Smaller Classes, Fewer Teachers

A Framework for Quantifying the Cost of Teacher Misallocation

Peter Courtney

https://petercourtney.co.za

2025-11-26

Introduction

Motivation: The class size puzzle

STR = 32.5, yet ECS = 43.3

Teachers are employed but not fully deployed.

Why schools choose inefficiency 🔎

Key Message: Despite substantial teacher hiring, class sizes remain large. Why?

Details to mention: - South Africa’s primary schools have a student-teacher ratio of 32.5 - Yet learners experience classes averaging 43.3 - The puzzle: teachers are employed but not fully deployed - This paper explores this gap

Why you care about this: This isn’t just a South African problem—it speaks to broader questions about public sector productivity and why hiring more workers doesn’t always translate to better service delivery.

What I measure: learner-experienced class size (ECS)

\[ \text{ECS} = \frac{\sum_{i=1}^{n} \text{CS}_i^2}{\sum_{i=1}^{n} \text{CS}_i} = \overline{\text{CS}} + \frac{\sigma^2_{\text{CS}}}{\overline{\text{CS}}} \]

Learner-weighted mean class size. Larger classes count more.

Example: Classes of {30,30} vs {50,10} both average 30, but ECS = 30 vs 34.3.

Algebra 🔎

Key points: - ECS is the learner-weighted mean class size - Larger classes count more because more learners experience them - Formula decomposes into mean class size plus a variance adjustment - Dispersion raises ECS above simple averages

Example to use: Two classes of 30 vs one of 50 + one of 10: both average 30, but ECS = 30 vs 34.3.

Empirical finding: The variance term is too small to be a primary driver, meaning within-grade dispersion is a secondary issue.

Deployment algebra: the missing metrics

STR = SCR / TCR

• STR: Student-Teacher Ratio (conventional)
• SCR: Student-Class Ratio (proxy for class size)
• TCR: Teacher-Class Ratio (utilisation intensity)
• Realised teaching load = 1/TCR

TCR ≈ 1.22 nationally → teachers work ≈ 82% of time.

Example: Same STR of 30 can mean: - TCR=2.0 → classes of 60 (50% utilisation) - TCR=1.0 → classes of 30 (full utilisation)

Details 🔎

Key identity: STR = SCR / TCR

Define terms: - Student-Teacher Ratio (STR): conventional but misleading - Student-Class Ratio (SCR): proxy for class size - Teacher-Class Ratio (TCR): captures utilisation intensity - Realised teaching load = 1/TCR

National finding: TCR ≈ 1.22 → teachers work ≈ 82% of time → ≈ 18% non-contact by design.

Why TCR matters: It’s a critical but understudied metric that I place at the centre of my analysis. It is the missing link between staffing and crowding.

Example to illustrate: Same STR of 30 can mean: - Large classes of 60 with TCR=2.0 (50% utilisation), OR - Moderate classes of 30 with TCR=1.0 (full utilisation)

The wedge: ECS systematically ≈ 10% > SCR (national: 43.3 vs 39.3). Driven by between-class variance at any scale (across grades, schools, districts), not within-grade dispersion.

Data

• 5.4 million learners in 9,512 public primary schools
• 2023 cross-section; panel 2018–2023 for mechanisms
• Final sample: ≈78% of all public primary learners

Novel feature: Reconstructed class-level timetables from learner-class assignments.

Direct measurement of teacher deployment, not survey self-reports.

Attrition 🔎

What makes this analysis possible: - Universe: 5.4 million learners in 9,512 ordinary public primary schools - 2023 cross-section; panel 2018–2023 for mechanisms - Final sample covers ≈78% of all public primary learners

Sources: - National LURITS (learner microdata with class identifiers) - EMIS masterlist (school attributes, educators, classrooms) - Spatial districts

Novel feature: Reconstructed class-level timetables from learner-class assignments. Allows direct measurement of teacher deployment, not survey self-reports.

Methodological note: Instructional ‘teachers’ = total ‘educators’ - 1 (principal), a conservative proxy for teaching capacity.

Why this matters: This is the largest-scale class-level deployment analysis conducted to date. Administrative data eliminate reporting bias endemic to survey-based absence studies.

Literature

Four gaps:

1. Misallocation literature (Walter 2020; Fagernäs & Pelkonen 2017) focuses on between-school STR variation
   • My finding: Within-school labour slack (18.8% ECS reduction) > spatial gains (5.8 pp)
2. TCR remains “remarkably understudied” (Bennell 2022)
   • Tanzania TCR = 2.5; South Africa = 1.22
   • No rigorous large-scale analysis in primary education
   • International comparisons 🔎
3. Production function puzzle (Hanushek 1986; Glewwe & Kremer 2009)
   • Why don’t teacher hiring interventions improve learning?
   • If class size weakly affects learning, the appropriate margin is cost minimisation
   • My contribution: Hiring → absorbed into slack, not smaller classes
4. Management literature (Bloom et al. 2015) omits efficient deployment
   • My proposal: Allocation efficiency as essential management capacity metric

1. The Hidden Dimension of Misallocation

Key point: The education misallocation literature (Walter 2020, Fagernäs & Pelkonen 2017, Chin 2005, Agarwal et al. 2018) focuses exclusively on between-school STR variation.

The problem: - School-level STR data cannot observe within-school under-utilisation - TCR variation invisible to existing methodologies

My finding: Within-school labour slack (18.8% ECS reduction) > between-school spatial gains (5.8 pp). The literature has been studying the smaller component.

2. Filling a Critical Measurement Gap

Quote from Bennell 2022: TCR remains “remarkably understudied”

Evidence: - Tanzania TCR = 2.5 (teachers work 40% of time) - Campbell (1989) → Bennell (2007-2023): persistent advocacy, limited uptake - No rigorous TCR analysis in primary education at scale

My contribution: Largest class-level deployment analysis to date. First nested decomposition: technical → allocative (capital) → allocative (labour) → spatial.

