Marketing Science & Industrial Organization
Nerlove-Arrow Advertising Goodwill Stock Formula
Generalize Dorfman-Steiner to durable brand equity dG/dt = a - delta*G using Marc Nerlove and Kenneth Arrow’s (1962) dynamic goodwill-to-sales rule G* / (P*Q) = beta_G / (epsilon_P * (r + delta)).
Stop treating durable brand advertising as an instant perishable expense!
Static Dorfman-Steiner sets Advertising / Sales = beta_A / epsilon_P, implicitly assuming brand memory depreciates 100% inside the current period (delta = 1, r = 0). Marc Nerlove and Kenneth Arrow (1962) modeled Advertising Goodwill Stock G(t) as a durable intangible capital asset accumulating via dG/dt = I_adv - delta * G (where each $1 of ad spend buys 1 unit of goodwill that depreciates at rate delta > 0 under discount rate r >= 0). Their intertemporal optimality theorem proves Optimal Goodwill-to-Sales Ratio G* / Revenue = beta_G / (epsilon_P * (r + delta)) — where r + delta is Jorgenson’s user cost of brand capital — and Steady-State Replacement Ad-Spend-to-Sales Ratio I*_adv / Revenue = (delta / (r + delta)) * (beta_G / epsilon_P)!
Test on Annual_Revenue = 10,000,000 (B2), Goodwill_Elasticity beta_G = 0.12 (B3), Price_Elasticity_Magnitude epsilon_P = 2.00 (B4 > 1), Discount_Rate r = 0.10 (B5), and Brand_Depreciation_Rate delta = 0.20 (B6): User Cost of Brand Capital is r + delta = 0.30, Optimal Goodwill-to-Sales Ratio is 0.12 / (2.00 * 0.30) = 0.2000 (Optimal Goodwill Stock G* = 2,000,000.00), and Steady-State Annual Replacement Ad Spend is delta * G* = 0.20 * 2,000,000 = 400,000.00 (4.00% of revenue — vs static 0.12 / 2.00 = 6.00% if r = 0)!
| Nerlove-Arrow Brand Capital Metric (Rev=10M, beta_G=0.12, eps_P=2.0) | Formula | Verified Output |
|---|---|---|
| User Cost of Brand Goodwill (r + delta) | 0.10 + 0.20 | 0.3000 (30.00% / yr) |
| Optimal Goodwill-to-Sales Ratio (G* / Rev) | beta_G / (epsilon_P * (r + delta)) | 0.2000 (20.00% of Sales) |
| Target Brand Goodwill Stock (G*) | Revenue * 0.2000 | 2,000,000.00 |
| Steady-State Annual Ad Budget (delta * G*) | 0.20 * 2,000,000 | 400,000.00 (4.00% Ad-to-Sales) |
Optimal Goodwill-to-Sales Ratio & Target Goodwill Capital Stock Formula
Require monopoly markup condition epsilon_P > 1, goodwill elasticity beta_G >= 0, r >= 0, delta > 0, and r + delta > 0.
=LET(rv_an, B2, bt_gw, B3, ep_pr, B4, r_dsc, B5, dl_dp, B6, IF(OR(rv_an <= 0, bt_gw < 0, ep_pr <= 1, r_dsc < 0, dl_dp <= 0), "Invalid Nerlove-Arrow Input", ROUND(rv_an * bt_gw / (ep_pr * (r_dsc + dl_dp)), 2)))Steady-State Replacement Advertising-to-Sales Ratio Formula
Notice the exact limit: when discount rate r = 0 (B5 = 0), delta cancels out of (delta / (0 + delta)) * (beta_G / epsilon_P) = beta_G / epsilon_P = 0.0600 (6.00%), recovering Dorfman-Steiner in steady state!
=LET(bt_gw, B3, ep_pr, B4, r_dsc, B5, dl_dp, B6, IF(OR(bt_gw < 0, ep_pr <= 1, r_dsc < 0, dl_dp <= 0), "Invalid Nerlove-Arrow Input", ROUND((dl_dp / (r_dsc + dl_dp)) * (bt_gw / ep_pr), 4)))Try the example
Use the original values or change them to test your understanding.