This chapter is a bridge between the standard macro model and the

elaborations on heterogeneous labour and taxes. The concept of the

‘welfare state’ depends upon our concept of subsistence and the

elements that go into its index , and on the decisions that we take on

this at the national level.

In Book III we already regarded some indexation of subsistence and

taxes. Here we will refine indexation of net subsistence. Gross

subsistence will be _T_ -1[_B_] as determined by the tax system. A way

to understand this chapter is that it formulates conditions for the tax

system.

We already saw two possible indexation schemes for subsistence: (i) on

average net income or (ii) on gross average income. The latter presumes

that taxes are an indication of welfare too. This current chapter will

look an another way of indexation that takes an intermediate position

that might be better but that might also be needlessly complex.

We will find that if we adopt certain indexations, then we must accept

some divergence in development in other terms.

Definitions

Subsistence labour forms a special group within heterogenous labour.

The group only exists if we acknowledge heterogeneity. In the labour

supply density we already hypothised a ‘dromedary shape’ that partly

reflected the fact that a minimum income means longer hours when the

wage drops. Let us now discuss subsistence more extensively.

With man a social animal, sociobiological and social psychological

causes apply in general. Precisely what these causes are, and how they

apply, is a subject of serious study, see for example Aronson (1992a&b)

and Wilson (1993). A regularity for mankind seems to be, vide these

studies, that in certain cases people show a certain amount of care for

their fellows.

This care should not be overrated. Part of it may not be empathy, but

simply be precaution and an insurance for the event of personal

misfortune. Also, some care obviously reduces the chance of a violent

reaction of the disadvantaged. There are clear examples of empathy

breakdown. For example, archeologists found ancient mines with such

small shafts that these mines could only have been worked by children.

We need not have illusions about working conditions, especially since

it were lead mines. Nevertheless, whatever these clauses and contrary

cases, ‘normal conditions’ seem to provoke a distinct level of care.

A strong assumption is that people have views about the whole income

distribution. A simpler assumption is that people recognise a level of

subsistence - which for dynamics likely implies that they adjust that

subsistence to developments.

The strong assumption might well be that the income distribution is

lognormal for _social_ - and not ‘economic’ - reasons, and that the

economic process only is oriented at directing people to a fitting

place in that distribution. Economic productivity is essentially a

nominal concept. It is not just the technical amount of goods per hour

that can be produced, but also multiplied by the price of the product,

and the price is determined in a social situation where status

considerations apply. The assumption that economic agents have views

about the income distribution actually need not be overly strong. As

Tobin (1972, p122) states:

“(...) This observation led Keynes to his central explanation: Workers,

individually and in groups, are more concerned with relative than

absolute real wages.”

However, for our discussion, we narrow down the problem to the

subsistence or the net minimum wage, and disregard views on the whole

income distribution.

Suppose that a group recognises some subsistence. A group even might be

defined by its shared views on this. For example, members of a royal

family receive a certain allowance that meets their standard of living,

and their standard of living helps to show that they are members of

that royal family. The view oriented at the inner group thus is linked

to the exclusion of others. Others should have less, precisely to

distinguish them from the inner group. Being a royal family does not

amount to much, if you don’t have subjects. This process works all the

way down, so that even people in minimum conditions flatten out

differences among themselves, and seem to compare themselves to beings

of assumed lesser stature. (So the simpler assumption could be used to

build the strong assumption.) This discussion also clarifies that the

size of the group matters. There is only room for a national

subsistence floor if the simpler assumption allows for a large group.

So the simpler assumption properly reads that groups not only define

subsistence for the inner group, which is less controversial, but also,

more controversial, subsistence for society as a whole.

Note that any assumption, simple or strong, is not sufficient by

itself. Society also has the coordination problem of aggregating the

individual preferences on national subsistence, particularly since not

everyone who wants to raise the living standard of the poor has the

personal means to do so. Sometimes there are legal rules. Often labour

unions come in. For example in Holland collective bargaining results

into industry minimum wages that are on average at least 10% higher

than the legal minimum wage. More generally, subsistence is simply a

social convention. A certain level of living is regarded as

inacceptable, both by most employers and by the work floor in general.

One way to implement a welfare system would be to set social security

at _B,_ and leave it at that. There would be no need for a minimum

wage, since employers would have to offer at least _B_. In practice

government nevertheless create a minimum wage system too, and allow a

gap between the working wage and the benefit. One of the reasons is

better control, so that agents are less likely to both receive a

benefit and work on the side. One of the other causes undoubtedly

derives of the social forces that call for a decent minimum. [82]

Sometimes labour market regulators may be aware of the problem of the

minimum wage, and may opt for a lower indexation of _M_ even though it

results into a lower _B._ But the effectiveness of such policies that

reduce subsistence depends upon the strength of conventions in all

factories and sectors.

It is useful to note that conventions are sensitive to various

considerations. For example, the Dutch legal minimum wage holds for

fulltimers, but does not hold for parttimers. Holland now has a lot of

parttime work. [83] It is also interesting to observe that tax

exemption _x_ is established within the bureaucratic realm where there

is no direct confrontation with the standard of living. For its own

historical reasons, exemption is generally indexed on inflation. These

matters, while also being evidence that human care for other people

should not be overrated, again clarify that our subject matter is not

simple in itself. Subsistence itself is very simple, especially to

those who are subject to it, but it can be made complex, especially by

those who govern.

Economic literature

Economic theory has long been aware of notions of empathy, vide Adam

Smith (1759, 1984) on moral sentiments.

Some tax theorists suggest that the social subsistence level should be

exempt from taxation. Hofstra (1975) recalls the Cohen Stuart 1889

analogy, that a bridge must hold its own weight before it can be used.

In his 1980 presidential address to the American Economic Association,

Solow (1980) discussed his reading of Pigou’ work, and writes:

“The last comment of Pigou’s that I want to cite is especially

intriguing because it is so unlike the sort of thing that his present

day successors keep saying. Already in the 1933 _Theory of

unemployment_ he wrote: “... public opinion in a modern civilized State

builds up for itself a rough estimate of what constitutes a reasonable

living wage. This is derived half-consciously from a knowlegde of the

actual standards enjoyed by more or less ‘average’ workers ... Public

opinion then enforces its view, failing success through social

pressure, by the machinery of .. legislation” (p.255). A similar remark

appears in _Lapses_ [Pigou 1944 _Lapses from Full Employment_]. Such

feelings about equity and fairness are obviously relevant to the

setting of statutory minimum wages, and Pigou uses them that way.” (p5)

Solow in the next sentences also emphasises the power of social

pressure, and shows himself aware that the minimum wage need not be a

special application since social pressure is abundant:

“... it is even more surprising ... that employers so rarely try to

elicit wage cutting on the part of their laid-off employees, even in a

buyer’s market for labor. Several forces can be at work, but I think

Occam’s razor and common observation both suggest that a code of good

behavior enforced by social pressure is one of them.”

