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Migration equilibria/disequilibria and the natural rate of unemployment in a regional context.


In this article, we examine, in an explicitly system-wide context, the theory of regional equilibria in the presence of endogenous migration of the form specified in Layard et al. (1991) and Treyz et al. (1993), where net migration flows (relative to lagged labour force) are determined by real consumption wage and unemployment-rate differentials. Accordingly, we provide a theoretical analysis of the importance of both market-clearing and steady-state concepts of equilibrium for the regional labour market. In particular, we examine the impact of a stimulus to local demand, in the form of an injection of externally financed government expenditure. We focus on short- and long-run comparative static equilibria and the adjustment paths which link them. Some issues are dealt with analytically, while others require simulation.

We conduct the investigation over a number of different conceptual time periods. The first is an "impact interval", a period of time over which population and capital stocks are both fixed. This is characterized by flow equilibrium. We then consider a longer-term interval where we impose the zero-net-migration condition for (static) steady-state equilibrium. Population is optimally adjusted over this period, but we initially assume that the capital stock is fixed[1]. Finally, we consider an interval which allows optimal adjustment of both capital and population stocks.

In the second section we deal analytically with the case where the regional real-consumption wage is negatively related to the unemployment rate (or positively related to the employment-labour-force rate). We refer to this as the wage function, which is motivated in Layard et al. (1991) by bargaining considerations, but which can be equally thought of as a conventional aggregate labour supply function. Then we investigate the situation where the regional nominal wage is set exogenously, a state of affairs consistent with national wage setting, before reporting simulation exercises which quantify the impact of the demand injection on sectoral output and employment and detail the crucial issue of the speed of adjustment to long-run equilibrium. Finally, there is a short conclusion.

Regional real wage bargaining

The impact interval

In the impact period, a government expenditure injection shifts the regional aggregate labour demand curve outwards in employment-labour-force ratio, real consumption-wage space, from [D.sub.0] to [D.sub.1] in Figure 1. The wage function is associated with the upward sloping curve, S. Labour market equilibrium shifts from A to B, the real consumption wage rises, the employment ratio rises and the unemployment rate falls. The existence of the wedge generated by extra-regional imports, ensures that natural-rate results do not apply over this time period.

Zero-net-migration equilibria

In the absence of other natural population change, long-run equilibrium implies zero net migration for the region. The locus of combinations of real wage and employment rates which generate zero net migration is given by the negatively sloping curve ZNM in Figure 1. If the region is initially in long-run equilibrium, point A lies on this curve. Points to the north-east of ZNM imply positive net in-migration, points to the south-west net out-migration. In the impact period, the region's labour market is in equilibrium at point B. This implies positive in-migration (though such in-migration has no effect in this period). However, for long-run equilibrium it is necessary for the region to return to a point on ZNM, and for a given wage function, the region must revert to the original equilibrium real wage and unemployment rate combination A.

How is the long-run equilibrium achieved? As the labour force expands, the employment-labour-force ratio declines, and the demand curve (for employment relative to labour force) contracts. There is a partially offsetting influence through the impact of in-migrants on the general equilibrium demand for labour, but this is insufficient to alter the process, given the existence of exogenous (and endogenous) demands that are not tied to population changes. As migration proceeds, and the labour demand curve shifts inwards from [D.sub.1], the real wage falls and the unemployment rate increases as the region moves down the wage function S until it reaches A. This implies that across steady state equilibria, unemployment rates are invariant to demand disturbances, so that natural rate results apply.

This result has implications for the issue of "who benefits" from a local demand stimulus. Ultimately, the employment rate returns to its initial level, so it is "as if" migrants absorb the increase in the level of employment (although the model does not formally distinguish migrants from other households, once migrants are absorbed into the population). The natural rate result is important for policy too. Traditionally, UK regional policy has focused on reducing the spatial dispersion of unemployment rates. However, across steady-state equilibria, demand has no effect on unemployment rates if wages are determined in real terms. Yet it would be quite wrong to suppose that demand management policies are ineffective. Although the unemployment rate is invariant with respect to such policies, the level of employment and output is not. The result does, however, suggest the need for caution in the use of regional unemployment rates as an indicator of policy efficacy.

