Refinement of Signaling Theory in Labor Markets: Informational Frictions, Educational Overinvestment, and Equilibrium Fragility

Abstract

This paper develops a dynamic signaling framework to analyze how educational investment evolves under imperfect information and how the informational value of credentials changes over time. It addresses a central question: under what conditions do signaling equilibria become fragile, and how does this fragility generate educational overinvestment and credential inflation in equilibrium? The model features heterogeneous productivity groups and endogenous educational choices, in which education plays both a signaling and a productive role. Informational frictions and wage-setting mechanisms jointly determine equilibrium configurations, allowing for separation, pooling, and mixed equilibria. The analysis shows that separating equilibria are inherently fragile: when signaling costs decline or when the share of lower-productivity workers becomes sufficiently small, incentives for imitation intensify, progressively eroding informational differentiation. This fragility gives rise to a cascade mechanism of overinvestment, whereby individuals increase educational attainment beyond efficient levels to preserve relative positioning. As a result, signaling distortions propagate across educational levels, generating persistent credential inflation and weakening the informational content of degrees. The framework also identifies conditions under which mixed equilibria may dominate separating equilibria in terms of aggregate welfare, particularly when the proportion of low-productivity workers is limited. By incorporating a productive dimension of education, the model distinguishes between pure signaling rents and genuine productivity gains, providing a unified interpretation of overeducation, declining returns to credentials, and persistent wage dispersion. Finally, the analysis characterizes an optimal taxation scheme that eliminates inefficient signaling rents while preserving incentives for productivity-enhancing investment. Taken together, the results highlight how equilibrium fragility, informational distortions, and strategic educational measures provide a unified explanation for diploma inflation, equilibrium segmentation, and persistent deviations from socially optimal investment levels.

1. Introduction

Education plays a central role in shaping labor market outcomes, both as a productive investment in human capital and as a signal under imperfect information. When individual productivity is not directly observable, employers rely on educational credentials to infer worker ability, generating wage differentials that reflect a combination of productivity and informational content (Spence, 2002; Stiglitz, 1975). In this context, education affects not only earnings but also allocation efficiency, unemployment dynamics, and inequality.

A large empirical literature documents substantial returns to schooling. Using natural experiments and twin data, Angrist and Krueger (1991) and Ashenfelter and Krueger (1994) show that additional years of education significantly increase earnings, even after accounting for ability bias and endogeneity. At the same time, diplomas generate discrete wage premiums—commonly referred to as sheepskin effects—indicating that degree completion conveys information beyond acquired skills (Hungerford & Solon, 1987; Harmon & Walker, 1995). These findings provide direct empirical support for the dual role of education as both a productive input and an informational signal.

This duality gives rise to a fundamental tension between three mechanisms: human capital accumulation, signaling, and employer learning. Human capital theory attributes wage differentials to productivity gains, while signaling models emphasize informational asymmetries. Employer learning models introduce a dynamic dimension, showing that the informational value of education may decline over time as firms observe worker performance (Altonji & Pierret, 2001). This implies that the return to education is not fixed but evolves with experience and information acquisition, making equilibrium outcomes inherently dynamic (Benhamed, 2025).

These issues have become particularly salient in the context of the rapid expansion of higher education. As the supply of educated workers increases, the informational content of diplomas may weaken, reducing their screening power and generating credential inflation. Barro and Lee (1996) document the large-scale accumulation of human capital across countries, while more recent evidence points to declining returns for certain degrees and increasing mismatch between education and employment outcomes. These patterns raise a central question: under what conditions does educational expansion lead to efficient human capital accumulation, and when does it generate excessive signaling and misallocation?

This paper addresses this question by analyzing the fragility of signaling equilibria and its implications for educational investment. Standard signaling models typically focus on static separating equilibria, in which different types choose distinct education levels. However, these models provide limited insight into how such equilibrium responds to changes in costs, ability distributions, or institutional environments. They do not explain how small perturbations may destabilize separation and induce widespread imitation.

The main contribution of this paper is to develop a unified theoretical framework in which signaling equilibriums are inherently fragile and may endogenously break down. First, the model formally characterizes the conditions under which separating equilibria collapse, showing that small changes in signaling costs or population structure can trigger imitation behavior and eliminate informational differentiation. Second, it introduces a mechanism of endogenous overinvestment in education: individuals increase their educational investment beyond the socially efficient level to maintain relative advantage, generating a cascade effect consistent with observed diploma inflation. Third, the framework integrates productive and informational returns within a single structure, allowing for a clear distinction between human capital accumulation and signaling rents. Finally, it characterizes a global equilibrium in which mixing and separating regimes coexist and shows that mixing equilibria may dominate separating equilibria in welfare terms under certain conditions.

Labor market frictions and dynamics play an important role in this framework but are incorporated in a disciplined and directly relevant way. Matching frictions and wage dispersion affect the incentives to invest in education by altering expected returns (Shimer, 2012). Unobserved heterogeneity in productivity complicates the identification of signaling effects and reinforces the reliance on observable credentials (Hwang et al., 1998). Moreover, uncertainty about future employment outcomes influences educational choices, as individuals respond not only to expected wages but also to the informational value of diplomas (Falaris & Peters, 1998; Brodaty et al., 2014). These mechanisms are directly linked to the model’s structure and help interpret its predictions in empirical contexts.

An additional contribution of this paper is to reinterpret common empirical patterns—such as wage dispersion, worker mobility, and panel attrition—as equilibrium outcomes of fragile signaling systems rather than purely econometric issues. When educational signals lose informational content, workers adjust both their schooling and mobility decisions, generating heterogeneous employment durations and wage trajectories. This perspective provides a structural interpretation of observed labor market dynamics and connects theoretical predictions to empirical regularities.

The analysis extends the existing literature by moving beyond static representations of signaling and developing a dynamic and unified framework in which equilibrium structures evolve in response to informational frictions, strategic interactions, and institutional parameters. It thus provides a coherent framework for understanding educational overinvestment, credential inflation, and the coexistence of heterogeneous labor market regimes.

This paper proceeds as follows: Section 2 introduces a baseline signaling model with exogenous productivity. Section 3 extends the framework by incorporating productive effects of education. Section 4 develops a global equilibrium combining mixing and separating regimes. Section 5 derives welfare implications and characterizes optimal policy, including taxation schemes that eliminate signaling rents while preserving productive incentives.

2. The Simple Model of Signage on the Labor Market

The labor market is characterized by informational asymmetries regarding workers’ productivity. In this context, signaling plays a central role by allowing individuals to reveal otherwise unobservable characteristics. Education, denoted by $E$, is introduced as a purely informational signal that does not directly affect productivity but enables separation across worker types.

2.1. Model Setup

The economy consists of a continuum of individuals indexed by their productivity type $i\in \{1,2\}$:

• Type 1: productivity equal to $1$;
• Type 2: productivity equal to $2$.

Let $\alpha \in (0,1)$ denote the fraction of individuals belonging to group 1. The remaining share $\left(1−\alpha \right)$ belongs to group 2. Productivity is exogenous and independent of education.

The informational structure of the model is characterized by asymmetric information between workers and firms. Each worker perfectly observes their own productivity type, whereas employers cannot directly observe individual productivity. Instead, firms only observe the level of education $E$ chosen by workers and use it as an informational signal. Based on this observation, employers form beliefs $\mu \left(E\right)$ representing the probability that a worker with education level $E$belongs to the high-productivity type $\theta =2$. These beliefs play a central role in wage determination and in sustaining equilibrium outcomes.

