Book Iii Introduced the Bentham Tax Function _bentham_[_y_] = _r _(_y -

x) with exemption x and marginal rate r. _This function is linear but already results into nonproportional taxes. Governments in practice have nonlinear tax schemes that give stronger nonproportionality, reflecting political views on the redistribution of income.

Strong nonproportionality has a special effect. Since taxes in the 1960s were more nonproportional than nowadays, the tax structure combined with the lognormal shape of the employment function, and generated strong nonlinear effects and a strong upswing of the CWIRU in the early phase of stagflation.

It is useful to introduce a flexible tax function with one more parameter than Bentham’s function to incorporate some curvature. This new function allows us to give concrete examples whenever nonlinearity is useful. For clarity, it appears that this function can approximate the actual Dutch tax situation. The tax function is:

(y > x) > > > > > > > > with y the tax base and x the exemption or threshold, r the marginal rate in the limit when y goes to infinity, and c a curvature parameter. The ordered set of parameters is q = (r, x, c). We do not use Greek symbols for these parameters since we will regard them as key strategic variables. If governments would use this function for practical tax collection, they might note (1) that exemption would be determined by subsistence, (2) that r would follow from the limit marginal rate for the highest incomes, (3) so that curvature c would follow from required total revenue and the income distribution. Use of this function thus both allows for a decent degree of nonproportionality and would reduce much of political debate about positioning of tax brackets and rates.

A person’s average tax is:

The marginal rate on the marginal dollar can be approximated as T[y + $1] - T[y] so that the common tax payer will have no problem in determining it. The proper formula itself is not too simple. At y = x it starts with the value r x / (c + x) and in the limit it equals r. For the whole range:

(29.1)

Note that the tax function can be transformed into a linear format consisting of income, average tax and a constant:

Tax[y] = r.y - r.x - c.Tax[y] / y = a1.y + a2 + a3.ATR[y]

Colignatus (1992) used this relation for a simple linear least square estimation that neglects the error on the average on the right hand side, using 1988 Dutch data for 12 selected income levels. The result was:

(in 1988 $)

The equation can be plotted for two ranges, (H1) for a low income range till $25 thousand to show the curvature, and (H2) for a wider income range till $250 thousand to show the straightness in the limit. In a plot, the 45-degree line is usefully added to allow visualisation of net income. Since the Dutch estimate has a high marginal rate in the limit of 57.2 %, we add US-alike lines (U1) and (U2) with a r = 40 % limit. The two ranges are plotted in Figure 25.

Figure 25: Different tax regimes 1988 ($1000) (H) Holland, (U) US-alike

Exemption

Heterogeneous income

The nonproportional tax clearly becomes important when incomes differ, i.e. labour is heterogeneous in terms of productivity, labour costs and income. Lower income earners are affected disproportionally by the exemption level, not merely in terms of the income distribution but also in terms of their competitive position versus higher earners.

In Book III, equation (13.1a) already shows how the minimum wage consists of two elements. For above tax function:

Analytically solving for the minimum wage gives, due to the nonlinear curvature, two solutions for M[B, r, x, c]:

Note that the denominators are positive, so that the first solution is more adequate. If exemption is taken at x = B, then these two solutions degenerate into M B and M - c / (1 - r).

Figure 9 and Figure 8 in Book III plot the tax situation and the effect of M and B for curvature c = 0 (in the considered range), and for Holland 2002.

Indexation of exemption

We already mentioned the OECD (1986) report that taxes generally are indexed on inflation. This indexation though is not consistent over time. The Economist (1991:45-46) reported:

“the most intriguing proposal now doing the rounds in Congress (...) is to increase the personal tax exemption (the amount by which taxable income is reduced for each person in a household). In 1948 the exemption was set at $600 a person; in 1990 it was $2050. According to recent evidence before the House of Representatives select committee on children and the family, had the exemption been indexed from 1948 it would now be worth $7800.”

The Dutch data had already been given in Table 4. Indexation on inflation need not be optimal. We already looked at indexation of subsistence, and it might be wise to index taxes on the same base as gross income, as suggested by property (13.3e) and the discussion on subsistence in chapter 27.

