Taxable-Income Elasticities and What Kinks Identify
A kink in a tax schedule changes the marginal return to earnings without creating a jump in consumption. In a frictionless model, people who would have chosen different earnings levels can all choose the kink. The observed mass therefore measures how much counterfactual income lies in a compressed interval. Turning that mass into an elasticity requires both a behavioral model for the interval’s width and information about its counterfactual density (Saez 2010; Kleven 2016).
1. The structural calculation
A simple model isolates the behavioral part. Take quasi-linear utility
Under a linear tax with net-of-tax rate \(r=1-t\), the first-order condition gives \(z=\eta r^\varepsilon\). Thus \(\partial\log z/\partial\log r=\varepsilon\); quasi-linearity removes income effects in this illustration.
Let the marginal tax rate rise from \(t_0\) to \(t_1\) at \(K\), so \(r_0>r_1\). Individuals bunch if their preferred earnings under the lower rate exceed \(K\) but their preferred earnings under the higher rate fall below it:
Proposition 1. Writing counterfactual earnings under the lower rate as \(Z_0=\eta r_0^\varepsilon\), the bunching interval is
If its counterfactual density is \(h_0\), the bunching mass is
The kink maps every counterfactual choice in this interval to the threshold. Integrating the counterfactual density over the interval gives the mass of the resulting atom.
2. The familiar elasticity formula
For a small rate change \(dt=t_1-t_0\), a first-order expansion gives
Hence \(B/h_0(K)\) estimates a response width, which becomes an elasticity after dividing by \(K\,dt/(1-t_0)\). For a finite kink under the isoelastic model, use the exact relation \(\varepsilon=\log(1+\Delta z/K)/\log(r_0/r_1)\).
With heterogeneous elasticities, integrate the small-kink response width against the joint counterfactual density of earnings and elasticity. The resulting coefficient is the mean local compensated elasticity at \(K\), under the smoothness and behavioral conditions supporting that local expansion.
3. What still has to be identified
The density outside the kink cannot directly reveal the density over the interval that has been compressed. To recover it, one must impose extrapolation or distributional restrictions, or use comparable control observations. Smoothness without a quantitative restriction need not determine the missing area.
Measurement error or incomplete adjustment can also spread the observed peak. Its visible width need not equal the response interval’s width. The identification bounds state what information is needed about the missing density; the nonlinear-budget likelihood specifies how latent bunching is spread into observed earnings.
4. Why the kink compresses an interval to one point
On a smooth budget segment with net rate \(r\), substitute \(c=rz+A\) into utility. Differentiating with respect to earnings gives
Strict concavity makes the first-order condition sufficient on that segment. At a downward kink in the budget slope, a maximum at \(K\) requires the left derivative to be nonnegative and the right derivative to be nonpositive:
Solving for \(\eta\) yields \(K/r_0^\varepsilon\leq\eta\leq K/r_1^\varepsilon\). Multiplication by \(r_0^\varepsilon\) gives the counterfactual earnings interval in the proposition. The bunching atom is the probability of this entire interval of types. It is not evidence that the latent type distribution itself has an atom.
The budget’s continuity matters. The virtual income on the upper segment must offset its lower slope at \(K\). Otherwise the utility comparison includes a discrete loss or gain and becomes a notch problem.
5. The finite-change formula and its approximation error
Let \(x=dt/r_0\), so \(r_1=r_0(1-x)\). The exact response width is
Expanding the response width in the relative tax change gives
The usual formula keeps only the first term. There is also an approximation in converting mass into width. If the counterfactual density is differentiable near \(K\),
There are two approximations: linearizing the response to the tax change, and replacing the density over an interval by its endpoint value. Even under exact isoelastic preferences, density variation can make the second approximation inaccurate.
Combining the two expansions explicitly, with \(h=h_0(K)>0\), gives
The finite-tax-change term is positive for \(\varepsilon>0\). The density-slope term can have either sign. Thus even if \(B\) and \(h\) were known exactly, the usual first-order elasticity formula could overstate or understate the structural elasticity, depending on the density slope. A sampling standard error around that approximation does not account for this approximation error.
If the full counterfactual cdf \(H_0\) is known instead, recover \(z_0^*=H_0^{-1}(H_0(K)+B)\) and use \(\varepsilon=\log(z_0^*/K)/\log(r_0/r_1)\). At a point with \(h_0(z_0^*)>0\), implicit differentiation gives
The derivative is large when the tax change is small or the endpoint density is low. In either case, a modest error in estimated bunching mass can produce a large error in the recovered elasticity. The mass supplies an area; the density and tax change are needed to convert that area into a behavioral response.
For example, if the relevant counterfactual density really is constant at \(h\) and \(B/h=0.1K\), the exact elasticity is \(\log(1.1)/\log(r_0/r_1)\). Replacing \(\log(1.1)\) by \(0.1\) and the log rate ratio by \(dt/r_0\) are separate approximations. Their errors need not offset.
6. Which elasticity is averaged under heterogeneity?
Let \(e\) denote a heterogeneous elasticity and let \(h(z,e)\) be the joint density of counterfactual earnings and elasticity. For a small kink with \(x=dt/r_0\), the response width for type \(e\) is approximately \(eKx\). Integrating over the affected types gives
Under domination and local smoothness allowing the expansion through the integral,
The density normalization therefore produces a local average elasticity among individuals whose counterfactual earnings are at the kink. It does not generally produce the population average elasticity. Individuals far from \(K\) contribute no first-order bunching mass for a small change at that threshold.
In this illustration, quasi-linearity removes wealth effects, so the response has a compensated interpretation. With income effects, the budget adjustment and the compensated parameter must be specified explicitly before using the same formula.
7. What a density estimate must supply
Suppose an empirical procedure estimates an excess mass \(\widehat B\) and a counterfactual density \(\widehat h_0(K)\). Even in the small-kink approximation,
Estimating the excess mass also requires estimating what would have occurred in its place. A smooth fitted curve supplies the missing density only under an extrapolation assumption. Thus uncertainty about the counterfactual affects both parts of the ratio.
The frictionless model conserves mass: the atom at the kink equals the probability removed from the counterfactual interval. Measurement error can spread the atom, and incomplete adjustment can reduce it. A chosen spike window need not recover \(B\) until those effects are modeled or bounded.
The density-bound note addresses the missing counterfactual with explicit restrictions. The likelihood note instead specifies heterogeneity and measurement error jointly. These are different routes to the quantities in the elasticity formula.