Bunching at Notches and Extensive Responses
A notch imposes a discrete consumption loss when earnings cross a threshold. Some earnings choices can then require more work while delivering less consumption than staying at the threshold. (Kleven and Waseem 2013) uses this dominated region together with bunching below the notch to study both behavioral responses and incomplete adjustment.
1. Derive the dominated interval
Let the tax schedule be \(T(z)=tz+(\Delta T+\Delta t\,z)1\{z>K\}\), with a positive tax jump. Put \(r=1-t\) and \(r'=1-t-\Delta t>0\). Consumption is \(rz\) below the notch and \(r'z-\Delta T\) above it.
An earnings choice \(z>K\) gives no more consumption than staying at \(K\) whenever
If work is costly and consumption desirable, these choices are dominated by \(K\): they involve more work and weakly less consumption. This conclusion does not require an elasticity estimate.
2. The marginal buncher
Use the isoelastic, quasi-linear utility from the kink model. An unconstrained interior choice on the upper segment is \(z_H=\eta(r')^\varepsilon\), with utility
Utility at the threshold is
The marginal buncher’s type solves \(V_H(\eta^*)=V_K(\eta^*)\), with the relevant interior/feasibility conditions. Its counterfactual earnings are \(z_0^*=\eta^*r^\varepsilon\).
Proposition 1. In the frictionless single-crossing model, types whose counterfactual earnings lie between \(K\) and \(z_0^*\) choose the notch point. Their mass is \(\int_K^{z_0^*}h_0(z)\,dz\), and the indifference equation connects the response endpoint to \(\varepsilon\) and the tax parameters.
A discrete budget loss requires a comparison of utility levels. A derivative at the threshold describes only a local slope and cannot determine whether moving across the loss is worthwhile.
3. What the hole adds
Frictionless optimization leaves the dominated interval empty. Observations in that interval indicate incomplete adjustment, imperfect information, measurement error, or another failure of the model. The paper combines the missing mass there with bunching, under its friction specification, to distinguish observed from structural responses.
Both moments require a counterfactual density, and their interpretation depends on how frictions affect choices. They describe different parts of the response; two moments cannot identify an unrestricted friction model.
For an upward jump in the budget set, the geometry changes. Moving across the threshold increases consumption but also requires more earnings; the dominated interval derived above no longer follows. Its indifference condition must be written afresh, with the correct sign of the transfer.
4. The tax jump and the size of the dominated region
The loss in consumption immediately above \(K\) is
Assume \(J>0\). The earnings value at which upper-segment consumption first catches up with consumption at \(K\) is
For every \(z\in(K,z_D]\), consumption is weakly lower than at \(K\) and work is strictly higher. Under increasing utility in consumption and decreasing utility in work, the threshold strictly dominates that entire region. At \(z_D\) consumption is equal but work is still higher, so domination includes the upper endpoint.
The dominated region follows from the budget geometry and monotonicity of utility, without isoelastic preferences. Keep it distinct from the counterfactual interval: the former contains actual choices ruled out by optimization; the latter contains choices that bunchers would have made without the notch.
5. A dimensionless indifference equation
Let \(v=z_H/K>1\) be the upper-segment choice of the marginal buncher, and put \(\rho=r'/r\) and \(d=\Delta T/(rK)\). Since \(z_H=\eta(r')^\varepsilon\), the type is \(\eta=vK/(r')^\varepsilon\). Substituting into the utility equality and dividing by \(rK\) gives
Equivalently,
The first term is the upper-segment optimum’s utility after accounting for the fixed loss; the term in \(v^{-1/\varepsilon}\) is the work cost at the threshold for that same type. This rescaling is useful for numerical calculations because it separates the level \(K\) from the rate ratio and relative tax jump.
The counterfactual earnings endpoint is
The relevant root must describe a feasible upper-segment choice and satisfy the global utility comparison. A solver may also find algebraic roots outside that region. The nonconcavity introduced by the notch makes these checks necessary.
