Source: Demand Analysis and Optimal Pricing lecture, Texas A&M
Tags: margin, mark-up, markup pricing, MR equals MC, Lerner index, rule of thumb pricing, full-cost pricing, elasticity and pricing, optimal price, supermarket vs convenience store
Real-world firms rarely think in terms of MR = MC, but the mark-up they apply over costs is mathematically equivalent to the MR = MC rule once you account for elasticity. The margin formula (P – MC)/P = 1/|Ep| links a firm's pricing power directly to how elastic its demand is. More elastic demand means less pricing power and a thinner margin.
Margin (Lerner index)
(P – MC) / P. The fraction of the price that is profit above marginal cost. Sometimes called "margin over price."
Mark-up
(P – MC) / MC. The fraction added on top of cost. Sometimes called "mark-up over cost." Different from margin: same numerator, different denominator.
Marginal revenue (MR)
The additional revenue from selling one more unit. For a linear demand curve with inverse demand P = a – bQ, MR = a – 2bQ (same intercept, twice the slope).
Full-cost pricing
Setting price as a mark-up over average cost (rather than marginal cost). Common in practice because average cost is easier to measure. Less precise than margin pricing but still effective when the mark-up reflects elasticity.
Rule of thumb pricing
Using the margin formula P = MC / (1 – 1/|Ep|) to set price. Translates elasticity directly into a pricing multiple over marginal cost.
The profit-maximising condition is MR = MC.
It can be shown (via calculus on the revenue function) that MR = P(1 + 1/Ep), where Ep is the (negative) price elasticity.
Setting MR = MC:
P(1 + 1/Ep) = MC
Rearranging:
(P – MC)/P = 1/(–Ep) = 1/|Ep|
This is the Lerner index. It says the optimal margin over price equals the reciprocal of the absolute value of elasticity.
Margin = (P – MC) / P (denominator is price)
Mark-up = (P – MC) / MC (denominator is cost)
Numerical example: P = $5, MC = $3
Margin = (5 – 3)/5 = 40%
Mark-up = (5 – 3)/3 = 67%
The Lerner index formula gives you margin directly. There is no single clean formula that gives mark-up from elasticity alone, but once you know P and MC you can compute it.
Supermarkets face highly elastic demand (many competing stores). Suppose Ep = –10.
Margin = 1/10 = 10%
From (P – MC)/P = 0.1: rearranging gives P = (10/9) × MC ≈ 1.11 × MC
Mark-up = (P – MC)/MC = 1/9 ≈ 11.1%
Convenience stores face less elastic demand (customers value speed and proximity over price). Suppose Ep = –5.
Margin = 1/5 = 20%
P = (5/4) × MC = 1.25 × MC
Mark-up = 25%
Intuition: convenience store shoppers have fewer substitutes at the moment of purchase, so the store can charge a higher mark-up.
A convenience store earns more per item than a supermarket.
But the supermarket sells far more units.
Profit = margin per unit × quantity. The supermarket can earn greater total profit despite thinner margins.
If Ep = –1/2, the margin formula gives (P – MC)/P = 1/(1/2) = 2, which implies P – MC = 2P, so P = –MC.
A negative price is nonsensical. This is the formula's way of telling you that you have not yet raised price enough.
On the inelastic portion, raising price increases revenue (quantity barely falls) and reduces costs (fewer units to produce). Both effects raise profit. Keep raising price until you reach the elastic region.
Marginal cost is difficult for most firms to calculate. In practice, firms use average cost (ingredient cost, unit cost) as the base.
They then add a percentage mark-up.
This is called full-cost pricing. It is less precise than the Lerner-index approach but works well when the mark-up percentage reflects the elasticity of the product.
Example: the mark-up on steak is larger than the mark-up on ground sirloin, partly because of higher input costs, but also because steak faces less elastic demand (fewer close substitutes at the same quality level).
MR for linear inverse demand P = a – bQ: MR = a – 2bQ
MR in terms of elasticity: MR = P(1 + 1/Ep)
Lerner index (margin over price): (P – MC)/P = 1/|Ep|
Price as a function of MC and elasticity: P = MC / (1 – 1/|Ep|)
Equivalently: P = MC × |Ep| / (|Ep| – 1)
Mark-up over cost: (P – MC)/MC (no direct single-variable elasticity formula; compute from P and MC)
⚠️ Margin and mark-up use the same numerator (P – MC) but different denominators. Exam questions often test whether you know which is which. Margin divides by P; mark-up divides by MC.
⚠️ The margin formula only applies on the elastic portion of the demand curve (|Ep| > 1). If |Ep| ≤ 1, the formula produces nonsensical results, which is your signal that the firm is not yet at its optimal price.
⚠️ "Rule of thumb" pricing is not a separate theory. It is MR = MC expressed in a form that business owners can use: take your cost, apply a multiplier derived from elasticity.
⚠️ Full-cost pricing uses average cost, not marginal cost. It is a practical approximation. The lecture notes that margin pricing (based on MC) is theoretically superior.
Q: A firm has MC = $20 and faces Ep = –4. Using the rule of thumb, what price should it set? What are the margin and mark-up?
A: Margin = 1/|–4| = 25%. From (P – MC)/P = 0.25: P – 20 = 0.25P, so 0.75P = 20, P = $26.67. Mark-up = (26.67 – 20)/20 = 33.3%.
Q: Two firms sell similar products. Firm A has Ep = –2 and Firm B has Ep = –8. Which firm has the higher margin, and why?
A: Firm A. Margin = 1/|Ep|. Firm A: 1/2 = 50%. Firm B: 1/8 = 12.5%. Firm A's customers have fewer substitutes (less elastic demand), so Firm A has more pricing power.
Q: A restaurant owner says "I just mark up my ingredient costs by 300%." Is this consistent with economic theory?
A: Yes, if the mark-up implicitly reflects the elasticity of demand for restaurant meals. The 300% mark-up corresponds to P = 4 × MC, which implies a margin of 75%, and therefore |Ep| = 1/0.75 ≈ 1.33. Restaurant diners have some alternatives but not many immediate ones, so mildly elastic demand is plausible.
Q: Why does the margin formula break down when |Ep| < 1?
A: It produces a margin greater than 100% (or a negative price), which is impossible. This means the firm is pricing in the inelastic region. It should raise its price: doing so increases revenue and reduces costs, so profits rise. The firm has not yet reached the optimal point.
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