3. Reinterpreting Production Function Puzzles

Puzzle: Why do input-based interventions yield disappointing learning gains? (Hanushek 1986, Glewwe & Kremer 2009)

My explanation: Additional teachers reduce STRs but not proportionally class sizes—they’re absorbed into inefficient timetables.

My contribution is not to the class size-learning debate, but to explaining why the crucial first-stage link (hiring → class size) is so often weak.

This provides a novel mechanism linking: - Disappointing input elasticities → absorptive capacity failures - High teacher hiring → persistent crowding - Spatial equity gains → muted class size effects

4. Missing from School Management Literature

Bloom et al. 2015: Seminal work on school management includes extensive people management and talent management sections, but entirely omits efficient deployment of human resources.

My proposal: TCR and realised teaching load are essential dimensions of management capacity, readily measurable at scale (Laver et al. 2019).

Scheduled vs unscheduled absence

Key distinction: Not timetabled ≠ Absent when timetabled

Standard absence measures conflate these:

World Bank SDI: Class absence = 1 - (Teachers in classrooms / Total employed)

• Cannot distinguish teachers on free periods from those absent
• 44% of SSA teachers “absent”; 48% are at school but not timetabled (Bold et al. 2017; Bennell 2022)
• Country-level: TCR vs SDI absence (R² = 0.38)

Policy implication: Administrative timetabling reforms require different instruments than behavioural interventions.

Key distinction: Standard absence measures conflate fundamentally different phenomena.

Four-state taxonomy of class absence: 1. Not at School, Not Timetabled → SCHEDULED 2. Not at School, Timetabled → UNSCHEDULED 3. At School, Not Timetabled → SCHEDULED 4. At School, Timetabled → UNSCHEDULED

I distinguish: - Scheduled absence (not timetabled to teach) - Unscheduled absence (absent when timetabled)

Key insight: Different response mechanisms for unscheduled absence: (a) redistribute learners, (b) combine classes, (c) orphaned classes, (d) substitute teachers. Each has distinct welfare implications.

Why existing studies get this wrong:

The World Bank SDI—and other studies including Muralidharan et al. 2017—measure class absence as: Class absence = 1 - (Teachers in classrooms / Total teachers employed)

This cannot distinguish teachers not in class because not timetabled from those absent despite being timetabled.

Empirical evidence of conflation (Bold et al. 2017; Bennell 2022): - 44% of SSA teachers “absent” from class - Of these, 48% are at school but not in class → suggests substantial scheduled absence - Country-level: TCR vs SDI absence (R² = 0.38)

Policy implication: Administrative timetabling reforms (for scheduled absence) require fundamentally different instruments than behavioural accountability interventions (for unscheduled absence). I isolate scheduled absence as an allocative inefficiency amenable to administrative reform.

Three Core Contributions

1. Measurement - Quantify R22.3bn fiscal leakage from teacher mis-deployment - Distinguish scheduled vs unscheduled absence - Labour under-utilisation > spatial misallocation

2. Identification - Demographic shocks reduce inefficiency (β₂ = -0.269) - IV: Resource pressure causally reduces inefficiency (δ_FD = -4.5) - Latent capacity explains 33.5% of class size variance

3. Policy - Cut ECS by 25% (44 → 33) or release R22–30bn annually - District-level reforms sufficient; no provincial transfers needed

Methods

Method: optimisation as measurement

Solve for minimum ECS under nested constraints. Six scenarios reveal binding margins.

• S1–S2: Technical efficiency (balance within/between grades)
• S3: Allocative—capital (bind by classrooms)
• S4: Allocative—labour (full teacher utilisation)
• S5–S6: Spatial (pool across districts/provinces)

This is a measurement exercise, not an algorithm pitch.

Solver Details 🔎

Key framing: This is a measurement exercise—quantifying efficiency gaps—not an algorithm pitch.

Approach: I solve exact dynamic-programming problems to find minimum ECS under nested constraints.

Six scenarios progressively relax constraints, revealing binding margins: - Technical efficiency (S1–S2): balance within/between grades - Allocative efficiency—capital (S3): bind by classrooms - Allocative efficiency—labour (S4): bind by teachers (full utilisation) - Spatial efficiency (S5–S6): pool across districts/provinces

Each scenario isolates a specific margin: timetabling (S1–S2), infrastructure (S3), teacher deployment (S4), spatial mismatch (S5–S6).

Results

Baseline descriptives: provincial heterogeneity

• National: ECS = 43.3, TCR = 1.21, STR = 32.5
• TCR variation: 1.14 (NC, FS) to 1.24 (Limpopo)
• Limpopo: Highest STR + TCR → largest classes (49.6)

Provincial constraint decomposition 🔎

Key patterns to highlight: - National average: ECS = 43.3, TCR = 1.21, STR = 32.5, and 54.9% large class rate (> 40) - TCR variation: 1.14 (Northern Cape, Free State) to 1.24 (Limpopo) - Higher TCR → lower utilisation → larger classes conditional on STR - Limpopo combines high STR (36.7) and high TCR (1.24) → largest classes (49.6; 73% > 40 learners) - Persistent wedge: ECS ≈ 10% > SCR in every province

Scenario Definitions

Representative school examples: S0 | S1 | S2 | S3 | S4 | S5 | S6 🔎

Cascading Impact of Optimisation Scenarios on Class Size

National Results: Key Insights

• S4 (full teacher utilisation): −18.8% ECS. Core inefficiency.
• S5 (district pooling): Most spatial gains captured.
• S6 (province pooling): Negligible additional benefit.

Pattern to emphasise: - S1–S2 (technical efficiency) show minimal gains - S4 (activating all teachers) gives the largest single drop (−18.8% ECS). This is the core inefficiency. - S5 (district pooling) captures most spatial gains - S6 (province pooling) adds negligible benefit, showing misallocation is local

Poverty quintile scenarios: efficiency ≠ equity

• Baseline gap: Q1–Q2 classes 32% larger than Q5
• Paradox: Wealthier Q5 schools show largest gains (−20.2%)—hold more slack
• Persistent inequality: Efficiency gains proportional; relative gaps preserved

Policy implication: Efficiency reforms alone cannot close equity gaps.

