Book VI. Structural Models

Structural models

Chapter 23 gives a textbook macro-economic model so that we better appreciate the point of reference of ‘existing economics’. Chapter 24 clarifies heterogeneity and nonlinear taxation. There is nothing new here yet either. The subsequent chapters then take up the same subject matter, and gradually add elements and interpretations that support the novel analysis.

23. A textbook macro-economic model

Our textbook model is a very simple and unpretentious first year undergraduate model. It is not interesting for itself, but for our later discussion.

The IS-LM model

We follow Dornbusch & Fischer (1994), chapters 1 - 4. The basic macro-economic identity for annual real values is:

C + G + I + NX YR YD + (RTAX - TRF) C + S + (RTAX - TRF)

C = consumption

G = government consumption

I = investment (incl. unintended stocks)

NX = exports minus imports

YR = real gross domestic product

YD = YR - RTAX + TRF = C + S = disposable income

TRF = government transfer payments

RTAX = real tax revenue

DEF = G + TRF - RTAX = S - I - NX = government deficit

S = saving

We take G, TRF and NX as exogenous and known. We are now only interested in expectational equilibrium. Aggregate demand is YR* = C* + G + I* + NX. With the rate of interest i and the marginal tax rate r, behavioural relations are:

C* = TRF + c (YD* - TRF) + C0

I* = I0 - b i*

RTAX* = r YR*

In equilibrium C = C* gives YR* = YR - since C = C* iff YD = YD* iff I* = S* = I = S. This can be represented by the IS curve:

YR = TRF + c (YD* - TRF) + C0 + G + I0 - b i + NX

i = (C0 + G + I0 + NX + TRF - (1 - (1 - r) c) YR ) / b (IS)

For the money and bond market:

L + DB WN / P MX / P + SB

L = demand for real balances

DB = demand for real bond holdings

SB = real value of the supply of bonds

WN = nominal financial wealth

P = price level

MX = money stock (M1, M2 or M3)

Liquidity demand is:

L = k (1 + h / (i - imin)) YR

Equilibrium on the money market L = MX / P gives the LM curve:

__ (LM)

Intersection of the IS and LM curves gives equilibrium for YR and i, and from these the other variables can be solved, in particular the price level P = MX / L[YR, i].

Note that we also use:

Y = P YR

While the IS-LM model already tells us something about inflation - via the quantity of money - there is also the labour market where wages drive up costs and prices. The IS-LM sectors of the economy and the labour market are linked via Value Added Y.

The production function

For our purposes we can use a Cobb-Douglas function with employment LE and capital KE:

YR = Y0 LE a KE 1 - a

Y P YR = W LE + i PK KE,

We assume that firms maximise profits - and since we assume constant returns to scale, there is no surplus. If firms accept wage W, then the marginal productivity of labour equals the real wage W / P, and then this determines LE which must be at most labour supply LS. Unemployment then follows as u = 1 - LE / LS. If companies also accept the rental price of capital, then the marginal productivity of capital must equal i PK / P, and this determines the employed real capital stock KE, which must be at most total stock KS.

The additional equations from these marginal conditions are (and we assume expectational equilibrium on these too):

LE = Y / W

KE = (1 - ) Y / (i PK)

With YR, P and i given from above, there is one degree of freedom from either PK or W. It is customary to close the model with a relationship that sets the average wage W.

YR = real income

LE = employment

KE = employed real capital stock

KS = total real capital stock

LS = labour supply

u = rate of unemployment

W = average wage

WT = W LE = total wage sum

In a full model, the price of capital must relate to investments I and to wealth WN. Also, apart from a theory on unemployment, we also need a theory on idle capital KS - KE. We could also include intermediate goods, as these appeared to have been important in the Oil Crises. These alternatives however lead too far for our purposes.

Important for our purposes however is inflation. We already indicated that the price level P is relevant for inflation. The crucial thing to note is that inflation is the relative change of the price level, so that it is a dynamic concept.