A note on partners

Statutory taxes generally take account of the household situation. Sometimes tax terminologies suggest an individual treatment. Regard for example the Dutch tax code. This states that partners can ‘transfer their exemption’ to the money earning partner. You may check that Table 4 on the Dutch situation indeed shows an exemption for partners, in the 1997 column, that is double the exemption for singles. The situation in 2002 is a bit more complex due to an EITC.

Note, though, that the Dutch minimum wage roughly is set at the income level for partners. Singles have less net income since their exemption is lower, but they are not allowed to work at a lower gross minimum wage that might be feasible, with the same net income by assigning them the same exemption as for couples. The Dutch concoction of ‘exemption transfer’ in fact is extremely silly. It is even more surprising that it has been introduced while all Dutch tax specialists kept a straight face. The concoction also complicates the Dutch policy debate, since a proposal to raise exemption to subsistence now associates, in Dutch minds, with exemption for couples of double subsistence (which is exorbitant).

The best tax format would start with exemption at subsistence for singles.

Secondly, for partners with a single earner, a measure of ‘individual taxation’ can be introduced in the following manner. The basic ideas are:

· Home maintenance produces a product, this product is real income, and income should be taxed. However, part of home maintenance also can be part of subsistence.

· We may allow for a degree of spillover of income from one partner to the other. This is the public good argument, i.e. that more people can benefit while the cost is constant.

· Not all interaction is just spillover. Part of the interaction concerns an economic transaction. While the single person has to work for his home maintenance, he also buys it from himself. The single earner out partner buys it from the home partner. Revenue from this transaction should be taxable, i.e. on the side of the person that receives the payment.

Let yh stand for the income of the home partner, and yo for the income of the out partner. Let us use the Bentham tax, and apply it individually. Assign virtual income H to parttime home maintenance activities - and we are ignorant about the required hours. Let parttime virtual home maintenance income be part of exemption x = B’ = B + H, with B money subsistence or the net minimum wage on the market. The situation is neutral for a single person, who’s exemption is x = (B + H) while his income is y + H. The couple however is treated as follows:

· The out partner earns on the market y, buys Ho from the home partner, and has spillover yh of the income of the home partner. Buying something does not add to income however. Income thus is yo = (y + yh), and the tax thus is found to be r (y + yh - B - H)

· The home partner has own virtual income Hh, earns income Ho from the out partner, and has spillover yo of the income of the out partner. Income thus is yh = (Hh + Ho + yo) = (2H + yo) since Ho = Hh = H (we used the indices only for the origins). The tax thus is r (2 H + yo - B - H) = r (H + yo - B)

· Combined income thus is yo + yh = (y + yh) + (2H + yo) which consists of earned income, home production and spillover (yh + yo)

The equations solve as:

In the special case that the tax authority thinks that spillover is zero, then the out partner gets a tax rebate of rH in comparison with the single person. The home partner would not have to pay taxes when H would be less than B (half a day home maintenance work would be less than a day at a minimum wage). In this case the couple has more net income than the single person, and the products of another persons work, though on a pro-person base they would have less. Conversely, if home maintenance is a highly priced good, then there could be a case to levy taxes.

If spillover is a nonzero constant, then there is an income level y where the taxable income of the home partner H + yo - B will become positive. A person will have to pay taxes ‘just because’ he or she forms a couple with a high income earner. If spillover is nonzero but variable, then the value of that makes taxable income of the home partner exactly zero follows from H + yo - B = 0, and appears to be a function of income y:

If B = 2H (i.e. home maintenance gets the minimum wage), then for y = B, = 1/3. This means that the partner remains exempt from taxes as long as spillover is limited to a third of income. Interestingly, at that point also the taxable income of the out partner is yo = (B - H) / = 3 H so that he does not pay taxes either (since x = B + H = 3H here).

Above relationships show that individual taxation is possible that takes into account household spillover effects. For us the issue is primarily interesting for complications about subsistence. We find that there are no great complications, and we thus will further neglect the issue of partners.

