Endogenous age discrimination

Abstract

This paper shows that hiring discrimination against old workers occurs in imperfect labour markets even if individual productivity does not decrease with age and in the absence of a taste for discrimination. Search and informational frictions generate unemployment, with less productive workers facing higher risks of unemployment. Therefore, the employment status provides a signal for expected productivity. This stigma of unemployment becomes stronger with individual age and reduces the hiring opportunities of older workers. Political measures such as a reduction in dismissal protection can help to restore efficiency.

Appendix

1.1 Proof of Proposition 1

In order to prove Proposition 1, I derive a sufficient condition such that both $L_0(\textrm {y,y})$ and $L_1(\textrm {y,y})$ are strictly larger than all other thresholds for any $\boldsymbol {\omega }$. Under Assumption 1, at the thresholds for two young workers $L_1(\textrm {y,y})$ and $L_0(\textrm {y,y})$, L is so high that no cooperation involving at least one old worker and unobservable productivity is possible even for the most extreme expectations, so we know that $L_1(\textrm {y,y})=L_1(\textrm {y,y})(0,0,0,1)$ and $L_0(\textrm {y,y})=L_0(\textrm {y,y})(0,0,0,0)$. Inversely, at levels of dismissal protection if two old workers or workers in mixed matches are indifferent between cooperation and non-cooperation, two young workers must strictly prefer to cooperate. That is, $L_1(\textrm {y,o})(\omega _{\textrm {oy}},\omega _{\textrm {yy}})=L_1(\textrm {y,o})(\omega _{\textrm {oy}},1)$, $L_1(\textrm {o,o})=L_1(\textrm {o,o})(1)$ and $L_0(\textrm {o,o})=L_0(\textrm {o,o})(1)$.

This is easy to check for $L_1(\textrm {y,y})$: the incentives to cooperate with an agent whose productivity cannot be observed are the highest if two young workers are matched. Young unemployed agents are on average more productive ($P_{\textrm {y}}>P_{\textrm {o}}$($\omega _{\textrm {yy}}$)), and a match of this kind promises additional income $2G-U_{\textrm {o}}({\boldsymbol {\omega }})>G$. Therefore, cooperation between two young workers ($\omega _{\textrm {yy}}=1$) can occur even if dismissal protection is too tough to allow for any other kind of collaboration, so $L_1(\textrm {y,y})$ is strictly larger than any other threshold.

However, it holds for $L_0(\textrm {y,y})$ only if

$$ \gamma[2G-U_{\textrm{o}}(\boldsymbol{\omega})]>G \qquad\quad \forall \;\;\boldsymbol{\omega}. $$

(20)

Since $U_{\textrm {o}}(\boldsymbol {\omega })$ is strictly decreasing in L and $\omega _{\textrm {yy}}$ and I am only interested in a sufficient condition, I can assume $L=0$ and $\omega _{\textrm {yy}}=0$. For $L=0$, $U_{\textrm {o}}(\boldsymbol {\omega })$ is strictly increasing in $\omega _{\textrm {oo}},\omega _{\textrm {yo}}$ and $\omega _{\textrm {oy}}$, so I use $U(1,1,1,0)$. Now, I can rewrite Eq. 20 as

$$ \gamma>\frac{2-P_{\textrm{y}}+O(0)}{4-3P_{\textrm{y}}+O(0)}. $$

(21)

The right-hand side of Eq. 21 is increasing in $O(0)$, so I substitute $P_{\textrm {y}}$ for $O(0)$ as $P_{\textrm {y}}>O(0)$ to obtain the sufficient condition

$$ \gamma>\frac{1}{2-P_{\textrm{y}}}. $$

(22)

This inequality holds if $\gamma$ is sufficiently large and $P_{\textrm {y}}$ is sufficiently small. Figure 3 shows the combinations of $P_{\textrm {y}}$ and $\gamma$ that satisfy Assumption 1.

Fig. 3

Illustration of Assumption 1

This sufficient condition is very loose (assuming $L=0$, etc.); thus, even a higher fraction of productive workers should support this equilibrium once I impose restrictions on $G/L$.

1.2 Proof of Proposition 2

To determine $L_1(\textrm {y,o})(\omega _{\textrm {oy}})$ and $L_1(\textrm {o,y})$, first note that $L_0(\textrm {o,o})\le L_1(\textrm {y,o})(\omega _{\textrm {oy}})\le L_1(\textrm {o,o})$ (because $L_1(\textrm {y,o})(0)=L_0(\textrm {o,o})$ and $L_1(\textrm {y,o})(1)=L_1(\textrm {o,o})$) and $L_1(\textrm {o,y})>L_0(\textrm {o,o})$. The threshold for an old worker in order to cooperate with a young one depends on the expected behaviour of that young agent (and vice versa). That is, the position of one threshold has an impact on the position of the other. There could be three different cases:

• 1. First consider $L_1(\textrm {o,y})<L_1(\textrm {y,o})(\omega _{\textrm {oy}})$. This implies $L_1(\textrm {o,y})=L_1(\textrm {o,y})(1)$ and $L_1(\textrm {y,o})=L_1(\textrm {y,o})(0)$, so

  $$ L_1(\textrm{o,y})(1)=\frac{P_{\textrm{y}}G}{1-P_{\textrm{y}}} \textrm{ and } $$

  (23)
  $$ L_1(\textrm{y,o})(0)=\frac{P_{\textrm{o}}(1)\gamma G}{1-P_{\textrm{o}}(1)}. $$

  (24)

  It is easy to check that then $L_1(\textrm {o,y})(1)>L_1(\textrm {y,o})(0)$, which contradicts the assumption $L_1(\textrm {o,y})<L_1(\textrm {y,o})(\omega _{\textrm {oy}})$.

