Education as a Pooling Signal

A worker is of high ability (probability $2/5$) or low ability (probability $3/5$); the worker knows his own type, firms do not. High workers have productivity 10 and low workers 3. A worker chooses a level of education $e\ge 0$ (a real number), which has no effect on productivity but costs $e$ for a high worker and $2e$ for a low worker. Two or more competing firms observe $e$ and then each offers a wage equal to the expected productivity given their beliefs, so the wage equals expected productivity. A worker's payoff is the wage minus the education cost. Consider **pooling** perfect Bayesian equilibria in which both types choose the same education level $e^*$. Beliefs after any other level are free to be whatever sustains the equilibrium best (the most pessimistic belief, that the worker is low-ability, is allowed). What is the largest $e^*$ that can be sustained in such an equilibrium? Give a reduced fraction.

• combinations
• conditional probability
• nash equilibrium
• optimal stopping

Source: Job-market signalling after Spence (1973), with numbers chosen for AxiomIQ. Statement written for AxiomIQ.