6.3 Competition on Price

In the Cournot model, firms must decide how many units to produce before the price is determined. In many markets, however, production capacity can be quickly adjusted to meet market demand, and the strategic variable becomes the price. This is the case, for example, with mobile phone and internet connectivity services, where new customers can be served instantly. The same applies to streaming platforms, which can take on new users without immediate physical limits.

Bertrand's Paradox

The Bertrand duopoly represents The paradox is named after mathematician Joseph Bertrand, who in 1883 criticized Cournot’s assumption that firms compete on quantities, arguing instead that real competition is on price. such a context as a simultaneous-move game between two firms offering a homogeneous good. Each firm announces a unit price at which to sell the good or service; consumers buy from the seller with the lower price or, in case of a tie, split between the two firms. As illustrated in the following figure — where we assume market demand $Q=5000-1000P$ and marginal cost $MC=2$ equal for both firms — this mechanism leads to a surprising outcome: the equilibrium coincides with that of perfect competition.

Several mechanisms help resolve the paradox and explain why, in reality, in oligopolistic markets prices are above marginal cost and firms earn positive profits. Among these mechanisms we find capacity constraints, product differentiation, and repeated interaction over time.

Capacity Constraints

The Bertrand paradox Airlines, as well as ferry operators and cruise lines, compete on price but cannot sell beyond the capacity of their seats. assumes that each firm can serve the entire market demand at the lowest price. This is not realistic: only in certain markets can firms produce unlimited quantities. In many other markets, firms face limits to their productive capacity. In the presence of such constraints, if a firm lowers its price, it still cannot capture the entire market. This reduces the incentive to engage in price wars, since the benefit of a lower price is limited by available capacity. As a result, in equilibrium firms maintain prices above marginal cost and earn positive profits. The equilibrium outcome then resembles that of the Cournot model.

As an example, suppose market demand is given by $Q=5000-1000P$ and that each firm can produce, at a constant marginal cost equal for both, $MC_1=MC_2=2$, any quantity not greater than $800$ units. It is easy to see that, in this context, setting prices equal to marginal cost does not constitute a Nash equilibrium. Consider firm 1. If it assumes that firm 2 is choosing $P_2=2$, its best response would not be $P_1=2$. By announcing a higher price, say $P_1=3$, it would still retain some market share. Firm 2 would sell its full capacity ($800$ units) at price $P_2=2$, leaving a residual demand, that is $Q=(5000-1000P_1)-800$, for firm 1. Firm 1 would also manage to sell its entire capacity, earning a profit equal to $(3-2)\times 800=800$.

What is the equilibrium in this case? Both firms choose the price that allows the market to absorb their total capacity. Since this is $800+800=1600$ units of output, the price both will choose in equilibrium is $P_1=P_2=3.4$, since $1600=5000-1000\times3.4$. Each firm then earns a profit of $(3.4-2)\times800=1120$. No firm will want to set a lower price, since it would still only sell $800$ units and not more, but at a lower price. Nor will it want to set a higher price. To understand why, suppose firm 1 sets a price of $3.4+\varepsilon$. Since this price is higher than that set by firm 2 ($P_2=3.4$), consumers will buy first from the latter, which sells its full capacity. This leaves to firm 1 the residual demand $Q=(5000-1000P_1)-800$, so firm 1’s profit will be:

\(\begin{gathered} (3.4+\varepsilon-2) \times \big[ 5000-1000(3.4+\varepsilon)-800 \big] \end{gathered}\)

But this is a decreasing function of $\varepsilon$!

Non-Homogeneous Goods

Another factor Although similar products, McDonald’s and Burger King burgers differ in preparation method, taste, and brand; consumers do not regard them as homogeneous. that limits price competition is product differentiation. If the goods produced by firms are not perfect substitutes for consumers, a price reduction by one firm does not automatically lead to a total loss of demand for its competitors. Each firm thus retains some market power and is able to earn positive profits. We now present a simple example in which price competition with differentiated goods leads to an equilibrium in which price exceeds marginal cost.

Suppose the goods produced by firms 1 and 2 are substitutes, but not homogeneous. To reflect this assumption, we assume that the market demand functions for the goods produced by firms 1 and 2 are respectively:

\(\begin{gathered} Q_1=5000-2000P_1+1000P_2 \\ Q_2=5000-2000P_2+1000P_1 \end{gathered}\)

where $P_1$ and $P_2$ are the prices chosen by the two firms. The two firms have constant and equal marginal costs: $MC_1=MC_2=2$.

To compute the Nash equilibrium, we write the firms’ profit functions:

\(\begin{gathered} \Pi_1=(P_1-2)(5000-2000P_1+1000P_2) \\ \Pi_2=(P_2-2)(5000-2000P_2+1000P_1) \end{gathered}\)

and maximize the first with respect to $P_1$ and the second with respect to $P_2$, obtaining:

\(\begin{gathered} 5000-4000P_1+1000P_2+4000=0 \\ 5000-4000P_2+1000P_1+4000=0 \end{gathered}\)

The best response functions are therefore:

\(\begin{gathered} BR_1: \quad P_1=(9+P_2)/4 \\ BR_2: \quad P_2=(9+P_1)/4 \end{gathered}\)

Solving the system we obtain the Nash equilibrium:

\(\begin{gathered} P_1=3 \qquad P_2=3 \end{gathered}\)

In equilibrium, the profits of the two firms are positive:

\(\begin{gathered} \Pi_1=\Pi_2=(3-2)(5000-2000\times 3+1000\times 3)=2000 \end{gathered}\)

Tacit Collusion

Finally, In 2015–2016 TIM, Vodafone, and Wind Tre switched from monthly billing to 28-day billing, increasing the number of bills from 12 to 13 per year (+8.6% in revenue).
In 2018, when required to return to monthly billing, they raised their monthly fees by the same 8.6% in almost identical fashion.
The Italian Competition Authority (AGCM) interpreted this parallel behaviour as tacit collusion and, in 2019, fined the three companies a total of 228 million euros. even in the presence of homogeneous goods and no capacity constraints, the Bertrand paradox can be overcome if firms interact repeatedly over time. In multi-stage games, which we do not discuss in these notes, firms do not make isolated decisions, but also consider the future consequences of their actions.

If a firm lowers its price to gain market share in the short term, it knows this could trigger an immediate reaction from the others, leading to a price war that would reduce everyone’s profits. Assuming that firms attach sufficient importance to their future profits, in equilibrium (in the multi-stage game) they may adopt tacit collusion strategies, keeping prices high and avoiding deviations that would compromise future gains. No explicit agreement is necessary: an implicit understanding, sustained by mutual observation and the possibility of retaliation, is sufficient.