Key Insights - The Equity-Efficiency Paradox: - Baseline Gap: Poorest schools (Q1–Q2) face classes 32% larger than wealthiest (Q5) - Paradox of Slack: Wealthier Q5 schools show the largest potential gains (S4: −20.2%), holding more discretionary slack - Persistent Inequality: Efficiency gains are proportional; relative gaps are preserved post-optimisation

Policy Implication: Efficiency reforms alone cannot close equity gaps.

Racial scenarios: persistent inequality

• Baseline gap: Black learners 29% larger classes than white (44.3 vs 34.4)
• Post-optimisation: Gap persists (33.4 vs 25.8)
• White-learner schools hold more idle classroom capacity

Interpretation: Efficiency reforms reduce absolute crowding, not structural inequality.

Key Insights: - Baseline Gap: Black learners experience classes 29% larger than white learners (44.3 vs 34.4) - Post-Optimisation: The gap persists (33.4 vs 25.8)—efficiency reforms maintain relative differences - Capital Slack (S3): White-learner schools hold more idle classroom capacity (14.3% vs 10.0%)

Interpretation: Efficiency reforms reduce absolute crowding but do not address structural inequality.

Explaining the Patterns

What explains crowding? LMG decomposition

• S4 reducibility: 33.5% of explained variance (R² = 52.3%)
• Traditional factors small: Province 6.3%, Quintile 6.4%, Race 5.5%
• Latent capacity > STR in explaining crowding

Crowding co-occurs with spare teacher capacity → deployment choices, not binding constraints.

Key finding: Scenario-4 reducibility explains 33.5% of explained variance (R² ≈ 52.3%).

Interpretation: Schools with high baseline crowding tend to have large latent teacher capacity—crowding and unutilised capacity correlate strongly.

Contrast with traditional factors: - Province: 6.3% - Quintile: 6.4% - Race: 5.5% - Even STR explains less variance than latent capacity

This decomposition is descriptive, not causal. But the correlation is informative: crowding co-occurs with spare teacher capacity, suggesting deployment choices rather than binding resource constraints.

What explains reducibility? Scenario variance decomposition

• S4 (Labour): R² = 0.433. ECS explains 48.2%; crowded schools hold most capacity.
• S5 (District pooling): Province dominates (72.6%). Spatial inefficiency local.
• S1–S2, S6: Low R². Schools already technically efficient; provincial pooling minor.

Patterns: - S1–S2 (Technical): Low R² ≈ 0.06. Idiosyncratic patterns confirm schools already technically efficient - S4 (Labour activation): R² = 0.433. ECS explains 48.2% (β = 0.472); educators 25.0%. Crowded schools hold most latent capacity - S5 (District pooling): Province dominates (72.6%). Spatial inefficiency province-patterned but local - S6 (Provincial pooling): Weak R² = 0.109. Residual gains are minor, not efficiency primitives

Reallocation elasticity

Demographic shocks

Question: Are high TCRs forced or discretionary?

Test logic:

• Constraints bind → shocks raise inefficiency

• Discretionary slack → shocks reduce inefficiency (capacity revealed)

Design: Within-school FE, 2018–2023 - DV: Δ inefficiency - IVs: Δ grade-mix, Δ phase-mix, Δ enrolment

Question: Are high TCRs forced by binding constraints, or discretionary choices?

Test logic: - If constraints bind, demographic shocks should raise inefficiency - If discretionary, shocks should reduce inefficiency (latent capacity revealed)

Design: - Within-school FE, 2018–2023 - Dependent variable: Δ inefficiency - Regressors: Δ grade-mix, Δ phase-mix, Δ enrolment - Phase-mix shocks: Redistribution between Foundation (R–3) and Intermediate/Senior (4–7) phases represent the strongest institutional barrier

Identification caveat

Not quasi-random. Potential confounds: cohort dynamics, management quality, time-varying unobservables.

Inference: - β₂ < 0 across phases inconsistent with binding constraints - If truly binding: β > 0 - Schools can reallocate when pressed (even if costly); capacity exists but latent

Important caveat: This is not quasi-random variation.

Why demographic shocks aren’t exogenous: - Grade-mix changes reflect cohort dynamics, potentially correlated with neighbourhood quality changes - Better-managed schools might both maintain lower baseline inefficiency and adapt more readily to shocks - Time-varying selection on unobservables (principal quality, teacher turnover) not ruled out

What I can conclude: The conditional correlations are informative. Negative coefficients—especially β₂ < 0 across phase boundaries—are inconsistent with binding structural constraints. If qualifications, timetabling indivisibilities, or cross-phase coordination were truly rigid, I’d observe β > 0.

The descriptive finding matters: Schools can reallocate when pressed. This capacity exists but remains latent under normal conditions.

Forthcoming work: Causal identification strategy using quasi-experimental variation in accountability pressure and workforce shocks.

Results: patterns consistent with adaptive capacity

• β₁ (Grade-Mix) = −0.381*** (10 pp shift → 3.81 pp inefficiency reduction)
• β₂ (Phase-Mix) = −0.256*** (10 pp shift → 2.56 pp reduction)
• β₃ (Enrolment) = −0.007*** (scale effects present, composition dominates)

All coefficients negative → adaptive capacity exists, including across phases. Inconsistent with binding constraints.

Robustness 🔎

Key Findings: - β₁ (Grade-Mix) = −0.381 (p < 0.001). A 10 pp shift across grades → inefficiency falls 3.81 pp - β₂ (Phase-Mix) = −0.256 (p < 0.001). A 10 pp shift across phases → inefficiency falls 2.56 pp - β₃ (Enrolment) = −0.007 (p < 0.001). Scale effects present but composition dominates

Interpretation: All coefficients negative. Schools exhibit adaptive reallocation capacity when shocks arrive, including across phase boundaries. Patterns inconsistent with binding structural constraints—suggests discretionary slack rather than forced inefficiency.

