who are competent, that is, given the properties of the licensing test, the number of teachers who are competent is also determined.^{10}

Figure E-4 considers the case of a perfect licensure test, incorporating market demand into Figure E-2. Recall that a perfect test has the property that all those who pass it are competent and all those who do not pass are not competent. *D*_{0} represents market demand for teachers when there is no licensure test. The equilibrium wage is *w*_{0}, the equilibrium number of teachers is *T*_{0}, and the number of competent teachers is *E*_{0}*. S*_{1} represents the supply when the test is costless. As seen, if the wage remained at its original equilibrium, the number of teachers would be *E*_{0} and the fraction of competent teachers would be one (all teachers are competent with a perfect test). However, given that the fraction of competent teachers is one, independent of the wage, the demand curve may shift either up or down. The model tells us only that the demand will not shift down so much that the number of competent teachers falls below what it was in the baseline no-licensure case. Regardless of the direction in which demand shifts, the wage paid to teachers will rise, but this new wage simply reflects the appropriate scarcity of competent teachers. Because the number of competent teachers rises, student achievement also rises.

With a costly test, the supply of teachers shifts further to *S*_{2}*.* There is no further shift in the demand for teachers (the proportion of competent teachers is still one). In this case it is possible that if the test is costly enough the number of competent teachers will actually fall, for example, from *E*_{0} to *E*_{1?}. Even in a less extreme case in which the number of competent teachers rises and thus so does student learning, it is possible that the municipality would be better off without the test given the higher wage that must be paid to teachers.^{11} However, recall that in the case of a perfect test, only group C individuals take the test and the effective cost is the same as the actual cost. If the cost of the test to the competent group can be considered small, a perfect test would not contain this inefficiency. The more imperfect the test, the more likely the test is to reduce community welfare.

The same analysis also applies to the case where the test classifies all group I individuals correctly but some of those from group C are mistakenly classified