Definition & Reality in the General Theory of Political Economy

The tax wedge at the minimum is caused by differential indexation, and makes for a higher gross minimum wage. This has been clarified above. A second point is curvature. Due to curvature, the wedge comes close to its limit value for already low levels of productivity growth. Thus, the negative effects of the wedge occur primarily at the onset of economic growth, and are less noticeable when stagnation has already set in. This already has been indicated above, but the argument can be developed by giving formulas and plots. Especially, it are the plots that may help us to understand that the major distortionary effects took place in the 1960s and 1970s. People looking only at the events in the 1990s are less likely to see the root of the problem.

In the following we first derive the formulas and then give plots for the average tax rate (ATR) and the gross-to-net ratio (GNR). The latter ratio may better express the effect on the gross minimum wage. We find that the ATR and the GNR at the minimum rise faster than for other incomes, since the minimum itself moves faster than those other incomes. For ease of exposition we use the Bentham tax.

The average tax rate (ATR) and the gross to net ratio (GNR) are:

ATR[y]  =  Bentham[y] / y  = r (1 - x / y)

GNR[y] =  y / (y - Bentham[y])  =  y / ( (1 - r) y + r x)  =  1/ (1 - r  + r x/y)

Examples work best. Let subsistence B be exempt from taxation so that x = B, and let the marginal tax rate be 50%. The average tax rate (ATR) of a subsistence worker then is 0, and the gross to net ratio (GNR) is 1. At twice subsistence, the tax is 50% (2 B  - B ) = B / 2,  and thus the average tax is 25% and the gross to net ratio of 4/3. In the limit, i.e. when exemption has been reduced to a negliglible proportion, then the average tax equals the marginal rate of 50% while the gross-to-net ratio is 2.

Next, notice two points. First, the formulas by themselves do not quite show how quickly the limit values are approached. To answer this question we can best look at some graphs. Secondly, these examples are static, i.e. at one point in time for different incomes. Thus, when we make graphs, then we can use a static index, and compare an income level 1 to an income ten times as large. In dynamics, i.e. when incomes rise, things are a bit complicated.

In dynamics, and concerning the current practice of adjusting exemption for inflation, we can take exemption as constant, and look at real incomes (adjusted for inflation). It seems as if we can take the formulas and graphs of the statics case, and compare real incomes regardless of the time. However, in dynamics, ‘minimum income’ is not just ‘income’ but is a mechanism. The concept of M is that it picks out one income as the minimum, but it can pick that income at a different rate of growth depending upon the mechanism. The interaction between indexation, net subsistence, the tax parameters cause a multiplier effect. Before we make plots we have to develop on this.

Let us first regard a general formula for dynamics, and see that it seems as if there were no difference with the formula for the statics case. Let exemption x be adjusted for inflation with index P, then x = P x[0]. Here we assume that x[0] can differ from subsistence in the base year B[0]. Let y be adjusted for the real level of income, with index rwi, too; then y = P rwi y[0]. Define  f = x[0] / y[0]. Then:

ATR[y]  = r (1 - x / y) =  r (1 - x[0] / (y[0]  rwi)) =  r (1 - f  / rwi) = ATRwi[f, rwi]

It must be noted that y[0] depends upon y, so that  f  may take continuous values. ATRwi[f, rwi] expresses that if we have a value of y, then we could interprete this as deriving from various combinations of f and rwi as long as rwi x[0] / f = y. The dynamic ATRwi[f, rwi] thus seems no different from the static ATR[y]. The complication however comes from subsistence. We cannot regard M as a normal case of y = P rwi y[0].

First we plot the static ATR and GNR for values of a real net wage index from 1 till 10. Figure 31 plots the paths for various marginal tax rates: 10%, 20%, ..., and even 70%, all assuming x = B = 1. These plots show the point made earlier, that the ATR is close to the marginal rate at already low income values, e.g. 2 or 3 times subsistence.

