The Edgeworth duopoly model, also known as Edgeworth solution, was developed by Francis Y. Edgeworth in his work “The Pure Theory of Monopoly”, 1897. It is a duopoly model similar to the duopoly model developed by Joseph Bertrand, in which two firms producing the same good compete in terms of prices. Edgeworth’s model presents a slight modification as it also includes constraints in the production capacity of the firms. In this market structure, firms have two potential options, to collude or not.

As shown in the adjacent figure, when firms choose to collude they will split and share the market and the production of the good. Firm₁ will produce from O to F and firm₂ from O to G, in this way the supply is limited and prices will be set at p. Revenues of each firm correspond to the rectangle above FO and OG, and each firm would enjoy an equal share. Note that d₁ and d₂ are parts of total demand, each part being supplied by one of the firms.

Collusion is not always possible as firms have incentives to break cooperation in their search for higher profits. Collusion is also considered an illegal business practice in many countries. Eventually one of the firms will decide to lower their prices and increase production in order to gain market share from the other competitor.  Consequentially the other firms will do the same. This process will escalade up to the point in which the maximum production of both firms is achieved. When this point is reached (OD for firm₁ and OE for firms₂), price will not be reduced any further and will remain at p’, as the increase in demand that follows price reduction will not be satisfied with a larger amount of production. On the contrary, prices will start to rise little by little so firms will be able once again to increase their profits. Overtime this process will be repeated and prices will oscillate from p to p’.

The following figure shows how all this translates into market demand (D), and how quantities sold will oscillate from Q and Q’.

Overall the Edgeworth’s solution is a more realistic one than Bertrand’s and it answers Bertrand’s paradox. However, it does not give a definitive solution.

In some cases, competition in terms of price changes seems more logical than quantity competition, especially in the short run. Besides, one of the assumptions of Cournot’s duopoly model is that firms supply a homogeneous product. Considering this, Bertrand proposed an alternative to Cournot. Considering Bertrand’s model from a game theory perspective, it can be analysed as a simultaneous game where the strategic choice is on prices, rather than quantities.

In this model, consumers will buy from the firm that offers the lowest price, so we can easily have the intuition that the Nash equilibrium is going to be the two firms setting the same price.

Assuming equal and constant cost functions, the demand for each firm is as follows: 

Bertrand’s equilibrium occurs when P₁=P₂=MC, being MC the marginal cost, yielding the same result as perfect competition.

The logic is simple: if the price set by both firms is the same but the marginal cost is lower, there will be an incentive for both firms to lower their prices and seize the market. Therefore, the only equilibrium in which none of the firms will be willing to deviate is when price equals marginal cost. 

The result of the model creates a paradox, known as Bertrand’s paradox: in a case of imperfect competition (here, a duopoly), where there is a strong incentive to collude, we end up with the same outcome as in perfect competition. The equilibrium does not hold with asymmetric cost functions since the firm with the lowest marginal cost would seize the entire market and become a monopoly. A great analysis of this paradox, known as Edgeworth duopoly model or Bertrand-Edgeworth duopoly, was developed by Francis Y. Edgeworth in his paper “The Pure Theory of Monopoly”, 1897.

Stackelberg duopoly, also called Stackelberg competition, is a model of imperfect competition based on a non-cooperative game. It was developed in 1934 by Heinrich Stackelbelrg in his “Market Structure and Equilibrium” and represented a breaking point in the study of market structure, particularly the analysis of duopolies, since it was a model based on different starting assumptions and gave different conclusions to those of the Cournot’s and Bertrand’s duopoly models.

In game theory, a Stackelberg duopoly is a sequential game (not simultaneous as in Cournot’s model). There are two firms, which sell homogeneous products, and are subject to the same demand and cost functions. One firm, the leader, is perhaps better known or has greater brand equity, and is therefore better placed to decide first which quantity q₁ to sell, and the other firm, the follower, observes this and decides on its production quantity q₂. To find the Nash equilibrium of the game we need to use backward induction, as in any sequential game. That is, start analyzing the decision of the follower.

