What is this?

This article contains worthwhile insights for those striving towards an anarcho-capitalistic ideal, those designing protocols as microeconomies or those who would vote with their wallet to loosen the grip of institutional capital on our beloved Internet. To give you an idea of what you’re getting yourself into:

Bonding Curves of the Continuous Token Model: Revisited as Mechanisms for Price Stabilization

might have been a more appropriate title for the information presented here under more rigorous academic settings. But this isn’t one of those settings.

I’m really just here to fulfill a penchant for shitposting.

For those unfamiliar, CTMs (Continuous Token Models) entered public discourse during the 2017 bull cycle. They’ve since fallen by the wayside, discarded as experimental primitives, largely due to a lack of understanding of how and why to apply them. Five years later, recent advancements have made them practical to implement on-chain. But it appears the industry as a whole still fails to grasp their significance.

I’m here to proselytize on a hidden gem of sorts, relevant to both builders and all who want of stability in their bleeding-edge virtual economies. I write for collaborators beyond this budding petri dish. And for those who would help sculpt the invisible hands of the Internet of Ownership.

So this article isn’t written like a traditional paper. It’s written to be read, to entertain even. Because God knows, I could not have found the love in me to finish, if the content here were to fall on deaf ears. I hope this more expressive format, interwoven with colorful (and perhaps excessive) shitposting, can bring some life to the dismal science. It’s certainly made the writing more tolerable.

Naturally, this looser format is much longer than it could have been in academic prose. So I’ve marked the trail with blazes. Cut through any sections you deem unnecessary. Expect mistakes. And feel free to reach out if there’s anything I can help clarify.

Current Market Dynamics

Price Instability

cycle observations, c. 2020-2022

Tokens today are launched with little regard for their price stability. Teams attempt to manifest value into existence, launching economies out of thin air, backed by nothing save the wispy dreams of electric sheep. By the most popular playbook, teams use emissions to bootstrap their communities and incentivize the creation of liquidity pools around a token. It’s not uncommon for emission schedules to double the circulating supply over the period of months to the benefit of holders and stakers. But without sufficiently large supply sinks to offset these emissions, the growth of token supply outpaces that of protocol utility (i.e. token demand).

The result is a devaluation of currency.

Trends during the crypto bull market have made it clear: retail investors.. don’t do math, at least not far beyond the spreadsheeting of APRs across yield farms.¹ Those that do are in the minority, but even the math-pilled pile into projects with little regard for “fundamentals” during periods of exuberance. It’s the dominant strategy—to don the WAGMI banner—in the upswing of a bull run.

However, some classes of tokens require more attention than others. In particular, unpermissioned stablecoins (when under-collateralized) involve some level of sophistication beyond that of your average memecoin to successfully peg the price of a token against shifting demand. Protocol investors should have a strong grasp of tokenomics before making the leap. Protocol designers even more so.

But clearly that hasn’t been the case.

Based on how much was lost through Terra-centered products (~$40B), many failed to evaluate the design of $UST and $LUNA, even locking significant capital atop a stablecoin protocol pegged using a single speculative asset. Even cursory analysis would have revealed a printing press vulnerable to inflationary feedback loops when run in reverse. Yet investments poured in at institutional scale.

Maybe it was excessive exuberance, maybe overconfidence in a narrative peddled by the Luna team. After all, social signals pointed towards Do Kwon’s balls being massive enough to hold it all together, even through a bank run.

But alas, we inevitably bore witness to one of our (supposedly) sophisticated founders nuking the collective bank account. What does this say about the rest of us?

We’re mecha-broke, founders and investors alike. The result is a broad mis-pricing of speculative assets in the short term where the price history becomes a meme in itself. And though meme investments can mint millionaires to the beat of a printer, they tend to implode when the music grinds to a halt. Bear market dynamics bring it all back down to earth. Minting one financial tragedy after another 🎭 for those unfortunate enough to volunteer as exit liquidity.

Bear Market Dynamics

skip this section if you’re familiar with the current macroeconomic landscape

We live in interesting times. It doesn’t take much effort to find yourself in informational overload and a self-induced whiplash. Technological advancements through a firehose: genomics, energy production, natural language models, text-to-generative-art, satellite imagery, financial markets independent of central authority. Widespread instability threatening the opposite: increasing wealth disparity within borders, the curtain call of a long term debt cycle, impending global recession, supply chain shortages, potential stagflation, an end to an era of globalization. And a global military pillar threatening the reintroduction of tactical nukes on the battlefield to top it all off.

Oh by the way: the Federal Reserve of the World’s reserve currency expanded its money supply by several trillion dollars in the face of a global pandemic.

Seems almost mundane by comparison, doesn’t it?

