1 indicates that country c has a comparative advantage in the female-
.
intensive good F . The comparative advantage can be decomposed into a technological or
Nc
Ricardian component c and an occupational or “factor-proportions” component 11 N c,
which can exacerbate or attenuate technological di↵erences. We rewrite the two equations
(7) and (8) as a system of two equations with two unknowns {✓c , ✓ c } given exogenous model
parameters and “pre-determined” values {N c , N c }:
⌘ (1 ⌘ ) ✓c ↵) ⌘ (1 ⌘ ) ✓ c
c ↵ + ( c )⌘(1 = 0 (9)
(1 + ✓ ) (1 + ✓ c )↵
✓ c
⇢c c = 1 (10)
✓
7
Equation (9) implicitly deﬁnes a downward-sloping “goods market-clearing curve” in the
space (✓ c , ✓c ) and is just⇣a rearrangement
⌘⌘ of equation (7), keeping in mind that normalization
M c F cM c ⌘
(1) implies that M c = M c F c = ( c ) (1 ↵) . Since goods produced by the two countries
are perfect substitutes, market clearing implies a negative relationship between the size ✓c
of the F -sector in country c and its size ✓ c in country c. On the other hand, the upward-
sloping “factor market-clearing curve” in the space (✓ c , ✓c ), deﬁned by (10), implies that
F -sectors have to be of comparable size in the two countries (i.e. the larger sector F gets in
country c, the larger it needs to be in country c as well), otherwise the return to capital will
diverge across the F - and M -sectors in each country. Thus, allocations of capital between
two sectors in the two countries {✓c }c2{X,Y } are uniquely determined by the system of two
equations (9) and (10).
Proposition 1: Production and trade equilibrium Consider the endowment struc-
ture K ¯ c , Lc c2{X,Y } = {1, 1, 1
¯ c, L N c }c2{X,Y } . The unique production and consump-
M F
c c2{X,Y }
tion equilibrium is characterized by the vector of prices {pi , rc , wi }i2{M,F } deﬁned by (3)-(6),
2 { }
and capital allocations {✓c } c X,Y
that solve (9)-(10).⌅
The proof of Proposition 1 establishes existence of an intersection of the two “factor
market-clearing” and “goods market-clearing” curves, which is therefore unique since the two
curves have opposite slopes.
2.3 Fertility Decisions
The analysis above is carried out under an exogenously ﬁxed fertility rate or, equivalently,
an exogenously ﬁxed level of female labor force participation. We now turn to endogenizing
households’ fertility decisions. To pin down equilibrium fertility N c , we proceed in two steps.
c
First, for a given N c , wF and N c are jointly determined by labor supply and demand. Thus,
we must ensure that labor supply is upward-sloping and the female labor market equilibrium
is well deﬁned. Second, fertility in the other country a↵ects the labor market equilibrium by
shifting female labor demand and hence fertility in country c. We therefore look for a ﬁxed
point in {N c , N c } such that the female labor markets are in equilibrium in both countries
simultaneously.
Fertility choices and female labor supply Taking N c as given and anticipating the
production equilibrium prices and quantities, households make fertility decisions accordingly.
8
Namely, they take prices as given and choose N c to maximize their indirect utility:
c
V c (N ) = r c + wF (1 c
N ) + wM + v (N ) . (11)
The ﬁrst-order condition for the representative household’s fertility decision is necessary and
su cient and given by 8