“Now, as a nation, we don’t promise equal outcomes, but we were founded on the idea everybody should have an equal opportunity to succeed. No matter who you are, what you look like, where you come from, you can make it. Where you start should not determine where you end up.”Barack Obama.

## Abstract

Is it always worth implementing an open enrollment policy? And implementing policies that pursue equity in school supply? What is the impact of these two policies on the labor market? Do they produce efficient aggregate outcomes in both the labor and educational markets? This paper theoretically provides answers to these questions by studying the link between distortionary school supply policies and labor market performance. We build a two-sector labor market matching model, where the skilled segment of the economy is composed of workers who differ in the quality of the school they attended. We show the impact of government interventions to eliminate educational supply policy distortions within this theory. We demonstrate that both open enrollment and school equity policies have ambiguous effects on the labor market. Whenever their impact on the measure of workers choosing to become better educated is stronger than the additional school quality gains generated by the policy, the effects on the economy are negative. We also study the central planner solution, emphasizing the existing inefficiencies.

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## Notes

Notice that this sorting mechanism emerges whenever house prices respond to this optimal residential decision. See Nguyen-Hoang and Yinger (2011) for a literature review on the capitalization effect of school qualities into house values. See Chyn (2018) and Chetty and Hendren (2018) for empirical evidence for the U.S.

Belonging to this group are open enrollment policies, school vouchers, magnet and charter schools. See Reback (2008), Deming (2014) and Abdulkadiroglu et al. (2018) on the impacts of school vouchers, open enrollment policies and charter schools on student achievement in the United States. In turn, Park et al. (2015) present evidence on the impact of magnet schools on students’ performance in China. Machin and Salvanes (2016) evaluate the impact of an open enrollment educational policy reform in Norway. See Hoxby (2003) for an extensive literature review on school choice programs.

An additional advantage of promoting equity in schooling provision is that it reduces the burden of future compensatory income transfers to reduce wealth inequalities and improve social welfare. See Woessmann (2008) for a discussion of the benefits of equity policies in education.

This policy is increasingly being used by many education systems around the world. See Stasz and van Stolk (2007) and references therein on this point.

We call these two sectors interchangeably as skilled and unskilled or qualified and non-qualified.

We mean, with this assumption, that there is a random assignment of school places to students in case of oversubscription. We come back to this point later.

This occurs whenever the minimum school quality required by those who value education most is lower than the degenerate school quality provided by the government.

Notice that the increased demand for education implies that the best unskilled agents in our economy decide, after the school supply policy is announced, to be educate. Now, as the best unskilled individuals become the least educated agents in our economy, there is a reduction in the average quality of skilled workforce in our model. The average productivity of the unskilled sector does not change. This is the demand side effect present in our model.

In a closely related paper, Charlot et al. (2005) found that the individual returns to human capital accumulation are below the efficient outcomes. However, they argued that this was not due to a hold-up problem but instead due to wage and congestion externalities.

There is also a literature that studies the impact of signaling on schooling investments at the decentralized and decentralized equilibrium. Please, refer to Perri (2019) and references therein.

As previously mentioned, we are considering the worst students as the set of agents with higher schooling costs in the economy. The best students, in turn, are the group of individuals with lower educational costs.

Notice that the evidence that the public supply of school vacancies is not homogeneous, especially among individuals with different educational costs, and that the government uses lotteries as a mechanism for assigning school places to students in case of excessive subscription is abundant in the literature. See for instance Benabou (1996a), Benabou (1996b), Fernandez and Rogerson (1996), Nechyba (2003), Levitt (2006), Staiger (2014), Zhang (2016) and references therein on these two points.

Note that the average productivity of the skilled sector depends on both aggregate demand (\(Q_d\)) and supply of education (\(Q_o\)). We return later, in the aggregation subsection, to these variables.

The assumption of decreasing returns is becoming more and more common in search theory. See, for instance, Kaas and Kircher (2015) and references therein on this point.

Note that these are the stationary versions of the Hamilton–Jacobi–Bellman equations. We assume this simplified version because we are only interested in the steady state equilibrium of our model economy.

Notice that \(w_N^{'}(l_N) = \frac{\partial w_N(l_N)}{\partial l_N}\) and \(w_S^{'}(l_S) = \frac{\partial w_S(l_S)}{\partial l_S}\).

Workers who do not study receive \(b_N\) units of the consumption good as unemployment insurance and they move to employment status at a rate \(z(\theta _N)\).