The scale is difficult to conceptualize. Not exactly a head-turner outside of fin-pilled circles. But its effect on the markets was hard to miss, manifesting the V-shaped market recovery so many wished for, if only for a fever dream.

Much of this money went directly into the largest securities markets in the form of Quantitative Easing (i.e. the Fed went on a shopping spree). Back at the start of the pandemic, the Fed poured an initial $700B then $120B/mo into treasury and mortgage-backed securities, only tapering off their purchases at the turn of this year with a balance sheet nearly $5T higher than when they started.

These markets, the once beloved playgrounds of yield-guaranteeing hedge funds, grew artificially inflated, pricing out investors who could no longer expect to generate the Guaranteed Returns they promised their clients.

You can analogize the effect as a fat man (fed) eying a hot tub (corporate bond market). He waddles over and sits down, raising the water level (price) around him. He slumps in deeper, and the water (money) begins to pour over the sides.

Well, this money had to find a new home. And it flooded into the smaller neighboring hot tubs like the public stock markets in search of alpha, cascading even further to venture capital, private equity and cryptocurrencies. Couple this with COVID relief funds distributed across the board and the maturation of financial products centered around crypto assets. The result was DeFi summer. A perfect summer storm of otherworldly gains, bequeathing freedom to the rare breed of degenerate that takes profit.

But the recent pullback of the Fed under the shadow of persistent inflationary signals has brought financial markets to capitulation.

“The fat man is exiting the hot tub!” you might have heard from the town crier. And with him, the waves of excess capital seemingly ebb and vanish into financial aether. Now he dances hawkishly on the horizon, ratcheting up interest rates in one motion and divesting corporate bond holdings in the next, investment firms readjust their growth valuation models, pull out of liquid investments and search for reentry into a market where the new normal is defined at a lower level. And in this transition, (3, 3) inevitably flips to (3, -3) in a cruel chapter of stakeholder misalignment.

For several projects, this market shift coincides with the end of lockup periods where teams and investors have vested significant portions of the token supply. From a purely financial perspective it’s in their best interest to dump their bags. After all, there’s no glory in a graveyard. No collateral value from a golden heart.

Collateral as Resilience

Stablecoins, a Brief

skip this if you’re familiar with the prominent stablecoin projects

One class of tokens has been largely immune to the onset of the bear market: collateralized stablecoins. $USDT, $USDC and $BUSD, respectively, are the leaders by market capitalization. The first two are collateralized by obfuscated assets, largely traditional, while the third is collateralized directly with USD. Pegged to the price of tradFi’s premier reserve currency, they offer a safe haven from the volatility native to our emergent ecosystems.

However, the three can hardly be considered decentralized given their permissioned model of issuance. Rather, they’re a necessary stopgap for stability in the transition to a decentralized financial system. And in that regard, $DAI and $FRAX are more relevant to the price analysis of collateralized stablecoins.

Maker’s $DAI is interesting because a significant portion of its collateral is comprised of volatile on-chain assets. It stays solvent through price fluctuations of its collateral, by requiring borrowers to over-collateralize into a protocol-controlled “Maker Vault” when minting/borrowing $DAI.

Holders of $DAI can rest easy knowing that even if their collateral loses value their $DAI will not, since the protocol automatically liquidates collateral in its Vaults to stay solvent. At the time of writing, the protocol holds ~$10.6B of collateral to back an outstanding supply of ~6.9B $DAI, a collateral ratio (CR) of ~150% in this period of volatility.

$FRAX on the other hand is under-collateralized. It achieves greater capital efficiency than its peers through a fractionally algorithmic approach to stabilization. The team’s core innovations in their first version of the protocol were (1) a sliding CR for $FRAX according to price pressures on the open market and (2) incentivized swapping between $FRAX and $FXS + $USDC, based on the market price of their governance token $FXS.

If this sounds familiar, it’s because (2) is a similar mechanic to Luna’s printer, only with significant $USDC as stable collateral and fewer memes about testicular gravity wells.²

Lessons in Price Controls

skip if you understand the core mechanisms of both $DAI and $FRAX

Despite their permissionless issuance, both $DAI and $FRAX have held their peg to fractions of a cent through the recent implosion of prominent peer protocol Luna.³

They’ve managed this through what one could roughly describe as collateral-based supply controls. In $DAI’s case a borrower’s collateral is automatically exchanged for the outstanding balance of $DAI on the open market when a Vault breaches the lower bound on the acceptable CR. The purchased $DAI is then burned, decreasing the total supply of $DAI, and any remaining collateral is returned to the borrower in its original form. With sufficient buffer room, collateral locked in the protocol always stays above the outstanding balance and can be used to contract the supply of $DAI as needed.