• 2. Now consider the case $L_1(\textrm {o,y})=L_1(\textrm {y,o})$:

  $$ \frac{P_{\textrm{y}}(\gamma+(1-\gamma)\omega_{\textrm{yo}})G}{1-P_{\textrm{y}}}=\frac{ P_{\textrm{o}}(1)(\gamma+(1-\gamma)\omega_{\textrm{oy}})G}{1-P_{\textrm{o}}(1)} $$

  (25)

  This can only be satisfied if $\omega _{\textrm {oy}}$ at the right-hand side is to a sufficient degree larger than $\omega _{\textrm {yo}}$ at the left-hand side in order to outweigh $P_{\textrm {o}}(1)<P_{\textrm {y}}$. When does a combination of strategies $\omega _{\textrm {yo}}$ and $\omega _{\textrm {oy}}$ exist such that $L_1(\textrm {o,y})=L_1(\textrm {y,o})$ can hold? I substitute $P_{\textrm {o}}(1)=1-\frac {\sqrt {1-P_{\textrm {y}}}}{\sqrt {1+P_{\textrm {y}}}}$ and obtain

  $$ P_{\textrm{y}}[\gamma+(1-\gamma)\omega_{\textrm{yo}}]=\left[\sqrt{1-P_{\textrm{y}}^2}-(1-P_{\textrm{y}})\right] [\gamma+(1-\gamma)\omega_{\textrm{oy}}]. $$

  (26)

  This can only hold if $\omega _{\textrm {oy}}>\omega _{\textrm {yo}}$. If even for $\omega _{\textrm {oy}}=1$ and $\omega _{\textrm {yo}}=0$ the left-hand side is larger than the right-hand side, no combination of $\omega _{\textrm {oy}}$ and $\omega _{\textrm {yo}}$ exists such that the two thresholds can coincide.

  Result:

  For $\gamma \le \frac {\sqrt {1-P_{\textrm {y}}^2}-(1-P_{\textrm {y}})}{P_{\textrm {y}}}$, some combinations of $\omega _{\textrm {oy}}$ and $\omega _{\textrm {yo}}$ exist with $\omega _{\textrm {oy}}>\omega _{\textrm {yo}}$ such that $L_0(\textrm {o,o})<L_1(\textrm {o,y})=L_1(\textrm {y,o})(\omega _{\textrm {oy}})\le L_1(\textrm {o,o})$. There can be different thresholds with $L_1(\textrm {y,o})=L_1(\textrm {o,y}) \in [L_0(\textrm {o,o}),L_1(\textrm {o,o})]$ because there are different combinations of $\omega _{\textrm {oy}}$ and $\omega _{\textrm {yo}}$ that support this condition.

• 3. Assume $L_1(\textrm {o,y})>L_1(\textrm {y,o})(\omega _{\textrm {oy}})$. This implies $L_1(\textrm {o,y})=L_1(\textrm {o,y})(0)$ and $L_1(\textrm {y,o})=L_1(\textrm {y,o})(1)$. Then,

  $$ L_1(\textrm{o,y})(0)=\frac{P_{\textrm{y}}\gamma G}{1-P_{\textrm{y}}}\textrm{and} $$

  (27)
  $$ L_1(\textrm{y,o})(1)=\frac{P_{\textrm{o}}(1)G}{1-P_{\textrm{o}}(1)}. $$

  (28)

  So $L_1(\textrm {o,y})>L_1(\textrm {y,o})$ is an equilibrium if

  $$ \frac{P_{\textrm{y}} \gamma G}{1-P_{\textrm{y}}}>\frac{P_{\textrm{o}}(1)G}{1-P_{\textrm{o}}(1)}, $$

  (29)
  $$ \gamma > \frac{\sqrt{1-P_{\textrm{y}}^2}-(1-P_{\textrm{y}})}{P_{\textrm{y}}}. $$

  (30)

  Result:

  For $\gamma > \frac {\sqrt {1-P_{\textrm {y}}^2}-(1-P_{\textrm {y}})}{P_{\textrm {y}} }$, $L_1(\textrm {o,y})=L_1(\textrm {o,y})(0)>L_1(\textrm {y,o})=L_1(\textrm {y,o})(1)=L_1(\textrm {o,o})$ is the only equilibrium (Fig. 4).

Fig. 4

Illustration of Assumption 2