Book Iii Gives a Development for the Bentham Tax Function, and Also

gives plots for regular numerical values. It appears that indexation and expectations about the growth of national income (relevant for indexation) again lead to other results than the conventional view on marginal rates.

Policy simulations

There is one area where the DMR cannot easily be overlooked. This is the area of policy simulation, where tax adjustment cannot be neglected. For sure, empirical analyses and government projections indeed deal with tax parameter changes. For example the well-known Reagan tax cuts were put into the forecasts at that time. However, we should wonder now whether the methods have been right. The analysis above focusses our attention on the impact on individual behaviour, where we regard the marginal calculation by agents themselves.

Let us regard policy simulations using common practical economic models. Let us for example regard the effects of a rise of government investments as financed by taxes, for a sustained period of 8 years (two presidential terms). To do a simulation properly, the tax function used must reflect government policy, which includes indexation. For example, exemption and other brackets are adjusted for last year inflation while the statutory marginal rates remain the same. The different investment paths result in different paths for the taxes. This is not just a model result, but also the agents in the economy would encouter different regimes. Thus the model generates different dynamic marginal rates, while the agents are assumed to react only to the same (static) rates. The situation gets even complexer when the alternative policy includes a different indexation scheme, such as indexation of taxes on national income. All this means, then, that we are justified in doubting the validity of current modeling practices. Modelers should start wondering about this kind of dynamic consistency (not to be confused with the ‘dynamic consistency of policy’ as another topic in economic literature on ‘credibility’).

It might even be, then, that the best way to understand the dynamic marginal rate is to see it as a solution to this kind of dynamic inconsistency.

Balanced growth

Under balanced growth, taxes will grow as fast as incomes, with a constant tax share TAX / Y, assuming proper indexation of the tax parameters. A result will be that the dynamic marginal equals the average tax rate, for all individuals. Book III already mentioned the key relationship here, in property (13.3e).

We use Tax[.] for an illustration. Here a solution for a balanced growth path is that parameters x and c are indexed on y. With the index for y as i = P ryi ( i > 0), we find for the (individual) average tax burden that the index drops from both numerator and denominator:

T[ i y; r, i x, i c] / (i y) = r (i y - i x) / (i c + i y) = T[y; r, x, c] / y

(Less relevant, (29.1) remains the same too.)

The situation of a constant dynamic marginal rate is depicted in Figure 30.

Figure 30: A balanced growth shift A-2A: constant frequency, A-C: the same average tax

Let us take the example of a doubling of income. Point A is an arbitrary point on the employment density. We scale the density so that A also lies on the tax function (H). For that arbitrary income at A we determine the average tax as a ray through A and the origin. Now, if all incomes double, then the employment frequency density shifts, and A becomes 2A. If tax parameters x and c double too, then the tax function becomes (2H). At 2A the individual pays tax C, which is the same average tax as in A (vide the straight line through origin, A and C).

Off balanced growth

Income growth means a shift of the employment density or the earnings distribution. Earlier we looked at income distributions for Holland 1950 and 1988, and the reader may now better understand why. The Dutch distributions could be approximated by lognormal distributions, but the mean, variance and the size of the labour force changed. Taxes also have been indexed on inflation instead of income. So we may surmise that there was no balanced growth.

How do agents react when there is no balanced growth ? Indexation to national income can be said to be “neutral to the income change”. The tax choices facing an individual, whose income grows as national income, are constant. The utility reaction thus depends on the change of income itself. It may be that an individual, whose income might grow as fast as national income, decides to grow differently, either more or less, depending upon his leisure-income utility. Since the context is that all individuals are adjusting, this may be reformulated as that individuals are determining their place within the income distribution.

Our analysis thus suggests that tax incentives primarily affect decisions about one’s place in the income density. Any individual change that differs from the national average can be interpreted, or defined, as the individual decision to accept another place in the income distribution. It would be interesting to reinterprete economic models on growth in these terms, and see whether elegant regularities can be found or constructed. However, it leads too far to really look into this matter, since it is not our proper subject.

We conclude that indexation and expectations about the growth of national income (relevant for indexation) lead to other results than the conventional view on marginal rates.

30. Dynamic curvature of the tax wedge

Introduction

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.

Formulas

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. Here we assume that x can differ from subsistence in the base year B. Let y be adjusted for the real level of income, with index rwi, too; then y = P rwi y. Define f = x / y. Then:

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

It must be noted that y 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 / 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.

Denote the average tax at the minimum wage as, ATR M [rwi]. We will use the suffix ‘M’ in general to signify this dynamic point of view.

In Book III we derived the real subsistence index rsi for the Bentham function when x = P x, so that B = rsi P B.