But $DAI is capital inefficient, even less-so than its centralized counterparts.

In $FRAX’s case there are multiple mechanisms, which you can read about here, that dictate the protocol’s CR and supply based on price pressures. When demand for $FRAX drops, sustaining price below peg, the protocol increases the target CR and contracts supply through a marginally incentivized exchange between $FRAX => $USDC + $FXS . $USDC comes from the protocol’s reserves and $FXS is minted according to market rates.

So how did the Luna-inspired market crash affect the price of of these two tokens? Well, it didn’t really. Instead, we saw sharp contractions in the supply of both tokens, as expected by the design of their respective protocols.

These protocols are able to maintain peg by contracting supply in times of reduced demand. For $DAI the influence of supply contraction is less obvious as the borrowed $DAI is overcollateralized, so Vault balances are always assumed solvent to a market redemption of $DAI at $1. For our undercollateralized counterpart however, supply controls are crucial. They’re used to balance the current supply of $FRAX against its market demand and maintain the price at peg.

Compare this with other classes of tokens, examples of which have dropped 80-95% from peak through this market cycle. Of course, price stability isn’t the highest priority for many of these tokens, and operating at or near full-collateral as stablecoin projects do would hinder a team’s ability to scale their networks quickly.

That being said, under no twisted narrative can a 90% drawdown in a protocol’s primary financial vehicle be a good thing for its community. Quite the contrary. Our bagholders now self-identify as down/tremendously.

If collateral-based supply controls can enable a protocol to maintain its token within half a percent of its price target in a major market meltdown, it’s not hard to imagine one could employ the same tools to design mechanisms meeting looser stability requirements.

Let’s explore the tradeoffs after a quick detour through the author’s childhood.

Inspired Mechanisms

The Drug Market of Neochibis (role of the shopkeep)

skip if familiar with price windows (tl;dr shopkeep = mint/burn contract)

There are several ways to augment an open market economy for greater price stability, and they derive inspiration from market structures common to game economies.

An example:

We can imagine the Orange Potions ($OP) from the above shop window circulating through the Maplestory economy. Without the presence of a shopkeep, our players are free to trade $OP with one another at any agreed upon price. The price of $OP can fluctuate unbounded: up, down and sideways, based on the balance of supply and demand of $OP on the open market. An0n, our savvy Maple investor, starts hoarding $OP as a modest increase in new players has her bullish on the projected shift in demand for these very mid potions.

But MMORPGs are at the mercy of their publishers, and Maplestory’s publisher Nexon decides they want to make the game more n00b-friendly by drastically increasing the potion drop rate for Orange Mushrooms, a spawn commonly hunted at lower levels.⁴

This is disastrous for An0n who took out a second mortgage on her studio flat in downtown Kerning City in an attempt to corner the supply of $OP on the market. Now the new cohort of players farms a surplus, rather than a deficit in their misadventures. More than enough to negate their personal consumption. Consequently, an accumulation of excess $OP reserves now floods the market, and if An0n hasn’t dumped her bags yet, they’re likely going to 0. Poore An0n.

She’s jolted awake in a cold sweat. Just a bad dream.

As she gains her bearings and recovers from shock, An0n remembers that Maplestory employs shopkeeps, shopkeeps who will always purchase $OP from any user at the price of 80meso. This price of 80meso is what we call a price floor on $OP: a market structure where a central authority acts as the buyer of last resort, preventing the market price of an asset from falling below a threshold.

If the fair market price of $OP were to fall to 75meso, for example, players would instead exchange their $OP with the shopkeep for 80meso, sustaining the open market price at 80meso.

Of course, this does nothing to recover the actual fair price of $OP and serves to suppress price discovery. Fortunately for our Maplestory shopkeeps, they’re in cahoots with Nexon and have infinite reserves of Fake Money at their disposal.⁵ They can’t be “bankrupt” buying back infinite supplies of $OP or $COPE like our web3 protocols can.

So An0n is always guaranteed an exit price of 80mesos per $OP. Maybe she should take out a second mortgage. After all, it’s 2007. Maplestory is new. Maplestory is HOT. And an influx of new players is inevitable.

But she checks the price of $OP on her toji-terminal and goes back to sleep. Why?

Well, $OP is currently trading at 150meso. And, yes, she assumes a fair amount of downside risk even with the price floor, but if her investment hypothesis on Maple hotness is correct this shouldn’t be cause for too much concern.

After all, it’s not like the world’s falling into financial crisis anytime soon. Just look at how hot the real estate market is. She expects $OP price to go up, hopefully moon, just as the price of her home has these last few years. Her bigger concern is the price ceiling. Our shopkeeps don’t just buy $OP at 80meso. They also sell $OP at 160meso. Rain or shine. All day, every day. dey some hustlers fr. nd business, is bussin.

