When the base protocol is not enough for field money usage, an off-block chain solution is necessary. To value a money whatever it is on or off block chain, always keep in mind it is all about cost to fake or rollback a commitment. This is why decentralized money is preferred compared with centralized money because a single point of centralization is easy to bribe or attack. Also, when a resource for handling deals and updating the account balance is not uniformly distributed and only available to some players, there must be some rent-seeking activity beneficial to that resource owners and balance the cost of production. However, due to technology constraint, it is the efficiency that an on-block chain coin cannot handle therefore an off-block chain is developed, the cost involved from centralization and rent-seeking (such as the fee of a license for operation of an exchange) is bypassed here for argument simplicity. Instead, an equivalent possibly higher effective energy power input is assumed in this reasoning; for example, an annual license fee of USD 1000 may be treated as an additional energy power resulting in annual electricity bill of USD 1000.

For this off-block chain coin, suppose the energy work per second of the whole off-block chain is K' and the volume is V'. Then its fee collected in one second is qV' and the price is P'=K'/qV' as explained here. The exchange rate with the on-block chain bitcoin K/qV can be easily deducted: one bitcoin shall equal to k = K/K' * V'/V off-block chain coin. If V' is around V and K' is lower than K, then the exchange rate k must be higher, in other words, the off-block chain coin is less valuable.

Some examples.

Fiat Exchange

The first example of off-block chain solution. People buy/sell daily economy activity in fiat as usual. Typically, the states-backed fiats may depreciate as time goes by but people can hold the majority of wealth safely in bitcoin block chain and keep only a fraction of wealth in the form of fiat to handle daily stuff. The price of the fiat is K'/qV' which is small due to low K' and depreciates due to increasing V'. Therefore, the exchange rate K/K' * V'/V tends to increase with the time. Hopefully, because the bitcoin can serve like a gold standard, people can watch the states' honesty in the local fiat currencies. When economics size increases or decreases, the real term price of bitcoin and the fiat appreciate or depreciate in the same ratio and the nominal fiat exchange rate remain intact, therefore all people enjoy or suffer the wealth change fairly with the wealth proportional principle. The crypto-to-crypto exchange is the same reasoning, to exchange bitcoin with some other lower price cryptocurrency for local purpose is just like to exchange with fiat.

LN network

In this context, the V' is the volume contributed by the channels providers who will charge a risk-free rate q because they put aside their coins V' to help the network. The design of an LN network is to use some cryptography tricks and game theory so that two parties of a deal will be honest to each other as long as they monitor each other. When two parties plan to deal by LN network, both open an LN channel by escrowing their on-block chain coins in an address. At the same time, a fix ratio, denoted by k, of LN coins will be created for 1.0 escrowed on-block chain coin. At the end of the deal, the channel is closed and the correspondent off-block chain LN coins are destroyed naturally. The party who submits the exceptional last-state LN coins (for example, the death of the other party) will wait for a period of time, 1000 blocks time for example, to receive the fund. All previous-state LN coins are effectively invalid because the other party or the hired watcher can take advantage of the time lag to take all fund as a penalty to the submit party. A typical scenario is that both parties agree the state of the deal all the way to the end and spend the original escrow on-block chain coins based on the final state, in between those LN coins serve as the agreed state of the deal only known to both parties and off-block chain. Since the monitoring effort of the LN nodes is much lower than mining effort of the miners, it is expected that K' is much lower than K. By k= K/K' * V'/V and typically k=1.0 by LN protocol design, this means the fee per second qV' and the volume V' in the LN network must be lower to maintain the equilibrium. Put in another way, it makes more sense that k is determined by the market and the given K' and V' of all off-block chain participants. LN network is in a sense fishy due to monitoring effort is supposed to be small and V' is supposed to be near k * V, it seems some rent-seeking activity to increase the effective K' is inevitable or its volume V' mustn't be great, not a clean and clear design as an exchange.