Educational Signaling in Two Different Education Systems

Proof of Proposition 1.

First of all, we rewrite equation (1) as the following function:

where F : t 0 , t n × 0,1 × 0 , + ∞ × 0 , + ∞ × 0 , + ∞ → R is the real valued function defined in equation (A.1). If we derive this function with respect to each variable, we obtain:

It is easy to see that ∂ F t ∂ t > 0 ∀ t ∈ t 0 , t n , F t 0 < 0 and F t n > 0 under assumptions 1, 2 and 3, which implies that there exists a unique t * ∈ t 0 , t n such that F t * = 0 . Likewise, ∂ F t ∂ t < 0 ∀ t ∈ t 0 , t n , F t 0 > 0 and F t n < 0 under assumptions 1′, 2′ and 3′, which implies that there exists a unique t * ∈ t 0 , t n such that F t * = 0 . Therefore, in the non-demanding and demanding educational systems, there is a unique t * ∈ t 0 , t n such that a separating equilibrium arises in which those worker’s types lower than t* choose e₀ and those types greater than t* choose e₁.

Additionally, since t 0 + t * 2 P 0 > 0 ∀ t * ∈ t 0 , t n , then ∂ F ∂ δ < 0 so that equation (A.1) is satisfied. Finally, as shown by equations (A.5) and (A.6), ∂ F ∂ ρ < 0 and ∂ F ∂ β > 0 . Thus, if we use the implicit function theorem, we obtain the desired results under Assumption 3:

Under Assumption 3′, these inequalities are reversed because ∂ F ∂ t * < 0 and this completes the Proof of proposition 1. QED.

Proof of Proposition 2.

From equation (A.4), we see that ∂ F ∂ P 0 > 0 when ρ = 0 and ∂ F ∂ P 0 decreases with ρ. Then, there will be a threshold, K ∈ R + , such that ∂ F ∂ P 0 > 0 when ρ < K, whereas ∂ F ∂ P 0 < 0 when ρ > K. Once again, we use the implicit function theorem in order to obtain the desired results. In particular, under assumptions 1, 2 and 3, we conclude that ∂ t * ∂ P 0 < 0 when ρ < K and ∂ t * ∂ P 0 > 0 when ρ > K. Likewise, under assumptions 1′, 2′ and 3′, we obtain that ∂ t * ∂ P 0 > 0 when ρ < K and ∂ t * ∂ P 0 < 0 when ρ > K. QED.

Proof of Proposition 3.

We start with the equilibrium with e₀. We only need to prove that no worker’s type wishes to defect. As e₁ is out of the equilibrium path, employers’ posterior beliefs are unrestricted. Imagine that employers assign probability one to t₀ after observing e₁. In this case, employers would pay a wage equal to P₁t₀ after observing e₁ and no worker’s type would deviate because t 0 + t n 2 1 + δ ρ P 0 + δ t 0 + t n 2 E ε > δ t 0 ρ P 0 + E ε − δ c t , e 1 ∀ t ∈ t 0 , t n . As this inequality is satisfied in both educational systems, we have proven that there is a pooling equilibrium in which all worker’s types choose e₀.

Now, we prove that the pooling equilibrium with e₁ may only arise in the non-demanding educational system. Specifically, we will show that no type deviates in the non-demanding educational system if employers respond to e₀ by treating the worker as t₀. Once again, this is possible because the out-of-equilibrium beliefs are unrestricted. Then, no worker’s type deviates providing that the following inequality is satisfied:

If we substitute t with t₀ in this inequality, it coincides with Assumption 1, which guarantees that t₀ does not deviate in the non-demanding educational system. As ς t decreases with the worker’s type, inequality (A.10) is also satisfied for all types greater than t₀, which implies that there exists a pooling equilibrium in which all worker’s types choose e₁ in the non-demanding educational system. However, Assumption 1′ is the opposite of inequality (A.10) when t = t₀, which implies that t₀ would deviate. Therefore, there cannot be an equilibrium in which all types choose e₁ in the demanding educational system. QED.

Proof of Proposition 4.

We start with the equilibrium in which all worker’s types choose e₀. In this equilibrium, a worker’s type t would deviate providing that the wage received in period one with e₁, w, is sufficiently high, that is, providing that the following inequality is satisfied:

As ς t decreases with t, it is clear that P t , e 1 ∪ P 0 t , e 1 ⊂ P t ′ , e 1 ≠ ∅     ∀ t ′ > t . This implies that the only divine posterior beliefs after e₁ is μ t n | e 1 = 1 . Given those posterior beliefs, a t-type of worker does not deviate from the equilibrium as long as:

As ς t decreases with t, inequality (A.12) will hold for all worker’s types as long as:

This inequality coincides with Assumption 2, which implies that no type would deviate and the equilibrium is divine in the non-demanding educational system. As Assumption 2′ is just the opposite inequality, it implies that t _n would deviate in the demanding educational system and therefore, this equilibrium is not divine in that context.

Next, we consider the pooling PBE in which all worker’s types choose e₁. Let us call w₀ and w₁ the wages paid by employers in periods zero and one, respectively, after observing the out-of-equilibrium level of education e₀. Then, a t-type of worker deviates providing that:

Now, as ς t decreases with t, it is clear that P t , e 0 ∪ P 0 t , e 0 ⊂ P t ′ , e 0 ≠ ∅       ∀ t ′ < t . This implies that the only divine posterior beliefs after e₀ is μ t 0 | e 0 = 1 . Given these posterior beliefs, a worker’s type t would not defect as long as:

As ς t decreases with t, we can guarantee that inequality (A.15) will hold for all worker’s types as long as it is satisfied for t₀, that is, as long as:

As this inequality coincides with Assumption 1, no worker’s type would deviate in the non-demanding educational system and the equilibrium is divine. As shown in Proposition 3, this pooling equilibrium does not arise in the demanding educational system. QED.

