If its size is not large, but the non-poor are scarce relative to the poor, the share of medium-skilled workers and the share of the modern sector increase, and the inequality between medium-skilled and low-skilled workers increases over time. time off. 10Note, however, that the economy can also converge to the second and third types of steady states, depending on the details of the initial distribution. Furthermore, they cannot capture with development the shift of production of goods for basic consumption from the traditional sector to the modern sector, which is universally observed in the real economy: in Yuki's models, the traditional sector remains even in the best steady state .

## Equilibrium educational choices and wages

### Critical equations determining educational choices and wages

As for Fh and Fm satisfying wfm(FFh . m) =P(Fh,Fm,B)AT (wl when P <θ), Lemma A1 of Appendix A examines its existence and properties. Therefore, the relative positions of P(Fh,Fm,B) =θ to these locations are important, which is investigated in Lemma A2 of Appendix A.

### Educational choices and wages

Hence, the demand for good T, its relative price, and the low-skill wage are low and so Lm=Fm holds. In contrast, when Fh is not small, the low-skill wage is equal to the net middle-skill wage and some of the poor do not take basic education.23.

## 3 Dynamics

### Dynamics of individual transfers

Fm< ϕ(Fh,B)Fh, the result is the same as in the previous case, but if Fh is higher, the demand for good T (and thus P) is high enough that wfm=wl holds. The equations show that the dynamics of transfers within the line depend on the time development of wages, which are determined by the dynamics of Fht, Fmt and Bt.

### Aggregate dynamics

*Dynamics of aggregate transfers*

Positions of Fht and Fmt relative to the three loci determine the directions of movement of the two variables. In areas with only horizontal arrows, only Fmt changes: for example, in the area under b∗(wfm) =em, b∗(wfm)< em and thus Fmt decreases. Arrows with slope −1 are present in the region above b∗(wfm) =eh and on or below b∗(wl) =em because b∗(wfm)> eh and b∗(wl)≤em and thus Fht increases with Fht+Fmt constant.

In the region above b∗(wl) =em and b∗(wfm) =eh (thus b∗(wl)> em and b∗(wfm)> eh) and below Fh+Fm= 1, both arrows with slope − 1 and the horizontal arrows are drawn, as Fht and Fht+Fmt increase, but the direction of movement of Fmt is unclear (Fht and Fmt move in the direction between the two arrows). Finally, both Fht and Fmt are constant and thus no arrows are present in the region above or below b∗(wfm) =eh and b∗(wl) =em and above or above b∗(wfm) =em. Thus, the dynamics of Fht and Fmt should be examined together with those of Bt.

Before examining the dynamics of the joint, the dynamic equation of Bt is derived, and then the direction of motion of Bt is examined for given Fht and Fmt. It is shown that the equation varies with Fht and Fmt, and for given Fht and Fmt, the direction of movement of Bt is determined by the ratio of the size of Bt to a fixed point: Bt increases (decreases) as it is smaller (larger). ) as a value at a fixed point.

### Joint dynamics of the aggregate variables

Bt+1=γb{wfhtLht+wgmtLmt+wlt(1−Lht−Lmt)+(1+r)Bt}, (24) where the expression within the curly bracket is the total income minus education costs, which can be expressed as a function of Fht, Fmt and Bt. The assumption states that the initial level of total transfers is lower than the fixed point level at (Fh,Fm) = (Fh0,Fm0), that is, the initial capital accumulation is not very large. Therefore, the shapes of these loci are similar to the case of constant B, and their positions on the (Fh,Fm) plane can be illustrated by a figure similar to Figure 4.

## 4 Main Results

*Characteristics of steady states**Relationship between initial conditions and steady states**Productivity growth**Policy implications**Discussions**Role of physical capital accumulation**Role of population growth*

Somewhat consistent with a finding by La Porta and Shleifer (2008), in SS 2 and SS 4, the share of traditional sector output increases with FFh. If the size of the extreme poor is not large, but the non-poor are few relative to the poor, i.e., the lack of adequate numbers of highly skilled workers usually inhibits the growth of the modern sector and thus the traditional sector continues to supply of goods for basic consumption (SS 2).

Hippe and Baten (2012) also find a negative relationship between land inequality and census development for European regions in the 19th and first decades of the 20th century. There are two types of steady states similar to SS 2 and SS 4 of the original economy, where convergence to the former type is more likely as Fh0 and Fm0 are higher. The productivity of the traditional sector is less affected by the progress of science and technology, but it would increase slowly in the real economy, so the assumption may not stand far in the past or in the future.