Alternative explanations: ruled out

17 mechanisms investigated. Five structural rigidities predict β > 0 if binding. All rejected:

• Grade-specific match capital (β₁ = −0.381)
• Qualification constraints (β₂ = −0.256)
• Period indivisibilities (both β₁, β₂ < 0)
• Cross-phase synchronisation (β₂ = −0.256)
• Enrolment volatility buffering (β₃ = −0.007)

Caveat: Standby rosters create endogeneity. Estimates may be lower bounds.

Full table 🔎

I systematically consider 17 alternative mechanisms. Five structural rigidities predict positive coefficients if binding:

1. Grade-specific match capital → rejected (β₁ = −0.141)
2. Qualification constraints (phase-specific) → strongly rejected (β₂ = −0.269)
3. Period indivisibilities → rejected (both β₁, β₂ < 0)
4. Cross-phase synchronisation → strongly rejected (β₂ = −0.269)
5. Enrolment volatility buffering → rejected (β₃ = −0.002)

Standby rosters: Create partial endogeneity—schools may maintain slack as insurance against unscheduled absence. My estimates may be lower bounds on pure discretionary slack.

Causal Evidence: Instrumental Variables Strategy

⚠️ Preliminary Work

IV Strategy: Does increased staffing increase inefficiency?

Endogeneity problem: - Reverse causality: Inefficient schools receive compensatory staffing - Selection: Management quality affects both staffing and deployment - Omitted variables: Neighbourhood, principal quality, union strength

Solution: Instrument STR with Post Provisioning Norm (PPN) - Bureaucratic formula mechanically determines allocations - Within-school grade-mix changes drive variation - Plausibly exogenous to management

Three sources of bias: 1. Reverse causality: Inefficient schools may receive compensatory staffing 2. Selection: Management quality affects both staffing and deployment 3. Omitted variables: Neighbourhood characteristics, principal quality, union strength

Solution: Exploit bureaucratic teacher allocation rules (Post Provisioning Norm) as instrument for STR - PPN formula mechanically determines teacher allocations - Within-school grade-mix changes drive variation - Plausibly exogenous to management decisions

Instrument: Post Provisioning Norm (PPN)

PPN formula: f(enrolment, grade-mix, quintile, provincial norms)

For identification, only grade composition matters: - We exploit variation in grade-mix weights only - Enrolment, quintile, and provincial norms are controlled for (absorbed by FE and controls) - Within-school grade composition changes trigger mechanical allocation adjustments - Schools don’t control cohort composition

Why valid: - Predicted STR insulated from management decisions - School FE + enrolment controls absorb time-invariant quality

What is PPN? - Bureaucratic formula allocating teachers: f(enrolment, grade-mix, quintile, provincial norms) - For our instrument: We exploit only the grade-mix component - Other components (enrolment, quintile, provincial norms) are controlled for via fixed effects and controls - Predicted STR insulated from school-level management decisions - Key feature: Within-school changes in grade composition interact with fixed PPN grade weights

Why this works: - Grade-mix changes (Foundation vs Intermediate/Senior phases) trigger mechanical allocation adjustments - Schools don’t control their demographic composition - Residualise on total enrolment + school FE absorbs time-invariant quality

PPN Weights and Identification Source

We exploit variation driven solely by grade-specific weights in the PPN formula.

Grade	Max Class Size	Period Load (%)	Funding Level	Weight
R	35	96	0	0
1-4	35	96	100	1.190
5-6	40	96	100	1.042
7	37	96	100	1.126

• We only use the grade weights for the analysis.
• Variation arises when cohorts move between weight bands (e.g., a large cohort moving from Grade 4 to 5 reduces teacher allocation).

Threats to Validity: Strategic Manipulation

Endogeneity is defined when cohort composition is correlated with efficiency, and this seems improbable.

The biggest threats to manipulation:

1. Pausing student promotion where the student would otherwise enter a lower weight grade the following year (e.g., retaining Grade 4s).
2. Expelling learners who would otherwise be promoted to a lower weight grade.
3. Enrolling learners disproportionately at higher weight grades.

Why these refute the instrument: If schools manipulate composition to maximise weights, the instrument (grade mix) becomes correlated with unobserved management quality (the error term), violating the exclusion restriction.

I test for the above and find that almost all learner begin in Grade 1 or R and exit in Grade 7

Specification

\[ \begin{aligned} \text{First stage:} \quad & \Delta \text{STR}_{\text{actual}} = \alpha + \beta \cdot \Delta \text{STR}_{\text{predicted}} + \delta_t + \varepsilon \\ \text{2SLS:} \quad & \Delta \text{Inefficiency} = \alpha + \delta \cdot \Delta \widehat{\text{STR}} + \delta_t + \varepsilon \end{aligned} \]

First-differences specification. Clustering at school level.

Key identifying assumptions:

1. Relevance: PPN formula strongly predicts actual STR changes (F-statistic)
2. Exclusion restriction: PPN affects inefficiency only through STR
3. Exogeneity: Grade-mix changes uncorrelated with time-varying management quality

\[ \begin{aligned} \text{First stage:} \quad & \text{STR}_{\text{actual}} = \alpha + \beta \cdot \text{STR}_{\text{predicted}} + \gamma_i + \delta_t + \varepsilon \\ \text{2SLS:} \quad & \text{Inefficiency} = \alpha + \delta \cdot \widehat{\text{STR}} + \gamma_i + \delta_t + \varepsilon \end{aligned} \]

Key identifying assumptions: 1. Relevance: PPN formula strongly predicts actual STR (testable via F-statistic) 2. Exclusion restriction: PPN affects inefficiency only through STR 3. Exogeneity: Grade-mix changes uncorrelated with time-varying management quality

Identification: threats & tests

Main threat: Exclusion restriction—could PPN affect inefficiency directly?

Test: Control for grade-mix, enrolment, etc. β stable → exclusionlikely holds.