Differential indexation

With subsistence indexed on income and taxes indexed on inflation, there is differential indexation, and due to the tax structure there is a multiplier increase in the minimum wage. Required gross minimum M shows a relative rise compared to other incomes, and it rises faster than both net minimum B and the general level of income Y/LE. In Figure 10 (in Book III), when we subtract the inflation component from x, B and M, then differential indexation shows up as: x stays fixed, B moves with the income density, M moves to the right, and M, as the intersection of the subsistence and tax lines, moves up more speedily. If productivity in the lower earnings scales doesn’t rise faster than general productivity or income, then ever more people grow unemployed.

For all clarity we shall prove this. This chapter uses the specific tax function (chapter 39 will give a proof independent of form). First we will show that M grows faster than B, and then we will show that M grows faster than productivity too, causing unemployment.

Let us first derive the real subsistence index rsi again, but now for the nonlinear tax. Recall the definitions of Book III. Let B = rsi P B with B subsistence in the base year. Let exemption x be adjusted for inflation with index P, then x = P x, with x the exemption in the base year that now may differ from subsistence in the base year B. Let also c be indexed on inflation as c = P c. Let the average wage index be W = P rwi W[0], with W[0] the average wage in the base year. Let h = x / W[0] and f = c / W[0].

rsi = Net[W] / Net[W[0]] / P = / / / / / / / which for f = 0 reduces to the Bentham-rsi deduced in Book III. For the limit, in general, we find:

which is normally below 1. Denote the denominator as F, and note that W[0] F = Net[W[0]] or F = 1 - ATR[W].

We use these properties for the following theorem.

Theorem T.1: With Tax[y, q], minimum wage setting M = B + Tax[M], and balanced growth, then: if B is indexed on the net average wage and x and c on inflation only, then M rises faster than other wages, and unemployment rises.

Note: That M rises faster than other wages is not inconsistent with balanced growth. For M is only the selection of one of the proper wages that is taken to be the minimum wage.

Proof:

For all clarity, parameter r will not be indexed. Let the price level index again be P. Again W = P rwi W. With real wage index rwi, the nominal index is wi = P rwi. For heterogeneous wages with wage density, we have w = wi w along the balanced growth path.

For a dynamic path we have starting position B giving M. In the base year the minimum level is taxed at an average rate less than r, implying that B > (1 - r) M.

We also use J as the index for the real minimum wage:

M = P J M i.e. J = M / (P M)

(1) We first prove that J > rsi in the limit. There are two relations for B, with rsi given by the relation above:

B = P rsi[rwi] B

B = M - Tax[M, (r, P x, P c)]

= M {1 - r (M - P x) / (M + P c)}

These equations define J as an implicit function of rsi. We also see that P falls away in the right hand side:

B = P rsi B = M {1 - r (M - P x) / (M + P c) }

rsi B = J M {1 - r (M - x / J) / ( M + c / J) } / / As rsi and J go to infinity, then rsi B ~ J M (1 - r). We had B > (1 - r) M. Thus J > rsi.

(2) We secondly prove that J > rwi in the limit. With limit ratio R:

using the fact that the denominator equals F defined above. We want to prove that R > 1. Note, then, that M < W[0], and that, due to the progressive character of the tax, the ratio of net income to total income must be higher at subsistence than at the average level, so that:

R = B / M / (Net[W[0]] / W[0]) > 1 > > > (3) Thirdly, we look at productivity and employment. For this theorem, the worst case to start from is full employment. When we start with full employment at M, then M provides the equilibrium of supply and demand. Let the supply price (or gross income or productivity) at the minimum be ms and let the demand price (labour costs) at the minimum be md. Then in the assumed start situation of full employment M = ms = md. Assuming balanced growth for demand and supply gives the development of the labour market situation at the bottom:

w = P rwi w in general, i.e. for all w

md = P rwi md & ms = P rwi ms

This means that the supplied (inherent) productivity of those at the (original) minimum grows as fast as the labour costs which employers could afford. However, the true supply price is not productivity but the (actual) minimum wage M that grows with P J and thus faster than the md. People in the class [ms, M) will not find jobs paying the social minimum. They become unemployed.

