Now consider the welfare (orreal wage) of industrial workers in both regions VH=wH
Analogously,
Comparative statics with respect toθ
Lemma 8. Welfare VH(θ) increases with respect to θ, while VF(θ) decreases, thus relative welfare VH
VF(θ) in-creases with respect toθ.
Proof. Representing welfare formula as follows
VH(θ) =
and recalling that both relative wage wwH
F(θ) and nominal wagewH(θ) increase with respect toθ (see Lemma 2 and Lemma 4), we obtain that welfareVH(θ)increases withθ. Analogously, the foreign welfare
VF(θ) =
decreases with respect toθ. Therefore, relative welfareVH
VF
(θ)increases with respect toθ.
Comparative statics with respect toλ
The relative welfare considered as function of industrial labor share is equal to VH
is a root of equation
(1−λ)(A−ϕ·x)−λx1−ρ(B−ϕx−1) =0,
wherex(λ)is implicit function defined by equation mentioned above and
−(2θ−1)2 then relative welfare function VH
VF
(λ,ϕ)increases with respect to industrial labor shareλ.
relative wage wwH
F(λ)is strictly positive non-decreasing function with respect to λ, while relative price index strictly decreases with respect toλ due to Lemma 6. Therefore relative welfareVH
VF increases with respect toλ.
Define the following threshold value that will be called “break-point”
ϕB(ρ,µ,θ) = ρ−µ
Lemma 10. i) Let Black-Hole conditionρ6µ holds, then relative welfare VH
VF
strictly increases with respect to industrial labor shareλ.
ii) Let No-Black-Hole conditionρ>µ holds andϕ>ϕB(ρ,µ,θ), then relative welfareVH
VF
strictly increases with respect to industrial labor shareλ.
iii) Let No-Black-Hole conditionρ>µ holds andϕ6ϕB(ρ,µ,θ), then there are real values
−(2θ−1)2 was considered in Lemma 9. Now assume that
ϕ<
Then
Q(x) = ρ
µ ·(ax−b)(c−dx) + (bd−ac)x6(ax−b)(c−dx) + (bd−ac)x=−(adx2−2bdx+bc)<0, due to Black-Hole condition ρ
µ 61 and the fact that discriminantD=4b2d2−4(ad)(bc) =4bd(bd−ac)<0. It
∂ λ <0. Moreover quadratic form is strictly negative if and only if its discriminant
D= abcd
It is light to see that this takes place if and only if 1< ac bd <
ρ+µ ρ−µ
2
. The first inequality is valid due to assumptionbd−ac<0 and the second one
ac following inequalities hold: ∂R
∂x <0 and ∂x
∂ λ <0 thereforeR(x(λ))strictly increases with respect toλ as well as VH
VF
(λ).
iii) Consider the following quadratic equation Q(x) =−ρ
µadx2+ (ρ−µ
µ ac+ρ+µ
µ bd)x−ρ
µbc=0.
Previous considerations imply that under assumptionsρ>µ andϕ6ρ−µ ρ+µ
s 1−(2θ−1)2 1+µ
1−µ
2
−(2θ−1)2 its discriminant is non-negative and there exists two real rootsξ26ξ1of equationQ(x) =0 coinciding each other only for
ϕ= ρ−µ
Moreover by definition signs ofQ(x)and derivative∂R
∂x coincide.
Therefore inverse function λ(x) (that may be explicitly obtained from equation (8)) decreases with respect to x. Define λ1=λ(ξ1),λ2=λ(ξ2), thenλ16λ2 and they are equal iffϕ =ρ−µ
in the caseϕ=ρ−µ
−(2θ−1)2 functionR(x(λ),ϕ)is strictly increasing becauseλ1=λ2and interval of decreasing is empty.
It is not guaranteed that bothλ1,λ2∈(0,1)and numerical simulations show that, for example, case 1<λ1<λ2 is quite possible. Note that, inλ =0 relative welfare always increases (see Lemma 11 below) which implies either 0<λ1orλ2<0. Yet the second case is impossible because in No-Black-Hole case for sufficiently smallϕ>0 we have VH
VF(0)>1 andVH
VF(1)<1 (see Lemma 13 in Appendix B), i.e. relative welfare should somewhere decrease betweenλ=0 andλ=1.
Let’s define one more threshold value that could be called “turn point”
ϕT(θ) =
VF(0,ϕ)>0. Relative welfare VH
Proof. Due to Lemma 9 it is sufficient to consider the caseϕ<
s 1−(2θ−1)2
2 instead ofθ we obtain, after some transformations, that
Note that the first multiplicand is positive due to assumptionϕ2< 1−α2 1+µ
Moreover,
f(α,ρ,µ,ϕ)<−(1+α)[(1−µ)((2+ (1−α)ρ)µ−(1−α)ρ) + ((1−α) + (1+α)µ)ρ] becauseϕ2>0, thusQ
A ϕ
<0 if and only if
g(α,ρ,µ) = (1−µ)((2+ (1−α)ρ)µ−(1−α)ρ) + ((1−α) + (1+α)µ)ρ>0.
Routine calculations show that under assumptions 06α61,06µ61,06ρ61 minimum value of polinomial functiong(α,ρ,µ)is equal to 0 and it is reached on the set of points(α,ρ,0)for arbitrary 06α 61,
06ρ61. Therefore for alladmissiblevalues (e.g.µ>0) the termg(α,ρ,µ)is strictly positive. The caseλ =0 is considered.