This caps An0n’s upside at just 10meso a pop. She runs some quick maths and decides it’s not worth the closing costs of refinancing. A stearn decision.

The market structure we see here is called a price window: bounded above by a price ceiling and below by a price floor. The market price of $OP can fluctuate to any price within this window, but it cannot breech the ceiling or floor set by the Nexon Pharma Cartel. For the sake of discussion, we’ll refer to these market structures more broadly as ‘mechanisms’ for their ease of incorporation into tokenomic models as composable building blocks.

Continuous Token Models: What, Why & What?

skip if hip to continuous token models

In the prior crypto bull run (c. 2017), several people floated the notion of Continuous Token Models (CTM). A method of token issuance open to the public, directly from a contract, infinitely scalable and price-pegged to a reserve currency(token). This model largely fell flat as founders preferred full mint and total control over their supply from the get go. Fixed supply dynamics were also more attractive to institutional investors, who pushed founders to mirror the model laid forth by L1s and coomed at the mere mouthing of the words Deflationary Token Model.

But what use is a “capped supply” or “deflationary model” when the circulating supply is aggressively expanded through emissions? All this serves to do is disproportionately reward those earliest to a protocol, with relatively tight timelines and rates arbitrarily chosen upfront by founders who I imagine have limited experience divining the future. That’s not the kind of hand we desire. Inadvertently gatekeeping and heavily penalizing the mass of adopters to come, more or less for having been born a bit later (sound familiar?).

Yes, as founders, we want to incentivize adoption, we want to reward those who are early. But handing out tokens to wallets simply for already having tokens is not only lazy; it’s self serving. Given you can plug “20% for founding team” to virtue signal on a pitch deck, then turn around to reap half the “60% for community” allocation through token-holder emissions. The game is almost comical at a distance. And disturbingly disingenuous under any scrutiny.

So it’s not clear yet whether the CTM can surface as the chosen modus operandi among protocol founders. It takes humility to decline a throne. An inhuman level of discipline when the option of accepting-absurd-amounts-of-capital-pre-product-for-a-shot-at-blitzscaling-on-network-effects exists. Understandable. So maybe this stagflation x crippled economy collab is just what we need, a ratcheting up of rates 1970s style to shed leverage and force a dry-up of institutional investment for risk-on ventures.

For communities, though, CTMs are the superior funding model, no doubt. They sidestep the steep token discounts VCs receive compared to the retail investor and offer greater price stability than the massive legal step-functions of liquidity granted to founding teams and their VCs today. In this early stage of the new internet, token markets cannot reasonably handle such massive fluctuations to circulating supply, not when they outright eclipse the morsels of utility offered by the protocol.

So what is this alternative? What is the bonding curve?⁶

Well, the bonding curve is the heart of the continuous token model. It’s the mapping between input and output. The redlining tachometer begging you to shift gears, to expand supply in times of over-exuberance and contract it when the market begs for mercy. The bonding curve is a cushion for the fall. A dampening function on the price action of shitcoins.

The bonding curve is a last line of defense for an unassuming market. A rate limit atop the shifting sands of demand. A countercultural force in the supply curve. A policy against economic violence. Padding for our overeager retail investor, the everyman, the laboring protocol participant. All discouraged by their imploding portfolios. The bonding curve is a solution to an acute problem. The bonding curve is..

The bonding curve is the shopkeep in your favorite MMORPG, selling $OP at 160meso and buying it back at 80meso. She sets the price window of a token by offering mint and burn prices in respect to a reserve token, meso in this case. Only on-chain, the shopkeep is free to update prices by her discretion, according to contract, guided by the invisible hand.

Discretionary pricing based on the time of day, market volatility or even the token balances of your wallet are all possible. Most sensibly though, most effectively and elegantly, in the name of the Holy Spirit, that which is dynamic price stability, price should be dependent on the current supply of the minted token. We’ll explore why in a moment. But given these axes, the X and the Y, the bonding curve’s mission—its purpose—is to bound the token within a price window, given current supply.

Technically Speaking

Monomial Curves, Shapes and Uses

skip if sublinear curve stan

Consider the monomial function p(x) = x^a as a candidate for our equation; the equation that maps token supply to price, the foundation of our CTM Bonding Curve.

The real mapping will look something like p(x) = k * x^a, or p_b(x) = r * k * x^a for the burn curve, where 0 < r < 1. But we can trim the fat to x^a for the sake of analysis since k and r are assumed constant scalars. Assuming also that a is positive, there are 4 distinct shapes our monomial curve can assume:

Our CTM Bonding Curve dictates the price window of the token underpinning our protocol’s entire economy. So though our expression may be simple, it speaks volumes. The nuances matter, and it’s worth hearing what each of these shapes have to say about themselves.

