Question. Under the uniswap pricing rule, what determines how much the price of an asset increases when you buy that asset?
Note. while this is a mathematical question, answering it requires an understanding of uniswap. For this reason, I apologise if this question is not best suited to this site!
My (not very successful) attempt.
My understanding of uniswap: Suppose there are two assets, yes shares and no shares (as in the linked post). The automated market maker (AMM) holds reserves of both, denoted $y$ and $n$. If I want to buy \$1 of yes shares, I will convert this into 1 yes share and 1 no share, send these shares to the AMM, and receive back more than 1 yes share. The AMM will give me back the number of yes shares that ensures that the product $y n$ remains equal to a constant $k$. In other words, I will give the AMM 1 no share, and the AMM will give me (on net) how ever many yes shares it needs to ensure that $y n$ doesn't change (recalling that $y$ and $n$ are the AMM's reserve of yes and no shares). These rules implicitly define the price of yes shares to be \begin{equation} p = \frac{n}{n + y} \end{equation}
Analysis. Let's consider a small purchase of yes shares (costing $\Delta$). This will increase $n$ by $\Delta$ and decrease $y$ by some amount. What amount? Since $yn$ is held constant, we have that \begin{equation} \frac{d}{dn} y(n) n = y(n) + n y’(n) = 0 \iff y’ = -\frac{y}{n} \end{equation} Following the change in $y$ and $n$, the price changes by \begin{equation} \frac{dp}{dn} = \frac{y(n)-n y'(n)}{(y(n)+n)^2} \end{equation} Plugging in our expression for $y'$, I get the fairly simple expression \begin{equation} \frac{dp}{dn} = \frac{2 y}{(n+y)^2} \end{equation} This seems to be correct. However, it shows that price sensitivity depends on $y$ and $n$, and I struggle to understand what these depend on (presumably, not just the initial constant $k$ but also the history of trades?) Also, in practice, price sensitivity seems to be depend a lot on the price: for example, if the price is near 1, it's hard to increase it. Is there an easy way to see this?