Modeling Funnel Entropy and Cost Propagation in Legal Lead Acquisition: An Empirical Study of U.S. Facebook Campaigns (June 2023 – October 2025)

Abstract

This paper presents a quantitative, empirically validated model of cost propagation within multi-stage digital funnels for legal advertising. Using data from a U.S. Facebook Ads account (June 2023–October 2025; total spend =$1,105,068.36; CPM =$28.30; CPC =$1.55; CTR = 1.83 %; CPL ≈ $300) spanning 17 states, we model the transformation of impression-level spend into retainer-level acquisition costs. We formalize funnel entropy—the compounded uncertainty that amplifies acquisition variance across transitions (impression → click → lead → MQL → SQL → retainer → won case)—and demonstrate that small shifts in mid-funnel qualification probabilities yield exponential cost compression. The observed expected Cost per Retainer (CPR) ≈ $1,940 and Cost per Won Case (CPW) ≈ $2,585 match predictive expectations from stochastic cost models. Findings integrate behavioral-response theories (Lewinski et al., 2014 – 2016) and auction-economic frameworks (Varian, 2007; Berman, 2020), offering a generalized function that links ad-auction density, entropy, and down-funnel efficiency to total client-acquisition cost.