## Introduction

Every currency faces the problems of establishing and retaining legitimacy. Traditional currencies are typically propped up by laws and regulations of legal authorities, who mandate how their currencies may be used. Cryptocurrencies are similar: mandates exist but, unlike traditional currencies, they are mostly regulated through code.

Various types of cryptocurrencies exist. The original Bitcoin is a great example of a well-designed token; last year's ignominous Luna debacle uncovered the truth behind a poorly designed token. This post is about yet another type of tokenomic model: the burn-and-mint.

In burn-and-mint, a token is "burned" *in nihilum* to pay for a service that a network provides; a token is "minted" *ex nihilo* to pay for a service that is provided to the network. Recently, I was suprised to learn that burn-and-mint has a bad reputation. Apparently, people are unsure whether a token has much value when it is subject to constant inflationary pressure. My own intuition about burn-and-mint, when I first heard of it when learning about Helium, was that it was similar to going to a foreign country and exchanging cash for the local currency: I suppose I *could* buy a chocolate bar in Japan using US dollars, but Japanese authorities insist that I do so in Yen. Similarly, I could try and pay for Helium's service in US dollars, but the protocol requires that I do so using direct credits, which must be bought using the Helium Network Token.

In any case, the question of how to value burn-and-mint is complex and requires stronger analysis than what has been made available thus far. Nowadays, if a project wants to use the burn-and-mint model, it needs to justify the use of a token and present a way to value its economy. This is why I wrote this post; herein I present a method of valuing a burn-and-mint economy, both deflationary (pioneered by Helium's HIP 20) and non-deflationary (the original model). To my knowledge, no one has yet rigorously tried to value a deflationary burn-and-mint economy. This post contains an attempt from first principles.

Specifically, the approach I took was to analyze burn-and-mint tokenomics as a discrete-time, full-information dynamic game. In this game, all participants are aware of future revenue, so there is no need for speculation and we can derive an optimal policy for each participant. The analysis follows below.

### Notation

List of important symbols:

\(V_k\) - value (NPV) function

\(M_k\) - circulating token supply

\(y_k\) - fiat operating cash flow

\(u_k\) - token outflow (burn)

\(w_k\) - token inflow (mint)

\(p_k\) - token price

\(\gamma\) - time discount factor

\(\gamma_d\) - token deflation factor

## Discrete-time analysis

Let us consider a dynamic game, played as an infinite sequence of discrete rounds \(k\). In a round, \(n > 1\) participants hold \(M_k^0,\dots,M_k^{n-1}\) tokens, respectively, and are prepared to exchange their holdings for incoming fiat. Furthermore, each participant \(i\) may receive an amount of tokens \(w_k^i\) that is generated via, e.g., mining, vesting, or staking. The amount of fiat available each round is given by the operating cash flow \(y_k\) and does not depend on any behavior of token holders; without loss of generality, we make the assumption that there is no outgoing fiat and that \(y_k\) is equivalent to network revenue. Since the incoming revenue is predetermined, the token price is determined solely by the number of tokens \(u_k\) exchanged for fiat,

$$p_k = \frac{y_k}{u_k}$$

where the amount \(u_k\) is the sum of all tokens that participants have traded for fiat in round \(k\). Given the price \(p_k\), we can say that value of the entire network \(V_k\) is given by,

$$V_k = p_kM_k$$

where \(M_k\) is the sum of tokens held by all participants.

### Optimal policy

An obvious question to ask is, how many tokens should a participant trade in a given round. To determine this, we consider the value of exchanging a token versus holding on to it for use in the next round. Exchanged, tokens \(u_k^i\) from participant \(i\) will return an amount of fiat proportional to the total number of tokens exchanged by all participants,

$$\frac{u_k^i}{\sum_j u_k^j}y_k$$

Leftover tokens, which could be exchanged in the next round, in addition to any tokens earned in the following round, are worth whatever the value of the network in the next round proportional to the total number of tokens held by all participants,

$$\frac{M_{k}^i-u_{k}^i+w_{k}^i}{\sum_{j} M_{k}^j-u_{k}^j+w_{k}^j} V_{k+1}$$

To maximize their gain, each participant should follow the optimal policy, determined by solving the following optimization, with decision variable \(u_k^i\),

$$\max_{u_k^i \leq M_k^i} \frac{u_k^i}{u_k}y_k + \frac{M_k^i-u_k^i+w_k^i}{M_k - u_k + w_k}\gamma V_{k+1}$$

where \(\gamma \in (0,1)\) is the time discount factor and \(w_k = \sum_j w_k^j\). Note that the constraint \(u_k^i \leq M_k^i\) will only be active in a scenario where the market has failed and participants are "rushing to the exits" (more on this later). Ignoring market failure, a bit of calculus gives us the following result,

\[ \begin{multline} \left(\frac{\gamma V_{k+1}}{(M_k - u_k + w_k)^2} - \frac{y_k}{u_k^2}\right)u_k^i - \frac{M_k^i+w_k^i}{(M_k - u_k + w_k)^2}\gamma V_{k+1} \newline = \frac{\gamma V_{k+1}}{M_k - u_k + w_k} - \frac{y_k}{u_k} \end{multline} \]

Summing over all participants and solving for \(u_k\),

$$u_k = \min\left\{\frac{y_k}{\gamma V_{k+1}+y_k}(M_k + w_k),M_k\right\}$$

where we have added the upper limit on \(u_k\) to remind ourselves of physical constraints.