Education is a costly signal:

• Cost for type 1: $C_1\left(E\right)=E$;
• Cost for type 2: $C_2(E)={\displaystyle \frac{E}{2}}$.

Thus, education is less costly for high-productivity individuals, generating the standard single-crossing property.

Firms are competitive and offer wages equal to expected productivity conditional on observed education:

$$w\left(E\right)=1\cdot (1-\mu (E\left)\right)+2\cdot \mu \left(E\right)=1+\mu \left(E\right)$$

If education is not used as a signal, employers cannot distinguish between types. The wage is equal to average productivity:

$$w=1\cdot \alpha +2\cdot (1-\alpha )=2-\alpha$$

However, this pooling outcome may not be sustainable:

• If $2-\alpha <2$, type 2 individuals are underpaid relative to their productivity.
• As a result, high-productivity workers may exit or deviate, destabilizing the pooling equilibrium.

We define equilibrium as a Perfect Bayesian Equilibrium (PBE):

• A signaling strategy $E_i$for each type $i$;
• A belief system $\mu \left(E\right)$;
• A wage schedule $w\left(E\right)$.

Workers choose their level of education $E$to maximize their net income, solving:

$$\mathrm{a}\mathrm{r}\mathrm{g} \underset{E}{\mathrm{m}\mathrm{a}\mathrm{x}}\{w(E)-C(E,\theta )\}$$

where wages depend on employer beliefs and costs depend on individual type. On the firms’ side, beliefs $\mu \left(E\right)$must be consistent with observed behavior along the equilibrium path, in the sense of Bayesian updating. Wages are then set equal to expected productivity conditional on these beliefs, ensuring that $w\left(E\right)$reflects the inferred type of distribution. Finally, beliefs must support optimal worker strategies so that no agent has an incentive to deviate given the anticipated responses of firms. This joint consistency between strategies, wages, and beliefs formally corresponds to a Perfect Bayesian Equilibrium and captures the notion of “self-validated beliefs,” where expectations are confirmed by the very behavior they induce.

Consider a separating equilibrium in which workers of different productivity types choose distinct levels of education. In this configuration, low-productivity individuals ($\theta =1$) select $E_1=0$, while high-productivity individuals ($\theta =2$) choose a strictly positive level $E_2=E^{*}>0$. Observing these choices, firms form perfectly revealing beliefs such that $\mu \left(0\right)=0$, implying that a worker with no education is identified as low-productivity, and $\mu (E^{*})=1$, implying that a worker with education $E^{*}$is identified as high-productivity. Given these beliefs, wages equal expected productivity, leading to $w\left(0\right)=1$ for low-education workers and $w\left(E^{*}\right)=2$for those who acquire the signal. This configuration thus achieves full separation of types through education.

For the separating equilibrium to be sustained, incentive compatibility constraints must ensure that each type of worker prefers their designated education level. Low-productivity workers must not find it profitable to imitate high-productivity workers. This condition requires that the payoff from not acquiring education remains at least as high as the payoff from mimicking, which implies:

$$1\ge 2-E^{*}\Rightarrow E^{*}\ge 1.$$

Conversely, high-productivity workers must prefer acquiring education rather than pooling with low-productivity workers. This requires that their net payoff from signaling remains at least as high as the pooling payoff, leading to:

$$2-{\displaystyle \frac{E^{*}}{2}}\ge 1\Rightarrow E^{*}\le 2.$$

Taken together, these two conditions define the range of admissible separating equilibria. A separating equilibrium therefore exists if and only if the education level satisfies:

$$1\le E^{*}\le 2.$$

This interval reflects the set of education levels that simultaneously deter imitation by low-productivity workers and ensure participation by high-productivity workers.

Although the model admits a continuum of separating equilibria characterized by different values of $E^{*}$, this multiplicity arises from the flexibility of off-equilibrium beliefs. To discipline these beliefs and select a unique outcome, we apply the Intuitive Criterion of Cho and Kreps (1987). This refinement requires that any off-equilibrium deviation be attributed to the type that has the strongest incentive to undertake it. In the present context, any deviation to a lower education level $E^{\prime }<E^{*}$is more attractive for high-productivity workers than for low-productivity ones. Consequently, firms must interpret such deviations as originating from type $\theta =2$, which eliminates equilibria sustained by non-credible beliefs. As a result, only the least-cost separating equilibrium survives, corresponding to $E^{*}=1$. This refinement restores uniqueness by selecting the equilibrium that minimizes signaling costs while preserving full separation.

Stability of the equilibrium: The separating equilibrium is stable in several complementary senses. First, it satisfies best-response stability, as neither type of worker has an incentive to deviate from their prescribed education choice given the prevailing beliefs and wage schedule. Second, beliefs are robust, in that they remain consistent with observed behavior and are continuously validated by equilibrium outcomes. Third, the equilibrium exhibits dynamic stability, since small perturbations in beliefs or strategies do not lead to its collapse but instead induce adjustments that bring the system back to the separating configuration. By contrast, pooling equilibria are inherently unstable: high-productivity workers, being underpaid relative to their productivity, have a strict incentive to deviate by acquiring education, and such deviations trigger belief revisions by firms that ultimately unravel the pooling outcome.

This model underscores that education operates purely as a signal rather than as a productive input, with its value arising entirely from its impact on employer beliefs through the wage function $w\left(E\right)=1+\mu \left(E\right)$. Labor market outcomes are therefore shaped by the consistency between beliefs, strategies, and wage formation, highlighting the central role of expectations in environments with asymmetric information. While the model allows for multiple equilibria, the introduction of a refinement criterion selects a unique and economically relevant separating outcome that minimizes signaling costs. In this equilibrium, the structure is self-confirming: observed education choices validate employer beliefs, and these beliefs in turn sustain the wage differentials that incentivize signaling.

2.2. Scenario with Three Groups

We extend the signaling framework to a setting with three productivity groups $i\in \{1,2,3\}$, where productivity remains exogenous and independent of education. Workers differ in their output levels, with type $1$producing $1$, type $2$producing $2$, and type $3$producing $4$. As in the baseline model, education $E$serves purely as a signal, allowing firms to infer worker productivity from observed educational choices. The objective is to sustain a separating structure in which each type selects a distinct level of education that credibly reveals its productivity.

To achieve separation, we consider three education levels associated with each group. Low-productivity workers choose no education, $E_1=0$, intermediate types select a positive level, $E_2=E_1^{*}$, and high-productivity workers choose a higher level, $E_3=E_2^{*}$, with $E_2^{*}>E_1^{*}$. Based on these choices, employers form beliefs that map education into expected productivity, generating a piecewise wage schedule. Wages are determined as follows:

$$w\left(E\right)=1 \mathrm{if} E<E_1^{*},$$

$$w\left(E\right)=2 \mathrm{if} E_1^{*}\le E<E_2^{*},$$

$$w\left(E\right)=4 \mathrm{if} E\ge E_2^{*}.$$

This structure ensures that each interval of education is associated with a distinct productivity level, thereby sustaining full separation across the three types.

