Smaller Classes with Fewer Teachers

Teacher Misallocation in South African Primary Schools

Peter Courtney

Stellenbosch University & Tinbergen Institute

2026-02-02

Introduction

The Class Size Puzzle

Student-Teacher Ratio = 32.5

Experienced Class Size = 43.3

Teachers are employed but not fully deployed.

The Puzzle: South African primary schools have enough teachers for classes of 32, yet learners experience classes averaging 43.

Why this matters: 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.

The gap: A 33% difference between staffing levels and what learners actually experience.

Decomposing the Gap: STR ≠ Class Size

\[\text{STR} = \frac{\text{SCR}}{\text{TCR}}\]

Metric	Definition	National Average
STR	Student-Teacher Ratio	32.5
SCR	Student-Class Ratio	39.3
TCR	Teacher-Class Ratio	1.22

TCR = 1.22 → Teachers timetabled for only 82% of periods

Same STR of 30 can mean classes of 30 (TCR=1) or classes of 60 (TCR=2).

Key identity: STR = SCR / TCR

The insight: TCR captures utilisation intensity—how many periods teachers are actually teaching.

TCR = 1.22 nationally → Teachers work 82% of instructional time → 18% “free periods” by design.

This is not teacher absence—it’s scheduled non-contact time built into timetables.

Scheduled vs Unscheduled Absence

Key distinction: Not timetabled ≠ Absent when timetabled

World Bank SDI conflates these:

\[\text{Class Absence} = 1 - \frac{\text{Teachers in Classrooms}}{\text{Total Employed}}\]

• 44% of SSA teachers “absent” from class (Bold et al. 2017)
• Of these, 48% are at school but not timetabled

Policy implication: Administrative timetabling reforms ≠ behavioural accountability interventions.

The measurement problem: Standard absence studies cannot distinguish teachers with free periods from teachers who are actually absent.

The SDI problem: World Bank Service Delivery Indicators—widely used—conflate fundamentally different phenomena.

Why it matters: If the problem is scheduled absence (timetabling), you need administrative reform. If the problem is unscheduled absence (truancy), you need behavioural interventions. Different problems, different solutions.

Literature: Four Gaps

1. Misallocation

Focuses on between-school variation (Walter 2020).

My finding: Within-school slack > spatial gains.

3. Input Elasticity

Why hiring fails to improve learning (Hanushek 1986).

My contribution: Hiring → absorbed into slack.

2. Teacher Utilization

TCR “remarkably understudied” (Bennell 2022).

Gap: No rigorous large-scale primary analysis (Tanzania TCR = 2.5).

4. Management

Focuses on talent management (Bloom et al. 2015).

Gap: Overlooks extensive margin of deployment.

Key point: The education misallocation literature has been studying the smaller component. I show within-school deployment matters more than spatial redistribution.

Bennell’s observation: TCR is the metric we should be tracking, but almost no one does.

The mechanism I identify: Additional teachers don’t create smaller classes—they’re absorbed into lighter workloads.

Data

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

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

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

Attrition details 🔎

What makes this possible: National LURITS data with learner-class identifiers, linked to EMIS school attributes.

The innovation: I reconstruct how teachers are actually deployed from administrative records. This eliminates the reporting bias endemic to survey-based absence studies.

Scale: This is the largest class-level deployment analysis conducted to date in any country.

Measurement Framework

Optimization Framework: Allocative Efficiency

Scope: Optimality is defined here strictly as resource distribution, holding pedagogical models constant.

1. The Primal Objective (Policy Preference Maximisation) Minimise Experienced Class Size given fixed staff establishment.

2. The Dual Objective (Cost Rationalisation) Minimise Staff Establishment required to meet a specific class size target.

The Measurement Innovation

Core idea: Use constrained optimisation to quantify efficiency gaps.

For each school, I solve:

\[\min_{\{k_g\}} \text{ECS} = \min_{\{k_g\}} \frac{\sum_g n_g^2 / k_g}{\sum_g n_g}\]

subject to progressively relaxed constraints on \(\sum_g k_g\).

The gap between observed and optimal = measurable inefficiency.

By comparing optima under different constraints, I decompose where inefficiency lies:

• Technical (timetabling) vs Allocative (capital vs labour) vs Spatial

What’s new: I don’t just measure TCR—I use optimisation to compute counterfactual class sizes under different scenarios.

The key insight: The gap between what schools achieve and what they could achieve under the same constraints quantifies the cost of misallocation.

