Products can be used either as intermediate inputs or for final consumption. We assume a CobbDouglas production function for each product _ω_ from labor market _k_ of country 1 at time _t_ where _M_ 1<sup>_k_</sup> _,t_<sup>isacompositeintermediateinput,and</sup><sup>_B_</sup> 1<sup>_k_isatime-invariantfixedfactorusedinlabor</sup> market _k_ of country 1. To focus on the dynamics from the worker side, we simplify the production by assuming that all producers in country 1 consider the aggregate price index _P_ 1 _,t_ as the price of composite intermediate inputs, instead of explicitly modeling an input-output structure.

In the production function, _L_<sup>�</sup><sup>_k_</sup> 1 _,t_<sup>isexpressedintermsofthenumberofefficiencyunitsprovided</sup> by all workers in labor market _k_ , which differs from _L_<sup>_k_</sup> 1 _,t_<sup>,theactualnumberofworkers.Asdescribed</sup> in Section 3.1, producers in labor market _k_ decide the optimal mass of tasks _Tt_<sup>_k_andpaythemarginal</sup> cost of task operation _c_ ̃ _>_ 0. From the producers’ point of view, workers in the same labor market are equally productive regardless of the task to which they are assigned, and thus are paid the same wage as long as they are in the same labor market. Therefore, producers will choose the total mass of tasks to operate, _Tt_<sup>_k_,aswellasthetotalefficiencyunitsoflabortheyuse,</sup><sup>_L_�</sup><sup>_k_</sup> 1 _,t_<sup>,takingintoaccount</sup> the fact that it is optimal to evenly allocate the labor supply _L_<sup>_k_</sup> 1 _,t_<sup>theyobserveintodifferenttasks.32</sup> For each labor market _k_ , we assume that _L_<sup>�</sup><sup>_k_</sup> 1 _,t_<sup>isaCESaggregateofallefficiencyunitsprovided</sup> by each task in labor market _k_ . Then, the total labor aggregate is where _L_<sup>�</sup><sup>_k_</sup> 1 _,t_<sup>(</sup><sup>_τ_)istheefficiencyunitstask</sup><sup>_τ_providesinlabormarket</sup><sup>_k_ofcountry1;and</sup><sup>_σ_ ̃isthe</sup> elasticity of substitution between tasks. Under the assumption of identical productivity across tasks, _L_ �<sup>_k_</sup> 1 _,t_<sup>canberewrittenas</sup><sup>_L_�</sup><sup>_k_</sup> 1 _,t_<sup>=</sup><sup>_Lk_</sup> 1 _,t_<sup>(</sup><sup>_T k_</sup> _t_<sup>)</sup> _σ_ ̃ _−_ <u>1</u> 1 . If _σ_ ̃ _>_ 1, then the number of tasks increases with the optimal demand for total labor force, conditional on the labor supply. This is the love for variety of tasks channel, which is analogous to the well-known love for variety of products in the Armington (1969) trade model.<sup>33</sup> Therefore, if there is a positive labor demand shock for labor market _k_ , then the number of tasks operated there will increase, conditional on _L_<sup>_k_</sup> 1 _,t_<sup>.Empirically,producersmayposta</sup> new type of task to operate a more differentiated production structure or open new establishments to meet the increased demand.<sup>34</sup> The exact magnitude of the response will be determined at the general equilibrium, as _L_<sup>_k_</sup> 1 _,t_<sup>respondstotheshockaswellfromthelabormobilitypartofthemodel.</sup> The effect of a labor demand shock on labor mobility through the number of job options is two-

> 32Note that the idiosyncratic task cost matters for individual worker’s utility but does not matter for producers.

> 33Feenstra (1994) further studies an endogenous change of the mass of varieties.

> 34Becker, Egger, Koch, and Muendler (2020) document that larger plants operate more specialized tasks.

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