Dec 18

Externality

Externalities are the benefits or costs that arise when the decision to consume or to produce generates some positive or negative impact on the environment affecting the welfare of others in a way that is not transmitted through prices or via market mechanisms. When the market price does not truly reflect the real price in which the externality should be considered, there is a loss of efficiency.

As previously stated, it is important to have in mind that externalities can be linked both to an improvement of the utility for individuals but also to its damage. The former are known as positive externalities and an example of one would be the vaccination of an individual that would prevent having a disease and at the same time avoid the spread of the disease to others. On the other hand negative externalities are those that reduce utility and a common example is the air pollution produced by many factories.

The diagram below shows the graphical representation of a negative externality problem using a polluting factory as an example. The horizontal axis or x-axis measures the amount of output produced by the polluting factory and the vertical axis or the y-axis measures monetary units. The marginal benefit curve (MB) represents the marginal benefit of the factory for each level of production which declines as the amount of output increases. The marginal private cost curve (MPC) shows the marginal cost of the factory as output rises. The more the factory produces, the more it pollutes and hence the bigger the negative effects, which is represented by the marginal damage curve (MD). Lastly, the marginal social cost (MSC) represents the total marginal cost for the whole society, and is constructed by adding together the private costs (which directly translate to higher prices) and social costs.

The socially efficient output Q_S in this case is located at a lower quantity and at a higher price than the actual output Q_A. Whenever market mechanisms do not properly transmit price there will be a loss of efficiency. If it were a positive externality being produced, the opposite would not necessarily occur. It is true that the optimum social would be to set a higher output but price could be lower, equal or even higher than the private optimum depending on the location and slope of the curves.

Several solutions have been proposed in order to reduce inefficiencies created by externalities:

Pigouvian tax: this corrective tax is levied on each unit of output an externality-generator agent produces. This tax is equal to the marginal damage at the efficient level of output (t on the adjacent diagram).

Subsidies: in a polluting industry, firms are paid not to pollute (the government pays a subsidy equal to s, the green rectangle). This method works in a similar way to a tax but this time the government spends money instead of collecting it, while it is the other way for firms.

Cap-and-trade: the basic idea behind this measure is to create a market, where there are a limited number of pollution permits, which allows the firms that pollute less to sell the permits they do not use to other firms that pollute more. This system of tradable pollutions permits will benefit low pollutant firms and penalize high pollutant ones.

Dec 9

Lope Gallego

General equilibrium

A market system is in competitive equilibrium when prices are set in such a way that the market clears, or in other words, demand and supply are equalised. At this competitive equilibrium, firms’ profits will necessarily have to be zero, because otherwise there will be new firms that, attracted by the profits, would enter the market increasing supply and pushing prices down. Following the first fundamental theorem of welfare economics, this equilibrium must be Pareto efficient. Both will have a fundamental relation as a mechanism for determining optimal production, consumption and exchange.

Initial approach:

Let’s consider an economy where there are:

Two factors of production: capital (K) and labour (L).

Two goods: good X and good Y.

Two agents: A and B

The economic problem that is faced needs to find the most adequate allocation of factors of production in order to produce goods X and Y and how these goods will be distributed amongst consumers A and B. This configuration will be such that there will be no other feasible configuration that will allow an increase in any individual’s welfare without decreasing the other individual’s welfare.

In order to achieve Pareto optimality, a certain set of assumptions need to be held.

-The production function needs to be continuous, differentiable, and strictly concave. This will result in a convex set of production possibilities, also known as production possibility frontier Its shape shows an increasing opportunity cost as we need to use a higher number of resources in order to produce a larger amount of a certain good.

-Consumers’ preferences need to be monotonic, convex and continuous, showing how individuals’ welfare increases with a greater amount of goods, but with a decreasing marginal utility.

–Perfect and free availability of information.

-There has to be an absence of externalities and public goods so the utility of individuals depends directly and uniquely from their possession of goods X and Y.

