Platform Economics & Network Effects
Katz-Shapiro Network Compatibility Equilibrium Formula
Solve Michael Katz and Carl Shapiro’s (1985) Fulfilled-Expectations Cournot Equilibrium under open standard compatibility q* = (A - c)/(N + 1 - N*v) versus proprietary incompatibility q* = (A - c)/(N + 1 - v).
Quantify exactly how much an open adapter standard expands industry demand and per-firm profit!
Michael Katz and Carl Shapiro (1985) pioneered modern platform economics by introducing Fulfilled-Expectations Cournot Equilibrium (FECE) for goods whose consumer willingness-to-pay rises by v in [0, 1) per expected user on the compatible network (P_i = A + v * y_i^e - Q, unit cost c < A, N >= 1 symmetric firms)! When all N firms adopt an Open Compatible Standard (y_i^e = Q^e = N * q^e), Cournot best response given consumer expectations Q^e is q = (A - c + v * Q^e) / (N + 1), and imposing Fulfilled Expectations (Q* = N * q*) yields Compatible Per-Firm Output q_compat* = (A - c) / (N + 1 - N * v) and Profit Pi_compat* = (q_compat*)^2! By contrast, under Proprietary Siloed Incompatibility (y_i^e = q_i^e), each firm’s network is only its own sales, shrinking per-firm output to q_incompat* = (A - c) / (N + 1 - v) and Profit to (q_incompat*)^2!
Verify on exact integers: Symmetric_Firms N = 2 (B2), StandAlone_Intercept A = 120 (B3), Unit_Cost c = 20 (B4 -> A - c = 100), and Network_Externality_Slope v = 0.50 (B5): Under Siloed Incompatibility, each firm sells 100 / (2 + 1 - 0.50) = 40.00 units (Industry Q = 80.00) and earns 40^2 = 1,600.00! Under Open Standard Compatibility, each firm sells 100 / (2 + 1 - 2 * 0.50) = 50.00 units (Industry Q = 100.00) and earns 50^2 = 2,500.00 — generating a Private Adapter Profit Gain of +900.00 per firm (+56.25% higher profit, justifying any adapter R&D cost F_adapter <= 900.00)!
| Katz-Shapiro Network Metric (N=2, A=120, c=20, v=0.50) | Formula | Verified Output |
|---|---|---|
| Siloed Incompatible Firm Output | (A - c) / (N + 1 - v) = 100 / 2.5 | 40.00 units (Industry = 80) |
| Compatible Open-Standard Firm Output | (A - c) / (N + 1 - N * v) = 100 / 2.0 | 50.00 units (Industry = 100) |
| Incompatible Per-Firm Profit | q_incompat^2 | 1,600.00 |
| Compatibility Profit Gain per Firm | q_compat^2 - q_incompat^2 | +900.00 (2,500 vs 1,600!) |
Fulfilled-Expectations Open-Standard Per-Firm Profit Formula
Validate integer N >= 1, A > c >= 0, v >= 0, and stability condition N * v < N + 1 (e.g., 0 <= v < 1).
=LET(n_fm, B2, a_in, B3, c_un, B4, v_nw, B5, IF(OR(n_fm < 1, INT(n_fm) <> n_fm, c_un < 0, a_in <= c_un, v_nw < 0, n_fm * v_nw >= n_fm + 1), "Invalid Katz-Shapiro Input", ROUND(((a_in - c_un) / (n_fm + 1 - n_fm * v_nw)) ^ 2, 2)))Per-Firm Compatibility Adapter Profit Gain Formula
Notice that for a monopolist (N = 1) or zero network effects (v = 0), N + 1 - N*v = N + 1 - v identically, so adapter compatibility gain equals 0.00!
=LET(n_fm, B2, a_in, B3, c_un, B4, v_nw, B5, IF(OR(n_fm < 1, INT(n_fm) <> n_fm, c_un < 0, a_in <= c_un, v_nw < 0, n_fm * v_nw >= n_fm + 1), "Invalid Katz-Shapiro Input", LET(q_cm, (a_in - c_un) / (n_fm + 1 - n_fm * v_nw), q_in, (a_in - c_un) / (n_fm + 1 - v_nw), ROUND((q_cm ^ 2) - (q_in ^ 2), 2))))Try the example
Use the original values or change them to test your understanding.