Bertrand Competition
Simultaneous price-setting competition among firms producing a homogeneous good.
Definition
Firms simultaneously post prices ; demand goes entirely to the lowest price (split if tied). Profit is $\pi_i=(p_i-c)D(p_i)\cdot \mathbf 1\{p_i\le p_{-i}\}\cdot\text{share}.
The "Bertrand paradox" arises with identical goods and marginal cost : the unique Nash equilibrium is for all .
Intuition
Price undercutting drives price to marginal cost with even as few as two firms.
Tiny asymmetries in cost or differentiation change the picture dramatically.
Worked example
Two symmetric firms with constant marginal cost both price at in equilibrium, earning zero profit.
Differentiated Bertrand with logit demand yields positive equilibrium markups.
The math
With capacity constraints (Edgeworth-Bertrand) or product differentiation, equilibrium prices exceed marginal cost.
Kreps-Scheinkman: Cournot outcomes emerge as the subgame perfect equilibrium of a two-stage capacity-then-price game.
Where it is used
Explains why concentrated markets still exhibit competitive pricing if products are homogeneous.
Motivates product differentiation as a profit-protection strategy.
More in Game theory
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