Optimize SaaS pricing and profit

A software business charges for one seat per customer each month. Demand follows $q(p)=10{,}000-50p$ units/month, variable cost is $c=\$5$ per seat, and fixed monthly cost is $F=\$100{,}000$. 1) Find the profit-maximizing price $p^*$ and quantity $q^*$, then compute the maximum monthly profit. Show the calculus. 2) With a capacity limit of 6,000 seats/month, determine the price that maximizes profit under that constraint. 3) If a 10% price increase raises monthly churn from 2% to 4% in a subscription setting, explain how your model changes and how you would estimate the LTV-optimized price instead of static monthly profit.

Overview: This question tests pricing optimization, linear demand modeling, capacity-constrained profit maximization, and the effect of subscription churn on lifetime value in an Analytics & Experimentation data-science context.

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