deposit amount as
\begin{align}
D_\tau = \frac{f_p}{N_\tau} \frac{\sum_{t=1}^{\tau} \frac{Y_{DOT}(t)}{\prod_{i=1}^{t}(I_{DOT}(i)+1)}}{\left[\prod_{t=1}^{\tau}(I_{DOT}(t)+1) - 1 \right]}.
\end{align}

We stress that this part of our model can easily be replaced with real time auction data or a simulation providing the values of $D_p(t)$ and $C_p(t)$, and does not affect the rest of the microeconomic model.

## 7. Token Dynamics

We use a utility function based decision mechanism for the growth of number of clients in the network. We describe how collators are incentived via collecting fees, getting shares of the profits and daily bonuses from the rewards reserve account. We also explain how tokens are relaesed from the rewards reserve account as well as how the treasury token reserve gets funded from profit shares. We use the Baumol-Tobin model to determine the token price similar to the DOT price described in Sec. 6.1.

### 7.1. Token Price

We use the Baumol-Tobin model for the token price dynamics as described in Sec. 6.1, where we replace the DOT variables with the TKN variables. We stress that at a given time the money supply is the total amount of liquid TKNs, namely the sum of the token treasury and all collator, client accounts, and the total income is the total the revenues of the network. Here we assume that the value creating in the network only occurs from providing services to the clients, and hence, the revenues will serve as the basis of the aggregate income of the network à la Gross Domestic Product (GDP) in macroeconomic terms.

### 7.2. Number of Clients

We use the Cobb-Douglas utility function to model the utility that a client receives by using the parachain network
\begin{align}
U_{DC}(N_{t}^{client},N_{t}^{collator}) = (1+N_{t}^{client})^a (N_{t}^{collator})^b.
\end{align}
Note that when the number of clients and collators increase, the client utility increases. Since we need at least a few collators to propose blocks in order to record these transactions, and we do not expect the clients to get a large utility incerase for more collators, we set $b < a$.

We also assume that if the service fees in USD increase, a client's utility decreases, which we capture with the utility function
\begin{align}
U_{P}(P_{TKN/USD}(t),ServiceFee_t) = \alpha~ ServiceFee_t (P_{TKN/USD}(t-1) - P_{TKN/USD}(t)).
\end{align}

The total utility a client receives from the network is then
\begin{align}
U = U_{DC} + U_{P}.
\end{align}

In order to capture the dynamics of the demand for services, we model the time evolution of the number of clients as
\begin{align}
N_{t}^{client} = N_{t-1}^{client} + \beta~ \left[U(N_{t-1}^{client},N_{t-1}^{collator}) - U_{benchmark}\right],
\end{align}
where $\beta$ is the coefficient of adjustment, and $N_{t}^{client}$ is the number of clients at time $t$. The demand grows if clients receive utility larger than a given benchmark $U_{benchmark}$, which we leave as a parameter in our model.

We assume that both collators and clients come from the initial crowdfunding participant accounts. We fix the number of collators in the network for simplicity and assum that the remaining accounts are potential clients, a subset of which is chosen as the initial clients.

### 7.3. Revenues, Costs and Profits
We assume that the revenues only come from fees charged to the clients as they use the network:
\begin{align}
Revenues_t = N_{t-1}^{clients}~ ServiceFee_t,
\end{align}
and there are two sources of costs for running the network: i) Cost of parachain slots, ii) Collator fees. However, the cost of parachain slots are charged to the treasury DOT reserve and the remaning costs in TKN are given by
\begin{align}
Costs_t = N_{t-1}^{clients}~CollatorFee_t.
\end{align}
In what follows, we set
\begin{align}
ServiceFee_t = f_{service}~CollatorFee_t,
\end{align}
where $f_{service} >1$ and $CollatorFee_t$ is a constant.

The profits are calculated simply as
\begin{align}
Profits_t = Revenues_t - Costs_t.
\end{align}

Then, the collators receive a fraction, $f_{profit-collator} \in (0,1)$, of the total profits and the remaining profits go to the token treasury account.

### 7.4. Rewards and Treasury Reserves
In order to incentivize collators in the beginning of the network, reserve tokens from the rewards account are slowly distributed to the collator everyday as a bonus. Assuming all the funds will be distributed within a period T, a collator receives the bonus
\begin{align}
Bonus_t = f_{rewards} \frac{w_t^{rewards}}{T},
\end{align}
where $f_{rewards} \in (0,1)$ is the fraction of funds to be distributed in period t, and $T$ is the period of the distribution.

The treasury token account starts out as zero and slowly accumulates a share of the profits:
\begin{align}
w_t^{treasury-TKN} = w_{t-1}^{treasury-TKN} + (1-f_{profit-collator})~Profits_t.
\end{align}

## 8. Research Questions
Some of the questions we want to answer with this version of the microeconomic model of parachains are as follows.
1. Is the business model sustainable?
- Do the number of clients and the demand on the services increase over time?
- Do profits remain positive?
- Are the service and collator fees low and competitive?
- Are collators incentived enough to provide the services?
- Is collator wealth increased over time?
2. How decentralized the network is in the beginning and how does it evolve?
- What is the time evolution of the Gini coefficient for the wealth of collators and the treasury?
3. Is the token price stable and growing over time along with the number of clients?