Education costs preserve the same monotonic structure as in the two-type model, reflecting the idea that higher-productivity individuals face lower marginal costs of signaling. Specifically, the cost functions are given by:

$$C_1(E)=E,C_2(E)={\displaystyle \frac{E}{2}},C_3(E)={\displaystyle \frac{E}{4}}.$$

This cost ordering satisfies the single-crossing property, which guarantees that higher-productivity workers have stronger incentives to invest in education. As a result, the signaling mechanism remains credible, allowing firms to correctly infer productivity from observed education levels while sustaining a separating equilibrium across all three groups.

2.2.1. Incentive Compatibility Conditions

Separating equilibrium with three groups requires that each type of worker prefers its designated education level, which imposes a set of incentive compatibility constraints. For low-productivity workers (group 1), the absence of education $E_1=0$must yield a payoff at least as high as mimicking intermediate types. This condition implies:

$$1\ge 2-E_1^{*}\Rightarrow E_1^{*}\ge 1.$$

This ensures that the threshold $E_1^{*}$is sufficiently high to deter group 1 from imitating group 2.

For intermediate-productivity workers (group 2), two conditions must be considered. First, the participation constraint requires that choosing $E_2=E_1^{*}$ yields a payoff at least as high as that of pooling with group 1, which implies:

$$2-{\displaystyle \frac{E_1^{*}}{2}}\ge 1\Rightarrow E_1^{*}\le 2.$$

Combining this with the previous condition yields:

$$1\le E_1^{*}\le 2.$$

Second, group 2 must not find it profitable to imitate high-productivity workers. This no-upward-deviation condition requires:

$$4-E_2^{*}\le 2-{\displaystyle \frac{E_1^{*}}{2}}\Rightarrow E_2^{*}\ge 2+{\displaystyle \frac{E_1^{*}}{2}}.$$

This inequality ensures that the education level required to signal type 3 is sufficiently costly to deter imitation by type 2.

For high-productivity workers (group 3), the incentive constraint requires that choosing the highest education level $E_3=E_2^{*}$ yields a payoff at least as high as that of mimicking group 2. This implies:

$$4-{\displaystyle \frac{E_2^{*}}{4}}\ge 2-{\displaystyle \frac{E_1^{*}}{2}}\Rightarrow E_2^{*}\le 4+{\displaystyle \frac{E_1^{*}}{2}}.$$

This condition guarantees that group 3 has a strict incentive to separate upward by acquiring the highest level of education.

2.2.2. Equilibrium Characterization

Combining all incentive compatibility constraints yields a characterization of the separating equilibrium in the three-type model. The admissible values of education thresholds must satisfy:

$$2+{\displaystyle \frac{E_1^{*}}{2}}\le E_2^{*}\le 4+{\displaystyle \frac{E_1^{*}}{2}},$$

Together with:

$$1\le E_1^{*}\le 2.$$

Taken jointly, these conditions imply the ordering:

$$1\le E_1^{*}\le E_2^{*}\le 5.$$

More precisely, given that $E_1^{*}\in [1,2]$, the bounds on $E_2^{*}$ can be expressed as follows:

$$2.5\le E_2^{*}\le 5.$$

These inequalities define the set of separating equilibria in which all three productivity types are perfectly distinguished through their education choices, ensuring that signaling remains credible and incentive-compatible across the entire distribution of worker types.

2.2.3. Economic Meaning of the Thresholds

Each education threshold admits a precise economic interpretation within the signaling structure. The condition

$$E_1^{*}\ge 1$$

acts as an entry barrier, ensuring that low-productivity workers (group 1) are deterred from imitating intermediate types. Conversely, the condition

$$E_1^{*}\le 2$$

is a feasibility constraint that guarantees participation of group 2 by keeping the signaling cost sufficiently low. At the upper level, the constraint

$$E_2^{*}\ge 2+{\displaystyle \frac{E_1^{*}}{2}}$$

serves as a deterrence condition preventing group 2 from mimicking group 3, while

$$E_2^{*}\le 4+{\displaystyle \frac{E_1^{*}}{2}}$$

ensures that high-productivity workers (group 3) have an incentive to separate. Together, these conditions define a hierarchical incentive structure in which each signal level must simultaneously repel lower types, retain its own type, and prevent upward imitation, thereby sustaining full separation across all groups.

2.2.4. Structural Interpretation: Interdependence of Signal Levels

A central implication of this framework is the interdependence of signal levels across the hierarchy. In particular, the condition

$$E_2^{*}\ge 2+{\displaystyle \frac{E_1^{*}}{2}}$$

shows that the education threshold required to separate groups 2 and 3 depends directly on the lower-level threshold $E_1^{*}$. This dependence is not a mere artifact of the chosen parameterization but reflects deeper structural features of the model. It arises from the single-crossing property, whereby signaling costs decrease with productivity, combined with discrete productivity gaps across types $\left(1→2→4\right)$. More generally, in any multi-level signaling environment with ordered types and heterogeneous costs, separation constraints become nested, implying that higher thresholds must adjust to maintain consistency with lower-level distinctions. This reveals a fundamental property of hierarchical signaling systems: upper-level signals are only credible if lower-level separations are themselves sustained.

2.2.5. The “0.5 Mark-Up” and Signal Spacing

The structure of the equilibrium also implies a minimum spacing between education levels. From the condition

$$E_2^{*}\ge 2+{\displaystyle \frac{E_1^{*}}{2}},$$

and given that $E_1^{*}\in [1,2]$, it follows that

$$E_2^{*}\ge 2.5.$$

This generates a minimum gap of approximately $0.5$ between consecutive signal levels. Economically, this “mark-up” reflects the need for increasing separation margins as one moves up the productivity hierarchy. Higher signals must provide stronger informational content to distinguish increasingly productive types, which requires larger differences in education levels. As a result, the signal structure becomes convex, with spacing between thresholds increasing at higher levels, capturing the idea that informational precision becomes more demanding as the hierarchy deepens.

2.2.6. Fragility of the Hierarchy

The signaling hierarchy is inherently fragile and depends critically on the lower-level separation. Consider the case where $E_1^{*}=0$. In this situation, the constraints reduce to:

$$2\le E_2^{*}\le 4.$$

This implies that the distinction between group 1 and group 2 collapses, as both types effectively choose the same education level. Therefore, the entire hierarchical structure weakens, since the credibility of higher-level signals relies on the existence of lower-level separation. Moreover, the disappearance of the minimum gap eliminates the “0.5 mark-up,” reducing the informational content of education levels and making the system more susceptible to imitation and pooling. This illustrates that the stability of multi-level signaling equilibria is conditional on the integrity of each layer in the hierarchy.

2.2.7. Key Insight

A fundamental insight from the multi-level signaling model is that the breakdown of a lower-level signal undermines the credibility of all higher-level signals. Signaling systems are inherently cumulative: each level builds upon the separation established at the levels below. When a foundational threshold disappears, the incentive structure that supports higher-level distinctions erodes, propagating instability upward through the hierarchy. This systemic fragility underscores the importance of maintaining credible separation at the base of the hierarchy to ensure the overall robustness and informational integrity of the signaling mechanism (

Table 1. Summary table.