The effect of endogenous investment

The qualitative analysis of the impact interval is little affected by allowing for endogenous investment. The demand stimulus typically increases capital rental rates above user costs. This stimulates the demand for capital which in turn increases investment and the demand for labour. However, over the longer term, as capital stocks are ultimately reequilibrated, dramatic results again arise. The interest rate facing the region and extra-region prices are taken to be constant, and we know also that the zero net migration condition implies that the real-consumption wage will be unchanged. Therefore, from Samuelson's non-substitution theorem (Samuelson, 1966) relative prices and the capital intensity of production in all sectors is fixed. Essentially, the perfectly elastic long-run supply of labour and finance generates a model of the regional economy with properties which we have never seen combined before: the model is a natural-rated-input-output model. This model exhibits a natural rate of unemployment which is invariant to demand. However, unlike aggregate natural rate models, it also demonstrates the sensitivity of input-output systems to demand disturbances.

A fixed nominal wage

It is often argued that regional wage rates are determined outwith the region, through some form of national bargaining procedure (Harrigan et al., 1992). Where this is the case, in the impact interval, real-consumption wages will fall, as regional prices rise as a result of the demand stimulus. Labour market equilibrium will be at a point such as C on the demand curve [D.sub.1]. Note that employment is higher than with the wage function. In the example in Figure 1, while the real wage has fallen, the reduction in unemployment is large enough to stimulate in-migration, which will continue until the zero-net-migration condition is achieved. With a fixed factor, as in-migration proceeds, regional prices will rise further, adding to the reduction in the real-consumption wage implied by a fixed nominal wage. Steady-state equilibria will be reached at a point such as E on the ZNM curve. Note that in this case, the natural-rate result does not hold. The unemployment rate is below its initial position, but only because there has been a reduction in the real consumption wage, requiring a reduction in the unemployment rate to maintain the steady state equilibrium. However, if we allow the capital stock to be endogenous, the natural rate input-output model outlined in the previous section applies. The reason is that where there is a perfectly elastic supply of finance and a fixed nominal wage, there will be no change in regional prices, implying a fixed real consumption wage. Note that all models with a wage function or a fixed nominal wage converge on a unique model of the economy in the long run providing there is full population and capital stock endogeneity.


In the last two sections we have derived qualitative results for the impact of a demand disturbance on the regional unemployment rate and real consumption wage over various time intervals and wage-setting regimes. However, this analysis is silent on the quantitative impact on employment and activity, the sectoral distribution of these effects and the rapidity with which long-run equilibrium is attained. In this section we tackle these issues through simulation using a regional computable general equilibrium (CGE) model parameterized on Scottish data[2].

In Table I we report the percentage changes in key economic variables which result from simulations of an across-the-board 10 per cent increase in government expenditures (financed outwith the region[3]). The simulations replicate the analytical results given earlier. They also show that with our "best guess" parameter values, total employment, output and the sector composition of the change in activity are sensitive to the wage-setting procedure. In the impact period, the increase in aggreate activity with a fixed nominal wage is [TABULAR DATA FOR TABLE I OMITTED] almost three times as large as with the wage function. Also, while with the fixed nominal wage all sectors expand, with the wage function there is extensive crowding out of exports in the traded sectors as a result of increased regional prices, and increased activity is confined to the non-traded sector. Over the interval where zero net migration is imposed, activity under both forms of wage setting is increased, though the results from the two forms of labour market closure are now qualitatively closer. This is because in-migration puts downward pressure on the nominal wage rate in the wage-function simulation. Finally, while the adoption of full capital and population stock equilibrium generates the expected zero change in the unemployment rate, it also results in a large stimulus to regional employment and output in all sectors. Note the input-output characteristics of this simulation: prices are unchanged and in each sector capital and labour inputs increase by the same proportionate amount.

It is clear that migration potentially plays a very important role in determining the impact of demand disturbances on the regional labour market In particular, some of the characteristics of the zero-net-migration equilibria differ quite markedly from the impact equilibria. However, an important issue is the time period over which the long-run equilibria are likely to be attained. We have therefore performed period-by-period simulations where population is updated between periods using the migration function of Layard et al. (1991), which is estimated on UK data. The speed of adjustment varies across the different wage setting regimes. With a fixed nominal wage, a period of ten years is required to fully restore a zero-net-migration equilibrium, with 93 per cent of the adjustment being made by the fifth year. With the wage function, full equilibrium takes 28 years and around 70 per cent of the adjustment is made after five years. Where capital stock endogeneity is also incorporated, full convergence on the zero-net-migration, optimal capital stock equilibria is even more extended, although all the closures (including the fixed nominal wage case) are now converging on the same steady state. Indeed, it transpires that neither of the models have converged over a 40-year time horizon, though the fixed nominal wage case is extremely close. Again the picture is of a system which takes an extremely long time to return to static steady-state equilibrium, and this serves to question the practical policy relevance of the natural rate properties of the regional economic models we have examined[4].

