TL;DR: In this article we present the relationship between the AppCoins token economics and the classical von Neumann growth model. We also analyse its relationship with modern extensions from the work of Samuelson and Ryuzo Sato, that employ the Noether Theorem and Lie Group theory. We also integrate product portfolio models such as McKinsey/GE, innovation diffusion and network effects into the analysis. In particular these are used to analyse the optimal growth the stability of the income-wealth ratio. We conclude that the AppCoins token economics approximates an economic ‘turnpike’, resulting in increased efficiencies for all players in the app economy.

Introduction

Jon von Neumann was one of the greatest mathematical minds that ever existed. He had a lasting impact not only on computer science (where the von Neumann architecture — the foundation of pretty much every modern digital device — was named after him), but also in many other fields including physics and economics.

The circular economy concept described in the AppCoins crowdsale document is consistent with the optimal growth model as envisioned by Jon von Neumann. Markets that behave similarly to the model are healthy economies. AppCoins may be conceptualised as a real-life example of such an economy.

AppCoins as a von Neumann Economy

The von Neumann growth model is a model of a circular economy that aims to be a “toy model” of real economies. We believe however that the in the case of the token economics built by the AppCoins protocol, we are approximating this toy model. According to Neumann:

“Goods are produced not only from “ natural factors of production,” but in the first place from each other. These processes of production may be circular, i.e. good G1 is produced with the aid of good G2, and G2 with the aid of G1.”

Imagine we have three commodities: Prospects, Leads, and Churners. When a new user joins AppCoins they increase the supply of Prospects for every app. Apps are looking to reach out to Prospects to convert them to Leads. This is done through advertising. They are willing to invest capital to pay for the attention of these Prospects, transforming them into Leads. The AppCoins conversion process ensures that exactly one Prospect will be transformed into one Lead. This assurance is possible through the Proof-of-Attention system.

After obtaining Leads, they have the opportunity to monetise them. Some apps are better at this than others. An in-app purchase (IAP) monetisation process results in different amounts of active leads and churners for different apps. This happens because different apps have different retention and resurrection rates.

Let’s assume that the IAP process generates AppCoins for the developer according to the value of the input commodities. Over time Churners will be very common. This happens because most apps have high churn rates and low resurrection rates. Churners have a low value. Leads will have a high value. These commodities cannot be exchanged between apps. Apps will have to decrease their churn rates over time or perish. Developers will need to have a portfolio of apps in order to reduce risk. Some apps will have more market maturity than others. Growing apps require an advertising investment for growth, and mature ones are ripe for harvest through IAP monetisation. This is consistent with traditional product portfolio models such as the BCG Matrix and the McKinsey/GE Model. Unless your app is highly mature (high α, possibly high γ), you’ll produce little capital in the IAP monetisation process. It also means that you have to invest in R&D efforts to develop new apps, which may not be successful apps.

Continuous innovation is your only choice.

One of the reasons so many apps are not able to retain users is the current app economy life cycle. The number of mature apps versus the total number of apps will be very small. This is because of the current early stage of the app economy life cycle in most emerging markets. Public data suggest that IAP revenue will grow strongly once emerging markets reach maturity. This is consistent with the conclusion that globally, the app economy is still on a growth stage.

This explains why industry reports consistently show that the churn rate for most apps is extremely high. The current percentage of apps capable of efficiently capturing user attention is very small. Especially when compared to the overall number of apps available in the app stores. This is consistent with the von Neumann assumptions.

Here’s why:

The average app retention (α) and resurrection rates (γ) will be very small (based on the current app economy lifecycle stage).

Developers have to invest in advertising before being able to monetise with IAP. Eventually, developers will need to reinvest a part of their earnings to support the new and growing apps.

Assuming that around half of the developer profit is reinvested in advertising, a large portion of the capital is retained in the system because the AppCoins given to the users for their attention need to be invested in IAP.

AppCoins ensures that user acquisition depends on the user experience, removing friction from the ad tech value chain.

Based on this we can argue that von Neumann’s model requirements are met. We can therefore assume that AppCoins should have a balanced growth following a path close to the “von Neumann Ray”.

We are assuming that the Advertising and IAP processes in the AppCoinseconomy should not be bound by ‘sectorial’ production limits (which would impose a non-balanced, cyclical growth). With the Proof-of-Attention system, the developer should be able to control its own ‘Leads’ production level (the basic commodity of the system) and should be aware of their own user retention (and resurrection) rates. Taken this into account, plus the fact that AppCoins is a decentralised economy, we can assume that the economy should have a balanced growth — also known as growing according to a “von Neumann Ray” or turnpike.

This ensures the optimal valuation growth of the AppCoins token and value for the investors. While many of these mechanisms already exist in the current app economy, AppCoins technology reinforces it. It reduces the app economy’s attritions substantially and reinforces natural market forces. This means increased economic efficiency for all the stakeholders.

AppCoins Optimal Growth

As Samuelson predicted by employing the Noether Theorem, the von Neumann model implies that at the “maximal growth” stage, the ratio between income and wealth is constant. Income is the value of tokens being used in transactions. Wealth is the total value of the AppCoins economy minus the income. Under the von Neumann assumptions, this indicator should become stable over time. This is the income-wealth conservation law of the von Neumann Model:

To measure how close we are to the optimal path, AppCoins may look to the variance of the income-wealth ratio. The lower the variance of this ratio on a given time interval, the closer we are to the optimal growth path on that interval.

