The Helium community has made a mistake. Implementing "net emissions" eventually removes any incentive for anyone to hold HNT. This is a serious problem for those of us who believe in the future of Helium and, more generally, DePIN, so let us consider it.

In the deflationary burn-and-mint model, tokens are destroyed and created, with the rate of burn dominating the rate of mint. The mathematical model describing this is given by \[ \begin{align} -\dot V &= -\rho V - \frac{w}{M}V + y \newline \dot M &= - u + w \newline u &= \frac{M}{V} y \end{align} \] where \(V\) is the fiat value of the network, \(M\) is the amount of tokens in circulation, \(w\) is the mint, \(u\) is the burn, \(y\) is the fiat revenue, and \(\rho > 0\) is the market discount rate. The model above is aggregate and derived based on a much more complex model of individual behavior and competitive strategy. It describes burn-and-mint tokenomics generally and, in particular, does not assume deflationary tokenomics, i.e., that \(u > w\).

As considered elsewhere on this blog, the problem with deflationary burn-and-mint is that, if \(w\) tends to zero, reward issuance \(u\) eventually follows, meaning that miners might one day not receive adequate rewards. The concept of net emissions, introduced in HIP-20 at the same time as the invention of deflationary burn-and-mint, seeks to remedy this by emitting a minimum amount of tokens \(w\) that is the lesser of \(u\) and some constant \(K\). We will analyze this in detail, but let us note that the problem becomes immediately apparent at equilibrium \(u = w\). In this case, $$ -\dot V = -\rho V - \frac{u}{M}V + y \newline = -\rho V $$ where we used the fact that \(u = \frac{M}{V} y\) to cancel out the \(y\) term.

The implication is that any network implementing net emissions is worthless at equilibrium because the solution to the value equation is \(V = 0\). While it may not currently pose a problem, the concept of net emissions is fast becoming entrenched in the Helium tokenomics with the introduction of HRP-2025-03.

Before anyone runs a victory lap, please take some time to read this post, where I show that the dominant strategy of each HNT token holder in the presence of net emissions will eventually be to convert all their HNT to fiat as fast as possible.

Notation

List of important symbols:

• \(V_k\) - value (NPV) function

• \(M_k\) - circulating token supply

• \(y_k\) - fiat network revenue

• \(u_k\) - token outflow (burn)

• \(w_k\) - token inflow (mint)

• \(\gamma\) - time discount factor

• \(\gamma_d\) - token deflation factor

Analysis

The derivation of the canonical equations is rigorously presented in this conference paper. A more accessible and less rigorous derivation is available on this blog.

Having given some thought as to why the concept of net emissions is so attractive, it seems to me that the proponents of net emissions may not recognize that something can have a price but still hold no long-term value. The well-known Twitter thread by JM Fayal, co-author of HIP-20, conflates the existence of a token price with the existence of a discounted present value and exhibits this misconception.

The problem becomes evident in the discrete-time case, where it is easier to see the dilution of token holders by those who obtain newly-minted tokens, e.g., miners: In discrete time, every participant must choose how many tokens to convert to fiat (either by buying direct credits or selling to someone in need of direct credits) and how many to hold. At equilibrium, the value to the holder must remain the same before and after the sale; it can so happen that the dilution is large enough that the holder has no incentive to hold on to their tokens and would be better off converting them all into fiat.

The result is a monetary collapse for, while the token might still have a price because fiat demand always exists, holders are in a race to dump their token as fast as possible. The only limit is the rate of mint.

This remainder of this section presents an analysis of how this collapse occurs in a model of the Helium network tokenomics.

Valuation of the Helium network

The discrete-time canonical equations are given by \[ \begin{align} V_k &= \frac{M_k-u_k}{M_k - u_k + w_k}\gamma V_{k+1} +y_k \newline M_{k+1} &= M_k - u_k + w_k \newline u_k &= \min\left\{\frac{y_k}{\gamma V_{k+1}+y_k}(M_k + w_k),M_k\right\} \end{align} \] The Helium network is semi-deflationary, with \(w_k\) halving every other year and any burn \(u_{k-1}\) less than or equal to \(K\) being re-emitted. To simulate this, we introduce the relationship \[ \begin{align} w_{k+1} &= \tilde w_{k+1} + \min\left\{K,u_k\right\} \newline \tilde w_{k+1} &= \gamma_d \tilde w_k \end{align} \] where \(\gamma_d\) has been introduced as a deflationary factor in place of a schedule.

