### Assumptions

Assume there are only two chains, Ethereum and chain X (representing the combination of all other PoW chains the can feasibly be mined with GPUs), where there are no costs to switching mining power, and that the total hash power of both chains is constant both before EIP-1559 $H=H^_{E}+H^X$ and post-EIP-1559 $H=H^*{1559}+H^*_X$. Also, assume costs to run mining hardware are directly proportional to the hash power used and the same for all chains, and that the total revenue extracted by all miners in any chain is constant. Finally, we make the assumption that miners are profit maximisers and sell their mined tokens immediately in a liquid market.

### The Model

The total profits of all mining activity per unit time on chain i $P_i$ is made up of revenues $R_i$ and costs $C_i$.

$$P_{i}=R_{i}-C_{i}$$

Since the total revenues are constant in any chain, the revenue per hash rate is simply inversely proportional to the total hash rate of a chain.

$$r_{i}=\frac{R_i}{H_i}$$

As asserted before, the cost per hash rate is constant for all chains under all circumstances

$$c=\frac{C_E}{H_E}=\frac{C_{1559}}{H_{1559}}=\frac{C_X}{H_X}$$

Assuming miners are profit maximizers, we should expect them to allocate their hash rate in such a way that no one has any incentive to switch their mining power between chain X and Ethereum. This implies that the expected profit per hash rate is the same, meaning:

$$\frac{P^_{E}}{H^E}=\frac{P^*{X}}{H^*X} \textrm{ and }\frac{P^{**}{1559}}{H^{}_{1559}}=\frac{P^{}_{X}}{H^{**}_X}$$

Using $H=H^_{E}+H^X$, $H=H^*{1559}+H^*X$ and $P{i}=R_{i}-C_{i}$, we can find the equilibrium hash power of both chains.

Starting with $H^*_E$:

$$\frac{R_E}{H^_E}-c=\frac{R_X}{H^_X}-c$$

Simplifying and rearranging

$$\frac{R_E}{H^_E}=\frac{R_X}{H-H^_E}$$

$$R_E(H-H^_E)=R_XH^_E$$

$$H^*_E=\frac{R_E}{R_X+R_E}H$$

Similarly for $H^{**}_{1559}$:

$$H^{**}{1559}=\frac{R{1559}}{R_X+R_{1559}}H$$