Group	Productivity	Cost Function	Education Choice	Constraint Type
1	1	$C_1\left(E\right)=E$	$E_1=0$	$E_1^{*}\ge 1$(Deterrence)
2	2	$C_2\left(E\right)=E/2$	$E_2=E_1^{*}$	$1\le E_1^{*}\le 2$, $E_2^{*}\ge 2+E_1^{*}/2$
3	4	$C_3\left(E\right)=E/4$	$E_3=E_2^{*}$	$E_2^{*}\le 4+E_1^{*}/2$

).

The extended three-group signaling model uncovers several important economic mechanisms. First, education levels no longer act as isolated signals but instead form a hierarchical structure, creating a ladder of credibility across productivity groups. Second, the spacing between successive education levels is endogenously determined by incentive constraints, ensuring that each type finds it optimal to choose its designated signal. Third, there is clear interdependence across groups: the behavior of each group is influenced not only by lower-level types but also by higher-level types, as their potential for imitation and upward or downward deviation shapes the equilibrium thresholds. Finally, the model emphasizes systemic fragility, as the weakening or collapse of a single signal at a lower level can destabilize the entire signaling hierarchy, illustrating the cumulative and delicate nature of multi-level signaling systems.

2.2.8. Equilibrium Perspective

The equilibrium in this extended framework remains a Perfect Bayesian Equilibrium, with firms’ beliefs mapping observed education to expected productivity and wages reflecting these expectations, while each group chooses its education level in accordance with incentive compatibility. Compared to the simpler two-group case, however, the equilibrium becomes multi-dimensional and interdependent: feasible education levels are constrained by nested incentive conditions, and each group’s optimal strategy depends on the thresholds set by other groups. This interdependence highlights that, in multi-level signaling environments, equilibrium outcomes are defined not only by individual optimization but also by the structural consistency of the entire hierarchy of signals.

2.3. Efficient Equilibrium and Welfare Analysis

We now characterize the efficient equilibrium of the signaling model and evaluate it from a social welfare perspective.

Let social welfare be defined as the sum of net incomes across all groups, net of education costs:

$$W=\sum _i{q}_i{N}_i$$

where $q_i$ is the population share of group $i$, and $N_i$ denotes net income (wage minus education cost). Since education is assumed to have no direct productive effect in this section, all education expenditures represent pure signaling costs and therefore constitute a deadweight loss from a social standpoint.

2.3.1. Separating Equilibrium

Consider the separating equilibrium characterized by the education levels:

$$E_1^{*}=1+\delta ,E_2^{*}={\displaystyle \frac{5+3\delta }{2}}$$

where $\delta >0$ is arbitrarily small.

The corresponding wages and net incomes are:

• $W_1=1$, $N_1=1$;
• $W_2=2$, $N_2=2-{\displaystyle \frac{E_1^{\mathrm{*}}}{2}}={\displaystyle \frac{3-\delta }{2}}$;
• $W_3=4$, $N_3=4-{\displaystyle \frac{E_2^{\mathrm{*}}}{2}}={\displaystyle \frac{11-3\delta }{4}}$.

This equilibrium satisfies incentive compatibility constraints, ensuring that each type prefers its designated education level. Hence, it is a Perfect Bayesian Equilibrium (PBE) where employers’ beliefs are consistent with observed education choices.

2.3.2. Marginal Returns to Signaling

$$N_2-N_1={\displaystyle \frac{1-\delta }{2}}<1$$

$$N_3-N_2={\displaystyle \frac{5-\delta }{4}}>1 \mathrm{and}<2$$

• Returns between levels are distorted by signaling costs;
• Lower transitions (1 → 2) yield compressed gains;
• Higher transitions (2 → 3) remain more rewarding.

This reflects a key inefficiency: private incentives exceed social returns due to signaling costs.

This reproduces a central result of the signaling literature: separating equilibrium generates informational efficiency at the cost of resource waste.

2.3.3. Efficiency Properties

Although this separating equilibrium is informationally efficient—as it perfectly reveals productivity types—it is generally not socially efficient. The reasons are that:

• Education does not increase productivity;
• Yet, individuals invest in education to signal their type;
• Generating excessive private investment relative to the social optimum.

This leads to a classic signaling externality: individuals do not internalize the fact that their education decision imposes costs on others by forcing them to invest more to maintain differentiation.

2.3.4. Mixing Equilibrium and Welfare Comparison

An alternative is a pooling equilibrium, where all individuals choose the same education level and receive a common wage equal to average productivity:

$$\stackrel{-}{w}=q_1\cdot 1+q_2\cdot 2+q_3\cdot 4$$

In this case:

• Education costs are minimized;
• But the informational content of education disappears.

The welfare comparison highlights a fundamental trade-off:

• Separating equilibrium: high informational efficiency, high cost;
• Pooling equilibrium: low informational efficiency, low cost.

Importantly, when the proportion of low-productivity individuals is sufficiently small (i.e., $q_1,q_2\to 0$), the informational gains from separation become negligible, and the pooling equilibrium may Pareto-dominate the separating equilibrium.

Separating equilibrium is preferred by high types if:

$$q_1+2q_2+4q_3\ge {\displaystyle \frac{11-3\delta }{4}}$$

This result aligns with the literature on signaling (notably Spence-type models), where equilibria may be privately optimal but socially inefficient. It also illustrates that equilibrium selection depends critically on population composition, reinforcing the idea that signaling systems may become inefficient in highly skilled economies.

2.4. Taxation and Welfare Improvement

We now introduce a tax on education investment as a policy tool to correct the signaling externality. The tax is proportional to the education cost, generating total revenue $k$that is redistributed lumpsum to all individuals. The objective of this intervention is to reduce excessive signaling while preserving the separation of types.

2.4.1. Tax Structure and First-Level Signal

Let $t$ denote the tax rate on education cost. The total tax revenue is $k,$ which is returned to all individuals in a lumpsum manner. The tax is designed to discourage over-investment in signaling without collapsing the information conveyed by education choices. The incentive constraints for the first-level signal are adjusted for taxation. Group 1 does not deviate if:

$$2-E_1^{*}-t{E}_1^{*}+k\le 1+k\Rightarrow E_1^{*}\ge {\displaystyle \frac{1}{1+t}}$$

Group 2 participants if:

$$2-{\displaystyle \frac{E_1^{*}}{2}}-t{E}_1^{*}+k\ge 1+k\Rightarrow E_1^{*}\le {\displaystyle \frac{1}{0.5+t}}$$

Thus, the feasible range is:

$${\displaystyle \frac{1}{1+t}}\le E_1^{*}\le {\displaystyle \frac{1}{0.5+t}}$$

The efficient first-level signal is then:

$$E_1^{*}={\displaystyle \frac{1+\delta }{1+t}}$$

2.4.2. Second-Level Signal

Similarly, for the second-level signal, group 2 does not mimic group 3 if:

$$4-E_2^{*}-t{E}_2^{*}+k\le 2+k\Rightarrow E_2^{*}\ge {\displaystyle \frac{2}{1+t}}$$

Group 3 separates if:

$$4-{\displaystyle \frac{E_2^{*}}{2}}-t{E}_2^{*}+k\ge 2+k\Rightarrow E_2^{*}\le {\displaystyle \frac{2}{0.5+t}}$$

