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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 $\Theta^* = \{\theta:r(\theta) \leq w^*\}$.

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  • $\begingroup$ I think it would be helpful to define what all these terms are. What is $r(\theta)$? What is positive selection? What is $\theta$? $\endgroup$ – Walrasian Auctioneer Nov 17 '20 at 5:07
  • $\begingroup$ @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. $\endgroup$ – CASIOfx-991EX Nov 17 '20 at 10:37
  • $\begingroup$ 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. $\endgroup$ – Bayesian Nov 17 '20 at 12:35
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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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