Types of indexation

We already have encountered these indexes of subsistence:

· The graphs in Book III are based on indexation on the net average

wage _Net_[_W_] = _W - T_[_W_]. This presentation has been chosen since

its approach is more conservative.

· Another indexation is on _W _itself, which thus considers taxes a

part of well-being. Property (13.3e) however shows this equivalent to

the first, for the Bentham tax, provided that exemption is properly

indexed too.

Indexation on gross income (i.e. on _W_) agrees better with economic

intuition, since taxes need not be a real burden, when they generate

goods that enter the utility function. However, some taxes can be

wasteful or can be discarded for other reasons. In the following we

will take a middle position, adding and substracting income elements.

In particular:

· some public goods _Q _are provided by nature: breathing air and the

berries in the field

· taxes go into public goods _Gp,_ that subsistence workers get for

free too (as licensed free riders)

· some government expenditure _Gs _may benefit only special interest

groups (wastefully)

· some government expenditures _Gn _actually benefit the average tax

payer, and should be considered part of ‘net income’

· some taxes go to the support of the unemployed - _B U_ - which the

unemployed cannot provide for themselves

· there is the possibility of different consumption baskets (different

deflators)

· it is recognised that people at subsistence tend to have more sweat

and less leisure

· tax revenue can change disproportionally with income.

Considering these element, it seems that the adoption of a detailed

index would likely cause little difference with gross income

indexation. Many of the additions compensate for many of the

substractions. Also, if subsistence were to lag behind average income,

then it might well happen that subsistence is increased at some point

anyway.

It nevertheless remains useful to develop the detailed index formally.

If your interest in the subject is not very strong, you are advised to

skip the remainder of this chapter. The reader who studies this section

will notice that we do not achieve very much. Some of the formulas look

complex, but on close inspection only say the obvious.

Formal development

We assume a ‘basic insurance’ setup for social security. The unemployed

get a benefit of _B. _At higher earning levels they may have additional

insurance, and be paid on top of _B. _But this is of no concern for our

issue. Also, who is on benefit but gets a job offer, accepts this, on

the penalty of losing the benefit anyhow. This means that nominal

transfer payments are _NTRF = B U_. We also take _b = NTRF / LE = B u

_(redefining the symbol _b _- no longer the IS curve). Similarly_ q = Q

/ LE._

Let _g = NG / LE_ be average nominal government expenditure per worker,

with _g = gn + gp + gs. _We will assume Ricardian equivalence, so that

government budget deficits are regarded as part of taxes, so that there

effectively is no deficit. [84] Hence _TAX = NG + NTRF._

Then the average wage tax rate _AWTR _ _TAX / WT = (TAX/LE) / (WT/LE)

= (g + b) / W. _

For the special interests we distinguish two kinds of situations.

· When average income itself is the special interest, then _gs_ can

also be regarded as net income, part of _gn_, and then this case is

equivalent to _gs_ = 0. Note that we could include _gn_ in _Net_[_W_]

mathematically anyhow (but don’t do this for clarity).

· Alternatively _gs_ 0. In particular, the average income group could

be a victim of a coalition of the poor and the rich, the first getting

a high _B_ and the second a large _gs_. [85] In a democracy with voting

population _LS,_ a majority of _LS_/2 + 1 indeed can levy high taxes on

the other _LS_/2 -1. In that case it would not be fair to regard the

tax on the average wage as beneficial to the common good. (Note that

this analysis for _gs_ 0 is weak, since not all possible

redistributive schemes are considered.)

Price indices for the average and subsistence workers are _P_ and _Pb.

_Real positions thus are _W / P_ and_ B / Pb. _Government prices are

_Pgn, Pgp _and_ Pgs, _giving _gnr _and _gpr _and _gsr. _Similarly _Pq_

and _qr._

The difference in leisure and sweat will be compensated here by

choosing a suitable Real Income Ratio RIR.

All together, we have:

· Net position of the average worker _Net_[_W_] + _gn + gp _+ _q_

· Net position of the subsistence worker _B_ + _gp_ + _q_

· The real income ratio RIR (_B/Pb_ + _gpr_ + _qr_) / (_Net_[_W_]_/P_

+ _gnr + gpr + qr_)

The government would set RIR at a specific value, and then determine _B

_from the other values:

_B = Pb_ { RIR_ (Net_[_W_]_ / P + gnr) - _(1 - RIR)_ (gpr + qr)_ }

(27.1)

One thing to show is that _B _has a small multiplier on itself because

of _b._ We can use the average tax rate difference _Z_ between

national and private average:

_Z _ _TAX / WT - T_[_W_] / _W_

_Z = (g + b) / W - T_[_W_] / _W_

_T_[_W_] = _g + b - Z W _

_Net_[_W_]_ = W - T_[_W_]_ = W + Z W - g - b = (1 + Z) W - g - b_

Using this for the RIR:

· Net position of the average worker _(1 + Z) W - gs - b_ + _q_

· Then RIR (_B/Pb_ + _gpr_ + _qr_) / (_(1+Z) W/P_ - _gsr - B/Pb u +

qr_)

(27.2)

The first term of (27.2) contains a small (negative) multiplier of _B_

on itself. In full employment, _u_ 0.02, and with RIR 0.30 the

multiplier might easily be neglected. That is, neglected in (27.2) but

not for the determination of the RIR in the base year - since _B u_

cannot be neglected for the base of the RIR. Since (27.1) and (27.2)

are mathematically the same, using (27.1) makes that the question of

neglecting that small multiplier does not arise.

Another point is that the index becomes simpler if all price indices

are the same. Taking _P = Pi _gives RIR (_B_ + _gp_ +_ q_) / ((1_+Z_)_

W_ - _gs - B u + q_).

Let us consider a numerical example. Suppose that _gn = gs_ = _q =_ 0

and that prices are equal. Suppose also that _AWTR = TAX/WT_ = 0.30. We

also take the Bentham tax _T_[_y_] = _Bentham_[_y_] = 0.5 (_y - B_).

Let us consider the path that subsistence is half of average income,

i.e. _B/W_ = ½, and then compute the various ratios. Then:

· Indexation on gross average income gives _B / W_ = 0.5.