Constant: King

Let’s start with the constant curve, where a = 0, simplified to p(x) = x^0 = 1. A flat line across supply. By analogy, the constant curve is our King. By divine right, the King decrees the price of a token. She is absolute and holds steadfast to her decrees. Her exchange rate is law, regardless of how the supply may grow or shrink.

Kings are most common in video games. Take our Maplestory shopkeep, for example. She sets her price window at 160/80 and doesn’t look back. Rain or shine, surplus or squeeze, those are her rates. Take it or leave it.

In the real world, Kings are rare. Kings are iconic. What’s the going rate for a can of Arizona Green Tea? How about a Costco Hot Dog? Did you have to search up either? No, you didn’t. Because these are Kings, steadfast and absolute, unlike fallen Kings of old such as McDonald’s 20pc Chicken McNuggets or dollar stores that once held some sliver of integrity.⁷ But King status in the physical world is always temporary. Even the mighty Costco Hot Dog will eventually bow to price hikes. After all, these goods are denominated in inflationary fiat. It’s not that they’ll necessarily “cost” more. It’s just.. your money will be ✨ worth.less ✨

On-chain, however, this pattern is more common. In fact, one major class of tokens leans into the constant curve as a personality trait, a core identity even. Stablecoins, as discussed above, are the quintessential example. Though nearly all stablecoin protocols innovate beyond the simple constant curve, at their core, they aspire towards a shared philosophy: X dollars in, X dollars out. Rain or shine, surplus or squeeze, those are their rates. Take it or leave it. What we refer to as a “peg”, quite simply put, is a very narrow price window, codified either as law or by market forces.⁸

Linear: Jack

Next up is the Linear curve, our Jack, our Knave. Here a=1, p(x) = x. A simple soldier, really. Easy to understand. Easy to implement. With the Jack, price increases in consistent lockstep with supply. He cares that our early adopters are rewarded for the risk they assume. And not much else.

Our Jack is the first of the bunch whom one would describe as monotonically increasing, a shmancy way of saying that as X increases, so does Y. Under the context of our mapping between supply and price, all this means is that the earlier you are, the less it costs to mint. Unlike our constant curve, this positive price mapping allows token price, rather than just token supply, to absorb shifts in demand. In essence, price is allowed to increase and encourage earlier investment. ~A reason to ape in~

But we can do more than just incentivize early investment. The nuance of how much our curve increases along the mint is important.

Sublinear: Ace

And for that reason, the Sublinear Curve is our Ace. It’s a curve that adapts its qualities depending on the needs of a protocol. The derivative of price over supply is relatively high early on in a project’s lifetime, when utility is low; we see large increases to the token’s price with each new mint. This encourages early adoption and offers teams a steep discount on their initial batch of tokens, so long as they don’t mint too many.⁹

As we further expand supply, the rate of change decreases. The slope levels off to near zero as the project matures and reaches a larger supply. This expansion to near zero means that, once matured, our sublinear curve begins to look a lot like our constant curve. Our Ace requires larger and larger shifts in supply to meaningfully impact the price window. And supply, rather than price, absorbs the fluctuations to token demand.

However, that isn’t to say that a mature protocol under the continuous token model would be unviable for investment. The token’s price-growth may be slowed, but the price is free to scale to the limits of code. The beauty of the monomial is that the shape looks the same no matter how much you zoom in or zoom out. That is to say, a X% increase in supply will always result in a Y% increase in mint price, no matter the current supply, given a set of curve parameters.

Superlinear: Joker

You expected a Queen, but it’s hard confer any sense of reverence to the superlinear curve. So instead, we present the Joker. Why? Because the Joker is hard to understand, in the now, why the fuck would you do that? kinda way.

The Joker speaks to your greed, lulls you in with promises of riches, only to shoot you in the back of the head and burn the massive pile of cash you’ve donated to his heist. The Joker may be easy to implement on chain, as simple as a single EXP opcode, elegant even. But don’t be fooled. your dance with the court fool only ends in disaster. And it’s utterly bewildering why anyone would trust the Joker, and how anyone could endorse this maniac in their writing. ..

Our Joker takes everything good about our Ace and inverses the play. Starting with a painfully slow creep in token price, only to shoot price volatility through the roof as a project matures. With an increasingly volatile price and a promise of riches, one can intuit the inevitable rug and resulting price history for any token achieving “adoption” through this approach. It exudes a smell reminiscent of pre-PMF high-burn blitzscale attempts. And though you may enjoy some smokiness to your dishes, it’s ill-advised to accept excessive charring in your portfolio.

