Uniswap vs Curve

Uniswap's constant-product formula is elegantly simple. A pool holds reserves of two tokens (x and y). Their product must always equal a constant k:

When a trader swaps dx of token X into the pool, the new reserves become (x + dx) and y'. The invariant gives us:

The trader receives y − y' tokens. Because this is a hyperbola, larger trades move the price further along the curve, creating slippage proportional to trade size relative to reserves. A 1% pool trade yields roughly 1% price impact; 10% yields roughly 10%.

Uniswap V3: Concentrated Liquidity

V3 lets LPs concentrate their capital within specific price ranges. Instead of spreading liquidity from 0 to infinity, an LP providing liquidity between prices p_a and p_b acts like a much larger V2 position. This multiplies effective depth by up to 4000x for tight ranges, dramatically reducing price impact for trades within that range. The trade-off: if the price moves outside the LP's range, they earn zero fees and hold 100% of the depreciating asset.

Curve's StableSwap invariant is a hybrid between constant-sum (x + y = D, zero slippage) and constant-product (x × y = (D/2), Uniswap-like). The full invariant for n=2 tokens:

The amplification parameter A controls the blend:

• A = 0: Behaves exactly like Uniswap's constant-product curve
• A -> ?: Approaches constant-sum (x + y = D), meaning near-zero slippage
• Typical A: 50-200 for stablecoin pools, creating a flat region near the peg where trades have minimal impact

This is why Curve dominates for stablecoin swaps. Near parity (USDC ? USDT ? 1.00), the curve is nearly flat - meaning massive trades create almost no slippage. But if the peg breaks, the curve steepens rapidly to protect LPs from total loss.

Why Not Use High-A for Everything?

A high amplification parameter assumes the assets should be priced equally. For volatile pairs like ETH/USDC, a high A would cause catastrophic losses during price movements. Curve V2 introduced a "crypto-pool" invariant that dynamically adjusts the concentration point, but Uniswap V3 concentrated liquidity remains more flexible for volatile pairs.