Inter-Vivos Transfers and Dynastic Bequest Model

[alanlujan91]

Summary

Most intergenerational transfers happen during the parent's lifetime, not at death: gifts for down payments, education funding, business financing, and strategic estate tax planning (McGarry 1999). HARK's BequestWarmGlowConsumerType includes a terminal warm-glow bequest motive, but all its transfers occur at death. These living transfers interact with estate tax policy (annual exclusion, lifetime exemption), the child's consumption and saving decisions (moral hazard), and the parent's own retirement security. A dynastic model with both inter-vivos transfers and bequests would address intergenerational wealth transmission, inequality persistence, and the efficiency of gift/estate tax policy.

Connection to Financial Services Practice

The wealth-management plugin financial-plan skill (wealth-management/skills/financial-plan/SKILL.md) includes a full estate planning module: estate tax exposure analysis (federal and state), gifting strategy (annual exclusion of $18,000 per donee, lifetime exemption of $13.61M), trust structures, and beneficiary review. The skill identifies estate document updates and gifting strategy optimization as top-priority action items. The financial-plan skill also models education funding through 529 plans, a tax-advantaged inter-vivos transfer to children. A HARK dynastic model would answer the questions these heuristics leave open: when should parents give during life vs. at death? How do gift and estate taxes affect the timing and size of transfers? How does the child's saving respond to expected parental transfers?

Mathematical Model

Environment

Time is discrete. We model overlapping generations of dynasties, each consisting of a parent and a child. The parent lives for $T^p$ periods (ages $t^p_0, \ldots, T^p$), and the child enters the model at the parent's age $t^{overlap}$ (the overlapping period begins). During the overlap, the parent observes the child's state and can make transfers. At the parent's death, remaining wealth is bequeathed.

State Variables

The family state during the overlapping period is $(a_t^p, a_t^c, z_t^p, z_t^c, t)$, where we denote by $a_t^p$ the parent's wealth, by $a_t^c$ the child's wealth, by $z_t^p$ and $z_t^c$ their respective income states, and by $t$ the parent's age.

Before the child enters the model ($t < t^{\text{overlap}}$), the parent solves a standard lifecycle problem. After the parent dies ($t > T^p$), the child solves independently with the inherited bequest as initial wealth.

Preferences

Parent's utility:

$$U^p = \mathbb{E} \sum_{t=t^p_0}^{T^p} \beta^{t - t_0^p} \varsigma_t^p, u(c_t^p) + \sum_{t=t^{\text{overlap}}}^{T^p} \beta^{t - t_0^p} \varsigma_t^p, \phi^{iv}, v(g_t) + \beta^{T^p+1 - t_0^p} \phi^{bq}, \Phi(a^p_{T^p+1})$$

The parent derives utility from own consumption $c_t^p$, from inter-vivos gifts $g_t \geq 0$ with weight $\phi^{iv}$ and gift utility function $v(\cdot)$, and from the terminal bequest with weight $\phi^{bq}$ and bequest utility $\Phi(\cdot)$. Here $\varsigma_t^p$ is the parent's conditional one-period survival probability at age $t$, and discounting is normalized so that $t_0^p = 0$ in practice.

The gift utility function follows De Nardi (2004):

$$v(g) = \phi_1 \frac{(g + \phi_2)^{1-\rho}}{1-\rho}$$

where the shifter $\phi_2 > 0$ ensures that the marginal utility of giving is finite at $g = 0$ (so that transfers are zero for sufficiently poor parents). The bequest utility is similarly:

$$\Phi(a) = \phi_b \frac{(a + \underline{a})^{1-\rho}}{1-\rho}$$

Child's utility: The child maximizes their own lifecycle utility independently. A key modeling choice is whether the parent is altruistic (cares about the child's utility $V^c$) or paternalistic (cares about the gift itself via $v(g)$). The paternalistic / "joy of giving" specification is more tractable because it avoids the need to solve the child's problem within the parent's optimization.

Gift and Estate Tax System

Annual gift tax exclusion: Each year, the parent can give up to $\bar{g}$ (currently $18,000) per donee without tax consequences. The taxable gift in period $t$ is $\tilde{g}_t = \max(g_t - \bar{g}, 0)$, which is the amount exceeding the annual exclusion. Actual gift tax is owed only when cumulative taxable gifts exceed the lifetime exemption.

