with each optimal level of (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> )), we can recover log( _n_ ) = log( _z_<sup>_∗_</sup> ) _−_ log( _h_<sup>_∗_</sup> ) _−_ log( _e_<sup>_∗_</sup> (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> ))). Finally, we can recover _α_ from optimal hours worked by netting out the substitution effects of different effort wages _n_ as before using:<sup>24</sup> In summation, the intuition behind Proposition 4.2 has three steps. First, optimal effort per hour is related to optimal hours worked through the equation: log( _e_<sup>_∗_</sup> ) = _<u>ξz</u>_<sup>_c_</sup> _ξ_<sup>_−_</sup> _h_<sup>_<u>cξ</u>_</sup> _<u>h</u>_<sup>_c_</sup> log( _h_<sup>_∗_</sup> ); intuitively, optimal effort varies with optimal hours worked in the ratio of the effort elasticity w.r.t. the tax rate to the elasticity of hours worked w.r.t. the tax rate. Hence, we can infer optimal effort per hour from optimal hours worked. Second, once we know optimal effort per hour, we can recover productivity using _z_<sup>_∗_</sup> = _nh_<sup>_∗_</sup> _e_<sup>_∗_</sup> . Third, once we have recovered productivity, we can subtract out the component of hours worked due to productivity effects; the remaining component allows us to recover preferences.

To solidify ideas, recall our example of the engineer making $30/hour working 60 hours/week and the mechanic making $10/hour working 40 hours/week (assume both work 50 weeks/year). For purposes of illustration, suppose that the hours elasticity is half as large as the income elasticity so that the hours and effort elasticities are equal: _<u>ξe</u>_<sup>_c_Hence:</sup> _ξh_<sup>_<u>c</u>_= 1.</sup> Thus, _e_<sup>_∗_</sup> _eng_<sup>=</sup> 2<sup><u>3</u></sup><sup>_e_</sup> _mech_<sup>_∗_,i.e.theengineerexerts1.5timesasmucheffortperhourasthe</sup> mechanic. Normalizing _e_<sup>_∗_</sup> _mech_<sup>_≡_1andusingthefactthathourlywageisequalto</sup><sup>_ne∗_,</sup> we can deduce that the mechanic’s productivity is 10 and the engineer’s productivity is 20; hence, once we account for effort differences, we infer that the engineer is only twice as productive as the mechanic as opposed to three times as productive if we assume productivity is equal to hourly wage. We could then find _α_ for both individuals by netting log( _h_<sup>_∗_</sup> <u>)</u> _−ξh_<sup>_u_log(</sup><sup>_n_</sup><sup><u>)</u></sup> out substitution effects using log( _α_ ) = _ξh_<sup>_<u>c</u>_</sup> .

We have shown in this section that we can still use labor supply elasticities to infer individuals’ productivities and preferences even if individuals make unobservable effort decisions in addition to choosing how many hours to work. Importantly, there are four key elasticities we need to recover the inverse function used to infer productivities and

> 24While the formulas are slightly different, this intuition still goes through with heterogeneous elasticities. We can still solve the system of differential equations 8 and 9 to find optimal effort as a function of optimal labor supply decisions ( _z_<sup>_∗_</sup> _, h_<sup>_∗_</sup> ). We can still find log( _n_ ) = log( _z_<sup>_∗_</sup> ) _−_ log( _h_<sup>_∗_</sup> ) _−_ log( _e_<sup>_∗_</sup> ). Finally, we can solve differential equations 2 and 3 (replacing the function arguments as (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> )), which is again without loss due to invertibility) to find log( _h_<sup>_∗_</sup> ) as a function of (log( _n_ ) _,_ log( _α_ )), and then invert this function to find log( _α_ ) as a function of log( _h_<sup>_∗_</sup> ) and log( _n_ ).

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