Introduction
I previously posted how one might value a burn-and-mint economy. Since then, there has been some further work done on the subject.
One of the main topics I took a look into is the problem of adequately rewarding contributors, specifically miners, for their contribution to the network. In the previous work, we found that "to avoid collapse, [you should] make sure to introduce sufficient deflation in your burn-and-mint economy." The necessary condition on deflation was presented as a strict inequality, i.e., the deflation rate has to be strictly less than the discount rate. However, this leaves something lacking: when reward deflation is strictly stronger than the prevailing interest rate, miners will eventually be inadequately rewarded because the design ensures that reward will tend to zero faster than price will appreciate. A similar concern first appeared in the discussion surrounding Helium's HIP20 and the result was the proposal of net emissions, which proposes to recycle the Helium token in order to ensure an adequate amount of reward.
Therefore, on the one hand, deflating reward too slowly leads to monetary collapse, i.e., hyperinflation. On the other hand, deflating too quickly leads to economic collapse, i.e., insufficient miner participation. In fact, since discount rates are subjection to perception and vary between market participants, there is no way to choose a deflation rate and guarantee economic stability.
Let me then present a middle ground. My conclusion in this post is that the revenues of a burn-and-mint economy, whether current or future,1 should be consistently shared with miners as this will lead to a stable economy. I particularly believe that this design principle should be followed by decentralized physical infrastructure (dePIN) projects.
Notation
List of important symbols:
$V$ - value (NPV) function
$M$ - circulating token supply
$y$ - fiat revenue
$z$ - fiat expense
$u$ - token outflow (burn)
$w$ - token inflow (mint)
\(z_d\) - desired fiat reward
$p$ - token price
\(\rho\) - time discount rate
\(\rho_d\) - reward deflation rate
Analysis
In this post, we consider the continuous-time burn-and-mint model, given by,
\[
\begin{align}
-\dot V &= -\rho V - \frac{w}{M}V + y - z \newline
\dot M &= - u + w \newline
u &= \frac{M}{V} (y-z)
\end{align}
\]
where $V$ is the value of the economy, $M$ is the amount of tokens in circulation, \(\rho\) is the discount rate, $y$ is incoming fiat, $z$ is outgoing fiat, $u$ is token outflow, and $w$ is token inflow.
Rewarding in contributors' units of account
When considering the options around rewarding contributors to the economy, we must to keep in mind their units of account. If one's unit of account is different than the network's currency, i.e., the token, we must consider the exchange rate between the two. This is because it is impossible to sustainably reward contributors in tokens without guaranteeing an adequate value for the token in the first place.
Suppose \(z_d\) is an adequate fiat reward to a network's contributors. One option would be to set \(z = z_d\) and route a part of revenue $y$ to reward miners. This can be done by burning \(y-z_d\) worth of tokens and sending the rest \(z_d\), in tokens, to miners. If $y$ is always greater than \(z_d\), then this approach is certainly sustainable. Nevertheless, this is just not possible when there is not enough revenue, which is mostly a concern at network launch, when system designers must rely on minting tokens, i.e., debasing the value of the token, to ensure reward. Given the desired fiat amount \(z_d\), the desired token amount is then, \(w_d = \frac{z_d}{p} = \frac{M}{V}z_d\) where \(p = \frac{V}{M}\) is the token price.
Now, suppose the only outflow of tokens is the desired amount, i.e., \(w = w_d\). Then,
\[
\begin{align}
-\dot V &= -\rho V - \frac{\frac{M}{V}z_d}{M}V + y \newline
&=-\rho V+(y-z_d)
\end{align}
\]
The above shows that there is no theoretical difference between paying \(z_d\) out in fiat or tokens, i.e., setting \(w_d = \frac{M}{V}z_d\), when the payout is a targeted reward. According to the model, both are theoretically the same.
However, it is somewhat remarkable that targeting a fiat-fixed reward results in a net present value $V$ that depends solely on interest rate \(\rho\), and net cash flow \(y-z_d\), and not at all on expansion of the monetary base $M$.2
Reward design implications
In the beginning, a token economy's value is very uncertain. The value $V$ tends to be depressed due to uncertainty about future cash flows; at the same time, it tends to be supported by the possibility of business growth. Nevertheless, the value of a crypto economy tends to be subject to the former. Due to self-selectoin bias, crypto projects are typically beset by uncertainty because, if future revenues were more certain, it stands to reason that it would be more lucrative for a project to pursue a more traditional business model.
