Chapter 7: Welfare theory II





1st Two preliminary notes

2nd The value premises of the welfare theory

3rd The two Gossen's laws

4th Welfare maximisation at income equality?

5th The paretian welfare theory

6th The compensation criteria

7th The pension concept

8th The theory of the second best

9th The significance of the competition for the welfare

              10th  Externalities

              11th  The Cost-Benefit analysis

              12th  Pareto-optimal redistribution



4th Welfare maximization at income equality?


The Welfare theory, however, is primarily interested in the welfare of the society as a whole, and the welfare of individual households or individuals is only of interest insofar as presumably the overall economy depends on the welfare of its individual members. Therefore, let us ask ourselves when we can speak of a maximization of the overall welfare.


Based on the individualistic approach, it is clear that a steady increase of the benefit of all individuals has also led to a general increase in the overall welfare, just as a steady reduction of the welfare of each individual member also means a reduction of the overall welfare. But how is it, when the welfare of the single individuals develops differently, when the welfare of some members increases, but the of the other members decreases?


Within the scope of the older welfare theory, an attempt has now been made to prove that a community maximizes its overall welfare especially when all incomes have the same level. If we assume a differentiation of incomes, we can, according to this consideration, increase the overall welfare by reducing the income of the richer (e.g. by a tax) and increasing the income of the poorer (e.g. by granting transfer incomes).


According to the law of the diminishing marginal utility of income, the marginal utility of the last income unit of the richer is less than the marginal utility of the poorer. If, therefore, the income is reallocated in favor of the poorer, the lost welfare loss among the rich is lesser than the achieved welfare gains among the poorer. Thus - it is concluded- the welfare of society as a whole has increased.


This welfare gain due to a reallocation of income is to be expected as long as the marginal utility of the taxed member is less than the marginal utility of the poorer, and this is the case as long as the income of the taxed members is higher than that of the beneficiaries. So one can conclude that only with a total equality (everyone receives an equally high income) the benefit of the society as a whole is maximized.


We like to make these thoughts plain to us by means of a diagram. For the sake of simplicity, we only assume two persons (A and B). In the initial state, person A has a higher income than person B. For both persons applies the law of the diminishing marginal utility of income, it is further assumed that both individuals have the same demand structure, and that therefore an identical marginal utility function applies to both. On the abscissa, we draw the income of both persons, on the ordinate, the height of the respective marginal utilities. The collective marginal utility function will indicate the extent of the marginal utility at alternative income heights.





The graphic shows that the utility loss of the rich (-) is less than the utility gain of the poor (+). Thus, the total utility increases due to a levelling measure and this is correct as long as the incomes still have different levels.


Now these conclusions apply only under a variety of assumptions, which are by no means fulfilled always, and therefore we want to examine them critically in the following.


First assumption: The law of the diminishing marginal utility of income is valid. We have already seen above that this assumption is not shared by all welfare theorists, and sometimes it is assumed that the law of the diminishing marginal utility is only partially valid to the individual goods, but not with respect to changes in income. Nevertheless, most economists assume that usually a diminishing marginal utility course can be assumed for the entire income.


Second assumption: identical demand structures are assumed. This assumption is likely not present generally in reality. On the contrary, it is generally assumed that the individual persons differ very much in the utility which they achieve maximally from individual goods and from the total income, that these differences are also regarded as desirable and in no way as questionable; but this results in a different course of the marginal utility curves of the single individuals.


Let us now assume that person A (the richer) has, for example, on account of his higher education, a demand structure at which the marginal utility curve is above the marginal utility curve of person B. The following graphic shows that in this case the maximum of total welfare is not achieved at complete equality, but with a differentiation of the marginal utility curves according to the differentiation of the incomes.




As it can be seen in the graphic, certainly the marginal utility of the richer is possibly lesser in the beginning than that of the poorer. However, an equality of the marginal utilities and thus a maximization of the total utility is achieved with a very specific income differentiation (E10 / E20). A further levelling would lead to a reduction of the overall welfare.


Third assumption: In our previous considerations, we proceeded implicitly from the assumption that the redistribution from the rich to the poor would be achieved without further additional costs. This is, of course, a completely unrealistic assumption. In reality, a resources consuming bureaucracy is necessary to carry out this redistribution.