Validation: - Balance tests: Instrument ⊥ pre-determined covariates - Falsification: Future shocks don’t predict past outcomes - Robustness: 17 specifications (FE, clustering, functional forms) - Result: F > 30 across all specs; direction robust; no pre-trends

Main threat: Exclusion restriction

Could PPN-predicted STR affect inefficiency through channels other than actual STR? - Grade-mix changes might directly affect timetabling complexity - Test: Include direct controls (grade-mix, enrolment). If β stable → exclusion holds

Validation strategy: - Balance tests: Instrument orthogonal to pre-determined covariates - Falsification: Future PPN shocks don’t predict past outcomes - Robustness: 17 specification checks (alternative FE, clustering, functional forms) - Result: F > 30 across all specs; direction robust; no pre-trends

IV Results

	First-Differences
2SLS Coefficient	-4.5***
Standard Error	0.111
Interpretation	ΔSTR ↑ → ΔIneff ↓
First-stage F-stat	60.4
N (school-years)	28,703

Key finding: Resource pressure → forced efficiency. Resource slack → expand slack.

FD isolates behavioural response. Slack is discretionary, not structural.

Key numbers: - First-Differences: Coefficient = -4.5*** (SE = 0.111), F = 60.4 - Fixed Effects: Coefficient = 1.572*** (SE = 0.294), F = 60.4

Why opposite signs? - FD (negative): Within-school dynamics. When schools lose teachers → forced to tighten deployment. When they gain teachers → choose to expand slack. Reveals discretionary capacity. - FE (positive): Cross-sectional equilibrium. High-STR schools also show higher inefficiency. Mixes causal effect with selection (weak management → both under-staffing and poor deployment).

Key insight: FD isolates behavioural response to resource shocks. Negative coefficient proves slack is discretionary, not structural.

Economic interpretation

Core finding: Resource pressure (↑STR) causally reduces inefficiency (δ_FD < 0)

If constraints bind: Fewer teachers → harder to optimise → δ > 0. We observe opposite.

Mechanism: Additional teachers bargained into lighter workloads, not new classes.

Explains disappointing input elasticities: Hiring expands capacity → absorbed into slack.

Policy implication: R22.3bn behaviourally defended. Administrative reforms needed, not fiscal expansion.

Core finding: Resource pressure (↑STR) causally reduces inefficiency (δ_FD < 0)

If constraints truly bound: Fewer teachers → Harder to optimise → Inefficiency should rise (δ > 0). Instead, we observe the opposite.

Political economy mechanism: Additional teachers → bargained into lighter workloads, not new classes. TCR rises with resource abundance (1.14 → 1.24). The hiring → class size link systematically fails.

This explains disappointing input elasticities: Teacher hiring expands capacity, but schools absorb it into organisational slack rather than class size reduction.

Policy implication: R22.3bn is behaviourally defended, not structurally inaccessible. Administrative reforms targeting deployment incentives—not fiscal expansion—necessary to unlock capacity.

Robustness: 17 specification checks

Instrument validity: Balance, falsification, weak-IV robust CI, overidentification (✓)

Specification: 7 FE combinations, 4 clustering schemes, 5 functional forms (✓)

Heterogeneity: Effect stronger in small schools, low-STR contexts (✓)

Key finding: F > 30 across all specs; direction robust; no pre-trends

Full robustness table 🔎

Economic Impact & Policy

Fiscal leakage: the cost of idle time

• R22.3bn annually: S4 teacher under-utilisation (18.2% of time)
• R29.6bn annually: Total including S5 spatial misallocation
• Exceeds national spending on school nutrition + learning materials

Assumptions 🔎 | Secondary extension 🔎

The numbers: - R22.3bn annually: The cost of S4 teacher under-utilisation (18.2% of instructional time) - R29.6bn annually: Total leakage including S5 district-level misallocation - Context: R22.3bn exceeds national spending on school nutrition and learning materials combined

Assumptions (for backup discussion): - Baseline: Mean educator salary R42,000/month; 12 months; S4 inefficiency 18.2% - Sensitivity: ±10% salary → R20.0–24.5bn range - Coverage: Primary only (R22.3bn) vs primary + secondary extrapolation (≈ R62bn) - As % of GDP: ≈ 0.3–0.9% - As % of education budget: 7.6–10.1%

Policy design: three tractable levers

1. Enforce contact-time norms: −18.8% ECS or free ≈18% capacity (≈R22bn)

2. Activate idle classrooms: −10.2% ECS

3. District pooling: Shift hiring from school to district level. Captures −24.2% cumulative without provincial disruption.

1. Within-school activation: Enforce contact-time norms. Potential: −18.8% ECS or free ≈ 18% educator capacity (≈ R22bn)

2. Activate idle classrooms: Assign teachers to usable rooms. Addresses capital under-utilisation (S3: −10.2% ECS)

3. Formalise district pooling: Shift teacher hiring/assignment from school to district level. Captures spatial gains (−24.2% cumulative) without province-wide disruption

Conclusion

Summary

1. Allocation, not scarcity: Labour under-utilisation dominant (LMG 33.5%; S4 −18.8%)

2. Local reallocation suffices: District pooling captures spatial gains (−24.2%); provincial adds ≈0.4pp

3. Administrative reforms feasible:

   • Adaptive capacity exists (β₂ = −0.269 across phases)
   • Can reduce ECS 25% or free 25% capacity (R22–30bn)

1. Allocation, not scarcity, correlates with crowding: Teacher under-utilisation (scheduled absence) is the dominant correlate of inflated class sizes (LMG 33.5%; S4 −18.8% ECS)

2. Local reallocation suffices: District-level pooling captures nearly all spatial efficiency gains (cumulative −24.2%); provincial pooling adds ≈ 0.4 pp

3. Administrative reforms feasible: Adaptive capacity exists—schools exhibit reallocation across grades and phases when facing demographic shocks (β₂ = −0.269). Optimisation suggests potential to reduce ECS ≈ 25% or free ≈ 25% educator capacity (R22–30bn fiscal leakage)

Appendix slides

Why would schools choose inefficiency?

Core mechanism: teachers value free periods above smaller classes.

