Above we noted that the structural form of western welfare states is quite complicated. We would like to have a more enduring result than awareness of complexity, and therefor we adopt the Definition & Reality methodology. As said, a proposition - as a statement on reality - can be regarded as a mathematical theorem about/within a model of stylized facts. When there is a tautology, we attain truth by definition. So we now (a) restate what we consider to be the stylized facts, (b) define our concepts, (c) develop theorems and proofs, (d) link back to conclusions about reality.

The reduced form that is most relevant concerns the (long run) comparative statics of the regimes of full employment (1950-1970; Japan/Sweden) and unemployment (1970-2005).

This kind of comparative statics should not induce us to think that we abolish dynamics, though.  Stagflation has both a dynamic (inflation) and a static or stationary (unemployment) aspect. When we skip proper dynamics and discuss regime switches in which unemployment features as an important switch variable, then Phillipscurve processes are included in the switching process, even though they don’t feature explicitly in the reduced form.

To attain the necessary level of generality, we use a reduced form where the economy is mapped into a model with three types of agents. One type is the net receiver; and two types are net tax payers. Since the latter two points give a line, that single line represents the state of the economy. The regime switch depends upon the choice of tax parameters.

There are regimes of full employment (1950-1970; Japan/Sweden) and unemployment (1970-2005).

In the welfare state, it is more efficient to have full employment. Unemployment causes lower income - not only directly as in old-fashioned capitalism but also, more noteworthy, by the additional benefit burden. Unemployment can have an adverse effect on inflation when it causes a shift of the Phillipscurve.

It turns out that the propositions that are most interesting, from the viewpoint of political economy, do not require continuity, and can be formulated by assuming dichotomous High and Low productivity labour, combined with one class of Benefit recipients. This assumption allows for a reduced form formulation that allows for generality. For expository reasons we can take social subsistence and productivities as purely constant. In the simple mathematical model the dichotomy gives fixed numbers, in actual observation they are subgroup averages which depend upon general equilibrium processes. The benefit level is rather not an average but a threshold, like the surface of the sea at Scheveningen beach. The words Benefit, High, and Low give letters BHL, and this abbreviation may be pronounced - converged upon after many walks - as ‘beachly’.

It is a stylized fact that welfare states are BHL. Checking this requires next definitions.

Theorem BHL.1:            For a BHL economy, both full employment and unemployment are possible.

Proof:

The structure of this proof is, that we determine the accounting equations, find the reduced form tax relations that are implicit in these, and then deduce the critical tax parameters that determine the regime switch.

Looking at the BHL concept, the only possibility for variation is in category (iii-b). The recipients in that class all move together, and thus there are only two regimes (in or out of  benefit dependency). Given that gross productivity has been fixed, the only possible variation concerns net income. We assign the term “tax regime” to the possible states in net income. We find, in other words, that these regimes are implicit in the BHL concept. Let t be the index for tax regime 0 (unemployment) or 1 (full employment).

Given BHL-ness, we thus have: t is 0 or 1, and:

            b permanent benefit recipients;

            h persons with gross productivity H and net N(t);

            l persons with gross productivity L << H, and net K(t).

The regimes are characterized by net income conditions K(0) < B and K(1) B:

(0)        In regime 0, K(0) < B and l are eligible for benefit B, and they don’t work.

(1)        In regime 1, K(1) B and l don’t get benefit B, and they work and earn L.

On benefit, the welfare rule is strict on not-working, while by assumption the black economy can be neglected. Off benefit, the l have no other means of support and thus work, and earn gross L. Since net income cannot be larger, L K(1) B.

In the following equations, personal income y takes values H and L. Relation (1-t) below gives the implied tax system, where the personal tax T(y, t) depends upon personal income y and the tax regime t:

                        T(H, t)   H - N(t) ;      T(L, t) L - K(t)            (1-t)

Two points share a line. Hence, the tax system can be represented by a straight line, with an intercept and a marginal tariff. These implied ‘parameters’ (actually: reduced form variables) are defined in (2-t), with 2 pairs of 2 equations & 2 unknowns, giving tax exemption X(t) and marginal rate R(t). The line is the reduced form representation, while the statutory system which guides people’s actions could be anything. Each regime gives a set of reduced form lines; our interest concerns the boundary line.

                        R(t) (y - X(t)) T(y, t)                                      (2-t)

Relation (3-t) defines national income Y(t), where the personal incomes are multiplied by the numbers of persons involved. Revenues h H + b 0 = h H are regime independent. Depending upon the regime the l bring in L or not.