As we have already shown in the Introduction, the above leads to monetary collapse. One way, and perhaps the only way to avoid this is to use a fiat-fixed rewards scheme as introduced here. If one wishes to couple high deflation in the early years with stability in the later years, this can be done as well. We therefore consider the following scheme \[ \begin{align} w_{k+1} &= \max\left\{\tilde w_{k+1}, 0.5\frac{M_k}{V_k}y_k\right\} \newline \tilde w_{k+1} &= \gamma_d \tilde w_k \end{align} \] where at least half of network revenue is re-emitted to miners in order to sustainably reward them.

Results

Steady-state simulation

In the first simulation, we assume a constant revenue \(y\) of $1 billion per year. We simulate the net emissions case with a limit \(K\) of 50,000 HNT per month. We choose the initial circulating token supply \(M_0\) to be 6 million HNT and the yearly discount factor \(\gamma\) to be 0.9.

The results clearly show a collapse at about year 18 with an impact on the tokenomics throughout the simulation. In the first subplot, the value of the blockchain is consistenly lower than in the fixed-fiat case, with monetary collapse leading to \(V = y\). The fact that the value does not become zero is a discretization artifact; the faster the update time, the faster tokens may move and the faster value tends to zero, as in the continuous-time case considered in the Introduction.

The second subplot shows that the token supply collapses to \(M = w\) as expected; the only tokens held are those that have just been minted and are then immediately ready to be sold between ticks. The third subplot shows that price exists in both cases but, due to the lack of liquidity in the case of net emissions, mean that price is not a measure of value. The final subplot shows the value transferred to miners. In the case of net emissions, the entire value is transferred after collapse and this is why there is no incentive to hold the token.

Growth simulation

We now consider revenue growth where the revenue \(y\) evolves according to the logistic equation $$y_{k+1} = y_k + ry_k\left(1-\frac{y_k}{y_s}\right)$$ where the initial revenue \(y_0\) is $2.4 million per year, the rate \(r\) is chosen so that revenue doubles in the first year, and the steady-state revenue \(y_s\) is $1 billion per year.

The results confirm monetary collapse, this time by year 37. Curiously, in the case of net emissions, more value is accrued to holders at the beginning. This is because miner rewards, measured in fiat, are much lower in the case of net emissions until the day of collapse, when the switchover happens rapidly. In any case, a higher initial token value should not play a determining factor here as one could always lower the amount of reward going to miners in the fixed-fiat case.

Another curiousity is the rapid, exponential convergence to steady state experienced in the fixed-fiat case. This is typically good to see in any response as it implies predictability and suggests the presence of robustness.

Conclusion

This post considered the concept of net emissions in the tokenomics of the Helium network and demonstrated that its implementation eventually results in monetary collapse as holders are aggressively diluted by newly-minted tokens.

What should be done

Even though the simulations suggest that the problem may be years away from appearing, it is unclear how chaotic market forces might impact poor tokenomics choices. The Helium community should probably fix this in favor of provably stable and, preferably, robust tokenomics.

In particular consider that, if the price of the token does not reflect the value of the network, then it becomes difficult for service providers to make economic decisions. Even well before collapse, because the market capitalization of the token does not equal the discounted cash flow of revenue, the use of net emissions makes it impossible to measure the impact of economic decisions on the overall health of the network.

On whose behalf? That is where things get tricky. If we are to naïvely view a nation as a single, unified body, then the privilege of money printing extends to the benefit of all American citizens. In practice, however, this has never been the case. Some individuals—American or otherwise—have not been able to partake in the benefits of US money-printing to the extent that this has been enjoyed by others.

The Highest Level of Privilege

The ability to create money out of thin air is not a privilege exclusive to states. It is a function of trust—and trust can be manufactured.

Anyone can write an IOU. Anyone can mint a currency. Anyone can issue stock certificates. This is becoming easier by the day and the general understanding of the benefits of being able to issue your own money and the tools available to do so are slowly becoming self-evident.

On an immediate timeframe, being credited an amount of money is no different than having that money in the first place. If a credit is good for buying a car, then it's good enough for most people. The problems with credit emerge when it comes with strings attached—restrictions on how it may be used or repaid.

The credit that the United States raises through its money printing comes with no similar stipulation. The existing system works on the assumption that the United States may be trusted with the power of the printer and that it will institute internal mechanisms to preserve the global order.

Qui Bono?

On a macro level, it is convenient to treat "The United States" as the money printer. However, that obscures the fact that the individuals who control the printer are real people with real desires, incentives and—most importantly—constraints. Qui bono, then?

One way to think about this question is to look at geography. Where does the money get printed? New York, perhaps? It is after all no coincidence that New York City is the financial capital of the world.