Proof of Proposition 5.

Now, the worker’s indifference equation shown in (4) is equivalent to:

where F : t 0 , t n × 0,1 × 0 , + ∞ × 0 , + ∞ × 0 , + ∞ → R is the real valued function defined in equation (A.17). In our setting, ∂ F ∂ t * > 0 under Assumption 6 and ∂ F ∂ t * < 0 under Assumption 6′. As U t 0 + t * 2 P 0 > 0 , the indifference condition (A.17) holds as long as ∫ ε ̄ − Δ ε ̄ + Δ U t 0 + t * 2 ρ P 0 + ε − U t * + t n 2 ρ P 0 + ε − ς t * − β f ε d ε < 0 . This implies that ∂ F ∂ δ < 0 . Next, it is straightforward to see that ∂ F ∂ β > 0 . Then, using the implicit function theorem, we obtain that ∂ t * ∂ δ > 0 and ∂ t * ∂ β < 0 under Assumption 6 (non-demanding educational system), whereas ∂ t * ∂ δ < 0 and ∂ t * ∂ β > 0 under Assumption 6′ (demanding educational system). This completes the proof of part i of Proposition 5.

Now, we analyze the effect of a change in ρ by obtaining the following derivative:

Then, ∂ F ∂ ρ ⋚ 0 if, and only if the following condition holds:

when θ* < 1, t * + t n t 0 + t * > 1 > θ * and ∂ F ∂ ρ < 0 . As ∂ F ∂ t * > 0 in the non-demanding educational system, then ∂ t * ∂ ρ > 0 when θ* < 1. Likewise, as ∂ F ∂ t * < 0 in the demanding educational system, then ∂ t * ∂ ρ < 0 when θ* < 1. This proves part ii of Proposition 5.

Since t * + t n t 0 + t * = 1 + t n − t 0 t 0 + t * , we can rewrite (A.19) as:

when θ* > 1, then ∂ t * ∂ ρ ⋛ 0 if t n − t 0 ⋛ θ * − 1 t 0 + t * in the non-demanding environment, whereas ∂ t * ∂ ρ ⋚ 0 if t n − t 0 ⋛ θ * − 1 t 0 + t * in the demanding educational system. Then, we have proven part iii of Proposition 5.

Next, we study the effect of a change in P₀ on the worker’s incentives to invest in education. To this end, we obtain the following derivative:

We can rewrite this derivative in the following way:

If ρ = 0, then ∂ F ∂ P 0 = U ′ t 0 + t * 2 P 0 t 0 + t * 2 > 0 . As the right-hand side of equation (A.22) is a continuous function with respect to ρ, there is a threshold K > 0 such that ∂ F ∂ P 0 > 0 when ρ < K. Then, ∂ t * ∂ P 0 < 0 under Assumption 6 and ∂ t * ∂ P 0 > 0 under Assumption 6′ when ρ < K. Then, we have proven part iv of Proposition 5.

In order to study the sign of ∂ F ∂ P 0 when ρ > K, we divide the right-hand side of (A.22) by m = ∫ ε ̄ − Δ ε ̄ + Δ U ′ t * + t n 2 ρ P 0 + ε − ς t * − β f ε d ε > 0 and obtain:

Then, ∂ F ∂ P 0 ⋚ 0 if, and only if:

This is equivalent to:

Once again, as ∂ F ∂ t * > 0 in the non-demanding educational system, then ∂ t * ∂ P 0 ⋛ 0 if t n − t 0 ⋛ θ * λ * δ ρ + θ * − 1 t 0 + t * , and as ∂ F ∂ t * < 0 in the demanding educational system, then ∂ t * ∂ P 0 ⋚ 0 if t n − t 0 ⋛ θ * λ * δ ρ + θ * − 1 t 0 + t * . This proves part v and completes the Proof of proposition 5. QED.

Proof of Proposition 6.

To start with, we rewrite the indifference equation (8) as:

If we derive this function with respect to the indifferent type, we obtain:

Under Assumption 9 (Assumption 9′), ∂ F ∂ t * > 0 ( ∂ F ∂ t * < 0 ).

Similarly, we obtain the following derivatives:

Now, we determine the signs of these derivatives. The indifference condition (8) is:

The left-hand side of this equation is positive because E t | t < t * P 0 ≥ t 0 P 0 > 0 . Therefore, the right-hand side of the equation is also positive, which implies that ∂ F ∂ δ < 0 . It is also clear that ∂ F ∂ ρ < 0 and ∂ F ∂ β > 0 . By using the implicit function theorem, we conclude that ∂ t * ∂ δ > 0 , ∂ t * ∂ ρ > 0 and ∂ t * ∂ β < 0 in the non-demanding scenario, whereas ∂ t * ∂ δ < 0 , ∂ t * ∂ ρ < 0 and ∂ t * ∂ β > 0 in the demanding scenario. Finally, ∂ F ∂ P 0 in equation (A.30) is positive when ρ = 0 and strictly decreases with ρ. Hence, there exists K ∈ R + such that ∂ F ∂ P 0 ≷ 0 when ρ≶K, which implies that ∂ t * ∂ P 0 ≶ 0 when ρ≶K in the non-demanding scenario and ∂ t * ∂ P 0 ≷ 0 when ρ≶K in the demanding scenario. QED.