                        Y(t)   h H + t l L + b 0                                     (3-t)

Relation (4-t) states the condition of a balanced budget. National income equals the sum of net incomes after redistribution. The condition may be called “Walras’ Law”.

                        Y(0) = h H = h N(0) + (l + b) B                        (4-0)

or           h T(H, 0) = (l + b) B

                        Y(1) = h H + l L = h N(1) + l K(1) + b B           (4-1)

or         h T(H, 1) + l T(L, 1) = b B

The budget condition implies that the tax ‘parameters’ are functions of each other. Per regime, a higher exemption means a higher marginal tariff, and vice versa. The regime switch itself might, but need not, be the exception. Given that marginal rates R are generally regarded as policy variables, we solve for X.  With X(1) L:

            (4-0)     h R(0) (H - X(0))  =   (l + b) B      

                        X(0) = H - (l + b) B / (h R(0))                           (5-0)

            (4-1)     h R(1) (H - X(1)) + l R(1) (L - X(1))  = b B        

                        X(1) = (h H + l L - b B / R(1)) / (h + l)              (5-1)

There is a set of critical levels of gross income M(t) = M(R(t), t), such that unemployment results iff earnings L are less than M(t). This follows directly from rule (iii-b). This critical income solves from:

                        M(t) - T(M(t), t)   B

                        M(t) = M(R(t), t) = (B - R(t) X(t)) / (1 - R(t))                            (6-t)

Under unemployment, the benefits cause additional taxes l.B which are levied on a smaller tax base. Given that l are unemployed anyway, the tax exemption X(0) can be lowered, so that the marginal rate is as low as possible. This has the effect that M(0) shifts to the right, so that the gap between the possible wage L and the wage ‘required for a decent living’ widens. There is obviously hysteresis, of a ‘catastrophic’ kind. Conversely, M(1) can range in B M(1)   L and allow for larger R(1) though this could have little effect since also X(1) rises (see below). While these properties apply to the reduced form, the same mechanisms apparently apply to the structural form too (as they concern the same reality).

Substituting (5-t) in (6-t) gives M(t) as an explicit function of R(t). The regime switch occurs at M(1) = M(RS, 1) = L with switch marginal rate RS and implied exemption XS:

                                        bB - (h + l) (L - B)
                        RS =     -----------------------------                                  (7-RS)
                                             h (H - L)

                                        bB - (hH/L + l) (L - B)
                        XS = L   ----------------------------------               (7-XS)
                                           bB - (h + l) (L - B)

Rewriting conditions K(0) < B and K(1) B gives:

                        {L - T(L, t) < B} { X(t) XS   & L < M(t)}                          (8-t)

                        {L - T(L, t) > B}   {X(t) XS  & L > M(t)}                (9-t)

Now consider the regimes, and determine whether they can exist:

Full employment: Given that L > B, it follows from (9-1) that the tax exemption can be chosen on or above the critical value XS. Hence XS X(1) < H. A prime example is X(1) = B. Hence (iii-b) is empty.

Unemployment: L is given as the market clearing wage for low productivity persons. If X(0) < XS, then taxes on these persons are increased, and their net income drops below B. Given that K(0) < B, they are eligible for benefits, and apply. Hence (iii-b) is not empty.

It has been shown that both cases are possible. Q.E.D.

Remark: This exposition may seem an overly complex translation of the Cohen Stuart 1889 quote (above) to the welfare state situation. The proof might have said “self-evident” after the first paragraph. Given the record of unnecessary unemployment, this author may however be excused for driving the point home. The usefulness of the BHL concept may be, that officials now can report, “we have diagnosed l people on benefit who should be able to earn L > B on the market, so let’s try to find out how we are stopping them from doing so”.

Remark: A more didactic exposition may start with a structural tax relation, e.g. with R(t) replaced by r in (2-t); see for example the Bentham tax. Then one can show that a ceteris paribus reduction of the tax exemption will increase unemployment. Hence, for the return of full employment it is necessary (but not sufficient) to increase income tax exemption - or something from the ceteris paribus part. Then, the second step in the exposition (as we have done here) is to rename the axis into compounded variables (including VAT, regulations, subsidies, excises, charity, etcetera), and then consider (2-t) as the reduced form. Then we find necessary and sufficient conditions. This however only works satisfactorily for an accepted model of a real economy.

Remark: The theorem doesn’t establish that unemployment has only one cause. Various kinds of unemployment have various causes. But, when various causes are mapped into the world of BHL-ness, then the theorem applies. For example, a long term unemployed academic would be categorised as unskilled labour, even though his employed colleagues earn much more. (The BHL concept thus is drastic. The reasons for applying it have been explained elsewhere.)

