In 1881, Francis Y. Edgeworth came up with a way of representing, using the same axis, indifference curves and the corresponding contract curve in his book “Mathematical Psychics: an Essay on the Application of Mathematics to the Moral Sciences”. However, the representation given, using as an example the work being done by Friday and wages being paid by Robinson Crusoe, was not the one we commonly know nowadays.

It was Vilfredo Pareto, in his book “Manual of Political Economy”, 1906, who developed Edgeworth’s ideas into a more understandable and simpler diagram, which today we call the Edgeworth box.

This diagram is widely used in welfare economics, game theory or general equilibrium theory, to name a few. It is easy to draw and can be easily explained. In the adjacent image, we can see two examples of an Edgeworth box, and how it is drawn.

The first example is mainly used for welfare economics and distribution matters. As we see, this “box” is formed using two sets of typical indifference maps, which in this case represent the indifference curves of agents A (green) and B (red), who must choose quantities of goods x and y. When the indifference map of agent B is rotated, and put on top of the map of agent A, the box is formed. When indifference curves are tangent to each other, which is the case in this example, a contract curve (blue) can be drawn using these tangency points.

Our second example is mainly used to explain Ricardian trade theory graphically. In this case, we draw the production-possibility frontier for countries 1 (green) and 2 (red). When we rotate the diagram of country 2, we end up with an Edgeworth box, which here will help understand how great the gains of international trade are and therefore helps illustrate how trade is not a sum zero game.

Video – Edgeworth box:

This efficiency criterion was developed by Vilfredo Pareto in his book “Manual of Political Economy”, 1906. An allocation of goods is Pareto optimal when there is no possibility of redistribution in a way where at least one individual would be better off while no other individual ends up worse off.

A definition can also be made in two steps:

-a change from situation A to B is a Pareto improvement if at least one individual is better off without making other individuals worse off;

-B is Pareto optimal if there is no possible Pareto improvement.

This can be easily understood using an Edgeworth box. Starting from point C, two Pareto improvements can be made:

-from C to D: individual 1 would increase its utility, since a further indifference curve would be reached, while individual 2 will remain with the same utility;

-from C to E: individual 2 would maintain its utility while individual 2 increases theirs.

Once we are at point either D or E, no further Pareto improvements can be made. Therefore, D and E are Pareto optimal.

Following the same steps for every indifference curve, we can say that every point in which indifference curves from different individuals are tangent is Pareto optimal. The curve that links these infinite Pareto optima is called the contract curve.

Video – Edgeworth box:

Pareto was an economist and sociologist of Italian origin, born in Paris (1848-1923), who taught at the University of Lausanne, as well as previously did his mentor, Léon Walras. They both were part of the Lausanne School, which is considered, along with the Austrian School, as the birthplace of marginalism and neoclassical economics.

His chief works were “Course of Political Economy”, 1896-97, and “Manual of Political Economy”, 1906. Among Pareto’s contributions, we can highlight the graphical development of Edgeworth box, as well as an improvement on the way indifference curves are drawn, and studies on the personal distribution of income.

In his studies of the late nineteenth century, Pareto found some regularity in the personal distribution of income in different countries. From that regularity, he determined a series of economic and sociological conclusions, which became known afterwards as Pareto’s Laws. He also studied some regularities in businesses, where concentration was the explanation for the fact that many companies get 80% of their revenue from only 20% of its products.

Nowadays, however, he is mostly known for the Pareto optimal, a notion widely used in game theory and welfare economics.

Pigou was a British economist (1877-1959), disciple of Alfred Marshall, whom he succeeded as a professor at Cambridge. Pigou is remembered above all as a precursor of welfare economics, for his books “Wealth and Welfare”, 1912, and “The Economics of Welfare”, 1920, in which he used measures of national income and its distribution in order to understand how wealth and welfare are related. He is also remembered for making a distinction between different degrees of price discrimination.

Being part of the Cambridge school, Pigou used common tools derived from neoclassical economics, such as marginalism, amongst others.

Welfare economics are a part of normative economics which objective is to evaluate different situations of a given economic system, in order to choose the best one.

Its study can be traced back to Adam Smith, who related an increase of welfare with an increase on production, and to Jeremy Bentham, whose utilitarian views made him think that welfare was equal to the sum of individuals utilities or, in other words, to a “social” utility.

Traditional welfare economics is based on the work of three neoclassical economists. Alfred Marshall stated that consumer’s welfare was the consumer’s surplus, and therefore was measurable in monetary units. Vilfredo Pareto would criticize this cardinal view, and would be the economist who built a true theory of welfare economics in his book “Manual of Political Economy”, 1906: based on the principles of unanimity and individualism, he designed what nowadays is known as the Pareto Optimality, which would become the core of welfare economics. Later, Pigou wrote “The Economics of Welfare”, 1920, stating that a definition of social welfare must include both efficiency and equity.

