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Question: Consider an economy with a competitive labour market in which firms pay a wage equal to the expected productivity of the employee. There are two types of employees: Good (with a productivity of $100$ dollars) and Bad (with a productivity of $60$ dollars). The proportion of Good employees is $p$. Each employee knows his type but no one else knows it. There is also a government-run skill test that potential employees can take. It costs $15$ dollars for a Good employee to successfully prepare for this test and it costs $x>15$ dollars for a Bad employee to do so. An employee can submit the certificate of successfully passing the test with his application and the employer can set different wages for applicants with and without the certificate.

For which values of $x$ there is a separating equilibrium in which all Good employees will take the test and none of the Bad employees will do so.

My Attempt: If no signalling is presented, then the employer would pay everyone with $60+40p$ dollars (expected value). To have a separate equilibrium, we must have that Good employees would receive benefits for taking the test while Bad employees would not. However, I am confused over here: should I use the expected pay after all? With the presence of the test, employers would pay everyone who passes the test with $100$ dollars and those who do not with $60$. So should I conclude that the separate cost $x$ to achieve that is $x>40$? Or should I take into account the expected pay before signalling? (if not, when should we use that expected pay?)

Thank you in advance and I really appreciate your help!

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I think your reasoning is correct. There is no need to have an ''expected pay'' as in a separating equilibrium, everyone ''knows'' that good types will take the test and bad types will not. The only restriction is that no type has an incentive to mimic the other type.

Assume there is a separating equilibrium. Given perfect competition, this means that employees with a certificate should receive in total 100 dollars (as these will be High types and employees without a certificate receive 60 dollars (as these are the Low types).

The second condition is that the low skilled types do not mimic the High skilled types. If they would mimic the High types, they would receive 100 but it costs them $x$ dollars. If they do not get the certificate they receive 60. As such, a separating equilibrium requires that: $$ 100 - x \le 60. $$ The final condition is that high skilled types do not mimic the low skilled types. This means that: $$ 100 - 15 \ge 60, $$ this will always be satisfied.

Summarizing, we get the condition that $x \ge 40$.

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