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Jun 9

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.

Jun 7

Lope Gallego

Welfare economics

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.

Mar 29

Lope Gallego

Indifference curves

Indifference curves are lines in a coordinate system for which each of its points express a particular combination of a number of goods or bundles of goods that the consumer is indifferent to consume. This is, the consumer will have no preference between two bundles located in the same indifference curve, since they all provide the same degree of utility. The indifference curves, as we move away from the origin of coordinates, imply higher consumption and, therefore, increasing levels of utility.

An indifference map is a combination of indifference curves, which allows understanding how changes in the quantity or the type of goods may change consumption patterns.

Francis Y. Edgeworth, developed the mathematics concerning the drawing of indifference curves in his book “Mathematical Psychics: an Essay on the Application of Mathematics to the Moral Sciences”, 1881, from earlier works by William Stanley Jevons. However, Vilfredo Pareto was the first economist to draw indifference maps as we know them nowadays, in his book “Manual of Political Economy”, published in 1906.

The first example of indifference map showed in the adjacent graph is the most common representation. It shows four convex indifference curves (red), showing each curve what amount of a good or bundle of goods x₁ the consumer has to give up in order to be able to consume more goods, or bundles of goods, x₂. This relation gives us the marginal rate of substitution (MRS) between these goods, which is the slope of the curve in each of its points.

Our second example is an indifference map with four parallel lines (green). This is the case for goods or bundles of goods, y₁ and y₂, which are perfect substitutes, since the lines are parallel and MRS = 1, that is the slope has an angle of 45º with each axis. It can also be the case for goods or bundles of goods that are perfect substitutes but in different proportions. In that case, the slope will be different and the MRS can be defined as a fraction, such as 1/2 ,1/3 , and so on. For perfect substitutes, the MRS will remain constant.

Our third example shows an indifference map with four indifference curves (blue) that represent perfect complementary goods, z₁ and z₂. This is, there will not be an increase on the consumer’s utility unless both goods increase in the required proportion. The best example of complementary goods are shoes, since the consumer’s utility will not increase when he or she gets a new right shoe without getting a new left shoe. Notice that the elbows are collinear, and the line crossing them defines the proportion in which each good needs to increase in order to have an increase in the utility. In this case the horizontal fragment of each indifference curve has a MRS = 0 and the vertical fractions a MRS = ∞.

These explanations of indifference curves can also be applied to production. In that case, the MRS turns into marginal rate of technical substitution and marginal rate of transformation.
Video – Indifference curves:

Mar 27

Lope Gallego

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:

Jan 30

Lope Gallego

Ricardian distribution theory

The importance of David Ricardo‘s model is that it was one of the first models used in Economics, aimed at explaining how income is distributed in society.

Starting assumptions:

-there is only one industry, agriculture; only one good, grain;

-there are three kinds of people:

Capitalists: they start the economic growth process by saving and investing. In return, they receive profits (P), which is what is left once wages and rents have been subtracted from the gross revenue. Capital can be divided into fixed capital (machines, for example) and working capital (wage fund, WF).

Workers: they represent the labour force, in return for wages (w).

Landlords: they allow production(y) to take place in their lands in return for rent (R).

-law of diminishing returns: affects labour (variable factor of production) and land (fixed factor).

-principle of margin: marginal product of labour, which, along with the average product of land, is decreasing.

-principle of economic surplus: profits are determined as a surplus of production.

Analysis:

At a given initial situation, production is at a y₀ level, which we can divide into wages, w₀, and profits, P₀. Rent paid to landlords corresponds to R₀. From w₀ and the level of labour, L₀, we determine the wage fund at the initial situation, WF₀.

In the long term, wages will arrive at a subsistence level, w_s, which can be defined as the wage a worker needs in order to survive. From this, and the level of labour being employed, we determine the wage fund in the long run, WF^*. As this level is the same as labour’s marginal product, the capitalist will not obtain any profits. On the other hand, landlords will get higher rents, R^*.

Steady state:

Real wages will stagnate at subsistence level, the interest rate of capital will stay at 0 and rents will reach its maximum level.

Ricardo explains how this steady state is painful, especially for the working class. However, this steady state can be delayed with technological progress or international trade, as is shown in Ricardian trade theory.