Hence:

$${\displaystyle \frac{2}{1+t}}\le E_2^{*}\le {\displaystyle \frac{2}{0.5+t}}$$

The efficient second-level signal is:

$$E_2^{*}={\displaystyle \frac{2+\delta }{1+t}}$$

2.4.3. Tax Revenue, Redistribution and Net Incomes

Total tax revenue can be expressed as follows:

$$k_1=t{E}_1^{*}{q}_2=t{\displaystyle \frac{1+\delta }{1+t}}{q}_2$$

$$k_2=t{E}_2^{*}{q}_3=t{\displaystyle \frac{2+\delta }{1+t}}{q}_3$$

$$k=k_1+k_2={\displaystyle \frac{t}{1+t}}\left[q_2(1+\delta )+q_3(2+\delta )\right]$$

Net incomes accounting for taxation and redistribution is:

$$N_1=1+k$$

$$N_2=2-(0.5+t)E_1^{*}+k$$

$$N_3=4-(0.5+t)E_2^{*}+k$$

Limit result (high-tax case) as $\delta \to 0,$and $t\to \mathrm{\infty }$ the tax revenue approaches:

$$k\to q_2+2q_3=1-q_1+q_3=1-\alpha$$

Leading to equalized net incomes:

$$N_1=2-\alpha ,N_2=2-\alpha ,N_3=3-\alpha$$

Taxation reduces signaling costs while preserving the informational structure of education. It redistributes gains such that groups 1 and 2 experience equal net incomes. This aligns with the literature on correcting signaling externalities, optimal taxation, and mechanism design. The tax functions as a Pigouvian instrument: it internalizes the social cost of excessive signaling and compresses inefficient investments.

2.4.4. Robustness

The results do not require extremely high tax rates. For any finite $t>0$, education thresholds $E_1^{*}$ and $E_2^{*}$ decrease, signaling costs are reduced, and welfare improves. The mechanism also generalizes to any ordered productivity structure with convex cost advantages, ensuring that taxation compresses signals, reduces waste, and preserves separation. The analysis reveals a fundamental trade-off between information, costs, and welfare (

Table 2. Analyses of fundamental trade-off between information, costs, and welfare.

Dimension	Separating Equilibrium	Taxed Equilibrium
Information	Perfect	Preserved
Costs	High	Reduced
Welfare	Inefficient	Improved

).

The optimal policy does not eliminate signaling entirely but disciplines it. Excessive signaling leads to waste, while insufficient signaling results in a loss of information. Taxation ensures the second best optimum by balancing these forces. While Section 2 assumes education is purely informational, Section 3 introduces a productive role for education, allowing for the coexistence of signaling and human capital accumulation, richer welfare implications, and closer alignment with real labor markets.

3. The Model with a Productive Signal

3.1. Productive Effect of Education

We extend the baseline signaling framework by introducing a productive role for education. Unlike previous sections where education was purely informational, here, it directly enhances individual productivity. This hybrid approach bridges two central paradigms: signaling, where education reveals type, and human capital, where education increases productivity.

3.1.1. Model Setup

Let $S_i\left(E_j\right)$ denote the productivity (or value of output) of an individual of type $wi\in \{1,2,3\}$ ith education Level, $E_j$, where $j\in \{1,2\}$ the following ordering is assumed:

$$S_3\left(E_2\right)>S_2\left(E_2\right)>S_2\left(E_1\right)>S_1\left(E_1\right)$$

$$S_3^{\prime }\left(E_2\right)>S_2^{\prime }\left(E_2\right)>S_2^{\prime }\left(E_1\right)>S_1^{\prime }\left(E_1\right)$$

Higher types are more productive and benefit more from education. Let $C_i\left(E_j\right)$ denote the cost of education. We assume:

$$C_1\left(E_1\right)>C_2\left(E_2\right)>C_2\left(E_1\right)>C_3\left(E_2\right)$$

$$C_1^{\prime }\left(E_1\right)>C_2^{\prime }\left(E_2\right)>C_2^{\prime }\left(E_1\right)>C_3^{\prime }\left(E_2\right)$$

Higher types face lower marginal costs, preserving the single-crossing property. Net income is given by:

$$N_i\left(E_j\right)=S_i\left(E_j\right)-C_i\left(E_j\right)$$

• $S$is increasing and concave;
• $C$is increasing and convex;
• $N$is concave.

These conditions ensure well-defined interior solutions and stability.

• Job 1 is accessible to groups 1 and 2;
• Job 2 is accessible to groups 2 and 3.

This overlapping access creates strategic signaling incentives.

3.1.2. Hidden Deviations and Incentive Constraints

• First deviation (group 1 mimicking group 2)

Define:

$$V_1=S_2\left(E_1\right)-C_1\left(E_1\right)$$

The condition for separation is:

$${\widehat{e}}_1<E_2^{*}$$

The gain from higher productivity $S_2\left(E_1\right)$ is insufficient relative to the higher cost $C_1\left(E_1\right)$; so, separation is sustained.

• Second deviation (group 2 mimicking group 3)

Define:

$$V_2=S_3\left(E_2\right)-C_2\left(E_2\right)$$

The separation conditions are:

$${\widehat{e}}_2<E_3^{*}$$

Even though productivity increases, the cost-adjusted gain does not justify imitation, preventing upward deviation. Under these conditions, there are two separating equilibriums:

• Between groups 1 and 2;
• Between groups 2 and 3.

These equilibria are incentive-compatible, stable (no profitable deviation), and internally consistent.

3.1.3. Economic Implications and Contribution

This extended framework highlights that education plays a dual role, simultaneously signaling individual type and enhancing productivity, making part of the investment socially valuable. Separation between groups can therefore be fully efficient, as productivity gains $S_i\left(E_j\right)$ partially offset education costs $C_i\left(E_j\right)$. Overinvestment may still arise, but it is now understood as the interaction between informational imperfections—where firms cannot directly observe the divergence between private and social returns, occurring whenever private incentives exceed social marginal productivity. The structure of the labor market further shapes these dynamics: overlapping job opportunities intensify competition and amplify incentives for upward signaling, generating richer strategic interactions. By bridging pure signaling and human capital paradigms, this hybrid model demonstrates that efficiency is endogenous, depending on the alignment of productivity gains and signaling costs. Its novel contributions include multi-level separation (three groups, two signals), endogenous stability conditions derived from latent deviation functions $V_1=S_2\left(E_1\right)-C_1\left(E_1\right)$ and $V_2=S_3\left(E_2\right)-C_2\left(E_2\right)$, the interaction of occupational access with signaling incentives, and the characterization of hidden equilibria that determine stability even when deviations are not observed. While the model achieves an optimal allocation and fully reveals information, its efficiency critically depends on the correlation between cost differentials and productivity gains; if this link weakens, separation may collapse, giving rise to pooling or inefficient equilibria. Overall, introducing a productive component transform signaling from a purely wasteful exercise into a potentially efficient sorting mechanism, illustrating that the social value of education emerges only when incentive compatibility aligns with productivity enhancement.

3.2. Demolition of Separation and Overinvestment

We now analyze the fragility of separating equilibria by examining situations in which incentive constraints fail, leading to the collapse of separation and the emergence of overinvestment in education.

3.2.1. Definition of Overinvestment

Overinvestment in education arises when individuals choose an education level $E$ such as:

$$C_i\left(E\right)>S_i\left(E\right)-S_i\left(E^{opt}\right)$$

where $E^{opt}$ denotes the socially efficient level.