· Indexation on net average income gives _B / Net_[_W_] = _B_ / (2_B_ -

0.5 _B_) = 0.66.

· Then _T_[_W_] / _W = _0.5 (_W - ½ W_) / _W _= 0.25, and _Z _ 0.30 -

0.25 = 0.05.

· Since _gn = gs_ = 0, _g = gp, _and _AWTR = _(_gp + b_) /_ W = gp/W +

_½ _u = _0.30. If we assume full employment _u_ = 0.02, then _gp/W_ =

0.29.

· Then RIR = (_B / W + gp /W_) / ((_1 + Z_) - 0.01) = (½ + 0.29) /

1.04 = 0.76.

Note that the ratio numbers 0.50, 0.66 and 0.76 by themselves mean

little. In both cases _B _is set at half _W, _so the value of _B _is

not affected. The only point is that the bases are different each time,

and apparently smaller. These bases of course change again for other

assumptions on the various variables and functions. Where there is no

difference at a particular moment (base year), there however arise

differences over time. The following tries to find out more about this.

Progression factor

One way to trace developments over time is to make plots as we did in

Book III. Another approach is more formally, and a commonly used route

here is the assumption of a constant macro-economic progression factor.

This factor is the elasticity of tax revenue with respect to income

(Koopmans (1975:103)), thus _mepf = (Y / TAX) (_ _TAX_ / _ _ _Y). _The

factor is determined by tax parameters, their indexation, the income

distribution and its change. In this case, without a deficit, the

progression factor applies to expenditure too, which may be taken to

mean, effectively, that taxes are indexed such that tax revenue follows

expenditure.

We shall take the progression factor for the average wage, which is

exclusive of profits and the growth of employment. Thus our _ = (W /

g) (_ _g_ / _ _ _W). _We assume a nominal position, thus include price

developments in government expenditure relative to the average wage. We

set _gn = _0_ _now, since it can be included mathematically with _gp.

_We also assume that _ _is equal for _gs_ and _gp_, so that _gs_ =

_gs_[0] _W _ / _W_[0]_ _= _gs_0 _W _ and _gp_ = _gp_[0] _W _ /

_W_[0]_ _ = _gp_0 _W _ . Thus _g_ = _g_[0] _W _

/ _W_[0]_ _ with properly _g_[0] = _gp_[0]

+ _gs_[0].

Then _ _ _g_

/ _ _ _W g / W = NG / WT. _ This has

the specific property that _ = _1 implies that the quote _g / W = g_[0]

/ _W_[0] is constant, and thus _NG /WT _is constant too. We will use

this property below.

Taking _W_ separate:

and hence

(27.3)

Inclusion of the progression factor does not cause special observations

yet. If _ _ < 1 then in the limit of _W_ the indexation can be rather

simple, especially if _Pb qr / W_ goes to zero too. If _ _> 1, then

there could be a point where the markup on _W_ is zero, or subsistence

would have to be zero - which would suggest an unrealistic tax

function. The progression factor becomes more useful if we regard

special cases.

Special cases

Definition: A (democratic) state is “Madisonian”, iff _gs_ = 0. James

Madison remarked that a proper democracy with a majority rule actually

safeguards the interests of the minorities.

Definition: A “real welfare state” aspires at a constant RIR and takes

_q_ = 0. The idea on the latter is that breathing air is prerequisite

to utility and no source of it. The berries in the field are owned by

someone, and no longer free. (If they were free, then Coase’s Theorem

shows that they could be counted as part of income, and hence they

would no longer be free for all practical purposes.)

Definition: A “pragmatic” real welfare state sets _u _= 0 in the

determination of the benefit level and RIR. The factor _B u _really

does not amount to much.

Definition: “Uniform prices” means _P _= _Pb = Pgs = Pgb = Pgn = Pq_.

If this happens then one price index _P_ suffices.

Theorem B1: In a pragmatic Madisonian real welfare state with Ricardian

equivalence and uniform prices, (i)

RIR = _(B + g) / ((1 + Z) W) _(base year)

and

_B = W_ ((_1 + Z) _RIR_ - NG/WT_) (henceforth)

(ii) If RIR is constant, then: (1) A constant quote for government

layouts (or progression factor _ _ = 1) only allows for some variation

in _B/W _by variation in the average tax rate difference _Z. _(2) If _Z

_is constant, then _B _is fully indexed on _W._

Proof:

(i) For the base year: substitute the results of the definitions in the

RIR (vide (27.2)), note that the prices cancel and that _g = gp_. Then

find the base year result as stated, and then use _(NG /WT) W = g _to

get the annual expression.

(ii) For (1), we use _ _= 1 _NG /WT _= _g_[0] / _W_[0] from above.

Then simply rework the equation for a constant.

For (2), if _NG/WT _and _Z _are constant, write _B = c W._ Then _B

/ _ _W _= _c_

= _B / W. _Hence Log[_B_] = Log[_W_].

Q.E.D.

Theorem B2: In a pragmatic Madisonian real welfare state with Ricardian

equivalence and uniform prices, net income indexation is only feasible

for special tax functions.

Proof: To see what happens if _B _is indexed on _Net_[_W_], write _n =

_ _ Net_[_W_] / _ _ _W. _Note that 1- _n_ is the marginal tax rate for

_W_, and that _B / _ _W = _ _B / _ _Net_[_W_] _n._

With _B = W (1 + Z) _RIR_ - g _(theorem B1)_ _use _W (1 + Z)_ =

_(Net_[_W_]_ + g + b) _and get:

_B _= RIR _Net_[_W_] _ _- (1 - RIR) _g + _RIR_ b_

Note that _b_ 0, since we have set _u_ = 0 only in the determination

of the RIR. Then:

_B / _ _W = _ (RIR _Net_[_W_] _ _- (1 - RIR) _g + _RIR _b_) _/ _ _W _

_= _RIR _n _- (1 - RIR) _ g / W + _RIR _u_ _B

/ _ _W_

/

/

/

/

_B / _ _W_= (RIR _n_ - (1 - RIR) _ g / W_)

/ (1 - RIR _u_)

/

/

/

/

/

/

/

/

We again find a small multiplier. Dividing by _n _gives the transform

to _Net_[_W_]:

_B / _ _Net_[_W_] = (RIR - (1 - RIR) _ NG / WT / n_) / (1 - RIR _u_)

Log_B / _ Log[_Net_[_W_]] = _Net_[_W_] / _B_ (RIR - (1 - RIR) _ g /

W / n_) / (1 - RIR _u_)

Indexation on _Net_[_W_] means that the left hand side is 1, and that

_Net_[_W_] / _B_ is some constant. Setting net income ratio _B /

Net_[_W_] = NIR[0]:

NIR[0] = (RIR - (1 - RIR) _ g / W / n_)

/ (1 - RIR _u_)

/

/

/

We want to find the conditions under which RIR is a constant (for the

‘real welfare state’). Solving above expression for RIR gives:

A special case has _ = _1 and thus _NG/WT = g / W _constant, and _n

_constant, i.e. for the Bentham tax function _n _= 1 - _r._ This is

only feasible if _u _is constant too. There is a more general class

when _ g / W

/ n _is some constant, but _u _must be constant here too. In other

cases the RIR is implicitly adjusted to make _B_ / _Net_[_W_]

constant. But nonconstancy of the RIR conflicts with above definition

of the welfare state (that must have constant RIR).