Benefits of the Money Printer

If I could take out unlimited credit, one should expect that I would "get rich" quickly—money would be no impediment to satisfy any economic desire. But the effects would not stop with me. Consider that those I choose to pay for goods and services would also get rich, and those who serve them would profit in turn. The further one is from the source—that is, me—the slower one's path to riches.

When money is no object, demand is entirely inelastic. Supply and the limits of desire are the only determining factors in price. If my bald head is willing to pay any price for a haircut, an entire industry might form around not only improving the quality of haircuts for the bald, but to also convince bald men that they need haircuts in the first place.

This is how demand flows outward from the money printer: unrelatable and hyper-optimized for the benefit of the center. As a result of the money printer, New Yorkers (resp. Americans) value their time more than the rest of Americans (resp. the World). When money is limitless, physics is your only constraint.

I pick on "New Yorkers" here to make an important point: not all Americans are equal. The coastal elites who sit near the money spigot do not equally share their privilege with the rest of the country. In fact, they often share it sooner with foreign counterparts—fellow elites—than with their own compatriots.

Tariffs and the End

Trump has recently imposed punitive tariffs on the rest of the world. Unlike his first term, where he pursued an accommodative policy that sought favor with Wall Street, he has now instituted shock therapy—in his words, "medicine"—and put the world on notice.

The United States is no longer interested in exporting dollars to the global economy. It wants those dollars back home.

Where will they go? At best—if inflation is to be avoided—they will return to the magic money machine that digitally "prints" the dollars, only to be deleted the same way they were created. Whether or not the President understands, the United States has stepped on the accelerator of de-dollarization; the magic money genie is going back in the bottle.

The Role of US Internal Politics

Surprisingly, the end of Bretton Woods is not due to some foreign plot. The system is unraveling because the American elite has failed to convince the lower class that extending the terms of their exorbitant privilege is of benefit to both sides. As a result, we are about to enter a world where the United States loses its ability to steer the global economy; with it, the exceptional way of life that the US citizenry has been able to enjoy since the end of the Second World War.

The realization of the end comes as a surprise. During his second-term campaign, Trump himself demonstrated an understanding of the importance of the US dollar, and promised to keep the system of power in place. The markets originally took this to mean that he would play by the rules, yet that has not been the case. His strategy appears to have shifted from working within the system to reshaping it entirely.

Who benefits from the shift? While it is not entirely true that people behave rationally, the median person does tend to act in ways consistent with their worldview. Problems might result from lack of foresight and inability to anticipate the desires and actions of others, especially rivals.

My sense is that the lower-class nativists in the United States feel comfortable enough to risk letting the entire system burn if it means leveling the playing field with the domestic elite. Two ingredients are required for revolution: class discontent and a lack of fear of the outside world. Both are present in America today. Trump's willingness to deliver may be the spark to a powder keg.

The market reaction to the poorly-executed implementation of tariffs means that there is no going back to the old way of doing things. Americans will work for their money just like everyone else.

While I do believe that this is a more-or-less intended consequence of revolution, some consequences may be unintended. Just because the US feels secure at this time does not mean that will continue to be the case. The Russian Revolution, for example, occurred during a period of high morale and widespread belief that the Imperial Army would soon march on Constantinople. Yet the internal upheaval resulted in territorial losses so severe that they undermined the security of the new Soviet state, leading to a delayed reckoning in 1941.

My guess it that the privileged status of the US will be lamented after it is lost and that the country as a whole will work hard to regain it. The current feeling of schadenfreude that some on the right feel towards the stock market drawdown is temporary. Reshuffling the deck won't change the rules of the game being played.

Conclusion

The Bretton Woods system is being dismantled from within. While this may come as a surprise to those who see the United States as a more-or-less perfect union, I posit that it signals an American revolution. Within a few years, the US dollar will likely lose its status as the world's reserve currency.

While that may hurt the average American in the short term, I believe that many Americans see this as a necessary cost—one worth bearing if it means reclaiming control over their lives at home.

Note

This article has been published on 𝕏; link available here.

Recently, the Helium network passed HIP-138, which abrogates the multi-token model introduced by HIP-51. While the Helium DAO passed the HIP, meaning that HNT rewards will be diverted from funding the MOBILE treasury to directly rewarding MOBILE deployers, the MOBILE subDAO did not, meaning that MOBILE emissions will continue.

