What you pay for, and why loops cost more

You build an agent. In testing it takes 10 steps and costs a few cents. Then you let it run on harder tasks, 40 steps, and the cost is not four times higher but about twelve times higher. Why?

For one call: cost = input tokens x input price + output tokens x output price. A call with 2,300 input and 150 output tokens costs 2,300 x 3 / 1,000,000 + 150 x 15 / 1,000,000 = $0.00915 at the example prices. A test checks this to the cent.

Why a loop re-sends everything

An agent works in a loop: call the model, run a tool, add the result to the conversation, call the model again. The model has no memory between calls, so every call re-sends the whole conversation.

Suppose the system prompt is 2,000 tokens, the user's task is 300, each model output is 150 and each tool result is 800. Step 1 sends 2,300 tokens. Step 2 sends those plus 950 (one output and one tool result), which is 3,250. Step i sends 2,300 + (i - 1) x 950 tokens. Add them up (real output of the example):

Going from 20 to 40 steps raises the cost 3.6 times, and from 40 to 80 steps 3.8 times. The total input follows n x (S + U) + R x n(n - 1) / 2, where S + U is the system prompt plus task (2,300) and R is the growth per step (950). That is a quadratic: the second term grows with the square of the steps. A test confirms the formula for ten steps (65,750 tokens), and I confirmed it for 80 (3,186,000).