Dynamics versus statics

Let p be an arbitrary price.

Statics assumes a timeless dimension. With supply S[p] and demand D[p], equilibrium (in expectations) is given by S[p] = D[p] and it solves for the equilibrating price p·.

Dynamics concerns developments in time. The price movement p’ = dp/dt is related to excess demand D[p] - S[p], so that p’ = dp/dt = f[D[p] - S[p]]. The solution of this differential equation gives the movement towards equilibrium. Dynamics causes different concepts of equilibrium: depending upon the specification of variables and function, the equilibrium can be market clearing (p°) or the fulfillment of expectations (p*). Economic agents generally have different speeds of reaction when expectations are not fulfilled. When there are surprises, there can be a ‘trade-off’ between prices and quantities.

Phillipscurve

For the labour market, dynamics implies a relationship between unemployment and the change in wages. This relationship is called the (wage-) Phillipscurve. Sometimes there is an additional assumption of a strong relationship between wages and product prices, and then the (price-) Phillipscurve gives the relationship between unemployment and prices.

The existence of a Phillipscurve thus follows essentially from the concept of dynamics itself. For the labour market, the price is the wage w and excess demand is represented by unemployment u (thus negative excess demand; with vacancies neglected partly because of unreliable measurement), so that w’ = f[u]. Much debate in macro-economics about whether the Phillipscurve ‘exists’ or not, could have been cut short by noting that it is a standard market adjustment equation. The true debate is about the proper form and stability of its parameters.

In the simplest model we choose inflation, and have, with u = 1 - LE /LS:

dLog[P] = f[u]

and this would add another restriction that closes the model. For example:

dLog[P] = dLog[P]* - 0.1 Log[ u / u* ]

would give an expectations augmented form, and when u = u* then expectations will be fulfilled, and LE = LS (1 - u*).

It is useful to note that above model does not yet contain an explicit reaction function of the monetary authorities with regarding to inflation. Money can be fixed or chosen to grow at a predetermined rate. In practice there will be a flexible reaction, and then part of the ‘Phillipscurve regression between dLog[P] and u’ will reflect that reaction function.

Macro-economic interactions

The textbook relations are simple in themselves, but the interactions already can be rather complicated. Figure 18 presents some common macro-economic interactions.

Figure 18: Some macro-economic interactions

The influence of income in that figure is stated in terms of growth dLog[YR], and the influence of prices is stated in terms of inflation dLog[P]. Positive transmissions are in black and explained in Table 5, negative transmissions are dashed in red and explained in Table 6.

Table 5: Positive impulses

Positive

Cause

Prime effect

Then

Then again

YR P

growth

increases demand

adds to inflation

u DEF

more unemployment

less income, less tax revenue

more expenditure on benefits

higher deficit

P i

more inflation

the Central Bank (CB) raises interest rates to fight it

possibly, though, inflation means more profits and a reduced demand on loans

and thus a lower rate of interest: but then the CB will maintain the level of interest

i DEF

higher interest rates

the government has a higher interest bill

higher deficit

DEF i

a higher deficit

more demand for loans, more supply of bonds

thus a higher rate of interest

DEF YR

a higher deficit

sustained expenditure

and thus sustained growth (at least by that channel)

Table 6: Negative impulses

Negative

Cause

Prime effect

Then

u P

more unemployment

lower wage demands

and thus less inflation

P DEF

more inflation

more tax revenue

and thus a lower deficit

i YR

a higher rate of interest

makes investments more costly

and thus lower growth

YR u

more growth

more demand for labour

lower unemployment

24. Heterogeneity and nonlinear taxation

Heterogeneity versus homogeneity

Homogeneity assumes that S[p], D[p] and p are real variables, while heterogeneity assumes vectors or densities. This book takes the density approach. In fact, employment e[w] = Min[s[w], d[w]] also provides the earnings or income distribution, i.e. the function that gives the number of people earning a level of income w, for labour supply s[w] and labour demand d[w].

