<p>In my interpretation of the model, competitive firms imply that the wage always equals the expected productivity, which depends on the beliefs. Clearly, in any pooling equilibrium the on-path beliefs are equal to the prior such that the wage is simply <span class="math-container">$E[\theta]$</span>. If, in your example, a worker sends the off-path messags <span class="math-container">$e=1$</span>, the wage must be <span class="math-container">$w(e=1)=w_1=b_1 \theta_H + (1-b_1) \theta_L$</span>, where <span class="math-container">$b_1$</span> is the belief in the information set of <span class="math-container">$(e=1)$</span> that the type is high. Since <span class="math-container">$b_1 \in [0,1]$</span>, it must be that <span class="math-container">$w_1 \in [\theta_L,\theta_H]$</span>.
This information set is not reached on equilibrium path such that we are free to determine the out-of-equilibrium belief <span class="math-container">$b_{1}$</span>. Only <span class="math-container">$b_{0}=p_H$</span> is a PBE-consistency requirement.</p>
<p>The set of all no-education pooling PBE is then given by the following: <br>
<span class="math-container">$e_H=e_L=0$</span>, <br>
<span class="math-container">$w_{0}=E[\theta]$</span>, <br>
<span class="math-container">$w_{1}=b_{1}\theta_H + (1-b_{1})\theta_L$</span>, <br>
<span class="math-container">$ b_{0}=p_H$</span>, <br>
<span class="math-container">$b_{1}: b_{1}\theta_H + (1-b_{1})\theta_L -c_H\leq E[\theta].$</span></p>
<p>Note that the beliefs are part of the equilibrium. To get rid of equilibrium multiplicity, have a look at the numerous refinements such as the intuitive criterion that kills unreasonable out-of-equilibrium beliefs (and thereby unreasonable equilibria). The restriction on belief <span class="math-container">$b_1$</span> originates from the incentive constraint of the high type. We want to make sure that <span class="math-container">$\theta_H$</span> does not want to deviate get eductation, and because the wages are competitive they are implied by the off-path belief which we can set freely.</p>
<p>The idea to set <span class="math-container">$w_1&lt;\theta_L$</span> would violate the competitive nature of firms.
This sentence can be interpreted in two ways.</p>
<ol>
<li>Wages are just exogenously set this way.</li>
<li>There are two firms (having the same beliefs) who both set a wage menu <span class="math-container">$(w_1,w_2)$</span> and a la Bertrand the worker always goes to the firm setting the higher wage.
For some off-path beliefs, one firm would deviate from <span class="math-container">$w_1&lt;\theta_L$</span> and may attract workers by setting a wage <span class="math-container">$w_1=b_{1}\theta_H + (1-b_{1})\theta_L - \varepsilon$</span> and thereby make a profit. Because you did not specify the off-path belief, we have no idea whether this deviation on the wage would be profitable.</li>
</ol>

