Appendix A: Supplementary analysis

A.1 Critical equations determining educational choices and wages

This section examines critical equations determining educational choices and wages, in par- ticular,F_handF_msatisfyingwf_m(_F^Fh

m) =P(F_h,F_m,B)A_T ⇔F_m=ϕ(F_h,B)F_handP(F_h,F_m,B) = θ. Remember that (_F^Fh

m)_hm is _F^Fh

m satisfying wf_h(_F^Fh

m) = wf_m(_F^Fh

m), which exists and is unique since wf_h (wf_m) decreases (increases) with _F^Fh

m and wf_h>(<)wf_m at _F^Fh

m = 0(= +∞) from (11) and (12), and (_F^Fh

m)_ml,θ is _F^Fh

m satisfying wf_m(_F^Fh

m) =θA_T (w_l when P=θ).

Lemma A1 shows the existence of F_h and F_m satisfying wf_m(_F^Fh

m) =P(F_h,F_m,B)A_T when

γB

1−γB(1+r)B < θA_T and describes its shape and its relation with (_F^Fh

m)_hm and (_F^Fh

m)_ml,θ. (When

γB

1−γB(1+r)B≥θAT, P(Fh,Fm,B)> θ from (15) and thus P=θ.)

Figure 9: Lemma A1

Lemma A1 Suppose ₁₋^γB_γ

B(1 +r)B < θA_T. Then, positive F_h and F_m satisfying wf_m(_F^Fh

m) = P(F_h,F_m,B)A_T exists and is expressed as F_m=ϕ(F_h,B)F_h, where ϕ(·) is a function satis- fying lim_Fh→0ϕ(F_h,B) =ϕ(B)≡

·

(1−α)AM

(1+r)(_1−γB^γB B+em)

¸¹

α

. When _F^Fh

m ≤(_F^Fh

m)_hm, ϕ(·) is a decreas- ing function of its arguments, and, for given B, there exists a unique F_h >0 satisfying [ϕ(F_h,B)]⁻¹= (_F^Fh

m)_hm, denoted F_h^‡(B), and the one satisfying [ϕ(F_h,B)]⁻¹= (_F^Fh

m)_ml,θ, denoted F_h^†(B), where F_h^‡(·) and F_h^†(·) are decreasing functions and F_h^‡(B)> F_h^†(B).

Figure 9 illustrates F_m=ϕ(F_h,B)F_h (wf_m(_F^Fh

m) =P(F_h,F_m,B)A_T), _F^Fh

m= (_F^Fh

m)_hm, and _F^Fh

m= (_F^Fh

m)_ml,θon the (F_m, F_h) plane. F_h^‡(B) andF_h^†(B) are unique intersections ofF_m=ϕ(F_h,B)F_h with _F^Fh

m= (_F^Fh

m)_hm and _F^Fh

m= (_F^Fh

m)_ml,θ, respectively. As F_h →0, F_m satisfying F_m=ϕ(F_h,B)F_h approaches 0 (since lim_Fh→0ϕ(F_h,B) =ϕ(B)<∞). _F^Fh

m=_ϕ(F¹

h,B) increases withF_h, thusF_m increases with F_h on the curve for low _F^Fh

m, but the relationship turns negative for high _F^Fh

m. AsB increases,ϕ(F_h,B) decreases, thus the curve shifts leftward and F_h^‡(B) andF_h^†(B) fall.

Lemma A2 describes the shape ofP(F_h,F_m,B) =θ and its relation withF_m=ϕ(F_h,B)F_h. Lemma A2 Suppose ₁₋^γB_γ

B(1+r)B < θA_T. When _F^Fh

m∈[[ϕ(0)]⁻¹,(_F^Fh

m)_hm] ([ϕ(0)]⁻¹ is the small- est _F^Fh

m satisfying F_m=ϕ(F_h,0)F_h), P(F_h,F_m,B) is an increasing function of its arguments.

Given B, for any _F^Fh

m∈[[ϕ(0)]⁻¹,(_F^Fh

m)_hm], F_h and F_m satisfying P(F_h,F_m,B) =θ exist and are unique, and for _F^Fh

m>(<)(_F^Fh

m)ml,θ, Fm<(>)ϕ(Fh,B)Fh when P(Fh,Fm,B) =θ.