Remark: The theorem is strongest in the t = 1    t = 0 part. Given full employment, it is easy to mess it up; and it is easy to see that you can mess it up. The other way around is less obvious. Here, both the requirement L B and Lemma II are crucial. For expository reasons those are sufficient, but not as sharp as they could be. For example, we might accept a small loss in H(1) H(0), as long as net N(1)   N(0).
However, even then the analytical structure remains, that productivity L is assumed, so that it doesn’t come as a big surprise that employment is possible. This actually is similar to the Arrow-Debreu setting, where endowments are assumed, and full employment appears to be possible. The modern reader might be inclined towards assumptions that generate the impossibility of full employment. (See for example the Grandmont (1983) setting of expectatory mismatch.) However, each impossibility can be questioned too. It is up to reality what model applies. Stated differently: the value of above tautological theorem is that it helps us to understand what is implicit in our concepts, so that we may be more aware in observing whether these concepts apply. This fits in with our concept of a proposition.

Remark: The reduced form also captures the ‘physical tax’. The lack of infrastructure, machines or tools may ‘tax’ people - and once these have been provided, they could start earning income, and their earnings would, crucially, be larger than needed to pay for the equipment. Economists of course understand this concept of a physical tax - as the lack of efficient capital markets, or the frustration of those by taxes - but the crucial point is the abstract one. When people don’t earn anything, and the economist suggests to abolish some tax, then a listener may become upset, since how can you abolish something that people don’t pay ?

Diagrams help understanding the analysis. Figure 42 shows two tax regimes, T(y, 0) and T(y, 1), characterized by different exemptions X(0) and X(1), and different critical incomes M(0) and M(1). The main difference is net income at L. In regime 0, net income at L falls below subsistence, causing unemployment and higher taxes to pay for benefits.

Chapter 40 showed the technical possibility of full employment for a welfare state. Chapter 34 showed that social choice is feasible, in the sense that there are consistent and reasonable constitutions that society might deem attractive. In particular, there is the example of a constitution that uses the efficiency criterion (Pareto optimality, PO) to select its policy. There still remains one issue to settle. This is the issue of information. Society might have a consistent preference, and consistently prefer full employment above unemployment, but when people don’t know that it is possible, and instead even have theories that tell them that full employment is impossible, then society might still choose for unemployment as the best of all evils. The issue of information already featured in our discussion of Arrow’s Theorem, and now returns for our discussion of unemployment.

Recorded full employment situations may have been caused by ‘chance’. Policy makers in 1950-1970 may have thought that functional finance was effective, while it also was the tax exemption level. A re-evaluation of the history may however also show that leading economic advisers in the 1950s may have been wiser than those of the 1960s.

It remains a stylized fact that much of the subject matter on employment is well-known. For example in Holland, CPB economists Van Schaaijk (1983), Bakhoven (1988) and Colignatus (1990) pointed the way to full employment. The state of knowledge turns out to be part of the model.

There is a Pareto Optimizing Change (POC) iff some advance and none suffer. A change from unemployment to employment need not be strictly POC. Note that we already have resolved that we don’t need high unemployment to keep inflation in check. So the CWIRU is no argument against a POC. There are other clear reasons that pose a problem. First these two:

·         Some bureaucrats have plush jobs administrating the unemployed, and would lose their job and sense of power.

·         The unemployed would lose their leisure. For some, the combination of low benefit B and leisure might be preferable to work at a higher income.

We can overcome these barriers by going back to basics, i.e. to our definitions. First of all, the bureaucrats are reminded that they are there to serve the public cause (‘res publica’) - and thus they have signed a contract - before they got the job - that they will welcome full employment and raise no anti-POC objections. In the same way, the people on the dole have signed a contract - before they got the benefit - that they will accept a job at a living wage, and will not raise anti-POC objections either.

A final observation is that the power elite, those who determine the SWF, might enjoy unemployment of a section of the population for some strange other reason. They might not care about the increase of income, freedom and welfare from a change towards full employment, but they would prefer the idea of people in helpless positions and the warm gratitude they show for their benefits. A king needs subjects. We resolve this problem by proper formulation of the theorem.

Note that we use the symbols of chapter 39 (that forms a unity with this chapter).

Above theorem on the technical possibility of full employment is essentially incomplete. It has not been specified how the tax regime comes about. The tax regime is an expression of the social choice already made, but it has not been explained how a particular choice has been caused. What is required is a power distribution on the b + h + l agents in the economy. In conventional terms the power distribution is expressed as a social welfare function SWF, and the tax regime is the result of the maximisation subject to the state of information I:

                        maximise SWF(h, H, N, l, L, K, b, B, t; I)                    (40.1)

Using a SWF serves expository purposes. When turning to practical application we could use the Drissen & Van Winden (1990) approach. But the logic of both approaches is the same.

