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while the diěerences between the rural job with the equivalent wage policy (XE) and the Addis job are
dXEA
0 −1 w − w w − w A A E E X RH h = XE − X A = E = X R − 1 −1 XT − 2 0 RP XR − 1 s − 1
In the base aĴribute-only model, the shares of respondents taking job P over job A, and job E over job A, are respectively PPA = F(β T (dXPA )) and PEA = F(β T (dXEA )). T T If these are set equal, then β (dXPA ) = β (dXEA ), or
− β L + β W ( wP − w A ) = −β L
+ β H XPH + β E (XPE − 1) + β T (XPT − 2) + β P (XPP − 1) + β W (wE − wA ) + hβ H − β E + β P (s − 1)
or
β W (wP − wA ) + β H (XPH − h) + β E XPE + β T (XPT − 2) + β P (XPP − s) = β W (wE − wA ). Alternatively, ∆w = wE − wP β H XPH − h XE 1 βE . = W T ⋅ PT β β XP − 2 P P β XP − s This is the extent to which a simple wage bonus would need to exceed the wage in the rural bundle to have the same impact on labor supply. Note that in general, the wage equivalent of a policy change that improves a single non-wage aĴribute will not be the same as the marginal value of, or marginal willingness to pay for, that aĴribute. This is because the MRS (marginal rate of substitution) between two aĴributes is calculated holding all other aĴributes constant, whereas the wage equivalent compares two jobs with diěerent aĴribute levels. In a model with characteristic interactions, the same aĴribute vectors are used, but to find the wage equivalent for a given type of person we now equate the diěerences in mean latent utilities for that person type. The shares of respondents with characteristics Z taking job P over job A, and job E over job A, are now respectively PPA = F(β T (dXPA ) + δ 䊟 dXPAZT ) and PEA = F(β T (dXEA ) + δ 䊟 dXEAZT ) These are equal if
β T (dXPA ) + δ 䊟 dXPAZT = β T (dXEA ) + δ 䊟 dXEAZ or
β T (dXPA − dXEA ) + δ 䊟 dXPA − dXEA ZT = 0.
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