Optimal Tax Policy Without Skill-specific Taxes

In this case, the government has limited information, cannot implement skill-specific tuition fees and has to rely on a linear income tax. FOC (32) is to be canceled and the parameterf_B is equal to zero throughout equations (33) to (35).

Then, let us define the net social marginal value of income (including the income effects on the tax base) of a household of typej as

b^j = α^j

λ +t·w_j·Z_j^∗·∂Z_j

∂T +t·(w_H ·H^∗−w_L·L^∗)·p_e· ∂e

∂T, j =H, L, (36) where Z_j = H, Lforj = H, L. The second summand on the RHS of equation

(36) represents the loss in tax revenue due to an income-effect induced decrease in labor supply and the third summand incorporates the revenue effect from taxing the skill premium, when the household adjusts its learning effort and therefore its probability of getting employed as a skilled worker.

Theexpectednet social marginal value of income is given from (36) by

¯b = p^∗·α^H + (1−p^∗)·α^L

λ +p^∗·t·w_H ·H^∗· ∂H

∂T (37)

+(1−p^∗)·t·w_L·L^∗· ∂L

∂T +t·(w_H ·H^∗−w_L·L^∗)·p_e· ∂e

∂T. Slightly rearranging FOC (33) and inserting the definition of¯b from equation (37), it is straightforward to show that for the expected net social marginal value of income it must be

¯b = 1. (38)

Next, we define the insurance characteristic as the negatively normalized co-variance of net social marginal value of incomeb^jand labor incomew_j·Z_j, being analogous to Feldstein’s distributional characteristic and measuring society’s con-cern of avoiding risk. Hence, the insurance effect is given by:

χ=− Cov(b^j, w_j ·Z_j)

¯b·(p^∗·w_H ·H^∗+ (1−p^∗)·w_L·L^∗) >0, (39) being positive, because the social net marginal value of income is decreasing in income.

Moreover, we define

²_HH = p^∗·w_H ·H^∗

p^∗·wH ·H^∗+ (1−p^∗)·wL·L^∗ · (1−t)w_H

H^∗ ·S_HH >0, (40)

²LL = (1−p^∗)·wL·L^∗

p^∗·w_H ·H^∗+ (1−p^∗)·w_L·L^∗ · (1−t)wL

L^∗ ·SLL >0, (41) as weighted compensated elasticities of labor supply with respect to its net wage, whereS_jj > 0represents the substitution effect in labor supplyZ_j. The weights are the share of skilled respectively unskilled labor income in aggregate labor income.

The compensated elasticity of learning effort with respect to a change in the expected net wage(1−t) ¯wis displayed as

²_ew¯ = (1−t) ¯w

e ·S_ew¯, (42)

wherew¯ =p^∗·w_H + (1−p^∗)·w_L.

Applying equations (38) to (42), some covariance rules and the Slutsky-decomposition, FOC (34) can be transformed in order to receive¹⁷

t

1−t = χ

²_HH+²_LL+ _p∗·w^p∗H^(·w·H^H∗·H+(1−p^∗−w∗L)·w^·L∗L⁾·L^∗ ·²_pe·[²_ew¯−ψ·η_eT] ≥0, (43) where ²_pe = _p∗(e,E)^e ·p^∗_e > 0 is the elasticity of the probability function p(e, E) concerning a change in learning efforte, ψ = Cov(wj,Zj)

¯

w·[p^∗·H+(1−p^∗)·L] >0, j = H, L, is the coefficient of correlation of labor supply and wages, being positive as long as labor supply is not backward-bending, andη_eT = ^(1−t)·_e^w·¯Z¯ ·_∂T^∂e <0is the income elasticity of learning effort with respect to a change in expected net wage income.

As expected, the optimal labor tax rate is increasing in society’s concern for in-suranceχand the only role of taxation is to insure against educational risk and ex post inequality, because the tax rate would be zero, if the insurance characteristics vanishes (χ= 0). Furthermore, the tax rate is decreasing in induced distortions in skilled and unskilled labor supply, measured by the elasticities²HHand²LL.

Last, but not least, wage taxation has a negative effect on compensated invest-ment in learning, as¹⁸

²_ew¯ −ψ·η_eT >0. (44) This effect translates via ²_pe in a change in quality of education, i.e., the success probability, and, weighted by the expected skill premium in wages (relative to expected income), the third summand in the denominator of the RHS in equation (43) then measures the income- (or revenue-)relevant effect of wage taxation on learning effort. The higher these distortions are, the lower the optimal tax rate should be as well.

17See Appendix A.1 for an explicit derivation of equation (43).

18See Appendix A.2 for a proof of the inequality in equation (44).