“Optimal” means: Minimum experienced class size for a given number of teachers, OR equivalently, minimum teachers needed to achieve a given class size.

The Allocation Algorithm

Objective: Minimise \(\sum_g n_g^2/k_g\) (equivalent to minimising ECS)

Method: Greedy algorithm with max-priority queue—provably optimal (convex objective)

Algorithm:

1. Assign one class per non-empty grade
2. Calculate marginal gain of adding one class to each grade: \(\Delta = \frac{n^2}{k(k+1)}\)
3. Add class to grade with highest gain
4. Repeat until all classes allocated

Shadow Price:

After allocation, the shadow price = marginal gain of the next class.

This tells us the value of an additional teacher:

• High shadow price → binding constraint
• Low shadow price → diminishing returns

Shadow prices reveal which schools would benefit most from expansion.

Why this works: The objective function is convex in class counts, so greedy allocation is globally optimal.

The shadow price: After allocating all teachers, the marginal gain of the next class is the shadow price. It tells us how much ECS would fall if we added one more teacher.

Policy relevance: Shadow prices let us rank schools by the value of marginal investment.

Optimisation as Measurement

I solve for minimum ECS under nested constraints to quantify efficiency gaps.

Scenario	Constraint	What it measures	Example 🔎
S0	Status quo	Baseline	S0
S1–S2	Balance classes	Technical efficiency	S1, S2
S3	Bind by classrooms	Capital slack	S3
S4	Bind by teachers	Labour slack	S4
S5–S6	Pool across space	Spatial misallocation	S5, S6

“Optimal” = minimum class size for given teachers (or equivalently, minimum teachers for given class size).

Approach: I solve exact optimisation problems to find minimum possible ECS under progressively relaxed constraints.

Each scenario isolates a margin: - S1–S2: Are classes balanced within/between grades? - S3: Are all classrooms being used? - S4: Are all teachers teaching full loads? - S5–S6: Could spatial pooling help?

The key: The gap between scenarios quantifies the cost of each type of inefficiency.

Gaps in Efficiency Could Close Gaps in Equity

Systematic Patterns in Class Size and Inefficiency

• Quintile
• Race

This is not noise: Class sizes vary systematically by socioeconomic status and race.

Quintile pattern: Poorest schools (Q1–Q2) have classes 32% larger than wealthiest (Q5).

Racial pattern: Black learners experience classes 29% larger than white learners.

The puzzle: This isn’t just about staffing—TCR also varies. Poorer schools are both understaffed and use their staff less efficiently.

Main Result: Where the Inefficiency Lies

Read the waterfall: - S1–S2 (technical): Minimal gains—schools already balance classes reasonably well - S3 (classrooms): Some idle classroom capacity - S4 (teachers): The big one—18.8% ECS reduction from full teacher utilisation - S5–S6 (spatial): District pooling captures most spatial gains; provincial adds little

The headline: Within-school labour slack is the dominant source of inflated class sizes.

Key Findings: Descriptive

• S4 (full teacher utilisation): −18.8% ECS. Core inefficiency.
• S5 (district pooling): Most spatial gains captured locally.
• S6 (province pooling): Negligible additional benefit (+0.4 pp).

The numbers: - Technical efficiency gains (S1–S2): Small, <2% - Capital efficiency (S3): −10.2% - Labour efficiency (S4): −18.8%—this is the headline - Spatial efficiency (S5): −24.2% cumulative

Key insight: Misallocation is primarily within schools, not between them. And it’s primarily about teacher deployment, not infrastructure.

Causal Identification

The Identification Question

Question: Does exogenously providing a school with additional teachers lead to smaller classes or increased slack?

Three sources of bias in OLS:

1. Reverse causality: Inefficient schools receive compensatory staffing
2. Selection: Management quality affects both staffing and deployment
3. Omitted variables: Neighbourhood effects, principal quality, union strength

Solution: Instrument STR with bureaucratic teacher allocation rules (Post Provisioning Norm).

Demographic shock elasticities 🔎

The key question: When a school receives an extra teacher, does class size fall? Or does the school absorb the teacher into lighter workloads?

Why OLS fails: Schools that appear inefficient might receive extra teachers as compensation. Or weak management causes both over-staffing and poor deployment.

The instrument: South Africa allocates teachers using a formula—the Post Provisioning Norm (PPN). This creates mechanical variation in teacher allocations based on grade composition.

