Scenario

You have three user-acquisition platforms: Live Phone Calls, Social Media Ads, and Email Campaigns. You are given a total marketing budget B. For each platform i, you know its cost per acquisition (CPA_i) and revenue per acquisition (RPA_i). Each platform also has a maximum spend cap Cap_i.

Additionally, you have funnel metrics for the Live Phone Calls channel: click-through rate (CTR), click-to-call rate (q_call|click), and call-to-purchase (acquisition) rate (q_acq|call). Media buying cost may be given as cost per click (CPC) or cost per thousand impressions (CPM).

Assumptions (explicit):

• CPAs and RPAs are average marginal values and remain constant within the relevant spend ranges (no diminishing returns within caps).
• Budget is divisible (fractional expected acquisitions are acceptable).
• For the phone-call funnel, if CPC is not given but CPM is, use CTR to derive CPC.

Questions

1. If you allocate the entire budget B to a single platform i, compute its expected net profit.
2. Using click-through and conversion rates for the Live Phone Calls funnel, estimate the average marketing cost incurred and expected profit earned from a single inbound phone call.
3. Each platform has a maximum spend cap. Given the same budget B and caps Cap_i, determine the optimal allocation across platforms to maximize total profit.

Hints

• Units acquired = budget / CPA.
• Profit = units × (RPA − CPA).
• For allocation with caps, rank platforms by profit per dollar of spend and fund greedily until caps or budget are exhausted.