Nonlinear versus proportional taxation

The proportional tax is r Y. A linear but non-proportional tax is Bentham[w, x] = r (w - x), though proportionality comes back again by assuming x = 0. A nonlinear tax adds curvature (see chapter 29), and then interacts with heterogeneous labour.

Some literature

The following references put the argument into perspective.

In his presentation of the IS-LM model, John Hicks (1937) could disregard differences in labour as being of secondary complication. For our purposes, however, the case of heterogeneous labour causes a crucial difference. Policy co-ordination then involves three distributions:

1. the gross income distribution that corresponds to the productivity distribution,

2. the net income distribution aspired by the policy maker (‘society’),

3. the actual net income distribution, resulting from taxes imposed (including e.g. the social security ‘insurance’ payroll tax) and from expenditure.

There is early recognition in the literature of the need for heterogeneous labour in discussing dynamics. For example, 20 years ago, Solow (1976:152), occasionally but not consistently using the more accurate term ‘surface’:

“George Perry, who was one of the earliest quantifiers of the Phillips surface, has recently produced an alternative explanation of great interest [reference]. Perry’s basic insight is that the aggregate unemployment rate may be an ambiguous measure of pressure in the labor market when the composition of the labor force and of the group of unemployed is changing. (...) In other words, the Phillips curve would have shifted upward. (...) Perry quantifies this observation by making the plausible assumption that an unemployed body generates downward pressure on the wage level proportional to the amount of “unemployed labor” he or she represents. In turn, the amount of unemployed labor can be measured by the number of dollars of wages it represents.”

No economist working in the field and worth his salt will have neglected Solow’s paper. Issues of the substitutionability of one kind of labour for another, and of dispersion measures for the differences in responses, can found even earlier in the literature.

Van Praag & Halberstadt (1980) present a continuous productivity distribution.

Bruno & Sachs (1985) give a standard reference for stagflation. Their formal analysis uses homogeneous labour and proportional taxes, though some of their statements allow for an interpretation of heterogeneity and nonproportionality.

The need for modelling heterogeneous labour and nonproportional taxation is clearly recognized in the literature, see e.g. Beenstock et al. (1987) and Minford & Ashton (1993). Layard, Nickell & Jackman (1991), another standard, allow for heterogeneous labour, yet tend towards proportionality in taxation.

In addition, these references use dynamics but do not explicitly discuss the consequences of changes in tax parameters. Auerbach & Kotlikoff (1987) give a wealth of information on fiscal dynamics but do not specifically tackle stagflation.

Other references which put the Phillipscurve in perspective are Okun (1981), Blanchard & Fischer (1989), Friedman (1991), The Economist (1994) and Phelps (1994). Extensive theoretical and empirical work has been done by the Central Planning Bureau (1992a&b), Gelauff (1992) and Colignatus (1992b).

25. Summary of current views

It is useful to recognise some current views on the labour market and the influence of taxes. This allows us to better see the impact of our new analysis.

A simple view

There exists a simple popular view that makes two errors:

· it is static and not dynamic

· it assumes homogeneity and not heterogeneity.

This model is the comparative statics model with homogeneous supply and demand for labour. Borjas (1996:159), Mankiw (1998:125) and The Economist of February 26 1994 present that model. As a model it of course is consistent and it can help us to get our thoughts started, but as a representation of real markets it is erroneous.

Figure 19 gives the wage W on the vertical axis and supply and demand quantities on the horizontal axis. (Note the causal order.) It must be mentioned that marginal tax rates have played a role in the deduction of the supply and demand curves.

In this Marshallian model, the original equilibrium is attained at the intersection of the LS and LD curves, at wage W° and employment LE°. An income tax causes workers to demand a higher wage, and supply shifts up, to LS1. Premiums that raise wage costs for employers cause these employers to offer a lower direct wage, and demand shifts down, to LD1. The new equilibrium of LS1 and LD1 is LE < LE° where employers pay direct wage W1 > W° and where workers receive net W2 < W°.

