A tax-based incomes policy that removes sectoral labour market imbalances

(c) Ian P. Hare, October 2003

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ABSTRACT: A modified version of a tax-based incomes policy (TIP) incentive tax
is proposed that retains the aggregate wage-restraining effect of the tax
proposed by Jackman and Layard, but in addition should help to speed the 
elimination of unemployment that results from the slow response of relative 
wages to sectoral labour surpluses or shortages. This effect is illustrated 
using a two-sector model with downwardly inflexible nominal wages.
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Notation:

^{ }, _{ }   = contents of { } is superscript or subscript
               (or lower and upper limits of summation (Sigma))
~            = approximately equal to
inf          = infinity
Delta, delta = Upper- and lowercase Greek delta, etc.
| |          = absolute value
*italics*
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1. Introduction

Tax-based incomes policies (TIPs) have been proposed as devices for 
counteracting inflationary pressures on wages, thereby permitting the operation 
of an economy at a lower level of unemployment than would otherwise be possible 
without giving rise to accelerating inflation. Under a TIP, a firm is permitted 
to grant a nominal wage increase in excess of a government-determined norm 
provided that it also pays an associated tax, most simply one in proportion to 
the amount of the excess wage increase. Thus, unlike under a simple wage freeze, 
it remains possible for firms to adjust the relative wages they pay in response 
to the respective conditions of the various sectoral labour markets. 

At best, however, this response would appear to be no more rapid than in the 
absence of a TIP. The mismatch of available unemployed labour to job openings 
seems to be a significant source of unemployment, and this mismatch is 
presumably due at least in part to the slowness of the response of relative 
wages to sectoral labour surpluses and shortages. That being the case, it should 
add to the effectiveness of a TIP if the policy were able not merely to restrain 
wage increases in an aggregate sense, but to speed up the equilibrating response 
of relative labour costs to high sectoral unemployment or labour shortages. This 
paper shows how a modified TIP can accomplish this. Perhaps surprisingly, this 
approach does not require the regulatory authority to demarcate the various 
sectors of the labour market and to apply differential tax rates to them in 
accordance with their relative unemployment or job vacancy levels. This seems a 
desirable feature of an incomes policy from an administrative point of view. 


2. A modified TIP formula

As a starting point we take the incentive-tax formula of Jackman and Layard 
(1986a):

   net tax on firm = t [ w - (1+n) w_{-1} ] N - s N                      (1)

where t is the tax rate, w is the current nominal wage per worker-hour, 
w_{-1} is the previous period's wage, n is the government-determined
economy-wide norm for nominal wage increases, N is the level of employment at
the firm in worker-hours, and s is a (typically small) flat per-worker-hour wage 
subsidy,^{1} identical for all firms, and chosen to achieve *ex post* overall
revenue neutrality for the scheme. Jackman and Layard suggest that n would be
chosen to approximate the inflation rate, in which case  the tax becomes a tax
on real wage growth, and a small positive value of s is necessary for fiscal
balance; but n could equally well represent the *ex post* average rate of wage
growth, in which case s~0.  Under this formula, the penalty for local wage
increases in excess  of the norm (or reward for wage increases below the norm)
is restricted to a single period immediately following the wage increase. 

For an average firm, because s has been chosen to balance t[w-(1+n)w_{-1}] in
the aggregate, this tax provides no direct incentives to vary the employment
level: overall, it acts solely to restrain wages. On the sectoral or single-firm
level, on the other hand, above-average wage increases will *temporarily*
produce a positive net tax in proportion to the employment level for the period
in which the wage has been raised. This means that there is an incentive to
reduce employment temporarily for those firms granting above-average wage
increases. Similarly, there is an incentive to expand employment temporarily for
firms granting below-average wage increases. The temporary nature of the
response of the incentive tax to a change in wages means that the long-run local
employment level will not be altered by the tax associated with a sudden above-
or below-average change in wages that remains in effect permanently (or by any
temporary fluctuation in the firm's wages). 