During the XXth century, welfare economics developed quickly. Nicholas Kaldor and John Hicks’ compensation criteria, and its following critics by Scitovsky, Little and Paul Samuelson, which aim was to find some way of classification of different optima. Also Bergson’s social welfare function, and Kenneth Arrow’s impossibility theorem, proving the former could not be identified. The theory of second best, developed by Lipsey and Lancaster, aimed at finding an optimum when Pareto optimality could not be found. Finally, the increasing use of cost-benefit analysis marks the validity of welfare economics nowadays.

The Shaked-Sutton model derives from a series of papers written by Avner Shaked and John Sutton. This model is centred in studying vertical differentiation and its role when discriminating the market, in order for firms to absorb as many consumers’ surplus as they can and the consequences this has for the market.

Vertical differentiation between products means discriminating goods as a result of their different qualities. Quality is a characteristic that all consumers want. Given the same price, higher quality products will always be chosen. However due to different income levels that vary from a to b, consumers will have different willingness to pay. This way, individuals with higher incomes will buy goods of higher quality paying a higher price, while lower income individuals purchase lower quality goods at a lower price. Given a higher spread of income between consumers, there will be a higher number of firms in the market each producing a different quality, and thus, making the number of firms in the market subject to the dispersion of incomes.

Firms will have to go through two key decision stages. In the first stage firms will have to choose the quality of their products, taking into consideration that higher quality goods have associated a higher production cost but are also preferred by consumers. Due to this firms are encouraged to produce higher quality goods as it will increase the competitiveness of their product. On the second stage firms will set their prices. Once the quality of the product is determined, firms will want to maintain the greatest differentiation for their product in order to raise their market power.

In markets with a great number of firms, competition between firms will be translated into a lower price for higher quality goods and will end up driving lower quality goods out of the market. A finite number of firms will exist in equilibrium depending on the size of the market and of the cost function associated to the firm.  If the market is in equilibrium it will only allow as many firms as different demands of quality are.

Salop’s circular city model is a variant of the Hotelling’s linear city model. Developed by Steven C. Salop in his article “Monopolistic Competition with Outside Goods”, 1979, this locational model is similar to its predecessor´s, but introduces two main differences: firms are located in a circle instead of a line and consumers are allowed to choose a second commodity. Consumers will have to choose between buying one or none of the differentiated goods, and spending the rest of their income in the second, non-differentiated good.

In this circular model all firms that offer the differentiated goods are located on a circle of perimeter 1, equidistant from each other, and it is an outside firm that offers the undifferentiated good. Consumers purchase goods taking into consideration several aspects such as their preferred brand specification, the distance and transportation cost and their price, and so will act accordingly in order to maximize their utility.

We will designate x’ as the consumer’s preferred good, and x will be the least valued good. Consumers enjoy a surplus S when x’ is consumed instead of x. Knowing that consumers can buy x at location x_i for a price of p_i, we can equate a decision formula:

Taking into consideration transportation cost,

where x_i-x’ is the shortest arc length between x_i and x’. If we reformulate the first equation,

This way the effective reservation price v is given by,

Changing now our attention towards the firm’s perspective, there is a total of n firms, and we know that they are at an equidistant distance of 1/n from each other, spread around in a circle. Suppose firm A charged p while the rest of firms charged p*. In the absence of competition the maximum distance consumers are willing to travel T, which is equal to:

However competition is a condition that is given in this market and consumers will consume from the firm that reports the highest surplus. The quantity sold by firm A will be equal to,

It is a logical outcome that firms that have their price set above v will not sell their goods. Only when they lower prices below v will they start attracting consumers.

There will be a point where the price is low enough to overlap with neighbouring firms. If a firm keeps lowering its price it will even capture its competition’s customers. 

The choice amongst firms is indifferent to client when the price is equal to,

To sum up, the number of firms in this market will depend on the degree of differentiation between goods, as well as price. We have here similar conclusions to those seen in the linear city model: there are two opposite effects. On one hand, there is an incentive for both stands to locate at the centre of the beach in order to increase their market share by reaching out to the greatest amount of customers, in what is known as the demand effect. And, on the other hand, there is an incentive for both stands to locate at opposite extremes in what is considered to be the strategic effect. While the first effect will reduce differentiation between the stands, the second one will increase it.

This model constitutes an important development from its predecessor, since it can be used to understand contestable markets and entry and exit barriers. It is considered, along with Harold Hotelling’s model, an important part of monopolistic competition theory.

Hotelling’s linear city model was developed by Harold Hotelling in his article “Stability in Competition”, in 1929. In this model he introduced the notions of locational equilibrium in a duopoly in which two firms have to choose their location taking into consideration consumers’ distribution and transportation costs. Hotelling’s model has been source of inspiration for a great amount of fruitful literature which does not only constrain to industrial organization theory but also to other sciences, such as politics, as some of its conclusion can be directly applied to them.

Initially the model was developed as a game in which firms first chose a location and after a selling price for their products. In order to set their business in the best location to maximize profits, the firms will have to evaluate three key variables: competitors’ location, customers’ distribution and transportation costs.