Production optimisation

The optimisation problem in production relies in the maximisation of total output production taking into consideration that it is subject to a limited amount of capital and labour. Analytically,

We can start by looking at the production of goods X and Y as two different optimisation problems. The firm will have to decide what quantity of capital and labour allocate to the production of good X, as shown on the left side of the diagram below, but also what quantity of capital and labour assign to the production of good Y, as shown on the right. These curves are the isoquants corresponding to each production process.

These two diagrams can be plotted together using what is known as the Edgeworth box, which makes it easier to compare quantities of capital and labour used, while also comparing quantities of goods X and Y being produced. Indeed, it’s not only easier to analyse, but also makes more sense, since the total available quantities of capital and labour are given.

The solution to this problem is related to the marginal rate of technical substitution (MRTS). A higher efficiency will be achieved if the reallocation of a unit of labour or capital from one good to another leads to a higher production of the former. When the marginal rate of technical substitution is equal for both goods, it means that all available inputs are being used, which translates into a purely efficient production process.

Graphically, if we plot all these points we construct what is known as the contract curve (blue curve in the Edgeworth box). These represent all Pareto efficient distributions, such as F, G or H. I is not Pareto efficient, since going from I to either G or H would result in an increase in the production of one of the goods without giving up the production of the other. From this curve we can derive the production possibility frontier, which shows the quantities of goods X and Y being produced, as shown in the following diagram. It must be noted that both the contract curve and its derivative, the production possibility frontier, show all the solutions that are Pareto efficient from the firm’s point of view. Only when considering input and output prices will we be able to determine a unique solution (because of the concavity of the production possibility frontier).

Consumption optimisation

Bundles of goods cannot be ranked in a reliable way without knowledge of the distribution of the products, especially if a bundle has different amounts of each good. There may be some bundles that have more products of a good but less of another. The optimisation problem will be to maximise the utility of individuals A and B subject to a limited total amount of goods X and Y. Analytically,

In this case we have to achieve the optimal distribution of two, already produced goods (X and Y) between two individuals (A and B). We can follow the same step by step method used before. Here, we’ll plot indifference curves corresponding to the amounts of goods X and Y consumed by A (on the left), and the amounts of goods consumed by B (on the right).

Again, we use the Edgeworth box to graph the different distributions that can be given between two individuals, A and B, and two goods, X and Y. The further the indifference curve is from the origin, the higher the level of utility enjoyed by the consumer.

Although all the points in the graphic are feasible, not all are efficient, given the utilities and preferences of consumers. The indifference curves join all the points that give consumers the same level of utility. By connecting all points of tangency between the indifference curves of both individuals, the contract curve is constructed and represents all Pareto efficient allocations. The tangency between indifference curves is the point where both consumers have an equal marginal rate of substitution for goods X and Y, and are therefore not willing to trade between them, as it would result in a lower utility.

Global optimum

Until now we have only considered different parts of the economy, and not the economy as a whole. The optimisation problem faced this time is similar to the previous one, although this time an additional restriction is added, since we are here considering both production and consumption: the production level also needs to be efficient.

As this optimisation problem is based on the previous one, we have the same marginal rate of substitution equalisation, but also these two must be equal to the marginal rate of transformation, the PPF’s slope,

These solutions are multiple, since there are various points where the condition holds. However, if we consider output prices (given by the consideration of input prices mentioned before), we are able to consider a unique solution. In the adjacent diagram, if output prices were to be P_X and P_Y, the equilibrium would be point E. However, if output prices were instead P’_X and P’_Y, the equilibrium would be point E’.

Let’s say that prices are set at P_X and P_Y, and that the equilibrium point is E, as seen in the diagram below. Consumers A and B will consume both goods X and Y in different amounts. These amounts are given by the equilibrium in consumption, point E on the contract curve. We have also equilibrium in the production process, given by point E on the production possibility frontier. We know this is a general equilibrium because the marginal rate of substitution is equal to the marginal rate of transformation; or, in other words, the slopes of the indifference curves are equal to the slopes of the production possibility frontier.

