Welfare economics II

 The analysis of welfare economics is built around the concept of Pareto efficiency. However, this efficiency criterion does not always represent a satisfactory answer. Other times, certain optimality conditions cannot be satisfied, and therefore Pareto efficiency simply cannot be reached. In order to solve this problem, and to find a new way to establish which allocation is best, economists have been since searching for new criteria to make a more informed decision.

In this Learning Path we’ll learn about some of these criteria, in order to understand them and being able to use them. In this LP we’ll learn about:

 Pareto efficiency, the cornerstone of welfare economics.

 Compensation criteria

Definition: what they are and how to use them. More precisely, we’ll talk about:

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

 Theory of the…

Second best, a theory by economists Kelvin Lancaster and Richard Lipsey.

Pareto efficiency

Summary

The analysis of welfare economics is built around the concept of Pareto efficiency. However, this efficiency criterion does not always represent a satisfactory answer. In order to solve this problem, and to find a new way to establish which allocation is best, economists have been since searching for new criteria to make a more informed decision. In this Learning Path we learn about some of these criteria.

his 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 efficiency is great, no doubt about it, but sometimes it is impossible to reach. That’s why we need other compensation criteria. Next, we’ll see a definition of compensation criteria: what they are, how they work, and what to expect of them.

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.

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.

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.

Some inconsistencies appear when using both Kaldor’s and Hicks’ criteria, known as the Scitovsky paradox. In order to solve it, we use what is known as Scitovsky’s criterion.

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

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.

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.

Now that we know everything we need about compensation criteria, let's learn about another way to avoid looking for Pareto optimality: Lancaster and Lipsey's Second Best theory.

Second best

Kelvin Lancaster and Richard G. Lipsey, in their article “The General Theory of Second Best”, 1956, following an earlier work by James E. Meade, treated the problem of what to do when certain optimality conditions (which must be considered in order to arrive at a Paretian optimum solution in a general equilibrium system) cannot be satisfied. The main idea in this article is that, when a constraint prevents the fulfilment of one of these conditions, the other conditions are in general no longer desirable. The optimum situation in this case can be attained only by neglecting the other conditions. Indeed, this new optimum is called “second best” because a Paretian optimum cannot be attained.

This can be easily understood using the diagram depicted in the article. We start by considering a typical optimization problem, with a given production possibility frontier (PPF) considered as a boundary condition, indifference curves (green curves, in this case representing a welfare function, ω) and the optimum where the PPF is tangent to ω (point P). Since this points lies on the transformation line and an indifference curve, it defines the production and consumption optima.

When we draw a new constraint condition (red curve), it can be easily seen that point P is no longer attainable. Q could be a second best solution, since it lies both on PPF and NewCC. However, as the authors point out, the second best point would be R, inside the transformation line. This is so because an improvement on welfare can be attained by moving to point R, since it lies on a further indifference curve (ω’’), and therefore means higher welfare.

The segment MN is technically more efficient than R, but since the points on this segment cannot be attained, R is the second best solution.

In this Learning Path we've learned about compensation criteria, used in order to avoid looking for Pareto efficiency when none can be reached. We've seen how Kaldor's and Hicks' criteria work well, except when Scitovsky's paradox appear. We've also learned about Little's and Samuelson's criteria, which keep in mind redistribution of wealth. Finally, we've seen how Lancaster and Lipsey's Second Best theory works.

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Welfare economics I: Efficiency and optimal allocation

 

Pareto efficiency

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.

Edgeworth box

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:

Production possibility frontier

The production possibility frontier (PPF) represents the quantity of output that can be obtained for a certain quantity of inputs using a given technology. Depending on the technology, the PPF will have a certain shape.

As you can see on the adjacent figure, this PPF (blue curve) slopes downwards. This slope, which equals the marginal rate of transformation between X and Y, shows us how, in order to increase the output X, the quantity of Y must decrease. In fact, the marginal rate of transformation measures the tradeoff of producing more X in terms of Y.

This frontier determines the maximum output (of both X and Y) that can be obtained given the technology. Production at point A will produce more quantity of Y and less of X than production at point B. However, both are technically efficient, since they maximize the output. For example, production at point C is technically inefficient because, at any point on the PPF, more combined output is produced using given the technology. Also, point D is unattainable given the technology, being this is the reason why it is outside the PPF.

The PPF can be derived from the contract curve on an Edgeworth box. In this box, we see the quantity of inputs (K, L) being used in the production of each good (X,Y). In fact, we can see how, for each quantity of each product, the quantity of each input can change. The isoquants (green curve for X, red for Y) determine how much a certain input has to increase in order to compensate the decrease in the other input, maintaining the quantity of output produced unaltered. The slope of these curves is given by the marginal rate of technical substitution of each output.

The points where the isoquants of different outputs combination intersect, which are Pareto-optimal, allow us to draw the contract curve, from which the PPF can be derived. Since the technology is given, only one PPF can be derived from the contract curve (as opposed to the case of the utility possibility frontier).

 

Video – Production possibility frontier:

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 PX and PY, 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 PX and PY, 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.

Fundamental theorems

There are two fundamental theorems of welfare economics.

 

-First fundamental theorem of welfare economics (also known as the “Invisible Hand Theorem”):

any competitive equilibrium leads to a Pareto efficient allocation of resources.

The main idea here is that markets lead to social optimum. Thus, no intervention of the government is required, and it should adopt only “laissez faire” policies. However, those who support government intervention say that the assumptions needed in order for this theorem to work, are rarely seen in real life.

It must be noted that a situation where someone holds every good and the rest of the population holds none, is a Pareto efficient distribution. However, this situation can hardly be considered as perfect under any welfare definition. The second theorem allows a more reliable definition of welfare

 

-Second fundamental theorem of welfare economics:

any efficient allocation can be attained by a competitive equilibrium, given the market mechanisms leading to redistribution.

This theorem is important because it allows for a separation of efficiency and distribution matters. Those supporting government intervention will ask for wealth redistribution policies.

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