One implication of this limitation of employment incentive effects to a single
period is that the value of the tax can readily be manipulated if the firm is
able to vary its employment level temporarily, as for example in the absence of
employment security laws. A firm wishing to grant an unusually high wage
increase, for example, might first reduce its employment level N, then raise its
hourly wages, pay the incentive tax on the reduced N, and finally restore its
original N. An obvious way around this problem of temporary manipulation of
employment levels would be to use some previous long-term average N. It is more
interesting, however, to consider the implications of the alternative of
spreading out the tax over a long period--most simply, an infinite one--*after*
the wage increase, at any time during which period the value of N can be taken
at that time, rather than at the time of the wage increase. In order for this
permanent tax not to dwindle into relative insignificance thanks to inflation or
real wage growth, we let it grow at a proportional rate of n (which will be
assumed to represent the aggregate nominal wage growth rate, so that we can set
s=0). That is, suppose that associated with a wage increase from period 
j-1 to period j (j = ...,-3,-2,-1,0) there is a subsequent tax, in the current 
period (period 0) of

   net tax = N_{0} (delta t)_{j} [w_{j} - (1+n) w_{j-1}] (1+n)^{-j}        (2)  

The present value of the tax (2) associated with a present change in the wage
level, when summed over all future periods, is equivalent to that of the tax of
eq. (1), assuming that the discount rate delta is chosen appropriately and that
employment is unchanging over time; thus the tax has the same effect as the
Jackman-Layard tax in restraining aggregate wages. On the other hand, the
incentive temporarily to manipulate the employment level at the time of a wage
increase now disappears. In addition, the lingering incentive tax now associated
with an earlier wage increase provides a continuing incentive to modify the
local employment level: earlier above-average wage increases imply a present
incentive to reduce employment, while earlier below-average wage increases imply
a present incentive to expand employment. These are the same present employment
incentives one would see, in the absence of the incentive tax, from an
additional previous wage change equivalent to
(delta t)_{j} [ w_{j} - (1+n) w_{j-1} ] (1+n)^{-j}. That is, the incentive tax
now *permanently amplifies the local employment impact of above- or below-
average local wage increases, in proportion to the tax coefficient
(delta t)_{j}. The incentive tax or subsidy now serves, from the employer's
point of view, as a wage tax or subsidy permanently added to the cost of each
(existing or new) unit of labour, in accordance with the relative reductions or
increases in the wages per worker-hour previously received by the firm's
employees.

Summing over the contributions from the wage changes in all periods up to the 
current period, the cumulative wage tax on the firm according to eq. (2) is, 
dropping the 0 subscript representing the current period,

   net tax = N Delta w                                                    (3)

where the wage tax per worker-hour, Delta w, is

   Delta w = 

      Sigma_{j=-inf}^{0} (delta t)_{j} [w_{j} - (1+n) w_{j-1}] (1+n)^{-j}

and in the preceding period

   (Delta w)_{-1} =

      Sigma_{j=-inf}^{-1} (delta t)_{j} [w_{j} - (1+n) w_{j-1}] (1+n)^{-j+1}

so that

   Delta w = (1+n) (Delta w)_{-1} + (delta t) [w - (1+n) w_{-1}]          (4)

(with Delta w set to zero, presumably, when the TIP is first implemented). 

In this formula, n is ideally chosen so as to achieve a long-term budget
balance. More realistically, there will be some small budget imbalance (even if
n is gradually adjusted in light of changes in the wage growth trend) which must
be made up by conventional taxes. A TIP tax must in any case be combined with
aggregate demand regulation of a conventional variety, with the associated net
injections and withdrawals from the economy, so that it does not seem crucial to
ensure exact revenue neutrality.