Competitive markets result in an equilibrium position such that it is not possible to make a change in the allocation without making someone else worse-off. In reality there are many Pareto optimums and we cannot state that one is better than the other. Even if one consumer got all of the production and the other one none, we cannot say it is an inefficient distribution if all resources are being used efficiently. This is the reason why some economists believe it is an incomplete criterion. However, there are others, such as Milton Friedman and the advocates of the Chicago School, for whom this proves that the economy will act efficiently without the need of government intervention.

Nov 28

Lope Gallego

Shaked-Sutton model

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.

Nov 22

Lope Gallego

Samuelson’s criterion

The Samuelson criterion, sometimes referred to as the Samuelson condition, was raised by the economist Paul A. Samuelson in his paper “Evaluation of Real National Income”, 1950, and belongs to the theory of welfare economics and used as a condition for the efficient provision of public goods. This critique provides a way to avoid intransitivity problems: state X will be preferred to Y if the alternative of X, X’, is preferred to the alternative of Y, Y’.

This criterion however, brings some limitations, since it is very similar to Pareto optimality. Samuelson explains that previous compensation criteria, such as Kaldor’s, Hicks’ or Little’s, hold just because they consider partial redistribution.

Nov 21

Lope Gallego

Little’s criterion

The Little criterion was developed by Ian M.D. Little in his paper “A Critique of Welfare Economics”, 1949, and it constitutes a further step for compensation principle theory. Little criticises the separation between efficiency and distribution and he demands as in Scitovsky’s criterion, for the Kaldor’s and Hicks’ criteria to hold. Furthermore, this criterion also requires that the income distribution is not worsened by the change of states.

This criterion however, brings some limitations, as a result of its implicit value judgement. The criterion will be met, if by a change of states the positively affected individual (winner) is poorer than the negatively affected individual (loser). As an example, let’s analyse the following graph, where we consider the utility of two individuals (A on the x-axis and B on the y-axis), which we will compare using the utility possibility frontier of two different moments.

Kaldor’s criterion is met when going from X to Y, Y to X or Y to Z, but not when going from Z to Y. However, Hicks’ criterion is only met when going from Y to Z. Therefore, when comparing state Y to Z, winners can compensate the loss of the losers, but losers cannot compensate the other part in order to avoid the change. This is the only case in our example where the Scitovsky criterion is met, making Z preferred to Y. However, Little’s criterion is only met if individual B is poorer than individual A.

Nov 20

Lope Gallego

Scitovsky’s criterion

The Scitovsky criterion was developed by Tibor Scitosky in his paper “A Note on Welfare Propositions in Economics”, 1941, in order to solve the inconsistencies, -known as the Scitovsky paradox-, that Nicholas Kaldor’s and John Richard Hicks’ criteria presented. In order to solve these inconsistencies, he required the fulfilment of both criteria simultaneously. As an example, let’s analyse the following graph, where we consider the utility of two individuals (A on the x-axis and B on the y-axis), which we will compare using the utility possibility frontier of two different moments.

Kaldor’s criterion is met when going from X to Y, Y to X or Y to Z, but not when going from Z to Y. However, Hicks’ criterion is only met when going from Y to Z. Therefore, when comparing state Y to Z, winners can compensate the loss of the losers, but losers cannot compensate the other part in order to avoid the change. This is the only case in our example where the Scitovsky criterion is met, making Z preferred to Y.

Scitovsky considered the possibility of changes in Pareto terms caused by state changes. This justified the dual requirements. Analytically,

Although this criterion brings some positive contributions, there are still only minor changes that furthermore need to meet conditions. The estimation of a potential Pareto improvement is yet to be answered. Nevertheless, the Scitovsky criterion contributes to an intransitive organisation of different states

Nov 19

Lope Gallego

Hicks’ criterion

The Hicks criterion is a compensation criterion developed by John Richard Hicks in his paper “The Valuation of the Social Income”, 1940. It is similar to that of Kaldor’s, with different implications although with the same limitations. In this criterion, state Y is preferred to X, if there is not a potential reassignment that turns X into Xˈ, that is at least as good as Y in Pareto terms. In the following graph we consider the utility of two individuals (A on the x-axis and B on the y-axis), which we will compare using the utility possibility frontier of two different moments.