Q.E.D.

28. Phillipscurve

This chapter deals with the confrontation of labour supply with labour

demand, and the equilibrating dynamics. With high unemployment, wage

growth may be reduced. With low unemployment there may be ample room

for wage demands, and wage inflation can rise.

Chapter 25 already provided a background discussion on the

Phillipscurve, and for example pointed to Graaflands c.s. derivation

from a Nash maximising framework. In this chapter we take that possible

development for granted, and concentrate on concepts: what variables

are relevant for a Phillipscurve, and how do we characterise

equilibrium.

It appears to be useful to first develop some concepts of dynamics.

Concepts

The Phillipscurve reflects the hypothesis that (wage) inflation is

influenced by unemployment. Of course other factors are important too,

such as (price, wage) expectations and forward shifting of taxes.

Whatever other influences, the key notion of the Phillipscurve remains

the influence of the employment situation. Wage adjustment now is

considered to be the dependent variable while normally the price would

be the independent variable. Wage adjustment will consist of a shift

along a curve and a shift of the curve, and for both we still use the

term ‘Phillipscurve’.

As remarked, labour supply is relatively fixed. Utility maximisation

and rational calculation will primairily be directed at finding a

competitive wage (competition not necessarily meaning _full_

competition - as we e.g. referred to a Nash equilibrium). An individual

who sets his wages too high will become unemployed. Even the

probability of becoming unemployed will have a sobering effect. Given

this framework, the model must concern a dynamic process of

unemployment (threats) and wage adjustment.

First consider a homogeneous market with price level _P. _Price

adjustment towards the market clearing equilibrium price _P° _depends

upon excess demand, and since excess demand is determined by the price

level, we get a differential equation:

_P’ _= d_P _/ d_t _= _f_[ _D_[_P_] - _S_[_P_] ] = _f °_ [ _P° - P _]

Note that the choice of ‘excess demand’ as the explanatory variable is

arbitrary. We might as well take excess supply, or allow demand and

supply to react differently, or have a different sensitivity to prices

and quantities. Similarly, we can also take the quantity as the

explained variable. And we can also formulate the equation in

expectational variables.

Some authors hold that above relationship for price dynamics is an

hypothesis that needs further clarification. I think that this is too

cautious. Admittedly, it might be too simple to only presume that

agents know that they are involved in a market ‘tatonnement’ process,

and further explanations can be helpful. Agents have various tools

available, and the choice of offering and accepting prices and

quantities can be described, using an optimising framework. The speed

of adjustment in markets depends upon characteristics like the size of

the market, the historical relationships between agents, ‘menu costs’,

and the like. It is also useful to distinguish ‘normal’ periods and

‘shocks’. However, the level of detail depends upon the use of the

model, and above relationship suffices our goal.

Inflation is the rate of growth of prices, i.e. _p _= dLog[_P_] / d_t_

= _P’ / P._ The change in inflation is d_p _/ d_t = P”/ P - (P’)2 /

P2_ in terms of the original price level. Acceleration of inflation

would be d2 _p _/ d_t2_.

We need to clarify a term. The economic literature uses the term

“Non-Accelerating-Inflation Rate of Unemployment” (NAIRU) for that rate

of unemployment that causes d_p _/ d_t _= 0.

This term thus should be “non-accelerating prices” or “non-changing, or

constant, inflation”.

Secondly, it appears that the formulation in terms of differentials is

less useful for practical economics than the formulation in

differences. So we will use differences instead. Inflation then is _p

_= (_P /P_[-1] - 1) (often expressed as a percentage).

Thirdly, we regard wage inflation rather than product price inflation,

thus _ _= (_W /W_[-1] - 1). Please note that we use the different

letter font _ _for wage inflation, since we use _w _for the level

variable in densities like _e_[_w_]. Properly we should substract

productivity growth, but for our purposes we may now assume that

productivity is constant. Note that wage inflation can be different

from price inflation, since productivity is determined in terms of the

output price, and output will not be only consumer goods but also

exports, investments and intermediates.

We will use the term “Constant Inflation Rate of Unemployment” (CIRU)

for that rate of unemployment that causes _p = p_[-1]. Similarly, the

Constant Wage Inflation Rate of Unemployment (CWIRU) gives that rate of

unemployment that causes _ = _[-1]. [86]

We use the term “Equilibrium Rate of Unemployment” (ERU) for that rate

of unemployment that causes wages to adjust to their equilibrating or

market clearing level _° = _(_W° /W_[-1] - 1). The CWIRU might be a

special kind of ERU. The idea is that once inflation has been constant

for a long while, you start expecting it. Table 8 contains an overview

of the concepts.

Table 8: Concepts for wage inflation

_REH: white noise surprise = * + _

_Non-REH: other surprises_

_ _

_CWIRU_

_

= [-1]_

_uf = ERU[FE] _

CWIRU = ERU[REH] = ERU[FE]

Maybe temporarily, but impossible in the long run

_Other_

CWIRU = ERU[REH]

Maybe temporarily, but impossible in the long run

_Non-CWIRU_

_

[-1]_

_uf = ERU[FE] _

_° _= _h_[_uf_, _u_[-1]] + … if expected …

_° _= _h_[_uf_, _u_[-1]] + …

_Other_

ERU[REH]

No equilibrium in any of these senses

Note: We use ° to indicate market clearing equilibrium, and * or E[.]

for expectations and expectational equilibrium. We use · when we

allow for either.

We can recognise at least two equilibria:

· FE: full employment, when all labour resources are used except for

friction unemployment _uf_ = ERU[FE]. Normally _° _is a direct function

of _uf_, for example _° _= _h_[_uf_, _u_[-1]] + dLog[_Money_]. It may

be that people’s expectations on nominal wages are not fulfulled, so

that _° _E[__] _. _A FE policy is only successful if _ = ° _and _u =

uf_.

