OPTIMAL REGIONAL FISCAL POLICIES WITH EDUCATION, LABOR
MOBILITY, AND OCCUPATIONAL CHOICE
by
Geraint Johnes*
Centre for Research in the Economics of Education
The Management School
Lancaster University
Lancaster LA1 4YX
United Kingdom
Voice: +44 1524 594215
FAX: +44 1524 594244
E-mail: G.Johnes@lancaster.ac.uk
May 1998
ABSTRACT
This note concerns the determination of optimal rates of region-specific taxes in a two region
model which allows for occupational choice and spatial migration. The tax system assumed in
this model exists to finance education which allows workers access to jobs requiring skills. It
is shown that the optimal tax rate differs across regions. Moreover, it is demonstrated that
central co-ordination of regions' tax policies is needed in order to arrive at the social welfare
optimum. The response both of migration and of optimal tax rates to region-specific
productivity shocks is shown to be ambiguous.
JEL Classification: H21, J61, J62, R23
Keywords: Regional policy, Migration, Occupational mobility, Optimal taxation
* The author acknowledges invaluable comments from participants at the 1997 EEEG
conference at Royal Holloway and from Bob Rothschild, but retains sole responsibility for
remaining deficiencies.
2
A NOTE ON MODELLING OPTIMAL REGIONAL FISCAL POLICIES WITH
EDUCATION, LABOR MOBILITY, AND OCCUPATIONAL CHOICE
1. Introduction
This note concerns the derivation of an optimal tax structure for a two region economy in
which (instantaneously acquired) education (or training) is funded out of local tax revenues.
Regional productivity disturbances can lead to an imbalance across space in the demand for
labor of various skill characteristics. This imbalance can be corrected by migration or by
regionally differentiated fiscal policies; the latter allow regions to differ in the extent to which
they invest in the skills of their labor force. The model developed below shows that there is
therefore a case for fiscal policy to discriminate between regions. Moreover, it is shown that
the socially optimal pattern of fiscal measures does not coincide with the vector of tax
measures which would freely be chosen by independent local governments.1 These findings
are of relevance in the applied context, either of regions within a single country or of countries
within a federation such as the European Union.
The remainder of the paper is structured as follows. The model is presented in section 2.
While parsimonious, it nevertheless does not yield to analytical solution. Section 3 therefore
contains details of the numerical solution of the model. Conclusions are drawn in section 4.
2. The Model
Consider a two region economy, each region i containing a firm which employs unskilled
labor, Lui, and skilled labor, Lsi. Workers may move across skills and regions in response to
disequilibrium wage differentials. In equilibrium all workers are employed at a wage of w.2
Suppose that the initial allocation of labor in each region is L0. Then
1This
last result contrasts with the findings of a Canadian school of researchers, obtained in the somewhat
different context of property taxes with land as a fixed input. This work is discussed later. Another part of the
literature which is relevant to the present paper is exemplified by Jagdish Bhagwati and Koichi Hamada (1982)
and some of the references therein. Rather than adopting the general equilibrium approach advocated here, these
authors consider the determination of equilibrium and the maximisation of welfare only in the spatial unit out of
which labor migrates. A recent contribution by John Leach (1996) investigates occupational and spatial mobility
in a model where training costs are borne by workers, where subsidies for training or migration might be
available, but where the distinct roles of national and regional governments are not considered.
2This is a common assumption in models of occupational mobility (see, for example, Ravi Kanbur, 1977). Note
that the costs of training skilled workers are borne by firms through the tax system, thus complicating the
relationship between wages, employment, and the productivity of each worker type. Wage equalisation is also
characteristic of many of the models of migration surveyed by Michael Greenwood (1985) - models where
3
(1)
Ls1 = 2L0 - Ls2 - Lu1 - Lu2
Firms pay a region-specific tax on the base of their wage bill, the tax rate being denoted by i (
 0 i). The revenue from this tax is used to train unskilled workers, at a cost of 1/ each, so
that they become skilled.3 Hence
(2)
Ls2 = {w[1(Ls1+Lu1) + 2Lu2] - Ls1}/(1-w2)
Assume a productivity disturbance4, 0 <  < 1, so that, given inputs, output in region 1
exceeds that in region 2. Specifically, assume a simple Cobb-Douglas production function
such that
(3)
Y1 = (1+)Ls1Lu11-
(4)
Y2 = (1-)Ls2Lu21-
Setting price to unity by choice of units of measurement, profits in each region may be defined
by
(5)
i = Yi - w(1+i)(Lsi+Lui)
Imposition of the zero profit condition implies
(6)
Lui = Yi/w(1+i) - Lsi
Define social welfare5 in each region by
'interregional wage differentials ... would presumably encourage migration from low to high wage regions which
would, in turn, contribute to a narrowing of the differential'.
3It is assumed that tuition is bought in from outwith the economy. The tax revenues are not reinjected into our
system.
4The assumption that  is symmetric across regions is inessential.
5The quadratic term in equation (7) is intended to capture the impact of inter-regional equity on social welfare. In
the absence of a curvature of this kind in the welfare function, the optimisation problem would be constrained
only by the requirement that real roots should exist for equations (9) and (10) below; such a constraint would be
4
(7)
Wi = Yi - (Lsi+Lui)(Y1 - Y2)2/2L0
and let the objective function of the national authorities be given by
(8)
W =  iW i
This is to be maximised by choice of i, i=1,2.