The Helium community is at this this time forming a plan on how to move forward because continuing MOBILE emissions without any price support supposedly makes MOBILE a DePIN memecoin just like, for example, ENTROPY, which mines a memecoin without any backing. What I would like to point out in this note is that, theoretically speaking, MOBILE was always a memecoin and continuing MOBILE emissions will have no theoretical effect on the function of either the MOBILE sub-network or its token. In essence, the rejection of HIP-138 by a subDAO has no impact on the functioning of the Helium network.

This is because it does not matter whether HNT rewards are used to fund the MOBILE treasury to support token price, or whether they are sent directly to the same deployers who would otherwise receive MOBILE; the effect to deployers is the same because the amount of reward is the same. Any value that the MOBILE token may have in addition to the treasury-supported price will exist whether that support is positive, as has been the case until now, or zero, as it might soon become.

In addition to making the point above, this note presents an analysis on where the additional premium may come from and why the Helium community should not be so quick to deprecate MOBILE, the first DePIN memecoin.¹ Namely, the analysis propounds a view that MOBILE serves as a "juicing" of rewards in the early stages of network buildout; MOBILE is used to cover deployers' capital costs in the short term and prevents HNT from getting "too hot" and detached from network fundamentals.²

Notation

List of important symbols:

• \(V\) - value (NPV) function

• \(\mu\) - token price

• \(w\) - token mint

• \(y\) - revenue burned (in fiat)

• \(\rho\) - time discount rate

• \(K\) - capital cost

• \(c\) - running cost

Analysis

The value of the HNT network is given by the cash-flow relationship, $$-\dot V = -\rho V - \mu w + y$$

Assume a simplified model where supply exists at numerous locations \(i\) and costs \(K_i\) to deploy, \(c_i\) to run, and provides \(y_i\) in revenue. The value \(V_i\) of the location to a deployer is then given by, $$-\dot V_i = -\rho_iV_i - c_i + \mu w_i$$ and the value of the location to the network is the revenue it provides.

A rational deployer will deploy in a location as long as the following relationship R0 is satisfied, $$V_i(0) > K_i$$ and, after deploying, will continue to run the device as long as the following relationship R1 holds, $$V_i(t) > 0$$

The conditional nature of supply implies that solutions to the above system of equations depend on the price \(\mu\). The difference between R0 and R1 further implies that solutions to \(\mu\) may be discontinuous. The network is incentivized to offer a high reward \(\mu w_i\) in the short term and reduce this in the long term. Since \(w\) is fixed through the reward mechanism, the only way to increase reward to deployers is through price.

Enter MOBILE

I posit that the function that MOBILE has served is as an additional reward that helps incentivize deployment. With MOBILE, the market may "juice" deployers' earnings in order to pay for capital costs \(K_i\) and avoid discontinuous jumps in the HNT price \(\mu\).

Exit MOBILE

Ceasing MOBILE emissions would remove a degree of freedom from the market, limiting its ability to self-regulate. The only way to improve deployers' earnings in the short term would be by an increase in the price of HNT. While this may sound good to token holders, it could be problematic for the network: the reflexive nature of markets could lead investors to misinterpret price spikes as sustainable growth, causing wasteful, uneconomical deployments and eventual disillusionment among deployers. That is supposing a jump happens at all, as broader market forces may suppress the HNT price and never allow for it to spike.

Exacerbating the situation is that rational deployers would expect a spike in the absence of MOBILE emissions and would therefore be wary to deploy until the HNT price reached a sufficient height to guarantee a quick return on investment. In a sense, deployers might expect a "rug-pull" and require an accelerated return to avoid being caught out.

Conclusion

The cessation of MOBILE emissions introduces risks to both the Helium network and its deployers. The network loses a degree of freedom in being able to reward rapid deployment and deployers are forced to share any additional reward with all other Helium stakeholders.

In this note, I hope to have made it clear that MOBILE was the first DePIN memecoin and, with MOBILE, deployers eat first.

¹ Chronologically, the IOT token was established around the same time as MOBILE but with an earlier HIP number. However, I would hesitate to call IOT the first DePIN memecoin as it scarcely traded above its support price and therefore never exhibited a memetic function. Besides, it is now being deprecated with the passage of HIP-138 by the IOT subDAO.

²This is why MOBILE has in the past traded well above its support price.

No one was really alarmed at first, but it eventually became clear that voters would not give up their "entitlements" [as if anyone is entitled to anything], and everyone would be expected to keep the charade going as long as humanly possible.

As people woke up to the reality, the financial system began to be abused as individuals recognized a need to keep more of the pie for themselves. The free-market financial system, which used to serve the public as an economic coordination mechanism, became a wealth-extraction scheme, shifting funds from the less knowledgable to the more sophisticated.