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2021-02-24T16:22:28.940Z
Ooops, I forgot about the cost of education $c_H$.
Bayesian
CC BY-SA 4.0
2021-02-25T10:28:50.373Z
"Your idea in the comments": this was in the main text and in fact the central question. To quote from the main text: "In particular, can a pooling equilibrium wage schedule specify a wage of zero following $e=0$?"
afreelunch
CC BY-SA 4.0
2021-02-25T11:18:59.917Z
I thought in the comments you referred to $w_1=w(e=1)=0$ and the main text has $w_0=w(e=0)=0$. However, there also cannot be $w_0=0$ with $\theta_L\geq0$ as a firm can always make a profit by setting a wage $w_0=E[\theta]-\varepsilon$.
Bayesian
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2021-02-25T11:41:30.423Z
Many apologies, there was a typo in the question! Now fixed.
afreelunch
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2021-02-25T11:45:47.407Z
On a separate note, I don't see why $w(e = 1) \leq \mathbb{E}[\theta]$. On seeing somebody get education (an impossible event, given by prior beliefs) why can't I conclude that the person getting education was a high type?
afreelunch
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2021-02-25T14:34:28.323Z
I edited in the cost, which I forgot for some reason. $w_1$ follows from the beliefs and the restriction on it is more or less a restriction on the off-path belief, and theses beliefs have to be such that the high type does not want to deviate.
Bayesian
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2021-02-25T14:36:08.323Z
Of course these off-path beliefs are "unreasonable" - they have to be such that the firms believe that the deviation  to get education comes from low types. For that reason, we may want to use the intuitive criterion to rule out such beliefs.
Bayesian
CC BY-SA 4.0
2021-02-25T15:10:51.847Z
So just to clarify again, my question was whether necessarily $w(e = 1) \in [\theta_L, \theta_H]$ in any pooling equilibrium where both types choose $e = 0$. (I understand that this was not clear earlier due to a typo!) Perhaps you could revise your answer to more directly answer this question?
afreelunch
CC BY-SA 4.0
2021-02-24T16:13:57.450Z
2021-02-25T20:39:33.083Z
2021-02-25T20:39:33.083Z
Bayesian
<p>Consider the simplest version of the <a href="https://www.jstor.org/stable/1882010" rel="nofollow noreferrer">Spence signalling model</a>. There are two types of worker, with either productivity <span class="math-container">$\theta_H$</span> or instead productivity <span class="math-container">$\theta_H &lt; \theta_L$</span>. The proportions of the workers are <span class="math-container">$p_H$</span> (share of high types) and <span class="math-container">$1 - p_H$</span> respectively. Risk-neutral and competitive firms pay wages that may depend on a worker's education level <span class="math-container">$e \in \{0, 1 \}$</span>. Finally, getting education (<span class="math-container">$e = 1$</span>) costs <span class="math-container">$c_H$</span> for the high types and <span class="math-container">$c_L &gt; c_H$</span> for the low types.</p>
<p>My question is simple:</p>
<blockquote>
<p>What are the possible wage schedules that sustain a pooling
equilibrium? In particular, can a pooling equilibrium wage schedule specify a wage of
zero following <span class="math-container">$e = 1$</span>?</p>
</blockquote>
<p>I will briefly elaborate on my understanding of the problem and what is unclear to me. In any pooling equilibrium, workers must be paid their expected productivity. So following <span class="math-container">$e = 0$</span>, the specified wage must be
<span class="math-container">$$
w(e = 0) = \mathbb{E}(\theta) = p_H \theta_H + (1 - p_H)\theta_L
$$</span>
Furthermore, to ensure that nobody wants to deviate (by getting education), we only need to ensure that the high types won't deviate. (If they won't want to deviate, then neither will the low types). This requires that
<span class="math-container">$$
w(e = 1) - c_H \leq w(e = 0) = p_H \theta_H + (1 - p_H)\theta_L
$$</span>
So the wage following <span class="math-container">$e = 0$</span> is uniquely determined in the pooling equilibrium (the equation). Moreover, the wage following <span class="math-container">$e = 1$</span> must be sufficiently low to ensure that nobody gets education (the inequality). I am wondering if there are any <em>additional</em> constraints that the wage schedule in a pooling equilibrium must satisfy, or whether any wage schedule satisfying these constraints will sustain pooling (as a PBE).</p>
<p>To make the issue a bit more salient, suppose that we propose some wage <span class="math-container">$w(e = 1) &lt; \theta_L$</span>. For example, suppose you make the wage for those who get education zero. Clearly, nobody will choose education, i.e. the inequality holds. Moreover, wages don't need to be set optimally following <span class="math-container">$e = 1$</span> since this happens with zero probability in equilibrium. On the other hand, no possible type could have a productivity of zero, so this wage schedule does not follow from <em>any</em> possible firm beliefs. This suggests that the wage following zero education needs to also satisfy
<span class="math-container">$$
w(e = 1) \in [\theta_L, \theta_H]
$$</span>
Is this indeed a further constraint (in any pooling equilibrium where both types choose <span class="math-container">$e = 0$</span>)?</p>