Two distinct notions must be distinguished. Private overinvestment occurs when individuals invest beyond their productive optimum to improve their relative position in the labor market. Social overinvestment arises when total investment exceeds the level that maximizes aggregate net output due to signaling externalities. In this framework, overinvestment is fundamentally driven by strategic imitation incentives.

3.2.2. Four Cases of Separation Breakdown

We characterize deviations by comparing equilibrium thresholds $e_2^{*},e_3^{*}$with optimal deviation levels ${\widehat{e}}_1,{\widehat{e}}_2$.

Case 1: partial breakdown (first barrier only):

$$e_2^{*}<{\widehat{e}}_{1 }\mathrm{and}{ e}_3^{*}>{\widehat{e}}_2$$

Group 1 imitates group 2, causing the first barrier to collapse, while the second barrier remains intact. Groups 1 and 2 pool, whereas group 3 stays separated.

Case 2: Cascade breakdown (sequential imitation):

$$e_2^{*}<{\widehat{e}}_1 \mathrm{and} {\widehat{e}}_2<e_3^{*}<{\widehat{e}}_1$$

Group 1 imitates group 2 and group 2 imitates group 3. The first barrier collapses, and the second becomes fragile, generating a cascade of upward imitation.

Case 3: full breakdown (pooling of all groups):

$$e_2^{*}<{\widehat{e}}_1 \mathrm{and} e_3^{*}<{\widehat{e}}_1$$

Both barriers collapse: group 1 imitates higher types and group 2 imitates group 3. All groups pool, resulting in a complete loss of informational content.

Case 4: upper-level breakdown only:

$$e_2^{*}>{\widehat{e}}_1 \mathrm{and}{ e}_3^{*}<{\widehat{e}}_2$$

The first barrier holds while the second collapses. Group 1 remains separated, whereas groups 2 and 3 pool, revealing selective fragility at higher levels.

3.2.3. Fragility, Cascade, and Overinvestment Dynamics

A separating equilibrium breaks down whenever the marginal gain from imitating a higher type exceeds the marginal signaling cost, that is, when:

$$S_{i+1}\left(E\right)-S_i\left(E\right)>C_i\left(E\right)-C_i\left(E_i\right),$$

implying that separation critically relies on a strict ordering of net returns across types. Any violation of this condition triggers strategic imitation, which does not remain localized but propagates hierarchically through the structure of signals. This gives rise to a cascade mechanism: the collapse of a lower barrier (between groups 1 and 2) endogenously increases pressure on the upper barrier (between groups 2 and 3), inducing generalized upward imitation and generating systemic instability. Anticipating such dynamics, individuals adjust their behavior ex ante by increasing their education levels to preserve differentiation, leading to an equilibrium in which:

$$E^{observed}>E^{efficient}.$$

In this context, education ceases to be purely a productivity-enhancing investment and instead becomes a defensive strategic instrument, aimed at avoiding downward pooling and escaping congestion. The result is a self-reinforcing process in which fragility of separation, cascade effects, and anticipatory behavior jointly produce persistent overinvestment in education.

3.2.4. Private vs. Social Overinvestment, Population Structure, and Systemic Implications

From a private perspective, overinvestment in education constitutes a rational response to competitive pressures, as individuals choose higher education levels to maximize their own net income and avoid adverse selection. From a social standpoint, however, this behavior generates inefficiencies whenever additional educational investment does not translate into sufficient productivity gains, leading to excessive resource allocation and a misallocation of talent. These inefficiencies are further shaped by the population structure: when the distribution of types satisfies:

$$q_1,q_2\ll q_3,$$

The high-productivity group dominates, raising average productivity but simultaneously reducing the informational value of signaling. In such a configuration, separating equilibria tend to collapse, and the economy converges toward a mixing equilibrium, as even high types may rationally cease signaling when marginal benefits fall below marginal costs. This mechanism provides a natural theoretical foundation for diploma inflation as an equilibrium outcome: as more individuals acquire higher levels of education, signals lose their discriminatory power, prompting the emergence of new, higher thresholds and generating a self-reinforcing dynamic in which increased education reduces differentiation, thereby inducing further investment. More broadly, these results yield three central insights. First, separating equilibria is endogenously fragile, relying on finely balanced incentive conditions that can be disrupted by small shocks. Second, education decisions exhibit strong strategic complementarities, as everyone’s investment directly influences the incentives of others. Third, competitive signaling environments are prone to producing excessive education, inefficient sorting, and unstable equilibria. Ultimately, the breakdown of separating equilibria should not be viewed as an anomaly, but rather as a structural feature of signaling systems characterized by multiple types and overlapping incentives.

3.3. Determination of Total Income and Equilibrium Selection

We now examine how the distribution of population shares $\left(q_1,q_2,q_3\right)$ shapes both aggregate income and the nature of equilibrium (separating vs. mixing), in direct continuity with the fragility mechanisms identified in Section 3.2.

3.3.1. Aggregate Income, Welfare, and Equilibrium Selection

Aggregate income is defined as follows:

$$W\left(E\right)=q_1{S}_1\left(E\right)+q_2{S}_2\left(E\right)+q_3{S}_3\left(E\right),$$

which represents the economy’s total productive capacity as a weighted sum of type-specific outputs. Given the ordering $S_3\left(E\right)>S_2\left(E\right)>S_1\left(E\right)$, an increase in the share of high-productivity individuals $q_3$ raises total income, while an increase in $q_2$ has a more moderate effect and an increase in $q_1$ reduces aggregate output; accordingly, welfare is maximized when high-productivity types dominate, that is, when $q_3$ is large and $q_1,q_2$ remain relatively small. However, this welfare-maximizing configuration is intrinsically linked to the fragility of separating equilibria: when $q_1$ and $q_2$ are limited, imitation incentives become predominantly upward, and the prevalence of high-productivity individuals weakens the informational content of educational signals, thereby destabilizing separation. This reveals a fundamental trade-off between productive efficiency and the stability of informational structures. Within this environment, agents determine their education choices by comparing net returns across groups, namely $W-C_1$, $W-C_2$, and $W-C_3$, which extends individual optimization to an aggregate benchmark and ensures consistency with equilibrium selection.

3.3.2. Population Dynamics and Equilibrium Outcomes

Different configurations of population shares generate distinct equilibrium outcomes.

Case 1: increase in $q_2$:

As the share of intermediate types rises, the total income becomes more dependent on intermediate productivity. The relative return to pooling for high types declines, implying that group 3 no longer benefits from a mixing equilibrium. As a result, separation at the upper level is restored.

Case 2: increase in $q_1$ (with low $q_2$):

A larger low-productivity group reduces average productivity and weakens the attractiveness of pooling for intermediate types. Consequently, group 2 prefers to remain differentiated, restoring separation at the lower level.

Case 3: joint increase in $q_1$ and $q_2$:

When both low and intermediate groups become quantitatively important, incentive constraints are reinforced at both margins. The economy converges toward a configuration with two separating thresholds, restoring full separation.

