Book IX. Reduced Form

Reduced form

39. The possibility of full employment in the welfare state

Introduction

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.

Stylized facts

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.

Concepts

Here we will redefine variables such as H, Z, b, n etcetera. Also the reduced tax function will be T(.) as opposed to structural T[.]. These redefinitions hold for this chapter 39 and chapter 40 - that together form a reduced form unity.

Definition: Biological subsistence, for survival, is S.

Definition: An economy is a welfare state iff people without income are not left to charity, stealing or death, but get a benefit B. The benefit B has the following properties: i. the net benefit has the social subsistence level B S, ii. people on benefit may not work, iii. eligible are: iii-a. permanent benefit recipients (e.g. ‘the elderly’) iii-b. people able to work but currently unable to earn at least net B (these people are called ‘the unemployed’).

Remark: it is useful to have category (iii-a) in the model. It introduces a degree of sufficient complexity. When there are levies even under full employment, then it is easier to understand that wrong co-ordination may cause a switch to unemployment. But (iii-a) might count zero people.

Remark: Property (iii-b) has the effect of a legal minimum wage. It sets a floor in the market. We might introduce a benefit threshold (for workers) XB such that S XB < B, but for expository reasons, we take XB = B.

Remark: The reservation wage effect is as follows. When vacancies with net income higher than B are registered, then the relevant unemployment benefits are simply scratched. This mimics the array of measures needed for continuous reality.

Remark: This definition implies that people working with subsidies in the Swedish/Japanese case are not on ‘benefit’. Such subsidies thus must be accounted differently, basically as part of taxes.

Remark: The black economy (another form of working while on welfare) is neglected. We neglect also the case that some people hate being on welfare, and thus continue working even when their net earnings are below the benefit threshold (S < net earnings < XB ).

Definition: A welfare state is bhl iff it remains meaningful to trisect its membership into the economic classes of Low and High productivity workers and permanent Benefit recipients.

Definition: A welfare state is nonrevolutionary, iff its economic classes and their data are stable across the change of employment regime.

Definition: A welfare state is BHL iff it is bhl and nonrevolutionary.

Remark: Denote High and Low gross productivity as H and L. Note that B is net. Also bhl-ness technically implies H >> L B.

Remark: L may be associated with a minimum wage and H with some average income including profits.

Remark: An example of ‘meaningful’ are subgroup subperiod averages.

Remark: Stability can sometimes be found by normalizing, e.g. take subperiod H(t) as the subperiod numeraire.

Remark: A person’s benefit is often related to the former period working wage. However, anything can be clustered into a social subsistence average. People ‘between jobs’ could be taken to be basically in the employed cluster, people with serious unemployment could be in the other cluster. Don’t object that this makes the matter tautological - since that is exactly what we try to do. (We try to find the definitions that make our understanding tautological.)

Remark: A nonrevolutionary welfare state still allows for politics and economic change.

Lemma I: A welfare state is BHL iff there is stability over the regimes for the variables B, H, L and the associated numbers of agents.

Proof: Self evident. Q.E.D.

Remark: The relevant notion is that the change from unemployment towards full employment (or vice versa) does not destroy the productive base of the economy. Instead of taking this notion explicitly, we have taken a stronger property of nonrevolutionarity, that allows, if bhl-ness applies too, to take (approximate) constancy of the variables.

Remark: At first glance these definitions seem self-defeating for the effort to apply the mathematical method to employment regime switches. When 35 million, nowadays unemployed in the OECD, are supposed to find a job, then apparently the policy maker is supposed to be able to judge on the ‘stabilities’ involved. That seems an impossibly strong assumption. We may however remind about the regime switch from 1950-1970 to 1970-2005. In addition, as modellers we discuss equilibrium states of various paths. Also, it is possible to give the variables an incremental interpretation, e.g. take 34 of the 35 (million) as permanently on benefit, and only look at 1 million on the margin (giving “local-BHL-ness”).

Lemma II: For a welfare state, the (apparent) existence of people with a productivity L’< B, does not block the application of BHL-ness.

Proof: Consider the pathological case of people with productivity L’< B, i.e. so low that (in whatever regime) their net market income is lower than B. Take the dentists, who in a regulated market cannot start a practice, and who are very bad at farming in a flowerpot (which could be done with a Cobb-Douglas production function). These people can be treated as:

(1) society is willing to classify them as (iii-a)

(2) like the Swedish/Japanese approach, they may keep on working with some employer subsidy Z; in that case L = L’ + Z

(3) society lowers B to B = S or B = L’, and reconsiders the problem

(4) if regulations are the bottleneck, then changing these regulations redefines ‘given’ productivity L’. Similarly, if Keynesian methods solve unemployment, then only if people’s effective productivity is restored. So the reduced form applies anyhow. (In that case the regulation or lack of a policy measure is a tax in terms of the reduced form, and ‘real productivity’ is higher than L’.)

(5) they get charity, steal or die, and hence there is no welfare state.

Hence BHL-ness implies that these cases can be ‘averaged out of the discussion’ or be left out for expository reasons.

Q.E.D.

Remark: In other words, BHL-ness is sufficient for discussing employment in the welfare state (but not necessarily for other topics, for example, how regulations affect productivity).

The theorem

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 ?

Graphical presentation

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.

Figure 42: Tax regimes

It can be seen that T(y, 0) is above T(y, 1), or that average tax rates are lower under full employment. On the left section of the horizontal axis, X(0) < X(1). On the right section, since taxes in regime 0 are higher and levied on a smaller tax base, T(H, 0) > T(H, 1). Thus the effect on the average tax rate is clear. The effect on the marginal rate depends upon the numbers. The case depicted here, with a higher marginal rate in regime 1, is only one possibility; but it shows that a higher marginal rate can combine with actually lower taxes.

40. The possibility of co-ordination