In Figure 33 we regard the dynamic GNR_M ’s, now plotted for various values of r. We can see that the rise is largest in the lower reaches of the graph. For example the 50% rate already reaches the level 1.6 around the index value of 4, and 1.6 does not differ much from the limit value of 2.

Some sectors of the economy are exposed to foreign competition and some are sheltered from it. These exposed and sheltered sectors are likely to have a different composition of their labour force, notably different rates of dependency on the minimum wage. If a national incomes policy does not respect these differences, a country can have both unemployment and a surplus on the trade account.

The two Oil Crises in the 1970s created a problem for the Dutch economy which has become known in the literature as the so-called “Dutch Disease”. When the price of a nationally produced but internationally traded resource rises - and this happened since Holland is rich in natural gas and a free rider of OPEC - then this causes the exchange rate to rise, and then this indirectly causes a reduction of the other exports and an increase in competing imports. Thus the original increase in national wealth paradoxically combines with an increase in unemployment - and eventually a lower growth path.

This chapter concerns the Dutch policy reaction to that Dutch Disease. If policy is not targetted at stabilisation of the exchange rate by monetary means and capital flows, but at tinkering with the labour market, then the situation - the disease - can grow worse.

Our analysis will use the distinction between the ‘exposed’ and the ‘sheltered’ sectors of the economy - a distinction that originates from Swedish analysis in the 1950s (Meidner c.s.).

Regard a general equilibrium model with 15 units of highly productive labour (h), 75 units of modally productive labour (m) and 10 units of lowly productive, minimum wage workers and possible benefit recipients (l). The economy has exposed and sheltered sectors that produce output y_E   and y_S, while a social welfare function (SWF) determines the optimal combination. In an open model, the y_E would be traded for y_Foreign, but here we assume that exports are directly equal to imports for consumption. The SWF will here be a Constant Elasticity of Subsitution (CES) function that neglects the distribution of income:

Output of the sectors is determined by production functions that depend upon the allocation of the labour factors h, m & l. Since we will compare two regimes, one with l and one without l, this factor cannot be complementary (necessary), and hence it is substitutable to some degree with the other factors. The sheltered sector is a one level CES with all factors substitutable:

The exposed sector is a two-level CES where highly and lowly productive labour are complementary, but both are substitutable with minimum wage labour:

The coefficients have been chosen so that these outcomes resemble a real economy. We should refrain from making our conclusions too specific though, since the coefficients are arbitrary.

The following tables give the numerical outcomes of the two regimes. When M is binding, the subsistence workers l are unemployed and dependent on a benefit. Since they do not work, output and social welfare are lower. Though there is no explicit social security in this model, we however can presume that part of earnings of the workers is channeled to the unemployed, leaving consumption from those earnings unaffected.

The social optimum is found as in Table 9. The associated allocations are in Table 10 - left and right side. When you compare the two regimes, please note that the prices are normalised per regime to a unit price for the sheltered sector, and thus are not comparable over regimes.

Note: All prices are scaled so that the product price of the sheltered sector = 1.
This is also done per regime, so that the price levels over the regimes are not comparable.

In Table 10 we see that the share of the highly productive in national income rises. Most of the share of the l go to the m, but this is generally viewed as an internal redistribution, and most attention goes to the share of ‘the rich’.

By proper choice of functions and parameters we have succeeded in reproducing and hence illustrating the Van Schaaijk observation & analysis of the differential reaction of the exposed and sheltered sectors on incomes policy. As Van Schaaijk found, the sheltered sector loses most, and it would be optimal to have wages reflect productivity. And similarly, this can be supported by tax policy. Whereas Van Schaaijk commented on the Dutch policy of the uniform containment of wage growth, we have concentrated on the minimum wage - as is more applicable for the OECD. Indeed, if the whole of the OECD would try to copy the ‘Dutch model’, then this would amount to trying to export unemployment to each other, and a thing like that surely would not work.