3.3.3. Equilibrium Selection, Welfare, and General Implications

The previous results can be synthesized into a unified proposition: a mixing equilibrium dominates a separating equilibrium if and only if the shares of low- and intermediate-productivity groups $\left(q_1,q_2\right)$ are sufficiently small. This highlights a fundamental trade-off between efficiency and identification. When the proportion of high-productivity individuals $q_3$ is large, aggregate production is maximized, but the informational value of signaling diminishes, making separation difficult to sustain; conversely, as $q_1$ and $q_2$ increase, heterogeneity becomes more pronounced, identification gains importance, and separating equilibria re-emerge. In this sense, mixing equilibria arise endogenously in environments where heterogeneity is limited or high types dominate, precisely because signaling costs then exceed informational benefits, rendering pooling Pareto superior. This also establishes a direct connection with overinvestment: when separation is fragile, individuals engage in excessive educational investment to preserve differentiation, whereas when mixing prevails, such overinvestment disappears, implying that inefficiency is itself equilibrium-dependent. From a welfare perspective, separating equilibria provides more precise allocation but at the cost of higher signaling expenditures, while mixing equilibria economize on costs at the expense of less accurate sorting; when $q_1$ and $q_2$are sufficiently small, aggregate welfare satisfies.

$$W^{mixing}>W^{separating},$$

so that pooling becomes socially optimal. The key implication is that the nature of the optimal equilibrium is not solely determined by technological conditions or cost structures but fundamentally depends on the distribution of types within the population. More broadly, this section completes the overall logic of the model: while Section 3.1 shows that productive signaling can be efficient, and Section 3.2 demonstrates that separation is inherently fragile and may generate overinvestment, Section 3.3 establishes that population composition ultimately governs equilibrium selection. Taken together, these mechanisms imply that educational congestion and diploma inflation are not anomalies, but rather endogenous equilibrium outcomes arising from the interaction between productivity, signaling incentives, and demographic structure.

4. A Global Equilibrium Model with Imperfect Information and Employer Heterogeneity

4.1. Motivation and Economic Environment

This section develops a unified equilibrium framework that endogenizes the coexistence of separating and mixing regimes, extending Section 3.1, Section 3.2 and Section 3.3. Labor markets are characterized by heterogeneous firms facing different informational environments: some invest in screening technologies (tests, interviews, probation, AI tools), while others rely on coarse signals or do not screen at all. This heterogeneity is central in the employer learning literature and reflects the coexistence of distinct hiring technologies within the same market.

We consider two types of firms. Type $S$ firms (separating firms) imperfectly observe ability $A$ and offer wages:

$$W_S=A-C,$$

where $C$denotes the cost of detection. Type $M$firms (mixing firms) do not observe ability and pay a uniform wage:

$$W_M=\widehat{A}.$$

The detection cost $C$captures screening technologies, information acquisition costs, and organizational friction; economically, it represents the price of information in the labor market.

4.2. Distribution of Abilities and Sorting Rule

Abilities are distributed over $A\in \left[A_{\mathrm{m}\mathrm{i}\mathrm{n}},A_{\mathrm{m}\mathrm{a}\mathrm{x}}\right]$with density $f\left(A\right)$and cumulative distribution $F\left(A\right)$, while the detection cost satisfies $C>0$. Individuals sort across firms according to:

$$A-C\gtrless \widehat{A}.$$

Indifference defines a threshold ability:

$$A^{*}=\widehat{A}+C.$$

Consistency in mixing firms requires:

$$\widehat{A}=\mathbb{E}[A\mid A\le A^{*}]={\displaystyle \frac{{\int }_{A_{\mathrm{m}\mathrm{i}\mathrm{n}}}^{A^{*}}Af\left(A\right) dA}{F\left(A^{*}\right)}}.$$

The resulting equilibrium structure is characterized by pooling for $A\le A^{*}$ and separation for $A>A^{*}$. This sorting rule is self-fulfilling: wages determine sorting, and sorting determines wages, providing a microfoundation for partial separation absent in standard signaling models.

4.3. Characterization of Equilibria

Combining the threshold and consistency conditions yields a fixed-point equation:

$$A^{*}=\mathbb{E}[A\mid A\le A^{*}]+C.$$

Under standard regularity conditions, the right-hand side increases in $A^{*}$, while the left-hand side is linear, ensuring the existence of a unique equilibrium threshold. Economically, a higher detection cost $C$ raises $A^{*}$ and expands pooling, whereas a lower $C$increases separation; thus, $C$ governs the degree of market transparency.

4.4. Extreme Cases

A pure mixing equilibrium arises if:

$$A_{\mathrm{m}\mathrm{a}\mathrm{x}}-C<\mathbb{E}\left[A\right],$$

in which case no agent benefits from separation and all individuals pool. Conversely, a pure separating equilibrium emerges if:

$$A_{\mathrm{m}\mathrm{i}\mathrm{n}}-C>\mathbb{E}\left[A\right],$$

so that all agents prefer screening. These cases illustrate that labor market structure is entirely driven by the cost of information acquisition.

4.5. Role of Signaling

This framework connects directly with the signaling model. When education is highly informative, it effectively reduces the detection cost $C$, leading firms to rely more on signals, expanding the separating region and increasing incentives for education, thereby generating overinvestment (as in Section 3.2). Conversely, when signals are weak or screening dominates, the mixing region expands and signaling incentives decline. The key implication is that signaling and screening act as substitutes in equilibrium.

4.6. Educational Investment and Individual Optimization

We embed a generalized Spence-type model where education $Y$, ability $a$, and employer valuation $b$define a type of index $n(a,b)$. Productivity is given by $S(n,Y)$, and costs satisfy $C(Y,n)$ with $C_Y>0$ and $C_{Yn}<0$. Wages depend on education, $W\left(Y\right)$, and net income is:

$$N(Y,n)=W\left(Y\right)-C(Y,n).$$

Individuals choose education by solving:

$$\underset{Y}{\mathrm{m}\mathrm{a}\mathrm{x}}\hspace{0.25em} W(Y)-C(Y,n),$$

leading to the first-order condition:

$$W^{\prime }\left(Y\right)=C_Y(Y,n),$$

and the second-order condition:

$$W^{″}\left(Y\right)-C_{YY}(Y,n)<0.$$

Equilibrium requires consistency between wages and productivity:

$$W\left(Y\right)=S(n,Y).$$

4.7. Equilibrium Characterization and Stability

Letting $n=N(W,Y)$, the equilibrium satisfies the differential equation:

$$W^{\prime }\left(Y\right)=C_Y(Y,N(W,Y\left)\right).$$

This defines a family of non-intersecting wage schedules $W(Y,k)$, where $k$indexes equilibria through beliefs, institutions, or screening technologies. Net income satisfaction is:

$${\displaystyle \frac{\partial N}{\partial n}}>0,$$

implying that higher ability increases returns to education. Differentiating yields:

$$W^{″}=C_{YY}+C_{Yn}{\displaystyle \frac{dn}{dY}},$$

so that stability holds since:

$$W^{″}-C_{YY}<0.$$

4.8. Private vs. Social Returns and Equilibrium Shifts

From wage determination:

$$W^{\prime }(Y)=S_Y+S_n{\displaystyle \frac{dn}{dY}},$$

implying:

$$W^{\prime }\left(Y\right)>S_Y\Rightarrow S_Y-C_Y<0,$$

which establishes overinvestment in education. Across equilibria, we obtain:

$${\displaystyle \frac{\partial N}{\partial k}}=W_k>0,{\displaystyle \frac{dY}{dk}}={\displaystyle \frac{W_k}{S_Y-C_Y}}<0,W_{Yk}<0.$$

Thus, higher equilibrium levels reduce educational investment, flatten wage schedules, and generate catch-up dynamics among lower types.