Q.E.D.

Above theorem and proof may be regarded as a bit simple. However, they help to highlight some useful aspects:

· Differential indexation can have surprising consequences compared to conventional ideas.

· Instead of thinking that productivity growth reduces employment for the lowly productive, we grow aware that it is likelier that technology creates so many job possibilities that employers can finance even higher costs than subsistence. But the multiplier effect from wrongly indexing taxes can be even faster.

· There is the combination of nonlinear tax and lognormal productivity, which causes an upswing of the CWIRU in the early phase of stagflation.

· This holds for a wide class of tax functions, even some very nonlinear ones.

· Where the term ‘income tax’ is used, it also applies to VAT and insurance for old age, disability and the like, as long as part of these are considered to be part of subsistence and thus should be included in exemption.

· This theorem and proof are for a structural form, and inspire the theorem and proof for the reduced form that we discuss later.

Raising exemption

Our analysis points to the suggestion of ‘waiving taxes for the lowly productive’, which can be translated as ‘raising exemption’. Interestingly, this latter translation appears to provoke some terminological confusions.

The notion of ‘raising exemption’ is often taken to imply that all other brackets shift along with exemption. This causes a huge loss of tax revenue. E.g. Gelauff (1992), who uses the official general equilibrium model of the Central Planning Bureau to compute the economic impact of raising exemption, adopts this expensive approach. (His scenario also includes the Dutch concoction of the ‘transfer of exemption’ by partners, so that his implementation is even more expensive.)

However, there are some alternative implementations. Their common feature is that taxes above the current minimum wage are essentially unchanged.

The issue can be clarified by the following two graphs. In Figure 26, the function with an exemption (bold line) can be compared to a function without an exemption (thin line) but with a tax credit (bold line again). The tax credit is given as c = r1 x where r1 is the rate of the first bracket (taking that as defined by the tax credit). The two systems are mathematically identical, when seen as a vertical translation while keeping the bracket positions fixed.

Figure 26. Piecewise linear tax function with more brackets

A dubious and horizontal transformation is given in Figure 27, where the assumption of ‘fixed bracket lengths’ has been assumed rather than ‘fixed bracket positions’. When we now substract a fixed sum from the line through the origin, the original function cannot be retrieved, and the higher incomes pay more tax. It now seems as if the tax credit is ‘fairer’. However, the true cause is that taxes have been raised by shifting the bracket positions.

Figure 27. Horizontal translation

The Dutch Government “Tax Plan for the 21st Century” used this misleading horizontal translation to argue that tax credits would be more just than plain old exemption. See Colignatus & Hulst (2003:32) for the misleading statements.

Useful approaches are:

1. Introduce a new separate ‘tax group’ that only holds for workers below the current minimum wage. Let this group have a high exemption at the new minimum wage and a normal marginal rate of 50%. Clearly, there could be jump in taxes at the current minimum wage. However, the high exemption can be said to apply to all citizens - and many simply don’t qualify since they do not fall in the new group. (The latter is only unfortunate for them, if they prefer a high exemption above their current high income.)

2. One might opt for a 100% marginal rate from subsistence (the new minimum wage) up to the current minimum wage. In this case there is no tax jump. High exemption again applies to all citizens, but its effect is undone by an intermediate high marginal rate region. Whether this is considered to be a bad situation, depends upon the analysis of marginal tax rates: see below.

3. Introduce a nonlinear trajectory from subsistence to some place in the current regime. Since reduction of wage costs generates employment, the state saves on benefit payments, and some revenue can be used to reduce taxes also above the current minimum wage. This reduction can be done in a nonlinear way that allows for a fluent change, without jumps and without new tax groups. Figure 28 gives an example of such nonlinear trajectory, where the function Tax[.] has been estimated to fit the 1997 Dutch tax code (inclusive of premiums) but with a nonlinear repair towards subsistence. The special point is that this estimated Tax[.] has a negative curvature parameter. The 1988 income distribution has been used to approximate tax revenues. The currency here still is Dutch guilders.