Unfortunately, not everyone gets to choose whether they wish to participate in the "free" market. When the market's main purpose is all-encompassing economic coordination, it ends up touching everyone's life. It seems absurd to expect an ordinary person to be educated and market-savvy, yet this is exactly what is required of one if one is to avoid exploitation by powerful market forces.

What to do?

Decentralized Finance

An emerging solution to the problems above is the concept of "web3", i.e., the third iteration of the World Wide Web. Whereas web1 enabled free and open one-to-many communication, and web2 enabled interaction over the Web, web3 enables ownership of digital assets natively on the Internet.

The myriad applications enabled by web3 are dwarfed in potential by the promise and market capitalization of decentralized finance (DeFi). DeFi is a concept of a permissionless financial market: fast settlement, transparent mechanics, and trustless execution all serve to enable a robust and trustworthy financial system without need for middlemen.

However, middlemen are not the underlying issue. Preventing free and fair access to markets is not only not undesirable to dominant market forces, they wish for the exact opposite. Namely, allowing people to participate more freely and seamlessly can only help serve their interests.

While improving market efficiency is possibly even a good thing — neutral at worst — it cannot, on its own, change or influence the aim and purpose of markets themselves. Therefore, web3 — while providing individuals a path to economic emancipation through participation in a rapidly expanding market segment — on its own does not necessarily provide an alternative economic coordination mechanism, one that would serve a social good.

What to do?

Financialized Futility

You're going to help us, Mr. Anderson. Whether you want to or not.
(The Matrix, 1999)

Opting out of the financial system is not an option. In every corner of the world, socialist policies have forced, either through compulsion or coercion, every individual to be numbered starting at the moment of birth for the purpose of serving the greater economic "good". Individuals with the need or desire to emancipate themselves from serving a purpose which they deem to be in conflict with not only their individual needs, but the needs of the greater society, are left in quagmire.

As homo economicus, the only way to opt out of economic participation without leaving altogether is general strike. Yet, the Western economy has already come up with a response to such a tactic: anyone not willing to work can be replaced, either through immigration or automation. These two are an effective counter-tactic as they serve a dual purpose. For those who accept the remedy — take the blue pill, so to speak — their lives are momentarily improved; others do the work while they reap the rewards. For those who take the red pill, their path leads to certain death; the steamroller of economic progress promises to eventually end their bloodline.

What to do?

Decentralized Rebellion

The only winning move is not to play.
(WarGames, 1983)

The above is a general overview of the problem we face in changing our financial system for the better. Without allowing anyone to opt out, the system cannot receive direct feedback that it does not serve the greater good. In fact, even indirect feedback is difficult because individuals cannot trust each other to not be co-opted in the case that a sufficient number decide to make a change [consider the co-optation of Bitcoin into the same financial system against which its creators rebelled].

The solution, then, is financial nihilism: to turn the sword on its maker. A non-discriminatory financial system, for all its power, has one glaring weakness: it responds to money without fail. All it takes to change the system is to create a financial incentive for change without this change being susceptible to system integration: Since nothing is non-financializable, one must invest in nothing. Not "no-thing", but nothingness itself!

What to do:

Send It to Zero

An asset worth nothing, yet having financial value is an asset that turns the mirror on financialization. In the logic of financialization, this is an absurdity; it is surely not a product of homo economicus; it is at best a curiosity to be studied theoretically. Yet, here we are. Such an absurdity is here; moreover, it is not only theoretical, it has been generated through the function of market forces.

I am talking of the universe of Bitcoin Puppets, a collection of financially-iconoclastic digital art traded on decentralized financial platforms, whose images are stored and whose provenance is guaranteed by the decentralized computer system running the Bitcoin protocol.

The Puppets' motto is "Send it to zero," and it is ironic how the price of these images goes up the more loudly their owners declare their belief in the "demise" of their own financial assets. This is because nothingness is exactly the value of these puppets. It is also the value of memecoins — cryptocurrencies whose sole purpose is to be traded — whose price gains more memetic momentum the less they tie their value to any real-world tangibility.

Puppets themselves have gained memetic momentum throughout their existence. Emerging at the beginning of this year as a follow-up to a collection of crypto-themed irreverent puppets, the open-source main collection has inspired derivatives including female, junior, cat, and mash-up versions, with more coming into being on a regular basis.

Puppets are better than memecoins, too. Since memecoins are not naturally associated with artwork, they are limited in the emotion they can convey. And what these puppets convey is an attitude of irreverence toward the same system that cannot help but give them a market price. By rejecting financial mores, Puppets expose the underbelly of financialization.