The employer is most directly affected by changes in the total unit labour 
cost, (w+Delta w), in response to changes in the wages, rather than by 
changes in the wage tax. If we rearrange (4) in terms of the changes in the 
total unit labour cost and in the wage, relative to the growth norm, we obtain

   (w+Delta w) - (1+n) (w+Delta w)_{-1} = (1+delta t) [w - (1+n) w_{-1}]

This formula is open to the objection that it leads to an arbitrarily scaled
employment incentive if (Delta w)_{-1} eventually takes on a magnitude
significant compared with the wage itself, as a result of long-term increases or
declines in local relative wages. Consequently it seems preferable to multiply
the preceding  expression for labour cost growth by a factor
(w+Delta w)_{-1}/w_{-1} (which has no significant effect in the case of small
|Delta w|) in order to bring about an equal proportional response of relative
total labour costs to proportional relative wage growth, independent of the size
of the accumulated wage tax. The resulting tax formula can be written

          w + Delta w
    ----------------------- - (1+n) 
    w_{-1} + (Delta w)_{-1}
                                             w
                            = (1+delta t) [------ - (1+n)]                (5)
                                           w_{-1}
       
This formula specifies that the percentage rate of growth in the total unit
labour cost, relative to the growth rate norm n, should equal the percentage 
rate of growth in wages received, also relative to that normal growth rate, but 
amplified by the incentive factor (1+delta t). Regardless of the size of the 
incentive coefficient, a wage growing at the growth norm will thus yield a total 
unit labour cost which grows at the same percentage rate. Also, if the incentive 
coefficient (delta t) is set to zero after a  wage tax has already been 
accumulated, the wage tax and the total unit labour cost will thereafter remain 
in fixed proportions to the wage, regardless of how the wage then changes. 
Changes in the wage tax as a fraction of the wage must reflect current rather 
than previous relative wage changes and incentive tax rates. 


3. The modified tax's effect on equilibrium wages

The modified tax given by (5) has essentially the same wage-restraining effect
in an overall sense as the Jackman-Layard tax given by eq. (1), provided that
the discount rate delta is appropriately defined. On the other hand, the 
modified tax will in general have different effects on wages at the sectoral 
level, as a result of the existence of a nonzero sectoral wage tax or subsidy, 
Delta w, before any wage increase relative to the norm is put into effect.

To demonstrate the equivalent overall effects of the two taxation schemes on 
wages, we assume homogeneous firms, constant N, n, t, and delta, and an
equilibrium wage growing at a constant exponential rate, w_{j} = (n+1) w_{j-1}. 
There is then no wage tax on wages growing at the equilibrium growth rate under
either of the two tax formulas, assuming that Delta w is initialized at zero in
the modified formula (5). Since the tax is to be operated with n chosen so as to 
achieve an approximate long-term aggregate budget balance, this last assumption 
is reasonable at the aggregate level. We then compare the taxes incurred by a 
typical firm as a result of introducing a perturbation in the firm's wages, in 
the form of a uniform percentage increase or decline in wages in all periods 
following a certain period, taken for convenience as period number 1. (One can 
construct an arbitrary perturbation of the wage function from a combination of 
such permanent shifts in the wage level.) That is, in place of the equilibrium 
wage function w_{j} = (n+1)^{j} w_{0}  (j= 0,1,2,3,...), we assume a perturbed
wage, in periods 1,2,3,..., of w'_{j} = (n+1)^{j} w'_{0}. 

In the case of the Jackman-Layard tax, this perturbation introduces a nonzero 
tax confined to period 1, with a value of

   tax per unit of labour = t (n+1) (w'_{0} - w_{0})                       (6)

Under the modified tax formula we have, in period  1, assuming Delta w_{0} = 0, 

   Delta w_{1} = delta t (n+1) (w'_{0} - w_{0})

and in subsequent periods 

   Delta w_{j} = (n+1) Delta w_{j-1},
 
or in general, 

   (1-delta_{N})^{j-1} Delta w_{j}

      = delta t (1-delta_{N})^{j-1} (n+1)^{j} (w'_{0} - w_{0})

