How constant-product pools price a swap

A constant-product AMM constrains two token reserves with an invariant commonly written as x × y = k. Selling one token increases its reserve and removes an amount of the other consistent with the rule. Larger inputs move further along the curve.

Calculate a fee-free swap

Take a hypothetical pool with 2,000 A and 6,000 B. Its product is 12,000,000. If a trader adds 100 A with no fee, the new A reserve is 2,100. The B reserve becomes 12,000,000 ÷ 2,100, or approximately 5,714.286 B.

The trader receives 6,000 − 5,714.286 = 285.714 B. Although the initial marginal relationship was 3 B per A, the average result is roughly 2.85714 B per A. The trade did not exchange every unit at the initial reserve ratio.

The general fee-free output formula is output = y × input / (x + input), where x is the input-token reserve and y the output-token reserve.

Introduce an input-side fee

For an illustrative fee f, the amount used to price the exchange is effective input = input × (1 − f). Using an invented fee of 0.2%, the 100 A input contributes 99.8 A to the pricing formula. Output is then 6000 × 99.8 / 2099.8, approximately 285.170 B.

This is a simplified mathematical model, not a current fee schedule. In a design that retains the fee in the pool, actual post-trade reserves include the full deposited input, so the reserve product can increase. “Constant product” names the pricing invariant; it does not imply that k remains numerically unchanged forever despite fees and liquidity changes.

Uniswap's v2 pricing guide describes reserve-based pricing and fee-adjusted invariant checks. Other AMM families use different invariants, so this formula should not be applied indiscriminately to stable or weighted pools.

Why a router cares

The output function is nonlinear. Doubling the input does not double the output at an unchanged average rate. A router can compare several pools and allocate input according to the curve each offers.

A pool's current ratio is therefore useful context but insufficient for quoting a finite trade. The calculation requires reserves, direction, size and the applicable fee rule. Real contract arithmetic also introduces integer rounding, which these decimal examples intentionally simplify.