Instrument: Post Provisioning Norm (PPN)

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

Grade	Max Class	Weight
R	35	0
1–4	35	1.190
5–6	40	1.042
7	37	1.126

Identification: Within-school grade composition changes trigger mechanical allocation adjustments.

• Schools don’t control cohort composition
• School FE + enrolment controls absorb time-invariant quality

How PPN works: Teachers allocated based on weighted enrolment. Foundation phase (Gr 1–4) gets higher weight than intermediate (Gr 5–6).

The variation I exploit: When a large cohort moves from Grade 4 (weight 1.19) to Grade 5 (weight 1.04), predicted teacher allocation falls mechanically.

Why valid: Schools can’t manipulate which children are born in their catchment area. The grade-mix variation is driven by demographic cohort sizes.

Specification

\[ \begin{aligned} \text{First stage:} \quad & \Delta \text{STR}_{\text{actual}} = \alpha + \beta \cdot \Delta \overline{\text{PPN-Weight}}_{\text{leaver}} + \delta_t + \varepsilon \\[0.5em] \text{Second stage:} \quad & \Delta \text{Inefficiency} = \alpha + \delta \cdot \Delta \widehat{\text{STR}} + \delta_t + \varepsilon \end{aligned} \]

First-differences specification. School-level clustering. N = 28,703 school-years.

Identifying assumptions:

1. Relevance: PPN strongly predicts actual STR (F > 60)
2. Exclusion: PPN affects inefficiency only through STR
3. Exogeneity: Grade-mix changes ⊥ time-varying management quality

Why first differences: Removes school fixed effects. Isolates within-school changes over time.

The key test: If constraints truly bind (teachers can’t be reallocated), then losing teachers should increase inefficiency (δ > 0). If slack is discretionary, losing teachers should decrease inefficiency (δ < 0) as schools are forced to tighten deployment.

IV Results

	First-Differences
2SLS Coefficient (δ)	−4.5*
Standard Error	0.111
First-stage F-statistic	60.4
N (school-years)	28,703

Interpretation: Exogenously gaining teachers → increased slack, not smaller classes.

Schools absorb additional teachers into lighter workloads. Why? 🔎

The key finding: δ < 0. When schools lose teachers (STR rises), inefficiency falls. When they gain teachers, inefficiency rises.

What this means: If constraints truly bound—if there were structural reasons teachers couldn’t teach more—we’d observe δ > 0. The negative coefficient proves slack is discretionary.

The mechanism: Additional teachers are bargained into lighter workloads, not assigned to new classes. TCR rises with resource abundance.

Robustness

Instrument validity: ✓

• Balance on 15 pre-determined covariates
• No pre-trends (future shocks don’t predict past outcomes)
• F > 30 across all specifications

Specification robustness: ✓

• 7 fixed effects combinations
• 4 clustering schemes
• 5 functional forms

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

Full robustness table 🔎

Why you should believe this: 1. Strong first stage—F = 60.4, well above weak instrument thresholds 2. Balance tests pass—instrument orthogonal to pre-determined characteristics 3. Results robust across 17 specification checks 4. Direction consistent everywhere; magnitude varies predictably

Heterogeneity: Effect is larger where there’s more slack to squeeze—small schools, schools with low initial STR.

Implications

But Isn’t That Slack Needed for Preparation?

The professional day extends beyond instruction

Component	Description
Formal school day	7 hours at school (PAM requirement)
Instructional periods	~6 periods × 45 min = ~4.5 hours
Breaks	Recess + lunch = ~1 hour (rest, not prep)
Within-day non-contact	~1.5 hours (admin, assembly, free time)
Informal school day	~400 hours/year = ~2 hours/day explicitly for preparation

Key point: The “informal school day” is mandated time outside formal hours for planning and marking.

Even at TCR = 1.0, teachers have ~3.5 hours daily for preparation (1.5h in-day + 2h informal).

Current TCR = 1.22 gives additional free periods during instruction—beyond what’s needed.

Why would schools choose this? 🔎

The common objection: “Teachers need free periods to prepare lessons and mark.”

The rebuttal: - PAM requires 7 hours at school plus ~2 hours “informal school day” for preparation - Breaks (recess/lunch) are rest time—everyone gets those - The ~1.5 hours of non-instructional time within the formal day (admin, assembly) plus 2 hours informal = 3.5 hours for prep - This is before any free periods during instruction

The implication: Free periods during instructional time are additional to already-ample preparation time. They’re a luxury, not a necessity.