The introduction of regime indicator t as a separate variable in the SWF means that it stands as a proxy. The economy is not simply a collection of individuals maximizing utility over consumption and labour. There are some institutional aspects too. An example of an institutional influence is that some social security officials might benefit from unemployment, since it keeps them in attractive jobs. All such (Public Choice) phenomena can be collected on their point of relevance: the employment regime t.

Secondly, there is information I. Ever since Keynes and Tinbergen, or even earlier, but for some economists more acutely since Muth and Lucas, economists have given attention to the information sets that guide the activity of agents. This concerns not just plain knowledge, but rather what people believe about the state of the world. The information sets may contain individual and social aspects, like own prices and the (announced) general price level.

Note that a conscious choice is made when one does not use
the information to switch to the other state.

When we apply Lemma III, which is about information handling in general, to our subject matter of employment, we get what for this area amounts to a theorem. The first theorem is special since it assumes the BHL property.

Definition:                There is wrong co-ordination if a SWF optimal change is blocked only by ‘lack of knowledge’ of the power elite while the information actually is available. (Co-ordination can go wrong on other counts too.)

Theorem BHL.2:       Given theorem BHL.1:
(i) full employment results from conscious choice or chance
(ii) unemployment results from conscious choice or from wrong co-ordination

Proof:

Theorem BHL.1 shows that full employment for the BHL welfare state is a controlable dichotomous state. The theorem is sound and compact. Thus Lemma III applies.

Possible states of sound compact knowledge and employment (I, t) are:

(1)        (1, 1): having the knowledge, full employment results;

(2)        (0, 1): lacking the knowledge, full employment results; thus there is a hidden cause; thus it is by chance;

(3)        (1, 0): having the knowledge, unemployment results; thus, the explanation comes from the power distribution, so that full employment is not to the advantage of those in power, and the choice for unemployment is conscious;

Definition:         There is wrong co-ordination if a SWF optimal change is blocked only by ‘lack of knowledge’ of the power elite while the information, though not yet available, still could be found rather quickly by not much effort. (Co-ordination can go wrong on other counts too.)

Theorem G.1:    If full employment is a controlable dichotomous state for which sound compact information exists in principle, that also can become available rather quickly by not much effort, then:
(i) full employment results from conscious choice or chance
(ii) unemployment results from conscious choice or from wrong co-ordination

Proof:

Work along the proof of theorem BHL.2. Note that BHL.2 fills in the properties that are now provided by hypothesis: controlability, soundness, compactness, and availability. Note that controlability means that the information exists in principle, while it need not be available yet.

Q.E.D.

Remark: Theorem BHL.2 thus gives an existence proof for this general theorem, i.e. shows that it is not vacuous.

Remark: The value of the theorem is that it focusses our attention on the perceptions that we have to deal with when judging the arguments in this book. Some questions to be answered are: (1) Do we still believe in full employment (only friction unemployment), or do we think that there are serious bottlenecks - or do we even think that we live in a probabilistic universe ? (2) Do we seriously believe that governments have done their best, or at least a reasonable effort, for (a) using available information, (b) finding additional solutions ? (3) Do we really think that the BHL-concept is useless, and that governments have been right to neglect the papers on them ? (4) Do we seriously believe that the PO-changes that seem so likely, are not POC ?

The mentioning of ‘chance’ in the lemma and theorems induces a short discussion on randomness.

Let Queen Q fall in love with Prince Random PR. Q especially adores PR when he goes about the court with an attractive air of responsibility. To this end she gives him the job of Treasurer. However, PR does not know much about taxes, and true to his name he chooses tax exemption at random. Hence, any regime is ‘subject to approval by official royal authority’, and in this sense there is a SWF and maximisation. And only economists think that the economy or economic theory are relevant. On the other hand, this is an incomplete sense of optimality. If PR happened to choose regime 0, then teaching PR about taxes would have Pareto Optimizing effects. In this sense, only one case is really optimal. This example shows that we can discuss cases with random elements, and that we can maintain our classification of cases. In fact, Y(1) - Y(0) would be the ex post implicit price paid, in regime 0, by the Queen for decentralizing decisions to a nitwit. If PR has ex ante probability p of choosing regime 0, the ex ante expected loss is (1 - p).(Y(1) - Y(1)) + p.(Y(1) - Y(0)). It is not very useful, however, to indulge in the notion of randomness, when considering the theorem. The stylized fact is that it is the (deliberate) lack of knowledge that is crucial here.