Appendix: Proof of Propositions
Proof of Proposition 1:
To write down the market clearing condition by aggregating all consumers’ consumptions and incomes, we have that aggregate consumption is:
and aggregate income (production):
where the first integral on the right hand side is income of the young generation and the second one is the income of the old generation. Since in steady state ht and ct, t+τ grow at the same rate gC (for τ=0 or 1), we can write the equilibrium condition as follows:
On the other hand, note that the resource constraint or goods market clearing condition of this economy can be also derived from the sum of budget constraints of all individuals. Integrating equation (17) over all agents we have that:
Comparing equations (A.4) with (A.3), it can be seen that both expressions hold only if gc=r. ∥
Proof of Proposition 2:
The first part of the proof establishes that ∂(Δg)/∂m > 0 within the appropriate range of the parameters. To decide the range, the following information is used. First, it is obvious that b, which represents a variance, is non-negative. Second, since we assume all individuals in the non credit market have a positive investment on education, the following inequality holds; 12/
It is easy to verify that for any b the above expression valued at the minimum m (=b/2+1/β) is greater than zero. Now it can be also shown that, for any given b, ∂χ/∂m > 0. Hence χ is always positive within the above parameter range. Given that Δg is increasing in m, to complete the proof that the growth effect is positive it is enough to show that for the minimum values of m (depending on b) the expression Δg is greater than zero. For this we first consider b=0, which is the minimum b. In this case the minimum value of m is 1/β, and hence the last term at the right hand side of δ g is zero, and the expression in square brackets is greater than 1+b, which implies that Δg valued at the minimum m and b is positive. It can be also shown that Δg valued at the minimum m is increasing in b, which implies that Δg valued at the minimum m’s (for any b) are positive.
Then it follows that for any b the growth effect of credit markets is positive. ∥
Proof of Proposition 3:
Given the same initial conditions, figure 2 shows that individuals allowed to borrow and lend have higher (or equal for j*) utility than those not allowed. In addition, the economy with credit market has a higher rate of growth (proposition 1) consequently all current and future generations are better off with credit markets, except individual j* in the first period. ∥
Proof of Proposition 4:
(i) For 0 < j ≤ j*,
Condition (36) insures that the “normalized” indirect utility
(ii) For j*≤ j < 1,
Using the expressions for consumption of agent j, above j*, we can derive the following:
After some algebra, it can be shown that the above expression is monotonically increasing in j and negative at j*. Since
it follows that there exists a unique value j’ ∊ (j*, 1) that satisfies the following relationships:
For j < j’
and for j>j’
which implies that the slope of the Lorenz curve in the case of credit market is flatter around j = j* (up to j’) and steeper around j = 1, and furthermore, that the Lorenz curve in the case of credit market is below that of non-credit market.
It follows from (i) and (ii) that Lorenz curve in the presence of credit markets is below that in the absence of credit markets. Then, utility inequality indices, such as Gini coefficients, are larger in the presence of credit markets. ∥
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We are grateful to Mohsin Khan, Ken Kletzer, Don Mathieson, Gian-Maria Milesi-Ferretti and Federico Sturzenegger for valuable comments.
For overviews of the empirical and theoretical literature see Fry (1993), Greenwood and Smith (1993), Pagano (1993), De Gregorio and Guidotti (1993), and King and Levine (1993). The last two papers also present empirical evidence.
The model assumes that in the absence of credit markets individuals cannot save and cannot borrow. The existence of a storage technology, through which individuals could save, could be easily introduced in the model, but would unnecessarily complicate it.
Galor and Tsiddon (1994) call this type of externality global technological externality, and distinguish it from home environment externality, by which parents affect directly the ability of their children to increase skills. They show that the two types of externalities have opposite implications for the evolution of income distribution.
In order for all individuals to have positive νj we assume that β(m-b/2)>1. Relaxing this assumption does not change the main results of the model. If we suppose there are some individuals with δj<1/β, they will decide not to accumulate human capital in both cases, with and without credit markets. In this case, however, the positive growth and welfare effects discussed in sections IV and V still hold.
The transitional dynamics off the balanced growth path are complicated. However, we can show that under some mild assumptions the steady state is stable.
Since νj=1 for j>j* and zero otherwise, total time spent in education is equal to 1-j*.
For presentation purposes we have rescaled all simulations to start at 100 when b equals 0. Nevertheless, as we formally show in proposition 1 growth differentials are increasing in m for all values of b, and hence, the figure only reveals the shapes of the relationship between b and Δg, and not the levels.
The analysis for j* close to 1 is analogous, and it will not be discussed further.
One alternative, for example, is that as countries develop and integrate to the world economy m and b may catch-up the world levels.
Relaxing this assumption does not change the main results of the proposition. For example, under the very weak assumption that education does not have a negative effect on human capital accumulation (that is, all δj≥0), it can be shown that the proposition still holds.