A.2 Effects of F

h

, F

m

, and B on welfare, output, and sectoral

composition

This section examines effects of F_h, F_m, and B on aggregate income net of education costs (NI ≡ wf_hL_h+wf_mL_m+w_l(1−L_h−L_m) + (1 +r)B), average utility, aggregate output (Y = Y_M+P Y_T), the share of the modern sector in production (^Y_Y^M), and the sector’s share in basic consumption when P =θ (^C_{P C}^BM

B), where C_BM denotes the amount of good M used for basic consumption. Proofs of the following two propositions are provided in Appendix D posted on the author’s website (http://www.econ.kyoto-u.ac.jp/˜yuki/english.html).

Proposition A1 (Net aggregate income and average utility) Suppose F_h>0.

(i)If _F^Fh

m≥(_F^Fh

m)_hm, NI and average utility increase with F_h+F_m and B.

(ii)Otherwise, (a)If _F^Fh

m∈((_F^Fh

m)_ml,θ,(_F^Fh

m)_hm), they increase with F_h, F_m, and B.

(b)If _F^Fh

m≤(_F^Fh

m)_ml,θ, 1.When ₁₋^γB_γ

B(1+r)B < θA_T and F_h< F_h^†(B), if F_m≥ϕ(F_h,B)F_h, they increase with F_h and B; otherwise, same as (a).

2.Or else, they increase with F_h and B.

Both net aggregate income and average utility increase with B and the proportion(s) of individuals accessible to education for jobs with higher net wages, i.e. F_h+F_mwhenwf_h=wf_m, F_h and F_m when wf_h>wf_m> w_l, andF_h when wf_m=w_l. As for NI and average utility when P=θ, this is because the negative effect throughwf_h orwf_m (except whenwf_h=wf_m> w_l=θA_T or wfh>wfm=wl=θAT) is dominated by positive effects through other wages (except when f

wh=wfm> wl=θAT), proportions of workers with higher net wages, and B. When P < θ, increases in these variables raise P and thus have a negative effect on average utility, but the positive effect through net aggregate income dominates.

Proposition A2 (Aggregate output and sectoral composition) Suppose Fh>0.

(i)When _F^Fh

m≥(_F^Fh

m)_hm, if F_h+F_m<h ^{(1−γB)θAT−γB(1+r)B} γBwgm((_Fm^Fh)_hm)+(1−γB)θAT

i, Y increases with F_h+F_m andB, and ^Y_Y^M increases with ^Fh+F_B ^m; otherwise, they increase with F_h+F_m, and ^C_{P C}^BM

B increases with F_h+F_m and B.

(ii)When _F^Fh

m<(_F^Fh

m)hm, (a)If _F^Fh

m∈((_F^Fh

m)_ml,θ,(_F^Fh

m)_hm), whenP(F_h,F_m,B)≤θ (possible only when ₁^γ₋^B_γ

B(1+r)B < θA_T), Y increases with F_h, F_m, and B, and ^Y_Y^M increases with F_h and F_m and decreases with B; otherwise, they increase with F_h and F_m, and ^C_{P C}^BM

B increases with F_h, F_m, and B.

(b)If _F^Fh

m≤(_F^Fh

m)_ml,θ, 1.When ₁₋^γB_γ

B(1 +r)B < θA_T and F_h< F_h^†(B), if F_m≥ϕ(F_h,B)F_h, Y increases with F_h and B, and ^Y_Y^M decreases with B (depends on Fh too); otherwise, same as (a) when P(Fh,Fm,B)≤θ.

2.Or else, Y and ^Y_Y^M increase with F_h, and ^C_{P C}^BM

B increases with F_h and B.

When P < θ, aggregate output increases with B and the proportion(s) of individuals accessible to education for jobs with higher net wages, as NI and average utility do. In the case of F_m< ϕ(F_h,B)F_h, this is because the increased proportion(s) raises L_h and L_m and shifts production to the more productive modern sector (an increase in Y_M is greater than a decrease inY_T), plus they and B increaseNI, thereby raising the demand for good T and thusP.³⁴ The modern sector’s share in production increases with the proportion(s) (except the caseF_m≥ϕ(F_h,B)F_h of (b) 1, where the effect is ambiguous) but decreases with B.