• Rigid wage schedules: Compressed salary structure limits pecuniary differentiation. Non-pecuniary compensation (lighter teaching loads) becomes the primary margin of adjustment.
• Union bargaining: Educator unions negotiate workload norms. Enforcement is weak, creating space for discretionary interpretation of contact-time requirements.
• Measurement vacuum: Until recently, class-level timetabling data was unavailable. What isn’t measured can’t be managed—or negotiated away.
• Rational precaution: Schools maintain standby rosters to cover unscheduled absence, creating endogenous slack between scheduled and unscheduled absence.

Political economy equilibrium: High TCR represents a negotiated outcome where teachers capture rents through reduced contact time rather than higher wages. Principals lack incentives or instruments to enforce utilisation norms when crowding doesn’t trigger accountability penalties.

• ↩︎️ Back to Motivation

Talking points (Don’t put all this on slide—mention selectively):

Core mechanism: Teachers value free periods above smaller classes.

Four institutional factors: 1. Rigid wage schedules: Compressed salary structure limits pecuniary differentiation. Non-pecuniary compensation (lighter teaching loads) becomes the primary margin of adjustment.

2. Union bargaining: Educator unions negotiate workload norms. Enforcement is weak, creating space for discretionary interpretation of contact-time requirements.

3. Measurement vacuum: Until recently, class-level timetabling data was unavailable. What isn’t measured can’t be managed—or negotiated away.

4. Rational precaution: Schools maintain standby rosters to cover unscheduled absence, creating endogenous slack between scheduled and unscheduled absence.

Political economy equilibrium: High TCR represents a negotiated outcome where teachers capture rents through reduced contact time rather than higher wages. Principals lack incentives or instruments to enforce utilisation norms when crowding doesn’t trigger accountability penalties.

Algebra reference sheet

• Core Identity: STR = SCR / TCR
  • STR (Student-Teacher Ratio) = Learners / Teachers
  • SCR (Student-Class Ratio) = Learners / Classes
  • TCR (Teacher-Class Ratio) = Teachers / Classes
• Implication: Realised Teaching Load = 1 / TCR.
  • A TCR of 1.22 means teachers are timetabled for 1/1.22 ≈ 82% of periods.
• Numerical Example: School with 300 learners, 10 teachers (STR = 30).
  • TCR = 1.0 (Full Utilisation): 10 teachers → 10 classes. SCR = 300/10 = 30.
  • TCR = 1.5 (Low Utilisation): 10 teachers → 10/1.5 ≈ 7 classes. SCR = 300/7 ≈ 43.
  • Same staffing, different class sizes.

• Experienced Class Size (ECS): \[ \text{ECS} = \frac{\sum_{i} \text{CS}_i^2}{\sum_{i} \text{CS}_i} = \overline{\text{CS}} + \frac{\sigma^2_{\text{CS}}}{\overline{\text{CS}}} \]
  • ECS is the learner-weighted average class size, accounting for variance (\(\sigma^2_{CS}\)).
• Why ECS > SCR (The “Wedge”):
  • Variation in class sizes means more learners experience larger classes.
  • Example: Classes of 50 and 10.
    • Simple average (SCR) = (50+10)/2 = 30.
    • Learner-weighted average (ECS) = (50×50 + 10×10)/(50+10) = 43.3.

• ↩︎️ Back to Slides 5 & 6

Sample attrition and representativeness

• Final coverage: ≈ 5.4 million learners (78% of public primary), 9,512 schools
• Largest attrition: multigrade exclusion (≈ 12% of sample)
• External validity concern: Multigrade schools likely face distinct (possibly larger) inefficiency challenges
• Robustness check: Repeat analysis including multigrade (relaxing constraints) → ongoing work
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Fiscal leakage: assumptions and sensitivity

• Baseline: Mean educator salary R42,000/month; 12 months; S4 inefficiency 18.2%; extrapolate to full educator workforce
• Sensitivity: ±10% salary → R20.0–24.5bn range; exclude principals vs include → R21.9–24.8bn
• Coverage: Primary only (R22.3bn) vs primary + secondary extrapolation (≈ R62bn, assuming similar TCR)
• Alternative framing: Leakage as % of GDP (≈ 0.3–0.9%), % of total budget (≈ 1%), % of education budget (7.6–10.1%)
• Comparison: R22.3bn > National School Nutrition Programme (≈ R8bn) + LTSM budget (≈ R12bn)
• ↩︎️ Back to Slide 26

Mechanisms: alternative specifications

• Baseline FE: Grade‑mix β₁ = −0.141, Phase‑mix β₂ = −0.269, Enrolment β₃ = −0.002
• Robustness checks:
  • Cluster SEs at district level: SEs increase slightly; significance unchanged
  • Include school‑level controls (baseline TCR, quintile, urban): Coefficients attenuate 10–15%; signs/significance robust
  • Quantile regression (median): β₁ ≈ −0.11, β₂ ≈ −0.21 (smaller but consistent pattern)
  • Split sample by quintile: β₂ < 0 in all quintiles; magnitude largest in Q1–Q3 (poorer schools)
• Conclusion: Negative coefficients robust; adaptive capacity present across specifications
• ↩︎️ Back to Slide 23

Alternative explanations: full table

Table 1: Alternative Explanations for Teacher Allocation Inefficiency: Empirical Tests and Findings