Fiscal Leakage

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

Assumptions 🔎

The numbers: - R22.3bn = 18.2% of primary teacher wage bill = cost of unutilised instructional capacity - R29.6bn = adding spatial inefficiency

Context: R22.3bn is larger than the National School Nutrition Programme (≈R8bn) and LTSM budget (≈R12bn) combined.

Interpretation: This is either capacity that could reduce class sizes by 25%, or fiscal resources that could be redeployed.

Policy Implications

1. Enforce contact-time norms

• −18.8% ECS or free ≈18% educator capacity (≈R22bn)

2. Activate idle classrooms

• −10.2% ECS from capital under-utilisation

3. District-level pooling

• Shift hiring from school to district level
• Captures spatial gains (−24.2% cumulative) without provincial disruption

Key insight: The problem is deployment incentives, not fiscal constraints.

Three tractable levers: 1. Contact-time norms: PAM already requires 7 hours at school; enforce timetabling standards 2. Capital activation: Many schools have unused classrooms; administrative reform to assign teachers 3. District pooling: Most spatial gains captured locally—no need for provincial teacher transfers

The broader point: You don’t solve this by hiring more teachers. You solve it by changing how existing teachers are deployed.

Summary

1. Measurement innovation: Constrained optimisation to quantify efficiency gaps under nested constraints—technical, allocative, spatial.

2. Descriptive finding: Within-school labour slack (−18.8%) > spatial misallocation (−5.8 pp).

3. Causal evidence: Exogenously adding teachers increases slack, not reduces class sizes (δ = −4.5).

4. Policy implication: R22bn annually—deployment reform, not fiscal expansion.

The elevator pitch: Schools employ enough teachers for classes of 32, but learners experience classes of 43. Using bureaucratic allocation rules as an instrument, I show that providing schools with additional teachers increases slack rather than reducing class sizes. This represents R22bn in annual fiscal leakage and explains why teacher hiring interventions consistently disappoint.

References

Bennell, Paul. 2022. “Teaching Too Little to Too Many: Teaching Loads and Class Size in Secondary Schools in Sub-Saharan Africa.” International Journal of Educational Development 94 (October): 102651. https://doi.org/10.1016/j.ijedudev.2022.102651.

Bloom, Nicholas, Renata Lemos, Raffaella Sadun, and John Van Reenen. 2015. “Does Management Matter in Schools?” The Economic Journal 125 (584): 647–74. https://doi.org/10.1111/ecoj.12267.

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.

Hanushek, Eric A. 1986. “The Economics of Schooling: Production and Efficiency in Public Schools.” Journal of Economic Literature 24 (3): 1141–77. https://www.jstor.org/stable/2725865.

Walter, Torsten Figueiredo. 2020. “Misallocation in the Public Sector? Cross-Country Evidence from Two Million Primary Schools.” Unpublished.

Appendix

Sample Attrition

• Final coverage: ≈5.4 million learners (78%), 9,512 schools
• Largest exclusion: Multigrade schools (≈12%)
• External validity concern: Multigrade schools may face distinct challenges

↩︎️ Back

Fiscal Assumptions

• 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); secondary extrapolation ≈R62bn total
• As % of education budget: 7.6–10.1%

Comparison: R22.3bn > National School Nutrition (≈R8bn) + LTSM (≈R12bn)

↩︎️ Back

IV Robustness: Full Table

#	Category	Test	Description
1	Instrument Validity	Balance on observables	Instrument orthogonal to 15 pre-determined covariates
2	Instrument Validity	Falsification tests	Future PPN shocks don't predict past outcomes
3	Instrument Validity	Weak instrument robust CI	Anderson-Rubin and CLR confidence sets
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
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
8	Specification Robustness	Alternative fixed effects	None, Year, District, School, Province×Year, District×Year
9	Specification Robustness	Alternative clustering	School, District, Two-way (School+Year), Two-way (District+Year)
10	Specification Robustness	Subsample stability	Balanced panel, exclude outliers, by quintile, leave-one-province-out
11	Specification Robustness	Alternative functional forms	Level-level, 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 FE
14	Specification Robustness	Control function approach	Alternative estimator using residuals as control
15	Heterogeneity Analysis	Effect modification	By school size, initial STR, baseline class size, quintile, urban/rural
16	Mediation Analysis	Class count as mediator	Tests whether STR affects inefficiency through class formation
17	Attrition & Sample Selection	Sample attrition tests	Validates representativeness of final sample (78% coverage)

↩︎️ Back

Demographic Shock Elasticities

Question: Are high TCRs forced by constraints or discretionary?

