Stackelberg Leader-Follower Duopoly Output & Profit Formula

Internalize the follower’s best-response slope via backward induction

In Heinrich von Stackelberg’s (1934) sequential quantity game under linear inverse demand P(Q) = a - b * (q_L + q_F), Leader L commits irrevocably to output q_L first, and Follower F observes q_L before choosing q_F(q_L) = MAX(0, (a - c_F - b * q_L) / (2 * b)). Because every extra unit produced by the Leader crowds out 0.50 units of Follower output, an interior Leader chooses q_L* = (a - 2 * c_L + c_F) / (2 * b), leaving Follower output q_F* = (a + 2 * c_L - 3 * c_F) / (4 * b) and equilibrium price P* = (a + 2 * c_L + c_F) / 4!

Build a hand-verifiable duopoly table with intercept a = 100 (B2), slope b = 1 (B3), and symmetric marginal costs c_L = 20 (B4), c_F = 20 (B5). A common first-draft mistake is plugging Cournot’s denominator 3*b instead of Stackelberg’s 2*b for the Leader! At symmetric costs, q_L* = (100 - 40 + 20) / 2 = 40 units (exact monopoly output!), q_F* = 20 units, total Q* = 60, price P* = 40, Leader profit Pi_L* = 800.00 (12.50% above Cournot’s 711.1111!), and Follower profit Pi_F* = 400.00. Test asymmetric costs c_L = 16, c_F = 24 to verify q_L* = 46, q_F* = 15, and P* = 39.

Interior Stackelberg Leader Output q_L* and Follower Best Response q_F* Formula

Enforce positive demand margin (a > c_L, a > c_F) and interior positive follower output (a + 2*c_L - 3*c_F > 0; if the Leader’s cost advantage is so large that 3*c_F >= a + 2*c_L, limit-pricing / entry deterrence takes over!).

=LET(a_int, B2, b_sl, B3, c_ld, B4, c_fl, B5, IF(OR(b_sl <= 0, c_ld < 0, c_fl < 0, a_int <= c_ld, a_int <= c_fl), "Invalid Demand Input", IF(a_int + 2 * c_ld - 3 * c_fl <= 0, "Follower Deterred (Corner Limit Output)", ROUND((a_int - 2 * c_ld + c_fl) / (2 * b_sl), 4))))

Stackelberg Leader Equilibrium Profit Pi_L* Formula

Evaluate (P* - c_L) * q_L* = (a - 2 * c_L + c_F)^2 / (8 * b) across any interior cost pair (c_L, c_F).

=LET(a_int, B2, b_sl, B3, c_ld, B4, c_fl, B5, IF(OR(b_sl <= 0, c_ld < 0, c_fl < 0, a_int <= c_ld, a_int <= c_fl), "Invalid Demand Input", IF(a_int + 2 * c_ld - 3 * c_fl <= 0, "Follower Deterred (Corner Limit Output)", ROUND(((a_int - 2 * c_ld + c_fl)^2) / (8 * b_sl), 4))))

Further reading

• Arte Sheets Lesson — Cournot Oligopoly Equilibrium Output, Price, and HHI-Lerner Rule
• Microsoft Support — LET function