Optimal Price Calculator
From the unit cost and the price elasticity of demand, find the profit-maximising price using the monopoly pricing formula.
Input Data
Results
At a glance:The monopoly markup rule sets the profit-maximising price from demand elasticity. With elasticity e (negative), optimal price = unit cost × e ÷ (e + 1), equivalently price = cost ÷ (1 + 1/e). The absolute value of e must be greater than 1 for the formula to give a positive price above cost.
Formula
Optimal price = unit cost × elasticity ÷ (elasticity + 1).
Equivalently, optimal price = unit cost ÷ (1 + 1 / elasticity).
$$P^{*} = \\dfrac{MC \\times \\varepsilon}{\\varepsilon + 1}$$How to Use
- Enter the unit (marginal) cost.
- Enter the price elasticity of demand (a negative number, |e| > 1).
- Read the optimal price.
FAQ
Why is the price positive when elasticity is negative?
The price elasticity of demand is usually negative (price up, demand down). With elasticity −2 and cost HK$40: optimal price = 40 × (−2) ÷ (−2 + 1) = −80 ÷ −1 = HK$80. Both numerator and denominator are negative, so the division gives a positive number. As long as |elasticity| > 1 (demand is elastic), (elasticity + 1) is negative and elasticity is negative, so two negatives divide to a positive, sensible price.
Why can't a price be computed when elasticity is −1?
When elasticity is exactly −1, the denominator (elasticity + 1) is zero, giving a divide-by-zero with no mathematical solution (the optimal price tends to infinity). Economically, elasticity = −1 is unit elasticity, where total revenue does not change with price, so there is no clear profit-maximising markup point. In practice, firms with pricing power usually operate in the elastic range (|elasticity| > 1).
Does this formula apply to every business?
No. It assumes the firm has pricing power (monopoly or monopolistic competition) and can influence demand by changing price. In perfect competition the firm is a price taker and must sell at the market price, so the formula does not apply. It also approximates cost with unit marginal cost and assumes stable elasticity, whereas real elasticity shifts with price and time. Treat the result as a markup-pricing starting point, then adjust with market testing and competitive conditions.
How do I estimate the price elasticity to enter, and how much does a wrong estimate matter?
The optimal price depends entirely on the elasticity you enter, so estimating it well is the key practical challenge. The best method is an actual market test (A/B pricing): try different prices across time, regions or customer groups, record the resulting sales volumes, then back out the elasticity with the midpoint method — for which you can use this site's demand-price-elasticity calculator. If testing is impossible, fall back to historical data (compare sales before and after past price changes) or to typical elasticity values for similar products (necessities and goods with few substitutes tend to have low |elasticity|; luxuries and goods with many substitutes tend to have high |elasticity|). A wrong estimate matters a lot and non-linearly: because elasticity sits in the denominator (elasticity + 1), near −1 the denominator approaches zero and the optimal price shoots up — a small error (say −1.5 vs −1.2) can change the price dramatically — whereas in the elastic range (−3, −4) errors matter far less. So: treat the output as a starting point, run optimistic/neutral/conservative elasticity scenarios to see the price range, and finalise with competitors, brand positioning and cost structure. Pair it with this site's demand-price-elasticity and marginal-cost calculators.
Why does 'less elastic demand mean a higher optimal markup', and what does that mean for pricing strategy?
This is the single most important business intuition behind the formula. In P = MC × ε ÷ (ε + 1), the closer |ε| is to 1 (less elastic), the closer the denominator (ε + 1) is to zero, so the optimal price and markup rise; the larger |ε| (more elastic), the closer the price is to cost and the smaller the markup. The reason is the 'cost of raising price': a firm with pricing power faces two opposing forces when it raises price — each unit earns more gross margin (good), but it loses some buyers (bad). The optimal price balances them, and elasticity measures exactly how many sales a price rise loses. If demand is inelastic (customers are price-insensitive), a price rise loses almost no sales, so the benefit of extra margin far outweighs the cost of lost volume, and a big markup is justified; if demand is elastic (very sensitive), a small rise loses many customers, so the firm must keep the markup low, near cost, to retain volume. Strategy implications: (1) to charge high prices, reduce elasticity — through branding, differentiation, uniqueness and loyalty that make customers feel they have no alternative (the logic behind luxury goods and patented drugs); (2) goods with many substitutes and little differentiation can only run thin margins, relying on volume or cost cuts; (3) the same product may have different elasticity across customer segments, enabling differential pricing (business vs leisure travellers, peak vs off-season). In short, elasticity is the core of pricing power — actively manage it rather than blindly adding a fixed percentage to cost.
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References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.