· REH: the rational expectations equilibrium, when expectations are

fulfilled except for random error. Thus _* _= E[__], it so develops

that _ _= _* + _, and this optimality is only in terms of expectations.

In ERU[REH] unemployment may be far from _uf_ = ERU[FE]. The situation

can be stable if people only regard the price signals (and whatever

else is in the specification), and are satisfied as long as their

expectations are fulfilled.

A homogeneous Phillipscurve

A linear format

Let the change in wage inflation be sensitive to wages with degree _ _

and sensitive to quantities with a function _f_[_u_], with _u _the rate

of unemployment_._ The following gives a rich (wage) Phillipscurve that

contains not only the rate of unemployment but also past and (forward

looking) equilibrating wage inflation. [87]

_ - _[-1] = _ _ (_ - _[-1]) + _f_[_u_] (28.1)

_ _= _ _ __ + (1 - _ _) __[-1] + _f_[_u_]

(28.2)

Equilibria

Generally for the CWIRU from (28.1):

0 = _ _ (_ - _[-1]) + _f_[CWIRU]

CWIRU = _f_ -1[ - _ _(_ - _[-1]) ]

According to the Rational Expectations Hypothesis (REH):_ * _= E[__] =

__. Then from (28.2) - interpreting REH as ‘model consistency’:

_*_ = E[__] = _ _ _*_ + (1 - _ _) __[-1] + _f_[E[_u_]]

_*_ = __[-1] + _f_[E[_u_]] / (1 - _ _) (28.3)

We can also prove that _u_ = E[_u_] and then define E[_u_] = ERU[REH].

[88] Hence: [89]

_ _= _ _[-1] + _f_[E[_u_]] / (1 - _ _)

E[_u_] = _f_ -1[ (1 - _ _) (_ - _[-1]) ] = _u _

In this specification, the CWIRU can be ERU[REH], and ERU[REH] can be

CWIRU. Namely, when _* = _[-1], or when expectational equilibrium is

associated with constant wage inflation. Some ERU[REH] however can

exist with nonconstant inflation that is not CWIRU. Since equilibrium

wage inflation _* _is determined also by other factors such as money,

the ERU need not be constant. Even when _u _= ERU[REH] for each

separate year, then _ _might still have an erratic development over the

years. Similarly, the CWIRU can be an ERU[REH], but need not be. It can

even be that _ = _E[__] but expectations are not REH - since the error

is not white noise.

For full employment, policy is successful, if and only if _u = uf_ and

_ = *, _so that:

ERU[FE] = _uf_ = _f_ -1[ (1 - _ _) (_ - _[-1]) ] (28.4)

This equation has the same format as ERU[REH]. It follows that _uf_ can

be REH, and REH could be _uf_. However, they need not be, since, though

we have used the same symbol _f, _in practice there can be different

functions and also additional variables depending upon the FE or REH

assumption. [90]

Similarly, with this specification there might be constancy, and of

course there might be not. And as said, constancy might not be the real

issue, as small fluctuations in a stable range might be acceptable too.

[91]

Selection of _f_[_u_]

In the selection of _f_[_u_] we have to take account of the fact that

_u _can shift as a result of the minimum wage. Workers below the

minimum wage are not relevant for the labour market, and do not exert a

downward pressure on wage inflation. Above we saw that _u _= _un + um._

Let _fu_[_un_] give the fundamental _nonshifted _relationship for that

part of unemployment that still affects the development of wages.

Conforming to empirical regularity:

_fu_[_un_] = _ - _Log[_un + _]

Here _ _ is a parameter for horizontal adjustment, __ gives the slope,

and _ _is a constant shift in _u._ Note that _fu_[_un_] may be very

sensitive to low values of _un _and _, _since the logarithm from 0 till

1 is very steep, and _un _commonly is measured in percentages and thus

covers that range. Now, for _f_[_u_], an endogenous shift in _u_ then

can be included by:

_f_[_u_] = _f_[_un + um_] = _fu_[_un_] = _fu_[_u - um_] = _ - _Log[_u

- um + ___]

Note that _f_[_u_] here is also acceleration, since 1/(1-__) disappears

in __ and __. Figure 24 gives two regimes, plotted for both the

_f_[_u_] in the left part and the Phillipscurve in the right part.

Parameters are _ _=_ = 5,_ _ = _0, and _um _= 0 [case (a)]

respectively _um _= 6 [case (b)]. It is assumed that _* _ = __[-1] =

2 respectively 5, so that the minimum wage unemployment of 0 associates

with an equilibrium wage inflation path of 2, while the high minimum

wage unemployment of 6 associates with a high wage inflation path of 5.

Since _* _ = __[-1] the CWIRU’s can be found when _f_[_u_] = 0, and

these result in values of 2.7 and 8.7 (= 2.7 + 6).

Figure 24: Dynamics: unemployment and inflation

Given the assumption of _* _ = __[-1] it also follows that the

Phillipscurves are just horizontal translations of the _f_[_u_], and

one can see the values of 2, respectively 5, for the assumed wage

inflations at the CWIRU’s.

The cases (a) and (b) in Figure 24 reflect the developments in the OECD

in the 1950-2005 period. Case (a) gives the situation somewhat like the

1950s. The trade-off of inflation and unemployment then took place at

low rates along the long drawn line. The trade-off of wage (price)

acceleration and unemployment gives the CWIRU. At that point price

acceleration is zero, and inflation remains at a low and constant

value. Case (b) gives the situation of stagflation, where both the

CWIRU and the trade-off-process around it have worsened. The move from

(a) to (b) can be called ‘stagflationary’. In the 1960s and 1970s

authorities targetted for low unemployment at the cost of rising and

eventually high inflation. In the 1980s and 1990s the authorities

targetted against inflation and accepted high unemployment.

The short term Phillipscurve concerns the direct trade-off of

unemployment and (wage) inflation and is given by the long drawn

curves. This trade-off has only limited explanatory value. Nowadays

unemployment is concentrated at the low income section of the income

distribution, and it is not likely that this can be battled with high

wage inflation. This phenomenon is rather explained by the shift of the

CWIRU or the long run relationships between equilibrium unemployment

and wage acceleration, which are given in the left diagram.

It is useful to note:

· The CWIRU need not be constant. It could be if e.g. the relation

indeed is linear and if the coefficients are fixed. But neither need be

the case. The CWIRU in all likelihood is itself a variable that traces

out a path. (Which is another reason why the name ‘natural rate’ is

unfortunate.)