Turning to optimal resource investment, this can be derived from multiplying FOC (35) byE, rearranging and recognizing thatt^∗ >0from (43) as

E^∗ =p^∗·

µV^H −V^L

λ ·²_pE+t^∗·[w_H ·H^∗−w_L·L^∗]·[²_pE+²_pe·η_eE]

¶ , (45) where²_pE = _p∗(e,E)^E ·p^∗_E >0is the elasticity of the success probability with respect to a change in public spendingE, and whereη_eE = ^E_e · _∂E^∂e >0is the elasticity of learning effort with respect to public educational expenditure.

Thus, there are three effects, determining optimal resource investment: The first summand in the bracket on the RHS of (45) represents the welfare increase, net of financing costs λ, by an additional household getting into the skilled sec-tor due to an improved quality of the educational secsec-tor. Note that, contrary to the First-best solution, the skill premium in (indirect) utility, V^H −V^L, must be positive, in order to have positive effort investmente >0by households.

The second term,t^∗·[wH ·H^∗−wL·L^∗]·²pE, measures increased tax revenue and, accordingly, the self-financing effect, because increasing educational invest-ment will increase the number of skilled tax payers by²_pE, paying each additional taxes on the skill premium in wage income.

Finally, the third term, t^∗ ·[w_H ·H^∗−w_L·L^∗]·[²_pe·η_eE], makes clear that investing in the quality of the educational sector is another way to foster learning effort e (due to the complementarity in η_eE) and to increase the probability of success. On the one hand, this increases ceteris paribus the number of skilled tax payers again, but on the other hand, and being more important, the increased public spending can mitigate partially distortions in learning effort, caused by implementing a wage taxt^∗ >0.

Summarizing our discussion in this subsection so far, we can conclude:

Proposition 2. If skill-specific tuition fees are not available (f_B = 0), the govern-ment will implegovern-ment a positive wage tax ratet^∗ >0, balancing income insurance on the one hand and distortions in labor supply and learning effort on the other hand.

Optimal resource investment in the educational sectorE^∗ aims to exploit the skill premia in utility respectively income by increasing the success probability

and it mitigates tax-induced distortions in learning effort by an increased quality in the educational sector.

In order to describe the full tax policy, we have to determine the optimal lump-sum transferT^∗. From the governmental budget constraint (30), we have in case offB = 0

T^∗ =t^∗·w_L+L^∗+t^∗·p^∗·[w_H ·H^∗−w_L·L^∗]−E^∗. (46) Inserting the optimal value ofE^∗from (45), we end up with

T^∗ =t^∗ w_LL^∗−p^∗²_pE V^H −V^L

λ −t^∗ p^∗(w_HH^∗−w_LL^∗) (²_pE+²_peη_eE −1). (47) From equation (47), we reason

Proposition 3. If either the elasticity of the success probability with respect to educational investment²_pEor its complementary effect via learning effort²_pe·η_eE or at least the sum of both effects is elastic, ²_pE +²_pe ·η_eE > 1, more than the entire tax revenue from the skill premium is invested in the educational sector and there are no direct (net) transfers from the skilled to the unskilled. Consequently, unskilled workers are net tax payers.

Proof. First, from the conditions on²_pE+²_peη_eE >1in Proposition 3 and equa-tion (47) it follows that the maximum lump-sum transfer is T = t^∗ w_LL^∗, thus all tax payers are maximally entitled to be reimbursed for their payment on un-skilled labor income. Hence, the entire revenue from taxing the skill premium w_HH^∗ −w_LL^∗ must be spent onE^∗. Moreover, there are deductions from this maximum transfer, which have to be invested into quality of education as well in order to guarantee a balanced budget of the government. This proves the first statement in Proposition 3.

Due to this spending in the educational sector, there is no money left for cash-transfers from the skilled to the unskilled. In fact, even if the lump sum transfer T^∗ should be positive, unskilled workers are net tax payers, because their overall

tax paymentT P_Lis equal to

T P_L = t^∗·w_L·L^∗−T^∗ (48)

= p^∗ ²_pE V^H −V^L

λ +t^∗p^∗(w_HH^∗−w_LL^∗) (²_pE +²_pe η_eE−1)>0, given the conditions met in Proposition 3. This proves the last statement.

No matter, whether the lump-sum transferT^∗remains (somewhat) positive or turns even into a lump-sum tax, Proposition 3 implies that all households have to pay for the educational system, at least as long as the success probability is sufficiently elastic. Unskilled workers are made better off ex ante by increasing the quality of education due to investing (strongly) into universities instead of providing income transfers ex post.

Thus, the optimal tax and education policy in case of unavailable skill-specific tuition fees can be summarized as decreasing the variance in wage income by implementing a labor tax and trying to get as many households employed in the skilled sector as possible – given direct resource costs and excess burden caused.

This provides ex ante insurance by increasing the success probability and ex post insurance by decreasing income variation. Both increase the expected utility of each household. However, net cash transfers from skilled to unskilled do not take place, because they are too expensive and ex post income transfers matter less than ex ante improving quality of education.

In the next subsection, we are going to extend the instruments of the govern-ment for skill-specific tuition fees and we will show that this fosters the focus on quality of education even more.