Figure 28: Nonlinear repair Holland 1997 (Dutch guilders)

4. Figure 29 uses euro’s and the new Dutch tax code and minimum wage of 2002. Using a 75% first bracket allows the minimum wage to shift from M1 to M2. The shaded area gives the tax revenue lost, which would be compensated by saved benefits.

Figure 29: Linear repair Holland 2002

We will discuss the optimal regime later, and return to the issue of raising exemption. This paragraph here was useful to clarify some terminological confusions. It also indicates that marginal rates will feature strongly in the discussion about the repair. A marginal rate of 100% or the marginal rates associated with negative curvature seem prohibitive for practical implementation. At least, in the conventional wisdom.

A note on the negative income tax

A common topic in the subject of taxation is the concept of a negative income tax (NIT). A person below a certain threshold receives money instead of paying it. The negative income tax can be presented as a ‘basic benefit’: all members of society receive allowance A from the state, and pay taxes only on their additional income. The negative income tax or basic benefit is often presented as a solution to the current unemployment problem. The Central Planning Bureau (1992a&b) in fact shows that this can work.

It is useful to clarify the following. We can distinguish three groups with different effects:

· for the currently employed the NIT has no effect, since they already are employed and in fact already earn their own basic benefit

· for the people in the Tax Void, the NIT effectively only means the increase of exemption, and thus one might as well increase exemption

· for workers with sub-subsistence productivity, the NIT indeed provides additional revenue.

The second effect cannot properly be regarded as a positive effect of a NIT. Only the last effect is the NIT proper. However, proponents of the NIT often include the second group when they claim good results. In the current situation of mass unemployment, the employment effect will also be largest for the second group, so the effects of the NIT are grossly overstated. You may be familiar with the joke of the mouse and the elephant walking on a bridge, and the mouse proclaiming: “We make quite a lot of noise together, don’t we ?”

It must be noted that proposals on the NIT generally state huge sums of money. The NIT is very ‘expensive’ since all spouses would apply, causing the need for more changes in the tax code.

The NIT complexities, and huge sums, also obscure the fact that abolishing the Tax Void would be for free. Proponents of the NIT thus can be compared to people at Amsterdam Schiphol airport wanting to go to Washington, and waiting at the ticket booth till they have enough money to buy the expensive ticket, while they overlook that, due to circumstances, the plane to New York flies for free.

The concept of a NIT, intended to do good, generally seems to cause people to do a lot of harm. The Central Planning Bureau (1992a&b) study assumed the gradual introduction of a NIT in the course of 25 years, keeping subsistence fixed at a constant inflation adjusted value of 1990, and the NIT fully introduced at that value in 2015. This scenario thus has the drawbacks of (a) achieving full employment only in 2015, (b) not indexing subsistence to general welfare.

It may well be that the Ministry of Finance is less equipped to deal with employment policy including the measurement of potential productivity. It would be better to quickly abolish the Tax Void, index subsistence properly, and restore the normal processes of social security and workfare to assist the sub-subsistence group.

The following equations clarify the relation between the NIT, exemption and subsistence. With market income y, the Bentham tax function Bentham[y], allowance A from the state, then net income and implied tax are:

net[y] = y - Bentham[y] + A = y - r (y - x) + A

implied tax[y] = y - net[y] = r (y - x) - A = r (y - (x + A/r)) = r (y - x)

So by taking x = (x + A/r) the allowance in fact means adjustment of exemption, with the subtle difference that x now just stands for the intersection with the horizontal axis, and not with exemption proper. Normally A would be chosen such that net income at subsistence y = B equals B, so that we might as well raise exemption to subsistence:

B = B - r(B - x) + A A = r (B - x) x = B

The marginal rate

The problem

The economic literature shows a conceptual problem, or paradox, on marginal rates. Statutory marginal rates are important in popular understanding, but not in the empirical data. Research, as witnessed by the existing literature such as Gelauff (1992), deals better with the data, but doesn’t convince the popular view. The following analysis suggests a solution.

