<p>In PO, you want all types with opportunity cost <span class="math-container">$r(\theta)\leq \theta$</span> to trade, because the firm gets more productivity than the worker has to give up on home productivity (&quot;opportunity cost&quot;).</p>
<p>By your assumptions <span class="math-container">$r(\theta_H) &lt; \theta_H$</span> and <span class="math-container">$r(\theta_L)&gt;\theta_L$</span> such that in PO, L shall stay at home and H shall work. However, you need to offer H a wage of at least <span class="math-container">$w\geq r(\theta_H)$</span>.
The highest wage a firm would offer is the expected <span class="math-container">$\theta$</span> of a worker who is willing to accept this wage. Suppose that <span class="math-container">$\mathbb E[\theta]=\lambda \theta_H + (1-\lambda)\theta_L &gt;r(\theta_H)$</span> and, by definition,  <span class="math-container">$\theta_L&lt;\mathbb E[\theta]&lt;\theta_H$</span>.</p>
<p>Now, it could be that <span class="math-container">$w=E[\theta] &gt; r(\theta_L)$</span> such that L would also want to accept this wage, but this is inefficient as
<span class="math-container">$$w &gt; r(\theta_L) &gt; \theta_L.$$</span>
Low types should not work, but because there are so few of them (<span class="math-container">$\lambda$</span> is large), the firms would still profit if they also accepted <span class="math-container">$w$</span>.</p>
<p>An uniformed third party could Pareto improve the allocation by changing the wage to <span class="math-container">$w' \in (r(\theta_H),r(\theta_L))$</span> such that L types would reject <span class="math-container">$w'$</span>.</p>

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2020-11-17T12:31:27.043Z
2020-11-17T12:42:06.547Z
2020-11-17T12:42:06.547Z
Bayesian
<p>I'm reviewing some question from Mas-Collel and I am stuck on a chapter 13 question related to adverse selection.</p>
<p>Consider a model of positive selection in which there are workers of two possible productivity types, <span class="math-container">$\theta_H$</span> and <span class="math-container">$\theta_L$</span>, with <span class="math-container">$\infty &gt; \theta_H &gt; \theta_L &gt; 0$</span> and <span class="math-container">$\lambda = Prob(\theta=\theta_H)  \in (0,1)$</span>.
A worker of type <span class="math-container">$\theta_i$</span> can produce <span class="math-container">$\theta_i$</span> for a firm in exchange for a wage <span class="math-container">$w$</span> or work at home and gain <span class="math-container">$r(\theta_i)$</span>.
r(.) is strictly decreasing and r<span class="math-container">$(\theta_H) &lt; \theta_H$</span> and that <span class="math-container">$r(\theta_L)&gt;\theta_L$</span>. Show that the highest-wage competitive equilibrium need not be a constrained Pareto optimum. [13.B.9]</p>
<p>If the equilibrium is not a constrained Pareto optimal, then it could be a Pareto optimal, this occurs where all types are full employed when <span class="math-container">$w = \theta$</span>, i.e. the condition that <span class="math-container">$\Theta^* = \{\theta:r(\theta) \leq w^*\}$</span>.</p>

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2020-11-17T05:07:17.900Z
I think it would be helpful to define what all these terms are. What is $r(\theta)$? What is positive selection? What is $\theta$?
Walrasian Auctioneer
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2020-11-17T10:37:43.683Z
@WalrasianAuctioneer It refers to the type of worker, in this case, the productivity level, it's standard in most adverse selection literature when referring to screening. It probably requires some familiarity with Akerlof (1970) Market for Lemons.
CASIOfx-991EX
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2020-11-17T12:35:03.143Z
I made some edits. I hope I changed the question in way that you had in mind. I still don't know what you mean by $\theta$ without any index.
Bayesian
<microeconomics><pareto-efficiency><asymmetric-information><adverse-selection><mwg>
Adverse Selection: Positive Selection of Worker Types (Mas-Collel)
CASIOfx-991EX
# Adverse Selection: Positive Selection of Worker Types (Mas-Collel)

(Asked by: username_0 on 2020-11-17)

I'm reviewing some question from Mas-Collel and I am stuck on a chapter 13 question related to adverse selection.

Consider a model of positive selection in which there are workers of two possible productivity types, $\theta\_H$ and $\theta\_L$, with $\infty > \theta\_H > \theta\_L > 0$ and $\lambda = Prob(\theta=\theta\_H) \in (0,1)$.
A worker of type $\theta\_i$ can produce $\theta\_i$ for a firm in exchange for a wage $w$ or work at home and gain $r(\theta\_i)$.
r(.) is strictly decreasing and r$(\theta\_H) < \theta\_H$ and that $r(\theta\_L)>\theta\_L$. Show that the highest-wage competitive equilibrium need not be a constrained Pareto optimum. [13.B.9]

If the equilibrium is not a constrained Pareto optimal, then it could be a Pareto optimal, this occurs where all types are full employed when $w = \theta$, i.e. the condition that $\Theta^\* = \{\theta:r(\theta) \leq w^\*\}$.

**Tags:** microeconomics, pareto-efficiency, asymmetric-information, adverse-selection, mwg
> **Comments:**
> **username_1**: I think it would be helpful to define what all these terms are. What is $r(\theta)$? What is positive selection? What is $\theta$?  
> **username_0**: @WalrasianAuctioneer It refers to the type of worker, in this case, the productivity level, it's standard in most adverse selection literature when referring to screening. It probably requires some familiarity with Akerlof (1970) Market for Lemons.  
> **username_2**: I made some edits. I hope I changed the question in way that you had in mind. I still don't know what you mean by $\theta$ without any index.  

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## 1 answers

### Answer by username_2 on 2020-11-17. Score: 5

In PO, you want all types with opportunity cost $r(\theta)\leq \theta$ to trade, because the firm gets more productivity than the worker has to give up on home productivity ("opportunity cost").

By your assumptions $r(\theta\_H) < \theta\_H$ and $r(\theta\_L)>\theta\_L$ such that in PO, L shall stay at home and H shall work. However, you need to offer H a wage of at least $w\geq r(\theta\_H)$.
The highest wage a firm would offer is the expected $\theta$ of a worker who is willing to accept this wage. Suppose that $\mathbb E[\theta]=\lambda \theta\_H + (1-\lambda)\theta\_L >r(\theta\_H)$ and, by definition, $\theta\_L<\mathbb E[\theta]<\theta\_H$.

Now, it could be that $w=E[\theta] > r(\theta\_L)$ such that L would also want to accept this wage, but this is inefficient as
$$w > r(\theta\_L) > \theta\_L.$$
Low types should not work, but because there are so few of them ($\lambda$ is large), the firms would still profit if they also accepted $w$.

An uniformed third party could Pareto improve the allocation by changing the wage to $w' \in (r(\theta\_H),r(\theta\_L))$ such that L types would reject $w'$.


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