Dynamics of female labor force participation and welfare with multiple social reference groups

Applying the implicit equation theory for (9), using (5) and (6), and imposing the stability condition (16):

Direct consequence of (5), (6) applied to (18) and (19)

Applying the implicit equation theory for (9), using (5) and (6), and imposing the stability condition (16):

Using (3) and (17):

Using (4) and (17):

Using (3) and (17):

Using (4) and (18):

Using (3) and (19):

Using (4) and (19):

As stay-at-home wives produce their own “home goods” using (7) it is inferred that their welfare is equal such that:

As working wives buy their “home” good and have heterogeneous preferences for producing it themselves, using (7) and (6) their average welfare can be written as:

dAverageWelfarew¯dx=dα1Chdx+dα2ln(wm+wf−pCh)dx+dα3ln(wm+wfwm+μ~wf+(1−μ~)pCh)dx+dα4ln(wf(1+μ~)wm+μ~wf)dx−α1Ch12dμ~dx=dWelfarew¯(μi)dx

for any μ_i if μ~=1and\;thus dμ~dx=0

Given that C_h is exogenous, dChdx=0.

Using (7), (5), and (17):

Using (7), (5), (6) and (17):

If dμ~dp=0thendWelfarew¯dp<0

Using Lemma 3, assumptions (5) and (6), and (17):

If dμ~dp=0\;then\;dAverageWelfarew¯dp<0.

Using (7), (5) and (18):

Using (7), (5), (6) and (18):

Using Lemma 2, (5), (6) and (18):

Using (7), (5) and (19):

Using (7), (5), (6) and (19):

If dμ~dwf=0 then dWelfarew¯dwf>0.

Using Lemma 2:

If dμ~dwf=0then dAverageWelfarew¯dwf>0.

Stability condition for μ^t

Using (14) I derive the slope of w_f,t as a function of μ~t−1 as:

And thus

In steady state μ~t−1=μ~t

I rewrite (19) to derive the slope:

and thus

If there exist an interior solution forμ^t the stability condition for it is that

Plug (14) into (9) to determine (μ^,w^f\;)as well as the partial derivatives that reflect the effect that changes in the exogenous parameters wmandp have on female labor force participation and female wages

where

Using (33) and (32) it is thus shown that if a stable interior solution exists

and

Using (14), (35) and (36):

Using (3), (36), (38) and assumptions (5) and (6):

Using (4), (36), (38) and assumptions (5) and (6):

Using (3), (35), (37) and assumptions (5) and (6):

Using (4), (35), (37) and assumptions (5) and (6):

Using (7), (5), (36) and (38):

Using (7), (5), (6), (36) and (38) it can be proven that:

Using (7), (5), (35) and (37):

Using (7), 5), (6), (35) and (37):