1 Introduction

Tax systems, social insurance programmes, and regulatory rules are riddled with discontinuities. Income tax schedules change slope at bracket thresholds; benefit programmes phase out at earnings ceilings; environmental regulations impose caps. These features of policy design are often treated as identification nuisances. Yet a body of work pioneered by Saez [2010] and synthesised by Kleven [2016] has turned them into a powerful identification strategy: the bunching estimator.

The insight is simple. If a convex budget set develops a kink at a threshold z*, utility-maximising agents near z* respond by clustering their choices at z*. The amount of bunching-how many more people locate at z* than would be predicted by the smooth counterfactual distribution identifies the structural parameter governing behavioural response. For tax applications, this parameter is the elasticity of taxable income.

Bunching methods have now been applied across economics: to estimate labour supply elasticities [Saez, 2010], to study firm size responses to regulatory thresholds [Garicano et al., 2016], to identify housing demand [Kopczuk and Slemrod, 2003], and to evaluate retirement savings incentives. This article explains the method, its assumptions, its extensions to notches and kinks in a unified framework, and its limitations.

2 The Basic Setup

‍Consider an individual with earnings z choosing over a budget set with a kinked tax schedule. Below the kink z*, the marginal tax rate is t_1; above it, the rate rises to t₂ > t₁. The budget constraint has a kink at z*.

A standard model of labour supply with a constant elasticity of taxable income (ETI) predicts that agents with unobserved ability n optimally choose:

where e is the elasticity of taxable income with respect to the net-of-tax rate (1 - t). At the kink, there is a range of abilities [n₁, n₂] for whom it is optimal to locate exactly at z*, regardless of their precise type. This generates a spike in the density of earnings at z*.

The Identifying Equation

Let B denote the excess mass at z*, measured as the number of individuals at z* in excess of the counterfactual density h₀(z*) (the density that would prevail absent the kink). The range of types that bunch is Δn = n₂ - n₁. For a first-order approximation,

where Δ(1 - t) = t₂ - t₁ is the change in the marginal net-of-tax rate at the kink. Rearranging,

The key objects on the right-hand side are: (i) the excess mass B, which is observed; (ii) the counterfactual density h₀(z*) which must be estimated; and (iii) the policy parameters (z*, t₁, t₂), which are known.

3 Estimating the Counterfactual Density

‍The critical empirical challenge is constructing h₀(z*) the earnings density that would prevail at z* in the absence of the kink. The standard approach, due to Saez [2010] and refined by Chetty et al. [2011], is polynomial extrapolation:

1. Bin earnings into small intervals of width δ.
2. Define the bunching region [z* - L, z* + U] as the support of the spike.
3. Fit a polynomial of degree p to the density outside the bunching region.
4. Extrapolate this polynomial through the bunching region to obtain ĥ₀(z).
5. Compute excess mass as B̂ = ∑_{z ∈ [z* - L, z* + U]} [h(z) - ĥ₀(z)].

The researcher must choose the degree p, the bin width δ, and the bunching region [z* - L, z* + U]. Sensitivity analysis over these choices is essential; Kleven [2016] recommends reporting estimates for a range of polynomial degrees and bunching windows.

4 Notches versus Kinks

‍A kink changes the slope of the budget set at z*; a notch creates a discrete jump in tax liability at z*. Notches are common in benefit programmes: earning one pound above the threshold can trigger a loss of benefits worth hundreds of pounds. The distinction matters enormously for identification.

At a notch, the dominated region where no rational agent should locate is [z*, z* + Δz^dominated]. Agents who would have chosen z in this region instead bunch at z*. The estimator is [Kleven, 2016]:

where ΔT is the discrete tax increase at the notch. Because the dominated region is observable (the density is zero there), notches provide sharper identification of the elasticity than kinks in many settings.

(Image Reference: Figure 1 illustrates the difference between a kink and a notch in the earnings density.)

5 Key ApplicationsElasticity of taxable income.

Saez [2010] applied the bunching estimator to US tax return data, finding clear bunching at the first kink of the Earned Income Tax Credit (EITC) and among the self-employed at the top of the first bracket. The implied ETI for the self-employed was around 0.5-0.7, substantially larger than estimates for wage earners, consistent with greater flexibility in reporting income.

Firm size and regulation. Garicano et al. [2016] used bunching to identify the cost of employment regulation in France: firms bunch just below size thresholds that trigger additional labour market obligations (50 employees under French law). The excess mass below 50 identifies the elasticity of firm size with respect to regulatory costs.

Housing markets. Kopczuk and Slemrod [2003] used bunching at estate tax thresholds to estimate behavioural responses to inheritance taxation. Best and Kleven [2018] used bunching in UK housing transaction data at Stamp Duty Land Tax thresholds to estimate the housing demand elasticity with respect to transactions taxes.

6 Identification Assumptions and Limitations

‍The bunching estimator rests on several assumptions worth stating precisely.

1. Smooth counterfactual density. The density h_0(z) in the bunching region must be well approximated by the polynomial extrapolated from neighbouring regions. If there is a pre-existing kink or step in the density at z* -for example, if a second policy threshold coincides with z* the counterfactual will be misspecified.
2. ‍Optimisation. The model assumes agents optimise over a smooth utility function. Frictions, inattention, and fixed adjustment costs attenuate bunching and cause the estimator to understate the structural elasticity [Chetty, 2012]. The bunching estimate is therefore a lower bound on the frictionless elasticity.
3. ‍No manipulation. The bunching must reflect genuine behavioural responses (labour supply, savings decisions), not measurement artefacts or strategic misreporting. In contexts where avoidance is easy (e.g., self-employment income), the estimate conflates structural labour supply with avoidance elasticity.
4. ‍Partial identification at kinks. Bertanha et al. [2023] show that point identification of the ETI at a kink requires the analyticity of the counterfactual density, a stronger assumption than smoothness alone. Under weaker restrictions, the method delivers partial identification of the elasticity, with bounds that widen as the polynomial degree increases.

7 Recent Developments

The bunching literature has expanded rapidly in recent years. Kleven [2016] provides a unified framework and a review of more than 50 applications. Bertanha et al. [2023] develop econometric theory for the bunching estimator with heterogeneous agents, covariates, and formal hypothesis tests. Stata and R implementations are now available (the bunchr package in R; the bunching command in Stata). Goff [2025] extend the framework to allow for causal effect identification in quasi-experimental designs where the kink or notch is generated by a policy intervention rather than optimising behaviour alone.

8 Conclusion

‍Bunching estimators transform institutional features of policy design tax bracket kinks, benefit notches, regulatory thresholds into identification strategies. By measuring how much agents cluster at policy thresholds and comparing to a smooth counterfactual density, researchers can identify structural elasticities without instruments or experimental variation. The key assumptions smooth counterfactual density, optimisation, no manipulation-are often defensible and testable. As economists engage more deeply with the behavioural responses to complex institutional environments, bunching methods will remain a central part of the applied econometrician's toolkit.

References

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2. Best, M. C. and Kleven, H. J. (2018). Housing market responses to transaction taxes: evidence from notches and stimulus in the UK. Review of Economic Studies, 85(1):157-193.
3. Chetty, R., Friedman, J. N., Olsen, T., and Pistaferri, L. (2011). Adjustment costs, firm responses, and micro vs. macro labor supply elasticities: evidence from Danish tax records. Quarterly Journal of Economics, 126(2):749-804.
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