s e asegurará que en cada cenTro haya maTeriales didácTi -

In the first stage every individual i makes his occupational choice to maximize his utility, ui. This stage is based on a Roy model, which is the most used

framework in the field of work self-selection.7

5_{A positive correlation between teaching and non-teaching skills is implicit in the fol-}

lowing works. Hanushek et al. (2015) study the private returns to cognitive skills using the PIAAC survey, and find a positive effect of cognitive skills on salaries. On the other hand, Hanushek et al. (2014) study the effects of teacher cognitive skills, also measured based on the PIAAC survey, on student achievements. This work finds a positive effect of teacher skills on student outcomes. Therefore, the same skills present positive effects in both, the teaching and non-teaching sectors.

6_{This assumption captures the empirical fact that the variance between teacher salaries}

is very low compared to the variance of salaries across the whole economy.

7_{Roy models have been successfully used to study the selection process in labour mar-}

The individual utility function is given by:

ui =

ln(w_b0) + ln(btτ) if be a teacher

ln(w_bθ) otherwise,

(2.3)

where ln(btτ_{) captures the non-pecuniary utility of teaching. The parame-}

ter τ measures the elasticity of the non-pecuniary utility with respect to the individual teacher quality.

The utility obtained as a teacher depends not only on the wage but also on the individual teacher’s ability. The idea behind this is that teaching skills must be positively correlated with the vocation of becoming a teacher. Thus, the more motivated the teacher, the larger the non-pecuniary utility of teaching is.8 The Appendix I discusses the sensitivity of the model’s equilibrium to changes in the parameter values. Particularly, the case where utility depends only on monetary compensation (τ = 0) is analysed.9

Given the utility function of Equation (2.3), individuals choose to become teachers if:

ln(w_b0) + τ ln(bt) > α ln(bθ). (2.4)

Defining ln(_bx) = x, using equation (2.4) and regrouping, the condition above can be written as:

αθ − τ t < w0, (2.5)

(Borjas, 2002), manufacturing industries (Heckman, 1985) and entrepreneurs (Evans and Jovanovic, 1989).

8_{This interpretation could be replaced by assuming some wage dispersion within the}

teacher sector where the best teachers are better paid. That is, τ could be interpreted as the elasticity of teacher wages respect to the individual teacher skill without modifying any result of the model. If τ is lower than α the scenario is still compatible with the fact of lower wage variance within the teacher sector with respect to the whole economy.

9_{In the utility function, I do not consider the probability of not being selected for a}

teaching position because I assume that individuals who are not selected as teachers can apply for a job in the market sector under the same conditions as the rest of the population.

78 Teacher Quality and Student Achievements Additionally, define v = αθ − τ as the ”self-selection function”. Since v is a sum of two normal random variables, it is also distributed normally. Particularly, v ∼ N (µv, σ2v), with mean µv = αµθ − τ µt and variance σ2v =

α2_σ2

θ+ τ2σ2t− 2ατ σtθ. This self-selection function will help us to identify the

supply of teachers, the average quality of the teacher supply as well as the correlation of this self-selection function with the general skills.

Let define the following indicator function:

I = 1 if the individual choose to be a teacher

0 otherwise. (2.6)

The probability that an individual chooses to work in the teaching sector, P (I = 1) which is the teacher supply size, is given by:

P (I = 1) = P (v < w0) = P  v − µv σv < w0− µv σv  , = Φ w0− µv σv  = Φ (z) , (2.7)

where Φ is the cumulative distribution function of the standard normal distribution, and z = w0−µv

σv . Since, the cumulative distribution function of

the standard normal is an increasing function in z, and z is an increasing function of the teacher salary (w0), the teacher supply is also an increasing

function of the teacher salary.

The average quality of the teacher supply, E(t|I = 1), is computed as follows:10 E(t|I = 1) = E  t|v − µv σv < w0− µv σv 

10_{This expression corresponds to the mean of an incidentally truncated bivariate normal}

distribution. Just to simplify, I present the expected mean of t instead oft. Note that,b since the logarithm is a strictly increasing function, any change in the mean of t implies a change of the same sign in the mean quality of teachers (i.e. mean ofbt).

= µt+ ρvtσt

 −φ(z) Φ(z)



, (2.8)

where µt is the mean teacher skill of the total population in the economy,

z = w0−µv

σv , ρvtis the correlation coefficient between the self-selection function

and the teacher skill, and φ and Φ are the density function and cumulative distribution function of the standard normal distribution respectively. The expression σt

_−φ(z)

Φ(z)



in Equation (2.8) is always negative. Therefore, the average quality among the supply of teachers, E(t|I = 1), is lower than the population mean (µv) when ρvt is positive. And thus, a negative selection

bias in the teacher supply arises.11 _{On the other hand, when ρ}

vt is negative,

the supply of teachers presents a positive selection bias. The expression for ρvt is given by:

ρvt =

σvt

σtσv

(2.9) Since, σtand σv are always positive terms, the selection bias of the teacher

supply will depend on the sign of σvt. Moreover, σvt = cov(αθ − τ t; t) =

ασθt− τ σt2 . Hence, a negative selection bias arises when σθt > τ σt2/α while

a positive selection bias arises in the opposite case. Therefore, even discard- ing the less intuitive cases where σθt is negative, a positive selection bias is

possible. If the non-pecuniary utility of teaching is eliminated, τ = 0, and ρθt ≥ 0 holds, the model always predicts a negative selection bias in the

teacher supply.