3.5 Appendix: Robustness Checks Individual Data
4.2.1 Closed Economy: Exogenous Returns to Skills
I posit a representative agent maximizing life-time utility over two periods, t = 1,2, within a closed economy.4 In period 1 resource windfall gains, R, are easing the bud-get constraint and the individual trades off human capital investments, h, and labor supply, n1 = 1−h. Educational costs are made up of both forgone earnings and di-rect educational costs, C(h) > 0. Without loss of generality, following Rea Jr (1977), I postulate a linear cost function for human capital formation, C(h) = αh. Educa-tional investments translate into further productivity and labor income in the future, according to the following function, w2 = φ(h) while incomes in period 1 are totally exogenous, w1 ≤ w2. Similar to Eaton and Rosen (1980), I assume that the returns to human capital investments are positive, φ0(h) > 0, but decreasing, φ00(h) < 0. In period 2, time is exclusively devoted to labor supply,n2 = 1, in the first place. However, in a second scenario discussed below, the individual trades of labor supply and leisure in period 2, which implies that n2 = 1−l. In light of this framework, without loss of generality, individuals decide self-responsibly about educational investments rather than delegating educational decisions to parents.
Formally, the representative agent chooses consumption in period 1 and 2, ct, and
4Similar models were set out by Becker (1962), Heckman (1976), Eaton and Rosen (1980), Rea Jr (1977) and Acemoglu (2017).
4.2. Theory 149
educational investments in period 1, h, in order to maximize life-time utility,
maxct,h 2
X
t=1
βt−1logct (4.1)
subject to his life-time budget constraint
2 where βt−1 equals the discount factor and τ represents a proportional labor income tax rate which is time invariant. Apparently, human capital investments exclusively impinge on the earnings potential such that educational investments do not depend on the specific functional form of utility. This is commonly referred to as separation theorem, originally laid out by Hirshleifer (1970). The first order conditions are given by:
Equation 4.3 equates the intertemporal marginal rate of substitution and the relative price, whilst equation 4.4 indicates that the optimal investment in human capital is characterized by the equality of returns to educational investments and marginal ed-ucational costs. The latter are made up of direct educational costs, α, as well as opportunity costs of educational investments, (1−τ)w1.
In light of further fiscal capacity in the course of a resource boom, the state gov-ernment might lower educational costs or might ease the household budget constraint through unconditional transfers or a decline in proportional tax rates. The educational effects of these policy options are discussed below in the course of three propositions.
Proposition 1: A resource windfall which lowers educational costs promotes
edu-cational investments.
Proof: Totally differentiating equation 4.4 with respect to α while taking into account that n2 = 1 yields
∂h
∂α = (1 +r)
(1−τ)Φ00(h) <0 (4.5)
According to this inequality, resource windfall gains which are invested into the quality of the educational system are unambiguously conducive to human capital investments as long as marginal educational costs are reduced since Φ00(h)<0.
Proposition 2: A resource windfall spilling into unconditional transfers might lead to a decline in educational investments as long as labor supply is endogenous.
Proof: Since the individual simultaneously decides upon human capital invest-ments and labor supply, the effects of lump sum resource transfers on educational investments and labor supply have to be evaluated concurrently. After differentiating equation 4.4 with respect to windfall gains while taking into account that the time devoted to work in period 2 is made up of the residual n2 = 1−l under endogenous leisure (l denotes the time devoted to leisure) and ∂U∂l >0, I wind up with the following equation:
Φ0(h) ∂l
∂R = Φ00(h)∂h
∂R(1−l) (4.6)
Accordingly, human capital investments and resource windfall gains are negatively as-sociated, ∂R∂h < 0, as long as the demand for leisure and resource windfall gains are positively related, ∂R∂l > 0. This holds under the sufficient condition that returns to skills are positive but decreasing. If leisure is a normal good, exogenous resource wind-fall gains lower labor supply and returns to skills in the future. While encountering lower returns to skills, individuals invest less in education at the present. Therefore, resource windfall gains serve as an impediment rather than a propeller for human cap-ital development.
4.2. Theory 151
Proposition 3: A resource windfall leading to a decline in proportional tax rates is neutral regarding educational investments under exogenous labor supply and conducive to educational investments under endogenous labor supply (if educational costs are fully deductible in both cases).
Proof: I have to separate two cases. In the first case, labor supply is exogenous in period 2, l = 0. Totally differentiating equation 4.4 with respect to τ yields
∂h
∂τ =− w1(1 +r)
Φ00(h)(1−τ)+ Φ0(h)
Φ00(h)(1−τ) (4.7)
As the first order conditions imply thatw1(1 +r) = Φ0(h) if educational costs are fully deductible, it directly follows that
∂h
∂τ = 0 (4.8)
Conspicuously, proportional labor income taxes are neutral regarding educational in-vestments. The neutrality of labor income taxation is due to the fact that the costs of educational investments, forgone wages, and the benefits of educational investments, gained wages, are equally affected through proportional labor income taxation. This result has been similarly derived by Eaton and Rosen (1980).
In the second case, labor supply is endogenous, n2 = 1−l. Again, totally differ-entiating equation 4.4 with respect to R while taking into account that n2 = (1−l) yields
Φ0(h)(1−τ)∂l
∂τ + Φ0(h)(1−l)−w1(1 +r) = Φ00(h)∂h
∂τ(1−τ)(1−l) (4.9) However, as long as educational costs are fully deductible, Φ0(h)(1−l) and w1(1 +r) coincide and ∂h∂τ as well as ∂τ∂l are negatively associated. Namely, under endogenous leisure, taxation unequally affects the opportunity costs of acquiring human capital at
the present and the returns to human capital acquirement in the future. Returns to skills are directly affected by labor income taxation and indirectly through a decline in labor supply. Hence, the abrogation of labor income taxes unfolds neutral (exogenous labor supply) or even positive (endogenous labor supply) educational effects. The lat-ter are strengthened even more if educational costs are not fully deductible (King and Rebelo (1990), Rebelo (1990)).
The policy interventions in response to resource booms discussed above are par-ticularly relevant for the specific case of Alaska. As a consequence of further fiscal capacity in the course of the oil boom, Alaska put in place the Alaska Permanent Fund which is equivalent to an unconditional transfer scheme. Moreover, the state govern-ment enacted several tax reforms which were supposed to abrogate all state income taxes. While the theory suggests a decline in educational investments as a consequence of the Alaska Permanent Fund, abolishing progressive income taxes is conducive to ed-ucational attainment as net returns to skills are increased in the future.
Thus far, I exclusively focused on a closed economy. However, in an open economy, resource windfalls lead to further dampening effects on educational investments. These effects are discussed in the following section.