In a non-failure scenario, the price \(p_k\) is given by,

$$p_k = \frac{y_k}{u_k} = \frac{1}{M_k - u_k + w_k}\gamma V_{k+1}$$

which, combined with \(V_k = p_kM_k\), gives a Bellman equation for the value function,

$$V_k = \frac{M_k-u_k}{M_k - u_k + w_k}\gamma V_{k+1} +y_k$$

We now have an equation for the value function and can determine how much each participant should sell each round. On aggregate,

$$u_k = \frac{y_k}{V_k}M_k$$

#### Interpretation as discounted cash flow

The value at initial time, expressed as a series, is given by,

$$V_0 = \frac{M_0-u_0}{M_0-u_0+w_0}y_0 + \frac{M_0-u_0}{M_0-u_0+w_0}\frac{M_1-u_1}{M_1-u_1+w_1}\gamma y_1 + \cdots$$

This expression is very close to the common expression for net present value (NPV), with a modification due to the fact that 1) one must spend tokens in order to extract value from the economy and 2) the monetary base is always being diluted. In effect, the ratio \(\frac{M_k-u_k}{M_k-u_k+w_k}\) itself has become a part of the discount factor.

### Valuing burn-and-mint at equilibrium

To determine the health of the economy, we analyze its behavior in steady state. It is instructive to understand the NPV \(V_k\) at the limit, when revenue \(y_k\) is stable. We begin by noting the equation for the evolution of the token supply,

$$\begin{align} M_{k+1} &= M_k - u_k + w_k \newline &= \left(1-\frac{y_k}{V_k}\right)M_k + w_k \end{align}$$

We furthermore assume a deflationary schedule for the incoming token supply,

$$w_{k+1} = \gamma_d w_k$$

where \(\gamma_d \in (0,1]\).

Assuming that \(\lim_{k\to\infty} y_k =: y^*\), there exist two steady-state solutions. One is given by,

$$\begin{align} u_k &\to M_k \to w_k \newline V_k &\to y^* \end{align}$$

The other is given by,

$$\begin{align} w_k &\to 0 \newline u_k &\to (1-\gamma)(M_k+w_k) \newline V_k &\to V^* := \frac{1}{1-\gamma}y^* \end{align}$$

We now consider the implications for each case.

#### Necessary deflation for a healthy market

Obviously, the first case above corresponds to a market failure since all tokens in circulation are exchanged for fiat. The question is then what leads to failure. It turns out that it is not enough to deflate the token supply, as the *deflation factor* \(\gamma_d\) *must outpace the discount factor* \(\gamma\). To see this, consider the dynamic equation for the token supply in both steady-state cases. The ratio \(\frac{y_k}{V_k}\) tends to \(1\) or \(1-\gamma\) depending on the case. Therefore, when \(\gamma_s \neq \gamma_d\),
$$M_k \to \frac{\gamma_s^k-\gamma_d^k}{\gamma_s-\gamma_d}w_0$$

where \(\gamma_s := 1-\frac{y^*}{V^*}\) can only have two values: \(0\) or \(\gamma\). Some further analysis shows that \(\gamma_d > \gamma\) implies that \(V_k\) does not tend to \(\frac{1}{1-\gamma}y^*\) and must instead tend to \(y^*\). Therefore, the ratio of incoming flow to current token supply satisfies,

$$\frac{w_k}{M_k} \to \begin{cases} 1 & \text{ if } \gamma_d > \gamma \newline 0 & \text{ if } \gamma_d < \gamma \end{cases}$$

Ignoring the case when \(\gamma = \gamma_d\) because the discount factor \(\gamma\) is market-dependent and therefore impossible to predict, this implies that a necessary condition for ensuring a healthy market, i.e., that the deflation factor \(\gamma_d\) must satisfy,

$$\gamma_d < \gamma$$

#### Impact of deflation on market health

The above analysis shows the necessity of deflation, or halving, for a healthy burn-and-mint economy. One way to think about the benefits of adequate deflation is that it ensures that inflation falls fast enough so as to ensure that NPV is equal to the present value of expected cash flow. Otherwise, similarly to how interest-rate differentials impact foreign exchange rates, if the deflation factor \(\gamma_d\) is not overcome by the discount factor \(\gamma\), then the economy will fail because the rate of incoming supply will outpace the rate of outgoing supply.

