Profit Maximization
Solve MR = MC on a linear demand curve with quadratic costs, with the markup and elasticity.
Demand: P = 100 − 2Q
Cost: TC = 50 + 20Q + Q²
Where marginal revenue meets marginal cost
From the demand curve
TR 977.78 − TC 494.44
Profit against quantity
| Marginal revenue at Q* | 46.6667 | a − 2bQ |
| Marginal cost at Q* | 46.6667 | c + 2dQ — equal to MR, which is the condition |
| Average cost | 37.0833 | TC ÷ Q |
| Average variable cost | 33.3333 | the shut-down comparison |
| Price elasticity at Q* | -2.75 | a monopolist always ends up where demand is elastic |
| Lerner index | 0.3636 | (P − MC) ÷ P, the markup — 0 for a price taker |
| Price-taking output | 40 | what the same costs would produce where P = MC |
A firm maximises profit where marginal revenue equals marginal cost. With demand P = 100 − 2Q and cost TC = 50 + 20Q + Q², that gives Q* = 13.33 at a price of 73.33, for a profit of 483.33.
Why the condition is MR = MC and not P = MC
A firm with any market power faces a downward-sloping demand curve, so selling one more unit means cutting the price — on that unit and on every unit before it. Marginal revenue therefore has two parts: the price of the new unit, minus the revenue given up on all the existing ones. For linear demand P = a − bQ that works out to MR = a − 2bQ, which falls exactly twice as fast as the demand curve.
Setting MR = MC and solving gives Q* = (a − c) ÷ (2b + 2d) for costs of the form TC = F + cQ + dQ². Notice what is absent: the fixed cost F. It shifts profit up or down but never changes the best quantity, because it does not appear in marginal cost. Only the shut-down decision — produce at all, or not — depends on it.
The price comes last
A common slip is to solve MR = MC and read the answer off the MR curve. The quantity is right, but the price is not: MR at the optimum is 46.67 in the worked case while the price is 73.33. The firm chooses a quantity, and the demand curve then says what price that quantity will clear at. Substitute Q* back into P = a − bQ, never into MR.
for demand P = a − bQ and cost TC = F + cQ + dQ²; b = 0 makes the firm a price taker
- 1 Write marginal revenue from the demand curve. P = 100 − 2Q gives TR = 100Q − 2Q², so MR = 100 − 4Q — twice the slope of demand.
- 2 Differentiate total cost for marginal cost. TC = 50 + 20Q + Q² gives MC = 20 + 2Q. The fixed 50 disappears.
- 3 Set them equal. 100 − 4Q = 20 + 2Q, so 80 = 6Q and Q* = 13.33.
- 4 Read the price off the demand curve. P* = 100 − 2(13.33) = 73.33. Do not use the MR curve for this.
- 5 Work out the profit. TR = 977.78, TC = 494.44, so profit = 483.33. Check MR = MC = 46.67 at that quantity.
The same costs under three market structures
Costs TC = 50 + 20Q + Q² throughout. Market power raises the price and cuts the quantity.
| Structure | Demand faced | Q* | P* | Markup (P − MC) ÷ P |
|---|---|---|---|---|
| Monopoly | P = 100 − 2Q | 13.33 | 73.33 | 0.36 |
| Weaker market power | P = 100 − 0.5Q | 26.67 | 86.67 | 0.15 |
| Price taker | P = 100 flat | 40.00 | 100.00 | 0.00 |
The markup, and what it is telling you
The gap between price and marginal cost as a share of price is the Lerner index, and it is not an arbitrary measure: at the profit-maximising quantity it equals exactly −1 ÷ ε, where ε is the price elasticity of demand. A firm facing elastic demand cannot mark up much without losing the sale; one facing inelastic demand can. The tool reports both so you can see the identity hold.
That identity also explains a result students often find surprising: a profit-maximising firm with market power always ends up on the elastic part of its demand curve. On the inelastic part, MR is negative — cutting output would raise revenue and lower cost at the same time — so no firm would ever stop there. Elasticity at the optimum is −2.75 in the worked case, comfortably elastic.
Two limits worth stating. Fixed costs do not change Q*, but they do decide whether producing is worth it at all; with the quadratic cost used here price always covers average variable cost at an interior optimum, so the classic shut-down case cannot arise — showing it needs the U-shaped average variable cost that comes from a cubic cost function. And this is the short run: in the long run entry competes profits away unless something blocks it, which is why a sustained markup is evidence about barriers to entry rather than about clever pricing.