Now letλ =1 and study the sign ofQ ϕ
B
. After some transformations we get
Q ϕ
B
=(1−µ)ϕ
(1−µ)2(1−α2)− (1+µ)2−(1−µ)2α2 ϕ2
4µ((1−α)(1−µ) + (1+µ+ (1−µ)α)ϕ2)2 h(α,ρ,µ,ϕ), where
h(α,ρ,µ,ϕ) = (1−α)((1+α)(1−µ)ρ−2µ)−(1+α)((1−α)ρ+ (2+ (1+α)ρ)µ)ϕ2. Note that the first multiplicand is strictly positive due to assumption
ϕ2< 1−α2 1+µ
1−µ 2
−α2
= (1−α2)(1−µ)2 (1+µ)2−(1−µ)2α2.
On the other hand, h(α,ρ,µ,ϕ) may positive or negative depending on parameter’s relations. More exactly, h(α,ρ,µ,ϕ)<0 if and only if
ϕ2>(1−α)((1+α)ρ−(2+ (1+α)ρ)µ) (1+α)((1−α)ρ+ (2+ (1+α)ρ)µ). Reverse substitutionα=2θ−1 transforms this inequality into
ϕ2>(1−θ)(θ ρ−(1+θ ρ)µ) θ((1−θ)ρ+ (1+θ ρ)µ). Note that under conditionµ > θ ρ
1+θ ρ numerator is negative and this inequality holds for allϕ.
B Appendix B
B.1 Stability of agglomerated equilibria
Direction of labor migration flow depends on welfare levels in both regions. For example,VH(λ)>VF(λ) for some 0<λ<1 causes in-flow, i.e. increasing ofλ and vice versa. As for limit casesλ =0 orλ =1 we interpret conditionVH(0)<VF(0)(orVH
VF
(0)<1) asstablestate, becauseλ=0 cannot be decreased further, while opposite inequality characterizes unstable situation. Analogously, λ =1 is stable ifVH(1)>VF(1) ⇐⇒ VH
VF
(1)>1.
Note that the same result may be obtained by linearizing ofad-hoc equation ˙λ =λ·(1−λ)(VA(λ)−VB(λ))in neighborhoods of steady statesλ =0 andλ =1.
Recall that by convention the home agricultural labor share satisfies conditionθ>1/2. Consider the following valueα=2θ−1∈[0,1)that may be interpreted asmeasure of asymmetryin agricultural labor.
Lemma 12. In the case ofµ>ρ (Black-Hole condition) both agglomerated statesλ =0andλ =1are stable regardless of other parameter values.
Proof. Note that
This equation allows explicit solution in the casesλ =0 orλ =1.
Forλ =0 we obtain that H(µ,ρ,ϕ,θ)<1. Under Black-Hole conditionµ>ρboth functionGandHincrease with respect toϕ. Moreover G(µ,ρ,1,θ) =H(µ,ρ,1,θ) =1 thusG(µ,ρ,ϕ,θ)<1 andH(µ,ρ,ϕ,θ)<1 for all 0<ϕ<1.
Lemma 13. Suppose thatNo-Black-Hole conditionµ<ρ holds.
Proof. Note that under No-Black-Hole conditionρ>µ we have lim
ϕ→0G(µ,ρ,ϕ,θ) = +∞,G(µ,ρ,1,θ) =1. The
(1−θ(1−µ))(ρ+µ).Moreover, the second derivative
∂2G holds for functionH(ϕ)except that the minimum point value is equal toϕ∗∗=q
(1−θ)(1−µ)(ρ−µ) (1−(1−θ)(1−µ))(ρ+µ).
Recall that we assumeθ>1/2 and considerα=2θ−1 as a measure of agricultural population’ asymmetry.
Substituting θ = 1+α
2 we obtain the following terms of minimum points ϕ∗,ϕ∗∗ as functions of asymmetry measureα
Note thatϕ∗(α)increases with respect toα thus for allα<1 following inequalities hold
ϕ∗(α) = strictly decreases with respect toϕin interval(0,ϕ∗(α))consequently there exist the unique point
ϕ0S(α)∈(0,ϕ∗(α))such thatG(µ,ρ,ϕ0S(α),1+α
2 ) =1. Moreover for all 0<ϕ<ϕ0S(α)inequality G(µ,ρ,ϕ,1+α
2 )>1 holds whileϕ0S(α)<ϕ <1 implies G(µ,ρ,ϕ,1+α
2 )<1. Analogously, there exists the unique valueϕ1S(α)∈(0,ϕ∗∗(α))such thatH(ϕ)<1 if and only ifϕ1S(α)<ϕ<1.
Now consider sustain pointsϕ0S,ϕ1Sas functions of asymmetry measureα =2θ−1∈[0,1). Note that ϕ0S(0) =ϕ1S(0)because functionsG(µ,ρ,ϕ,1+α
2 )andH(µ,ρ,ϕ,1+α
2 )coincide forα =0. For allα ∈(0,1)
and fixedρ>µ sustain point functionϕ0S(α)is an implicit function defined by equation
ρ >0. It implies that an implicit function derivative ∂ ϕ0S
∂ α =−∂G
∂ α ∂G
∂ ϕ >0, i.e.ϕ0Sincreases with respect toα. Analogous considerations show thatϕ1Sdecreases with respect toα.
Now consider the caseµ < ρ
Thus forα<α∗all of the previous considerations still hold. In case ofα>α∗an inequality ϕ∗(α) =