Notice that the bargaining between workers and firms is on the marginal surplus generated by the additional worker. See Pissarides (2000) on this point.

Considering the assumption of strict concavity of the production function, it can be shown that workers’ productivity decreases while the number of employed workers increases. Thus, there is a reduction in the marginal cost of an additional worker, as firms become larger. See Smith (1999).

Alternatively, we could see the timing assumptions in our model as having two periods only; the period of schooling, which occurs from the beginning to period

*T*, and the other, which begins from that period on.This condition excludes the following limit cases: only individuals with the lowest cost of education will study, \(Q_d(I_L)=q_H\), and all agents decide to study, \(Q_d(I_H)=q_L\). The first case arises, for instance, when the cost of education is so high that no one except the agent with the lowest education cost succeeds in studying. The opposite occurs when \(Q_d(I_H)=q_L\). In this scenario, the whole society would be studying. Thus the unskilled sector would disappear in our model. We return later to this point.

Notice that \(Q_d(I)\) is continuously differentiable in \([I_L,I_H]\) and \(Q'_d(I)<0\) and \(Q''_d(I)>0\). Also, the higher the skilled unemployment option, \(U_S(q^e(Q_d, Q_o))\), the bigger \(Q_d(I)\) is.

Notice that we do not specify a microeconomic mechanism that establishes the agent’s exact school choice on the previous set. It is random in our model in order to comply with the educational supply policies literature. However, it should be mentioned that different school assignment mechanisms could be chosen, provided that the set of individual choices coincides with the previous schooling sector equilibrium set.

We only need to apply L’Hopital’s Rule to prove these results.

Consider that the schooling distribution is non-degenerate such that \(\mu < q_H\). This result can be used to explain the stylized fact that countries with low education levels tend to pay higher wage rates to their educated workforce. See Avalos and Savvides (2006), Birdsall et al. (1995), Bils and Klenow (2000), Psacharopoulos and Patrinos (2004) and references therein on this topic.

Notice that in this first analysis, we maintain the assumption that there is a lottery that randomly assigns individuals to schools in this extended set. Next, we evaluate the impact of this friction.

This case is presented at the top of Fig. 4 (when \([Q_o - q^e(Q_d, Q_o)] g(Q_o) > [q^e(Q_d, Q_o) - Q_d] g(Q_d)\)).

As there is an increase in the average quality of schools available to each agent after the open enrollment policy, it becomes more advantageous to study and be part of the qualified sector. The measure of agents in the unskilled sector is then reduced, also bringing a reduction in the unemployment rate in this sector. There is also an increase in the wage inequality between the two sectors with this policy.

Notice that the measure of schooled agents in the economy is now defined by \(1- H(I({\bar{q}})) = \int _{I({\bar{q}})}^{I_H} dH(I)\) instead of \(H({\tilde{I}}) = \int _{I_L}^{{\tilde{I}}} dH(I)\) with the random supply policy.

Please refer to “Appendix 6” for the solution of the central planner problem.

Our strategy in this section is to impose some efficiency conditions and check if the equilibrium remains efficient. Thus, in addition to imposing the Hosios Condition, we also assume \(\epsilon \rightarrow \infty\) and constant returns to scale to eliminate the large firms’ externality. See Cahuc and Wasmer (2001) and Smith (1999) on this point.

In sum, the problem of the social planner consists of defining the measure of individuals who study, \([Q^P, q_H]\). Then, the school to be attended by each individual is, as before, randomly defined in this subset. The social planner also defines \(v_i\) and \(l_i\), for \(i = \{S, N\}\). Notice that the expected productivity in this subset is now given by \(q^e(Q^P, q_H)=E[q \mid q \ge Q^P]\). In “Appendix 4” we present the detailed solution to this problem.

Note que \(d{\tilde{I}}/d\varepsilon >0\) como demonstrado no Apêndice A.

The Hosios Condition states that \(\beta _i = 1 - \eta _i\), for \(i = \{S,N\}\) and \(\eta _i = \frac{z'(\theta _i)}{z(\theta _i)} \theta _i\) represents the elasticity of the job matching with regards to a job vacancy. Notice that by imposing the Hosios Condition in the social planner problem we expunge the trade externality of our model.

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## Acknowledgements

We thank Michele Boldrin, Etienne Lehmann, Federico Etro, Fernanda Estevan, Marcelo Silva, Cézar Santos, Eduardo Azevedo and seminar participants at PIMES-UFPe and 37th SBE Annual Meeting, Brazil. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (Capes)—Finance Code 001—PRINT/Universidade Federal do Ceará.

