discrete choice models.<sup>27</sup> In addition, we denote the probability of workers in labor market _k_ staying in the same job, thus in the same labor market _k_ , by _μ_<sup>_k_</sup> 0 _,t_<sup>;the probability of changingjobs but staying</sup> in _k_ by _μ_<sup>_k_</sup> 1 _,t_<sup>;andtheprobabilityofmovingfrom</sup><sup>_k_toanyothernon-residuallabormarket</sup><sup>_l_=</sup><sup>_k_(thus</sup> also changing jobs) by _μ_<sup>_k_</sup> 2 _,t_<sup>.Lastly,wedenotetheprobabilityofmovingfromlabormarket</sup><sup>_k_tothe</sup> residual labor market by _μ_<sup>_k_</sup> _I,t_<sup>,whichiseffectivelyequalto</sup><sup>_mkI_</sup> _t_<sup>.28Thejoboptionschannelallows</sup> for internal—i.e., within-market—churning between jobs, whose probability is measured by _μ_<sup>_k_</sup> 1 _,t_<sup>for</sup> labor market _k_ at time _t_ . Since we assume that the residual labor market _I_ offers a single job option, _μ_<sup>_I_</sup> 0 _,t_<sup>=</sup><sup>_μI_</sup> _I,t_<sup>shouldhold.</sup> With the new notations, we can re-write the labor-market-specific value as where _−ν_ log( _μ_<sup>_k_</sup> 0 _,t_<sup>) is an option value of moving to a different job.This option value can be decomposed</sup> into internal and external option values. The external option value is defined as the option value from alternative job options in a different labor market. By netting out the effect from switching to a different job in the same labor market, the external option value is derived as The internal option value is the option value from alternative job options within the same labor market, which is given as the difference between the total and the external option values,

# **3.3 Relative Welfare and the Number of Job Options**

Although the model features various determinants of mobility and welfare, we can derive simple formulae describing the relationship between changes in welfare as well as the number of job options, and changes in an underlying policy, conditional on two assumptions. First, we assume that the change in an arbitrary policy variable _z_ does not change the moving costs.<sup>29</sup> Second, any change in _z_ of period _t_ is realized after workers receive their wage. Under these assumptions, the change in the relative welfare of workers in labor market _k_ compared to workers in labor market _l_ can be derived as

> 27This conditional probability equation will allow us to identify structural parameters using labor mobility data.

> 28By construction, _μk_ 0 _,t_<sup>+</sup><sup>_μk_</sup> 1 _,t_<sup>+</sup><sup>_μk_</sup> 2 _,t_<sup>+</sup><sup>_μk_</sup> _I,t_<sup>= 1holdsforany</sup><sup>_t_,</sup><sup>_k_.Usingequations(6)-(9),each</sup><sup>_μ_canbeexpressedas</sup> _μ_ 0<sup>_k_</sup> _,t_<sup>=</sup> _θ_<sup>_k_</sup> 0 _,t_<sup>+</sup><sup>_θ_</sup> 1<sup>_k_</sup> _,tθ_<sup>+</sup><sup>_k_</sup> 0 _<u>,</u>_<sup>_θ_</sup> _t_ 2<sup>_k_</sup> _,t_<sup>+</sup><sup>_θk_</sup> _I,t_<sup>,</sup><sup>_μk_</sup> 1 _,t_<sup>=</sup> _θ_<sup>_k_</sup> 0 _,t_<sup>+</sup><sup>_θ_</sup> 1<sup>_k_</sup> _,tθ_<sup>+</sup><sup>_k_</sup> 1 _<u>,</u>_<sup>_θ_</sup> _t_ 2<sup>_k_</sup> _,t_<sup>+</sup><sup>_θk_</sup> _I,t_<sup>,</sup><sup>_μk_</sup> 2 _,t_<sup>=</sup> _θ_<sup>_k_</sup> 0 _,t_<sup>+</sup><sup>_θ_</sup> 1<sup>_k_</sup> _,tθ_<sup>+</sup><sup>_k_</sup> 2 _<u>,</u>_<sup>_θ_</sup> _t_ 2<sup>_k_</sup> _,t_<sup>+</sup><sup>_θk_</sup> _I,t_<sup>,and</sup><sup>_μk_</sup> _I,t_<sup>=</sup> _θ_<sup>_k_</sup> 0 _,t_<sup>+</sup><sup>_θ_</sup> 1<sup>_k_</sup> _,tθ_<sup>+</sup><sup>_k_</sup> _I,_<sup>_θ_</sup> _t_ 2<sup>_k_</sup> _,t_<sup>+</sup><sup>_θk_</sup> _I,t_<sup>.</sup>

> 29We will relax this assumption in counterfactual simulations where we quantify the changes in welfare, instead of relative welfare, using the full general equilibrium structure of the model.

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