Let us now return to the deflationary burn-and-mint model, in which the reward deflates according to,
$$\dot w = -\rho_d w$$
In the previous work, we showed that a necessary condition for a viable, deflationary economy is that,
$$-\rho_d < -\rho$$
Paradox
Herein lies a paradox. The price $p$ grows exponentially at a rate \(\rho\),
$$\dot p = \rho p$$
while the reward amount must deflate faster, implying that the fiat-valued reward amount $pw$, must go to zero because,
$$(pw)^{\cdot} = -(\rho_d-\rho)p < -\varepsilon pw$$
for some \(\varepsilon > 0\).
The implication is therefore that, eventually, the network will have to switch to targeting a fiat amount and no longer pursue reward deflation; otherwise, it will suffer degradation due to a lack of contributors.
Implications for dePIN
The burn-and-mint model has become quite popular among dePIN projects. I think that the reason for this is that dePIN revenue is priced in fiat but projects require upfront investment before being able to realize revenue; therefore, a mechanism is required to link fiat to token value.
In dePIN schemes, miners are set up out in the field and cannot be easily replaced by other miners.3 The implication is that switching costs are high and, therefore, it is preferable to keep miners maintained versus letting them leave and running the risk of losing coverage out in the field.
To keep maintenance high, it is obvious to almost anyone that a designer must adequately incentivize miners. What is not so obvious is that a designer must also avoid overincentivizing miners. This is because overincentivization can lead to complacency and dissatisfaction with future reward amounts, since these must necessarily decrease in a deflationary approach.
Compare this to the Bitcoin network, which gives all rewards to block miners and none to developers, or node operators. Bitcoin is not necessarily unsustainable for a few reasons: firstly, specific physical locations of miners are not important to the network; secondly, the phenomenon of Bitcoin is so large that it is able to adequately incentivize developers and operators through indirect network effects; and, finally, Bitcoin can scale miner difficulty according to the number of participating miners, disincentivizing competition from new miners.
None of the above is typically true in the dePIN space: physical location is a defining characteristic; projects tend to be small and require highly specialized contributions; and miner difficulty is constant.
My own prescription for reward mechanism design in dePIN is to follow a fixed-fiat amount. I imagine that this amount could be based on a target amortization schedule that could vary through time and across geographic regions, among other things.
Resolving the paradox
We now come to the final result: a fiat-fixed reward deflates the economy at the prevailing discount rate. To see this, note that when \(w_d = \frac{z_d}{p}\):
$$\dot w = \frac{d}{dt}\left(\frac{z_d}{p}\right) = -\rho \frac{z_d}{p} = -\rho w$$
implying that rewards decrease at the same rate that price appreciates.
The paradox that we faced earlier has been resolved at the expense of having to consistently share some (future) revenue with miners. As long as the desired reward \(z_d\) is eventually less than network revenue $y$, i.e.,
$$z_d < y$$
the economy remains viable.
Numerical simulation
We consider a modification of the example from the previous post. The discount rate is 10% per year, i.e., \(\rho = -\log(0.9)\). The revenues in this economy are zero for the first two years and then ramp up over eight years to \(y_{\text{ss}} =\) m$200 per year. We set the initial token supply \(M(0) = V(0)\), so that the initial token price is $1. We let target amount to miners be 20% of stable revenues, i.e., \(z_d = \) m$40 per year.
The results of the simulation are given below. The NPV rises steadily before reaching equilibrium at \(\frac{y_{\text{ss}}}{\rho} =\) b$1.519. The token supply rises at first, before revenue starts to arrive and the rate of burn overtakes the rate of mint. The price rises exponentially at a rate \(\rho\) so that, after 20 years, it is equal to \(\exp(20\rho) =\) $8.23.
Conclusion
The previous post on valuing a deflationary burn-and-mint economy provided a prescription on avoiding overincentivization of miners. Unfortunately, the prescription provided in that post necessarily leads to underincentivization. The current post provides a middle ground, suggesting that designers should implement a fiat-fixed miner reward scheme.
Acknowledgments
My fellow ono Thomi Nigg for conceptual guidance, and Michael Chiu and T. Nigg for discussions
1 Transferred in the form of tokens which represent net present value of the network.
2 Pretty much like any other economy 🙂
3 This is according to my own definition of dePIN; some definitions are broader but the stricter definition is important here.