When P=θ, by contrast, P does not depend onNI and thus Y and ^Y_Y^M are independent ofB (and increase with the proportion(s)). The modern sector too produces goods for basic consumption, i.e. CBM >0, in this case. The proportion of basic consumption supplied by the sector increases with B as well as the proportion(s), because ^C_{P C}^BM

B =^{P CB}_{P C}^{−P YT}

B = 1−_γ^θY_BNI^T and thus it increases with NI and decreases withY_T=A_T(1−L_h−L_m).

A.3 The dynamic equation of B

_t

and its fixed point

This section examines the dynamic equation of B_t,(24), of Section 3.2 and its fixed point.

When _F^Fht

mt≥(_F^Fh

m)_hm, if F_ht+F_mt< ^{(1−γB)θAT−γB(1+r)Bt}

γBwgm((_Fm^Fh)hm)+(1−γB)θAT

and thus P_t< θ, the equation is:

B_t+1= γb

1−γ_B{wf_m((_F^Fh

m)_hm)(F_ht+F_mt)+(1+r)B_t}. (29)

γb

1−γB(1+r)<1 is assumed so that the fixed point for givenF_ht+F_mt exists, which equals:

Bb^∗(F_ht+F_mt) = γ_b

1−γ_B−γ_b(1+r)wf_m((_F^Fh

m)_hm)(F_ht+F_mt). (30) Clearly, when B_t<(>)Bb^∗(F_ht+F_mt),B_t+1>(<)B_t. If F_ht+F_mt≥_γ ^{(1−γB)θAT−γB(1+r)Bt}

Bgwm((_Fm^Fh)hm)+(1−γB)θAT

and thus P_t=θ, the dynamic equation and its fixed point equal:

B_t+1=γ_b{wf_m((_F^Fh

m)_hm)(F_ht+F_mt)+θA_T[1−(F_ht+F_mt)]+(1+r)B_t}, (31) Bb^∗(F_ht+F_mt) = γ_b

1−γ_b(1+r){wf_m((_F^Fh

m)_hm)(F_ht+F_mt)+θA_T[1−(F_ht+F_mt)]}, (32) where Bb^∗(F_ht+F_mt) is an increasing function.

When _F^Fht

mt∈((_F^Fh

m)_ml,θ,(_F^Fh

m)_hm), if P_t=P(F_ht,F_mt,B_t)≤θ, they equal:

B_t+1= γ_b

1−γ_B{[A_M(F_ht)^α(F_mt)^1−α−(1+r)(e_hF_ht+e_mF_mt)]+(1+r)B_t}, (33) B^∗(F_ht,F_mt) = γ_b

1−γ_B−γ_b(1+r){A_M(F_ht)^α(F_mt)^1−α−(1+r)(e_hF_ht+e_mF_mt)}, (34) where B^∗(F_ht,F_mt) is an increasing function. If P(F_ht,F_mt,B_t)> θ (thusP_t=θ), they are:

34In the caseFm≥ϕ(Fh,B)Fhof (b) 1, the effect ofFhonYM is ambiguous and that ofB is negative, but their effects onP YT are positive and dominate.

B_t+1=γ_b{A_M(F_ht)^α(F_mt)^1−α−(1+r)(e_hF_ht+e_mF_mt)+θA_T(1−F_ht−F_mt)+(1+r)B_t}, (35) B^∗(F_ht,F_mt) = γ_b

1−γb(1+r){A_M(F_ht)^α(F_mt)^1−α−(1+r)(e_hF_ht+e_mF_mt)+θA_T(1−F_ht−F_mt)}, (36) where B^∗(F_ht,F_mt) is an increasing function sincewf_ht>wg_mt> w_lt=θA_T.

When _F^Fht

mt ≤(_F^Fh

m)_ml,θ, ₁₋^γB_γ

B(1 +r)B_t< θA_T, and F_ht< F_h^†(B_t), if F_mt< ϕ(F_ht,B_t)F_ht, the equations are (33) and (34) above. If F_mt≥ϕ(F_ht,B_t)F_ht, the dynamic equation is:

B_t+1= γ_b 1−γ_B

©£A_M(ϕ(F_ht,B_t))^1−α−(1+r)(e_h+ϕ(F_ht,B_t)e_m)¤

F_ht+(1+r)B_tª

. (37) The next lemma shows that, given Fht, Bt converges monotonically to the unique fixed point of (37), B^∗(F_ht), and B^∗(F_ht) increases and ϕ(F_ht,B^∗(F_ht)) decreases with F_ht.

