89 In these four equations, black variables are either directly observed or previously estimated in a prior stage, red variables are the key unobserved variables that the system solves for, and blue variables are “intermediary” variables in the sense that they are functions of black and red variables only.<sup>117</sup> We solve this system of equations numerically to recover all unobserved variables ( **_T_**<sup>_M_</sup> , **_b_** , **_F_** , **_N_**<sup>_e_</sup> , _t_ , **_p_**<sup>_−_</sup> , **_P_** , **_R_** , **_E_** , **_X_** , **_β_** ), and then recover agricultural productivities ( **_T_**<sup>_A_</sup> ) through equation (6), which links observed crop production to underlying productivity, crop prices, and input prices.<sup>118</sup> We should note that the equation system is homogeneous of degree zero in the preference _b_ shifters ( _rice_ = 3 _×_ **_b_** ). 2Therefore, normalizations are necessary.1 _σ_ ˆ _−AσK_ ˆ _−_ 1 . Naturally, these particular normalization choices do not affect the results in anyWe set _bRoW,k_ = 2 for each crop _k_ , as well as meaningful way.

# **E.B Data Requirements for Inversion**

To solve the system of equations in Appendix E.A, we must plug in the “black” observed variables, which include both previously estimated parameters and data from various sources. In that process, it is first necessary to make decisions concerning the basic geographic units of analysis, the set of crops, and the time period for which to run the inversion. We use Sri Lanka’s 25 districts (i.e. _I_ = 25), which are the country’s second-level administrative divisions, because that is the most granular level of aggregation for which we have data on important variables such as wages and producer prices. We use the eight crops (i.e. _K_ = 8) for which we have data on both fertilizer intensity and producer prices, namely: cinnamon, cloves, groundnut, maize, onions, potatoes, rice, and tea. We choose to use 2019 for inversion because it is prior to the 2021 fertilizer import ban (and to other potentially influential events such as the Covid-19 pandemic and Russia’s invasion of Ukraine) while also offering rich household survey data.

Preference parameters ( _σA_ , _σK_ , _σM_ ) are taken from Table II and production parameters ( _{γk_<sup>_f_,</sup>

> 117Blue variables can be recovered from black and red variables by using equations (5), (8), (14), (15), (17), (18), (19), and (21).

> 118Region-specific fertilizer endowments ( _{Fi}Ri_ =1<sup>)arerecoveredbyapportioningtheglobalfertilizerendowment</sup> ( _F_<sup>_G_</sup> ) across regions in two stages. First, total domestic fertilizer endowment (<sup>�</sup><sup>_I_</sup> _i_ =1<sup>_Fi_) is set to 3.5% of total domestic</sup> fertilizer use, with the remainder of the global fertilizer endowment being apportioned to the RoW (i.e., _FRoW_ = _F_<sup>_G_</sup> _−_<sup>�</sup><sup>_I_</sup> _i_ =1<sup>_Fi_).Thischoiceof3.5%forSriLanka’sendowment-to-useratioisbasedonFAOdataonfertilizer</sup> production and use by country and year. Second, the total domestic fertilizer endowment (<sup>�</sup><sup>_I_</sup> _i_ =1<sup>_Fi_)isdistributed</sup> across Sri Lankan districts ( _{Fi}_<sup>_I_</sup> _i_ =1<sup>) in proportion to each district’s number of mobile workers (i.e.,</sup><sup>_Fi_= (�</sup><sup>_I_</sup> _j_ =1<sup>_Fj_)</sup><sup>_×_</sup> _Ni_<sup>_m/_(�</sup><sup>_I_</sup> _j_ =1<sup>_N_</sup> _j_<sup>_m_)).</sup>