Marginal analysis vs. Average roas: the profit-maximization fallacy in digital advertising budget allocation

Digital advertising budget allocation remains dominated by a single aggregate metric: return on ad spend (ROAS), computed as total revenue divided by total expenditure.Industry surveys confirm that the majority of advertisers treat ROAS targets as the primary criterion for scaling or contracting media budgets [1].This practice, while operationally convenient, conflates average and marginal returns and can produce allocation decisions that systematically deviate from profit-maximizing optima.The present work formalizes this divergence, derives the conditions under which average-based and marginal-based decision rules yield opposing recommendations, and demonstrates the magnitude of the resulting profit gap through a numerical scenario grounded in standard microeconomic theory [2].The analytical foundation rests on the concave response function.As expenditure increases, a channel first reaches its most responsive audience segments; subsequent increments target progressively less responsive populations, producing diminishing marginal returns [3].Following Ivitskiy [4], the revenue-expenditure relationship is modeled as 𝑅𝑅(𝑥𝑥) = 𝑎𝑎𝑥𝑥ᵇ, where 𝑅𝑅(𝑥𝑥) is revenue from expenditure 𝑥𝑥, 𝑎𝑎 is a scale parameter, and 𝑏𝑏 is the elasticity parameter satisfying 0 < 𝑏𝑏 < 1.The constraint on 𝑏𝑏 ensures strict concavity.This specification aligns with functional forms in Bayesian media mix models [5] and the classical Dorfman-Steiner theorem on optimal advertising intensity [6].Differentiating the revenue function yields marginal revenue 𝑀𝑀𝑅𝑅(𝑥𝑥) = 𝑎𝑎𝑏𝑏𝑥𝑥ᵇ⁻¹, strictly decreasing in 𝑥𝑥.Profit is 𝛱𝛱(𝑥𝑥) = 𝑅𝑅(𝑥𝑥) -𝑥𝑥 -𝐹𝐹 , where 𝐹𝐹 denotes fixed costs.The first-order optimality condition requires 𝑀𝑀𝑅𝑅(𝑥𝑥 * ) = 𝑀𝑀𝑀𝑀(𝑥𝑥 * ) , where 𝑀𝑀𝑀𝑀(𝑥𝑥) is the marginal cost of advertising.Under constant marginal cost 𝑐𝑐₀, the profitmaximizing expenditure is 𝑥𝑥 * = (𝑎𝑎𝑏𝑏 / 𝑐𝑐₀)^(1/(1 -𝑏𝑏)).The second-order condition 𝑑𝑑²𝛱𝛱/𝑑𝑑𝑥𝑥² = 𝑎𝑎𝑏𝑏(𝑏𝑏 -1)𝑥𝑥ᵇ⁻² < 0 holds for all 𝑏𝑏 < 1, confirming 𝑥𝑥 * as a global maximum [4].Optimal allocation is thus uniquely determined by channel elasticity, scale, and marginal cost, not by any threshold imposed on the average return ratio.A numerical illustration clarifies practical consequences.Scenario A allocates $10,000 and generates $50,000 in revenue (ROAS = 5.0).Scenario B allocates $25,000International scientific and practical conference "Science and education in the modern world"