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## Appendices

### Appendix 1

### Proof

For each \(\epsilon\), let \({\tilde{I}}(\epsilon )\) be the solution of \(Q_o({\tilde{I}}(\epsilon )) = Q_d({\tilde{I}}(\epsilon ))\). Now, consider an increase from \(\epsilon\) to \(\epsilon\)’ and let:

It follows that \(Q'_o(I) > Q_o(I)\), for all \(I \in (I_L,I_H)\). Since \({\tilde{I}} \in (I_L, I_H)\) for any \(\epsilon \ge 1\) we have that:

However, from the definition of \({\tilde{I}}(\epsilon ')\), we have that:

Adding the last two expressions and multiplying by \(\frac{1}{2}\) yields:

By the concavity of \(Q'_o\) and the strict convexity of \(Q_d\), it follows that:

From these two last expressions, we have that:

then, \({\tilde{I}}(\epsilon )<{\tilde{I}}(\epsilon ')\).

Now, we show that \(Q_d\) is decreasing in \({\tilde{I}}\). Consider that:

for:

and

Notice that changes in \(Q_d\), due to changes in \({\tilde{I}}\), occur through the term \(\nu _d\). We may rewrite \(\nu _d\) as:

where \(x = \frac{1}{I}\) and \({\tilde{x}} = \frac{1}{{\tilde{I}}}\).

Then, increases in \({\tilde{I}}\) are equivalent to reductions of \({\tilde{x}}\) and \(\nu _d\). This guarantees that the higher \(\epsilon\) is, the lower \(Q_d\) will be.

Now we verify the impact of \(\epsilon\) on \(Q_o\). Consider that:

for

As previously, consider that:

and

for all \(I \in [I_L,I_H]\). Then,

where \({\tilde{H}}(x)\) is the distribution of transformation \(x = (I-I_L)/(I_H-I_L)\).

Differentiating this last expression with respect to \(\epsilon\), we get

The sign of the previous expression is derived from: \({\tilde{x}}^{\epsilon } - \nu _o < 0\), \(0< x < 1\) and \(d {\tilde{x}}/d \epsilon \propto d {\tilde{I}}/d \epsilon\).

\(\square\)

### Appendix 2

### Proof

To simplify the proof, consider that \(q^{e}(Q_{d},Q_{o})=q^{e}\). We have that:

By using the fact that \(\frac{\partial q^e(Q_d, Q_o)}{\partial Q_d} > 0\), \(\frac{\partial q^e(Q_d, Q_o)}{\partial Q_o} > 0\) and Proposition 1, we have that all previous derivatives are positive except \(dQ_{d}/d\varepsilon\).

We now rewrite the previous expression as:

Therefore, we have that \(dq^{e}/d\varepsilon >0\) if and only if:

To complete the demonstration we need the following result.

### Lemma

For any values of \(Q_{d}\) and \(Q_{o}\) in \([q_{L},q_{H}]\) and \(\varepsilon \in [1,\infty )\), we have:

### Proof

By definition we have:

\(\square\)

where \(I(\varepsilon )\) is such that \(Q_{d}(I(\varepsilon ))=Q_{o}(I(\varepsilon ))\). Differentiating the previous expression with respect to \(\varepsilon\) yields:

It is easy to shown that for any \(\varepsilon \in [1,\infty )\), \(Q_{d}(I(\varepsilon ))<Q_{d}\). In addition, from the proof of Proposition 1 in “Appendix 1”, it is shown that \(dI/d\varepsilon >0\). Thus we have:

By using the definition of \(Q_{o}\), we have:

Differentiating this previous expression with respect to \(\varepsilon\) gives us:

Manipulating the previous expression:

By Adding and subtracting \(q_H\) in the term between brackets, we have:

Note that the above expression is positive, since \(Q_{o}(I(\varepsilon ))>Q_{o}\) for any \(\varepsilon \in [1,\infty )\) and \(\ln [(s-I_{L})/(I_{H}-I_{L})]<0\) for any \(s<I_{H}\). Finally, using (27) and (29), we have:

By definition \(Q_{o}(I(\varepsilon ))=Q_{d}(I(\varepsilon ))\) such that \(Q_{o}(I(\varepsilon ))+Q_{d}(I(\varepsilon ))=2Q_{o}(I(\varepsilon ))\). Thus, we have \(Q_{o}(I(\varepsilon ))+Q_{d}(I(\varepsilon ))=2Q_{o}(I(\varepsilon ))>2Q_{o}>Q_{o}+Q_{d}\). Therefore:

\(\square\)

Using the previous result in the proposition, we have that:

or equivalently:

\(\square\)

### Appendix 3

### Proof

The expressions that characterize \(l_{S}\) and \(l_{N}\) are respectively given by:

where

Therefore we have that

By definition we have:

Using (30) and (31) in (32) and (33) we concludes that:^{Footnote 41}

This shows the first part of the corollary.