In chapter 25, the ‘more sophisticated view’ section, we mentioned that Graafland (1990b) elaborated on Hersoug (1984), and recently again in Graafland & Huizinga (1999). The approach here is a Nash solution to wage bargaining. The approach causes that marginal tax rates penalize wage demands and increase employment - contrary to the common thought that statutory marginal tax rates reduce incentives and hence reduce employment.

We ourselves forwarded the novel insight of the ‘dynamic marginal tax rate’: saying that marginal tax rates should be better measured by also including expectations on parameter changes and economic growth.

The question now arises how these two approaches combine. The Nash approach uses partial derivatives, while the dynamic approach uses total derivatives. If we would take the total derivative of the Nash solution, it might well be that statutory marginal tax rates show an effect again that is more in line with the conventional view. The four possible combination cases are shown in Table 11.

The following has been in my mind since Colignatus (1989) but was not stated in the first edition of this book. One of the key points of Keynes in the General Theory was that the true, real, savings of an economy consist of what is invested. All the money that people save does not count as an investment or real saving. Whatever amount they bring to the banks or even hide under their beds, it is only money. One can have nominal saving S and price level P, but the division S / P is more psychological than real. What counts are the houses built, bridges constructed, lessons learnt, all that can be carried over to the next period. In fact, a company that produces but can’t sell and goes bankrupt might actually do society a favour, since at least some goods have been produced which otherwise might not have come into existence. The challenge is to get production and investment without such perceived incompetence or fraud. The economy should be designed so that those investments come about in an optimal way, where the optimum must be defined not only in terms of expectations and stability but also in terms of social welfare and full employment.

Governments, especially European ones, have been experimenting since World War II with all kinds of methods to control investments, but have been confronted with two major outcomes: (a) unemployment remained high, (b) many investments were considered failures. The economic paradigm since the Reagan years has been to let investments be determined by the market. Also Dutch social democrats like Wim Kok supported this approach, since it was thought that employment depended upon growth while growth depended upon the best investments that the market could provide. This paradigm led to reduced government outlays, less fiddling in the market, privatisation, and reduced taxes for the wealthy who were assumed to do the investing. The 1990s showed the boom associated with silicon valley - though should properly be associated also with this policy and the implementation of new financial instruments. But the boom went bust and the world was reminded of the logic of Keynes’s depression economics, see Krugman (1999).

Kenneth Arrow (1950, 1951, 1963) presented an Impossibility Theorem in which he showed that decisions about ‘the general welfare’ are impossible in certain cases or have to be left to a dictator. Arrow presented some five axioms that each seemed reasonable when considered by itself, and he argued as well that these axioms are morally desirable and fitting to the concept of ‘general welfare’. He also formulated the problem in general terms so that it concerns choices on goods or people. Subsequently, he derived a contradiction. This result caused quite some consternation, but eventually the mathematical rigour caused acceptance, and since then the Theorem forms the core of many books, such as Sen (1970) and Mueller(1989). The Theorem was also one of the reasons to award Arrow the Nobel Prize in economics.

A voting example is given by the US Presidential election of 2000. Apart from the problems around the ballot process itself, there was a more basic problem: with main contenders Bush, Gore and Nader, Bush got elected, but in another system, such as a run-off between the two ‘major’ contenders, the Nader vote apparently would have switched largely to Gore, making him the US President. So the choice depends as much upon the system chosen as on the preferences. Can we find a generally good system ? Arrow’s Theorem suggests ‘No’.