For this model, with supply and demand schedules derived with marginal analysis of utility and profits, there is an important role for statutory marginal tax rates. First best here are lump sum taxes and zero marginal rates.

Figure 19: Statics Marshallian model for the influence of the tax wedge

There are clear objections to this model:

· It is comparative statics, with homogeneous and flexible labour.

· It concerns any kind of tax, while some taxes are socially desired and generate employment. The model doesn’t distinguish between optimal and suboptimal taxes.

· Empirical research shows that labour supply elasticities are low. Elasticities are higher for partners, but that is less relevant here. People are very much in the position that they have to work for a living, and taxes generally pose no restraint on the availability for the labour market. This means that LS ~ LS1 ~ vertical. (Borjas (1996) shows this graph too.)

· The model does not really allow for unemployment. We might define U = LE° -LE, but LE° is an unobserved variable. Firms and workers react to observed variables, and in those terms there is full employment. Even if labour would be inflexible in this model, then there still would be no involuntary idleness at the net wage earned.

The use of this model thus is limited. Mankiw (1996) correctly presents the model as a ‘tax incidence’ model, and we should be hesitant of other conclusions.

The Simple View however regards this model as a real description of real labour markets, and it thus makes the category mistake of using arguments concerning the income distribution for issues of growth and employment.

The reader is advised to read again Chapter 2 of Keynes’s 1936 General Theory. The General Theory is in my perception an effort to seriously develop dynamics. Keynes’s precursors did discuss dynamic developments, but always ended up in static modelling. See also Patinkin (1976:140 footnote 4).

In the following quote, Keynes discusses a real wage reduction caused by prices. For our purposes, we might substitute a real wage reduction caused by taxes.

“To sum up: there are two objections to the second postulate of the classical theory. The first relates to the actual behaviour of labour. A fall in real wages due to a rise in prices, with money-wages unaltered, does not, as a rule, cause the supply of available labour on offer at the current wage to fall below the amount acually employed prior to the rise of prices. To suppose that it does is to suppose that all those who are now unemployed though willing to work at the current wage will withdraw the offer of their labour in the event of a small rise in the cost of living. Yet this strange supposition apparently underlies Professor Pigou’s Theory of Unemployment [voetnoot] and it is what all members of the orthodox school are tacitly assuming.” (Keynes (1936:12-13)).

Note, by the way, that the format of Figure 19 can always be used in terms of the average wage W. So the format of Figure 19 may be inviting to our intuition, in that we think that we indeed can draw a diagram like that, but we then should be aware that our true model is heterogeneous labour and not homogeneous labour.

A complex view

An alternative view is more empirical, thus inherently more dynamic, and builds on Keynes’ observation. Empirical research, see e.g. Ashenfelter & Layard (1986), Theeuwes (1988), Hum & Simpson (1991) and Gelauff (1992) shows that marginal tax rates have ‘surprisingly’ low elasticities. The reason for a lesser importance of marginal rates is that labour supply is not flexible, but rather fixed. That labour supply is primairily given by demographic factors, is for example a well known assumption of practical models developed at the Dutch Central Planning Bureau. In Western economies people will have to become active on the labour market in order to earn a living, and taxes hardly form a barrier. People are still very much like Marx’s proletariat, and they have little else to fall back on but to supply their labour. There is some choice for partners and for people on benefits, but this does not have a major impact. For the majority, if anything, the average wedge is more important than the marginal one, see Den Broeder (1989). Recently Minford & Ashton (1993) see scope for a larger effect of marginal rates, but, their study is still far from explaining stagflation, partly for the reason that it is not fully dynamic.

By consequence, the major equilibrating forces exert themselves on the wage and the related employment. Here arises the dynamic situation of (wage) inflation and unemployment, and thus the issue of the Phillipscurve. Thus, conceptually, tax rates have their major impact not on labour supply but on the Phillipscurve.

