Higher education expansion, tracking, and student effort

Abstract

This paper studies the effects of tracking on student effort and academic achievement during secondary school. Students under the comprehensive school system make effort to increase their academic achievement that determines the probability of gaining admissions to college. As college admissions become less competitive or it becomes easier to gain admissions, it increases the benefits of effort, encouraging more effort. Those students streamed into academic tracks at early ages under tracking are most likely to be admitted to college, and the competitiveness of college admissions does not affect their effort. As a result, tracking tends to perform worse in terms of average secondary-school academic achievement, as higher education expands and college admissions become less competitive.

Appendix

1.1 Derivation of the results in Sect. 5.1

With $Prob\;(\epsilon \ge t-b) = 1 - H(t-b)$, the first-order condition for a maximum of $U^C(e_1)$ is

$$\begin{aligned} U^{C'}(e_1) = [h (v - k) + (1 - H) w_b]b_e - c_1' = 0. \end{aligned}$$

(45)

The second-order condition is

$$\begin{aligned} U^{C''}(e_1)&= [h (v - k) + (1 - H) w_b]b_{ee}+ [2hw_b - h'(v - k) \nonumber \\&+ (1 - H)(w_{bb}+ {wbe}_{2}^{C'}(b)](b_e)^2 - c_1'' < 0. \end{aligned}$$

(46)

All terms in (46) are negative, except $hw_b (b_e)^2$. For the satisfaction of the second-order condition, all other negative terms are assumed to dominate the only positive term. For example, $c_1(e_1)$ is convex enough or $b(.)$ is concave enough. Differentiation of (45) gives the expression for $\partial e_1^C/\partial t$ in (23). If $\partial e_1^C/\partial t \le 0$ in (23),

$$\begin{aligned} \frac{d}{dt}[t - b(a,e_1^C(a,t))] = 1 - b_e \frac{\partial e_1^C}{\partial t} > 0. \end{aligned}$$

(47)

If $\partial e_1^C/\partial t > 0$, using (23) and (46),

$$\begin{aligned}&\frac{d}{dt}[t - b(a,e_1^C(a,t))] = 1 - b_e \frac{\partial e_1^C}{\partial t} = - \frac{1}{U^{C''}}\{-[h(v - k) + (1-H)w_b]b_{ee} \nonumber \\&\quad - [hw_b + (1-H)(w_{bb} +w_{be}e_2'(b))](b_2)^2 + c_1''\}. \end{aligned}$$

(48)

All terms in (48) are positive, except $-hw_b(b_e)^2$. As in the second-order condition in (46), all other positive terms are assumed to dominate the only negative term, for example, by assuming that $c_1(e_1)$ is convex enough or $b(.)$ is concave enough. Alternatively, combining this negative term with $-h(v-k)b_{ee}$, the first term of the numerator of (48),

$$\begin{aligned}&-hw_b(b_e)^2 - h(v-k)b_{ee} > - h'(v-k)(b_e)^2 - h(v-k)b_{ee}\nonumber \\&\quad = - h(v-k) \left[ \frac{h'}{h}(b_e)^2 + b_{ee}\right] = - h(v-k)\left[ - \frac{t-b}{\sigma ^2} (b_e)^2 + b_{ee}\right] \nonumber \\&\quad = h(v-k)\left[ \frac{t-b}{\sigma ^2} (b_e)^2 - b_{ee}\right] , \end{aligned}$$

(49)

where $\sigma$ is the standard deviation of the normal random variable $\epsilon .$ The inequality comes from $-hw_b > - h'(v-k)$ when $\partial e_1^C/\partial t > 0$ in (23). The second equality uses the normal density function with mean zero, $h(\epsilon ) = \frac{1}{\sigma \sqrt{2\pi }}exp(-\frac{\epsilon ^2}{2 \sigma ^2})$, and $h'(\epsilon ) = - \frac{\epsilon }{\sigma ^2}h(\epsilon ),$ so that $h'/h = - \epsilon /\sigma ^2 = - (t-b)/\sigma ^2.$ Equation (49), and hence (48), become positive if $\sigma$ is large or $b_e$ is small (effort is not too productive) or $b$ is concave enough.