Optimal Price Calculator
Enter your demand curve parameters or two price-quantity observations to find the optimal (profit-maximizing) price for your product. The calculator applies the classic MR = MC rule to compute monopoly price, output, profit, consumer and producer surplus, deadweight loss, and the Lerner Index. Switch between linear-demand mode and elasticity mode using the selector below.
Formula
Worked example
Inverse demand P = 100 - 0.5Q; MC = 20. MR = 100 - Q. Set MR = MC: 100 - Q = 20, so Q* = 80. Optimal price P* = 100 - 0.5 x 80 = 60. Profit = (60 - 20) x 80 = 3,200. Competitive output Qc = (100 - 20) / 0.5 = 160 at P = 20. Deadweight loss = 0.5 x (60 - 20) x (160 - 80) = 1,600.
What is the optimal price?
The optimal price (also called the profit-maximizing price) is the price at which a firm earns the highest possible profit. In a standard market model, profit is maximized where marginal revenue equals marginal cost (MR = MC). Charging more than this price reduces sales by more than it raises revenue; charging less sells more units but at a margin too thin to offset the volume gain. The result, under a linear demand curve, is that the optimal price always sits exactly halfway between the demand intercept and marginal cost: P* = (a + MC) / 2.
How the optimal price formula works
For a linear inverse demand curve P = a - bQ, the corresponding marginal revenue is MR = a - 2bQ (same intercept, twice the slope). Setting MR equal to MC and solving gives the profit-maximizing quantity Q* = (a - MC) / (2b). Substituting back into the demand curve gives the optimal price P* = a - bQ*. When you do not have demand-curve parameters but have observed how quantity responded to a price change, you can estimate the price elasticity of demand using the arc (midpoint) method, then apply the markup rule P* = MC x [PED / (PED + 1)], which produces the same result because MR = P x (1 + 1/PED) and MR = MC at the optimum.
Monopoly vs. competitive pricing
In a perfectly competitive market, price is driven down to marginal cost (P = MC), and deadweight loss is zero. A firm with market power sets price above MC, which raises profit but reduces the quantity exchanged below the socially efficient level. The gap between the monopoly quantity and the competitive quantity creates deadweight loss, a triangle of value that neither buyer nor seller receives. The Lerner Index L = (P - MC) / P measures how far above marginal cost the firm prices, ranging from 0 (perfect competition) to 1 (extreme monopoly power). Economic regulators often target Lerner Index reductions through price caps or antitrust action.
Consumer surplus, producer surplus, and deadweight loss
Consumer surplus is the area below the demand curve and above the price paid; it represents the benefit buyers get beyond what they paid. Producer surplus is the area above the marginal cost curve and below the price received; it represents the seller's gain. Together they form total welfare. Under competitive pricing (P = MC), the sum of consumer and producer surplus is maximized. Monopoly pricing transfers some consumer surplus to the producer and eliminates the triangle of transactions that would have occurred if the price had been lower, creating deadweight loss. This calculator shows all three areas for the linear-demand mode so you can see the welfare trade-off directly.
Lerner Index benchmarks by market structure
| Market structure | Typical Lerner Index | Pricing power |
|---|---|---|
| Perfect competition | 0.00 | None |
| Monopolistic competition | 0.10 - 0.30 | Low |
| Oligopoly | 0.30 - 0.60 | Moderate |
| Regulated monopoly | 0.40 - 0.70 | Significant |
| Unregulated monopoly | 0.50 - 0.90 | High |
| Pure monopoly (zero MC) | 1.00 | Maximum |
The Lerner Index (L) measures market power as the share of price that exceeds marginal cost. L = 0 is perfect competition; L = 1 is a pure monopoly with zero marginal cost.
Frequently asked questions
What is the profit-maximizing rule?
A firm maximizes profit by producing and selling the quantity at which marginal revenue equals marginal cost (MR = MC). At that quantity, selling one more unit would cost more than it earns, and selling one fewer unit would give up more profit than it saves. The price is then read off the demand curve at that quantity.
Why does the optimal price depend on elasticity?
Price elasticity tells you how sensitive buyers are to price changes. The markup rule P* = MC / (1 + 1/PED) shows that when demand is highly elastic (buyers are very price-sensitive), the optimal markup over marginal cost is small. When demand is inelastic (buyers are insensitive to price), a larger markup is sustainable without large quantity losses. A profit-maximizing firm always prices in the elastic range (|PED| > 1) because if demand were inelastic, raising price would increase revenue while quantity falls, so profit would be higher.
What is the Lerner Index?
The Lerner Index is L = (P - MC) / P, where P is the selling price and MC is marginal cost. It measures the fraction of the price that exceeds cost, which is a direct measure of market power. In perfect competition L = 0 because P = MC. A higher index means the firm can charge a larger premium. It equals -1/PED at the profit-maximizing point, so a firm with a Lerner Index of 0.5 faces a price elasticity of -2 at its optimal price.
What is deadweight loss and why does it matter?
Deadweight loss is the value of transactions that do not happen because the seller prices above marginal cost. In the linear-demand model it is a triangle with area 0.5 x (Pm - MC) x (Qc - Qm), where Pm and Qm are the monopoly price and quantity, and Qc is the competitive quantity. These are sales that would benefit both buyer and seller (the buyer values the unit above MC) but do not occur because the seller would have to cut the price for all existing customers to make them. Deadweight loss is the main welfare cost economists use to justify antitrust regulation.
Can the optimal price be below marginal cost?
Not in the standard model: the formula P* = MC x (PED / (PED + 1)) gives a price above MC whenever PED < -1, which is always true at the optimum. However, firms sometimes price below cost as a loss-leader (to attract customers who buy other profitable items) or during aggressive market-entry campaigns. These situations require a more complex multi-product or dynamic model, which goes beyond the single-price profit-maximization framework this calculator uses.
What are the demand intercept and slope, and how do I find them?
The inverse demand curve P = a - bQ says that the market price equals a (the maximum willingness-to-pay) minus b times quantity sold. You can estimate a and b from sales data: if you observe two price-quantity pairs (P1, Q1) and (P2, Q2), then b = (P1 - P2) / (Q2 - Q1) and a = P1 + b*Q1. Alternatively, use the elasticity mode of this calculator, which estimates the relationship from the two observations directly.
Does this calculator work for multiproduct firms?
This tool calculates the optimal price for a single product in isolation, which is the standard textbook case. Multiproduct firms must also account for cross-price elasticities (how the price of product A affects demand for product B) and joint costs. If your products are substitutes or complements, the single-product formula underestimates or overestimates the true optimal price. A multiproduct pricing model is a natural extension of this framework.