Calculating Price Monopoly No Variable Cost
Use this premium calculator to find the profit-maximizing monopoly price when variable cost is zero. Enter a linear inverse demand curve in the form P = a – bQ, add any fixed cost, and instantly see monopoly quantity, price, revenue, profit, and a chart of demand, marginal revenue, and marginal cost.
This is the price when quantity is zero.
Price falls by b for each extra unit sold.
Optional. Used only for profit, not for MR = MC.
Choose the symbol used in the results.
Expert guide to calculating price monopoly no variable cost
When economists talk about a monopoly with no variable cost, they are describing a firm that can produce additional units at essentially zero marginal production expense. This setup appears in digital products, software licensing, streaming content, online subscriptions, patents, and some information goods where the expensive part is the initial creation or platform build, not the next copy sold. In that setting, the core pricing question is not “what does one more unit cost to make?” but rather “what quantity maximizes revenue and, after fixed costs, profit?”
The logic is elegant. A monopolist chooses output where marginal revenue equals marginal cost. If variable cost is zero, then marginal cost is zero. That means the firm expands output until marginal revenue falls to zero. For a linear inverse demand curve written as P = a – bQ, the corresponding marginal revenue curve is MR = a – 2bQ. Setting MR = 0 gives the monopoly quantity:
Q* = a / (2b)
Substitute that quantity back into demand and you get the monopoly price:
P* = a – bQ* = a / 2
This is why a zero variable cost monopoly under linear demand often produces a remarkably simple result: the profit-maximizing price equals half of the demand intercept. Total revenue is then TR = P* × Q*, and profit is TR – Fixed Cost because variable cost is assumed to be zero.
Why the no variable cost assumption matters
Many business owners confuse zero variable cost with zero total cost. They are not the same. A SaaS company may spend heavily on engineering, servers, customer acquisition, legal compliance, and product design. Those are often fixed or quasi-fixed over a relevant range. But once the product exists, serving one more digital user may add almost no incremental cost. That is exactly why monopoly-style pricing models are useful in technology and intellectual property settings.
- No variable cost means marginal cost is zero for each additional unit.
- Fixed cost still matters because it affects profit even though it does not change the MR = MC output rule.
- Demand determines price because the monopolist faces the market demand curve.
- Revenue maximization and profit maximization align on output when variable cost is zero, though fixed cost still changes whether profit is positive.
The exact step by step calculation
- Write inverse demand as P = a – bQ.
- Derive marginal revenue as MR = a – 2bQ.
- Set MR = MC. With no variable cost, MC = 0.
- Solve a – 2bQ = 0 to get Q* = a / (2b).
- Insert Q* into demand: P* = a – b(a / 2b) = a / 2.
- Calculate total revenue: TR = P* × Q*.
- Calculate profit: Profit = TR – Fixed Cost.
This calculator automates each step. It also visualizes the demand curve, the marginal revenue curve, and the horizontal marginal cost line at zero. That chart is especially useful because it makes the monopoly solution intuitive: the optimal quantity is where the MR curve crosses the zero-cost line, and the price is then read upward from the demand curve at that quantity.
Worked example
Suppose demand is P = 100 – 2Q and fixed cost is 500. Because variable cost is zero, marginal cost is zero. Marginal revenue is MR = 100 – 4Q. Setting MR = 0 gives Q* = 25. Price is P* = 100 – 2(25) = 50. Total revenue is 50 × 25 = 1,250. Profit is 1,250 – 500 = 750.
Notice what happened: the monopolist did not produce until price was zero. It stopped earlier because selling more units would have forced price down enough to reduce marginal revenue below zero. That is the central insight in monopoly pricing with no variable cost. Even if one more unit is free to produce, it can still be unprofitable to expand because lowering price applies to all units sold.
Comparison table: monopoly outcomes under different demand curves
| Inverse Demand | Fixed Cost | Monopoly Quantity Q* | Monopoly Price P* | Total Revenue | Profit |
|---|---|---|---|---|---|
| P = 100 – 2Q | 500 | 25.00 | 50.00 | 1,250.00 | 750.00 |
| P = 120 – 3Q | 600 | 20.00 | 60.00 | 1,200.00 | 600.00 |
| P = 80 – Q | 900 | 40.00 | 40.00 | 1,600.00 | 700.00 |
| P = 60 – 0.5Q | 300 | 60.00 | 30.00 | 1,800.00 | 1,500.00 |
The statistics in the table show how the two demand parameters shape the result. A larger intercept raises the feasible monopoly price, while a steeper slope reduces the optimal quantity. Because there is no variable cost, revenue is the main driver of output choice. Fixed cost then determines whether the business model generates attractive economic profit.