6. A pure fixed notch gives a closed-form example
Set \(r'=r\), so \(\rho=1\), and take \(\varepsilon=1\). The equation reduces to
Of the two roots, the feasible one above the threshold is
The dominated region ends at \(z_D/K=1+d\), while the marginal buncher’s unconstrained upper choice is \(v>1+d\). The type indifferent between the notch point and the upper optimum must earn strictly beyond the dominated region before the extra work is worthwhile.
For a small fixed loss, \(v-1=d+\sqrt{d(2+d)}\sim\sqrt{2d}\). The leading response is proportional to the square root of the relative notch size in this example, rather than linear in it. Applying the small-kink formula to a notch would therefore miss even the order of the response.
The square-root response is not peculiar to \(\varepsilon=1\). For a pure fixed notch, put \(v=1+w\) in \(v+\varepsilon v^{-1/\varepsilon}=(1+\varepsilon)(1+d)\). A Taylor expansion gives
After multiplication by \(\varepsilon\) and addition of \(1+w\), the linear terms cancel. Consequently
For fixed \(\varepsilon>0\), the expansion applies as \(d\downarrow0\) along the feasible root. At a smooth optimum, the first-order utility cost of displacement is zero, leaving a second-order cost. Balancing that cost against a fixed consumption loss produces the square-root response. A small kink instead changes the marginal incentive and yields a first-order displacement.
Since \(z_0^*=Kv\) for this pure fixed notch, a locally flat counterfactual density gives \(B\sim h_0(K)K\sqrt{2\varepsilon d}\). The dominated interval has width \(Kd\), a smaller order. Bunching mass and the hole above the threshold thus encode different behavioral information even for arbitrarily small notches. This is the reason the notch design can use the hole to learn about incomplete adjustment while using the bunching mass to learn about the structural response.
7. What bunching and missing mass identify under a friction model
For illustration, suppose a fraction \(\lambda\) adjusts optimally and the remaining fraction follows the no-notch choice distribution, independently of its latent earnings within the relevant intervals. Let \(B^*=\int_K^{z_0^*}h_0(z)\,dz\) be frictionless bunching mass and \(H_D=\int_K^{z_D}h_0(z)\,dz\) the counterfactual mass in the dominated region. Then this particular mixture model gives
where \(M_D\) is missing mass in that region. If \(h_0\) is identified, \(M_D/H_D\) identifies the common adjustment fraction and \(B_{\rm obs}/\lambda\) recovers the structural mass.
The constant-adjustment mixture explains how the two moments can separate response and adjustment in this illustration. If adjustment depends on latent earnings, they instead integrate different weights. Measurement error may also put observed earnings inside the dominated interval despite optimal latent choices. The friction model must distinguish these possibilities.
An upward budget jump changes the sign of \(J\) and removes the dominance argument used above. The feasible choices and their utility levels must be compared again; the same response formula cannot be reused by changing a transfer’s sign.
Selecting the unique feasible indifference root
The dimensionless equation has a unique root on the economically relevant region \(v>1\) under the positive-jump condition. Define \(F(v)=v+\varepsilon v^{-1/\varepsilon}\). Then
Moreover, \(J>0\) is exactly \(1+d>\rho\), so \((1+\varepsilon)(1+d)/\rho>F(1)\). Continuity and strict monotonicity prove existence and uniqueness of the root \(v>1\). At \(v\le z_D/K\), the proposed upper choice would be dominated by \(K\), so it could not solve the indifference equation; thus the feasible root lies strictly above \(z_D/K\). For the standard downward-slope case \(0<\rho\le1\), types whose upper-segment unconstrained choice is at most \(K\) prefer the threshold to every feasible upper choice. For larger types, the utility difference has the sign of \(F(v)-(1+\varepsilon)(1+d)/\rho\), so it crosses once. This establishes the threshold sorting used in the bunching-mass calculation.