For firm 2 (follower), the problem is similar to the Cournot’ model. The reaction as a function of q₁ (blue lines) is as follows:

Firm 1 (leader) anticipates the follower’s behavior and takes it into consideration to make the strategic choice of q₁:

Therefore, the quantities sold by each firm at equilibrium are:

The perfect equilibrium of the game is the Stackelberg equilibrium. In this game, the leader has decided not to behave as in the Cournot’s model, however, we cannot ensure that the leader is going to produce more and make more profits than the follower (production will be larger for the firm with lower marginal costs). Total production will be greater and prices lower, but player one will be better off than player two, which serves to highlight two things: the importance of accurate market information when defining a strategy, and the interdependence of each player’s strategies, especially when there is a market leader (with the benefit of moving first) and a follower.

When it comes to economic efficiency, the result is similar to Cournot’s duopoly model. The Nash equilibrium is not Pareto efficient (isoprofit curves, green curves, are not tangent to each other) and therefore, there is a loss in economic efficiency. Nevertheless, the loss is lower in the Stackelberg duopoly than in Cournot’s.

Stackelberg and Cournot equilibria are stable in a static model of just one period. In a dynamic context (repeated games), the models need to be reconsidered.

Comparison with Cournot duopolies:

-Stackelberg’s model is a sequential game, Cournot’s is a simultaneous game;

-In Stackelberg duopolies, the quantity sold by the leader is greater than the quantity sold by the follower, while in Cournot duopolies quantity is the same for both firms;

-When comparing each firm’s output and prices, we have:

Leader: q^S₁ > q^C₁ and π^S₁ > π^C₁

Follower: q^S₂ < q^C₂ and π^S₂ < π^C₂

-With regard to total output and prices we have the following:

Q_M < Q_C < Q_S < Q_PC

P_M > P_C > P_S > P_PC = MC

with:

Q_C: total Cournot output
Q_S: total Stackelberg output
Q_PC: total perfect competition output
Q_M: total monopoly output
P_C: Cournot price
P_S: Stackelberg price
P_PC: perfect competition price
P_M: monopoly price
MC: marginal cost

Cournot duopoly, also called Cournot competition, is a model of imperfect competition in which two firms with identical cost functions compete with homogeneous products in a static setting. It was developed by Antoine A. Cournot in his “Researches Into the Mathematical principles of the Theory of Wealth”, 1838. Cournot’s duopoly represented the creation of the study of oligopolies, more particularly duopolies, and expanded the analysis of market structures which, until then, had concentrated on the extremes: perfect competition and monopolies.

Cournot really invented the concept of game theory almost 100 years before John Nash, when he looked at the case of how businesses might behave in a duopoly. There are two firms operating in a limited market. Market production is: P(Q)=a-bQ, where Q=q₁+q₂ for two firms. Both companies will receive profits derived from a simultaneous decision made by both on how much to produce, and also based on their cost functions: TC_i=C-q_i.

So, algebraically:

In order to maximise, the first order condition will be:

And, if q_i=q_j, then both equal:

Therefore, the reaction functions (blue lines), where the key variable is the quantity set by the other firm, will take the following form:

What all this explains is a very basic principle. Both companies are vying for maximum benefits. These benefits are derived from both maximum sales volume (a larger share of the market) and higher prices (higher profitability). The problem stems from the fact that increasing profitability through higher prices can damage revenue by losing market share. What Cournot’s approach does is maximise both market share and profitability by defining optimum prices. This price will be the same for both companies, as otherwise the one with the lower price will obtain full market share, which makes this a Nash equilibrium, also known for this model the Cournot-Nash equilibrium.

If we consider isoprofit curves (those which show the combinations of quantities that will render the same profit to the firm, red curves) we can see that the equilibrium of the game is not Pareto efficient, since isoprofit curves are not tangent. The outcome is below that of perfect competition and therefore is not socially optimal, but it is better than the monopoly outcome.

Extending the model to more than two firms, we can observe that the equilibrium of the game gets closer to the perfect competition outcome as the number of firms increases, decreasing market concentration.

