Estimating Elasticities under Nonlinear Budget Sets
Latent bunchers all choose the kink, yet their observed earnings may form a broad peak because of measurement error or imperfect adjustment. Counting observations in a chosen window then misses some bunchers and includes some nonbunchers. The likelihood approach in (Aronsson, Jenderny, and Lanot 2024) models the peak and both tails together to recover behavioral and error parameters.
1. Convolve the optimal-choice distribution
Let optimal earnings \(Z^*\) have continuous density \(f_-\) below \(K\), continuous density \(f_+\) above \(K\), and an atom of mass \(B\) at \(K\). Suppose observed earnings are \(Z=Z^*E\), where \(E>0\) is independent with density \(g\).
Proposition 1. For \(z>0\), the observed density is
Conditional on \(Z^*=u\), the density of \(uE\) is \(g(z/u)/u\). Integrating this conditional density over the distribution of \(Z^*\) gives the formula. The atom contributes \(B g(z/K)/K\), not an integral with a continuous density. Its Jacobian is \(1/K\).
The formula combines three sources of observations: a blurred atom and two blurred continuous regions. They overlap in the observed density, even though they are distinct in the latent choice distribution.
2. What the likelihood adds
A log-linear labor-supply model gives optimal earnings on a linear budget segment as \(Z^*=\eta(1-t)^\varepsilon\). Distributional assumptions on \(\eta\) determine the continuous components and the kink mass. A model for \(E\) determines how those components are blurred together.
For example, \(\log E\sim \mathcal{N}(-\sigma_E^2/2,\sigma_E^2)\) implies \(\mathbb{E}[E]=1\). The variance is positive; the negative sign belongs only in the mean adjustment. Likelihood estimation then uses the density above to estimate behavioral and error parameters jointly, subject to identification of the chosen specification.
The assumptions make the likelihood model estimable only when its parameter-to-distribution map is identified. They provide restrictions on heterogeneity and error; they do not overturn nonidentification under unrestricted heterogeneity.
3. Extending beyond one kink
For a piecewise-linear budget, each segment has a net wage and a virtual income. A monotone scalar heterogeneity index partitions individuals into segment interiors and kink points. Expected labor supply is obtained by integrating the segment-specific choices over those regions and adding kink masses.
The representation in (Soren Blomquist and Newey 2002; Sören Blomquist, Hausman, and Newey 2023) reorganizes these integrals into a baseline final-segment expectation plus adjacent-segment corrections. The reason is telescoping: each interior boundary adds the contribution of the segment below it and subtracts the continuation of the segment above it.
The partition requires monotonicity in the type index and integrability. Zero-hours corners need an additional region unless excluded. Since type determines the chosen segment, the observed marginal rate is selected together with earnings; inserting it in a linear labor-supply regression need not recover this optimization model.
4. Checking that the convolution is a density
For any fixed \(u>0\), substitution \(e=z/u\) gives
Integrating the three components of \(h\) therefore yields
Normalization checks the atom term and every Jacobian simultaneously. Omitting \(1/u\) would generally change the integrated mass and misweight observations associated with different latent earnings levels.
In logs, the convolution is simpler. Let \(L=\log Z\), \(L^*=\log Z^*\), and \(E_L=\log E\). Then \(L=L^*+E_L\). If \(q\) is the density of \(E_L\), the kink atom contributes \(Bq(l-\log K)\) to the log-earnings density. Returning to levels divides that density by \(z\). Since \(g(e)=q(\log e)/e\),
The change from logs back to levels therefore gives exactly the earlier atom term.
5. Deriving the continuous components from latent heterogeneity
Suppose \(\eta\) has density \(f_\eta\), with net rates \(r_0>r_1\) and elasticity \(\varepsilon>0\). The types choosing below the kink satisfy \(\eta<K/r_0^\varepsilon\) and choose \(Z^*=\eta r_0^\varepsilon\). A change of variables gives
Above the kink the corresponding density is
and the bunching mass is
One type distribution and one elasticity determine the lower branch, upper branch, and mass at the kink. Estimating them as unrelated density components would remove the restrictions that connect the likelihood to the behavioral model.
Error variance affects both tails as well as the peak. Their joint behavior may distinguish a larger latent bunching mass from greater dispersion under the chosen specification. Parameter identification requires that distinction in the likelihood map; convergence of an optimizer does not establish it.
6. A likelihood-score identity explains how observations are assigned
Write the observed density as \(h(z)=h_0(z)+h_-(z)+h_+(z)\), where \(h_0\) is the blurred atom. For a differentiable parameter \(\theta\), whenever the components are positive,
The component weights are posterior probabilities and sum to one. A near-kink observation may come from a continuous branch, and a latent buncher may be observed far from the kink. The likelihood uses these probabilities instead of assigning every observation in a window to the atom.
This identity follows by differentiating the sum and dividing by \(h\). Differentiating the component integrals additionally requires domination and handling parameter-dependent cutoffs consistently. If a parameter changes the latent bunching thresholds, its effect on the mass and continuous regions must both be included; holding one part fixed would give the wrong score.
The moving-boundary terms can be checked in latent-type coordinates. Keep \(K\) fixed, set \(a=K/r_0^\varepsilon\), \(b=K/r_1^\varepsilon\), and write \(k(z,u)=g(z/u)/u\). The density is equivalently
Differentiate with respect to a behavioral parameter that moves \(a,b\), under conditions justifying differentiation of the integrals. The boundary terms from the lower region, atom, and upper region are, respectively,
They cancel. At either boundary, the smooth segment’s chosen earnings equal \(K\), so its measurement-error kernel agrees with the atom’s kernel. Mass moving between adjacent regions cannot create or destroy observed probability by itself. The remaining derivative comes from how choices, heterogeneity, or error densities change inside the regions.
For the score, all three moving-boundary terms must be considered together. Keeping only the derivative of the bunching mass gives the wrong result. Their cancellation here follows from matching kernels at the boundaries; it is not a general permission to omit moving-boundary terms.
The same elasticity changes choices along the continuous branches and changes which types bunch. The cancellation shows how the likelihood combines these movements without creating or losing probability at a boundary.
For \(\log E\sim \mathcal{N}(-\sigma_E^2/2,\sigma_E^2)\), the normal moment-generating function gives
Mean-one multiplicative error preserves \(\mathbb{E}[Z\mid Z^*]=Z^*\), but its median is \(e^{-\sigma_E^2/2}<1\). The visible peak need not be centered exactly at \(K\) in levels. Mean preservation does not imply preservation of modes or quantiles.
7. Telescoping across several budget segments
A simple three-region representation shows the telescoping argument. Suppose a scalar type \(\eta\) selects choice functions \(z_1(\eta),z_2(\eta),z_3(\eta)\) on regions separated by thresholds \(a<b\). Then, pointwise,
Below \(a\), both corrections apply and leave \(z_1\); between \(a\) and \(b\), only the first applies and leaves \(z_2\); above \(b\), neither applies. Integrating gives a final-region baseline plus adjacent-region corrections.
Kink regions can be included by inserting a constant choice \(K_j\) over the corresponding interval of types. Their integrals are \(K_j\) times the interval’s probability mass. Zero-hours corners require another region unless ruled out. The general nonlinear-budget representations organize these same cancellations under their monotonicity and integrability conditions.
Type determines the chosen segment, and hence the observed marginal rate. Treating that rate as externally assigned drops this selection step. Integrating the optimization model retains it.