The state officials must first determine the income of those who are to be taxed and the to be favored in order to determine who is to be taxed and who is to be favored, and further which tax amounts are necessary to achieve the desired welfare increase; taxes have to be collected and the transfer payments have to be paid out. Finally, it is necessary to check whether there are abuses, at the rich, the way that they do not declare all incomes, in order to pay less taxes; at the poor, the way that they pretend to receive too little income in order to receive higher transfer payments.


All these activities cause costs, so that only a part of the tax revenues are available for transfer payments, with the result that already at a certain income differentiation all possibilities for welfare increase have been exhausted due to redistribution. The following graphic makes these relations clear:




Despite considerable income differences, the differences in the utility changes of both persons are much lower because of the resource consumption, so there is also much less room for a welfare increasing redistribution.


Fourth assumption: Although we had assumed so far that the persons concerned perhaps might try to conceal their actual income levels, it was always assumed, though, that the actual amount of the factor incomes would not be affected by the redistribution. This assumption must be criticized also. At redistribution activities of the state, both the rich ones and the poorer are indeed very well interested in achieving less factor income than before the redistribution.


For the poorer person, the transfer payments mean incomes that are obtained without any additional performance; here, it can be quite utility increasing if the person concerned reduces his performance, thus gaining a greater amount of leisure time which may be perhaps greater than the marginal utility of the last income unit because of the larger total incomes (factor income plus transfer payments).


For the richer it is analogically valid that the marginal utility of the remaining net income is reduced due to the higher taxation; here, too, it will generally be utility increasing if more leisure time is taken. If so far the marginal utilities of economic activity and leisure time have been balanced before additional taxation, the marginal utilities of economic activity have now declined due to taxation, and a compensation can be achieved the way that more leisure time is taken.




The green arrows show the effect of the original redistribution, the blue arrows inform about the influence of the anti-performance incentives. The remaining total gain of welfare is therefore again considerably lower, again with the consequence that even less space remains for a welfare increasing redistribution.


A similar welfare-reducing effect consists in the fact that redistributions can generally not be handled in an allocation-neutral manner, that they distort the price ratios and, in this way, lead to an allocation of scarce resources onto the individual goods, which leads away from an optimal allocation.


Fifth and sixth assumption: In our previous considerations we have always implied that the marginal utilities can be measured cardinal, and in addition can be compared interpersonally. Exactly these assumptions are, however, doubted by Vilfredo Pareto and the followers of the Paretian welfare economics. According to Pareto, utility units can only be measured ordinal, but not cardinal.


Here, one speaks of an ordinal utility scale whenever one confines himself to indicate whether two goods bundles bring about an equal high utility or one bundle of goods brings about a higher utility than the other. On the other hand, one would speak of a cardinal utility scale if moreover, it could be indicated how many multiples higher the utility is that one bundle of goods could offer in relation to the other bundle of goods.


Although in literature there are some attempts to defend the possibility of a cardinal utility scale, it was not achieved to provide this evidence generally convincing, so that the majority of the welfare theorists assumes together with Pareto that utility can be measured only ordinally.


In this case, however, it is not possible to display the marginal utility curve in a diagram with a cardinal utility scale. We can, however, state that the marginal utility decreases with increasing incomes, but can no longer determine about how many units the marginal utility decreases.


Now this criticism is of rather formal nature. We could interpret a utility diagram in such a way that two points of different height on the ordinate indicate only that the higher point also reflects a higher marginal utility, but that the concrete distance between these points can not in indicate anything about how much the marginal utility of the higher point is higher than that of the lower point.


Much more crucial is the second criticism, that benefit units according to Pareto can not be compared interpersonally. If person A is not able to compare the benefit experienced by himself with the benefit of another person, and if, therefore, a third party (the state) can not compare the benefits of two persons with each other, then there is also no statement possible about whether the total welfare of a society can be increased by a levelling of the incomes.


The argumentation of the older welfare theory that equality produces a welfare maximum collapses. One could say at most that the benevolent dictator assesses the benefit of the persons concerned from his view, he can do this  of course; but such an approach would be contrary to the generally accepted self-determination criterion.