Elasticity and the monopoly rule
Another way to understand this topic is through elasticity. For a monopoly, the Lerner index connects markup to elasticity. In a more general setting, the firm prices where the percentage markup over marginal cost relates inversely to the absolute value of demand elasticity. With zero marginal cost, the markup can look extremely large because the denominator is zero. For linear demand, the monopoly point under zero marginal cost occurs exactly at the unit elastic midpoint of the revenue schedule. That is why marginal revenue becomes zero there.
| Concept | Value at Monopoly Optimum with MC = 0 | Interpretation |
|---|---|---|
| Marginal Revenue | 0 | Output expands until one more unit adds no extra revenue. |
| Price | a / 2 | The monopoly price equals half of the choke price under linear demand. |
| Quantity | a / (2b) | Output is half of the competitive quantity when MC = 0 and demand is linear. |
| Elasticity at optimum | |E| = 1 | The firm stops at the unit elastic point because total revenue peaks there. |
How to interpret the chart
The chart generated by this calculator plots three economically meaningful lines. The Demand line slopes downward because lower prices are required to sell more quantity. The MR line is steeper and lies below demand because cutting price to sell one more unit lowers revenue on all previous units too. The MC line sits at zero because there is no variable cost. The intersection of MR and MC gives Q*. Moving vertically from that quantity to the demand curve gives P*.
In practical business analysis, this visual helps answer strategic questions such as:
- How sensitive is optimal price to a change in willingness to pay?
- How much does quantity expand if demand becomes flatter?
- Can high fixed costs still be justified under a zero marginal cost model?
- What happens if the market becomes more competitive and the firm loses monopoly power?
Common mistakes when calculating monopoly price with no variable cost
- Using average cost instead of marginal cost. Fixed cost affects profit but does not determine the MR = MC output in this case.
- Forgetting to double the slope in marginal revenue. If demand is P = a – bQ, then MR is a – 2bQ, not a – bQ.
- Assuming zero variable cost means price should be zero. A monopolist restricts quantity to maximize profit, not to maximize output.
- Ignoring the domain of demand. Quantity cannot be negative, and the model only makes sense while price remains nonnegative.
- Overlooking regulation or antitrust limits. Real-world market power can be constrained by law, entry, contracts, and reputational issues.
Business applications
This framework is highly relevant for digital economics. Streaming platforms, e-books, software licenses, mobile apps, cloud-based tools, and AI subscriptions often involve very low incremental delivery costs. The initial investment may be substantial, but once the platform exists, the cost of serving the next user can be close to zero over a range. That makes monopoly pricing logic especially valuable when a company owns protected intellectual property, has strong network effects, or controls a niche market.
Still, executives should not treat the model as a complete pricing system. Real businesses also consider churn, customer lifetime value, versioning, competition, platform fees, legal risk, price discrimination, and capacity constraints. The zero variable cost monopoly formula is a powerful baseline, but it is still a simplified baseline.
Policy and market power context
Because monopoly pricing can raise prices above competitive levels and reduce output, it remains a central concern in antitrust economics and public policy. If you want to study how market power is assessed and why regulators care about concentration and pricing incentives, the following authority sources are useful starting points:
- U.S. Department of Justice: Herfindahl-Hirschman Index guidance
- Federal Trade Commission: Guide to antitrust laws
- MIT OpenCourseWare: Principles of Microeconomics
Final takeaway
If you need a fast and accurate method for calculating price monopoly no variable cost, remember the core chain of logic: start from inverse demand, derive marginal revenue, set MR equal to zero, solve for quantity, and then read price from demand. Under linear demand P = a – bQ, the answers are beautifully compact: Q* = a / (2b) and P* = a / 2. Revenue is P*Q*, and profit is revenue minus fixed cost. That is exactly what the calculator above computes.