Grocery Loyalty Program Economics — Year‑1 Incremental Profit and Premium Break‑Even

Context: You are evaluating the Year‑1 unit economics of a grocery loyalty program for individual customers. Assume one full year of participation (12 months) for costs. Baseline = behavior if the customer does not join.

Inputs (unless otherwise stated):

• Gross margin rate before discounts: m = 28%.
• Basic tier discount on eligible spend: 10%.
• Baseline annual spend before joining: S0 = $2,200.
• Uplift in annual spend when joining Basic: u = 6%.
• Program operating cost: $5 per member per month.
• Acquisition cost: $15 one‑time.
• No membership fee for Basic.

(a) Compute the Year‑1 incremental profit per Basic‑tier member versus the counterfactual of not joining. Use: Incremental profit = [m × S1 − (discount rate) × S1] − m × S0 − operating cost − acquisition cost, where S1 = S0 × (1 + u). State sign (profit or loss).

(b) Premium tier addition:

• Annual fee F = $60; additional benefits cost b =$ 24/year.
• Total discount for Premium d = 15%.
• Premium increases spend by u_p = 12% over the non‑member baseline.

Derive and compute the minimum baseline annual spend S0* at which upgrading a non‑member to Premium breaks even in Year‑1. Use: Incremental profit_premium = F − b − d × S1_p + m × (S1_p − S0), where S1_p = S0 × (1 + u_p). Solve for S0* such that Incremental profit_premium = 0.

(c) Sensitivity: By how much (in dollars of S0) does S0* change if m increases by +2pp or if u_p decreases by −3pp? Provide the formula you use and numeric answers.