Comparison with Stackelberg duopolies:

-Cournot’s model is a simultaneous game, Stackelberg’s is a sequential game;

-In Cournot duopolies quantity sold is the same for both firms, while in Stackelberg duopolies, the quantity sold by the leader is greater than the quantity sold by the follower;

-When comparing each firm’s output and prices, we have:

Leader: q^S₁ > q^C₁ and π^S₁ > π^C₁

Follower: q^S₂ < q^C₂ and π^S₂ < π^C₂

-With regard to total output and prices we have the following:

Q_M < Q_C < Q_S < Q_PC

P_M > P_C > P_S > P_PC = MC

with:

Q_C: total Cournot output
Q_S: total Stackelberg output
Q_PC: total perfect competition output
Q_M: total monopoly output
P_C: Cournot price
P_S: Stackelberg price
P_PC: perfect competition price
P_M: monopoly price
MC: marginal cost

Duopoly (from the Greek «duo», two, and «polein», to sell) is a type of oligopoly. This kind of imperfect competition is characterized by having only two firms in the market producing a homogeneous good. For simplicity purposes, oligopolies are normally studied by analysing duopolies. What strategies firms follow and their interactions are a key feature of this market structure.

In duopolies there are two variables of interest: the prices set by each firm and the quantity produced by each firm. Several models have been developed through time, from which we must highlight the Cournot, Stackelberg, Bertrand and the Edgeworth solution. The first two models seek the optimum quantity a firm should produce. Both have different conclusions as they have a different initial assumption. With time, and as the next two models proved, the focus changed to target the optimum price a firm should set in order to maximise profits.

There are also different perspectives in the analysis of duopolies, which deal with game theory. While the models by Antoine Cournot and Joseph Bertrand occur under a basis of simultaneous games, Heinrich von Stackelberg’s model depends on sequential games.

In the real world, firms interact with each other and have to consider the potential negative effects a price war may report in the long-run. Edward Chamberlin suggested that in an oligopolistic market, firms would soon recognise their interdependence and hold monopolistic prices without the implications of collusion, but that would give the same equilibrium point. Collusion equilibrium will be maintained for as long as a firm finds it more beneficial to do so. The moment a firm thinks it can achieve higher profits by deviating from it, it will do so.

Oligopoly (from the Greek «oligos», few, and «polein», to sell) is a form of market structure that is considered as half way between two extremes: perfect competition and monopolies. This kind of imperfect competition is characterized by having a relatively scarce amount of firms, but always more than one, which produce a homogeneous good. Due to the small number of firms in the market, the strategies between firms will be interdependent, thus implying that the profits of an oligopolistic firm will highly depend on their competitors’ actions.

Firms in oligopolistic market can have a wide range of behaviour patterns making it difficult to have a single model.  Static models are used as they present a simple way of analysing equilibriums in this market. However, the maximisation problem faced by the firm will be marked by the different strategic interdependence context in which that market works. Therefore, the firm must estimate and collect the reactions of its competitors in its optimisation problem to choose the best strategy to follow. As a result we must propose a conjectural variation on how competitors modify their behaviour as the firm varies strategies.

Let’s see how conjectural variation affects each firm’s behaviour and final output. We start with the profit (π_i) maximisation problem, where Q is total output, q corresponds to each firm’s output and C is its production costs:

Setting this to zero for maximisation, we have:

Therefore, rearranging the previous function we have: 

From this last function, we can assume the following:

-each firm’s profits are directly related to its competitors’ behaviour, which determines the conjectural variation (CV).

-each firm’s behaviour depends on the conjectural variation of their competitors: if CV>0, the firm will react by striking back with the same action (lowering prices, for example); if CV<0, the firm will choose the opposite strategy; if CV=0, the firm will accommodate its competitor’s strategy, as the Cournot duopoly model assumes.

-each firm’s behaviour, understood as reactions to each other’s strategies, define the market structure and the degree of competition.

For simplicity purposes, oligopolies are specially studied in cases in which there are only two competitors in the market. These are known as duopolies and its analysis and conclusions can be extrapolated for oligopolies.