## 9. Parameters

In our simulation, we use the following set of parameter values.

| Parameters | Values |
| -------- | -------- |
| Era | $t=$ 1 day |
| Total Time | $T=$ 10 years |
| Baumol–Tobin Parameter for DOT | $c_{DOT}=$ 1 |
| DOT Price Normalization | $N_{DOT/USD}$ = 35 |
| Fraction of Total Payouts Financed by Parachains |$f_p=$ 0.3 |
| Number of parachains| $N_\tau$ = 100 |
| Initial Project Valuation (USD) | $V_0$ = \$10 M
| Initial Token Supply | $N_{tkn}$ = 1 M|
| Initial Token Price (USD)| $P_{tkn}$ = \$10|
| Number of Agents | $N_{private} = 10$, $N_{ico}$ = 3000, $N_{founders} = 20$, $N_{ecosystem} = 50$, $N_{rewards} = 1$, $N_{future-fundraising}$ = 1, $N_{treasury}$ = 2, $N_{collators}$ = 200|
| Initial Number of Clients |$N_{initial-clients}$ = 200|
| Initial Token Distribution | Private Investors (10%), ICO (30%), Founders & Employee (20%), Rewards to Collators (10%), Reserved for Ecosystem Funding (15%), Reserved for Future Fundraising (15%)|
| Collator Fee (TKN)| 0.02|
| Service fee in terms of the collator fee |$f_{service}$ = 2|
| Profit share for collators | $f_{profit-collator}$ = 0.5|
| Reward factor for collators | $f_{rewards}$ = 0.1 |
| Douglas-Cobb Utility Parameters | $a$ = 0.8, $b$ = 0.2|
| Bechmark Utility| $U_{benchmark}$ = 10|
| Transaction demand adjustment parameter| $\beta$ = 0.01|
| Annual opportunity cost of capital| $r_{opp}$ = 0.2
| Baumol–Tobin Parameter for TKN | $c_{tkn}$ = 0.1|
**Table 2:** Parameters of the microeonomic model for parachains with native tokens.

## 10. Simulation

A python jupyter notebook implementation of the model can be found [here](https://github.com/w3f/NPoS-Economics/blob/master/Codes/3%20Parachain%20Economics/ParachainEconomicsV3.ipynb).

| | 6 Months | 12 Months | 18 Months | 24 Months |
| -------- | -------- | -------- | -------- | -------- |
| Parachain Cost (DOT) | 496.4 | 976.9 | 1,440.2 | 1,888.5 |
|Parachain Cost (USD) | 55,196.6 | 112,337.1 | 172,223.7 | 232,557.7 |
|Parachain Deposit (DOT) | 9,814.7 | 9,478.8 | 9,114.9 | 8,758.2 |
|Parachain Deposit (USD) | 1,091,386.3 | 1,089,981.9 | 1,089,976.6 | 1,078,531.9 |
**Table 1:** The Parachain Deposit and Cost in DOTs and USD for the specified leasing periods.

**Figure 1:** The time series evolution of the DOT Money Supply.

**Figure 2:** The time series evolution of the DOT Price in USD.

**Figure 3:** The time series evolution of the Daily Real Income in 1,000 USD.

**Figure 4:** The time series evolution of the Number of Clients.

**Figure 5:** The time series evolution of the Revenues and Costs.

**Figure 6:** The time series evolution of the Token Price in USD.

**Figure 7:** The time series evolution of the nominal Wealth of various agents.

**Figure 8:** The time series evolution of the Gini Coefficient for wealth inequality.

## 11. Results
We summarize our main results below:
1. Number of clients, and hence revenues increase over time.
2. Revenues exceed expenses, hence the network remains profitable.
3. Collators are well incentivized as they receive three sources of income: i) collator fees, ii) shares of the profits, iii) rewards paid out from the initial reserve. Their wealth increase in time.
4. Rewards reserve account gradually distributes its funds to collators over time in order to keep collators incentivized.
5. Token treasury receives the remaining profits and accumulates wealth over time. This can be used for funding the development and required infrastructure services.
6. DOT treasury is initially funded by the private and ico investors, and it maintains a non-zero balance over time. Hence, the parachain slot remains affordable by the network.
7. Token price initially drops reflecting the fact that only a fraction of the total token supply is liquid in the beginning since only ICO participants holding 30% of the initial token supply are assumed to contribute to the token economy by becoming clients and collators. We assume that there are 200 clients initially. The number of clients dynamically increases as a client's utility increases. Hence, the liquid token supply and demand increse as well. Therefore, increasing number of clients and demand on services is reflected on the increasing token price.
8. Finally, we observe that the Gini coefficient for the token accounts excluding clients decreases substantially over time. Therefore, the liquid part of the token supply is getting more decentralized in time. However, we note that in this version of the model, the remaning token holdings are assumed to be passive. As they start contributing to the netwerk activity, we expect more decentralization and higher token price to be achieved over time.

### References

Token Design
[1] https://www.zora.uzh.ch/id/eprint/157908/

Initial Token Distribution and Token Sale Structure
[2] https://medium.com/@plaurent789/fair-initial-token-distribution-for-optimal-decentralization-8363e83173ff
[3] https://blog.coinbase.com/the-perfect-token-sale-structure-63c169789491

ICO Structure
[4] https://blog.coinbase.com/the-perfect-token-sale-structure-63c169789491

ICO Investors
[5] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3419944

Crypto Economics
[6] https://medium.com/semadaresearch/crypto-economics-809b4d3f6d25
[7] https://medium.com/casperlabs/equilibrium-in-cryptoeconomic-networks-cc824c32118e
[8] https://hackernoon.com/utility-tokens-discussion-economic-model-and-simulation-in-r-798c0ff3d26c

Baumol-Tobin Money Demand Model
[9] https://en.wikipedia.org/wiki/Baumol%E2%80%93Tobin_model
[10] https://en.wikipedia.org/wiki/Demand_for_money