Model network-service unit economics and breakeven

A network service costs $40 per month, with the first 3 months free. Its costs include a variable service cost of $25 for each active month, a one-time $35 installation charge at activation, and a one-time $20 onboarding overhead per new customer. Fixed annual cost is $1,000,000.

a) Assuming an average customer tenure of 12 months, calculate the per-customer contribution (revenue minus variable and one-time costs) and state whether the unit economics are positive or negative; show the formula: $\text{contribution} = \text{revenue} - \text{variable costs} - \text{one-time costs}$.

b) If the result is negative, propose two quantitative levers (such as price changes, fewer free months, or cost reductions) and estimate how much each would need to move to make contribution non-negative.

c) Under a revised policy, customers sign a 21-month contract (the free months still apply); 10% cancel immediately and pay a $100 penalty, while the remaining 90% complete the full 21 months. Compute the expected average contribution per customer and the number of customers required in a year to break even on the fixed cost. Give a general formula first, then substitute the numbers.

d) Identify which single sensitivity—price, churn timing, or cost—most changes breakeven, and explain why.

Overview: This question tests a data scientist's ability to model unit economics, compute per-customer contribution and breakeven, and perform sensitivity and probabilistic churn analysis.