Metcalfe, Network Effects and Crypto Valuations
In the hope of being able to justify high price valuations for crypto tokens, many commentators resort to a discussion of ‘network effects’, citing the original Metcalfe ‘Law’ that the value of a network tends to N-squared as networks get large, because every agent can access n-1 other agents.
This discussion note examines this concept.
The general idea is that every participant in a network confers some non-negative utility on all other participants. Whether it is n-squared, or nlogn or some other variant doesn’t matter as long as it is positive or monotonic in the number of participants.
We can write the utility or benefit function for participant i consuming q units of service in a network with n participants as:
B(i,n) = b(q) +f(n) where b’(q)> 0 and f’(n)≥ 0
A simple case where the relationships are linear is shown below, where
f(n) = a.n and b(q) = b
In the diagram the blue line is the individual benefit, and the gray line is the marginal network benefit or marginal network revenue.
In a decentralized crypto utility network it is a first order assumption that the marginal cost of delivery of the service is constant, and this is shown with the orange horizontal line.
If this network existed in isolation, then the individual user would obtain the service provided the price was equal to the marginal benefit and the network provider would provide the service at the point where the marginal benefit was equal to the marginal cost. This is shown in the diagram below:
In this example the network would provide the service for 8 consumers at a price of P, and would make a marginal loss of (P* -P ).
However with rational expectations both the network provider and the network consumer could forsee that if the network grows, then in the future at a price of P* then the network is providing the optimal level of service and the consumers are achieving a competitive rational price for the service. This can be extrapolated to any period in the future, and so it is rational for consumers to have almost infinitely expanding views as to the existence of period by period equilibrium points that have future prices higher than current prices. In essence this is the ‘Metcalfe’ value proposition — it doesn’t really matter what the exact relationship is, as it foreshadows an infinite period of price inflation in the price per unit service of the utility token.
However, what if this network does not exist in isolation? And instead there are many competing ways in which the network utility could be provided?
In that case, any price above P*, the marginal cost of providing the service, would be competed away (essentially because the demand curve would fall due to substitution effects, even if they are not perfect substitutes).
Hence any service providing a utility with a price above P* would see demand fall, and the long run equilibrium is a static price of P* even with network effects. There is no price escalation unless it is a perfect monopoly with almost zero elasticity of substitution.
Lets relax some of the assumptions. What is the marginal cost of delivering the service was low?
In this case, where the network is very efficient, the value to the individual is higher than marginal cost no matter how many people are in the network.
In turn this means that in a competitive environment where people can create new services and new tokens at will, competition between networks will be fierce because every network, no matter what its size, is making a marginal profit. The logical conclusion is that there will be a proliferation of networks, all of modest size, and no large network will exist, because as soon as one exists it will be copied and its benefits competed away. An ICO explosion!
So what if we relax the constant marginal cost assumption and assume some form of network saturation effect?
In the diagram at the left the benefit curves follow and n-squared rule, and it is assumed that marginal costs start flat and then saturate due to some form of congestion effect.
Now competition between networks will see them set price P, and stabilize at a network size of 11. Consequently we will see a proliferation of networks at the saturation or congestion size, and the price level can be seen as a congestion pricing mechanism. Obviously if the marginal cost is always below the marginal benefit, there is no cross over point and networks will proliferate at any size.
Conclusion
The introduction of so called network effects as a justification for pricing of cryptographic utility tokens ignores the critical importance of pricing and competition on the market place and on consumer demand curves. If we allow for substitution and competition between networks, then the impact of network effects becomes almost irrelevant to the pricing calculation and the forward expectations of such prices.
If we make the bold assumption that there are ever increasing network effects (that is, the value benefit per individual is monotonically increasing in the number of participants), and that there is a perfect monopoly with no substitution, and that the discount rate is zero, we then get an explosive bubble in price expectations.
It is highly unlikely that these assumptions are realistic — the proliferation of ICO projects is reasonably indicative that the cost of launching a competitor is low, and for many areas there is already good evidence of healthy competition between projects. It is also highly likely that beyond a certain number, network effects are essentially flat.
With that said, calls to network effects to capture future prices in current prices is probably more wishful thinking than sound rational judgement.