The next question then is whether their effects are positive or negative. The common argument is that a higher marginal rate fuels inflation. Whether this is the case then becomes the next issue.

Efficiency wages intermezzo

Before we can continue the discussion, a note on the ‘efficiency wage theory’ is required. The idea is here that, though people are forced to work to earn a living, they still can choose whether they shirk or not. They take account of a probability of getting caught and getting fired, but supervision would be expensive, and, if fired, one eventually could find another job. Unemployment then is required to discipline the workers. Borjas (1996:459) provides an introductory discussion, and the graphs are quite similar to the supply and demand schedules of old.

I tend to regard this approach as an example of academic excess. This may be an error on my side, but let us look at some of the arguments: 10% of the European labour force is unemployed, hence Europeans apparently shirk a lot ! And employers are so dumb that they cannot think of cheap ways to determine productivity, like setting standards and such. Agreed, shirking is undoubtedly a phenomenon, and eventually the superior economic model will include a subtle relationship between wage, effort and productivity to determine the last digits, but all this is less relevant for the Great Stagflation and the need for an Economic Supreme Court.

A more sophisticated view

Graafland (1990) introduced another approach at the Dutch Central Planning Bureau, and he refers here to Hersoug (1984). The Phillipscurve here is derived using a model of wage bargaining between unions of employers and employees. The approach is adopted by Gelauff (1992) on the CPB model MIMIC, Gelauff & Graafland (1994). It recently is refined by Graafland and De Mooij (1998), Bovenberg, Graafland and De Mooij (1998), Jongen and Graafland (1998), Graafland & Huizinga (1999), Graafland and Nibbelink (1999), Oers, De Mooij, Graafland and Boone (1999), and De Mooij (1999). In this approach, a higher statutory marginal rate actually increases employment, instead of reducing it as the Simple View and many standard Phillipscurves would hold. The mechanism is as follows:

· A higher marginal rate (under constant average) penalizes wage demands, lowers such demands, reduces (wage) inflation and thus increases employment.

· A higher average rate (under constant marginal) causes compensating wage demands at the margin, and reduces employment.

These properties actually are well known, as they are consistent with analyses concerning a Tax-based Incomes Policy (TIP). For example the Congressional Budget Office (1977:119):

“In recent years there have been proposals to use tax incentives and other schemes to encourage more moderate price behavior. (...) Rather than overriding market forces, these newer proposals attempt to take advantage of market incentives by making moderate price and wage increases a matter of self-interest for firms and employees. The best known of these proposals involves tax incentives to reward or penalize wage decisions that deviate from some established standard.”

This view however still does not take account of the dynamic marginal rate. There are also the issues of labour heterogeneity and optimal taxation that we have encountered in discussing the Simple View, but that have not had sufficient attention. These issues will be discussed below.

Confusions

Given more than one view, there is scope for confusion. This has in fact occurred.

· The OECD policies referred to above, directed at lowering statutory marginal rates, have been advocated using the rhetoric of the Simple View even though economic advisers often are aware of the Complex View.

· If one would really think that high marginal rates reduce work effort and supply, then a situation of high unemployment would call for higher rates - that would reduce unemployment. Policy however has been to reduce rates.

Secondly, when these views are confronted with the effects of the policy of rate reduction, there again is ample scope for confusion.

When unemployment has been reduced, then this is being seen as corroboration of the Simple View. For example the data on the US now show the combination of a reduction of taxes on higher incomes and some reduction of unemployment, and it will now be difficult for policy makers to accept other lines of arguments. Actually, in so far as there has been some success in practice, it is because the policies have also lowered average rates. Higher budget deficits have been relied on to pay for additional benefits and average rate reductions for higher incomes. The reduction of marginal rates actually had a negative impact.

In most cases unemployment has remained high. In this case one should expect that policy makers would reconsider their views. They don’t seem to do this, and rather look at the few cases where there seems to have been success along the expected pattern.

