Effects of Migration in a Basic Labour Market Model

This paper tackles the textbook message that free migration of labour equa-lizes real wages between local labour markets, since nominal wages should rise and prices should fall in emigrating localities and vice versa in immigrating localities. Reverse price adjustments should thus help in stabilizing migration. The paper investigates the idea in a basic labour market model with sequential comparative statics, and gets conflicting findings: both decreasing prices in the emigrating end and increasing prices in the immigrating end foster emigration. Furthermore, common wisdom is that, if emigration forces the locality to elevate tax rates, people’s voting with feet should foster emigration. This paper shows that this is true only with notable tax increases. In the other end, induced emigration appears if the initially immigrating locality is forced to increase its taxes, even modestly.

The main idea of the labour market model is that real wage equalization produces a stable and efficient market solution. In the process, both nominal wages and consumption prices adjust so that welfare differences disappear and systematic migration ends. Thus, a decline in local prices should dampen emigration and an increase in local prices should dampen immigration. A common view also is that taxes affect welfare comparisons between localities thus inducing people to vote with their feet. In particular, the emigrating localities may be forced to elevate their taxes, which should then cause a further boost on emigration.
This paper examines these questions by taking a closer look on the basic labour market model (cf. [5]). The paper proceeds as follows. Section 2 constructs the model. Section 3 presents the migration equilibrium between two localities based on nominal wage adjustment, and then carries out sequential comparative static analyses of the effects of price and tax changes. By now, the sequential approach seems to be novel in the literature.
The findings from the sequential analysis somewhat contradict the common wisdom by showing that a price decrease in the emigrating locality, as well as a price rise in the immigrating locality rather foster than dampen emigration. Moreover, a reasonably modest tax increase in the emigrating locality does not foster emigration, whereas even a small tax increase in the originally immigrating locality turns the migration flow backwards. Section 4 concludes the findings.

The Basic Model
Following the usual procedure in the textbook literature (see [6], pp. 383-390; [7], pp. 158-179; [8], pp. 464-588), ignore non-labour income, normalize total available time to unity, and compress locally produced and consumed private and public goods into one consumption bundle as perfect substitutes. Write for the individual maximization problem. In Equation (1), q is consumption, l-l is leisure and l is time used in work, w is nominal wage, and p is consumption price, including the tax price of local public goods. From the budget constraint q = lw/p, where w/p is the real wage. Assume that the qualitative aspects of leisure, work and consumption are all included in the market information (the real wage), determined in competitive local labour markets. Local supply of labour derives from individual time use decisions, yielding the first order optimum condition w = pU 2 /U 1 , where the subscripts 1 and 2 indicate the derivatives against the first and second argument of the utility function, respectively. The aggregate labour supply can be written in inverse form (see [9], pp. 7-8) as where L denotes total labour time and g(L) describes people's market valuation where t is the tax rate. The production sector consists of private firms that produce private goods, and public organizations that produce public goods. Both operate efficiently, obeying Equation (3). Private firms get sales revenue from private goods, while public firms' revenue consists of taxes paid by the working residents. A practical interpretation of Equation (3) is that taxes are deducted from wages and channeled to finance public production. Thus, the tax rate t can be regarded as the public sector's share of the local economy. Moreover, assume that the tax system is fair so that the workers who pay the taxes also receive the corresponding tax financed benefits. The private and public firms use equal technology.

( )
, , where K denotes the local capital stock. The usual assumptions on the production function apply. Technically, the factors K and L are perfectly elastic between private and public production, but there may be some friction in the short term.
Keeping the capital stock is constant in the short term, optimization on labor use yields.

Initial Migration Equilibrium
The economy consists of two localities A and B with self-sustaining labour markets. Ignore trade in both private and public goods, and assume that production and capital are immobile between the localities and owned by the residents of each locality. That is, only people are perfectly mobile between the localities ac-  In the upper section of Figure 1, the labour demand graphs  willing to hire lower-cost labour from A. The middle panel presents these motives. The market supply curve S a (the horizontal gap between S A and D A ) describes the excess labour supply from A for wages higher than w A , and the market demand curve D b (the gap between D B and S B ) describes the excess labour demand from B for wages lower than w B . Since there are no changes in prices and taxes, wages rise in A and fall in B until the market equilibrium e in the middle panel is reached at w * . Employment is L a in A, and L b in B. Emigration from A is L a1 − L a and immigration to B is L b − L b1 , which displaces L B − L b1 of original workers. Thus, L a1 − L a = L b − L b1 = L e . In A, firms lose w A w * ae A , of which w A w * aa 4 goes to the staying workers, whose surplus is a 3 w * aa 2 . The net welfare loss in A is a 2 ae A . In B, firms gain w * w B e B b of which w * w B e B b 1 comes from the original workers so that the net gain is b 1 e B b. The emigrants' gain is a 2 aa 1 , carried from A to B. Since it overwhelms the welfare loss in A, aa 1 e A measures the net welfare effect of emigration. Total welfare gain is w A w B e in the middle panel, of which w A w * e equals aa 1 e A in the left panel and w * w B e equals b 1 e B b in the right panel. The economy wide resource allocation is efficient.
Note that the effects depend on local market conditions. First, the more capi-