3. Solution
For simplicity, let  = ½. The model will be solved subject to the requirement that unique
positive real roots of Lui, i=1,2, should exist which satisfy non-negativity restrictions on the
employment of each type of labor in each region. Routine manipulation yields6
(9)
Lu1 = Ls1(1+)2{1/w2(1+1)2 + [1/w4(1+1)4 - 4/(1+)4]}/2
(10)
Lu2 = Ls2(1-)2{1/w2(1+2)2 + [1/w4(1+2)4 - 4/(1-)4]}/2
The nonlinearities which remain suggest that the use of heuristic methods would be
appropriate in the search for a solution.7 In the sequel the solution to the problem defined by
(1)-(4) and (7)-(10) is found, for assumed parameter values, using the method of simulated
annealing (Scott Kirkpatrick et al., 1983).8
The main results of the note are presented in the following two propositions.
difficult to justify in economic terms. It seems natural to allow each region's government to be concerned with
issues of national equity; were representatives of each region not concerned with inter-regional equity there
would be little reason for the national government to be concerned with it either. To keep the model tractable, the
social welfare function in (7) is not explicitly derived from individuals' utility functions.
6Note that  and  each influences both L
1
2
u1 and Lu2 through their effect on Ls1 and Ls2 respectively.
7For an earlier example of an optimal tax problem in which numerical methods are used to find a solution, see
Geraint Johnes (1987).
8The FORTRAN code used in the solution is available from the author on request. The solutions for endogenous
variables, given i, i=1,2, are found using routine C05NBF in the National Algorithms Group library; simulated
annealing is then used to find the values of i, i=1,2, which maximise W.
5
PROPOSITION 1: Social welfare maximising tax rates may differ across regions.
Proof: The proof is made by way of example. Let the parameter values be L0 = 10000,  = 5,
 = 0.5,  = 0.0005 and w = 0.1. Solution of the model9 implies 1 = 0.0697 and 2 = 0.0267.
QED.
Proposition 1 implies that there is a need for a spatially differentiated fiscal policy. As shown
by Gordon Myers (1990) and Arman Mansoorian and Myers (1993), however, this does not
necessarily imply that a central authority needs to co-ordinate the fiscal actions of local
governments. In the models provided by these authors, the optimising behavior of local
governments coincides with that required for efficiency.10
PROPOSITION 2: The region-specific tax rates which maximise social welfare across both
regions may differ from those which would be set, in the absence of a central co-ordinator, by
regional authorities seeking to maximise region-specific welfare.
Proof: Denote by 1* and 2* the solutions obtained in the proof to proposition 1. If
Proposition 2 did not hold, (1*, 2*) would represent a stable equilibrium so that the
authorities in region i could not raise the region-specific measure of welfare, Wi, by setting i
 i*, given j, ji. Proceed by extension of the earlier example. Given 2 = 2*, the
authorities in region 1 could raise W1 by reducing 1. For example, setting 1 = 0.05 would
raise W1 (from 1015 at the global optimum) to 1163, while reducing W (from 2047 at the
global optimum) to 1974. QED.
It follows from Proposition 2 that there is a need for central government to play a coordinating role in the determination of spatially differentiated fiscal policy.11 Intuitively this
is because a change in local taxes in one region has, as an externality, an impact upon the
labor market (and hence also on output and welfare) in the other region.
The impact of  on Ls1+Lu1, and hence on migration, and the impact of  on (1*, 2*) are
further matters of interest. But unambiguous results are not obtainable. Consider the case
9The
solutions for Ls1, Lu1, Ls2 and Lu2 respectively are 48, 9493, 424 and 10035.
Burbridge and Myers (1994), however, suggest that involvement of the national government in regional
redistribuition might be justified by reference to regional differences in preferences for such redistribution.
11A formally equivalent interpretation would be to represent this as a case in favor of inter-regional transfers.
10John
6
where  = 0.6 and all other parameters adopt the values assumed in the example used to prove
Proposition 1 above. Then Ls1+Lu1 rises to 9877, 1* falls to 0.0248 and 2* rises to 0.1333.
But if, ceteris paribus, w assumes the value 0.05, an increase in  from 0.5 to 0.6 brings about
a fall in Ls1+Lu1 (from 11684 to 9654), and a rise in 1* (from 0.0099 to 0.0143); meanwhile
2* rises from 0.0349 to 0.0593.
4. Conclusion
Given a labor market in which the demand for skills is unevenly distributed across space, and
where skills are acquired by training financed through local tax collections, migration of labor
will in general result, and there is an efficiency case for regionally disparate fiscal policies.
Such disparity is essentially a second best solution to a welfare problem, the market failure
being sourced, in the present instance, in the externality effect imposed on the labor market of
one region by the other region's fiscal actions. Local governments, serving to optimise local
welfare objectives, cannot therefore be relied upon to set local tax rates which serve to
maximise global welfare.
REFERENCES
Bhagwati, J.N. and Hamada, K. (1982) Tax policy in the presence of emigration, Journal of Public Economics,
18, 291-317.
Burbridge, J.B. and Myers, G.M. (1994) Redistribution within and across the regions of a federation, Canadian
Journal of Economics, 27, 620-636.
Greenwood, M.J. (1985) Human migration: theory, models and empirical studies, Journal of Regional Science,
25, 521-544.
Johnes, G. (1987) Optimal tax progressivity under imperfect competition, Economics Letters, 22, 279-282.
Kanbur, R. (1979) Impatience, information and risk taking in a general equilibrium model of occupational
choice, Review of Economic Studies, 46, 707-718.
Kirkpatrick, S., Gelatt, C.D. and Vecchi, M.P. (1983) Optimization by simulated annealing, Science, 220, 671680.
Leach, J. (1996) Training, migration and regional income disparities, Journal of Public Economics, 61, 429-443.
Mansoorian, A. and Myers, G.M. (1993) Attachment to home and efficient purchases of population in a fiscal
externality economy, Journal of Public Economics, 52, 117-132.
Myers, G.M. (1990) Optimality, free mobility and the regional authority in a federation, Journal of Public
Economics, 43, 107-121.