Arrow’s Theorem has had a huge influence on scientific and political thought. Part of this influence is subtle, where skepsis arises about the concept of ‘democracy’. That shiny goal loses its appeal when we don’t know how representatives should be elected and when morally desirable rules would be impossible. Opting for the natural forces in the social process may be more pragmatic. The influence of the Theorem can sometimes be more explicit. Next to the model of the utility maximising individual, there is the model for society as a whole and then the maximisation of a Social Welfare Function (SWF). But when a morally acceptable SWF is impossible, what would be the use of research into such an inherently flawed concept ? Many nations co-ordinate their economic policy, and have created institutions for this, like the Council of Economic Advisors (US), the Commissariat du Plan (France), the Sachverständigenrat (Germany), and the Central Planning Bureau (Holland). Such an institution, given its role in the co-ordination of economic policy, could be expected to do reseach on the national SWF. However, those institutions tend to abstain from that kind of research, pointing to Arrow’s Theorem as one of the arguments, if not the major argument.

Over the years an ‘accepted view’ has grown in economics concerning the meaning of Arrow’s Theorem. This accepted view however has also implied a kind of moral stagnation.

There are two main reasons to reconsider the accepted wisdom on the meaning of the Theorem and to rekindle the debate on it. The first reason is destructive, since it rejects Arrow’s position; the second reason is constructive, since it provides an alternative.

These reasons are: (1) There is a distinction between the mathematical framework on one hand and its interpretation on the other hand. The Theorem holds, and the impossibility holds for Arrow’s axioms, but the questions of reasonableness and moral desirability are of a different kind. (2) The area of application of Arrow’s axioms seems rather static, while reality is dynamic. By considering the role of time, there is more scope for morality, and then one can identify a voting procedure that many would find attractive.

The two following chapters develop these arguments subsequently. Readers interested in more details are referred to Colignatus (2001), “Voting Theory for Democracy”. That book develops the theory of direct single seat elections from the bottom up while it also provides programs (in Mathematica) to eliminate the tedious work of the calculations of the various voting procedures.

Summary

Arrow’s Theorem holds that no constitution can satisfy certain properties. In annex to that theorem, Arrow claims that those properties are reasonable and morally desirable. In Arrow’s view there thus is the difficulty that people desire a constitution that cannot exist. While the Theorem stands as a mathematical result, the additional claims concern some other matters, namely the domains of reasonableness and morality. It are these claims that have caused much confusion in the literature. It is shown here that the claims are unwarranted, since inconsistent properties are neither reasonable nor morally desirable. It is shown too that Arrow’s axiom of Pairwise Decision Making (formerly known as the Independence of Irrelevant Alternatives) is not realistic, and thus unattractive. We show the existence of some constitutions without that axiom that are consistent and might be optimal to many. The major error made by Arrow and his students is to mix up the context of scientific discovery and learning with the context of application to the real world by educated people.

Arrow (1950, 1951, 1963) showed that if certain properties are postulated for a constitution, then such a constitution would not exist. This result has been checked by numerous scholars, is accepted by this author, and thus stands as a mathematical theorem. In fact, we will give a short proof below.

Arrow also claimed, annex to the theorem, and this will be at issue here, that those properties would be reasonable and morally desirable. He recently repeated that claim in the Palgrave (1988:125). He writes:

“(...) conditions to be imposed on constitutions (...)”

“(...) there is no social choice mechanism which satisfies a number of reasonable conditions”.

For clarity it is useful to introduce the following abbreviations for the theorem and its companion claims, and their conjunction:

            AT  = the Arrow Theorem

            ARC = the Arrow Reasonableness Claim = the properties are reasonable

            AMC = the Arrow Moral Claim = that they are to be imposed

            AGV = the Arrow General View = AT & ARC & AMC

Note that Arrow’s phrasing on ARC and AMC is a bit ambiguous. The “to be imposed” might not be moral but merely logical, in a sense that one needs at least some conditions to make a constitution. However, the topic of collective choice is distinctly a moral one. Secondly, Arrow emphasises what is to be imposed and what is reasonable, but he may not be in a position to impose his views and morals on us. The best interpretation of the situation likely is as follows. Presume that Arrow sees the Founding Fathers at work. He then retreats to his office, and conjectures: ‘If I interprete correctly what they want, then it are these properties.’ Thus the ARC and AMC are not quite Arrow’s personal ideas. Above quotes can best be interpreted as factual statements on what people apparently want and consider reasonable.