From Eqs. (20) and (21) we have:

Notice that the previous results depend on the assumption of decreasing returns to scale. \(\square\)

### Appendix 4

### Proof

Initially, it should be mentioned that:

Thus, it suffices to show that \({\bar{q}}>Q_{o}\). By definition, we have that:

As \(h(I)\ge 0\) for any \(I\in [I_{L},{\tilde{I}}]\), the mean value theorem for integrals guarantees that there exists \(\delta \in [I_{L},{\tilde{I}}]\) such that:

Therefore, if \({\bar{I}}<\delta\) then:

which implies the desired result. \(\square\)

### Appendix 5

### Proof

The condition \({\bar{q}}>q^{e}(Q_{d},Q_{o})\) implies that \({\bar{l}}_{S}<l_{S}\). On other hand we get \(F({\bar{I}})<F({\tilde{I}})\). Therefore by usin the definition of unemployment rate we see that

The components on the right side of (1) are both positives. The first one represents the loss of employment due to productivity gains and the other is the reduction of the mass of students required for reach that gains. The final effect on the unemployment rate depends on which the effect is dominant. Finally in the unskilled sector there is no effect on the employment but, due to the decreasing of the mass of students the unemployment rate rises. These arguments shows item a).

From the Eqs. (20) and (21), we have:

Using the facts shown in the item a) and decreasing returns to scale we have \({\bar{w}}_{S}-w_{S}>0\). The wage in the unskilled segment do not depends of the changes in the productivity such that \({\bar{w}}_{N}=w_{N}\). That finish the proof.

\(\square\)

### Appendix 6

### Proof

We initially solve the centralized problem. Consider again the problem presented in (26). Manipulating the previous constraints, the central planner problem can be redefined as:

Assume that the Hosios Condition is met.^{Footnote 42} The set of expressions that characterize the social optimum is given by:

where:

and \(\theta ^{P}_S\), \(\theta ^{P}_N\) and \(Q^P\) represent the efficient values of the market tightness in the skilled and unskilled sectors and the efficient mass of skilled individuals, respectively.

Notice from (30) that the cost of opening a new vacancy in the unskilled sector equalizes the social surplus generated by this sector. This expression defines the optimal steady-state value of \(\theta ^{P}_N\). The following equation characterizes the efficient value of \(\theta ^{P}_S\). Finally, expression (32) determines the optimal mass of the educated workforce that generates the efficient outcome. \(\square\)

### Appendix 7

Let

It can be seen that all previous expressions are increasing in \(\theta\). Furthermore, note that for any \(\theta\):

Consider that \(\alpha _N = \alpha _S = 1\), \(C_N = C_S = \frac{1}{(1+ \rho )}\) and the Hosios Condition are satisfied. In this case, the equilibrium expressions of \(\theta _N\) and \(\theta ^P_N\) satisfy, respectively:

By using (34) and (35) we have that:

At the limit, when \(\rho \rightarrow 0\) and using (33), we conclude that:

Then \(\theta ^P_N < \theta _N.\)

By using the same reasoning for the expressions that define \(\theta _S\) and \(\theta ^P_S\) we have that:

By definition, \(q^e (Q^P) = q^e (Q^P, q_H).\) Thus:

Therefore, if \(Q^P > Q_d\) then \(q^e (Q^P) > q^e(Q_d, Q^o)\). By using (36) we have that:

Finally, taking the limit when \(\rho \rightarrow 0\), we have:

Then \(\theta ^P_N < \theta _N.\)\(\square\)

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Benegas, M., Veras Corrêa, M. Educational supply policies: distortions and labor market performance.
*J Econ* **129**, 203–239 (2020). https://doi.org/10.1007/s00712-019-00686-4

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DOI: https://doi.org/10.1007/s00712-019-00686-4