Lemma A3When the dynamics of B_t follow (37), given F_ht, B_t converges monotonically to unique B^∗(F_ht), which is a solution to

B^∗(Fht) = γ_b

1−γ_B−γ_b(1+r){AM(ϕ(Fht,B^∗(Fht)))^1−α−(1+r)(eh+ϕ(Fht,B^∗(Fht))em)Fht}, (38) and when B_t<(>)B^∗(F_ht), B_t+1>(<)B_t. B^∗(F_ht) is increasing and ϕ(F_ht,B^∗(F_ht)) is decreas- ing in F_ht and lim_Fht→0ϕ(F_ht,B^∗(F_ht)) =ϕ(0)≡lim_Fht→0ϕ(F_ht,0).

When _F^Fht

mt≤(_F^Fh

m)_ml,θand either ₁₋^γB_γ

B(1+r)B_t< θA_T andF_ht≥F_h^†(B_t) or ₁₋^γB_γ

B(1+r)B_t≥θA_T, B_t+1=γ_b{fw_h((_F^Fh

m)_ml,θ)F_ht+θA_T(1−F_ht)+(1+r)B_t}, (39) B^∗(F_ht) = γ_b

1−γ_b(1+r){wf_h((_F^Fh

m)_ml,θ)F_ht+θA_T(1−F_ht)}, (40) where B^∗(F_ht) is an increasing function.

A.4 Welfare, output, and sectoral composition in steady states

The next proposition examines the steady states in terms of welfare, output, and sectoral composition, based on Propositions A1 and A2 and Proposition 3 of Section 4.1.

Proposition A3 (Welfare, output, and sectoral composition in steady states) (i)Aggregate net income and average utility are highest in SS 1. They increase with F_h in

SS 2 and SS 3, and with F_h and F_m in SS 4. Their maxima in SS 2 and SS 3 are strictly higher than in SS 4, and the infinima in SS 2 are strictly higher than in SS 3 and SS 4.

(ii)The same result as (i) holds for aggregate output, except that the magnitude relation of the maxima in SS 3 and SS 4 is unclear. In SS 1, ^Y_Y^M =^C_{P C}^BM

B = 1. In SS 2, if F_h< F_h^†, ^Y_Y^M increases (decreases) with _F^Fh

m= [ϕ(F_h,B^∗(F_h))]⁻¹ for [ϕ(F_h,B^∗(F_h))]⁻¹>(<)₁₋^α_α^e_e^m

h, where

1−αα em

eh >wf_m⁻¹

h1−γb(1+r) γb e_m

i

; if F_h≥F_h^† and _1−F^Fh

h≤(_F^Fh

m)_ml,θ, ^Y_Y^M and ^C_{P C}^BM

B increase with F_h; otherwise, ^Y_Y^M =^C_{P C}^BM

B = 1. In SS 3, ^Y_Y^M is constant. In SS 4, ^Y_Y^M increases (decreases) with

F_h

Fm for _F^Fh

m>(<)₁₋^α_α^e_e^m

h.³⁵

35CBM= 0 in the caseFh< F_h^† of SS 2 and in SS 3 and SS 4.

The proposition proves that SS 1 is the best in terms of aggregate net income, average utility, and aggregate output. Other steady states cannot be ranked definitely, but if they are to be ranked, SS 2 is the second best, SS 3 follows, and SS 4 is the worst: the maximum values of these variables in SS 2 and SS 3 (except aggregate output in SS 3) are strictly higher than the ones in SS 4, and the infinima in SS 2 are strictly higher than the ones in SS 3 and SS 4. The three variables increase with the proportion(s) of those accessible to education for jobs with higher net wages, i.e. F_h in SS 2 and SS 3, and F_h and F_m in SS 4.