AppCoins is a platform based on the blockchain. It has the potential to become a standard industry protocol — the universal language of the app economy. With this in mind, we expect its value to grow steadily with its adoption, which is an incentive for stakeholders. The value of successful blockchain technologies is known to follow the Metcalfe’s Law of network value. According to this law, the value of the network increases proportionally to the square of the number of its users (the direct network effect). We can estimate the value of AppCoins based on its user base on a given moment based on its direct value:

The growth of online networks comes from new user adoption and retention. Blockchain growth follows the pattern of known innovation adoption models such as the Bass diffusion model. The Bass diffusion model has been the gold standard of new product adoption forecasting of the last five decades. We’ll use this to forecast the growth of AppCoins. We only require mild assumptions such as potential market size and well known average parameter values:

Using this we can forecast AppCoins user base growth:

Through Metcalfe’s Law (using parameters from other cryptocurrencies), and using the previously obtained adoption values, we can then forecast AppCoins value (market cap). At the end of the 5-year period, its value should be around $20 billion, reaching close to $70 billion by April 2024.

While this explains direct network effects we also need to take into account indirect effects. Direct effects refer to the utility of the network to each user, which increases with the number of users (this is the effect measured by Metcalfe’s Law). Indirect effects are further divided into two subtypes: demand side and supply side. Demand side refers to the consumer benefits due to having more adopters. For instance, due to being able to send and receive money from users that adopt AppCoins. Supply side refers to the business benefits of having a larger market. They can provide additional complementary services to the network users. Because of this, we argue that AppCoins should exhibit both direct effects and indirect effects. Additional typologies of network effects have been presented by industry experts.These new typologies include up to eleven different types of network effect. We’ll analyse the ones we believe should be relevant in the AppCoins ecosystem:

Personal Utility — Suppliers can provide extra new services to AppCoins users, as AppCoins are developed using the ERC20 infrastructure. While harder to quantify, plug and play effects will also have a significant impact on the network value.

— Suppliers can provide extra new services to AppCoins users, as AppCoins are developed using the ERC20 infrastructure. While harder to quantify, plug and play effects will also have a significant impact on the network value. 2-Sided Platform — AppCoins is a protocol leveraging the Ethereum Blockchain that allows several actors to speak a common language. One side provides something by integrating their value “pipes”. The other side receives that, while also selling their output to other platform members. This is consistent with well-known platform effects.

— AppCoins is a protocol leveraging the Ethereum Blockchain that allows several actors to speak a common language. One side provides something by integrating their value “pipes”. The other side receives that, while also selling their output to other platform members. This is consistent with well-known platform effects. 2-Sided Marketplace — AppCoins is also an advertising marketplace. Advertisers (developers) are purchasing ad-space offered by publishers (app stores and other developers). Once publishers provide an interesting market for advertising, the advertisers’ demand increases, ad space increases with it, and so forth.

— AppCoins is also an advertising marketplace. Advertisers (developers) are purchasing ad-space offered by publishers (app stores and other developers). Once publishers provide an interesting market for advertising, the advertisers’ demand increases, ad space increases with it, and so forth. Bandwagon — Once AppCoins, designed to be a universal standard, reaches its critical mass, it then becomes an industry standard. This effect is consistent with observed bandwagon effects in social networks. Adoption from one user is contingent on the adoption of the other more influential (central) users in his social network. Once a threshold has been reached, even more users adopt. The effect should continue to be observed over the growth of AppCoins.

We assume the value of the platform will grow proportionately to the adoption, according to Metcalfe’s law. This however only quantifies the direct effect. As we’ve seen, indirect effects will also have a substantial positive impact on the valuation.

Turnpike Analysis

Under the assumptions of that model, optimal growth should follow the “von Neumann Ray” or “turnpike” at the equilibrium. At this stage, the ratio between the income (the capital output, or more precisely the value of the AppCoins used in advertising and IAP) and wealth (the cumulative wealth in all the AppCoins wallets minus the income), should be approximately constant over time.

We’ll assume that wealth can be approximated by the market cap minus the estimated added value of the economy in Table 6 of page 40 of the crowdsale document. To analyse if our business case is close to the ‘turnpike’ we compare the overall wealth with its corresponding income overtime. The figure below shows the result of the turnpike analysis for the AppCoins projected growth path:

It is apparent that under the assumptions of our business case, the AppCoins platform should approach the optimal growth path after its value reaches the $20 billion mark. This matches the end of our 5-year forecast (December 2022), where each AppCoin will be worth more than $125. We’ll also confirm if the income-wealth conservation law holds for our business case. For that, we’ll compute the income-wealth ratio using the theoretical expressions introduced in the annex to this document. We shall assume that developers will have a profit margin of 50%. App stores and OEMs have the AppCoins’ smart contract value of 15%. We assume, conservatively, that these profits will be leaving the AppCoins system (meaning that developers reinvest 50% of their IAP profit). We’ve computed the income-wealth ratio since month 12 until month 100. The results are presented below:

Conclusion

We can see that, over time, this macroeconomic indicator will tend to stabilise around 1.80. Its value will start to stabilise around the end of the 5-year period (in December 2022) which is consistent with the turnpike analysis and our business case. This behaviour is consistent with well-known empirical results that show that the fastest growing economies (mainly U.S.) always have income-wealth ratios that stay approximately constant over time.

This analysis suggests that our business case matches the theoretical predictions of the Von Neumann economic model.