Constant Product Formula

The constant product formula is the pricing rule at the heart of most AMM-based decentralized exchanges. A liquidity pool holds reserves of two tokens, and their product must remain constant through every trade.

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x * y = k

where x and y are the reserve quantities and k is the invariant. Uniswap V2, SushiSwap, and most early AMMs use this formula. Slippage and impermanent loss are both consequences of this single constraint rather than separate mechanisms.

Intuition#

Take a pool with 10 ETH and 25,000 USDC. The product is 250,000. Buying ETH means adding USDC and removing ETH, and the invariant is what bounds the result: the ETH reserve can never reach zero, because as it shrinks the price rises toward infinity along a hyperbolic curve. At the other end, a small USDC deposit into a deep pool barely moves the price at all.

This creates a natural supply-demand curve: scarce assets become expensive, abundant assets become cheap. No order book, no market maker, no off-chain infrastructure – just arithmetic.

Swap mechanics#

A trader wants to buy Token Y by sending dx of Token X to the pool. Before the trade, reserves are (x, y) with x * y = k. After the trade:

1. New Token X reserve: x' = x + dx
2. New Token Y reserve: y' = k / x' = k / (x + dx)
3. Tokens out: dy = y - y' = y - k / (x + dx)

Simplifying:

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dy = (y * dx) / (x + dx)

This is the output amount before fees. In practice, AMMs charge a fee (typically 0.3%) on the input. Uniswap V2 applies the fee by treating the effective input as dx * (1 - fee):

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dy = (y * dx * 997) / (x * 1000 + dx * 997)

The fee stays in the pool, slightly increasing k after every trade. This is how LPs earn income.

Price and marginal price#

The spot price of Y in terms of X is the ratio of reserves:

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price_Y = x / y

This is also the limit of dy/dx as the trade size approaches zero. For any finite trade, the effective price is worse than the spot price – this difference is slippage.

Slippage#

Because the curve is convex, larger trades move the price further. The price impact of a trade scales roughly with the trade size relative to the pool reserves.

For a small trade dx << x, the output is approximately:

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dy ≈ (y / x) * dx

which is just the spot price times the input – minimal slippage. For larger trades the effective price is worse than spot by a factor of 1 / (1 + dx/x). A trade taking 10% of the Token X reserve therefore pays about 9% above spot, and a $10K swap against a pool holding $100M of the input token pays about 0.01% above it — three orders of magnitude apart on the same formula.

That ratio is what a pool’s total value locked (TVL) figure feeds into. Depth is not a separate quality of a pool; it is the denominator in every price a trader gets.

Worked example#

Pool: 10 ETH and 25,000 USDC. k = 250,000. Spot price: 2,500 USDC/ETH.

A trader swaps 2,500 USDC for ETH (ignoring fees for clarity):

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new USDC reserve = 25,000 + 2,500 = 27,500
new ETH reserve  = 250,000 / 27,500 ≈ 9.0909
ETH received     = 10 - 9.0909 ≈ 0.9091

The trader paid 2,500 USDC for ~0.91 ETH, an effective price of ~2,750 USDC/ETH – about 10% worse than the spot price. That is the cost of trading a significant fraction of the pool in a single swap.

After the trade, the new spot price is 27,500 / 9.0909 ≈ 3,025 USDC/ETH. The price has shifted, creating an arbitrage opportunity for anyone with access to a cheaper ETH source.

Connection to impermanent loss#

The constant product formula forces the pool to rebalance on every trade. When the external price of ETH rises, arbitrageurs buy cheap ETH from the pool until the pool’s price matches the market. The pool ends up holding less ETH and more USDC than it started with. An LP who deposited at one price ratio and withdraws at another always ends up worse off than someone who simply held the original tokens. This is impermanent loss, and it is a direct consequence of the constant product invariant.

Limitations#

• Capital inefficiency. The formula spreads liquidity across all prices from zero to infinity. Most of that liquidity sits at prices far from the current market and is never used. Concentrated liquidity (Uniswap V3) addresses this by letting LPs specify a price range.
• Slippage on large trades. Thin pools produce poor execution for sizable swaps. Aggregators like 1inch mitigate this by splitting trades across multiple pools.
• Not ideal for correlated assets. Stablecoin-to-stablecoin swaps suffer unnecessary slippage under x * y = k. Curve’s StableSwap invariant handles this case better by flattening the curve near the 1:1 peg.

Other invariants for comparison#

External links#

• Uniswap V2 whitepaper
• Uniswap V2 docs – How Uniswap works
• Vitalik Buterin – On Path Independence (2017) – early exploration of AMM bonding curves