Arrow’s general view has been accepted in many places in the literature and textbooks, see Luce & Raiffa (1957), Johansen (1969), Sen (1986) or various other entries in that same Palgrave. For example, Tobin (1990):

“We know there is no way to aggregate individual preferences into social rankings (...). As if this were not obvious, Kenneth Arrow proved it rigorously years ago. The impossibility applies to aggregations across contemporaneous cohorts, a fortiori across generations living and unborn.”

In a much used book on Cost-Benefit Analysis (CBA), A.K. Dasgupta & D.W. Pearce (1980):

“(...) no escape route (...) seems yet to be available.”

Apparently feeling that Arrow's argument destroys the foundations of CBA, they find themselves forced, rather grudgingly, to reduce CBA to something like information gathering.

In an otherwise recommendable volume of Statistical Science, Gill & Gainous (2002) find:

“In fact, he proved that unless one is willing to violate one of a set of reasonable democratic norms, (…inconsisteny...) is an inevitability. (…) Therefore, collective social decisions cannot yield a truly democratic system in this sense.”

Jorgenson (1990), once president of the Econometric Society, concludes ‘more positively’ to dictatorship:

“The classic result of social choice theory is Arrow’s (...) impossibility theorem, which states that ordinal noncomparability of individual welfare orderings implies that a consistent social ordering must be dictatorial, corresponding to the preferences of a single individual.”

Not everybody falls for dictatorship. The impact of the AGV generally comes from the fact that people find themselves, either from moral obligation or from reasonableness, wanting the impossible. And many simply stay in that fixture.

Note the subtlety in that fixture. The impossibility is logical and not just empirical. An example may help. Let me confide that I want to found a new university on the island of Crete. However, I am not that rich, so I want something impossible. This however does not put me into a fixture, since I am used to the fact that I cannot afford some things that I want. However, the Arrow general view concerns a logical impossibility, which is something quite different.

We can usefully recognise:

            reasonable = rational & realistic

Reasonableness is the intersection of rationality and empirical realism. Nonexistence may derive from empirical circumstances or from logical impossibility. Irrationality however is always unrealistic. Inconsistency cannot exist, in the true empirical sense. For example a round square cannot exist. The nonexistence of the Arrowian constitution similarly derives not from empirical reality but from logical necessity.

Given the AGV, the question arises what the reasonableness and moral presumptions of Arrow’s claims actually are. Are these claims as strong as conjectured ?

My position is as follows:

1.       As has been said on ‘round tables’, it is not rational to postulate inconsistent properties. People involved in a learning process may indeed make inconsistent assumptions. However, once the inconsistency is discovered, it is no longer considered to be rational to adopt those assumptions. People may enjoy ‘roundness’ and ‘squareness’, but having both simultaneously is seen to be inconsistent, even inconceivable, and hence unreasonable. The Arrowian properties are unreasonable in the exactly same manner. Arrow’s pitfall is to confuse the learning process, his context of discovery, with real world applications by educated people.

2.       Similarly, one cannot be morally obligated to a logical impossibility. Hence Arrow’s properties are morally undesirable.

These points will be clarified below.

Note that people have in practice rejected some of Arrow’s properties. Even those scholars who seem to accept the general claim AGV, accept, a fortiori, the implied inconsistency, and thus in practice drop some assumptions to cope with the real world. Unfortunately, however, the literature has not converged to some agreement on which properties are best to drop. The position of this paper will be to forward the proposition that the Arrow axiom of Pairwise Decision Making (formerly known as the Independence of Irrelevant Alternatives) is the culprit to kill. It is a bad axiom for rational collective decision making, since it appears to be incongruent with that very notion itself.

In the following we develop the concepts, give a short proof and discussion of Arrow’s Theorem, construct the argument against the claims, reappraise the literature, and conclude.

Please note that we will have to redefine some symbols for this chapter only.