(13.3d)

Then:

M = B + Bentham[M] M = (B - r x) / (1 - r)

M = (P rsi B - r P x) / (1 - r)

m = M / P = (rsi B - r x) / (1 - r) = m[rsi]

ATR M [rwi]= ATR[m[rsi[rwi]]]

We can develop this a bit further, using j = x / B: / / GNR M [rwi]= M / B = (1 - r x / B / rsi) / (1 - r) = (1 - r j / rsi) / (1 - r)

ATR M [rwi]= Bentham[M] / M = 1 - 1 / GNR M [M] = r (1 - j / rsi) / (1 - r j / rsi )

Over time, rsi will rise to infinity, and limit values will be GNR[] = 1 / (1 - r) and ATR[] = r as for all incomes.

Graphs

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.

Figure 31: Average tax, in statics, for various marginal tax rates

We might interprete static Figure 31 in a dynamic way. Take B = x = 1, j = 1. We may take a theoretical example. If you have a period of 35 years, then a real growth of 2% per annum would suffice to double incomes. So in the standard unrefined analysis, the tax creep in 35 years would cause incomes to be taxed at average rates close to the marginal rate.

The more refined analysis for the minimum wage takes account of the multiplier effect. First of all, if real subsistence doubles from B = 1 to B = 2 B, the gross minimum wage would be M = (2 - ½) / ½ = 3, and hence we should look in Figure 31 at index 3 instead of index 2. This issue however is a bit more complex, since when rwi = 2, rsi is not 2 but 1.7.

In Figure 32 we compare the standard ATR and the dynamic ATRM. We regard only one marginal rate (a 50% rate) and a ‘peg average’ W[0] = 2 B or h = 0.5. It appears that the dynamic ATRM is steeper and higher than the static ATR. However, the difference is not that big. Note though that we would want an average tax rate of 0 for the minimum wage (subsistence) instead of something close to 30%.

Figure 32: Average tax rate, static and dynamic, for r = 50%

In Figure 33 we regard the dynamic GNRM ’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.

Figure 33: Gross-to-net ratio, in dynamics, for various marginal tax rates

31. Differential impact of the minimum wage on exposed and sheltered sectors

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.

Introduction

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.).

The Dutch policy reaction - though with some lag - was a general restraint of wage growth. This reaction was motivated by reference to the so-called Vintaf model developed by Den Hartog and Tjan at the Central Planning Bureau - see Driehuis & Van der Zwan eds. (1978) and Driehuis, Fase & Den Hartog eds. (1988). The direct assumption was that high wage costs cause the scrap of old vintages of the capital stock, resulting in an irreversible loss of capacity. The indirect presumption was that a relative reduction of production costs could compensate for the rise in the exchange rate, restoring competitiveness and employment.

However, in a quite brilliant exposition that up to now has been neglected to the shame of the Dutch economics profession, Marein van Schaaijk (1983) of the same Bureau showed that a general wage restraint neglects the fact that the exposed and sheltered sectors have a different composition of their labour force, with important effects. He noted that the exposed sector is industrial and has the larger share of well educated, highly productive or high value added labour; while the sheltered sector concerns services and has the larger share of lowly educated, lowly productive or low value added labour. A uniform wage restraint - targetted at reducing unemployment rather than balance on the external account - is too high for the exposed sector and thus subsidises exports; and the restraint is too low for the sheltered sector and thus generates unemployment. The restraint of incomes also means a restraint of imports, aggravating the situation. So Van Schaaijk noted in fact both the internal and the external imbalance, recognised that these mirrored each other, and that these were prolongued, now not by the original energy price hike but instead by policy.

Indeed, Holland since then has a strong external position - exporting unemployment to Europe - and a high internal unemployment - where the unemployment is hidden in ‘disability’ (and hence registered by dull statisticians as ‘low participation’). Some surplus of the external account is reasonable given the natural resource, and the capital flows for foreign investments are useful for when the resource is depleted. But the Dutch external surplus is excessive.

Van Schaaijk’s suggested remedy was standard and sound. It was and is to let wages develop in line with productivity. Since Dutch policy is oriented to maintaining a more equal distribution of income - which explains part of the policy drive to see a uniform development in wages - Van Schaaijk advised to use tax policy to correct the differential development of gross wages for its effect on net incomes.

However, as said, Van Schaaijk’s analysis has been neglected to this day, and Holland now suffers from a long period of unemployment and a trade surplus and a general restraint of wages and net incomes. There is a curious ‘consistency’ in the delusion with policy makers, that incomes restraint is required to maintain employment by generating a trade surplus, since, by restraining the home market, most Dutch employment growth seems dependent upon trade indeed. Strangely, economic developments caused the Central Planning Bureau to drop the Den Hartog & Tjan model in the mid 1980s, but the policy of wage restraint remained.

In the 1982-1991 period I worked at the Central Planning Bureau too, and had the opportunity to get acquinted - albeit around 1986 only - with Van Schaaijk’s analysis. Apart from being enlightening by it itself, it opened my eyes - even while it was standard - to the importance of tax policy for unemployment, and thereby led to my papers (Colignatus (1989-1996)) and this present book, on the solution to the current mass unemployment in the OECD countries in general.