(j=1,2,3,...), where the factor (1- delta_{N})^{(j-1)}, with delta_{N} = the
nominal discount rate, is introduced to discount the tax back to period 1. The
total discounted value of this geometric series of taxes, evaluated in period 1,
is

   discounted tax per unit of labour

                            delta (1-delta_{N}-n delta_{N}+n)
      = t (w'_{0}-w_{0}) ---------------------------------------
                         (1-delta_{N}) (delta_{N}+n delta_{N}-n)

For this to be identical to the value of the corresponding Jackman-Layard tax,
given by eq. (6), we need only set

   delta = 1 - (1-delta_{N}) (1+n)                                         (7)

in the modified tax formula (5). If real wages are growing at a zero rate, delta
is then equivalent to the real discount rate. 

On the sectoral rather than aggregate level, there may be an appreciable 
inherited wage tax or subsidy in the absence of any departure of wages from the 
norm, and there is no simple equivalence between the effects of the two taxes. 
According to the analyses of *e.g.* Jackman *et al.* (1986) of the effects of
wage subsidies (or taxes) on equilibrium unemployment under various models of
wage-setting, one would expect a sectoral subsidy to reduce equilibrium total
unit labour costs (taking into account the negative contribution from the
subsidy itself), and for a sectoral tax to increase  equilibrium total unit
labour costs.^{2} That is, a local subsidy (tax) might increase (decrease) the
equilibrium wage level, but not so far as to nullify the direct effect of the
subsidy (tax) on local employment at unchanged wages. This does not strike one
as unacceptable behaviour.


4. The cumulative wage tax as a labour-market-balancing device

In place of a tax reflecting current wage increases and employment levels, we 
now have a cumulative, compounded wage tax reflecting earlier increases in 
wages, relative to the average increases at those times. In effect, to each 
worker at a given firm is attached not only the usual wage w, but as well a wage
tax Delta w, both of these comparatively stable unit labour costs being 
payable by the employer as long as the worker remains employed, and extended to 
new workers as soon as they are hired. 

As suggested in the introduction, the typical response of relative sectoral 
wages to relative sectoral labour shortages or surpluses may be a gradual, 
rather than immediate, decline of relative wages toward a long-run equilibrium 
in the presence of unusually high unemployment, and a gradual increase of 
relative wages in the presence of unusually low unemployment. It is plausible 
that this type of relative-wage rigidity results, for example, from a perception 
of customary wage differentials *ipso facto* as "fair". Such differentials 
would be alterable only slightly without unduly demoralizing workers or arousing 
union opposition, until workers eventually became accustomed to the new, altered 
wage differentials. A related source of rigidity would be sticky nominal wages: 
the unacceptability to workers of nominal wage declines would imply that 
relative wages could decline in response to local unemployment no faster than 
the inflation rate plus the rate of economic growth per capita. In light of 
efficiency-wage theory, one does not expect this slow adjustment of wages ever 
to eliminate unemployment altogether, even in the absence of further shocks to 
sectoral labour markets. Nevertheless, a more rapid adjustment of labour costs 
in response to sectoral labour market imbalances should speed up the absorption 
of sectoral unemployment once it is generated by a sectoral shock, and thereby 
reduce the average unemployment rate, which is ascribable in part to randomly 
recurring sectoral shocks. 

The proposed form of the TIP apparently provides a means by which sectoral 
labour costs can be made to adjust more rapidly to local labour market 
surpluses. The net effect of the tax is somewhat uncertain, quantitatively 
speaking, because, besides amplifying the response of total sectoral labour 
costs to a given change in relative wages, the tax may also affect the speed of 
response of relative wages to sectoral unemployment or labour shortages. In the 
absence of a rigorous theory of wage rigidity, it does not seem possible to 
estimate this latter effect. We can, however, discuss the case of complete 
downwards nominal wage rigidity (and complete upwards flexibility) in precise 
terms. 