Quite simply, anyone playing the Joker is either a fool or a Ponzi masquerading as one, attempting to forcefully divine hypergrowth across the wrong dimensions.

But even the Joker has a redeeming trait: He promises founders cheaper token-capital than any of his counterparts, a lower relative mint price. It’s an offer hard to decline. After all, the hockey-stick curve triggers a pavlovian response from the best of us. It’s investor-primal. But it’s worth holding the drool for a moment to remember: hockey-stick success is something measured over time, not token supply. Price stabilization in maturity is a good thing, both for longterm investment and protocol health. Tokens aren’t just investment equity. They’re a stake in the ecosystem of value being created, a means of procuring and rewarding utility under the context of a protocol.

(supply) Based Price Controls

skip if gigabrain econ chad

So, great. We like the Ace. Makes sense. And we can mint new tokens along this sublinear price curve. But why bother with the whole window.. why burn tokens at all?

Good question, An0n. Remember our stablecoins from earlier? Remember how they maintained a strict peg, even through a market meltdown? How’d they manage that?

That’s right: our stablecoin protocols maintained price-peg by contracting supply. They were able to maintain peg by adjusting the supply of their token in accordance to the drop in demand. Of course, $FRAX and $DAI had their own mechanisms for doing so––less direct than the shopkeep’s––though similar in effect. Following the $UST debacle, the increase in market volatility and decrease in stablecoin confidence generated substantial sell pressure for these tokens. If their circulating supplies had remained constant through the ordeal, it’s unlikely either would have kept peg.

It’s simple economics, ackshually. At any point in time, the price of an asset is determined by the balance between supply and demand for the asset. Intuitively, we understand that when overall demand drops while supply remains constant, price also drops. Abstracted, this is a left shift in the demand curve, as seen above.

Our protocols have limited tools in their arsenal to influence demand in these situations. They can only list to Coinbase so many times, only conduct so many psyop campaigns until CT catches on. So assuming we have negligible control over the demand, we have one option: supply controls.

Assuming our whales are savvy™ they’ll dump their bags on the buyer with the highest offer. And in the case where market price breaches the current price floor, that would be the bonding curve contract. Our contract helps stabilize price by removing sellers from the market. They absorb some of the sell pressure by reducing circulating supply, as sellers choose to exchange their tokens with the shopkeep rather than with their peers. Buyers are then forced to engage with the remaining sellers at or above the new floor price. At this lower supply with the remaining sellers the market finds equilibrium at a higher price.

If it all sounds a bit complex, think about it this way: if all the sellers dump their bags on the shopkeep, who’s left to dump on the market? That’s right, nobody.

Now, for the red pill. This time we flip the script.

Consider what happens on the other side of the window: when excess demand for the token pushes the price beyond the ceiling. Buyers of course, would transact with a shopkeep offering tokens at lower-than-market rates, leaving our sellers to match at lower prices in the face of expanding supply. That is one major tradeoff of the continuous token model; it serves as a neutral party, reducing price volatility in both directions.

But though the market price is pushed back into the window, it’s worth noting that the price window itself shifts in the direction of volatility, allowing for progression of price in line with the market’s will.

This occurs passively from the perspective of the protocol. The shopkeep here is a contract that anyone can interact with, to exchange derivative tokens for a reserve token and vice versa. To this end, the shopkeep also serves as a treasury of funds (1) to honor redemptions across the burn curve and (2) to hold any balances in excess of the burn curve, the area between mint and burn curves.

Fractional Reserve Requirements

5% pleasure, 50% pain, 100% rant to remember the game

Let’s ignore the burn curve for a moment; or rather, set it equal to the mint curve. Consider the properties of the reserve being created here in relation to the bonding curve. Though I doubt the math-pilled among you would have trouble following along, we’ll skip the calculus for now as a show of mercy to the broader audience. So forget the math. Just take a look at these colorful charts.

The rectangular area is the market capitalization of the token at the existing supply. The area under the curve is the size of the reserve. The maximum reserve ratio then, given the shape of the curve, would be 100% for constant, 50% for linear and anything along this spectrum for the sublinear curves we could produce, following a ratio of 1/(α+1).

Where along this line do most protocols lie today?

That’s right, 0. They’re not on the spectrum, and that’s a problem.

Earlier, we disparaged the Joker for the absurd stabilization mechanism he offered. Zero is worse than that. Zero is a house of cards. Zero is free fall from a magic carpet ride when valuations are mismatched against reality.

Today we see protocols attempting to mint entire economies out of thin air. Why? Because training wheels don’t make for a great sales pitch. They limit speed. And investors like speed, not safety features. They don’t understand safety features. Because safety isn’t sexy. Safety doesn’t sell, blitzscale or lead to hypergrowth. Safety isn’t what VCs crave. What VCs crave is exit liquidity. Better-than-market deals. And we’ve been groomed to give it to them, in exorbitant amounts, at the cost our own (protocol) health.