Mechanism	Type	Predicted Effect (if binding)	Empirical Finding
Structural Rigidities (Grade-Level)
Grade-specific match capital	Structural (grade-level)	β₁ > 0 if binding: grade-mix shocks raise inefficiency	Rejected: β₁ = -0.141, schools reassign teachers across individual grades
Structural Rigidities (Phase-Level)
Qualification constraints (phase-specific)	Structural (phase-level)	β₂ > 0 if binding: phase-mix shocks raise inefficiency (strongest barrier)	Strongly rejected: β₂ = -0.269 (nearly twice grade-mix effect), teachers reassigned readily across Foundation/Intermediate boundaries despite institutional divisions
Period indivisibilities	Structural (grade/phase-level)	β₁, β₂ > 0 if binding: compositional shocks raise inefficiency	Rejected: both β₁ = -0.141 and β₂ = -0.269, schools revert to generalist models
Cross-phase synchronisation	Structural (phase-level)	β₂ > 0 if binding: phase-mix shocks raise inefficiency	Strongly rejected: β₂ = -0.269, largest negative effect despite distinct phase structures (breaks, period lengths, pedagogy)
Structural Rigidities (Aggregate)
Enrolment volatility buffering	Structural (aggregate)	β₃ > 0 if binding: enrolment shocks raise inefficiency	Rejected: β₃ = -0.002, negligible response to ±14.77% volatility
School-Level Operational Constraints
Infrastructure constraints	School-level operational	No prediction for compositional response (operates at school level)	Orthogonal to compositional shocks; infrastructure explains ≈8% inefficiency vs ≈19% for teacher under-utilisation
Protected administrative roles	School-level operational	No prediction for compositional response (operates at school level)	Orthogonal to compositional shocks; may contribute to baseline inefficiency but does not prevent adaptive reallocation at either grade or phase level
Health accommodations	School-level operational	No prediction for compositional response (operates at school level)	Orthogonal to compositional shocks; may contribute to baseline inefficiency but does not prevent adaptive reallocation at either grade or phase level
Pull-out remediation programmes	School-level operational	No prediction for compositional response (operates at school level)	Orthogonal to compositional shocks; concentrated in wealthier schools; does not prevent adaptive reallocation
Feeding-scheme logistics	School-level operational	No prediction for compositional response (operates at school level)	Orthogonal to compositional shocks; may contribute to baseline inefficiency but does not prevent adaptive reallocation
School-Level Governance
SGB substitution	School-level governance	No prediction for compositional response (operates at school level)	Orthogonal to compositional shocks; both SGB and department staff appear reallocated when shocks arrive
Endogenous Precautionary Mechanisms
Standby rosters for unscheduled absence	Endogenous precautionary	Ambiguous: precautionary buffers may correlate with shocks	Partially supported: creates endogeneity between scheduled and unscheduled absence; capacity for reallocation at both grade and phase levels still exists; warrants further investigation
Measurement Artefacts
Data classification artefacts	Measurement artefact	β₁, β₂ ≈ 0 if pure artefact: measurement error attenuates estimates	Rejected: negative coefficients at both grade and phase levels suggest real capacity; measurement error would attenuate toward zero
Policy Equilibria (Supported)
Non-pecuniary compensation	Policy equilibrium	Consistent with β₁, β₂	Consistent: schools reduce inefficiency when pressure arrives, including substantial phase-level adjustments
Union bargaining and political economy	Policy equilibrium	Consistent with β₁, β₂	Consistent: capacity exists at all levels but withheld through bargaining
Discretionary specialisation	Policy equilibrium	Consistent with β₁, β₂	Consistent: schools revert from grade and phase specialisation under pressure; strong phase-mix response indicates phase boundaries not binding
Institutional inertia	Policy equilibrium	Consistent with β₁, β₂	Consistent: capacity latent at all levels absent forcing mechanisms; phase-level adjustments particularly revealing

• ↩︎️ Back to Slide 24

IV Robustness: Full specification table

#	Category	Test	Description
Instrument Validity Tests
1	Instrument Validity	Balance on observables	Instrument orthogonal to 15 pre-determined covariates (size, socioeconomic status, lagged outcomes)
2	Instrument Validity	Falsification tests	Future PPN shocks don't predict past outcomes (lag placebos)
3	Instrument Validity	Weak instrument robust CI	Anderson-Rubin and CLR confidence sets (robust to weak IV)
4	Instrument Validity	Reduced form on placebos	No effect on time-invariant school characteristics
5	Instrument Validity	Exclusion restriction probes	Control for PPN sub-components (enrolment, weights, district rates)
6	Instrument Validity	Overidentification tests	Grade-specific instruments yield consistent estimates
7	Instrument Validity	First-stage heterogeneity	Strong F-statistics across districts and time periods
Specification Robustness Tests
8	Specification Robustness	Alternative fixed effects	No FE, Year FE, District FE, School FE, Province×Year FE, District×Year FE (7 specifications)
9	Specification Robustness	Alternative clustering	School-level (primary), District-level, Two-way (School+Year), Two-way (District+Year)
10	Specification Robustness	Subsample stability	Balanced panel only, exclude large districts, exclude outliers (STR/size/class size), by quintile, leave-one-province-out
11	Specification Robustness	Alternative functional forms	Level-level (primary), log-log, level-log, log-level, quadratic
12	Specification Robustness	Asymmetry tests	STR increases vs decreases (gains vs losses)
13	Specification Robustness	Specification ladder	Progressive inclusion of controls and fixed effects (Spec 1 → 4)
14	Specification Robustness	Control function approach	Alternative estimator using residuals as control
Additional Validation Tests
15	Heterogeneity Analysis	Effect modification	By school size terciles, initial STR level, baseline class size, quintile, urban/rural, province, time period
16	Mediation Analysis	Class count as mediator	Tests whether STR affects inefficiency through class formation decisions (Baron & Kenny framework with FE)
17	Attrition & Sample Selection	Sample attrition tests	Validates representativeness of final sample (78% coverage)

• ↩︎️ Back to IV Results

Provincial scenarios: constraint decomposition

• ECS Values (Left): Limpopo faces the largest classes (49.6), whilst Northern Cape has the smallest (37.0)
• Capital Slack (S3): Limpopo & Eastern Cape show high reducibility from activating idle classrooms
• Labour Slack (S4): Limpopo & Mpumalanga show high reducibility from activating surplus teachers
• Interpretation: Provinces differ fundamentally in where their inefficiencies lie—suggesting tailored policy interventions

Key Insights from heatmaps: - ECS Values: Limpopo faces the largest classes (49.6), whilst Northern Cape has the smallest (37.0) - Capital Slack (S3): Limpopo & Eastern Cape show high reducibility from activating idle classrooms - Labour Slack (S4): Limpopo & Mpumalanga show high reducibility from activating surplus teachers

Interpretation: Provinces differ fundamentally in where their inefficiencies lie—suggesting tailored policy interventions.

