Differentiating implicitely equation (10) yields the partial derivatives of . The derivatives with respect to
and
remain unchanged with the exception of the definitions of
and
. Therefore, only the derivatives with respect to
and
are calculated. They read
Equations (11) and (12) show that the contribution rates only depend on variables related to the respective region. Wages have no impact because both, revenues and expenditures, depend linearly on the respective wage. Solving for and differentiating partially yield:
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Model TR consists of the characteristic submodel described above, and of the submodels determining employment and wages in both regions. The relevant equilibrium conditions are thus equations (1) and (6) respectively for region 1 and region 2, and equations (10), (11) and (12). Figure 2 summarises the partial effects of the endogenous variables on one another. As the formal analysis shows, under the assumptions set above, all effects can be derived unambiguously. The only link between the regions is migration, symbolised by the variable . If the situation of workers in region 1 improves by lower UI contributions, higher gross wages, or higher employment, immigration from region 2 increases. This lowers the equilibrium UI contribution rate in region 2, which has an impact on the wage rate and consequently on employment. As can be seen, all variables mutually depend on each other. The complexity of the simultanous equations brings about that the total effects of variations of exogenous variables cannot be determined in general. Therefore, a comparison of the models TC and TR is undertaken only in the calibrated form of the models (section 5).