Adjustment of Prices
It is quite plausible and empirically reasonable that migration induces price changes in both ends of migration. As people exit a locality, local demand for goods decreases. Recalling the assumption of immobile production and exclusion of trade, market prices are due to fall. The opposite is reasonable in the immigration end. Note that the standard model in Figure 1 is constructed from Equations (2) and (5)    Second, consider the effects of a migration induced price rise in locality B, keeping prices in A fixed. Figure 3 illustrates such effects under the assumption that the real wage in B remains unaltered after the rise in prices.
In the right panel of Figure 3

Tax Implications
A common view is that taxes affect welfare comparisons between communities and thus induce people to vote with their feet (see [10]). It is also quite conceivable that while emigration erodes the local tax base, the evolution of service provision and other social and physical infrastructure may lag behind thus causing financial strain in the short term. Another angle is that also immigrating communities may face short-term budgetary problems because of increased need for local public services and investments in infrastructures thus urging elevation of local taxes.

Consider first the effects of an emigration induced tax increase in locality A,
keeping the price of the consumption bundle p fixed. Emigration from A means a decline in total production, consisting of private and public goods. Assume that the fall in total production treats private and public goods asymmetrically thus causing changes in the share of public production, measured by the tax rate t. This is reasonable, because public goods include inflexible physical and social infrastructures. Thus, due to emigration, the production factors must shift partially from private to public production, which makes the share of public goods (the tax rate t) rise. Recall that private and public goods are perfect substitutes, and the tax system is fair so that the tax payers (that is workers) also receive the corresponding benefits.  in A for a modest tax increase from t to t′ . In Figure 4, a small tax rise from t to t′ shifts the labour demand curve inwards from D A to A D ′ in the left panel. Since The finding is that a modest tax increase induced by emigration does not affect migration in the short term. Figure 5 presents the respective effects of a more notable tax increase to t′′ . In Figure 5, tax rate t′′ in locality A makes the labour demand curve shift from D A to A D ′′ . The local equilibrium settles to a′′ as the attainable wage adjusts to w′′ . Labour demand falls thus making a a L L ′′ − emigrate to B. The net wage w′′ is lower than w * , but the tax financed public provision means that purchasing power is w′ , which exceeds w * . Emigration depicted by the segment  inefficiency in resource allocation. The finding is that significant tax increases may foster emigration in the short run.
Note that the effects depend on market conditions. If locality A is small enough to face a flat sur-local demand curve at w * , emigrants' welfare gain is bigger and the welfare loss reduces. Furthermore, if A is also capital intensive in production so that D A is steeper, tax t′ would make its gross wage adjust more and net wage adjust less so that both emigration and welfare effects would be smaller. In a capital intensive locality, taxes can be high without big effects on employment, and the small fall in net wages induces only modest emigration.
Quite surprisingly, small capital intensive localities seem to be less vulnerable to taxation than large diverse ones.
Second, consider the effects of a migration induced tax rise in locality B. Figure 6 presents the analysis.
In the right panel of Figure 6, the tax t′ imposed in B shifts the labour de-  dead weight loss of taxation in the economy thus is w e ew * ′′ ′ in the middle panel. The finding is that even modest tax increases cause backward migration thus dampening immigration.
Note that the effects depend again on size and capital intensity. In Figure 6, the migration effect is the greater the smaller B is relative to the rest of the economy. If B is very small and the sur-local demand curve is horizontal, emigration from B grows, the whole welfare gain attaches to the emigrants, and the welfare loss of the whole economy shrinks (note that the welfare effect in B does not depend on its size). Thus, the smaller B the bigger the migration effects and the smaller the welfare effects and vice versa. The role of relative capital intensity can be analyzed by making B's labour demand curve steeper. Taxation would then cause smaller effects on migration, market wages and welfare. Thus, the more capital intensive the industrial structure in B the smaller the economic effects of taxation. The finding is that big immigrating localities with a capital-intensive industrial structure can use taxes without notable effects on migration and welfare.

Conclusions
The paper scrutinized common conceptions concerning the effects of migration by taking a closer look on the traditional theory of labour market migration. The comparative static analyses were conducted in a sequential manner. First, the migration equilibrium between two localities was constructed according to nominal wage adjustment, and second, price adjustment and the effects of taxation were studied sequentially in that equilibrium setting. The approach seems to be novel in the literature. in the immigration end. However, the sequential analysis showed that if local prices fall due to emigration, it rather enforces than dampens emigration in the short run. This contradicts the common sense that a decline in local prices should alleviate people's motives to emigrate. The same kind of an unorthodox finding rose from the immigration end: a local price rise rather fosters than dampens immigration.
Sequential comparative static analysis of taxes produced also some counterintuitive findings. In particular, an initially emigrating locality can impose modest tax increases without making the still remaining residents vote with their feet. There emerges no dead weight loss either. The tax increase must be decidedly high in order to trigger such effects. This is because past emigration erects a migration threshold, which means that the reservation wage of the remaining workers is considerably lower than the current market wage. The height of the threshold depends on local labour market conditions compared to the rest of the economy. There is no such threshold in immigrating localities, and even a small tax increase turns the migration flow backwards and causes dead weight losses.
In practice, the tax effects may be minor.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.