Most simply, suppose that there are two sectors, A and B, in the labour market,
also taken as corresponding to the sectors of a two-sector product market; both
of which are at full employment, employing equal numbers of workers at equal
wage rates and constant marginal products; and each of which is characterized by
perfect competition (zero profits), has no costs other than labour costs, and
consequently has an output of equal sales value, and so an equal weight in the
overall price index. Fluctuations in relative production by the two sectors are
assumed to be small enough that this equal weighting of each sector's products
in the price index does not have to be altered significantly. We also neglect
migration of workers between the sectors. Sectoral wages and prices are assumed
to rise immediately in response to any sectoral shortage of labour, sufficiently
to eliminate that shortage; on the other hand, sectoral nominal wages and prices
are assumed to be unable to fall in response to a sectoral labour surplus--
although even a small sectoral labour surplus is taken as sufficient to bring
about a freeze in the sectoral nominal wage level, *i.e.*, a slow decline in
real wages. The government is assumed to be able to vary overall product demand
at will, resulting in a common proportional variation of demand in both sectors;
the policy adopted in setting this overall demand is to maintain a fixed,
"creeping" inflation rate r (say, 2% per year) in the overall price index. The
total workforce is assumed to have a fixed number of members, and there is
assumed to be zero growth in labour productivity in both sectors, so that real
economic growth or shrinkage comes purely from changes in employment levels as
percentages of the two sectoral workforces.

Suppose now that in the absence of any incentive tax, there is a sudden shock in
relative product demand, giving a demand increased by a factor (1+alpha) in 
sector A and reduced by a factor (1-alpha) in sector B (alpha>0). This would
bring about an immediate increase in wages and prices in A by a factor of
(1+alpha), so that an employment level and volume of output unchanged from 
its original value remained consistent with zero profits in that sector. Wages
and prices cannot fall in B, however, so any increase in wages and prices in A
would constitute a discontinuous overall price increase, whereas only a
continuous inflation is acceptable. The price increase in A therefore has to be
prevented by a reduction in overall demand by the factor 1/(1+alpha), 
"simultaneous" with the relative demand shock, which is thereby converted into
an unchanged demand in sector A and a demand reduction in sector B by a factor 
(1-alpha)/(1+alpha) ~ (1-2 alpha). This demand 
reduction will cause the immediate layoff of a fraction of approximately 2 alpha
of the workforce in sector B, while sector A remains at full employment; there
will be no immediate change in any wages or prices. 

The economy will then gradually recover from this shock (in the absence of
further shocks) by virtue of the gradual decline in sectoral real wages and
prices, at a rate r, associated with the frozen nominal wages of sector B: this
decline in real prices leads to a gradual growth in the output and employment
level in that sector. At the overall inflation rate of r, the constant nominal
prices in sector B must be offset by a growth in nominal prices and wages in
sector A at a rate 2r (that is, a real wage and price growth rate equal to r)--
the result of an infinitesimal labour shortage in that sector, maintained,
notwithstanding the increasing real price for the product of that sector, by an
appropriate gradual growth in real overall demand. Meanwhile, this increasing
overall demand will also increase the recovery rate of output and employment in
sector B. The total rate of growth in output in sector B, and hence the duration
of the unemployment resulting from the shock, will depend on the elasticities of
sectoral product demand; but clearly this growth rate will initially be
proportional to r. (One cannot, of course, simply increase r to deal with the
unemployment arbitrarily rapidly, because permanent inflation above some low
level will make it necessary to pay workers inflation-indexed wages.) 

