Dorfman-Steiner Optimal Advertising-to-Sales Ratio Formula

Tie marketing budget intensity directly to gross Lerner margin and demand elasticities

How much of gross sales revenue P * Q should a differentiated brand reinvest in advertising A? Dorfman and Steiner (1954) proved that at joint price-and-advertising optimality, the optimal Advertising-to-Sales Ratio equals the ratio of advertising demand elasticity e_A > 0 to price demand elasticity magnitude |e_P| > 1: A* / (P * Q) = e_A / |e_P|. Moreover, because optimal Lerner markup satisfies (P - MC) / P = 1 / |e_P|, optimal ad spend at ANY fixed margin is A* / Revenue = ((P - MC) / P) * e_A!

Verify on a subscription software brand with unit price P = 50 (B2), marginal fulfillment cost MC = 30 (B3 -> Lerner margin 20 / 50 = 0.4000), price elasticity magnitude |e_P| = 2.50 (B4, consistent with 1 / 2.50 = 0.40), advertising elasticity e_A = 0.15 (B5), and revenue 4,000,000 (B6): optimal ad intensity is 0.15 / 2.50 = 0.0600 (6.00% of revenue), setting optimal ad spend at A* = 240,000.00!

Dorfman-Steiner Optimal Advertising-to-Sales Ratio e_A / |e_P| Formula

Require elastic brand demand |e_P| > 1 (since a firm facing inelastic demand |e_P| <= 1 has non-positive marginal revenue and would raise price before expanding ad spend!) and non-negative advertising elasticity 0 <= e_A < |e_P|.

=LET(e_pr, B4, e_ad, B5, IF(OR(e_pr <= 1, e_ad < 0, e_ad >= e_pr), "Require |e_P| > 1 and 0 <= e_A < |e_P|", ROUND(e_ad / e_pr, 4)))

Fixed-Price Margin-Weighted Optimal Ad Spend Revenue * ((P - MC) / P) * e_A Formula

When retail price P is contractually fixed, multiplying actual gross margin (P - MC) / P by e_A * Revenue yields the exact profit-maximizing ad budget.

=LET(px_u, B2, mc_u, B3, e_ad, B5, rev_s, B6, IF(OR(px_u <= 0, mc_u < 0, mc_u >= px_u, e_ad < 0, rev_s < 0), "Invalid Price-Margin Input", ROUND(rev_s * ((px_u - mc_u) / px_u) * e_ad, 2)))

Further reading

• Arte Sheets Lesson — Bass Diffusion Model Adoption Curve and Peak Time Formula
• Microsoft Support — ROUND function