The migration equilibrium

s,˜s^∗

factors, such that the skill-intensive good is relatively scarce in Foreign. Both effects (partially) offset each other and it is hence a priori not clear, in which country (relatively) more workers are employed in the labour- or the skill-intensive sector. Figure 6.1depicts the knife-edge case with ˜s= ˜s^∗, which results forσ= 1 and resembles the familiar result that sector-specific Hicks-neutral technological change has no effect on the allocation of workers and on relative factor prices with Cobb-Douglas preferences (cf. Krugman,2000;Xu,2001).

6.3 The migration equilibrium

The migration equilibrium again is derived in three steps. At first Lemma 6.3.1 addresses the selection into migration. Goods markets are assumed to clear at a global scale and the correspondinggoods market equilibrium for the integrated world economy is derived in Lemma

6.3.2. Finally, thegoods market equilibrium condition from Lemma 6.3.2and the (unchanged) labour market equilibrium condition from Lemma 6.2.1are combined and jointly determine the migration equilibrium and the terms of trade between Home and Foreign.

Migration is costly and the cost of migration are paid in units of the final goods, whose relative spending ratio is assumed to be the same as the relative consumption ratio implied by Eq. (6.6). As a consequence, the resource use associated with migration (γ ≥ 0) is evaluated at the aggregate price index P^W(p^W,1) and individual migration decisions do not depend on the terms of trade. Thus, if international migration and the associated reallocation of labour across sectors in Home and Foreign feeds back into the relative world market price, this does not distort individual migration decisions.

From the perspective of a skilled foreign worker with s^∗ > s˜ migration is beneficial if the access to Home’s superior production technology (A > A^∗ = 1) compensates for a compara-tively lower rank in Home’s income distribution and moreover covers the migration costγ >0.

Formally, a threshold-level s˘^∗ ∈[˜s,1] can be defined such that all workers with s^∗ ≥s˘^∗ prefer to work in Home, while workers with s^∗ < s˘^∗ prefer to stay in Foreign. The critical worker

˘

s^∗ ∈ [˜s,1] thereby trades off a higher absolute income, due to the access to Home’s superior technology, against the top rank in Foreign’s income distribution, which is associated with zero disutility from disadvantageous inequality aversion. High skills in this trade off not only ensure a higher absolute income but also a higher rank in Home’s income distribution, which renders migration to Home – ceteris paribus – more attractive. The corresponding indifference condi-tion then follows from V[p^W, wS(˘s^∗)]−P(p^W,1)γ = V^∗[p^W, w^∗(˘s^∗)], which evaluated at wages wS(˘s^∗) =p^WA˘s^∗≥w^∗(˘s^∗) = max{w^∗_S(˘s^∗), w_L^∗}with w_S^∗(˘s^∗) =p^Ws˘^∗ andw_L^∗ = 1 simplifies to

A[˘s^∗−βΩ (˘s^∗)]−γ = max{s˘^∗,s˜^∗}, (6.11) in which γ < A−1 is assumed to ensure that a migration equilibrium withs˘^∗<1exists.

For skilled workers in Home with s >s˜migration is considerably less attractive, which has two reasons. At first, these workers already start with Home’s superior technology (A > 1), which they have to give up when migrating abroad. And secondly, migration as such is costly (γ >0). The corresponding indifference condition follows from the comparison ofV[p^W, wS(˘s)] = V^∗[p^W, w^∗(˘s)]−P(p^W,1)γ, which evaluated at wages w (˘s) = p^WA˘s≥w^∗(˘s) = p^Ws˘simplifies

to

A[˘s−βΩ (˘s)] = ˘s−γ, (6.12) in which s˘assumes the role of the skill threshold above (below) which workers stay in Home (migrate to Foreign). Comparing Eqs. (6.11) and (6.12), it can be safely concluded that workers from Home are relatively more inclined to live in Home (i.e. s <˘ ˘s^∗) whenever γ > 0. A sufficient condition to rule out migration from Home to Foreign is γ > AβΩ (0), which states that the disutility from disadvantageous inequality aversion in Home should not exceed the cost of migration for any worker. Lemma6.3.1summarizes:

Lemma 6.3.1 For migration costAβΩ (0)< γ < A−1workers withs^∗ ≥s˘^∗ migrate to Home, while workers with s^∗ < ˘s^∗ stay in Foreign. Thereby the cutoff s˘^∗ ∈ [˜s^∗,1] is decreasing in A and increasing in β and γ.

Proof Analysis and formal discussion in the text.

Goods markets clear at a global scale. Hence, the relative world market supplyy_S^W/y_L^W with y_S^W = yS+y_S^∗ and y_L^W = yL+y^∗_L has to equal the relative world market demand in Eq. (6.6) evaluated at the (relative) world market price p^W. Thereby, it can be inferred from Lemma 6.2.1, that, at a common (relative) world market pricep^W =p=p^∗, Home – due to its superior technology – employs (relatively) more workers in its skill-intensive sector (A˜s = ˜s^∗ implies