A specific example is the Dutch 1990 tax reform (known as “Oort reform” ). This reform was supported by computations using the MIMIC model, see Gelauff (1992). The reform reduced both marginal rates and exemption. The reduction of statutory marginal rates reduced Phillips curve sensitivities, and induced larger wage claims and lower employment. The reform however also included a reduction of average taxes, and this caused employment to rise on balance. We may restate the situation in more mundane terms: the reduction of average taxes was sold on the political market as a reduction of marginal rates. Politicians had their eyes fixed on the reduction of marginal rates and the reduction of unemployment, and they got what they wanted to see, without realising that the mechanism in MIMIC was entirely different, and that proper exploitation of this mechanism would lead to even lower unemployment.

26. Heterogeneous labour

We will first discuss heterogeneous labour supply, and forward a hypothesis on its distribution. Note that supply is difficult to observe, since generally we only observe actual employment, which is the minimum of supply and demand. However, data on actual earnings do allow the encouraging conclusion that the earnings distribution can be approximated by a lognormal distribution. For an indication we look at Dutch data on the distribution of income in 1950 and 1988. We complete this chapter by a more thorough sets of definitions for earnings, cost and income accounting, and we construct integrals that are relevant for the minimum wage.

Dromedary supply

Let us first regard labour supply.

At a Dutch economists “Masterclass” session in Fall 1991, Orley Ashenfelter explained that labour supply was unresolved and actually some kind of a researcher’s nightmare. In a break I put my suggestion on the blackboard, and my ‘quiggly’ line (see below) at least drew the compliment of an amused smile. I almost put this suggestion into Colignatus (1994a), but backed away from that since it was not essential for that paper (and I used only the normal right hand side of the supply graph). However, to my surprise and pleasure I saw that same quiggly line in De Groot & Keuzenkamp (1995) who discuss results of Quah (1993).

De Groot & Keuzenkamp have another subject than labour supply. Their problem is whether international economic growth results into convergence, as Adam Smith’s “The Wealth of Nations” seems to imply. De Groot & Keuzenkamp refer to the results of Quah (1993) who has compiled the distribution of output per labourer per country, which turns out to be that quiggly line.

To understand the point, let me first explain my reasoning on labour supply. At low productivity, one has to work 24 hours around the clock in order to survive. For example, if subsistence is at B and productivity is y, then the hours are B / y. Hours thus quickly rise when y drops (the working poor). When productivity increases, one quickly starts working less hours, particularly since the kind of work at that level often concerns hard labour. At higher levels of productivity again, the kind of work is less exacting and pay is better, and one may work longer hours again. However, at the highest levels of productivity, labour again becomes a relative disutility. In summary, when plotted in a graph, the figure looks like a dromedary, starting high at the left, having a dip in the neck, then the bump, and sliding away towards the tail.

If labour supply is like this, then it likely affects the productivity distribution across nations. While every individual has his or her own parameters, aggregation may average things out, and as a result one nation then may stand for a certain income group. Thus Quah’s finding is consistent with my intuition and indirectly confirms it.

Figure 20 plots the quiggly line, for imaginary income y in thousands of dollars and subsequent working hours per week, for both long and short ranges of income so that the curvature can better be appreciated.

Figure 20: Supply in hours per week, depending upon income

Note: These are not observations, just give an hypothesis on shape

I’m still working on a correct form of the complete utility function. Barro & Sala-i-Martin (1995) give a recent discussion of the trade-off of work and leisure in the context of growth, and that might be a fruitful framework. However, for the present purposes, our development may stop here.

Dutch income distribution data

The literature on the distribution of income has resulted into a general impression that this distribution can be approximated by a lognormal distribution, see e.g. Pen & Tinbergen (1977). For the purposes of our exposition it is useful to test this impression. Also, since we will discuss long periods of indexation, notably from 1950 till 2002, it is also useful to look at the distribution in 1950 and a recent one. We then take the distribution data in the appendices for Holland 1950 and 1988.