Natural monopolies occur in those industries in which the total costs of production are lower if a single firm produces the whole output instead of having production divided amongst more than one firm. Although this is the usual definition, which is attributed to William Baumol, who provided it in his article “On the Proper Cost Tests for Natural Monopoly in a Multiproduct industry”, 1977, natural monopolies have been studied since the time of John Stuart Mill and Alfred Marshall. In 1923, economist J.M. Clark greatly contributed to the understanding of natural monopolies in his book “Studies in the Economics of Overhead Costs”, explaining how in manufacturing industries, since overhead costs are a significant fraction of total costs, a high number of firms will not mean lower prices.

Natural monopolies are brought by economic and technical reasons, such as high capital costs, economies of scale and other barriers to entry, and not by legal imperatives. Public utilities such as electricity and water services are some of the most common examples. Natural monopolies are characterized by subadditivity of a representative firm’s cost function. Subaddititvity will exist when 

which is the essence of the idea behind natural monopolies. The demand function plays a key part in subadditivity and hence, the existence of natural monopolies depends on its position, and whether it will be more or less efficient, to have a single firm or more in the market.

However, subadditivity is considered a necessary but insufficient condition for a natural monopoly to be considered optimal, whereas if economies of scale exist, this is a sufficient but not necessary condition for a natural monopoly to be sustainable. Consider the figure below. The cyan demand curve, within the economy of scale region, indicates a viable natural monopoly. The red line would indicate a non-sustainable natural monopoly. The black line indicates the frontier of a profit generating monopoly (which would attract competitors) and the green line indicates a viable duopoly.

As in most imperfect competition market structures and especially in monopolistic ones, a firm who is in a position of natural monopoly may practice an abusive behaviour, which will translate into a loss of welfare. In such cases, government intervention will be praised both by consumers and those firms that seek for lower prices and a profitable share of the market. Regulations such as price setting, taxation or subsidies may be used in order to restore and maximise the initial efficiency of natural monopolies.

Nevertheless, the government must be cautious when setting and applying regulations, as an incorrect comprehension of the market structure may bring a higher cost to the total social welfare instead of the expected benefits. In order to achieve an optimal regulation level, governments should analyse and determine if natural monopolies can be sustained whenever they ensure a lower total cost. If this is the case, the government will have to guarantee that the firm earns no excessive revenues, and that fair prices are maintained. If on the contrary, the total costs of the industry would diminish if new firms entered the market, the government should regulate their entrance. Essentially, what governments should do is to correctly balance the conflict between the industry’s efficiency and its profitability.

Subadditivity is an important concept because it is often used to justify imperfect competition, the classic nemesis of neo-classicists. The only real way to justify less than perfect competition is the kind of sector which lends itself to natural monopolies. Natural monopolies are those where barriers to entry are so high that it is worth making the investment only once, and where one producer will operate more efficiently than two. Traditionally, natural monopolies are state-run, generally in energy or communication sectors, although current trends are leaning towards privatisation.

When a large investment is required in order to begin production, these costs must naturally be passed on to the consumer. What differentiates natural monopolies is that generally, once the investment is made, marginal costs are infinitely small. In these cases, if there is only one company in the market, prices for consumers may actually be lower than if there are rivals, because of the way average costs drop with additional users.

So what exactly is the difference between subadditivity and economies of scale? Subadditivity is a more generic concept which encompasses economies of scale. We say subadditivity exists when:

That is, when the costs function (including initial investment) for the sum of all units at a given level of production is actually smaller than the sum of the various cost functions (given multiple companies) at the same given level. If we look at the diagram, we can see the traditional long run costs functions for two companies. We only consider economies of scale when additional units provide a lower average cost, but costs per unit begin to rise after a certain level. Subadditivity simply means that it is cheaper to produce the same production level when only company one is producing, and that the same production level is more expensive when company two joins the market.

 

This is why subadditivity is considered a necessary but insufficient conditionfor a natural monopoly to be considered optimal, whereas if economies of scale exist, this is a sufficient but not necessary condition for a natural monopoly to be sustainable. The cyan demand curve, within the economy of scale region, indicates a viable natural monopoly. The red line would indicate a non sustainable natural monopoly. The black line indicates the frontier of a profit generating monopoly (which would attract competitors) and the green line indicates a viable duopoly.