· There is a movement of the curve and a movement along the curve.

· The movement of the curve is not determined by the labour market

alone. Policy makers may neglect labour market measures, and may opt

for high inflation (1970s) or for high interest rates (1980/90s) to

fight minimum wage unemployment that is not affected by these.

On expectations

We may recall the 1995 Nobel Prize for Robert Lucas. The Swedish

Academy put the following text on the internet:

“The change in our understanding of the so-called Phillips curve is an

excellent example of Lucas’s contributions. The Phillips curve displays

a positive relation between inflation and employment. In the late

1960s, there was considerable empirical support for the Phillips curve;

it was regarded as one of the more stable relations in economics. It

was interpreted as an option for government authorities to increase

employment by pursuing an expansionary policy which raises inflation.

Milton Friedman and Edmund Phelps criticized this interpretation and

claimed that the expectations of the general public would adjust to

higher inflation and preclude a lasting increase in employment: Only

the short-run Phillips curve is sloping, whereas the long-run curve is

vertical. This criticism was not quite convincing, however, because

Friedman and Phelps assumed adaptive expectations. Such expectations do

in fact imply a permanent rise in employment if inflation is allowed to

increase over time. In a study published in 1972, Lucas used the

rational expectations hypothesis to provide the first theoretically

satisfactory explanation for why the Phillips curve could be sloping in

the short run but vertical in the long run. In other words, regardless

of how it is pursued, stabilization policy cannot systematically affect

long-run employment. Lucas formulated an ingenious theoretical model

which generates time series such that inflation and employment indeed

seem to be positively correlated. A statistician who studies these time

series might easily conclude that employment could be increased by

implementing an expansionary economic policy. Nevertheless, Lucas

demonstrated that any endeavor, based on such policy, to exploit the

Phillips curve and permanently increase employment would be futile and

only give rise to higher inflation. This is because agents in the model

adjust their expectations and hence price and wage formation to the

new, expected policy. Experience during the 1970s and 1980s has shown

that higher inflation does not appear to bring about a permanent

increase in employment. This insight into the long-run effects of

stabilization policy has become a commonly accepted view; it is now the

foundation for monetary policy in a number of countries in their

efforts to achieve and maintain a low and stable inflation rate.”

The Academy is a bit too assertive. The Phillipscurve need not be

vertical in the long run. It may well be that there is no fixed

solution, and that the long run gives a non-converging movement. Also

Phelps (1994) has reminded us that the CWIRU (in his words the NAIRU or

‘natural rate’) need not be constant.

Secondly, there can be other causes than expectations, and these might

be more important for understanding the present situation. One

important cause is the mechanism of the minimum wage. Hence the models

used by Lucas and his predecessors need not be the relevant models for

explaining the empirical shifts in the Phillipscurves and their

CWIRU’s.

Heterogeneous Phillipscurves

If labour is heterogeneous, then utility maximisation and rational

calculation are not only directed at demanding a competitive wage, but

they are also directed at selecting the kind of submarket (and its

associated wage). This complicates the situation. Can we say that a

dentist is ‘unemployed’ in the market for farmers ? Or closer linked,

that an assistant professor is ‘unemployed’ in the market for

professors ? However, we may note that an individual who sets his wages

too high will become unemployed in any submarket. This causes an

intuition that the selection of submarkets can still be represented by

wage schedules. There will be more equilibrating forces than wages

only, e.g. education or migration, but it can be reasonable to

concentrate on wages.

With heterogeneity, the unemployment that is relevant for a submarket

will have effects on the evolution of the wage in that submarket.

Aggregating, however, we get an effect of macro unemployment on the

average wage. Hence above simple relationship can be retained, but its

interpretation changes from homogeneity to aggregation of heterogeneous

submarkets.

More factors that cause a shift

Above we used _um _to show how the Phillipscurve can shift. Note that

this in fact has only been a didactic procedure. I wanted you to

understand the formulas, and it appeared very instructive to draw

graphs of shifting Phillipscurves. However, when there are _LS

_homogeneous labourers, we have some difficulty explaining why (1 -_

u_) _LS _ could work and _u LS _could not, even though they essentially

are the same. Hence minimum wage unemployment and the shift of the

Phillipscurve due to it, properly belong to the world of heterogeneous

labour.

We here can extend the list of factors that can cause a shift in the

aggregate Phillipscurve:

· The match of demand and supply above the minimum wage may cause

separate problems. We will discuss the issue of _crowding out on the

labour market _below.

· Vacancies will strengthen the position of employees and their unions.

Employers may nevertheless wait with filling vacancies in order to find

better opportunities later.

· There is ‘forward shifting’ of the tax burden _T_[_w_] / _w _from

employees to employers (and then into product prices).

· The Labour Cost Quotes _w _/ _y _ may not just affect the

equilibrating wage (or expectations) but may as well cause a shift.

· Poverty - see below.

We would basically model all submarkets - with minimum wage

unemployment of course only occurring at the bottom. However, let us

first look at the macro level only. Let _us_ be the summary shift

variable inclusive of all factors including _um. _Let _usr _be the

summary shift variable exclusive of _um. _Let _v_ the rate of

vacancies, _TAX/WT_ the tax burden. Let _History_ be the history of all

variables. Then redefine _f_[_u_]:

_us_ = _us_[_u, v, TAX/WT, WT/Y History_] = _um _+ _usr_[_u, v,

TAX/WT, WT/Y, History_]

_f_[_u_] = _fu_[_u - us_] = _ - _Log[_u - us + ___]

Crowding out

A crucial topic is crowding out on the labour market. Highly productive

labour can replace lowly productive labour more easily than conversely,

and this has effect on wage claims. This might be something like a

continuous version of the insider-outsider theory.

Unemployment among the higher skilled is not large. The analysis here

is that this is caused by crowding out on the labour market. When

potentially higher productive people face the choice between

unemployment and a comparatively lower paid job, they choose the latter

(noteably when they are tired of waiting or when the benefit runs out).

They thereby “take the places” of others - who repeat the process to

others below. The initial set-back in pay level tends to translate into

demand for pay rises. Who crowds out, has a stake in trying for pay

rises. A lot of crowding out will cause a mood for inflation. Who have

been crowded out towards unemployment, have some incentive not to

inflate, but have little countervaling power against the general mood

for inflation.