So resilience isn’t attractive, not when the money is sloshing around. Resilience is the antithesis of just-in-time manufacturing, the enemy of capital “efficiency” misaligned towards hypergrowth. And we’re leaving no slack for the swans. Instead we’re doing what’s sexy. Minting our entire supplies from the get go, signing criminal term sheets for funding and accepting praise when the temporarily constrained supply squeezes our tokens to the moon. Well, those criminal term sheets are coming back to haunt you, anon. And no amount of holy water or salt can exorcise the skeletons in your closet.

Of course, this isn’t to advocate for a 100% reserve ratio. A full reserve ratio is overkill in almost every circumstance. We’re not (all) out here architecting stablecoins after all. But wouldn’t it be nice to hit a veranda or two over Agrahbah? When hurdling towards Gaia at terminal velocity. Who knows. If one slows us down, maybe two can keep us out of her eternal embrace. Maybe then the palace guards won’t find us strewn across the Bazaar come daybreak.

We’ve discussed max reserve ratios for each of these shapes, but those assume the mint and burn curves are one and the same. The above setups leave no room for a token to trade on the open market, as buyers and sellers must matchmake within the gap of the shopkeep’s mint and burn prices. A gap must exist to allow for PvP trading.

For example, let’s say our burn curve were a 70% scale replica of our mint curve. With this setup, we could see a maximum 30% drop in a token’s price from peak before our supply controls kicked in to help dampen us from free fall. The market would be free to trade up, down or sideways anywhere within this 30% gap. But if price breaches either bound, players would trade with the shopkeep to shift the window once more.

For a linear bonding curve with this gap, the actual reserve ratio would be 35% (RR_actual = RR_max * BurnMintRatio = 0.50 * 0.70 = 0.35), meaning the treasury must maintain 35% of the token’s current market capitalization as a reserve requirement. This is only one example, of course, and both the shape of the curve and the gap between the mint and burn curves can be adjusted to fit a protocol’s needs.

Implementation

Project Funding through the CTM

the meme factory has run out of steam, this section is meant for founders

It’s worth reiterating this reserve isn’t necessarily bankrolled by founders, VCs, or any one party in particular. It’s built up from all our minters’ contributions, with the area under the burn curve allocated to the reserve and the area above available for discretionary funding. As a result, the scalar gap between the mint and burn curves, and not the shape of the curve, determines what proportion of the invested funds are available to our founders for discretionary spending. If our burn curve is a 0.6x replica of our mint curve, for example, our founders receive 40% of invested funds to manifest their dreams into reality. This holds true whether the curve is constant, linear, sublinear or clinically insane.

Additionally, founders get to mint their own tokens at discounted rates. As the area above the burn curve is unallocated and available as discretionary funding, founders effectively mint at the burn price, which is always at a discount to the market rate as the burn curve serves as our price floor. A neat consequence of the windowed setup.

So that’s cool and all, but what about the area under the burn curve?

Yes we still have the reserve requirement, the integral of the burn curve at the current supply. Unfortunately, we’re unable to manifest new tokens out of thin air using the model described so far. This is a strict requirement, a hard cash-money requirement for minting tokens even for our founders. Despite getting the best rate as the first to mint––and even a discounted rate in perpetuity––most founders would consider the early capital requirement a deal breaker, a fatal flaw in the model.

I’m inclined to agree. The reserve ratio isn’t negligible, after all. And founders need an initial balance of tokens to incentivize their community and align their teams to the success of the protocol.

I can speak to potential solutions in more rigorous writings to come. For now, though, let’s just say there are workarounds. Free minting during contract publication and postponed reserve requirements come to mind. Given this model will ultimately result in founders having access to smaller portions of their “fully diluted” token supply upon founding, it’s important we don’t increase the capital burden for founders looking to launch protocols with tokens stabilized by shopkeep.

Sublinear Curves in Practice

skip this unless you’re math-pilled and solidity-literate

One might ask, why monomial curves? In short, they’re easy to integrate and easy-ish to approximate for implementation. Just look at on-chain approximations for sigmoids and logarithms by comparison. My math may be a bit rusty, but approximating and implementing their integrals on-chain seems less than ideal. By comparison, the Babylonian Method is elegant and already employed to approximate square roots semi-efficiently on-chain (sqrtLibrary.sol). My take on this employs an extension to the Babylonian Method, which generalizes the property of quadratic convergence to arbitrary n-roots.

Monomials are also easy. Much easier to communicate to an audience. And their properties, such as point derivatives, are easy to reason about, so they’re easy to design around. The generalized equation for maximum reserve ratio, for example, is 1/(α+1). Elegant innit?