The article emphasizes the importance of the initial distribution of wealth in determining the accumulation of human capital and structural changes in the economy, which is supported by the empirical studies listed in chapter 4.2. If the size of the extreme poor is not large, but the non-poor is small relative to the poor, and the economy is therefore on the way to SS 2, subsidizing higher education should be prioritized, which lowers eh and shifts b* (wfm) = eh down (see Figure 6 ). In contrast, raising the productivity of the traditional sector is not very efficient, as the analysis in Section 4.3 suggests that growth in AT does not affect the rate of convergence to SS 1 (unless the initial condition is very good).

Because the relative productivity of the modern sector is low, the sector cannot generate a sufficient number of jobs for educated workers, and the traditional sector usually absorbs uneducated workers.

## 5 Conclusion

If population growth is fast and therefore γb is very low, the dynamics could be illustrated by a diagram similar to that of the low AM case, Figure 7, where the best equilibrium state does not exist. Therefore, full modernization of the economy may not be possible while population growth is rapid.

Vollrath (2009), "Land Ownership Inequality, the Emergence of Human Capital Promoting Institutions, and Great Divergence", Review of Economic Studies. Baten (2012), "Keep Them Ignorant." mimeo, University of Tuebingen. 23] McDonald, Stuart and Jie Zhang (2012), “Income Inequality and Economic Growth with Altruistic Inheritance and Human Capital Investment,” Macroeconomic Dynamics 16 (S3), 331−354.

1994), North-South Trade, Employment and Inequality: Changing Fortunes in a Skills-Driven World, Oxford University Press, Oxford.

## Appendix A: Supplementary analysis

*Critical equations determining educational choices and wages**Eﬀects of F h , F m , and B on welfare, output, and sectoral composition**The dynamic equation of B t and its ﬁxed point**Welfare, output, and sectoral composition in steady states**Relationship between initial conditions and steady states*

This section examines the effects of Fh, Fm and B on total income after deducting education costs (NI ≡ wfhLh+wfmLm+wl(1−Lh−Lm) + (1 +r)B), average utility, total output (Y = YM+P YT), the share of the modern sector in production (YYM), and the share of the sector in basic consumption when P = θ (CP CBM . B), where CBM denotes the amount of good M that used for basic consumption. Both net total income and average utility increase with B and the number of individuals accessible to education for jobs with higher net wages, i.e. B(1+r)B < θAT), Y increases with Fh, Fm and B, and YJM increases with Fh and Fm and decreases with B; otherwise they increase with Fh and Fm, and CP CBM.

When P < θ, aggregate output increases with B and the proportion(s) of individuals accessible to education for jobs with higher net wages, as do NI and the average society. Proposition A3 (Welfare, output and sectoral composition in the steady state) (i) Total net income and average utility are highest in SS 1. They increase with Fh in. Their maxima in SS 2 and SS 3 are strictly higher than in SS 4 , and the infinities in SS 2 are strictly higher than in SS 3 and SS 4. ii) The same result as (i) holds for aggregate production , except that the relation of the magnitude of the maxima in SS 3 and SS 4 is unclear.

Fht+Fmt rises initially or after a period, and the final state is SS 1. 3. After Fht+Fmt rises for a while, Fht becomes constant, Fmt rises, and the economy converges to SS 2.

## Appendix B: Proofs of lemmas and propositions

The first scenario is more likely because Fh0 and Fm0 are lower, and the second is more likely than the third than FFh0. Since Fh>0, an equilibrium with Lh, Lm>0 always exists from the form of the production functions. In the first case, Lh≤Fh, Lh+Lm=Fh+Fm and LLh. m≤FFmh, and in the latter Lh=Fh,Lm≤Fm and LLh. i) wfm =wl is not possible because wfh > wfm and LLh.

Since the LHS decreases with ϕ and the RHS and its denominator increase with ϕ, its numerator increases with Bt. Thus, the numerator of the RHS of (37) is positive at Bt = 0 and increasing in Bt. In the steady state, the relative positions of the critical loci that determine the dynamics of Fh and Fm and the magnitude ratio of P and θ are shown in Figure 5.

In the region satisfying b∗(wfm)> eh and b∗(wl)> em of the figure, Fh and Fh+Fm increase when Fh <1, so Fh <1 cannot be a steady state. In the region satisfying b∗(wfm)> eh and b∗(wl)≤em, Fh increases and Fm decreases when Fm >0, so steady states are such that Fm= 0 and Fh satisfies b∗(wl )≤em⇔ Fh≤. Levels of Lh, Lm and Ll, and wages are from Propositions 1 and 2 and the result onP. i) From Proposition A1 (i), total net income (NI) and average utility of SS 1 are strictly greater than those of SS 3, and it increases with Fh in SS 3 (B =Bb∗(Fh) from Proposition 3 . ).