The constant product formula, expressed as $x \cdot y = k$, is the foundational mathematical principle behind Automated Market Makers (AMMs) like Uniswap v1. This formula defines the relationship between the reserves of two assets in a liquidity pool.

How It Works

In this model:

• x represents the quantity of one asset in the pool (e.g., ETH)
• y represents the quantity of the other asset (an ERC20 token)
• k is the "constant product" or invariant, which must remain constant during a trade (before fees)

Price Curve

The formula creates a hyperbolic bonding curve for asset prices. When a trader swaps an amount Δx of asset X for an amount Δy of asset Y, the new reserves must satisfy:

$(x + \Delta x) \cdot (y - \Delta y) = k$

Key Implications

Infinite Liquidity: The curve asymptotically approaches the axes but never touches them, meaning a pool can theoretically never be fully depleted of one asset through trading alone.

Price Impact: Larger trades have increasingly worse prices due to the hyperbolic curve, creating natural slippage.

Fee Accumulation: A trading fee (e.g., 0.30%) is added back to the reserves after each trade, causing k to increase over time, which rewards liquidity providers.

Security Considerations

The constant product formula is deterministic and can be manipulated within a single transaction, making it unsuitable as a price oracle without additional safeguards like time-weighted average prices (TWAP).