Maximizing the natives’ wage sum

Justman and Thisse’s (2000) suggestion of a potential source of underinvestment in public education, namely high-skilled labor mobility, is not the only lesson to learn from their analysis. The demonstration that assumptions about local government objectives drive the results is just as important. In their model,

“[decentralization] leads to under-investment in education when political inter-ests are predominantly defined in geographic terms and local governments act to maximize regional output, but may lead to over-investment when the political in-terests of native-born highly educated are well represented” (Justman and Thisse, 2000, p. 255). Reassessing this issue in the context of the extended model in the present paper as well seems worthwhile.

Think of an objective function that considers only the native-born skilled popu-lation – which studies and works either in the home country or abroad – and the locally used immobile factor (or rather respective factor incomes). The optimiza-tion problem then becomes

maxsi

Φ^N = P_iⁱⁱsif1(hi, li) +P_i^ijsif1(hj, lj) +P_i^jisjf1(hi, li)

+P_i^jjs_jf₁(h_j, l_j) +l_if₂(h_i, l_i)−cs_i. (2.24)

TheP’s represent the numbers of workers who are born in i(subscript) and who have studied either in i orj (first superscript) and work either ini orj (second superscript). Note that

P_iⁱⁱ= m−s_i(w_j−w_i)

∆m Γ_i , P_i^ij = (w_j−w_i)s_i−m

∆m Γ_i ,

P_i^ji = (w_i−w_j)s_j −m

∆m (1−Γ_i) , P_i^jj = m−s_j(w_i−w_j)

∆m (1−Γ_i).

The first-order condition for the maximization reads

Calculating the derivatives of the P’s with respect to education policy (refer to the Appendix), evaluating the whole condition at a symmetric equilibrium, and using (dh_i/ds_i)equ. and (dh_j/ds_i)equ. from section 2.2 and the properties of

where p0 = Γequ. = m0/∆m0 reflects students’ propensity to stay in their home country, and 1−p0 = 1−Γequ. = −m₀/∆m0 refers to students’ preference to study abroad in equilibrium.

Different spillover effects are again likely to cause a deviation of decentralized from globally optimal policy. Note that now, the migration behavior of students, as mainly represented by p0, also plays a decisive role. Two different types of spillovers can be distinguished. The first one can be illustrated by inspecting especially the first term on the left-hand side of (2.26) and ignoring the wage-rate-related effects (second term) for a moment: if the wage differential between countries were not affected by a change in education policy, the local first-order condition would be

f₁p₀ =c. (2.27)

Since only those natives benefit from the increased expenditure in their home region who stay there as students, a stay rate p₀ smaller than one implies that

regions (exclusively interested in natives’ incomes) only receive part of the invest-ment’s total marginal benefit from a global point of view, while bearing the entire marginal cost. Compared to the centralized solution, this exclusive focus on na-tives generates the local underinvestment that technically follows from (2.27) due tof₁ > c and the assumptions on the production function. In other words, local governments ignore the spillover effect of education expenditures on foreign stu-dents who benefit from increased incomes in the future due to better education.

This result is basically in line with the one presented by B¨uttner and Schwager (2004) in a model with interregionally mobile students in a federation (graduates are assumed to stay in the region where they were educated) and a comparable local objective function. Tuition fees at the federal level would in their model mitigate this underinvestment, as local governments had an increased incentive to attract students by quality in order to raise revenue from fees.

The second type of spillover is related to the fact that local governments ignore the effect of a marginal policy change on the earnings of nonnative skilled workers and the immobile factor abroad: the second term on the left-hand side of (2.26), which is unambiguously nonnegative for the assumed parameter range 1/2 ≤ p₀, p ≤ 1, reflects a local incentive to overinvest in education. Overall, taking both kinds of spillovers into account, it is a priori – again – not clear in which direction local policy might deviate from the efficient solution, as the signs of the spillovers are opposing. For p₀ = 1 (i.e., no student migration in equilibrium), the result is unambiguous, as the B¨uttner–Schwager type of spillover described earlier vanishes. The local optimality condition then looks like the one presented by Justman and Thisse (2000, p. 256), implying local overinvestment:

f₁−sf₁₁

(1−p)

2 (dh_i/ds_i)_equ.−1

| {z }

(−)

=c. (2.28)

Note that p₀ = 1 does not mean there cannot be any student migration at all.

It only says that with equalized education qualities among regions, no student wants to study abroad. There can be migration in out-of-equilibrium situations.

Effectively p₀ = 1 also means that students do not have any other migration motives than income-related ones.

Finally, I present a numerical example illustrating the results for different param-eter values p₀ and p. As can be seen in figure 3, where the light-colored plane

indicates the globally optimal expenditure levels^∗ which is independent of both parameters, and the dark-colored surface represents the local equilibrium expen-diture s for different (p₀, p) combinations, again, in principle three scenarios – overinvestment, underinvestment, and globally optimal investment – may result from decentralized education policy.

Note: Specification: A= 1.4,α= 0.7, l=1, ∆m₀= 1, ∆m= 1,c=1

Figure 2.3: Local education investment with governments maximizing natives’

wage sum

As proved analytically by means of the condition (2.28), for p0 = 1, local gov-ernments overinvest for all given levels of p in the interval (1/2,1). The lower the stay ratep0 of students in equilibrium, however, the more likely there will be local underinvestment. This result can be traced back to the first type of spillover as discussed above (the one that also drives the result in B¨uttner and Schwager, 2004), which is enforced by a lower student stay rate.¹⁰

10 In order to really observe the illustrated effect, the absolute value of the wage elasticity must not be too high, because otherwise the opposing effect ofp0on the wage-rate-related spillover dominates:

sgn(ds/dp₀) = sgn{f1+sf₁₁[(2p−1)[2 (dh_i/ds_i)_equ.−1]]}= 1 only if

−wh<{(2p−1)[2 (dhi/dsi)_equ.−1]}⁻¹.

The curvature of the local expenditure surface along the p-dimension can hardly be explained intuitively any more. The wage-rate-related effect of a marginal policy change (second term on the left-hand side in (2.26)) is affected by the graduates’ stay rate through the actual residency of natives (as finally reflected in the factor 2pp₀ −p −p₀) and through its effect on the international wage differential’s sensitivity to the policy change (as represented byf₁₁[2 (dh_i/ds_i)_equ.− 1] = (^∂w_∂sⁱ

i −^∂w_∂s^j

i)equ.).

The important message to take away from the illustration is that the unambigu-ous local overinvestment in Justman and Thisse (2000, ch. 4.2) is reversed into underinvestment once student mobility exceeds some critical threshold.