Figure 23 already presented the stylized fact for labour demand and

supply, i.e. that vacancies tend to occur at higher income and

unemployment at lower income. [92]

There is a meaningful aggregation of vacancies and unemployment by

subcategory of low and high productivity workers, giving _Vl, Vh, Ul_

and _Uh_. When vacancies are asymmetrically relevant only for the

higher incomes (_V ~ Vh, Vl ~_ 0), and when there are always vacancies

for higher incomes due to crowding out (_Vh >>_ 0), then _V_ is not

that important. However, _V_ may become important again when _Vl_ is

made nonzero by proper tax policies. If low productivity labour has a

stronger position in the labour market, then the risk of unemployment

is spread more evenly, and trend-setting high productivity labour will

be cautious about wage claims. High values of _Vl_ and _Uh,_ i.e.

vacancies for the low productivity group and unemployment for the

highly productive group, have the largest wage checking effect. High

_Vl_ and _Uh_ make it difficult for the trend setting higher productive

workers to shift the risk of unemployment to the lesser productive

workers. We will not formally develop this point.

Crowding out on the labour market typically refocusses the policy

co-ordination problem to the lower end of the market. This phenomenon

tends to reduce the problem and our vocabulary in these pages to social

subsistence, tax exemption and (legal) minimum wage.

Poverty

A crucial difference between the United States and Europe is that the

US accept more poverty (e.g. by low controls on its minimum wage laws),

while Europe chooses high minimum wages and benefits to raise standards

of living. The shift of the Phillipscurve thus is more obvious and

stronger in Europe than in the US. In the US the working poor still

work, so unemployment is lower, and the shift of the Phillipscurve is

less strong. Sometimes the argument stops here. It remains a topic of

consideration though whether more than just this can be said about

poverty.

Poverty affects productivity directly. A clear case is medical care.

With less medical care, there are longer periods of illness, and more

chances for complications of a less well attended illness. Employers

are less likely to hire less healthy persons.

Poverty affects personal appearance. A shabbily dressed and badly

groomed individual has less chance of employment than a person of

average appearance.

Poverty affects social attitudes. Social seggregation and cultural

differences reduce the chances of employment.

Poverty affects capacities. Rich people need not study much, need not

read many papers, and may only watch soap operas. They are rich, and

can enjoy themselves. But those of the rich who would like to study,

read, watch serious tv programs, and drive out to educational events,

have the means to do so. Those who are not that rich, and those who

have to study to maintain a higher living standard, may work and still

earn enough to enable them to study. Those of the poor section that

might want to do the same, do not have those means.

One aspect of US poverty is crime. Poverty does not actually force

people to crime, as some people demonstrate, but for many it in fact

appears to be very seductive. Jacobs (1996:573), referring to Freeman

(1996:25-42), explains that about 2% of US males is in prison, about

the same rate as long term unemployment in Germany. Taking account of

women, the overall US imprisonment rate is about 1.2%. The highest rate

of European imprisonment is for the UK, with 0.3%. So for the US we

might add 0.9% to the unemployment rate.

Also, additional 5% of US males is on conditional leave etcetera from

the prison system. More have a criminal record. Those points reduce the

chance for employment.

Some of these points, like imprisonment, work directly as a minimum

wage. Some other points rather affect the employment or earnings

distribution, and cause a structural rise of _Ul_.

The submarket Phillipscurves

Here, for simplicity, we take the wage level _w_ instead of wage

inflation. The rates of change can be found by comparing to _w_[-1].

Wage _w, _a continuous vector for each market, depends upon the power

position of employers and employees, which is determined, amongst

others, by the relative situation of unemployment versus vacancies.

Since unemployment and vacancies have been expressed above as functions

of _w _we solve _w _as a fixed point. We also add the equilibrating

_w*_ (or expectations E[_w_]) that are a function of product _y,_ the

tax burden for forward shifting, the labour cost quote, macro variables

and the history of the variables. The submarkets Phillipscurves can

include influences of other submarkets and general developments

pertaining to all markets. A macro-economic hypothesis is that the

development within markets is not merely influenced but even _dominated

by general events_. The relationships are clearly dynamic, and we thus

read all variables as time dependent.

_w_[_y, T, Macro_] = _w_[ _w*_[_y_], _ud_[_w_], _vd_[_w_], _T_[_w_]

/ _w_, _w / y, Macro, History_ ]

Note that modern large models depend upon convergence techniques, and

that the computation of fixed points can be included into convergence

in general (though it would be computationally burdensome).

Shifting back

The stylized facts can be summarized as: [93]

· In the 1950-1970 period, welfare states generally had a _high _tax

exemption level and full employment.

· In the 1970-2005 period welfare states generally had a _low_ tax

exemption level. To ensure a decent stardard of living, required gross

income then rose and exceeded productivity in the low end of the

market, generating unemployment, while shifting the Phillips curve and

reducing its sensitivity.

· Even when the statutory tax system has a low exemption level, then

subsidies for the lowly productive keep them in work. And subsidies can

be at the firm or state level. This is crucial for the Japanese and

Swedish experiences, see e.g. Aoki (1990) and Standing (1990). Note

that, in a _reduced form, _subsidies turn up as ‘system-wide

exemption’. A subsidy is no ‘real’ subsidy if it compensates for wrong

taxes.

Measures to block crowding out boil down to giving the low productivity

group some guarantee for work at decent income. Such guarantees can be

collective/semi-private arrangements of the Swedish/Japanese type. For

the more common mixed economies, the guarantee is market-conforming,

and notably consists of tax exemption.

29. Tax basics

Taxes are relevant for the discussion of stagflation at least for the

following reasons:

(1) Taxes divert income and thus affect aggregate demand, especially

when tax revenues go to benefits and consumption instead of saving and

investments.

(2) Taxes are thought to cause forward shifting, i.e. that taxes are

shifted into wage costs, which then may cause inflation.

(3) Taxes reduce net wages, and might affect the supply of labour.

Statutory marginal rates are thought to have disincentive effects.

(4) If exemption is lower than subsistence, then a higher minimum

wage is required. Differential indexation widens the gap.

In the following we will first discuss the relation of social insurance

premiums to the _economic_ concept of a tax. Then we regard the common

tax structure of OECD countries, where the structure concerns both a

statute and the dynamic adjustment policy. We introduce a nonlinear tax

function and rules on indexation that captures this structure. We then

show the effects of differential indexation, and present our new

analysis on marginal rates.

Tax dynamics can be split into two components: the dynamics of the

short run - where a local temporal equilibrium is attained using the

calculations on the marginals - and the dynamics of the long run -

where the locus of possible equilibrium points is shifted by long run

effects on the levels of the variables. Both components appear to be

equally important for our understanding of the subject. The

observations on the long run can be usefully discussed in conjunction

with the theoretical developments.

