Cost per Call versus a Human Agent

Finance asks whether the voice agent saves money. Assumptions (not measured): ASR USD 0.006 per minute, TTS USD 15 per million characters at 170 characters per agent turn, telephony USD 0.014 per minute, LLM USD 1 per million input and USD 4 per million output tokens, 1,500 tokens of system prompt and tool schemas, history growing 95 tokens a turn, 6 turns a minute, 60 output tokens per call, 30 percent of turns making a second LLM call with about 250 extra tokens. A contained call lasts 4 minutes, an escalated call has 2.5 minutes of agent time, then a human. A human costs USD 30 an hour loaded at 75 percent occupancy (USD 0.681 a minute with telephony); average human handle time is 6 minutes (USD 4.08); escalated calls need 8 human minutes with a context packet. There are 300,000 calls a month.

1. Build the per-call cost of a contained call. Which line dominates, and how does the context grow?
2. Write the cost per call as a function of the containment rate and derive the break-even containment in closed form.
3. How sensitive is the saving to containment, handle time and recontact? What is the cost of a cold transfer?
4. What would you tell finance about the ASR vendor's price?