Worked Schooling Choice Model

Why Schooling Is a Dynamic Choice

A student deciding whether to continue school weighs an immediate cost against a stream of future consequences. Continuing can mean tuition, effort, and earnings forgone today, while also increasing the education level associated with later wages. Dropping out avoids another period of schooling but fixes the student’s education state and wage opportunities. A dynamic choice model represents these trade-offs period by period rather than treating schooling as a one-time comparison.

The model below is deliberately small enough to calculate by hand. It has two decision periods, an education state measured in completed schooling levels, and two actions: continue or drop out. Its wage schedule makes the role of both foregone earnings and future wage returns explicit. Choice probabilities arise from idiosyncratic shocks, so a higher return shifts the probability of continuing without making every student choose the same action.

Definition: Let \(t\in\{1,2\}\) index decision periods and let \(s\) denote completed education levels. At each period in school, the student chooses \(S\), continue schooling, or \(D\), drop out. Choosing \(S\) advances education to \(s+1\) in the next period. Choosing \(D\) is absorbing: the student receives wages associated with \(s\) and makes no further schooling choices.

A Finite-Horizon Schooling Model

Let \(w(s;\rho)\) be the wage available to a dropout with education \(s\), where \(\rho\) controls the wage return to an additional education level. Wages are received in each remaining decision period after dropping out. At the end of period 2, a student with education \(s\) receives a terminal value \(G(s)\), interpreted as the present value of post-decision-period earnings. In the numerical examples, \(G(s)=A w(s;\rho)\), where \(A\) is a fixed multiplier representing the post-horizon earnings stream.

Payoffs are expressed in utility-equivalent units, with the marginal utility of money normalized to one. A student who continues receives the current payoff \(-c\), where \(c>0\) is a schooling cost, and does not earn the dropout wage during that period. A student who drops out receives \(w(s;\rho)\) immediately. Thus, the current payoff difference includes both the schooling cost and the earnings forgone by remaining in school.

Model assumptions: The decision horizon and education transition are known; wages depend on education through a specified schedule; dropout is absorbing; and \(c\), \(\rho\), and the discount factor \(\beta\) are fixed and known for the choice calculation. At each active schooling decision, the student observes a choice-specific shock. Shocks are independent across choices and periods and follow normalized Type I extreme-value distributions with scale one and mean zero. Conditional on the state and these assumptions, the current education level and period are sufficient to calculate the choice values.

Write \(Q_t^S(s)\) and \(Q_t^D(s)\) for the choice-specific values before the current shocks are realized. These values include current payoffs and discounted future values. Let \(V_t(s)\) be the expected value before the period-\(t\) shocks. With the stated shock normalization, integrating over the shocks gives the log-sum expression below. The terminal value is \(G(s)\); there are no further schooling decisions after period 2.

\begin{aligned} Q_2^S(s)&=-c+\beta G(s+1),& Q_2^D(s)&=w(s;\rho)+\beta G(s),\\ V_2(s)&=\log\!\left[\exp\!\left(Q_2^S(s)\right)+\exp\!\left(Q_2^D(s)\right)\right],\\ Q_1^S(s)&=-c+\beta V_2(s+1),& Q_1^D(s)&=w(s;\rho)+\beta w(s;\rho)+\beta^2G(s). \end{aligned}

The period-1 dropout value includes the current wage, next period’s wage, and the discounted terminal value. There is no \(V_2\) term on that path because a student who drops out does not make another schooling choice. At any active decision, the conditional probability of continuing is

P(S\mid t,s)= \frac{\exp(Q_t^S(s))} {\exp(Q_t^S(s))+\exp(Q_t^D(s))} = \frac{1}{1+\exp\!\left[-\left(Q_t^S(s)-Q_t^D(s)\right)\right]}.

These probabilities describe the distribution of choices across students facing the same state and model parameters, because their shocks differ. They are not deterministic rules, and they are not probabilities that a particular value is correct.

Worked Example: A Period-2 Choice at a Lower Wage Return

Suppose \(w(s;\rho)=10+\rho(s-11)\), \(G(s)=5w(s;\rho)\), \(\beta=0.9\), \(c=2\), and \(\rho=2\). Consider a student at \(t=2\) with \(s=12\). Wages are \(w(12;2)=12\) and \(w(13;2)=14\), so the terminal values are \(G(12)=60\) and \(G(13)=70\). The period-2 choice values are

\begin{aligned} Q_2^D(12)&=12+0.9(60)=66,\\ Q_2^S(12)&=-2+0.9(70)=61. \end{aligned}

The value difference is \(61-66=-5\), so the conditional probability of continuing is

P(S\mid 2,12)=\frac{1}{1+\exp(5)}=0.006693.

Under these hypothetical values, a student in this state has about a 0.6693% probability of choosing another schooling period. The calculation accounts for the immediate wage available on dropping out as well as the terminal earnings value associated with each education level.

Tracing a Change in the Wage Return

A larger \(\rho\) raises wages more steeply with education in the example’s wage schedule. It can therefore increase the payoff from continuing, because schooling leads to a higher education state and a higher terminal wage value. But a wage increase at the student’s current education level also makes dropping out more attractive now. The net effect depends on the difference between these changes, not on the future wage gain considered in isolation.

Worked Example: Increasing the Return to Education

Keep \(w(s;\rho)=10+\rho(s-11)\), \(G(s)=5w(s;\rho)\), \(\beta=0.9\), and \(c=2\), but increase the return from \(\rho=2\) to \(\rho=4\). For a period-2 student at \(s=12\), the wages become \(w(12;4)=14\) and \(w(13;4)=18\). The terminal values are \(G(12)=70\) and \(G(13)=90\). Thus,

\begin{aligned} Q_2^D(12)&=14+0.9(70)=77,\\ Q_2^S(12)&=-2+0.9(90)=79,\\ Q_2^S(12)-Q_2^D(12)&=2. \end{aligned}

The schooling probability is now

P(S\mid 2,12)=\frac{1}{1+\exp(-2)}=0.8808.