5th The Paretian welfare theory


Which way out of the dilemma described in the first part of this article offers now the Paretian welfare theory? Vilfredo Pareto has formulated the following criterion, which is named after him: If the welfare of the community depends on the welfare of the individual members, it can be safely assumed that when all individuals have experienced a welfare increase, that also the general welfare has increased; just as incidentally a reduction of the welfare of all community members also means a reduction in the total welfare. We have already mentioned this conclusion.


But how is it with the evaluation of the total welfare if individual welfares develop differently, if only single individuals experience a welfare increase, the welfare of the remaining members remains either unchanged or is even diminished? Here, the Pareto criterion states: one can only speak clearly of a welfare increase (a measure, a change of state) if at least one individual of the community achieves a welfare increase, and if no other individual experiences a diminution of his welfare at the same time due to the planned changes.


The question of whether such changes of state are possible at all, whether there will not always be winners and losers in every planned change, will be discussed in detail below. At this point, let us consider the question whether the Pareto criterion can actually be considered as generally accepted. Take as an example a political measure that brings a super rich citizen (a billionaire) and only him a welfare increase, the rest of the population gets nothing, especially the poorer classes of the population would not benefit from this measure. Should one really speak here of the case that the common welfare has increased, although only someone has become richer, who has anyway lived in excess so far already?


Or to make a second example: Let us take the case that a supplementary tax will be demanded from the super rich (the billionaires), and that this tax would be used to provide the poorest of the poor with an income equally high to the minimum subsistence level, or perhaps to treat their diseases. According to the Pareto criterion, this measure could not be unequivocally classified as welfare increasing, since it is assumed that single individuals (the billionaires) suffer welfare losses by this measure. However, if we were to carry out a survey, a large percentage of respondents would surely say that this is a welfare increasing measure.


In order to prevent misunderstandings: the Pareto criterion does not mean that in this case there would occur no increase in welfare, but only that in this case a welfare increase could not be proved scientifically correct. Naturally, it remains to each citizen to speak nevertheless of an increase in welfare in this case. This constitutes an evaluation that can not be achieved with the help of scientific means - and is therefore not binding for all people.


Whenever distribution issues are concerned in the context of welfare changes, three different and competing positions are likely to be represented in the public: We have, firstly, the socialist maxim according to which every reduction in income differentiation is called desirable. According to the one (more radical) variant, a societal optimum would only be reached with a total income equalization, in other words, if all citizens were to have the same income.


According to a somewhat more moderate variant, certain differentiations of income are considered to be justified (whoever performs more, should also get more), but it is assumed that the actual differentiation in the incomes does not in any way reflect the performance differences of the citizens and had reached a no longer justifiable level. Here, it is desirable that the income differentiation will be reduced to such an extent that it only corresponds to the performance differences (and other justified differentiations such as, for example, a different disease level).


According to a liberal principle, by contrast, any growth in the domestic product is seen as a welfare gain, whereby we can differentiate between a radical and a more moderate variant here again. In the case of the radical variant, it is not asked how the distribution of income is changing as a result of politically introduced measures, the increase in production as such is classified as a welfare gain. In the more moderate version one assumes that the income distribution has not been altered by this measure in favor of the poorer, and the gain in this measure is also seen in the fact that socio-political aims can be addressed more intensively due to the increased production and thus also increased tax revenues.


By contrast, John Rawls has formulated the Maximin principle. An economic policy measure is thereafter recommended under justice aspects if it contributes to raising the income of the lowest income classes, even if by these measures the income of the rich increases more and thus the income differentiation increases.


Let us now realize the difference between these three principles by means of a graphic. The population shall be divided into two groups according to the income level: The recipients of the lowest income class (Epoor) and the remaining income classes (Erich). We draw the income on the ordinate: Epoor and on the abscissa the income: Erich. Under consideration would be a political measure, which would lead to an increase in the domestic product. The growth function, drawn in blue, indicates how the extension of this measure -starting with the origin of the diagram - affects the two income groups. The assessment of this measure is under consideration according to the three principles developed above (socialist, liberal principle and the Rawls's maximin criterion).





