Determine Optimal Marketing Budget Allocation for Maximum Profit

You are provided with performance metrics for three marketing acquisition channels: Phone Calls, Social Media Ads, and Email Blasts. For each channel, you have data on Click-Through Rate (CTR), Conversion Rate (CVR), Revenue-Per-Cost (RPC), and an individual spend cap.

If RPC is not directly provided, it can be derived from CTR, CVR, average revenue per conversion (R), and the channel's pricing model (e.g., CPM, CPC, cost per send).

Assume linear returns up to each channel's spend cap (no saturation within the cap) and that all rates are stable over the budget range considered.

Tasks:

1. Given a total budget B, compute the expected net profit for each channel if an amount s is allocated to that channel. For this part, consider the scenario where all of B is spent in a single channel, ignoring caps.
2. Using the conversion rate for the Phone Calls channel, estimate:
   • The average cost per inbound phone call.
   • The average profit per inbound phone call.
3. With a fixed overall budget B and considering individual channel spend caps, determine the optimal spending allocation across channels to maximize total profit. Explain your reasoning.

Constraints & Assumptions:

• Assume linear returns up to each channel's spend cap.
• Assume all rates are stable over the budget range.
• If RPC is not directly provided, it can be derived from CTR, CVR, average revenue per conversion (R), and the channel's pricing model.