If we now add an incentive tax to the model, of the form given by equation (5)--
with the initial taxes, Delta w_{-1}^{A} and Delta w_{-1}^{B}, allowed to take
on any values--this will not alter the assumed perfect downwards inflexibility
of nominal wages: whatever causes are at work in precluding nominal wage
declines will operate with undiminished effect when the benefits to the employer
of such declines are merely multiplied by some moderately large factor.
Similarly, any increases in nominal wages necessary to clear sectoral labour
shortages continue to occur immediately. Assume that n in the tax formula is set
equal to r. Following the same sudden initial reduction in demand by the factor
(1-alpha) in sector B and increase by the factor (1+alpha) in A, again
transformed into a reduction of demand by a factor of ~(1-2 alpha) in sector B
and an unchanged demand in sector A (with no immediate change in wages, wage
taxes, or prices), the relevant growth rates in the two sectors, which come
about in essentially the same way as in the absence of the
incentive tax, can be summarized by the following table:

Sector                                 A          B

Initial unemployment level             0          ~2 alpha
--------------------------------------------------------------

Nominal wage growth rate               +2r        0 (sticky
                                                  wage)

Wage growth rate relative to norm n=r  +r         -r

Total labour cost (w+Delta w) growth   +(1+       -(1+
rate relative to norm; also, real      delta t)r  delta t)r
price growth rate                      (for zero  (eq. 5)
                                       average)

Real output growth at constant demand  -k_{1}(1+  +k_{2}(1+
                                       delta t)r  delta t)r

Real demand growth                     +k_{1}(1+  +k_{1}(1+
                                       delta t)r  delta t)r

Real output (employment) growth        0          (k_{1}+
                                                  k_{2})(1+
                                                  delta t)r

(where k_{1} is the elasticity of sector A output with respect to sectoral
prices at constant aggregate real demand, and k_{2} is the corresponding
elasticity for sector B). The rate of recovery of employment in sector B,
following the sectoral demand shock, will thus be dependent on the sectoral
demand elasticities, but strictly proportional to 1+delta t (as well as to the
inflation rate r). 

In effect, the modified TIP is a method of directing employment subsidies into
depressed sectors of the labour market, "objectively" identified by the relative
wage changes among the sectors. Sectors where wages are failing to keep up with
the average are interpreted as being depressed; those where wage settlements are
exceeding the average are interpreted as overheated. The subsidies assist in
reducing unemployment in depressed sectors until unemployment has been reduced
to the point where wage settlements recover to the average rate of increase,
whereupon subsidies are frozen at their accumulated level (as a proportion of
wages). Unemployment associated with high sectoral efficiency wages or union
power does not, on the other hand, elicit subsidies, as there is no link between
this form of unemployment and gradually declining relative wages. This sectoral
balancing effect of the modified TIP is additional to the scheme's more familiar
restraining effect on aggregate wages, which should permit the operation of the
economy at a higher level of real overall demand. 


5. The wage tax as an income-insurance device

If a wage tax of the kind proposed here were as effective as claimed, it would
remove much of the need for wages to adjust to long-term sectoral shifts in
product demand or labour productivity: the labour market would balance after a
comparatively modest rise or fall in wages received, with the wage tax taking
the place of the remaining shift in wages that would otherwise be necessary to
balance the labour market. That is, the labour market would balance not only
more promptly, as we have just been arguing, but with less ultimate cumulative
disturbance to relative wages received. The proportional wage tax or subsidy
that had been accumulated by that point would then remain in effect indefinitely
in the absence of further shocks, as sectoral wages in the now-balanced sector
advanced at the average rate. 

A tax scheme of the proposed kind thus seems to offer some of the economic
advantages of a more flexible wage régime without any of the increased
insecurity in labour income implied by that régime: indeed, with a greater
income security. Unlike a true flexible wage, the wage tax will provide no
direct incentive to workers to migrate to low-unemployment sectors; but on the
demand side it substitutes for a more flexible wage, providing a substantial
motivation to employers to absorb surplus labour, or release labour in response
to a sectoral shortage. In other words, the scheme somewhat resembles an
efficient system of income insurance for a risk-averse workforce. 

