Monopoly Profit Maximization Calculator
Determine the profit-maximizing price and output level for a monopolist facing a linear demand curve. Useful for economics students, instructors, and analysts modeling imperfect competition scenarios.
About this calculator
A monopolist maximizes profit by producing where Marginal Revenue (MR) equals Marginal Cost (MC). For a linear inverse demand curve P = a − b×Q (where a is the demand intercept and b is the slope), total revenue is TR = a×Q − b×Q², and marginal revenue is MR = a − 2b×Q. Setting MR = MC and solving for the profit-maximizing quantity gives: Q* = (a − MC) / (2 × b). The monopoly price is then found by substituting Q* back into the demand curve: P* = a − b×Q*. Profit is calculated as: π = (P* − MC) × Q* − Fixed Costs. The factor of 2 in the denominator of the quantity formula reflects the fact that the MR curve has twice the slope of the demand curve — a key result in monopoly theory. Unlike competitive firms, monopolists set price above MC, creating a deadweight loss and transferring consumer surplus to producer surplus.
How to use
Suppose demand intercept a = 100, demand slope b = 2, marginal cost MC = 20, and fixed costs = $500. First, find the profit-maximizing quantity: Q* = (100 − 20) / (2 × 2) = 80 / 4 = 20 units. Next, find the monopoly price by substituting into the demand curve: P* = 100 − 2 × 20 = 100 − 40 = $60. Calculate profit: π = (60 − 20) × 20 − 500 = 40 × 20 − 500 = $800 − $500 = $300. The monopolist charges $60 per unit, sells 20 units, and earns $300 in profit after fixed costs.
Frequently asked questions
Why does a monopolist produce where marginal revenue equals marginal cost rather than where price equals marginal cost?
In a competitive market, firms are price-takers and set P = MC because they cannot influence price. A monopolist, however, faces the entire downward-sloping market demand curve — to sell more units, it must lower the price on all units, not just the last one. This means marginal revenue is always less than price for a monopolist. Profit is maximized where the additional revenue from one more unit (MR) exactly equals the additional cost of producing it (MC); producing beyond that point would cost more than it earns. Setting price equal to MC would actually cause the monopolist to earn less profit.
What is the relationship between the MR curve slope and the demand curve slope in monopoly analysis?
For a linear inverse demand curve of the form P = a − bQ, the marginal revenue curve is MR = a − 2bQ. The MR curve shares the same vertical intercept (a) as the demand curve but has exactly twice the slope. This means the MR curve bisects the horizontal distance between the price axis and the demand curve at any price level. Geometrically, this is why the profit-maximizing quantity Q* always lies at the midpoint of the demand curve's horizontal intercept — a useful shortcut for graphical analysis in economics textbooks.
How does a monopoly's profit-maximizing outcome differ from the socially optimal competitive outcome?
In a competitive market, equilibrium occurs where P = MC, maximizing total surplus (consumer plus producer). A monopolist restricts output to Q* and raises price to P*, both relative to the competitive benchmark. This creates deadweight loss — units that would have generated positive net value for society are not produced. Consumer surplus shrinks as some is converted to producer profit and some is simply lost. Regulators may respond with price ceilings, antitrust action, or public ownership to push outcomes closer to the social optimum. The magnitude of the welfare loss depends on the elasticity of demand and the gap between P* and MC.