The results of the numerical simulations rely on the assumptions about the functional form of the utility function and the production function, as well as on the choice of the parameter values. In this appendix the robustness of the results for changes in these assumptions are discussed.
The Utility Function
The choice of the log-linear utility function was made for simplicity only. The exact functional form of the utility function is not very important for the results. As long as the utility function satisfies the Inada conditions, changes in lifetime income result in changes in utility, because a higher (net) income allows to reach a higher isoquant curve. Furthermore, also for other specifications of the utility function, more income in the first period of life results in larger savings. But of course, the numerical results may differ if another functional form of the utility function is used. In particular, the speed of convergence of capital presented in Fig. 5 could be different, affecting the speed of convergence of other variables of interest.
Composite Goods Technology
The composite goods, which enter as arguments in the utility function are produced using a Cobb–Douglas technology [Eq. (4)]. If a more general function allowing for a constant elasticity of substitution is used, the allocation of capital to the countries described by Eq. (23) would not be constant any more. It would depend on the elasticity of substitution between the goods produced in the countries. If the goods produced in both countries are substitutes, immigration into a country leads to a smaller negative effects on the price, which induces a reallocation of capital to that country. As in this case capital flows follow labour, migration will be larger and thus the positive overshooting in Figs. 7, 8 and 9 can be larger. In the limiting case that the goods produced in the countries are perfect substitutes, the model may produce corner solutions where both labour and capital are concentrated in one of the countries. If the goods produced in the countries are complements, even a small migration flow to a country would lead to a large drop of price of goods produced there, inducing a reallocation of capital to the other country. As a result, the degree of migration would be smaller. However, the direction of the migration flows is unaltered. As the different allocation of capital has a very moderate effect on welfare, the welfare effects remain almost the same for realistic parameter values.
The Production Function
The effects of using a more general CES form instead of the Cobb–Douglas production function are exactly opposite to those of using a more general function for the composite goods technology. That is, if capital and labour are substitutes, migration to a country would lead to an outflow of capital from it, and if capital and labour are complements, capital an labour move in the same direction. However, the development of the capital stock, which drives most of welfare effects, remains similar to that presented in Fig. 5.
Fully Inflexible Pensions
If the inflexible country has a completely inflexible pension scheme, i.e., \(\eta = 1\) then in the case of imperfect mobility there is a positive spillover on the generation born at \(t = 0\) only, because at \(t = 1\) the interest rate is higher. However, the spillovers on the generations \(t\ge 1\) are definitely negative, because in this case pension benefits in country F are never reduced and the total amount of capital in the union does not overshoot. With perfectly mobile labour the presence of a completely inflexible neighbour is definitely negative, because it shares all the short run losses with the Home country and does not over-accumulate capital as it does with \(0<\eta <1\). With a large \(\eta \) (but \(\eta <1\)) the inflexible country F may negatively affect more generations, compared to the case discussed, but still after a negative short-run spillover future generations are affected positively.
The size of the migration flows depends on the parameter values. In both cases when labour is perfectly and imperfectly mobile, migration is larger when \(\xi \) is smaller, so the pension system is closer to DB, because then migration to country F reduces taxes there and raises taxes in country H more, which makes the Foreign country even more attractive. The case is symmetric for migration to country H. So, in panel b of Fig. 3, large migration to country F at time \(t = 3\) is followed by relatively small migration at time \(t = 4\). In the next period, migration is larger again and so forth. also When mobility is perfect (panel a), the overshooting in migration at time \(t = 2\) is even larger. So a lower value of \(\xi \) leads to a slower convergence to the steady state. However, the direction of migration is the same, and the further results remain unchanged. When \(\xi \) tends to unity, the overshooting disappears and the convergence to the steady state is very fast and monotonic. If \(\xi =1\) and the countries run pure DC pension schemes, migration is smaller but still exists. Different values of the parameters \(\alpha ,\,\rho ,\,\gamma \) and \(\phi \), affect the results quantitatively but not qualitatively.