Solution
Education as a Pooling Signal
Show the problem again
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.
Worked solution
The answer is 7/5. In a pooling equilibrium at $e^*$ the wage equals the prior mean productivity: $\frac25\cdot 10+\frac35\cdot 3 = \frac{29}{5}$. The harshest deterrent against deviations is the belief that anyone choosing another level is low-ability, which yields a wage of 3. The best deviation is then $e=0$, with payoff 3. The low type is the one who has most to gain from deviating, because his education cost is higher. So pooling at $e^*$ requires, for the low type, $\frac{29}{5}-2e^*\ge 3$, i.e. $e^*\le \frac{7}{5}$, and for the high type, $\frac{29}{5}-e^*\ge 3$, i.e. $e^*\le \frac{14}{5}$. The binding constraint is the low type's, so the maximum is $7/5$. At $e^*=0$ pooling is also an equilibrium, so the set of pooling levels is the interval $[0,7/5]$. A grid search over exact fractions gives the same threshold.
Source: Job-market signalling after Spence (1973), with numbers chosen for AxiomIQ. Statement written for AxiomIQ.