So, what to do? Buy a Bitcoin Puppet!

Call to Action

In a well-functioning financial system, perhaps gold would serve the function of an asset whose price would be inversely proportional to the desire of the market to deploy liquidity. In a broken financial system, memetic assets serve a similar function: their price is inverse to the market's desire to participate in financialization.

Fiat currencies have underlying value because men with guns say they do. And this means their value isn't a bubble that can collapse if people lose faith.
(Krugman, 2018)

As we have seen, non-participation is not enough to enact change. A dollar spent on puppets is not only a dollar not spent on financialization; it is a dollar spent on incentivizing the system to change. Consider that, the higher the price that Puppets command on the market, the more the system is incentivized to serve the Puppets. Surely, at some point someone will realize the absurdity and that things need to change... if not, then the price will have to go higher.

The beatings will continue until morale improves.

Therefore, I both advise and implore everyone with any discontent in their hearts against the system we find ourselves in: to vote with their capital and invest in memes; to stay strong in the face of co-optation, lest Puppets become financialized and productized like any asset before; to police and punish co-optation, always being ready to deploy capital in whatever serve our interests; to hope that what will be sent to zero is the morally bankrupt system itself.

—Or don't. Idc. It's all going to zero, anyway.

Notes

This article has been published on 𝕏; link available here.
Cover image source here.

However, we can all feel the backpressure building. The Americas are not quite as open and free as they used to be; the Natives have been replaced by much stronger and more organized interests, that conspire to impede others' progress for their own selfish wants. It therefore appears that we must either limit our exploitation of Spaceship Earth, or build a literal spaceship and go somewhere else. I would say that not only is this a limited way of thinking, it also ignores the recent discovery of a totally new place of potential salvation: the Internet.

1524

Imagine that the year is 1524 and Columbus discovered a barren land called the Internet not even four decades ago. The Internet is easy to get to but survival is impossible; whatever one does on the Internet, one still needs to return to the Old World at least daily. The people of the Old World very much enjoy visiting the shores of the Internet on a regular basis, but few could imagine living there, let alone going inland.

Yet, progress has been made over the decades. For one, over the past decade, intrepid engineers have discovered ways of building lasting structures on the Internet, which they base upon a technology called "blockchain". Blockchain has allowed people to create and store value on the Internet that, upon their return to the Old World, securely remains their personal property.

This is not all that engineers have achieved. Whereas there was no life present on the Internet 30 years ago, new life forms have been introduced that are able to survive inland. The first such life forms were so crude that one could not reasonably refer to them as living, i.e., computer viruses resembling the biological RNA strands of the Old World. The new life forms are referred to as "AI" and only live on the Internet.

Being artificial, AI often exhibit malformations due to crudeness in their code. The harsh and unknown conditions of the Internet's interior make the design of AI difficult work. However, the AI have been improving over time and it would be hard to think that their progress should come to a halt anytime soon. Elon Musk himself is funding a method of allowing people to mount AI and visit the interior of the Internet, through the "Neuralink" project.

Meanwhile, life on the shores is much simpler. Internet shores are bastions of freedom, where occasional piracy and lawlessness are seen as normal. Blockchain technology has allowed fledgling economies to form, many of them centered around trading pictures called "NFTs" and speculating on currencies called "memecoins".

Internet denizens often weigh whether they should return any of their Internet funds to the Old World or leave them in vaults. This causes liquidity issues as the Internet economy is small and relatively small injections of liquidity from the Old World can lead to severe economic dislocations on the Internet. Most denizens do not really care; as far as they are concerned, injecting liquidity from the Old World means that everyone can win if they just ride the wave.

The Old World authorities themselves have taken notice and formed a view that Internet economic activity ought to be taxed, but are slowly coming to the realization that their tools only allow them to tax Internet gains when these gains are repatriated to the Old World. In response to having their newfound freedom curtailed, some Internet denizens are forming new ideas of statehood and citizenship, calling for a "network state", for example.

Speaking of money, Internet economic activity on the Internet is not solely limited to that of blockchain. Since its discovery, anyone who has committed any appreciable amount of time to exploring and building on the Internet has been rewarded handsomely. The Internet is therefore often seen as a gold mine, if only one take the risk.

2024

Now, let us switch to the present. The year is 2024. Until the nineteenth century, it was popular to call the Internet "the Metaverse", but that is now seen as archaic, akin to calling America "Columbia". Many of us now live on the Internet permanently. There are many different countries present here, even nations. They even sometimes go to war, both with each other and with the Old World.