This solution hasn’t been seen in the wild, largely (i assume) due to a lack of interest in innovating in this direction. Founders and investors aren’t aligned in the short term to opt for the Continuous Token Model as the preferred mode of funding. And for those that have considered it, it’s not entirely obvious that the sublinear curve is something we prefer, as evidenced by the dialogue around CTM bonding curves to date. And if it is obvious, the sublinear option is certainly more challenging to conjure an implementation for than its counterparts.

So without clear reasons for pursuing this solution, I can’t imagine anyone would venture down this path less traveled. Some of you may have noticed the obscurity of the paper referenced. With 6 citations in its 23 years of life, it ain’t exactly pop. And it certainly wasn’t the first option explored. The path to its discovery was a winding and painful one, littered with dead ends and disappointments.

But this solution has promise. As many of you are probably aware, there are no generalized, low-level abstractions to represent floating points in Solidity. As a result, calculating a fractional exponent proves impossible. Bad news for anyone attempting to map sublinear monomials. That was our motivation for applying the Babylonian Method to the square-root approximation found in quadratic voting. Now, imagine if we had a tool to efficiently approximate arbitrary n-roots, not just square-roots. Well, we could use that primitive.

By representing our sublinear monomial as a fractional exponent x^(m/n) it simply becomes the n-root of x^m where m and n are both positive whole numbers. As each loop in this approximation is fairly expensive (~30 Gas), it’s important to tighten the initial estimate to minimize the number of loops before convergence. We can do this by bounding the subtotal x^m by a power of 2 with some bit shifting. Bounded tightly, the loop converges to the nearest whole number within 10 iterations for all cases covered in testing.

It’s not hard to imagine uses for these approximations on-chain. One critical use case that comes to mind is sublinear stake-to-vote-weight mapping. Sublinear mappings in voting, paired with sybil resistant practices (outside the scope of this article) are a prerequisite to shifting away from the purely plutocratic voting systems we employ today.

If even for that sole reason, I’m eager to see this implementation fleshed out to a higher standard. As is, this implementation would be sufficient for the CTM bonding curves described in this article, perhaps even on Ethereum mainnet for high-enough mint/burn balances. Further gas optimizations would enable higher-frequency use cases such as vote-mapping, at least on L2s and PoS systems.

There are currently limitations on the inputs allowed in order to avoid overflows: loosely m*n*log2(x) < 256. We can optimize this for real world use cases by focusing on the number of significant figures, rather than rounding to the nearest whole number. I’m also not the expert on gas optimizations, but I have uploaded a first-pass implementation of the Continuous Token Model here as an extension to the ERC20. I’ll be finishing the tests and cleaning up the code in coming weeks to have it available open source.

Having only slung(?) Solidity for a couple months now, I have a lot of catching up to do on the industry. And I’d like to take this as an opportunity to announce my existence. To attract collaborators much more capable than myself. And to identify some mentors willing to field some inevitably obvious-in-hindsight questions. Hopefully, some of you see the same value in sublinear mappings as I do. And hopefully, this CTM use-case is one compelling enough to to be adopted as a building block.

Again, my DMs are open. Let’s talk tokenomics :^)

in nuce (🔩 , 🥜)

1. the continuous token model enables better alignment between investors

2. reserve funds can enable supply controls to dampen price volatility

3. a sliding price window soft-pegs tokens while allowing for full scalability

4. sublinear monomial curve works best and is already cheap enough for an L2

5. available 🔜™️ at a repo near you

Notes

1. source: my TL

2. iykyk

3. $FRAX price is defined according to multiple chainlink oracles tracking exchange rates for FRAX in DEx pools. $DAI price is defined according to ______

4. The experts among you may recognize that, ironically, Orange Mushrooms never dropped Orange Potions in OG Maple, but I implore you to suspend your disbelief for the sake of example

5. the Fed and the Shopkeep. they’re the same picture

6. Those more DeFi-savvy among you will recognize these CTM Bonding Curves differ from the constant-product curves introduced by Uniswap. They’re both formulations used to describe the relationship between price and supply of assets. I’ll continue referring to them simply as “Bonding Curves”, though, for the sake of brevity; just know that I refer specifically to the monomial bonding curves explored in the 2017 era.

7. For real world consumable goods, the constant bonding curve is limited to just minting. To the extent of my knowledge, there is no standard burn rate for a Costco Hotdog.

8. an oversimplification, but hopefully, you get my point (insert something about frax)

9. Of course, this isn’t nearly the same level of financial freedom granted when minting the entire supply of your entire economy out of thin air. But it seems our founders have collectively shown how well they wield this great responsibility.