Figure 21 and Figure 22 plot the resuls of a (rough) estimation. It appears that we get the best fit when we transform the data into logarithms (and recompute the frequency densities - i.e. the transformation required to deal with different class sizes). The logarithmic data are approximately normal, as can be seen in the plot of log[income] versus its frequency density. We can transform the estimated distribution for a plot in the income-frequency format.

Figure 21: Dutch income distribution 1950

Figure 22: Dutch income distribution 1988

In the 1988 plot, the estimation has been done with the 1988 ‘parttimers’ dropped, but they are included again in the income-frequency plot so that we can better appreciate that their inclusion would confuse a discussion on fulltimers. But it is nice to see the dromedary shape returning.

We conclude that income can indeed be approximated as a lognormal distribution, and throughout time; at least as a stylized fact that we can use for propositions and illustrations.

Definitions and formulas

There are some useful definitions and formulas for heterogeneous labour markets. These hold for any distribution, not just the lognormal distribution. Let y and w be micro values that have a certain density. First of all, there are the following accounting definitions, for annual and nominal values:

· = the profit rate, expressed as a markup on labour costs

· y = labour costs + profit = w (1 + ) = product revenue = productivity

· labour cost quote = LCQ = w / y = 1 / (1 + )

· labour costs = w = (direct) wage + nonwage (but labour related) costs

· w = net labour income + (direct + indirect) taxes + premia + other nonwage costs

· tax = T[w] = (direct + indirect) taxes + premia

· gross labour income = labour costs - other nonwage costs = net labour income + tax

· Neglecting the “other nonwage costs” gives w = labour costs = gross labour income. (Thus the w are labour earnings only if the other nonwage costs are zero.)

Observed labour costs have a density fw[w]. Since the product is y = w (1 + ), equalisation of profit rates with respect to labour would give the labour cost density fw[w] as a shift of the productivity density fy[y]. Normally, though, the profit rates are equalised in terms of capital, which for example causes different Labour Cost Quotes (LCQ) per sector of industry, and then the relation between fw[w] and fy[y] is a more complicated affair.

The proper labour supply density sp[.] depends on net labour income (w - T[w]). But supply can, with the neglect of “other wage costs”, be regarded as a function of labour cost w, as:

s[w] = sp[w - T[w]]

Labour demand is a density d[w]. Total supply follows from the integral:

The employment density is the minimum of supply and demand, and equals the observed labour cost density:

e[w] = Min[s[w], d[w]] = fw[w]

For total employment we take account of a minimum wage M.

For the discussion below it is also useful to compute aggregate labour costs and its (nominal) tax revenue:

Important are the average wage W = WT / LE and the average tax rate ATXR = TAX / WT (when we can neglect other nonwage costs).

Densities for unemployment ud and vacancies vd follow from the difference between supply and demand and actual employment:

ud[w] = s[w] - e[w] & vd[w] = d[w] - e[w]

The aggregate unemployment and vacancy are U and V, and their rates are:

u = (LS - LE) / LS = u[M] & v = (LD - LE) / LS = v[M]

Figure 23 gives the stylized fact that vacancies tend to occur at higher income brackets and unemployment at lower ones. The figure is quite stylized, since it is a difficult issue to construct plausible s[w] and d[w].

Figure 23: Supply and demand of labour

If labour supply LS was homogeneous, we would have difficulty explaining that u LS would be unemployed, since these persons are similar by assumption. Basically then u is a probability.

For heterogeneous labour we could use characteristics and a mechanism that explains why some are employed and others not. This mechanism could be related to the shift of the densities over time due to aggregate demand, inflation, technology, job changes and the like. In fact, we would use such methods to determine ud[w] and vd[w] in practice - and perhaps we would not start with w as the defining characteristic, but start with other characteristics and work towards the wage. However, we will not look into this deeply. We will use heterogeneity mainly to explain the effect of the minimum wage. For a level of income above the minimum wage we again assume some probability, quite analoguous to the homogeneous case. Basically, an agent has offers for various kinds of jobs and incomes, and associated probabilities (and one for unemployment). The s[w] and d[w] thus have a stochastic base.