Let X be the commodity domain. An element in the commodity domain can be called an item or a candidate. An agent is a compound of various properties such as utility, wealth etcetera. Let S be the set of possible compounds on X.  With n agents, our interest concerns the function c: Sⁿ   S. which maps the society into an aggregate compound. This is generally called the ‘Arrow type of social welfare function’ or simply a constitution. 

A constitution differs from the ‘Bergson-Samuelson type of social welfare function’ (SWF) - and the latter is defined directly over X as SWF: X [0, ).

Arrow’s Theorem concerns Social Welfare Function Generating Mechanisms (SWF-GMs) like the c above. Thus, a constitution can be seen as a mechanism that uses the population as input and generates a SWF that orders all elements in the commodity space. This can be compared to a Social Decision Function (SDF) that selects only one element, namely the best of a budget set. This can be weakened further by considering preference orderings instead of functions. Constitutions generally associate better with SDF-GMs since parliaments generally don’t care ordering all proposals. However, these concepts can be translated into each other via varying the budget set. Since the SWF is the conventional concept in economics, the word “constitution” can remain associated with a SWF-GM.

It suffices to restrict S to preference orderings. These orderings satisfy reflexivity, transitivity and completeness. It is important to add that there is no cheating. Let R denote normal preference, P strict preference, and I indifference. When there is no confusion, we can also use the symbols , < and =. A suffix denotes an individual preference, otherwise it is the aggregate. An element in Sⁿ is called a profile, and R = c(R₁, ...R_n).

There are the following Arrowian axioms:

                        AWP     the weak Pareto principle

                        AU       universal domain (wide ranging preferences)

                        AD       no dictator

                        APDM pairwise decision making (the axiom

f.k.a. independence of irrelevant alternatives)

                        a          AWP & AU & AD & APDM.

The Arrow Theorem can be expressed in various equivalent logical forms:

                        AT        a   falsum

                        AT’      a   ~a

                        AT”      ~a

                        AT”’    (AWP & AU & APDM)   ~AD

with falsum a contradiction or falsehood and ~ the negation sign. If something leads to a contradiction, then we conclude to the falsehood of the assumptions themselves.

There is a Kantian distinction between technical, pragmatic and moral (categorical) imperatives. Utility, as commonly regarded by economists, likely is of the pragmatic kind. Interestingly, theorists on morality have developed something called ‘deontic logic’, which appears to give many similar results as economic theory. Deontic logic however applies to propositions and not to commodity domains. It is possible, though, to integrate all these kinds of preferences into an integral utility index, when we replace a point x in the commodity domain by a statement “The state of the world is x”. This integral utility index likely would be lexicographic, in that some moral and constitutional issues might dominate pragmatic results in the commodity domain. Thus, while we would use the same symbols R, P and I, we would need to look into the structure of the index to find the Kantian distinction as made by the particular agent. We conclude that we can usefully introduce and apply some terms from deontic logic. Define:

                        Ap (~p  p)  means that p is allowed (at least as good as ~p)

                        Op (~p < p)  means that p is a moral obligation (one ought to p)

An exemplaric deontic result is:

                        Op ~(A(~p))

Deontic logic allow us to translate:

                        AMC = Oa

The use of deontic logic allows a forceful restatement of Arrow’s difficulty in social choice:

                        Oa & ~a

Let us consider some more properties of morality and deontic logic.

The gap between Is and Ought (Sein und Sollen) means the rejection of p p Op (‘If something is, then it should be like that’) and, in principle, p Op p (‘what ought to be, is achieved’).

It appears very useful to discuss the example given by the Marquis de Condorcet 1785. Sen (1970) gives a simple example that appears to be presented first by Nanson 1882. A similar example is reproduced in Table 12, and I will refer to it as “the Condorcet case”. There are three parties and three topics A, B and C on ballot, and the numbers of seats and the preferences are such that, with pairwise voting and a majority rule, a cycle results: A < B < C < A.