An integral part of the present proposal, at least if we assume downwards
nominal wage rigidity together with slow economic growth, is the maintenance of
a "creeping" inflation rate designed to bring about gradual declines in real
wages in demand-deficient sectors. In the absence of this inflation, real wages
do not fall--except haphazardly, perhaps, in response to the threat of
bankruptcies at the local level--and wage subsidies never come into effect. From
the point of view of income security, this form of wage flexibility is greatly
preferable to the only other apparent alternative, of relying on a high rate of
bankruptcies or mass layoffs at high-wage firms to bring about relatively large
and highly localized reductions in otherwise inflexible wages. Even so, at first
sight, the need for creeping inflation, with its consequent slow erosion of
wages in those sectors with deficient demand, may seem to be an imperfection in
the income insurance system. In fact, however, the possibility of such income
erosion is logically necessary, given the usual practice of paying new workers
"non-discriminatory" wages, *i.e.*, keeping them on the same pay scale as the
senior workers in the same workplace, regardless of how uneconomic the pay scale
of the senior workers may have become since these workers were originally hired.
Wages permanently above the market-clearing level and applicable equally to new 
and existing workers would be reconcilable with sectoral full employment only
with the aid of permanent and arbitrarily redistributive sectoral employment
subsidies. 

A wage tax scheme of the present kind would warrant the objection that the wages
received by the permanently subsidized or taxed workers would never be brought
into line with the marginal product of those workers. Some categories of worker
would end up receiving permanent subsidies from others; and this preferential
treatment, arising out of the history of sectoral unemployment or labour
shortages, would bear no necessary relation to the wealth or poverty of the
subsidized workers. 

To avoid this prospect, it seems necessary to build some form of decay into the
equation for the evolution of the wage tax. Most obviously, one might add an
exponential decay term, as for example in
 
     w + (1+beta) Delta w
    ----------------------- - (1+n) 
    w_{-1} + (Delta w)_{-1}
                                             w
                            = (1+delta t) [------ - (1+n)]                (8)
                                           w_{-1}

with 0 < beta << 1. One has to be careful here, however, if one wishes to 
preserve the sectoral labour-market balancing function of the wage tax. Once a
formerly depressed sector has been balanced with the help of a substantial wage
subsidy, any too-rapid removal of that subsidy can be expected to induce new
unemployment in the sector. Not only should subsidies not simply be suddenly
eliminated after several taxation periods, for example, but even a smooth decay,
as implied by eq. (8), should be slow enough for the limited flexibility of the
relative wages in the sector to be able to cope with most of the ensuing
unemployment. 

In light of the difficulty in estimating an appropriate value of β in the
preceding equation, it may be preferable to use a bias of another kind. Suppose
that we define two unequal values, (delta t)_{H} > (delta t)_{L}, of the
coefficient used to calculate the wage tax at a given firm, and apply the higher
or lower value in equation (5) according to the following criterion:

                 /
                | (delta t)_{H} , sgn [w/w_{-1} - (1+n)] = -sgn (Delta w)_{-1} 
   (delta t) = <
                | (delta t)_{L} , sgn [w/w_{-1} - (1+n)] = +sgn (Delta w)_{-1}
                 \
                                                                             (9)
Since eq. (5) can be rearranged to give

   Delta w = (w/w_{-1}) Delta w_{-1}

             + delta t ( w_{-1} + Delta w_{-1}) [w/w_{- 1} - (1+n)]         (10)

(and w_{-1}+Delta w_{-1} > 0), we can see that the form (9) for delta t will
increase Delta w, where Delta w_{-1} is negative, and reduce Delta w, where
Delta w_{-1} is positive; *i.e.*, this form will reduce |Delta w|, relative to
the case where a single, all-purpose value of (delta t) is used. This reduction
in |Delta w|, which would be cumulative, should help to prevent either the
subsidies or taxes from becoming too large overall. At the same time, this
approach avoids removing subsidies, merely on the grounds that they have been
paid too long, in formerly depressed sectors where employment levels and
relative wages have been stabilized only with the help of high wage subsidies:
the subsidies begin to be removed only once relative wages begin to recover. 


