Understanding the Constant Product Formula in AMMs (Without Getting Tricked by k)

Source: Dev.to

The Core Idea: What Does x * y = k Actually Mean?

In a constant‑product AMM:

x * y = k

• x = reserve of token A
• y = reserve of token B
• k = a constant

During a swap, k must remain constant. The reserves change, but the product of the reserves stays the same—that’s the entire pricing mechanism.

Initial State

uint256 reserveA = 1000;
uint256 reserveB = 1000;

Compute k:

uint256 k = reserveA * reserveB; // 1_000_000

This k defines the curve the pool must stay on.

A User Swaps Token A for Token B

The user swaps:

uint256 amountAIn = 100;

Step 1: Use the Current Invariant

uint256 k = reserveA * reserveB; // 1,000,000

This is the only valid k for this swap.

Step 2: Add the Incoming Tokens

uint256 newReserveA = reserveA + amountAIn; // 1,100

We now know the new A reserve but not the new B reserve.

Step 3: Solve for the New B Reserve

To preserve the invariant:

uint256 newReserveB = k / newReserveA; // ≈ 909.09

Step 4: Calculate What the User Receives

uint256 amountBOut = reserveB - newReserveB; // ≈ 90.91

The user receives ~90.91 token B, and the pool stays on the same curve.

Verifying the invariant

1100 * 909.09 ≈ 1,000,000

The invariant holds.

The Common Mistake: Recomputing k

A lot of people instinctively try this instead:

uint256 newK = newReserveA * reserveB; // 1,100 * 1,000 = 1,100,000
uint256 newReserveB = newK / newReserveA; // = 1,000
uint256 amountBOut = reserveB - newReserveB; // = 0

The user gets nothing because this logic moves the pool to a new curve, violating the fundamental invariant.

The Real Swap Equation

The actual AMM equation during a swap is:

(reserveA + amountIn) * (reserveB - amountOut) = reserveA * reserveB

Solving for amountOut:

uint256 amountOut = reserveB - (reserveA * reserveB) / (reserveA + amountIn);

That is exactly what Uniswap‑style contracts implement—no magic, just algebra.

Why the Old k Must Be Used

• k represents the curve.
• A swap moves the pool along the same curve.
• Adding liquidity creates a new curve.
• Swaps do not create a new curve.

If you recalculate k during a swap, you’re jumping to a different curve, and pricing instantly breaks.

Mental Model That Helps

• Think of k as a rail track.
• Liquidity providers lay down new tracks (new k).
• Traders move along the existing track.
• You never redraw the track mid‑swap.

Takeaway

• k is not recomputed during swaps.
• The invariant defines the pricing curve.
• Swaps must start and end on the same curve.
• Using a “new k” destroys the AMM logic.

If this ever feels counterintuitive again, just remember:

Swaps move the pool along the curve — they don’t redraw it.