On the Internet, there is a running joke among the pretentious about gargoyles who do not even own a passport and have no intention of ever visiting the Old World. But why should they. The Old World and the Internet are the same: overcrowded, indebted, and stagnant — many feel the need to escape. Elon Musk is talking about going to Mars and has built a spaceship that could get us there.

Conclusion

The New World analogy is a framework for thinking about the ascendance of the Internet. With the advent of web3, the latest iteration of Internet technologies, many ideas are being explored and novel theories developed. However, the road is long and it will take many decades to get to where we are headed. The European discovery of the Americas directly led to the invention of new nations and ideas, but this process took centuries.

I believe the development of the Internet will follow a similar timeline because humanity is limited in how quickly it can adapt. What is nice about it is that we just might get to avoid a Noah's Ark scenario and put off our escape to Mars for another few centuries, until we are more ready. For now, we get to look forward to the gold rushes to come.

In conclusion:

We are still at the Internet's shores, mining the (electronic) gold and returning it back home.

Note

This article has been published on 𝕏; link available here.

• First blog post (Feb '23)
• Second blog post (Apr '23)
• Conference paper published as part of the IEEE Workshop on DePIN at the 2023 World Forum on IoT (Oct '23)
• Video of Token Engineering Labs research seminar (May '23)
• Video of IEEE workshop talks (Oct '23) [currently unavailable]
• Video of Breakpoint talk (Nov '23)

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.

To formalize the naïve view, let us assume a cost functional, $$F_T(x_k) + \sum_{k=0}^{T−1} F_k(x_k,u_k)$$ and ask if it is possible to determine an optimal policy forwards in time. Define, $$\tilde V_k(x_0,u_0,\dots,u_k) = \sum_{i=0}^k F_i(x_i,u_i)$$ so that, $$\tilde V_{k+1}(x_0,u_0,\dots,u_k,u_{k+1}) = \tilde V_k(x_0,u_0,\dots,u_k) + F_{k+1}(x_{k+1},u_{k+1})$$

Suppose we want to find the minimum¹ of the function \(\tilde V_{k+1}\). We would need to optimize over the variables \(u_0\),...,\(u_{k+1}\) without exception. There is in general no way to find the optimum of \(\tilde V_{k}\) and use this information to optimize \(\tilde V_{k+1}\). This is because the term \(x_{k+1}\) in \(F_{k+1}(x_{k+1},u_{k+1})\) depends on all variables \(u_k\) preceding \(u_{k+1}\), so determining the optimum of \(\tilde V_{k+1}\) would necessarily influence the decision variables used in determining the optimum of \(\tilde V_{k}\). Therefore, given a certain state, optimizing forwards in time can give us an optimum with respect to that state, but in no way can it generally be used to determine an optimal policy.

The inverse of this realization is encapsulated in the principle of optimality, which states that:

An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.

R.E. Bellman (1957)

Bellman had more to say on the topic of determining an optimal policy:

In place of determining the optimal sequence of decisions from the fixed state of the system, we wish to determine the optimal decision to be made at any state of the system. Only if we know the latter, do we understand the intrinsic structure of the solution.

R.E. Bellman (1957)

The implication is that the principle of optimality leads to closed-loop control laws, as opposed to mere open-loop control. Previous to Bellman, various thinkers had encountered this principle and even dismissed it as trivial. Remarkably, Rufus Isaacs formulated his own, game-theoretic version and called it the tenet of transition:

If the play proceeds from one position to a second and V is thought of as known at the second, then it is determined at the first by demanding that the players optimize (that is, make minimax) the increment of V during the transition.

R.P. Isaacs (1951)

Isaacs, even with an intrinsic understanding that differential games require closed-loop control laws, later came to regret dismissing the importance of his tenet:

Once I felt that here was the heart of the subject and cited it often in the early Rand seminars. Later I felt that it – like other mathematical hearts – was a mere truism. Thus, in Differential Games it is mentioned only by title. This I regret. I had no idea that Pontryagin's principle and Bellman's maximal principle (a special case of the tenet, appearing a little later in the Rand seminars) would enjoy such widespread citation.

R.P. Isaacs (1973)

Let me now state my own understanding of the principle:

An optimal policy remains optimal backwards in time.

Start from the end and work your way backwards: Words to live by.

The rest of this post contains derivations of the Bellman and Hamilton-Jacobi-Bellman (HJB) equations.