˜

s < s˜^∗). Hence for a given migration cutoff s˘^∗ ∈ [˜s,1] two possible patterns of specialisation exist. Suppose˜s <s˜^∗ <˘s^∗. Then both countries are incompletely specialised and produce both goods. Thereby the employment of domestic workers in Home is divided across sectors. Workers with skills s≥˜sare employed in the skill-intensive industry and workers with skills s <s˜end up in the labour-intensive industry. On the contrary, migrants in Home with skills s^∗ ≥ ˘s^∗ are solely employed in Home’s skill-intensive industry. In Foreign the remaining non-migrants find employment in both industries, such that all workers with skills s^∗ < s˜^∗ are employed in the labour-intensive sector, while those workers with skills ˜s^∗ ≤ s^∗ < ˘s^∗ are employed in the skill-intensive sector. Now suppose s <˜ s˘^∗ ≤ ˜s^∗. Then only Home produces both goods and Foreign specialises in the production of the skill-intensive good. While the sorting of workers according to the thresholds ˜s and s˘^∗ in Home is the same as before, in Foreign the migration

cutoff s˘^∗ replaces the industry cutoff s˜^∗ and all workers with skills s <˘s^∗ are employed in the labour-intensive industry. Aggregating up the sectoral outputs and replacingx^W_S /x^W_L =y^W_S /y_L^W in Eq. (6.6) then yields the global goods market equilibrium, which is summarised in Lemma 6.3.2.

Lemma 6.3.2 The global goods market clearing condition is given by p^W =

Proof Analysis and formal discussion in the text.

Replacing˜s^∗in Eq. (6.14) bys˜^∗=A˜s, which follows from Lemma6.2.1forp^W =p=p^∗, reveals thatp^W and ˜sare positively linked through the global goods market equilibrium. The intuition between this link is the following: A parallel rise ins˜and, hence, ins˜^∗ shifts resources from the skill to the labour-intensive sector and, thus, expands the relative supply of the labour-intensive good, which can only be absorbed by the market if at the same time the consumption of the skill-intensive good becomes less attractive, which is the case when the relative world market pricep^W goes up.

Given that s˘^∗ in Eq. (6.13) according to Eq. (6.11) only depends on exogenous parameters, the goods market equilibrium condition in Eq. (6.13) can be plotted as an upward sloping curve g(˜s,˘s^∗) in Figure 6.2, whereas g(0,s˘^∗) = 0, ∂g(˜s,s˘^∗)/∂s >˜ 0 as well as g(1,s˘^∗) → ∞ follow from Eq. (6.14), when replacing s˜^∗ by ˜s^∗ =A˜s. In order to find the equilibrium values of p^W and s, the goods market equilibrium condition˜ g(˜s,˘s^∗) can be combined with Home’s labour market equilibrium condition in Eq. (6.7), which is depicted as downward sloping curve in Figure 6.2. The intersection point of both curves then determines the equilibrium values of p^W and ˜s. Finally, the cutoffs˜^∗ can be read off from Foreign’s labour market equilibrium condition p^W = 1/˜s^∗ oncep^W has been determined.

Figure 6.2: The migration equilibrium

b

b bb

˜ 1

s^m s˜^∗m 0

1/A p^m

1

1/A˜s 1/˜s^∗ g(˜s,˘s^∗)

p^W

˜ s,˜s^∗

To explore how migration alters the allocation of factors (i.e. the cutoffs ˜sand ˜s^∗) as well as the terms of trade p^W, consider a reduction in migration cost γ > 0, which is reflected by a decline in the migration cutoff˘s^∗. From the inspection of Eq. (6.14) it follows that the influx of migrants is associated with a clockwise rotation of goods market equilibrium condition g(˜s,s)˘ in Figure 6.2. As an immediate consequence the relative world market pricep^W declines, while the industry cutoffs s˜and s˜^∗ go up. Interestingly, the intuition for these adjustments depend on the underlying specialisation patterns in both economies. If both countries are incompletely specialised (i.e. ˜s≤ s˜^∗ ≤s˘^∗), worker migrate between countries but within the skill-intensive sector. At a global scale then more workers are employed at Home’s superior technology, which – at a notionally fixed world market price – leads to an excess supply of the skill-intensive good and, hence, to a downward shift in the goods market equilibrium condition g(˜s,˘s^∗). For the world market to rebalance, the relative price of the skill-intensive goodp^W has to decline, which

–ceteris paribus – makes employment in both countries’ skill-intensive industries less attractive and, hence, results in an increase in s˜ and s˜^∗. For s˜ ≤ s˘^∗ < ˜s^∗ a different mechanism is in place. Since Foreign is specialised in the production of the skill-intensive good, workers migrate from Foreign’s labour-intensive sector into Home’s skill-intensive sector. Thus, at a global scale migration is associated with a shift of resources from labour to skill-intensive production. As before the skill-intensive production expands globally, which shifts the goods market equilibrium condition downwards and hence results in a smaller relative world market price p^W and higher cutoffss˜and s˜^∗. Summing up we can conclude:

Proposition 6.3.3 Migration from Foreign to Home is associated with a decline of the relative world market price and a reduction in the employment of domestic workers in the production of the skill-intensive good.

Proof Analysis and formal discussion in the text.

Note that the resulting migration pattern implicitly determines the trade flows between Home and Foreign. If a larger fraction of workers emigrate from Foreign, disproportionally more workers are employed at Home’s advanced technology in the skill-intensive sector. As a consequence the relative supply of the skill-intensive good from Foreign falls short of the corresponding relative supply from Home. With identical preferences, the free trade equilibrium then features skill-intensive exports from Home, and labour-intensive exports from Foreign.⁶.