Industrial Organization & Spatial Pricing
Quadratic Hotelling Maximal Differentiation Price Formula
Calculate d’Aspremont, Gabszewicz, and Thisse’s (1979) quadratic-transport spatial Nash equilibrium prices p_1* = c + t*(1 - a - b)*(1 + (a - b)/3) and prove maximal product differentiation.
Resolve Hotelling’s linear-city price-war breakdown with quadratic transport costs!
Harold Hotelling (1929) famously claimed competing firms cluster at the center of a [0, 1] linear city ("Minimal Differentiation") — until Claude d’Aspremont, J. Jaskold Gabszewicz, and Jacques-Francois Thisse (1979) proved that pure-strategy price equilibrium breaks down when linear-transport firms locate too close because each undercuts to steal the rival’s entire hinterland! Replacing linear disutility with Quadratic Mismatch/Transport Cost t * (x - x_i)^2 (where Firm 1 locates at a >= 0 from 0 and Firm 2 at b >= 0 from 1, with separation 1 - a - b > 0 and unit cost c) restores a smooth indifferent consumer x_hat and a unique Nash Price Equilibrium for EVERY distinct pair of locations: p_1* = c + t * (1 - a - b) * (1 + (a - b) / 3) and p_2* = c + t * (1 - a - b) * (1 + (b - a) / 3)!
Verify on Firm_1_Inset a = 0.10 (B2), Firm_2_Inset b = 0.10 (B3, coordinate 0.90 -> Separation D = 0.80), Quadratic_Mismatch_Rate t = 30.00 (B4), and Unit_Cost c = 10.00 (B5): Equilibrium Prices are p_1* = p_2* = 10 + 30 * 0.80 * 1 = 34.00, Market Share x_hat = 0.5000, and Equilibrium Profit Pi_1* = (34.00 - 10.00) * 0.50 = 12.00! Now test Maximal Differentiation (endpoints a = 0, b = 0 -> D = 1.00): price rises to p_1* = 40.00 and profit peaks at 15.00 — proving firms flee the center to soften price competition! Test asymmetric advantage a = 0.25, b = 0.05 (D = 0.70): centrally positioned Firm 1 charges p_1* = 10 + 30 * 0.70 * (1 + 0.20 / 3) = 32.40 vs peripheral Firm 2 p_2* = 29.60!
| Quadratic Spatial Competition Metric (a=0.10, b=0.10, t=30, c=10) | Formula | Verified Output |
|---|---|---|
| Firm 1 Nash Price (p_1*) | c + t * (1 - a - b) * (1 + (a - b) / 3) | 34.00 |
| Firm 2 Nash Price (p_2*) | c + t * (1 - a - b) * (1 + (b - a) / 3) | 34.00 |
| Indifferent Consumer Boundary (x_hat) | 0.5 + (a - b) / 6 | 0.5000 (50.0% Share) |
| Firm 1 Equilibrium Profit (Pi_1*) | (p_1* - c) * x_hat | 12.00 (15.00 at a=b=0) |
d’Aspremont-Gabszewicz-Thisse (1979) Nash Equilibrium Mill Price Formula
At Nash prices p_1* and p_2*, the indifferent consumer simplifies algebraically to x_hat = (3 + a - b) / 6 = 0.5 + (a - b) / 6, so Pi_1* = 0.5 * t * (1 - a - b) * (1 + (a - b) / 3)^2!
=LET(a_in, B2, b_in, B3, t_qd, B4, c_mc, B5, IF(OR(a_in < 0, b_in < 0, a_in + b_in >= 1, t_qd <= 0, c_mc < 0), "Invalid Spatial Location", ROUND(c_mc + t_qd * (1 - a_in - b_in) * (1 + (a_in - b_in) / 3), 4)))Equilibrium Market Share & Firm 1 Profit Formula
Notice dPi_1* / da < 0 for all interior a >= 0: moving inward steals demand (+1/6) but triggers a fiercer strategic price cut (-t), driving both firms to the endpoints a* = b* = 0!
=IF(ISTEXT(E2), E2, ROUND((E2 - B5) * (0.5 + (B2 - B3) / 6), 4))Try the example
Use the original values or change them to test your understanding.