Profit-Maximizing Price with Costs and a Demand Curve
You sell a single software product at one price P. You are given fixed cost F, variable cost as either constant marginal cost c or a known variable cost function, and an estimated price-demand relationship.
Derive the profit-maximizing quantity and price, illustrate with a small numeric example, and explain what additional data you would request before recommending a price change.
Constraints & Assumptions
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State whether demand is given as inverse demand P(Q) or demand Q(P).
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Fixed cost affects profitability and break-even but does not usually affect the unconstrained optimal price.
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Check second-order conditions and feasibility constraints.
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Treat the demand estimate as uncertain and validate before launch.
Clarifying Questions to Ask
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What is the demand curve form, and how was it estimated?
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Are there capacity, contract, regulatory, competitive, or fairness constraints?
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Are variable costs constant or changing with scale?
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Is the goal profit, revenue, market share, customer lifetime value, or long-term growth?
Part 1 - General Derivation
Derive the profit-maximizing quantity and price.
What This Part Should Cover
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Define profit as revenue minus fixed and variable costs.
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Use marginal revenue equals marginal cost for an interior optimum.
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Translate Q* into P* through the demand curve.
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Check feasibility, boundary cases, and second-order conditions.
Part 2 - Common Demand Forms and Example
Provide closed-form solutions for linear and constant-elasticity demand, then give a small numeric example.
What This Part Should Cover
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For linear inverse demand, derive MR and solve against MC.
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For constant-elasticity demand, use the markup rule when elasticity is greater than one in magnitude.
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Show the numeric steps clearly and verify profit, not just revenue.
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Mention that fixed cost does not move the unconstrained optimum but matters for whether the business is viable.
Part 3 - Additional Data and Validation
List the data or analyses needed to de-risk the pricing recommendation.
What This Part Should Cover
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Request price experiments, historical price variation, competitor prices, customer segments, churn, acquisition, costs, capacity, and elasticity uncertainty.
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Validate with randomized tests, geo tests, cohort analysis, or conjoint/survey evidence where appropriate.
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Include cannibalization, long-term retention, willingness to pay, and fairness or regulatory constraints.
Follow-up Questions
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What if the estimated elasticity is below one in magnitude?
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How would you price differently by customer segment?
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How would you handle a competitor response to the price change?