In my papers I have always referred to Van Schaaijk’s 1983 article whenever it was proper. However, in this chapter I have occasion to more specifically combine his analysis with my own. This chapter improves on Colignatus (1996g), and as I wrote there: this combination of our analyses has been in my mind for a long time, but there was no time to develop it, as, in fact, this chapter suffers from some time constraints too.

We shall use a general equilibrium model where the exposed and sheltered sectors have different combinations of labour as in the Van Schaaijk observation. But now we take my analysis on the minimum wage, and let the minimum wage have the differential impact. This is more relevant for the OECD in general. Note, though, that I do not want to imply that all OECD countries have a trade surplus; other conditions are relevant here too, of course.

Due to lack of time we use a closed model. Thus we cannot reproduce the external imbalance. But we can reproduce the difference in reactions of the two sectors. We may study situations with full employment (1950-1970) and without this (1970-2005). Below, we give a model, tables and graphs.

Model

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 yE and yS, while a social welfare function (SWF) determines the optimal combination. In an open model, the yE would be traded for yForeign, 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.

Graphs

We consider two regimes, one With l (i.e. the minimum wage M is not binding), and one Without l (with M binding, causing unemployment and lower national income). Subsequently, the model is run with the computer program listed in the appendix; see chapter 37 for another application of the computer routine (and additional explanations of terms).

Figure 34 plots the production possibility curves and the SWF indifference maps of the two situations. The regime with a binding minimum wage - and less workers - indeed has lower production and lower utility. The drop in production in the sheltered sector is larger than in the exposed sector.

Figure 34: Production Possibility Curves & Indifference Maps

Figure 35 plots the Edgeworth-Bowley diagram for factors h and m, with Sheltered in the lower left and Exposed in the upper right. The movement is upwards along the contract curve. The highly productive workers in the second regime become relatively scarce, and command a relatively higher share of national income.

Figure 35: Edgeworth-Bowley Diagram

Tables

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.

Table 9: Utility, production and national income for two regimes

Utility level

National income

Product prices Sheltered & exposed

Production S & E

With l

21.20

39.67

0.9579

24.93

15.38

Without l

18.16

32.37

0.840

20.74

13.85

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’.

Table 10: Allocations

Allocation with l

Allocation without l

High

Middle

Subsistence

High

Middle

Labour units Sheltered

6.53

53.08

9.57

7.07

54.73

Labour units Exposed

8.47

21.91

0.43

7.93

20.27

Labour units Total

Wage

0.88

0.33

0.19

0.74

0.28

National Income Share

0.33

0.62

0.05

0.34

0.66

Note: Using unrounded data on the wages, the high/low wage ratio in the first regime is 2.69, and in the second regime 2.60.

Conclusion

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.

32. Dynamic optimality

The Phillipscurve revisited

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.

Table 11: Two marginal approaches for two Phillipscurves

Phillipscurves

Marginal approaches

Traditional: only labour supply

Nash bargaining

Standard marginal analysis

(1) the marginal tax rate has a disincentive on labour supply and thus causes wages to rise

(2) the marginal tax rate has a disincentive on wage claims

Dynamic marginal tax rate

(3) the marginal tax rate has no disincentive, relevant is the average tax

(4) ?

I have not performed the analysis yet. By the next edition of this book I should have. My intuition however suggests - and I keep an eye on reality - that the two approaches only combine into a stronger argument against the conventional view. Doing this additional work thus currently is expected to be a bit overdone just now.

Investment, growth and productivity

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).

The point of criticism is that employment and growth are rather separate issues. Our own analysis in this book shows that a return to full employment is possible. The main instrument is to get rid of the tax void. Employment does not depend upon growth per se but employment depends upon a properly working system to allocate the work that is being done in an economy. Growth comes only into the story when we aspire at higher welfare by means of higher productivity. If we don’t want growth, we can easily imagine a stagnant economy. That said, most economies aspire at a growth in welfare. We can do this by designing new products or by material investments or by creative ways to reorganise production. Then the problem returns of optimising investments that define real savings. Since some sections of the economy are devoted to investments, there is also the Keynesian phenomenon that investments influence activity, income and nominal savings.

The paradigm to ‘minimize’ the role of government in investment was misguided since the relation between growth and employment was misspecified. Now that we know that the tax void was the main cause of stagflation we can reconsider the paradigm. The argument that remains is that government meddling supposedly caused failed investments. The answer to that argument is (i) that failures must be judged on a case-by-case manner, by Cost Benefit Analysis, and (ii) that one should include the concept of Keynesian recession and that some investments might seem a failure but actually are beneficial. Note that there is no need for a government deficit since the analysis on the dynamic marginal rate shows that progressive taxes need not be a drawback for the richer. If growth is the issue, then the true issue is its optimality in terms of level and composition and effects.

The line of thought that I would suggest is that this optimum requires competing investment banks that develop plans during the economic upswing that can be implemented during the economic downswing. Who worries about pensions and the EU Lissabon Strategy is advised to consider this approach. Since the market is an anonymous beast that may or may not generate such competition, it remains the challenge for governments to mastermind and manage it all.