In this hypothetical model, raising \(\rho\) from 2 to 4 moves the period-2 continuation probability at \(s=12\) from 0.006693 to 0.8808. This large change follows from the chosen numerical parameters; it is not a general claim that every wage-return increase produces such a large response.

Why Earlier Decisions Depend on Later Options

At \(t=1\), the value of continuing includes the expected value of the schooling decision available at \(t=2\). That value reflects the option to continue again or to drop out later. Consequently, the period-1 comparison cannot generally be obtained by comparing only the current wage and schooling cost. The dynamic model carries the later choice forward through \(V_2\).

Worked Example: Continuing from an Earlier Education State

Use \(\rho=4\) and the other parameters from the preceding example. Consider a student at \(t=1\) with \(s=11\). At \(t=2\), if this student continues now, the state will be \(s=12\). From the preceding calculation, \(Q_2^D(12)=77\) and \(Q_2^S(12)=79\), giving the inclusive value

V_2(12)=\log\!\left(\exp(77)+\exp(79)\right) =79+\log\!\left(1+\exp(-2)\right) =79.1269.

The period-1 continuation value is therefore

$$ Q_1^S(11)=-2+0.9(79.1269)=69.2142. $$

If the student drops out immediately, the current and next-period wages are both \(w(11;4)=10\), and the terminal value is \(G(11)=5(10)=50\). Hence

$$ Q_1^D(11)=10+0.9(10)+0.9^2(50)=59.5. $$

The period-1 value difference is \(69.2142-59.5=9.7142\), and the probability of continuing is

P(S\mid 1,11)=\frac{1}{1+\exp(-9.7142)}=0.99994.

The probability is high because continuing provides access to a valuable later choice and its associated education-dependent earnings. The calculation uses the same wage schedule, discount factor, terminal value, and shock assumptions at both periods.

A Short Comparative-Static Derivation

The period-2 value difference isolates the forces behind the schooling decision. For a general terminal-value multiplier \(A\), it is

\begin{aligned} \Delta_2(s;\rho) &=Q_2^S(s)-Q_2^D(s)\\ &=-c-w(s;\rho)+\beta A\left[w(s+1;\rho)-w(s;\rho)\right]. \end{aligned}

This follows by subtracting \(Q_2^D(s)=w(s;\rho)+\beta A w(s;\rho)\) from \(Q_2^S(s)=-c+\beta A w(s+1;\rho)\). The current wage and schooling cost are the immediate sacrifice from continuing; the discounted difference in terminal values is its modeled future payoff. Under \(w(s;\rho)=10+\rho(s-11)\), the wage difference between adjacent education levels is \(\rho\). Therefore,

\frac{\partial \Delta_2(s;\rho)}{\partial \rho} =-(s-11)+\beta A.

For the worked examples, \(s=12\), \(\beta=0.9\), and \(A=5\), so this derivative is \(-1+4.5=3.5\). Raising \(\rho\) by 2 increases the value difference by \(2(3.5)=7\), exactly matching the change from \(-5\) to \(2\) in the first two examples. Since the choice probability is the logistic function of \(\Delta_2\), its derivative is

\frac{\partial P(S\mid 2,s)}{\partial \rho} =P(S\mid 2,s)\left[1-P(S\mid 2,s)\right] \frac{\partial \Delta_2(s;\rho)}{\partial \rho}.

Thus, in this particular period-2 model, the probability rises with \(\rho\) when \(\beta A>s-11\), falls when \(\beta A<s-11\), and is locally unchanged when the two terms are equal. This condition makes explicit why a higher return does not automatically encourage schooling: the return also changes the wage that can be earned by dropping out at the current education level.

Common Pitfall and Model Interpretation

Counting only future benefits. It is incomplete to say that a higher education wage must increase schooling because it raises future wages. The dropout wage may rise too, increasing the earnings forgone by continuing. Compare the full choice values.
Treating a choice probability as a certainty. A continuation probability of 0.8808 does not mean every student in that state continues. It is the model-implied probability under the assumed distribution of choice shocks.
Ignoring the absorbing dropout path. Once dropout is chosen here, the student receives wages at the existing education level and cannot return to the schooling decision. Allowing re-entry would require a different transition structure.
Interpreting the numerical comparison causally without the model assumptions. The calculations are conditional implications of the specified wage schedule, costs, discounting, terminal value, and shock distribution. They do not establish that an observed wage-return change causes schooling behavior to change by the same amount.

Key takeaway: A schooling model compares the value of continuing—including future choices and education-dependent terminal earnings—with the value of dropping out—including current and subsequent earnings at the existing education level. A wage-return change shifts schooling probabilities through both sides of that comparison.

Check Your Understanding

Use the model definitions and calculations above to answer the following questions.

In this model, which state change follows a continuation choice, and what happens to the state after dropout?
For a period-2 student, explain why the difference \(Q_2^S(s)-Q_2^D(s)\) includes the current dropout wage.
Using the period-2 values \(Q_2^S(12)=79\) and \(Q_2^D(12)=77\), calculate the probability of continuing.
In the comparative-static derivation, what condition makes the period-2 value difference increase with the wage return \(\rho\)?
Why does the period-1 continuation value use \(V_2(s+1)\), while the dropout value uses wages at the existing education level?

What You'll Learn