Nevertheless, a project may still avoid economic collapse without having to fork to provide liquidity to the economy: a remedy might include switching to a fiat-payment scheme once the token experiment has reached end-of-life. However, it is a question whether a non-deflationary token should get off the ground in the first place: an investor would be cautioned against bootstrapping a burn-and-mint economy if it does not promise to provide sufficient deflation.

### Irrational behavior: Hodlers

A consequence of deflationary burn-and-mint is that there is constant pressure on the price \(p_k\) to rise. In fact, the price must theoretically rise exponentially due to the fact that tokens are constantly being removed from the economy,

$$p_{k+1} = \gamma^{-1}p_k$$

With this in mind, a naïve investment approach would be to just "hodl", i.e., refuse to capture any incoming revenue due to expected price appreciation, contradicting the conclusion of the preceding analysis. However, there is no contradiction here because the analysis was performed under the assumption that all participants act rationally. The implication is, therefore, that hodling is an irrational investment strategy.

To see the effect of hodlers on the rest of participants, we assume hodlers hold an amount of tokens \(M'\). The optimal policy for a rational participant is then given as the solution to,

$$\max_{u_k^i \leq M_k^i} \frac{u_k^i}{u_k}y_k + \frac{M_k^i-u_k^i+w_k^i}{M'+M_k - u_k + w_k}\gamma V_{k+1}$$

A simple change of variables \(M'+M_k \to M_k\) leads to a result where the optimal policy is for participants to sell fewer tokens than in the absence of hodlers,

$$u_k = \frac{y_k}{V_k}(M_k-M')$$

The implication is that hodlers cause a faster rate of price increase, which in turn leads to temptation for hodlers to turn rational and sell. Of course, the analysis ignores stubborn hodlers who refuse to sell at any price — they have willingly removed themselves from market participation and have almost no impact on price.

### Numerical example

Let us consider a simple example with a YoY discount \(\gamma = 0.9\), which corresponds to a 10% interest rate. The revenues in this economy are zero for the first two years and then ramp up over eight years to m$200. We set the initial token supply \(M_0\) at 50 million tokens and introduce a vesting period (for founders and investors) of three years with 75 million tokens vesting yearly. The inflow is given by \(w_k = \bar w_k+v_k\) where \(v_k\) is the amount of tokens that vest in round \(k\), and \(\bar w_0 = \gamma_d M_0\) and \(\bar w_{k+1} = \gamma_d\bar w_k\). We consider two different deflation factors: \(\gamma_d = 0.84\) (results in halving every four years) and \(\gamma_d = 1\) (no deflation).

The results of the first, deflationary simulation are given below. It is easy to see that the NPV rises quickly on its way to a value of b$2.

The results of the second, non-deflationary simulation are given below. Here, the NPV rises rapidly but begins to decrease around the fourth year, at the same time that the token supply begins to decrease. The market collapses by the fifteenth year, at which point there is no reason to hold on to a token once it has been earned.

The following figure compares prices in both cases. In the deflationary case, the price rises exponentially due to decreasing token supply. Note that there likely exist other market effects which would lead to a stabilization of the token supply, where the price would also stabilize. In the non-deflationary case, the price stabilizes and remains constant at \(\frac{y_k}{\bar w_0} = \) $4.

## Conclusion

This post presents an analysis of the burn-and-mint tokenomic model. The main results are the derivations of optimal policies for rational investors, in both the absence and presence of hodlers, and the discovery of a key result that the amount of incoming tokens should deflate faster than the prevailing interest rate so as to prevent market failure.

In the analysis, the burn-and-mint economy is treated as a discrete-time, full-information game that is independent of the rest of the market. For a more complete picture, these assumptions would have to modified. Since markets operate continuously, and future revenues are uncertain, one would have to modify the approach to consider continuous time and partial information in the form of expectations. Furthermore, one might consider how this model might fit into a larger, more complex market model where revenues depend on service quality and participants' behavior is affected by their expected returns.

### tl;dr

Below are the formulae for finding the net present value \(V_0\) of a burn-and-mint economy (under certain assumptions):

\[\begin{align} V_k &= \frac{M_k - u_k}{M_k - u_k+w_k}\gamma V_{k+1}+y_k \newline M_{k+1} &= M_k - u_k + w_k \newline u_k &= \frac{y_k}{V_k}M_k \end{align} \]

To avoid collapse, make sure to introduce sufficient deflation in your burn-and-mint economy:

$$\gamma_d < \gamma$$

With sufficient deflation, the steady-state value of burn-and-mint economy is:

$$V_k = \frac{1}{1-\gamma}y_k$$

Don't just hodl burn-and-mint tokens (capture revenue if you can)

## Acknowledgments

Mark Ballandies for proposing this problem; and Mark B., Michael Chiu, Marko Kalabić, and Kris Paruch for discussions

## References

K. Samani, New models for utility tokens, Feb 2018

P. Atanasovski, Burn and mint's pro's and con's, Apr 2019