Profit Decomposition, Attribution, Experiment Design, and Diagnostics

Context and Assumptions (to make the task self-contained)

We analyze why daily profit fell on a “mixed” coupon day even though the restaurant had more tables and higher average spend per table. We define daily profit as contribution margin = net revenue after discounts/commissions minus variable costs, ignoring fixed costs.

Assumptions (typical coupon economics; adjust if your prior scenario differs):

• Variable cost rate (food + variable labor) = 40% of gross spend.
• Non-coupon table revenue = 100% of gross spend.
• Coupon economics: customer receives 50% off; the platform takes 20% commission on the discounted amount. Net to the restaurant = 0.5 × (1 − 0.20) = 40% of gross spend.
• Baseline day: 20 tables, average gross spend $30, no coupons.
• Mixed day: 25 tables, average gross spend $36, with 10/25 coupon tables.

Tasks

1. Decompose the change in daily profit into volume effect, spend effect, and mix effect using a stepwise counterfactual:
   • (a) Baseline: 20 tables, $30, no coupons
   • (b) Change only tables to 25 (keep $30, no coupons)
   • (c) Change only spend to $36 (25 tables, no coupons)
   • (d) Apply observed mix: 10/25 coupon tables at $36 Quantify each step’s contribution and verify the final profit difference.
2. Attribute the loss primarily to which factor(s) with numeric justification.
3. Design an experiment to measure cannibalization and long-term value of coupon customers. Specify:
   • Randomization unit (e.g., day-of-week within store, ZIP-level or DMA holdout)
   • Treatments (e.g., discount depth, commission share, minimum spend, new-customer-only)
   • Primary metric (e.g., daily contribution margin), guardrails (e.g., utilization, service time), and sample-size/duration assumptions.
4. Propose a diagnostic dashboard: daily KPIs and derived ratios to catch issues early and prevent future surprises.