CC BY-SA 4.0
2021-02-24T14:22:20.893Z
It seems to me all you need is $w(e=1)\in[\theta_L,\, \mathbb E(\theta)+c_H]$. Not sure if this is what you're trying to get at.
Herr K.
CC BY-SA 4.0
2021-02-24T14:35:03.587Z
So we can't have, for example, $w(e = 1) = 0 < \theta_L$? This would definitely prevent any deviations from the pooling equilibrium! (But one could never find employer beliefs which lead to this wage...)
afreelunch
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2021-02-24T17:00:41.207Z
What is a PBE here?  With competitive firms, this is not really a game.
Michael Greinecker
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2021-02-25T15:15:29.153Z
@MichaelGreinecker I think the idea is that firms play Bertrand competition over wages. However, I do agree that this is not modelled rigorously at least in the original Spence paper
afreelunch
<microeconomics><signaling>
Signalling models and out of equilibrium beliefs
afreelunch
# Signalling models and out of equilibrium beliefs

(Asked by: username_0 on 2021-02-24)

Consider the simplest version of the [Spence signalling model](https://www.jstor.org/stable/1882010). There are two types of worker, with either productivity $\theta\_H$ or instead productivity $\theta\_H < \theta\_L$. The proportions of the workers are $p\_H$ (share of high types) and $1 - p\_H$ respectively. Risk-neutral and competitive firms pay wages that may depend on a worker's education level $e \in \{0, 1 \}$. Finally, getting education ($e = 1$) costs $c\_H$ for the high types and $c\_L > c\_H$ for the low types.

My question is simple:

> What are the possible wage schedules that sustain a pooling
> equilibrium? In particular, can a pooling equilibrium wage schedule specify a wage of
> zero following $e = 1$?

I will briefly elaborate on my understanding of the problem and what is unclear to me. In any pooling equilibrium, workers must be paid their expected productivity. So following $e = 0$, the specified wage must be
$$
w(e = 0) = \mathbb{E}(\theta) = p\_H \theta\_H + (1 - p\_H)\theta\_L
$$
Furthermore, to ensure that nobody wants to deviate (by getting education), we only need to ensure that the high types won't deviate. (If they won't want to deviate, then neither will the low types). This requires that
$$
w(e = 1) - c\_H \leq w(e = 0) = p\_H \theta\_H + (1 - p\_H)\theta\_L
$$
So the wage following $e = 0$ is uniquely determined in the pooling equilibrium (the equation). Moreover, the wage following $e = 1$ must be sufficiently low to ensure that nobody gets education (the inequality). I am wondering if there are any *additional* constraints that the wage schedule in a pooling equilibrium must satisfy, or whether any wage schedule satisfying these constraints will sustain pooling (as a PBE).

To make the issue a bit more salient, suppose that we propose some wage $w(e = 1) < \theta\_L$. For example, suppose you make the wage for those who get education zero. Clearly, nobody will choose education, i.e. the inequality holds. Moreover, wages don't need to be set optimally following $e = 1$ since this happens with zero probability in equilibrium. On the other hand, no possible type could have a productivity of zero, so this wage schedule does not follow from *any* possible firm beliefs. This suggests that the wage following zero education needs to also satisfy
$$
w(e = 1) \in [\theta\_L, \theta\_H]
$$
Is this indeed a further constraint (in any pooling equilibrium where both types choose $e = 0$)?

**Tags:** microeconomics, signaling
> **Comments:**
> **username_1**: It seems to me all you need is $w(e=1)\in[\theta_L,\, \mathbb E(\theta)+c_H]$. Not sure if this is what you're trying to get at.  
> **username_0**: So we can't have, for example, $w(e = 1) = 0 < \theta_L$? This would definitely prevent any deviations from the pooling equilibrium! (But one could never find employer beliefs which lead to this wage...)  
> **username_2**: What is a PBE here?  With competitive firms, this is not really a game.  
> **username_0**: @MichaelGreinecker I think the idea is that firms play Bertrand competition over wages. However, I do agree that this is not modelled rigorously at least in the original Spence paper  

