Compute profit and break-even for memberships

Consider a grocery loyalty program. Assume gross margin rate before discounts m = 28%. Members receive a 10% discount on all eligible spend. For a typical customer, baseline annual spend before joining is S0 = $2,200 and joining lifts their annual spend by u = 6% (so post‑join spend S1 = S0×(1+u)). Program operating cost is $5 per member per month and acquisition cost is $15 one‑time. There is no membership fee for the basic tier. (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. State sign (profit or loss). (b) The company adds a Premium tier: annual fee F = $60, additional benefits cost b = $24/year, total discount for Premium is d = 15%, and 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* where 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.