However, this model includes two different approaches, the first one being a static model consisting of a single stage were firms choose their location and prices simultaneously, and a dynamic model in which price is set after determining the location. The model is based on a linear city that consists of only a single straight street. For a better comprehension, Hotelling’s model is sometimes explained by using the example of a beach were two ice cream stands are trying to decide the best location.

Static model

In a beach going from west to east, of size [0,1] where consumers are distributed evenly, two identical ice cream stands (A and B) with a marginal cost of production, c > 0, try to determine their best location. A is located at a distance a from point 0, while B is located at a distance b from point 1.

It is indifferent for customers whether to go to one or the other stand, as long as their utility is maximized. For this reason, we must assume that both ice cream stands offer the exact same ice creams, and therefore consumers’ utility will be given only by the price of the ice cream and the distance to the stand. Despite differences in prices, the stand with the lowest prices will not necessary attract all the demand since consumers will also consider the distance to the stand.

If both stands’ prices were equal, the differential factor would be how close consumers are to each stand. All consumers located to the left of a would go to stand A, and all consumers located to the right of 1-b would go to stand B. The remaining consumers, located between both stands would go to whichever is the closest. In this first model,  is the exact middle of that beach, so consumers to its left would go to stand A while consumers to its right would go to stand B.

Two conditions are necessary for profits to be positive and maximized in both stands:

-Selling prices must be higher than marginal costs

–

This second condition implies that both stands can’t be located at the same point exactly in the middle of the beach. If this were the case it would be indifferent for the customers to go to either one. Each stand would decrease its prices in order to attract customers, and thus they would enter into what is referred to as a price war. If they had different marginal costs, the stand with the highest marginal cost would be in a clear disadvantage and would end up exiting the market, as the other stand would be able to push the prices further down, and so attract all customers. Depending on how prices are set it could lead to a Bertrand’s solution, in which the prices of both stands are equal to their marginal costs, thus achieving zero profits.

We can conclude by saying that in this model the key factor for product differentiation is location. Each stand will therefore set prices that will be higher than their marginal costs, and will choose a location other than the middle of the beach.

Dynamic model

In this case, the model has two stages. During the first stage, the stands chose their location and on the second stage prices are set.  With regards to the previous model there are two additional assumptions:

Both stands have equal marginal production cost.

Total cost for consumers depends both on the price of the goods and the distance to its location (t).

The point of division between the areas served by these two stands is determined by the condition that at a certain point it is indifferent for consumers between consuming in one stand or the other. Equating and equalising we get 

By solving, the demand function for each stand will be: 

By using backwards induction we can find the subgame perfect Nash equilibrium where both firms maximize their profits.

Stage 2: having already determined a location the stands will now have to set the price that will report the highest possible profits. This will lead us once again to the static model result, in which each stand sets its own price.

Stage 1: the stands anticipate and choose their location. Profits will be higher the less distance there is to the extremes, therefore maximum differentiation between stands will be given when A locates at 0 and the B at 1.

The conclusions that can be drawn from this model are two opposite effects. On one hand, there is an incentive for both stands to locate at the centre of the beach in order to increase their market share by reaching out to the greatest amount of customers, in what is known as the demand effect. And, on the other hand, there is an incentive for both stands to locate at opposite extremes in what is considered to be the strategic effect. While the first effect will reduce differentiation between the stands, the second one will increase it.

In politics we can observe how candidates follows this demand effect, as they tend to come from a specific ideology but tend to convey towards a moderate ideology with the objective of attracting as many voters as possible.

Product differentiation is a marketing process that has the objective of making customers perceive the product of a specific firm as unique or superior to any other product belonging to the same group, and so creating a sense of value. Differentiation does not always imply changing the product, sometimes it is enough just by simply creating a new advertising campaign or by changing its packaging. This term was introduced in economics by Edward H. Chamberlin in his book “Theory of Monopolistic Competition”, 1933.

What a firm achieves by differentiating its product from competitors is to create a market in which it can act as a monopoly, enabling them to have price-making power. However, since products are also similar enough and there are plenty of customers, this type of market structure is also considered as perfect competition. This market is therefore characterized by a group of related products that can be considered to be close substitutes amongst each other which is reflected by their high crossed elasticity. Nevertheless, product differentiation can also be a way to avoid or to create entry barriers and thus become a source of competitive advantage. As a product becomes more differentiated and unique for consumers, it will become more difficult to compare it to other products and it will move competition with other products to non-pricing factors.

There is an important distinction to be made between different differentiation strategies. Product differentiation can be subject to subjective or objective preferences. Two categories are distinguished from this division, horizontal and vertical differentiation. The former is given when consumers base their purchasing decision on subjective preferences when comparing products, e.g. colours or flavours. The latter occurs when a product can be evaluated against the other ones in terms of measurable and qualitative factors, e.g. technological differences or technical properties in engines. Several models have been developed to analyse these two strategies, the most famous being Hotelling’s linear city model and its extension, the Salop’s circular city model, for horizontal differentiation and the Shaked-Sutton’s model for vertical differentiation.