Continuous competition game / Bertrand pricing with perfect substitutes

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Continuous competition game / Bertrand pricing with perfect substitutes

• Players: Two players, Firm 1 and Firm 2.
• Strategies: Firm 1 chooses price $p_1$; Firm 2 chooses price $p_2$.

Rules

• Start with two firms selling perfect substitutes at constant marginal cost; Players simultaneously choose prices.
• The player who undercuts profitably captures demand.
• Market demand is $Q=120-p$; Marginal cost is $20$.
• Consumers buy from the lower-price seller because the products are perfect substitutes.
• If prices tie, firms split demand equally.

Payoff Details

• Firm 1’s quantity is:

• Firm 2’s quantity is defined symmetrically.

Diagram (Demand Jump at $p_1=p_2$)

• The graph fixes $p_2=60$ to illustrate the discontinuity.
• A tiny undercut moves Firm 1 from half the market to the whole market.

Derivation (Best Response Analysis)

• Firm 1’s profit is $\pi_1=(p_1-20)Q_1$.
• Calculus is not the right tool here because $Q_1$ jumps at $p_1=p_2$.
• If $p_2>20$, Firm 1 can set a slightly lower price $p_1=p_2-\varepsilon$ with $\varepsilon>0$ small, capture the whole market, and earn positive profit.
• If $p_2<20$, undercutting would require selling below marginal cost, so winning the market is unprofitable.
• If $p_2=20$, any lower price gives negative margin and any higher price yields zero demand.
• Therefore the only price that can survive best-response logic is the marginal-cost price.

Derivation (Nash Equilibrium)

• Suppose both firms charge the same price above $20$.

  • Either firm can undercut by a tiny amount, capture the whole market, and raise profit.
  • So no common price above $20$ can be a Nash equilibrium.

• Suppose one firm charges a higher price than the other.

  • The high-price firm sells nothing, so that profile is not stable against deviation.
  • Suppose both firms charge $20$.
  • Undercutting below $20$ gives negative margin, while raising price above $20$ loses all customers.

Nash Equilibrium

Result:

The Bertrand equilibrium is

Insights

Insight:

• Perfect substitutes create aggressive undercutting incentives.
• The discontinuity in demand means first-order conditions are not useful.
• Competition drives price down to marginal cost.