# A3.3 The Circumstance in Which Physical Bequests Are Zero

Suppose the government does not levy any taxes: т = *в _{в} = в_{м} =* 0. If 1

*—*S

*+ h> 1 + r*at M = 0, we have the corner solution where bequests are zero. From Eqs. (5. A4.2) and (5.A15.1), we have

Substituting Eq. (5.A16) into Eq. (5.A4.1^{0}), we have

However, from Eq. (5.A15.2) we have in the steady state

Hence, considering Eqs. (5.A7), (5.A17), and (5.A18), the steady-state physical capital/human capital ratio к is uniquely given as a solution of (5.A19). Thus,

The left-hand side of Eq. (5.A19) increases with k, while the right-hand side of Eq. (5.A19) decreases with k. When e increases, the left-hand side decreases, so that *к* increases. When p decreases, the right-hand side increases, so that *к* increases.

However, considering Eqs. (5.A8.1) and (5.A8.2), 1 — 5 + h> 1+ratM = 0 if and only if

or

When there are less incentives to leave bequests, we may well have the corner solution of M = 0. When e is larger and p is smaller, inequality is more likely (see Eq. (5.A21)).

The laissez-faire growth rate is given by

where *к* is given by Eq. (5.A19). An increase in the intragenerational preference for life cycle capital e raises the physical capital/human capital ratio к, leading to a higher rate of return on human capital and higher economic growth. An increase in the intergenerational preference p has two effects. It stimulates intergenerational transfer from the old to the young, inducing high growth. However, it reduces к* ^{к}* and the rate of return on human capital, h, depressing economic growth.

Considering Eqs. (5.A19) and (5.A22), we have

Thus, if _{1}^{1}^_{h}^{ah}g > p, then|P > 0 (and vice versa). In other words, if *a* and 8 are high, it is likely that |p > 0. However, it should be stressed that |P < 0 is also possible. In the bequest-constrained economy, an increase in the parent’s concern for the child’s welfare does not necessarily raise the growth rate.

Since 1 - 8 + h> 1+r, h>r. Hence, r*y*_{M=0} > *y*:* The laissez-faire growth rate in the constrained equilibrium is too high. If the externality effect is absent (8 = 0), 1 + h > 1 +h*, the laissez-faire growth rate is always too high.

The laissez-faire economy may not attain the first best solution because of two reasons. First, the externality effect in the accumulation of human capital is not considered by the parent. This means that the competitive growth rate becomes too low. Second, M_{t} cannot be negative because there is no institutional mechanism to enforce such a liability on future generations. Human capital is too little and the marginal return of human capital is too high, which means that the competitive growth rate becomes too high. When 8 is lower, it is more likely that the second effect predominates and the laissez-faire growth rate is too high.

# A3.4 The Circumstance in Which Physical Bequests Are Operative

When M > 0, Eqs. (5.A15.2) and (5.A15.3) have equality. Hence, we have

k is given by k, which is defined by the following equation:

Thus, the unconstrained growth rate is given by

I

From Eq. (5.A25), к is independent of p or e. Equation (5.A26) shows that the life cycle saving motive e does not affect the growth rate, while an increase in the transfer-saving motive p definitely raises the growth rate.

Since h > h* and r < r*, we always have

When physical bequests are operative, the competitive economy could be different from the first best solution only because of the externality effect of human capital. Thus, the laissez-faire growth rate is always too low.