denoted by *w* and, because competency is assumed to be indeterminate without a test or for institutional reasons, the wage offer is assumed to be the same regardless of whether the teacher is competent or not.^{2} However, the wages of individuals outside of teaching do differ. In particular, individuals who would be competent teachers have a wage distribution in alternative occupations with a given mean of *w*_{C}^{a} and individuals who would be incompetent teachers have a distribution with mean wage *w*_{I}^{a}.

Individuals choose teaching versus an alternative occupation depending on which provides the higher wage. Thus, an individual *k* in group C who would command a wage of *w*_{Ck}^{a} in a nonteaching occupation chooses teaching if and only if *w>w*_{Ck}^{a}*=w*_{C}^{a}*+ε*_{Ck}^{a}, where *ε*_{Ck}^{a} reflects the deviation of the individual’s wage from the mean wage of group C. Analogously, an individual *j* in group I would choose teaching if and only if *w>w*_{I}_{i}^{a}*=w*_{I}^{a}+ε_{I}_{i}^{a}, where ε_{I}_{i}^{a} reflects the deviation of the individual’s wage from the mean wage of group I. Given these rules for choosing an occupation, the probability that an individual in group C chooses teaching is given by α_{C}=Pr(ε_{Ck}^{a}<*w−w*_{C}^{a}). Likewise, the probability that an individual in group I chooses teaching is α_{I}=Pr(ε_{I}_{i}^{a}<*w−w*_{I}^{a}). From these definitions it is easy to see that the proportion of individuals who choose to be teachers at any given wage is *q*_{C}*α*_{C}+(1−*q*_{C})α_{I,} and the proportion of them who would be competent teachers is Π_{C}=*q*_{C}α_{C}/[*q*_{C}*α*_{C}+(1−*q*_{C})α_{I}].

As the wage paid to teachers increases, teaching becomes more attractive to both groups (both α_{C} and α_{I} are increasing in the wage, *w*). However, it is not possible to determine whether the proportion of teachers who are competent increases or decreases as the wage paid to teachers increases. The effect of the wage on the relative supply of competent teachers, that is, on Π_{C}, depends on the level of the wage paid to teachers and on how the distribution of wage offers in alternative occupations differs between the two groups. In the analysis that follows, it is assumed that the wage paid to teachers is in a range such that the proportion of competent teachers does not change with the wage, that is, Π_{C} is independent of *w*.

Figure E-1 shows the supply of teachers as a function of the wage paid to teachers. The number of teachers, denoted by *T*, is increasing with the teachers’ wage. The proportion of competent teachers (at wage *w*_{0} and, by assumption, at any wage) is given by *E*_{0}/*T*_{0.}

Assume now that all potential teachers must take a licensure test. The purpose and design of the test are to distinguish between the two groups, which it