If we follow the socialist principle of equality, then we reach the optimum of welfare where the 45° line from the coordinate origin is tangent to the growth function (red point of contact). On the other hand, if we apply the liberal principle, the welfare optimum has been achieved where the left-angled 45° line is tangent to the growth function (yellow point of contact). The maximin principle finally achieves its welfare optimum where the income of the poor reaches its maximum (green point of contact).


The lowest growth is achieved if we follow the socialist maxim, on the other hand, the highest growth is achieved at application of liberal principles, the maximin principle assumes a middle position in relation to growth. If, however, we mind the income of the poorer, this income reaches its maximum by following the maximin rule. It is then furthermore dependent on the course of the growth function whether the income of the poorer is higher in compliance with a socialist rule than when observing liberal principles.


In any case, the socialist rule leads to a greater equality of income. But since the socialist rule generally results in less growth than the liberal rule, the possibility has quite to be expected that the absolute income of the poorer will be higher when adhering to liberal principles than by observing socialist principles.


If one, though, assumes the validity of the Pareto criterion, then all attempts of redistribution can no longer be  assessed clearly, but nonetheless it seems to be possible to assess general policy measures whether utility gains are promised to at least one person and no utility reduction to any other person.


We would like to postpone this criticism and ask ourselves the question of which consequences in the context of the welfare theory result thereof that utility can only be measured ordinally. The solution lies in the development of the theory of choice, which was initiated by Pareto.


In a diagram, the consumption of two substitutive consumer goods is drawn on the coordinates. An arbitrary point of the diagram (P1) is selected, which coordinates indicate the consumption quantities of the two goods (x1) and (x2). This bundle of goods creates a certain utility (called degree of ophelimity).


Now the one good (x1) shall be replaced by the other good (x2) by subtracting a unit from good (x1), and by deploying of good (x2) so much goods until the loss of utility is replaced that was caused by the subtraction of one unit of the good (x1). In this way, a second point results which presumably has the same degree of ophelimity as the first point.


In this way, for every possible quantity of good (x1), a quantity of good (x2) can be named, which together have the same degree of ophelimity as the first two points. This way emerges an indifference curve with the degree of ophelimity (O1), which lists all combinations of good (x1) and good (x2) with the same degree of ophelimity (O1).


In the same way further indifference curves with other degrees of ophelimity can be constructed. If we start from point 1 and now use one unit more from both goods, we get a goods combination which is clearly superior to the combination (P1), thus has a higher degree of ophelimity. For this new point we can again determine all those combinations which lie on a new indifference curve. If we proceed in this way, we obtain a whole series of indifference curves, each individual indifference curve being assigned to a particular degree of ophelimity. Through every point of this diagram passes such an indifference curve. For the sake of clarity, however, we have drawn in our graphic only single, in no case all indifference curves.




If we compare the two indifference curves (O1) and (O2), we can clearly ascertain that O2 reflects a higher ophelimity than O1; but we can not indicate how much the ophelimity has increased from O1 to O2, since degrees of ophelimity represent only ordinal, but not cardinal variables.


The curvature of the indifference curves is given by the law of a diminishing marginal rate of substitution. This law refers to the same correlation as the law of diminishing marginal utility. If I replace successively good (x1) by good (x2), then I need more and more units of Good (x2) for a unit of the good (x1). This is, of course, precisely valid because the marginal utility of good (x1) increases at the removal of this good, while the marginal utility of the good (x2) diminishes because of the addition of this good.


Now, in order to determine by analogy with 2. Gossen's law at which allocation of income a household has its utility optimum, we have to draw the balance line in the graphic which indicates which goods combinations of (x1) and (x2) at given prices of both goods (p1 and p2) and also given income (e1) are actually possible.




The angle, which this balance line forms with the abscissa, measures the price ratio (p1/ p2). This angle is equal to the ratio x20/ x10, whereat x20 is the quantity of x2 which the household receives when it expends its total income for this good, thus e1/ p2. To the same degree, x10 corresponds to the quantity of the good x1 if the household outputs its total income for this good x1, thus e1/ p1. If we put these values into the ratio x20/ x10, we finally get the price ratio (p1/ p2).