Minimum wage unemployment differs from the ‘normal’ unemployment above the minimum. Thus:

u = um + un

Only part of um can be gainfully employed when the minimum wage would be abolished.

Only un will exert a meaningful pressure on wages. A major dynamic process is that um rises over time, contributing to the phenomenon of hysteresis. Labour market processes and wage settlements might stay stable in terms of un, i.e. the “normal” unemployment rate, but they shift in terms of u, the overall unemployment rate.

One may wonder why M is nonzero, when its abolition would create employment ume. The apparent reason for governments is that labour markets are not fully competitive and require some regulation. This issue is taken up again in the next chapter on subsistence.

These integrals don’t say how large the densities are. An indication of how much M ‘bites’ is difficult to find. An approach is the following. Let us define ms such that (for example) 1% of supply has an earning power of less than ms. Similarly, md for demand. Then Table 7 distinguishes six situations.

Table 7: Combinations of ms, md and M

ms < md

md < ms

Minimum wage irrelevant (M < md)

M < ms < md

M < md < ms

Minimum wage irrelevant (M < md)

ms < M < md

See point (b) below.

md < M < ms

See points (a) and (b) below.

ms < md < M

md < ms < M

There are some notable effects:

(a) On the supply side, if ms < M, then would-be earners of ms < w < M become eligible for benefits. When they accept these benefits voluntarily or from social pressure, they, in a sense, form no real supply. Yet they are supply, otherwise they would not be eligible for a benefit.

(b) On the demand side, if md < M, then there would be a real demand for md < w < M if government would reduce M. But this demand is not relevant when M exists.

A crucial point to see is that, as we here are concerned with productivity, that we can use subsidies to manipulate the densities, for example by subsidising a particular industry or profession. Doing this of course causes an accounting problem: does the w on the horizontal axis measure productivity before or after such subsidy ? The most practical approach is to use w inclusive of subsidies - because market measurements are always inclusive. Subsidising firms would allow them to hire at higher wages: this would shift d to the right. Subsidising workers would allow them to work for lower wages: this would shift s to the left. What happens to employment is not a priori obvious.

It turns out that the minimum wage is important in practice. Our analysis will strongly rely on minimum wage unemployment. In this we differ a bit from the original position taken by Keynes. As Tobin (1972: 122) states:

“But why is the money wage so stubborn if more labor is willingly available at the same or lower real wage ? Consider first some answers that Keynes did not give. He did not appeal to trade union monopolies or minimum wage laws. He was anxious, perhaps over-anxious, to meet his putative classical opponents on their home field, the competitive economy.”

In my view, Keynes’s argument (as further explained by Tobin) is to the point, and aggregate demand, sticky wages and the co-ordination failures on these are established concepts in macro-economics. However, the record of the Great Stagflation is very much influenced by the minimum wage problem, and thus it is that kind of analysis that merits our attention here.

Amendment to the textbook model on the Phillipscurve

With respect to the textbook macro-economic model in chapter 23, we can introduce a minimum wage component in unemployment uM that can rise gradually over the long run due to taxation. With u = uM + uR (R from ‘remainder’) a possible Phillipscurve with less dampening effect of uR is:

dLog[P] = dLog[P]* - _ Log[ (uM + uR) / u*_ ]

Alternatively, the two submarkets have their own curves. In both cases, it must be determined how the two submarkets develop and how they interact. The most obvious hypothesis is that high productivity labour sets the trend for the development of wages. When minimum wage unemployment rises stronger than general unemployment, then the higher educated have more scope for wage demands, and then there is an upward effect on wages and prices, even stronger so when price expectations come into play. This would show an unfavourable (upward or rightward) shift of the (aggregate) Phillipscurve.

27. Subsistence

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.

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. 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. 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. 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 / (2B - 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