---

## 1 answers

### Answer by username_3 on 2021-02-24. Score: 5 | Accepted answer

In my interpretation of the model, competitive firms imply that the wage always equals the expected productivity, which depends on the beliefs. Clearly, in any pooling equilibrium the on-path beliefs are equal to the prior such that the wage is simply $E[\theta]$. If, in your example, a worker sends the off-path messags $e=1$, the wage must be $w(e=1)=w\_1=b\_1 \theta\_H + (1-b\_1) \theta\_L$, where $b\_1$ is the belief in the information set of $(e=1)$ that the type is high. Since $b\_1 \in [0,1]$, it must be that $w\_1 \in [\theta\_L,\theta\_H]$.
This information set is not reached on equilibrium path such that we are free to determine the out-of-equilibrium belief $b\_{1}$. Only $b\_{0}=p\_H$ is a PBE-consistency requirement.

The set of all no-education pooling PBE is then given by the following:   
$e\_H=e\_L=0$,   
$w\_{0}=E[\theta]$,   
$w\_{1}=b\_{1}\theta\_H + (1-b\_{1})\theta\_L$,   
$ b\_{0}=p\_H$,   
$b\_{1}: b\_{1}\theta\_H + (1-b\_{1})\theta\_L -c\_H\leq E[\theta].$

Note that the beliefs are part of the equilibrium. To get rid of equilibrium multiplicity, have a look at the numerous refinements such as the intuitive criterion that kills unreasonable out-of-equilibrium beliefs (and thereby unreasonable equilibria). The restriction on belief $b\_1$ originates from the incentive constraint of the high type. We want to make sure that $\theta\_H$ does not want to deviate get eductation, and because the wages are competitive they are implied by the off-path belief which we can set freely.

The idea to set $w\_1<\theta\_L$ would violate the competitive nature of firms.
This sentence can be interpreted in two ways.

1. Wages are just exogenously set this way.
2. There are two firms (having the same beliefs) who both set a wage menu $(w\_1,w\_2)$ and a la Bertrand the worker always goes to the firm setting the higher wage.
   For some off-path beliefs, one firm would deviate from $w\_1<\theta\_L$ and may attract workers by setting a wage $w\_1=b\_{1}\theta\_H + (1-b\_{1})\theta\_L - \varepsilon$ and thereby make a profit. Because you did not specify the off-path belief, we have no idea whether this deviation on the wage would be profitable.

> **Comments:**
> **username_3**: Ooops, I forgot about the cost of education $c_H$.  
> **username_0**: "Your idea in the comments": this was in the main text and in fact the central question. To quote from the main text: "In particular, can a pooling equilibrium wage schedule specify a wage of zero following $e=0$?"  
> **username_3**: I thought in the comments you referred to $w_1=w(e=1)=0$ and the main text has $w_0=w(e=0)=0$. However, there also cannot be $w_0=0$ with $\theta_L\geq0$ as a firm can always make a profit by setting a wage $w_0=E[\theta]-\varepsilon$.  
> **username_0**: Many apologies, there was a typo in the question! Now fixed.  
> **username_0**: On a separate note, I don't see why $w(e = 1) \leq \mathbb{E}[\theta]$. On seeing somebody get education (an impossible event, given by prior beliefs) why can't I conclude that the person getting education was a high type?  
> **username_3**: I edited in the cost, which I forgot for some reason. $w_1$ follows from the beliefs and the restriction on it is more or less a restriction on the off-path belief, and theses beliefs have to be such that the high type does not want to deviate.  
> **username_3**: Of course these off-path beliefs are "unreasonable" - they have to be such that the firms believe that the deviation  to get education comes from low types. For that reason, we may want to use the intuitive criterion to rule out such beliefs.  
> **username_0**: So just to clarify again, my question was whether necessarily $w(e = 1) \in [\theta_L, \theta_H]$ in any pooling equilibrium where both types choose $e = 0$. (I understand that this was not clear earlier due to a typo!) Perhaps you could revise your answer to more directly answer this question?  

---

