# A2 The Endogenous Growth Model

## A2.1 Technology

A general feature of standard models of endogenous growth is the presence of constant or increasing returns in physical capital and human capital. Firms act competitively and use a constant-returns-to-scale technology.

where Y is output, K is physical capital, and H is human capital. A is a productivity parameter which is taken here to be multiplicative and to capture the idea of endogenous growth in accordance with Rebelo (1991).

## A2.2 The Three-Period Overlapping-Generations Model

In order to make the point clear, consider a three-period overlapping-generations model similar to those of Batina (1987), Jones and Manuelli (1990), Caballe (1995), and Buiter and Kletzer (1993). The number of households of each generation, n, is normalized to one. In period t — 1, when the household of generation t is young, the parent of generation t — 1 can choose to spend private resources other than time on human capital formation of her or his child, B_{t— 1}, and physical savings (bequests) for her or his child, M_{t— 1}.

The stock of human capital used by generation t during period t, H_{t}, is assumed to be a sum of a function of transfer input, *B _{t—}* 1, and the average level of human capital achieved by the prior generation,

*H*

_{t—1}.

where S *=* 1 — П. H_{t} is the ratio of the others’ human capital to the total number of people. n is the total number of individuals of each generation. The first term reflects the effect of the parent’s own human capital on the average human capital and the second term reflects the effect of the others’ human capital on the average level. When *n* !i, the parent would not recognize the externality effect of her or his own capital, and hence the externality effect of human capital is perfect. When n = 1, she or he considers her or his own capital and the average level as equivalent; the externality effect is absent. Thus, 5 may be regarded as the degree of externality. This extra term, *H* _{t—1}, embodies a similar kind of externality as in Romer (1986), and reflects the fact that production is a social activity.

Thus, we have

All human capital is inherited either genetically or through educational expenditure B by parents. The externality effect in the accumulation of human capital is not fully considered by parents when they decide how much to invest in their children’s education.

During middle age, the household choice of generation t concerns how much to consume, c^{1}; to save for old age, *s _{t};* to save for her or his child,

*M*and to spend on the human capital formation of her or his child,

_{t};*B*

_{t}. The entire endowment of labor time services in efficiency units

*H*is supplied inelastically in the labor market. Thus, wage income

_{t}*h*is obtained where

_{t}H_{t}*h*is the wage rate. In the last period of life (“old age” or “retirement”), households do not work or educate themselves, and consume c2

_{t}_{+1}.

The government imposes taxes on capital accumulation and tax revenue is returned as a lump sum transfer to the same generation. This is a standard assumption of the differential incidence. Otherwise, the tax policy would include the intergenerational redistribution effect such as debt issuance or unfunded social security.

Thus, the middle-age budget constraint is given by

I

Substituting (5.A3) into (5.A4.1), we have The old-age budget constraint is given by

I

where *6 _{B}* is a tax on income from human capital (a wage income tax),

*Q*is a tax on physical bequests, т is a tax on income from life cycle physical capital (an interest income tax), R

_{M}^{1}is a lump sum transfer to the young in period t, and R

^{2}is a lump sum transfer to the old in period t.

The government budget constraints with respect to generation t for the middle- age period t and the old-age period t +1 are given respectively by

Taxes on human capital accumulation are represented by taxes on wage income, *e _{B}h_{t}H_{t}.* Note that from Eq. (5.A2),

*H*holds in the aggregate economy.

_{t}= H_{t}The feasibility condition in the aggregate economy is given by

Physical capital accumulation is given by

Recall that both life cycle saving and bequests provide funds for physical capital accumulation in the aggregate economy. Note also that human capital accumulation is given by Eqs. (5.A2) and (5.A3). The rates of return on the two types of capital are given respectively by

where *к = K/H* is the physical capital/human capital ratio.

## A2.3 The Altruistic Bequest Motive

An individual born at time t — 1 consumes *cj* in period t and c^_{+1} in period t + 1, and derives utility from her or his own consumption. Thus,

I

Here, e reflects the private preference of old-age consumption or life cycle savings. For simplicity, we assume a log-linear form throughout this appendix. The qualitative results would be the same in a more general functional form.

In the altruism model, the parent cares about the welfare of her or his offspring. The parent’s utility function is given by

0

p reflects the parent’s concern for the child’s well-being.