The Theory of Market Leaders in a simple example

Imagine a market for homogenous goods where demand at price p is Q=a-p. Each firm can produce q units of the good at cost cq after investing F as a fixed cost of production. Each firm producing q has profits R=qp-cq-F, where the equilibrium price must be p=a-Q, that is decreasing in total production by all firms Q. As well known, in such a market a monopolist would choose production q to maximize profits R=q(a-q)-cq-F. Setting the first derivative with respect to q equal to zero we have a-q-q-c=0, which leads to the output q=(a-c)/2 and to the monopolistic price p=(a+c)/2. However, as long as positive profits can be made entering in such a market, there are incentives for other firms to enter and compete.

In the early XIX century the French economist Augustine Cournot studied a duopoly where two firms compete choosing how much to produce taking as given the production of each other. If one firm produces s and the other produces x, the former will choose s to maximize profits R=s(a-s-x)-cs-F and the latter will choose x to maximize R=x(a-x-s)-cx-F. Putting together the two optimality conditions (first derivatives with respect to s and x respectively) and solving for s and x,one obtains what, in modern terminology, we call a Nash Equilibrium. This implies that the production for each firm is the same and equal to q=(a-c)/3 and the price is p=(a+2c)/3. If other firms can enter in the market, they will enter as long as there are positive profits to be made. However, entry strenghtens competition, increases production and reduces the equilibrium price, lowering profits. Under free entry, the market can then accept a limited number of entrants whose profits are driven to zero: such an equilibrium concept is often associated with the analysis of the English economist Alfred Marshall at the turn of the XIX century. In modern terms a socalled Nash Equilibrium with Free Entry implies an endogenous number of firms n=(a-c)/f-1, where f is the square root of the fixed cost F, and each firm produces q=f, so that the equilibrium price is p=c+f.

In the first half of the XX century, a German economist, Heinrich von Stackelberg, studied another simple duopoly where one of the two firms is a leader with a first mover advantage in the choice of production over the follower. The leader maximizes its own profits taking in consideration the consequent behaviour of the follower, which leads to the production of the leader s=(a-c)/2 against a production of the follower x=(a-c)/4. In a Stackelberg equilibrium the price is now p=(a+3c)/4. Also in this case entry by other followers may be profitable and it may drive down the price and the profits. In particular, given some production of the leader s, free entry leads to n=(a-c-s)/f firms producing x=f each. But this generates expected profits of the leader R=s[a-s-(n-1)x]-cs-F=sf-F hence the leader will like to produce as much as possible until no other firms enter in the market. The Stackelberg Equilibrium with Free Entry implies only one firm active in the market, the leader, producing s=a-c-2f, which implies an equilibrium price p=c+2f.