Partial versus total derivative

Conventional theory, public discussion and empirical research generally use statutory rates as the “marginals”. With T[y] the tax associated with income y, the marginal rate commonly is computed as T[y]/ y. For our function this is the partial derivative as used in equation (29.1). However, the tax function is better understood not as T[y] but as the multivariate T[y, q] with q the (now arbitrary) tax parameters. Agents will tend to take account of parameter changes. So optimisation remains our paradigm - and it results into marginal rates - but the better marginal rate is the total derivative, or dynamic marginal rate (DMR):

dT[y, q] T[y, q] T[y, q] _ ----------- = ------------ + ------------ dq / dy

dy y q_

The topic of discussion is dq / dy. To proceed from this point, it appears didactically useful to first restate the conventional reaction to the DMR, and then develop the new analysis.

A conventional reaction

The conventional reaction is that tax parameters may be indexed to national income, but are not indexed to personal income. The individual agent in the economy will not think that his change in income can affect national tax parameters. Hence dq / dy should be zero.

Let us use the Bentham tax function again. Let us assume that only exemption is indexed on national income, and in continuous form the indexation reads as x = Y with as a fixed value for a base year. Thus:

T[y] = Bentham[y, Y] = r (y - Y)

It appears that is very small. For example, with LE the number of tax payers, and Y / LE average income, we may take exemption as a third of average income, so that = x / Y = 1 / (3 LE). But the small size does not invalidate the indexation method, since:

dLog[x] = dLog[ Y ] = dLog[Y]

Note that Y is the sum of all incomes. An income change for an individual does not affect the income changes of others. Assuming that other incomes stay fixed, we find for an individual income dY / dy = 1. If y rises and no other income rises, then the growth of national income dLog[Y] is equal to the growth for the single person weighted by its share in total income:

dLog[Y] = (y / Y) dLog[y]

It follows that the marginal tax for the individual is:

d T[y] / dy = r (1 - )

Now, since is such a small number, the marginal rate is virtually equal to r.

In general we find:

dq / dy = (dq / dY) . ( dY / dy) = dq / dY

Since dY / dy = 1. If parameters are indexed on national income, then dLog[q] = dLog[Y] and then dq / dY = q / Y so that

dq / dy = q / Y / / / which is close to zero since parameters q are generally much smaller than national income. We conclude that dq / dy = dq / dY is not quite zero, but practically zero, and this seems to corroborate the conventional reaction to the DMR.

Hence the conventional reaction to the DMR is that the DMR does not change the traditional analysis on marginal rates. Hence there is no hope for unemploment along these lines. With ongoing technological growth and competition of low wage countries, only the flexibility of labour markets will help to reduce unemployment, even if this means a reduction of net minimum wages. That, at least, is the conventional reaction.

The expectations revolution

However, Keynes (1936) explained that proper dynamic analysis inherently means that we have to consider expectations.

In this case the agent will be aware that parameters are indexed in some manner. Due to indexation, the term dq / dy can take significant values. Let q be indexed on national income growth Y. For many tax functions the indexation of parameters may take the form dLog[q] = dLog[Y] - as can be done for exemption and curvature of Tax[y]. If dLog[q] = dLog[Y] then

This again may reduce to the q / Y above. However, if we take expectations of the growth of national income, which means that the agent assumes that the other incomes do not remain constant, then:

Thus, next to knowledge about indexation, the agent will have expectations about the national income growth dLog[Y], and compare his own growth of income dLog[y] to this expectation. In terms of expectations, dq /dy does not vanish to zero. This is especially relevant when the parameter q gives exemption x that is a sizeable part of income.

So there is hope for the unemployed.

Discrete form

Above can also be formulated in discrete form. Indexation generally takes place with a lag, and then the discrete DMR is more adequate. This is:

DMR[y] = (T[y, q] - T[y-1 , q-1 ]) / (y - y-1 ) = T / y