When moving from state X to Y, individual A’s utility decreases while it increases for individual B. Due to this, individual A should compensate individual B so the change of states does not happen, going from X to X’, which will increase B’s utility as much as going from X to Y, while the drop in A’s utility would not be as large. The same would happen if moving from Y to X. Since this ex-ante compensation is possible, neither X is preferred to Y nor Y will be preferred to X.

When moving from state Y to Z, again individual A´s utility decreases while it increases for individual B. When going from Y to Z, there is no possible compensation from individual A to individual B, since to the left of Y the utility possibility frontier is always higher. Individual A therefore can not compensate individual B, so Z is preferred to Y in Hicks’ terms. However, when comparing movement from Z to Y, the opposite logically occurs. Individual A’s utility increases while individuals B’s decreases. Individual B would compensate individual A going from Z to Z’ , and hence Y is not preferred to Z.

If we compare this with Kaldor’s criterion we see some significant changes but still both criteria fall under the Scitovsky paradox. This paradox is centred in the phenomenon that while Y can be preferred to X the opposite can also be true. This does not give a truly asymmetric result as it could just mean that going back to the initial situation is preferred. The economy would therefore oscillate between both points.

Nov 18

Lope Gallego

Kaldor’s criterion

The Kaldor criterion is a compensation criterion developed by Nicholas Kaldor in his paper “Welfare Propositions of Economics and Interpersonal Comparisons of Utility”, 1939. This criterion is satisfied if state Y is preferred to state X and there is such a compensation and reassignment that Y turns to Yˈ that is at least as good as X in a Pareto sense. In the following graph we consider the utility of two individuals (A on the x-axis and B on the y-axis), which we will compare using the utility possibility frontier of two different moments.

When moving from state X to Y, individual A’s utility decreases, while individual B’s increases. Individual B is willing to compensate individual A and move to Yˈ where both increase their initial utility. The opposite, moving from Y to X, can also occur if the winner, this time individual A, compensates the looser, individual B, and is willing to relocate to Xˈ.

When moving from state Y to Z, the utility of individual A decreases, while individual B’s increases. Individual B is willing to compensate individual A and go to Zˈ where both increase their initial utility. On this case the opposite, moving from Z to Y, would not be feasible.

Tibor Scitovsky pointed out some inconsistencies and the consequent limitations of this criterion which are known as the Scitovsky paradox. This paradox is centred in the phenomenon that while Y can be preferred to X the opposite can also be true, as it was previously explained. This does not give a truly asymmetric result as it could just mean that going back to the initial situation is preferred. Economy would therefore oscillate between both points.

Nov 14

Lope Gallego

Compensation criteria

In welfare economics, compensation criteria or the compensation principle is known as a rule of decision for selecting between two alternative states. Two states will be compared; if one state provides an improvement for one part but causes deterioration in the state of the other, it will be chosen if the winner can compensate the loser’ losses until they situation is at least as good as in the initial situation. However, this compensation may not necessarily occur.

This neo-Paretian concept was developed in order to solve the dead end in which the Pareto criterion was at the moment due to its limitations. Although, in essence, the compensation principle reduces to the Pareto criterion, it values positively a wider set that allows a positive ordering without transgressing the Pareto optimal.

To this day there has not been yet a unique and definitive compensation criterion due to its limits and some of its paradoxical implication; on the contrary, a great number of similar criterions have been formulated. From them we must highlight:

Kaldor’s criterion, Hicks’ criterion, Scitovsky’s criterion, Little’s criterion and Samuelson’s criterion.

Nov 9

Lope Gallego

Salop’s circular city

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:

If L is the total number of customers in the market, the catchment area of a given firm in the absence of competition is defined by,

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.