Taxes and premiums

In our discussion we will take premiums as part of taxes in so far as

it is economically relevant to do so. This may need some clarification.

Premiums for old age, sickness, disability, unemployment and the like

are often regarded as insurances, and studied separately. In the

practical situation of empirical economies these provisions are often

indeed administered by separate institutions called ‘insurance

companies’. And there indeed exists the possibility to apply the

mathematics and economics of insurance to these topics. However, that

these provisions are called ‘insurance’ should not cause us to regard

them as only such. Part of these so-called insurances are provisions

for the efficiency of the labour market.

To understand this, let us take the case of a low wage labourer.

Suppose that he would have to pay such an amount of premiums, for only

a limited package of insurance, that his net wage would make him

eligible for benefits, or his gross wage would make him unemployed so

that he also gets a benefit. Once he relies on benefits, the mentioned

insurances are provided for him for free.

This thus shows the structural identity of the problem of exemption in

‘insurance’ with the problem of exemption in taxation. Hence, on

economic grounds, insurances here are lumped together with taxes, in so

far as they are provisions for the well functioning of the labour

market.

Note too that governments would be wise to follow a ‘basic insurance

policy’ which holds that workers can be insured up to a basic level but

without payment of premiums. This reminds of the ‘basic income

argument’, but only applies to the mentioned premiums. Similarly poor

people exempt from taxation receive public goods, without paying for

them.

Common structure

Most developed nations have nonproportional taxes, i.e. tax codes with

an exemption at the threshold and then a (rising) statutory marginal

rate. The latter parameters in fact concern the intercept and the slope

of the tax function. There is also a remarkable similarity in the

policy regarding these two parameters (or sets of parameters), see OECD

(1986):

· The policy feature concerning the intercept or exemption.

Exemption generally is low, also with respect to social insurance.

Tax parameters, and notably exemption, are generally indexed on

inflation. Since incomes tend to grow faster than inflation, exemption

lags behind incomes. There is a deliberate tax creep - measured by the

‘macroeconomic progression factor’.

· The policy feature concerning the slope or the statutory marginal

rate.

Both in theory and public discussion there is a consideration that high

marginal rates have disincentive effects. This has resulted in the

policy objective to reduce marginal rates. One way to reduce marginal

rates has been the switch from income tax to VAT.

Given the common notion of budget neutrality, these two features in

policy tend to complement each other. Budget neutrality requires that

the revenue loss due to slope reduction is compensated for by other

proceeds. These other proceeds will often come from the tax creep and

the reduction of exemption. At least, it is often thought that the

reduction of exemption generates additional revenue. This, however,

turns out to be a wrong assumption.

Nonlinear tax function

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_). [94]_ _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. [95] 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_) = (2_H + _ _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_)_ +_ (2_H + _ _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_[0]_ _with _B_[0]_ _subsistence in the base year. Let exemption _x

_be adjusted for inflation with index _P,_ then _x = P x_[0],_ _with

_x_[0]_ _the exemption in the base year that now may differ from

subsistence in the base year _B_[0]_. _Let also _c _be indexed on

inflation as _c _= _P c_[0]. Let the average wage index be _W = P rwi

W_[_0_], with _W_[_0_]_ _the average wage in the base year. Let _h =

x_[0]_ / W_[_0_]_ _and _f = c_[0]_ / 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_[0]].

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_[0]. With real wage index _rwi_,

the nominal index is _wi = P rwi_. For heterogeneous wages with wage

density, we have _w _ = _ wi w_[0] along the balanced growth path.

For a dynamic path we have starting position _B_[0] giving _M_[0]. In

the base year the minimum level is taxed at an average rate less than

_r_, implying that _B_[0] > (1 - _r_) _M_[0].

We also use _J _as the index for the real minimum wage:

_M_ = _P J_ _M_[0] i.e. _J_ = _M_ / (_P_ _M_[0])

(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_[0]

_ B = M - Tax_[_M_, (_r, P x_[0]_, P c_[0])]

= _M_ {1 - _r_ (_M_ - _P x_[0]) / (_M _+ _P c_[0])}

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_[0] = _M_ {1 - _r_ (_M_ - _P x_[0]) / (_M

_+ _P c_[0]) }

_ rsi B_[0] = _J_ _M_[0] {1 - _r _ (_M_[0] - _x_[0]_ / J_) / ( _M_[0]

+ _c_[0]_

/ J_) }

/

/

As _rsi_ and _J_ go to infinity, then _rsi B_[0] ~ _J_ _M_[0] (1 -

_r_). We had _B_[0] > (1 - _r_) _M_[0]. 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_[0] < _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_[0]_ / M_[0]_ / _(_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_[0], then _M_[0] provides the equilibrium of

supply and demand. Let the supply price (or gross income or

productivity) at the minimum be _ms_[0] and let the demand price

(labour costs) at the minimum be _md_[0]. [96] Then in the assumed

start situation of full employment _M_[0] = _ms_[0] = _md_[0]. Assuming

balanced growth for demand and supply gives the development of the

labour market situation at the bottom:

_w = P rwi w_[0] in general, i.e. for all _w _

_md = P rwi md_[0] & _ms = P rwi ms_[0]

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_ = _r_1 _x_ where _r_1 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. [97]

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, [98] 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_

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_[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]_._

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.

[99]

In Book III we derived the real subsistence index _rsi_ for the Bentham

function when _x = P x_[0], so that _B = rsi P B_[0]_._

(13.3d)

Then:

_M = B + Bentham_[_M_]_ __ M = _(_B - r x_)_ / _(1_ - r_)

_M = _(_P rsi B_[0]_ - r P x_[0])_ / _(1_ - r_)

_m = M / P _= (_rsi B_[0]_ - r x_[0])_ / _(1_ - r_) = _m_[_rsi_]

_ATR M _[_rwi_]_= ATR_[_m_[_rsi_[_rwi_]]]

We can develop this a bit further, using _j = x_[0]_

/ B_[0]_:_

/

/

_GNR M _[_rwi_]_= M / B = _(1_ - r x_[0]_ / B_[0] _/ 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_[0]_ =

x_[0]_ = _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. [100]

The more refined analysis for the minimum wage takes account of the

multiplier effect. First of all, if real subsistence doubles from

_B_[0] = 1 to _B_[35] =_ 2 B_[0]_,_ 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_[0] 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). [101] 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. [102]

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.

[103]

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

1

0.9579

24.93

15.38

Without _l_

18.16

32.37

1

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

15

75

10

15

75

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).