distribution is not the same as the population distribution, we expect this should not affect the comparisons between the relative importance of productivities vs. preferences for different assumptions on the elasticity parameters.

# **5.2 Baseline Estimates**

Using our baseline estimates from Chetty (2012), _ξz_<sup>_c_=</sup><sup>_ξ_</sup> _z_<sup>_u_=</sup><sup>_ξ_</sup> _h_<sup>_c_=</sup><sup>_ξ_</sup> _h_<sup>_u_=0</sup><sup>_._15,wecan</sup> recover the function _G_<sup>_−_1</sup> (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> )) using the inverse Jacobian from Proposition 4.2. Applying _G_<sup>_−_1</sup> (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> )) to the observed distribution of incomes and hours worked from the ATUS yields a value of ( _n, α_ ) for each individual in our sample. However, this distribution of productivities and preferences ( _n, α_ ) is not easily interpreted. Towards understanding the role productivities and preferences play in driving income inequality, we will construct the counter-factual income for each individual if (1) everyone had the same productivity or (2) everyone had the same preferences. Comparing these measures with actual income will help us understand the extent to which inequality is due to productivities vs. preferences.<sup>29</sup> First, for all individuals ( _n, α_ ) we will calculate _zn_<sup>_CF_</sup> 0<sup>=</sup><sup>_n_0</sup><sup>_h∗_(</sup><sup>_n_0</sup><sup>_, α_)</sup><sup>_e∗_(</sup><sup>_n_0</sup><sup>_, α_), the income</sup> they would optimally earn if they had productivity _n_ 0 and preferences _α_ . This is feasible because we have identified each person’s productivity _n_ and we know the manner in which both hours worked and effort per hour change with _n_ . Second, for all individuals ( _n, α_ ) we calculate _zα_<sup>_CF_</sup> 0<sup>=</sup><sup>_nh∗_(</sup><sup>_n, α_0)</sup><sup>_e∗_(</sup><sup>_n, α_0), the income they would earn if they had productivity</sup> _n_ and preferences _α_ 0; this exercise is possible because we have identified each individual’s preferences _α_ and we know how both hours and effort per hour change with _α_ .

In Figure 4a we plot average counter-factual incomes at each actual income level assuming all individuals had the same _n_ (the baseline level of _n_ 0 is chosen so that the mean income level in the counter-factual income distribution matches the mean income level in the empirical income distribution). In Figure 4b we plot average counter-factual incomes at each actual income level assuming all individuals had the the same _α_ (again, the baseline level of _α_ 0 is again chosen so that the mean income level in the counter-factual income distribution matches the mean income level in the empirical income distribution).

> 29While this counter-factual income exercise is nominally performed under the assumption that _α_ enters the utility function as _αu_ ( _c_ ) _− v_ ( _h, e_ ), all of the counter-factual income measures in this section are actually invariant to any functional form of preference heterogeneity for which income monotonically increases in preferences. Conceptually, as long as hours are increasing in preferences, our method recovers the correct preference rankings among all individuals (even if the nature of preference heterogeneity is wrong due to functional form mis-specification); we show in Appendix A.6 that the counter-factual incomes only depend on these ordinal preference rankings.

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