As for shares of the modern sector in production and in basic consumption, when P < θ (thus ^C_{P C}^BM

B = 0), ^Y_Y^M depends on _F^Fh

m and the relation can benon-monotonic: in the caseFh<

F_h^†of SS 2 and in SS 4, ^Y_Y^M decreases with _F^Fh

m for _F^Fh

m<₁₋^α_α^e_e^m

h (note ₁₋^α_α^e_e^m

h>wf_m⁻¹

h1−γb(1+r) γb e_m

i ) and the relation turns positive for _F^Fh

m > ₁₋^α_α^e_e^m

h if ₁₋^α_α^e_e^m

h <wf_m⁻¹

h1−γb(1+r) γb e_h

i

. That is, the production sharedecreases with _F^Fh

m when _F^Fh

m is relatively low. By contrast, when P=θ,i.e.

in the case Fh≥F_h^† and ₁₋^Fh_F

h≤(_F^Fh

m)ml,θ of SS 2, ^Y_Y^M and ^C_{P C}^BM

B increase with Fh. (They equal 1 in SS 1 and in the case ₁₋^Fh_F

h>(_F^Fh

m)_ml,θ of SS 2; ^Y_Y^M(<1) is constant and ^C_{P C}^BM

B = 0 in SS 3.)

A.5 Relationship between initial conditions and steady states

The next proposition presents the relationship between initial conditions and steady states.

Since the lengthy analysis of the dynamics is involved, the proof is provided in Appendix C posted on the author’s website (http://www.econ.kyoto-u.ac.jp/˜yuki/english.html).

Proposition A4 (Initial conditions and steady states) (i)When _F^Fh0

m0<wf_m⁻¹

h1−γb(1+r) γ_b e_m

i

a.If F_h0<F_h^♭, F_ht is constant, F_mt falls, and the economy most likely converges to SS 4.³⁶ b.If F_h0≥F_h^♭,when F_h0≥F_h^♭(B₀), F_ht is constant, F_mt increases, and the economy converges

to SS 2.³⁷ When Fh0< F_h^♭(B0), at first, Fht is constant and Fmt decreases, and it could converge to any type of steady states or cycle.³⁸

(ii)When _F^Fh0

m0∈h f w_m⁻¹

h1−γ_b(1+r) γb e_m

i ,wf_m⁻¹

h1−γ_b(1+r) γb e_h

ii

a.If b^∗(w_l)≤e_m at (F_h,F_m,B) = (F_h0,F_m0,B^∗(F_h0,F_m0)), F_ht and F_mt are constant and the final state is SS 4.

b.Otherwise, F_ht is constant, F_mt rises, and the economy converges to SS 2.

(iii)When _F^Fh0

m0>wf_m⁻¹

h1−γb(1+r) γ_b e_h

i

, F_ht increases and F_ht+F_mt non-decreases at first.

36 F_mt could ”jump over” the region _F^Fh

m∈h g w_m⁻¹

h1−γ_b(1+r) γ_b e_m

i ,gw_m⁻¹

h1−γ_b(1+r) γ_b e_h

ii

depending on the initial distribution, in which case it converges to another type of steady states, particularly SS 3.

37The exception is whenFh0=F_h^♭ andB0=B^∗(Fh0), in which case bothFmt andBtare constant.

38The economy possibly cycles between the region _F^Fh

m <gwm−1h

1−γ_b(1+r) γ_b em

i

andFh∈[F_h^♭, F_h^♭(B)) and the region _F^Fh

m ∈h g wm−1h

1−γ_b(1+r) γb em

i

,gwm−1h

1−γ_b(1+r) γb eh

ii .

a.If _F^Fh0

m0≥(_F^Fh

m)_hm and b^∗(w_l)≤e_m at (F_h,F_m) = (F_h0,F_m0)and B =Bb^∗(F_h0+F_m0), F_ht+F_mt is constant and the economy converges to SS 3.

b.If _F^Fh0

m0 < (_F^Fh

m)_hm and b^∗(w_l) ≤ e_m at (F_h,F_m) = (F_h0,F_m0) and B = B^∗(F_h0,F_m0), the following three scenarios are possible depending on details of the initial distribution.

1.The more likely is the same scenario as a.

2. F_ht+F_mt rises from the start or after some period and the final state is SS 1.

3.After F_ht+F_mt increases for a while, F_ht becomes constant, F_mt increases, and the economy converges to SS 2.

The first scenario is more likely as Fh0 and Fm0 are lower, and the second one is more likely than the third one as _F^Fh0

m0 is higher.

c.Otherwise, the same scenarios as 2. and 3. of b. are possible.

ドキュメント内 Education, Inequality, and Development in a Dual Economy (ページ 32-38)