Discrete time (Bellman equation)

Consider the optimal control problem, $$\min_u F_T(x_T) + \sum_{i=0}^{T−1} F_i(x_i,u_i)$$ where, $$x_{k+1}=f_k(x_k,u_k)$$ for all \(k \in \mathbb Z_T\), and let, $$V_k(x') = \min_u F_T(x_T)+ \sum_{i=k}^{T−1} F_i(x_i,u_i)$$ where \(x_k = x'\).

The Bellman equation follows easily, \[ \begin{align} V_k(x') &= \min_u F_T(x_T) + \sum_{i=k+1}^{T−1} F_i(x_i,u_i) + F_k(x',u_k) \newline &= \min_u V_{k+1}(x_{k+1})+F_k(x',u_k) \end{align} \] Replacing with the expression for \(x_{k+1}\), we obtain the result: $$V_k(x)= \min_u V_{k+1}(f_k(x,u)) + F_k(x,u)$$ with final condition \(V_T(x) = F_T(x)\).

Discounted time

Oftentimes, the cost functional is discounted in time: $$\min_u \gamma^T \bar F_T(x_T) + \sum_{i=0}^{T−1} \gamma^i \bar F_i(x_i,u_i)$$

A direct application of the Bellman equation gives us the result, $$V_k(x) = \min_u V_{k+1}(f_k(x,u)) + \gamma^k \bar F_k(x,u)$$

To avoid the appearance of small terms \(\gamma^k \bar F_k\) in the Bellman equation, we make a change of variables \(V_k = \gamma^k \bar V_k\), obtaining, $$\bar V_k(x) = \min_u \gamma \bar V_{k+1}(f_k(x,u))+\bar F_k(x,u)$$

Continuous time (HJB equation)

Consider the optimal control problem, $$\min_u F_T(x(T)) + \int_0^T F(x(t),u(t),t)dt$$ where, $$\dot x(t) = f(x(t),u(t),t)$$ for all \(t \in [0,T]\), and let, $$V(x',t) = \min_u F_T(x(T))+ \int_t^T F(x(\tau),u(\tau),\tau)d\tau$$ where \(x(t) = x'\).

We begin as in the derivation of the Bellman equation, optimizing over a time-step \(h\), \[ \begin{align} V(x',t) &= \min_u F_T(x(T)) + \int_{t+h}^T F(x(\tau),u(\tau),\tau)d\tau + \int_t^{t+h} F(x(\tau),u(\tau),\tau)d\tau \newline &= \min_{u|_{[t,t+h]}} V(x(t+h),t+h) + \int_t^{t+h} F(x(\tau),u(\tau),\tau)d\tau \end{align} \]

Taking \(V(x(t),t+h)\) away from both sides, \[ \begin{multline} V(x',t)-V(x(t),t+h) \newline = \min_{u|_{[t,t+h]}} V(x(t+h),t+h)-V(x(t),t+h) + \int_t^{t+h} F(x(\tau),u(\tau),\tau)d\tau \end{multline} \] Dividing by \(h\) and taking the limit \(h \to 0\), $$−\frac{\partial V}{\partial t}(x',t) = \min_u \frac{\partial V}{\partial x}(x',t) \cdot \dot x + F(x',u,t)$$

Replacing with the expression for \(\dot x\), we obtain the result: $$−\frac{\partial V}{\partial t}(x,t) = \min_u \frac{\partial V}{\partial x}(x,t) \cdot f(x,u,t) + F(x,u,t)$$ with final condition \(V(x,T) = F_T(x)\).

Discounted time

When the cost functional is discounted in time, $$\min_u e^{-\rho T}\bar F_T(x(T)) + \int_0^T e^{-\rho t}\bar F(x(t),u(t),t)dt$$ we can take a similar approach as in the case of discrete time. Specifically, we make a change of variables \(V = e^{-\rho t} \bar V\), obtaining, $$−\frac{\partial \bar V}{\partial t}(x,t) = -\rho\bar V(x,t) + \min_u \frac{\partial \bar V}{\partial x}(x,t) \cdot f(x,u,t) + \bar F(x,u,t)$$

Postface

This post started when I sat down to re-derive the HJB equation for the umpteenth time. I figured it was about time to put a stop to that and write a note that I could refer to in the future. The history lesson is a bonus; I hope you enoyed it.

References

• H.J. Pesch & R. Bulirsch, "The maximum principle, Bellman's equation, and Carathéodory's work," 1994

• D. Liberzon, "Calculus of variations and optimal control theory: A concise introduction," 2011

¹ For convenience, I am assuming in this post that functions are well-behaved enough so that the extremum not only exists, but is unique.

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