4.9. Synthesis and Central Contribution

The model yields three central relationships:

$$W_k>0,{\displaystyle \frac{dY}{dk}}<0,W_{Yk}<0.$$

Overall, this section delivers a unified theory in which: (i) market structure (mixing vs. separating) is endogenously determined by information costs $C$; (ii) signaling and screening interact as substitutable mechanisms; (iii) overinvestment arises endogenously from imperfect information; and (iv) multiple equilibria emerge, indexed by parameter $k$.

The central contribution is that labor market equilibrium is jointly determined by information frictions (screening cost $C$), signaling behavior (education), and population structure, providing a comprehensive framework to understand educational congestion, wage dispersion, and equilibrium instability.

5. Maximization of Total Net Income and Optimal Policy

5.1. Welfare, Constraints, and Social Optimum

We now turn to the normative dimension by analyzing how total net income can be maximized in an economy with signaling and imperfect information, building on the overinvestment result (Section 3) and the coexistence of mixing and separating regimes (Section 4). Let the unobservable characteristic $n$follow distribution $f\left(n\right)$; social welfare is defined as follows:

$$\int N(y,n)f(n) dn\mathrm{with}N(y,n)=W(y)-C(y,n),$$

where wages $W\left(y\right)$ include signaling rents while true productivity is $S(y,n)$, implying that private and social returns diverge. The economy is constrained by individual rationality:

$$W^{\prime }\left(y\right)=C_y(y,n),$$

and by consistency between wages and productivity:

$$\int W(y)f(n) dn=\int S(y,n)f(n) dn,$$

which ensures rational expectations. The social planner thus maximizes:

$$\int (S(y,n)-C(y,n))f(n) dn,$$

leading to the optimality condition:

$$S_y=C_y.$$

This result implies that education is chosen solely for its productive return, eliminating signaling motives; since in equilibrium $W^{\prime }\left(y\right)>S_y$, the decentralized outcome features excessive investment, and the social optimum effectively removes signaling rents by internalizing informational externalities.

5.2. Implementation and Optimal Taxation

The social optimum can be decentralized through appropriate wage design. Let $r\left(y\right)$ denote the equilibrium mapping from education to types; the incentive constraint becomes:

$$W^{\prime }\left(y\right)=C_y(y,r(y\left)\right),$$

which integrates to:

$$W(y)=W(0)+{\int }_0^y{C}_y(v,r(v)) dv.$$

Thus, wages reflect cumulative marginal costs rather than signaling rents, while $W\left(0\right)$ ensures aggregate consistency. To implement the optimum, define a tax function:

$$t\left(y\right)=S(y,r(y\left)\right)-W\left(y\right),$$

whose derivative satisfies:

$$t^{\prime }\left(y\right)=S_y+S_n{r}^{\prime }\left(y\right)-W^{\prime }\left(y\right)=S_n{r}^{\prime }\left(y\right).$$

The optimal marginal tax therefore coincides exactly with the marginal signaling component, removing informational rents and neutralizing inefficient incentives. As a result, overinvestment disappears and the decentralized outcome, characterized by:

$$Y^{\mathrm{decentralized}}>Y^{\mathrm{optimal}},$$

is corrected, restoring the first-best allocation.

5.3. Robustness, Literature, and Global Interpretation

The core condition $S_y=C_y$ is highly robust, holding under general cost functions, arbitrary type distributions, and across all equilibria in the family $W(y,k)$, although the precise tax schedule depends on institutional features such as observability of education and the estimation of $r\left(y\right)$. This framework connects three major strands of the literature: signaling theory (Michael Spence), optimal taxation (James Mirrlees), and employer learning (Edward Lazear), thereby unifying informational distortions, firm heterogeneity, and policy design. Overall, while the positive analysis (Section 3 and Section 4) shows that overinvestment and multiple equilibria arise endogenously, the normative analysis demonstrates that the social optimum eliminates signaling rents and can be implemented through taxation. The central insight is that educational overinvestment is a rational response to informational frictions, yet one that can be efficiently corrected through appropriate policy intervention.

6. Conclusions

This paper develops a unified framework in which education simultaneously operates as a signal and a productive investment in labor markets characterized by imperfect information and heterogeneous screening technologies. The analysis delivers a coherent set of theoretical results while preserving a tight link between equilibrium conditions, welfare properties, and policy design.

First, the model establishes the fragility of separating equilibria. We show that separation breaks down when incentive compatibility fails, i.e., when $S_{i+1}\left(E\right)-S_i\left(E\right)>C_i\left(E\right)-C_i\left(E_i\right)$, implying that separation is not robust and may collapse through strategic imitation and cascade mechanisms. Second, from the equilibrium condition $W^{\prime }\left(y\right)>S_y$, we derive $S_y-C_y<0$, showing that educational investment endogenously exceeds the socially optimal level due to signaling externalities. Third, using the global equilibrium condition $A^{*}=E[A\mid A\le A^{*}]+C$, we demonstrate that the labor market partitions endogenously into a mixing segment $\left(A≤A^{*}\right)$ and a separating segment $\left(A>A^{*}\right)$, depending on the information cost $C$. Finally, the planner’s problem yields $S_y=C_y$, while the optimal tax satisfies $t^{\prime }\left(y\right)=S_n{r}^{\prime }\left(y\right)$, implying that efficient policy eliminates signaling rents without distorting productive incentives.

Beyond these formal results, the model provides a structural interpretation of several empirical phenomena. Diploma inflation emerges as an equilibrium outcome driven by fragile separation, strategic imitation, and heterogeneous screening. The coexistence of firm types implies substitutability between signaling and screening, as improvements in screening technologies reduce reliance on educational signals. Moreover, the distribution of abilities plays a central role in equilibrium selection, determining whether mixing or separating regimes dominate.

These mechanisms highlight that information frictions generate inefficient educational investments, while strategic interactions amplify local deviations into systemic effects, and institutional structures shape equilibrium outcomes. The normative implication follows directly: efficient labor market allocations require correcting signaling distortions without eliminating the informational role of education, through instruments such as taxation of educational rents, improvements in screening technologies, and policies reducing informational asymmetries.

The analysis nevertheless remains subject to limitations. The framework is static and does not incorporate dynamic learning or career paths; the signaling function $S(n,y)$ is specified in reduced form, combining productivity and informational content; the implementation of the optimal tax requires knowledge of $S_n$ and $r\left(y\right)$, which may be difficult to observe; and general equilibrium feedbacks—such as labor demand adjustments, educational supply constraints, and wage bargaining—are not explicitly modeled.

These limitations open several directions for future research. Extending the model to dynamic settings with Bayesian employer learning would allow the interaction between early signals and realized productivity to be studied explicitly. Empirical validation could test the conditions $W^{\prime }\left(y\right)>S_y$ and the existence of the threshold $A^{*}$ using microdata on wages, education, and productivity. Structural estimation would help identify the relative importance of signaling versus productive components of education and quantify welfare losses. Finally, policy design could be extended to second-best environments with imperfect observability and institutional constraints.

The central insight of this paper is that educational investment is not purely driven by productivity considerations but is fundamentally shaped by informational frictions. These frictions generate fragile equilibria, endogenous inefficiencies, and a complex interaction between signaling, screening, and policy, providing a unified explanation for diploma inflation, equilibrium segmentation, and persistent deviations from socially optimal investment levels.