The extended adjustment period associated with capital and labour dynamics clearly reflects the particular functional form and parameter values of the migration equation used which are based on UK data. The corresponding migration equation estimated on US data in Table II (Treyz et al., 1993) gives parameter estimates approximately double those for the UK and we have investigated the impact of altering the migration response in this way in our simulations. The results are dramatic, with adjustment completed for both labour market regimes occurring within eight years. Accordingly, US responses imply a regional system which much better approximates the assumption of continuous equilibrium. However, within a time-frame of two or three years, the assumption of steady-state equilibrium would typically provide a poor approximation.


Migration has an important effect on the response to a regional demand disturbance. In particular, where there is full population and capital stock adjustment, a unique natural rate input-output model applies. However, the system generally takes a long time to return fully to static steady-state equilibrium, particularly if wages are sensitive to market forces (either for competitive or bargaining power reasons). Our multi-period simulations consequently put the striking "natural rate" property of the results reported in Table I in perspective, and raise serious questions about the practical policy relevance of the natural rate properties of the regional economic models we have examined.

The authors acknowledge the research assistance of Sarah Le Tissier and financial support from Scottish Enterprise National but remain solely responsible for the opinions expressed in this article.


1. This interval might be more generally interpreted as indicating the behaviour of the system where there is some fixed factor, even over a longer-term interval. In a regional context, land is the most obvious candidate for such treatment. However, Evans (1993, p. 92) argues that although the supply of land for development is inelastic in the short run, supply becomes much more elastic in the long run, especially in the USA where, in contrast to the UK, the land use planning system does not constrain the supply.

2. This model is AMOS, a macro-micro model of Scotland. A detailed description of the model is given in Harrigan et al. (1991). For the wage function we adopt wage and unemployment rate elasticities of 0.058 and -0.081 respectively, which are taken from Layard et al. (1991). In the full article we also report simulations using other wage functions.

3. Since the rest of the UK accounts for nearly 92 per cent of the entire income of the UK, the "small region" assumption which we employ here seems a reasonable approximation.

4. In the capital-endogenous, period-by-period simulations, investment is specified as a simple capital stock adjustment process with an adjustment parameter of 0.3. It is clear that the adoption of this relationship is partly responsible for the protracted adjustment period.


Evans, A.W. (1993), "The assumption of equilibrium in the analysis of migration and interregional differences", Journal of Regional Science, Vol. 30 No. 1, pp. 515-31.

Harrigan, F., McGregor, P., Dourmashkin, N., Perman, R., Swales, K. and Yin, Y.P. (1991), "AMOS: a macro-micro model of Scotland", Economic Modelling, Vol. 8 No. 4, pp. 424-79.

Harrigan, F. McGregor, P.G. and Swales, J.K. (1992), "Imperfect competition in regional labour markets: a computable general equilibrium analysis", Environment and Planning A, Vol. 24 No. 10, pp. 1463-81.

Layard, R., Nickell, S. and Jackman, R. (1991), Unemployment: Macroeconomic Performance and the Labour Market, Oxford University Press, Oxford.

Samuelson, P.A. (1966), "A new theory on nonsubstitution" in Stiglitz, J.E. (Ed.), The Collected Scientific Papers of Paul Samuelson, Vol. 1, MIT Press, Cambridge, MA.

Treyz, G.I., Rickman, D.S., Hunt, G.L. and Greenwood, M.J. (1993), "The dynamics of US internal migration", Review of Economics and Statistics, Vol. 75 No. 2, pp. 209-14.

P.G. McGregor, J.K. Swales and Y.P. Yin, University of Strathclyde, Scotland, UK
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Title Annotation:EMRU Conference Papers
Author:McGregor, P.G.; Swales, J.K.; Yin, Y.P.
Publication:International Journal of Manpower
Date:Feb 1, 1995
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