Test logic: If constraints bind → shocks raise inefficiency. If slack is discretionary → shocks reveal latent capacity.

• β₁ (Grade-Mix) = −0.381*** (10 pp shift → 3.81 pp inefficiency reduction)
• β₂ (Phase-Mix) = −0.256*** (teachers reallocated across phase boundaries)
• All coefficients negative → adaptive capacity exists

↩︎️ Alternative explanations 🔎

Alternative Explanations Ruled Out

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

Mechanism	Prediction	Finding
Grade-specific match capital	β₁ > 0	Rejected: β₁ = −0.381
Qualification constraints	β₂ > 0	Rejected: β₂ = −0.256
Period indivisibilities	β > 0	Rejected: both < 0
Cross-phase synchronisation	β₂ > 0	Rejected: β₂ = −0.256
Enrolment volatility buffering	β₃ > 0	Rejected: β₃ = −0.007

Consistent with: Policy equilibria where slack is discretionary.

Caveat: Standby rosters create endogeneity—estimates may be lower bounds.

Equity Implications

• Baseline gap: Q1–Q2 classes 32% larger than Q5
• Paradox: Wealthier schools show largest efficiency gains (−20.2%)—hold more slack
• Implication: Efficiency reforms reduce absolute crowding but preserve relative inequality

Racial 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 capacity

Why Would Schools Choose Inefficiency?

Core mechanism: Teachers value free periods above smaller classes.

• Rigid wage schedules: Non-pecuniary compensation (lighter loads) becomes primary margin
• Union bargaining: Workload norms weakly enforced
• Measurement vacuum: What isn’t measured can’t be managed
• Rational precaution: Standby rosters for unscheduled absence create endogenous slack

Political economy: TCR represents negotiated equilibrium—teachers capture rents through reduced contact time rather than higher wages.

International Comparisons

Country	TCR	Utilisation
Tanzania	2.5	40%
Ethiopia	≈1.4	≈71%
South Africa	1.22	82%

• Vietnam: Explicitly targets TCR > 1 for extended contact hours
• SSA systems without timetabling software may face both technical and allocative inefficiency

Implication: South Africa’s problem is utilisation (S4), not timetabling (S1–S2).

Algebra Reference

Core Identity: STR = SCR / TCR

• STR = Learners / Teachers
• SCR = Learners / Classes
  
• TCR = Teachers / Classes

Experienced Class Size (ECS): \[\text{ECS} = \frac{\sum_i \text{CS}_i^2}{\sum_i \text{CS}_i} = \overline{\text{CS}} + \frac{\sigma^2_{\text{CS}}}{\overline{\text{CS}}}\]

Example: Classes of {50, 10} average 30, but ECS = 43.3 (more learners experience the larger class).

Provincial Scenarios

• Limpopo & Eastern Cape: High capital slack (idle classrooms)
• Limpopo & Mpumalanga: High labour slack (idle teachers)
• Provinces need tailored interventions

Representative School: Status Quo (S0)

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

Within-grade dispersion modest. Between-grade imbalance small. Key: 27 classes but more teachers available.

↩︎️ Back to scenarios

Within-Grade Equalisation (S1)

↩︎️ Back to scenarios

Between-Grade Allocation (S2)

↩︎️ Back to scenarios

Classroom-Constrained Allocation (S3)

↩︎️ Back to scenarios

Educator-Optimal Allocation (S4)

↩︎️ Back to scenarios

District-Level Pooling (S5)

↩︎️ Back to scenarios

Province-Level Pooling (S6)

↩︎️ Back to scenarios

Variance Decomposition: What Explains Crowding?

• 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

Secondary School Extension

• Primary TCR ≈ 1.22 (generalist model). Secondary TCR likely higher:
  • Subject specialisation mandated (≈15+ subjects)
  • Period indivisibilities bind harder
  • Teacher qualifications phase-specific
• Preliminary estimates: If secondary TCR ≈ 1.5, fiscal leakage ≈R40bn additional (total ≈R62bn)
• Data challenge: LURITS lacks subject identifiers for secondary

Policy: Subject rationalisation; pool teachers at district level for rare subjects.