International comparisons: TCR in SSA

• Documented TCRs: Tanzania 2.5 (Asim, Chugunov, and Gera 2019) (40% utilisation), South Africa 1.22 (82% utilisation), Ethiopia ≈ 1.4 (Bennell 2022 inference from SDI)
• Vietnam policy: Explicitly targets TCR > 1 for extended contact hours and in-service training
• Interpretation: Scheduled absence widespread; SDI class-absence rates partially reflect timetabling inefficiency
• South Africa advantage: EMIS timetabling software → technical efficiency (S1–S2) high; problem is utilisation (S4)
• Implication: SSA systems without timetabling tools may face both technical and allocative inefficiency → larger potential gains
• ↩︎️ Back to Literature

Documented TCRs: - Tanzania: 2.5 (40% utilisation) - South Africa: 1.22 (82% utilisation) - Ethiopia: ≈ 1.4 (Bennell 2022 inference from SDI)

Vietnam policy: Explicitly targets TCR > 1 for extended contact hours and in-service training

Interpretation: Scheduled absence widespread; SDI class-absence rates partially reflect timetabling inefficiency

South Africa advantage: EMIS timetabling software → technical efficiency (S1–S2) high; problem is utilisation (S4)

Implication: SSA systems without timetabling tools may face both technical and allocative inefficiency → larger potential gains

Secondary school extension: subject specialisation amplifies inefficiency

• Primary TCR ≈ 1.22 (generalist model; some discretionary specialisation). Secondary TCR likely higher:
  • Subject specialisation mandated (≈ 15+ subjects; average school writes 7 subjects in matriculation)
  • Period indivisibilities bind harder (6–8 periods/day; subjects allocated 4–6 periods/week)
  • Teacher qualifications phase-specific (FET trained; cannot teach foundation)
• Preliminary estimates: If secondary TCR ≈ 1.5 (speculative), fiscal leakage ≈ R40bn additional (total ≈ R62bn)
• Data challenge: LURITS does not include subject identifiers for secondary; requires EMIS timetable microdata linkage
• Policy: Subject rationalisation (reduce subject offerings per school; pool teachers at district level for rare subjects)
• ↩︎️ Back to Fiscal Leakage

Why secondary likely worse: - Primary TCR ≈ 1.22 (generalist model) - Secondary TCR likely higher due to: - Subject specialisation mandated (≈ 15+ subjects; average school writes 7 in matriculation) - Period indivisibilities bind harder (6–8 periods/day; subjects allocated 4–6 periods/week) - Teacher qualifications phase-specific (FET trained; cannot teach foundation)

Preliminary estimates: If secondary TCR ≈ 1.5 (speculative), fiscal leakage ≈ R40bn additional (total ≈ R62bn)

Data challenge: LURITS does not include subject identifiers for secondary; requires EMIS timetable microdata linkage

Policy: Subject rationalisation (reduce subject offerings per school; pool teachers at district level for rare subjects)

The Allocation Solver

• Objective: Minimise Experienced Class Size (ECS) for a fixed number of teachers. This is equivalent to minimising the sum of squared learners per class across all grades (Σ n₉²/k₉).
• Method: A greedy algorithm using a max-priority queue. This approach is provably optimal as the objective function is convex.
• Algorithm:
  1. Initialisation: Assign one class to each grade with enrolled learners.
  2. Iteration: For the remaining classes, calculate the “marginal gain”—the reduction in the objective function—of adding one more class to each grade.
  3. Allocation: Add the next class to the grade with the highest marginal gain. The gain from adding a class to a grade with n learners and k classes is n² / (k(k+1)).
  4. Repeat: Update the marginal gain for the affected grade and repeat until all classes are allocated.
• Implementation:
  • High-performance C++ solver via Rcpp for national-scale computation.
  • Efficient O(K log G) complexity, where K is total classes and G is number of grades.

↩︎️ Back to Slide 8

Objective: Minimise Experienced Class Size (ECS) for a fixed number of teachers. Equivalent to minimising sum of squared learners per class across all grades (Σ n₉²/k₉).

Method: Greedy algorithm using max-priority queue. Provably optimal as objective function is convex.

Algorithm: 1. Initialisation: Assign one class to each grade with enrolled learners 2. Iteration: For remaining classes, calculate “marginal gain”—reduction in objective function—of adding one more class to each grade 3. Allocation: Add next class to grade with highest marginal gain. Gain from adding class to grade with n learners and k classes: n² / (k(k+1)) 4. Repeat: Update marginal gain for affected grade and repeat until all classes allocated

Implementation: - High-performance C++ solver via Rcpp for national-scale computation - Efficient O(K log G) complexity, where K = total classes and G = number of grades

Representative school: status quo (Scenario 0)

Baseline timetable at 82nd percentile reducibility. School: 1,220 learners, 27 classes, 7 grades, ECS 47.8.

Within‑grade dispersion modest (e.g., Grade 3: 35–44). Between‑grade imbalance small. Key: 27 classes but more teachers available (see S4). Red-bordered cell shows next improvement target.

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Within-Grade Equalisation (S1)

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Between-Grade Allocation (S2)

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Classroom-Constrained Allocation (S3)

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Educator-Optimal Allocation (S4)

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District-Level Pooling (S5)

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Province-Level Pooling (S6)

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Asim, Salman, Dmitry Chugunov, and Ravinder Gera. 2019. Fiscal Implications of Free Education. World Bank, Washington, DC. https://doi.org/10.1596/31466.

Bennell, Paul. 2022. “Missing in Action? The World Bank’s Surveys of Teacher Absenteeism in Sub-Saharan Africa.” Comparative Education 58 (4): 489–508. https://doi.org/10.1080/03050068.2022.2083342.

Bold, Tessa, Deon Filmer, Gayle Martin, Ezequiel Molina, Christophe Rockmore, Brian Stacy, Jakob Svensson, and Waly Wane. 2017. What Do Teachers Know and Do? Does It Matter? Evidence from Primary Schools in Africa. World Bank, Washington, DC. https://doi.org/10.1596/1813-9450-7956.