6. Some practical considerations

It would evidently at present be impossible, in practice, to guarantee the
continuation of wage subsidies and taxes for more than a few years, so that only
a "conventional," temporary TIP tax may seem realistic. Once a TIP became an
accepted part of a permanent employment-promoting economic régime, however,
there would seem to be no reason why a temporary tax on a given wage increase
should not be gradually spread out over a longer payment period, and thereby
transformed into an implicitly targeted employment subsidy as described in
section 2. Jackman and Layard (1986b) reject what they term a "fixed-base" tax,
claiming that a tax on a given wage increase could be imposed for no more than a
few years because to do otherwise would penalize firms that upgraded their
workers' skill level. While I agree that the tax's penalization of increased
skill (and incentive to substitute less skilled labour) is a serious problem, it
appears to be just as serious an objection to the version of TIP proposed by
Jackman and Layard, if the TIP is viewed as a permanent device for reducing
unemployment levels. The crux of the problem is not whether the tax is applied
immediately or spread out over a period of years, but that it is calculated on
the basis of average wages per worker, or per worker-hour, without reference to
the average skill level of the workers, and that any effort to estimate this
average skill level would involve the state in a much more complicated and
intrusive kind of regulation. 

The present form of the scheme can be interpreted as attaching a long-term wage
tax or subsidy to each worker. In the basic scheme, this tax or subsidy is
determined entirely by the identity of the worker's current employer. However,
one can readily envision versions of the scheme where more elaborate criteria
are used for determining a given worker's tax or subsidy level. If it is
feasible to identify particular groups of workers with high unemployment levels,
such as the young, it may be reasonable to give these groups increased wage
subsidies. One might also set initial taxes and subsidies to zero for new
entrants to the workforce, as a means of discouraging them from entering the
more heavily subsidized sectors--where it is in the public interest to reduce
the size of the sectoral workforce--while giving the new workers a compensating
advantage in obtaining jobs in the sectors with positive wage taxes. 

An important instance of this kind of extension to the basic wage tax is the
case of new firms. If new firms started off with zero wage taxes, they would
often be at a severe disadvantage to or advantage over their competitors,
depending on whether the competitors generally had negative or positive wage
taxes. Nor does one wish to encourage the entry of new firms in sectors with
labour shortages and (typically) high wage taxes on existing firms. It should be
possible largely to circumvent this difficulty, however, by basing the new
firm's initial wage tax level on the average wage tax level for the most recent
employers of the new firm's employees: these employers will ordinarily share a
single sector of the labour market with the new firm.


Notes

1. Actually, Jackman and Layard (1986) specify a fixed subsidy per capita 
rather than per person-hour of work. This is however inconsistent with their
taxation formula. Meanwhile, it should perhaps be stressed (see Jackman, Layard
and Pissarides, 1986, p. 125) that this relatively insignificant subsidy does
not correspond to the subsidy in the simpler alternative incentive taxation
scheme, under which net taxes would equal a fixed proportion of income minus a
fixed per-capita subsidy, which Jackman *et al.* also propose as an equally
effective means of restraining wages. In that scheme the subsidy is of a
substantial amount and has a crucial importance.

2. Jackman *et al.* consider the effects of flat subsidies combined with 
budget-balancing proportional wage taxes, but removing these proportional wage
taxes does not appear to alter their analyses, as noted for example on pp 116-
-117 of Jackman *et al.* (1986).


References

Jackman, R. and Layard, R. (1986a). "The Economic Effects of Tax-Based Incomes
Policy", in *Incentive-Based Incomes Policies: Advances in TIP and MAP*, David
C. Colander (ed.), Ballinger, Cambridge MA. 

Jackman, R. and Layard, R. (1986b). "Is TIP administratively feasible?", in
*Incentive-Based Incomes Policies: Advances in TIP and MAP*, David C. Colander
(ed.), Ballinger, Cambridge MA. 

Jackman, R., Layard, R., and Pissarides, C. (1986). "Policies for Reducing the
Natural Rate of Unemployment", in *Keynes' Economic Legacy: Contemporary
Economic Theories*, James L. Butkiewicz *et al.* (eds.), Praeger, New York.