Now, at which combination of goods does the household reach its maximum of utility at given income and given price ratios? Obviously, at the point at which the balance line is tangent to an indifference curve. The to be selected combination must necessarily be a point on the balance line. Each point, which is to the left or right of the tangent point (balance line with an indifference curve), is located on an indifference curve with a lower degree of ophelimity.


This means that the individual household reaches its utility maximum exactly when the marginal rate of the substitution (the tangent to the indifference curve) coincides with the angle of the balance line and thus also with the goods price ratio. Since the marginal rate of substitution represents the subjective exchange ratio, and the price ratio, however, represents the objective exchange ratio, one can also speak of the fact that the individual welfare is then the greatest when subjective and objective exchange relations coincide.


As in the context of the marginal utility theory, we must also here point out at which goods combination the entire community achieves its welfare potential. If we start from the diagram of the theory of choice shown above, we replace first the set of indifference curves by means of a set of collective indifference curves. Whereat the collective indifference curves indicate which combinations of goods grant the same ophelimity to the entire population, respectively, which combinations relative to a comparative combination have a higher or a lower degree of ophelimity. Also for these indifference curves a convex curvature is assumed, and its marginal rate of the substitution (tangent to an indifference curve) also decreases with increasing substitution.




Secondly, we replace the balance line by a transformation curve. This indicates how a predefined stock of resources can be distributed to the two goods (bundles of goods) by assumption of constant technology. This curve is concavely curved due to the regularity of decreasing marginal returns of the individual production factors. The marginal rate of the transformation (the tangent to the transformation curve) decreases with increasing use of a factor. If we move along the transformation curve, then there occurs also a substitution of the one good by the other good, whereat the respective inclination of the tangent to this curve indicates how much goods units of good (x1) can be produced additionally if the production of one unit of the good (x2) is spared.


As the graphic shows, a maximum of welfare is achieved in the goods combination in which the given transformation curve is tangent to a collective indifference curve. The objective ratio of the transformation must correspond to the subjective exchange ratio.


In this model, problems are mainly caused by the construction of collective indifference curves. According to the idea this theory, the collective indifference curves are derived from the individual indifference curves. But, how is this possible? Paul A. Samuelson has shown that collective indifference curves can be derived from the individual indifference curves only under very restrictive assumptions. Here, Samuelson mentions three alternatives:


Firstly, the construction of a collective indifference curve would be possible if we either assume only one single individual, or assume that a benevolent dictator with meritorious intention takes the decision of how the individual combinations of goods must be assessed respectively. This assumption contradicts, of course, the self-determination criterion and is thus not an acceptable solution.


Secondly, one could then derive collective indifference curves from the system of individual indifference curves, if one assumes that all individuals had identical demand structures. We have already seen in the review of the marginal utility theory that this assumption does not correspond to reality also, since the single individuals differ especially in the nature of their demand structure.


Thirdly, collective indifference curves could also be derived from the set of individual indifference curves if changes in the allocation, thus movements along an indifference curve, would not affect the income distribution. But this assumption also contradicts reality, since changes in the allocation are generally associated with changes in the goods- and factor-price ratios, but the prices - especially the factor prices - also determine the respective income distribution.


Only, if we could assume for all markets the market form of a bilateral monopoly, if we continued to assume that the market partners proceeded in the sense of a step-by-step negotiation strategy, and if finally homogeneous-linear production- and utility functions would be present, then could be assumed that the income distribution is broadly disconnected from the allocation.


Thus, we come to the conclusion that under realistic conditions no collective indifference curves can be derived directly from individual indifference curves. In spite of this criticism, collective indifference curves are used frequently, particularly in the context of real international economics theory, so that this actually destructive critique has not led to abandon this thinking tool.


Now, one should credit to this approach, that naturally a construction of collective indifference curves is quite conceivable, by simply asking for the evaluation of the individual goods combinations in the context of a survey of the individual households, and by using a rule (such as the majority principle) to determine the opinion the collectivity.


Against such an approach one can, of course, object that such a survey could not be carried out in reality by reason of a very high effort. But, of course, this accusation also applies to the development of individual indifference curves; we can indeed not expect individual households to develop their own indifference curves before their consumption decisions!


Continuation follows!