An important institutional detail: the gift tax is tax-exclusive (the donor pays $\tau^g$ on each taxable dollar given, so transferring $1 costs $1 + \tau^g$), while the estate tax is tax-inclusive (the tax comes from the estate, so a $1 estate yields $(1-\tau^e)$ to the heir). At the same statutory rate, gifts are marginally cheaper, as noted by Poterba (2001).

Lifetime exemption: Cumulative taxable gifts (above the annual exclusion) consume the lifetime exemption $\bar{E}$ (currently $13.61M). We track cumulative taxable gifts $G_t^{\text{cum}}$:

$$G_{t+1}^{\text{cum}} = G_t^{\text{cum}} + \tilde{g}_t = G_t^{\text{cum}} + \max(g_t - \bar{g}, 0)$$

Actual gift tax is owed only when cumulative gifts exceed $\bar{E}$:

$$\text{tax}_t(g, G^{\text{cum}}) = \tau^g \cdot \max\bigl(G_t^{\text{cum}} + \tilde{g}_t - \bar{E}, ; 0\bigr) - \text{gift tax previously paid}$$

Estate tax: At death, the estate tax uses the current $G^{\text{cum}}$ value to determine the remaining exemption:

$$\tau^{\text{estate}}(a^p, G^{\text{cum}}) = \tau^e \cdot \max\bigl(a^p - (\bar{E} - G^{\text{cum}}), ; 0\bigr)$$

The remaining exemption $\bar{E} - G^{\text{cum}}$ reflects lifetime gifts already made. This creates the fundamental trade-off: gifts made today reduce the estate tax exemption available at death, but the child earns returns on the transferred wealth in the interim.

Recursive Formulation (Parent, During Overlapping Period)

Paternalistic (joy-of-giving) specification:

$$V_t^p(a^p, z^p, G^{\text{cum}}) = \max_{c^p,, g,, a^{p\prime}} ; u(c^p) + \phi^{iv}, v(g) + \beta, \varsigma_t^p; \mathbb{E}\bigl[V_{t+1}^p(a^{p\prime}, z^{p\prime}, G^{\prime\text{cum}})\bigr] + \beta,(1-\varsigma_t^p), \phi^{bq}, \Phi\bigl(a^{p\prime} - \tau^{\text{estate}}(a^{p\prime}, G^{\text{cum}})\bigr)$$

subject to

$$c^p + g + \text{tax}_t(g, G^{\text{cum}}) + a^{p\prime} = R \cdot a^p + Y^p(z^p, t)$$

$$G^{\prime\text{cum}} = G^{\text{cum}} + \max(g - \bar{g}, 0)$$

$$g \geq 0, \quad a^{p\prime} \geq 0$$

where $\text{tax}_t(g, G^{\text{cum}}) = \tau^g \max(G^{\text{cum}} + \tilde{g}_t - \bar{E}, 0) - \text{tax previously paid}$. For most non-wealthy households, $G^{\text{cum}} < \bar{E}$ throughout life, so no gift tax is owed and gifts within the annual exclusion incur zero tax cost. In this regime, $G^{\text{cum}}$ can be dropped from the state.

The parent's optimization is self-contained: the parent chooses how much to give without solving the child's problem. The child receives $g_t$ as a lump-sum addition to their own cash-on-hand.

Altruistic specification (more ambitious):

$$V_t^p(a^p, a^c, z^p, z^c) = \max_{c^p, g, a^{p\prime}} ; u(c^p) + \phi^{alt}, V^c_t(a^c + g, z^c) + \beta, \varsigma_t^p; \mathbb{E}[V_{t+1}^p(\cdot)]$$

Here the parent internalizes how the gift $g$ affects the child's continuation value $V^c_t$. This creates a game between parent and child (the child's saving decision responds to expected transfers, and the parent anticipates this), leading to potential moral hazard and a Samaritan's dilemma. The altruistic specification requires joint solution of both problems.

Child's Problem

The child solves a standard consumption-saving problem, taking parental transfers as given:

$$V_t^c(m^c, z^c) = \max_{c^c,, a^{c\prime} \geq 0} u(c^c) + \beta \varsigma_t^c \mathbb{E}[V_{t+1}^c(m^{c\prime}, z^{c\prime})]$$

subject to $c^c + a^{c\prime} = m^c$ and $m^{c\prime} = R a^{c\prime} + Y^c(z^{c\prime}, t+1) + g_{t+1}$, where $\varsigma_t^c$ is the child's conditional one-period survival probability at age $t$. Cash-on-hand $m^c_t = R a^c_{t-1} + Y^c(z^c, t) + g_t$ includes the parental transfer.

Key Model Properties

Timing of transfers: The model predicts that transfers should be front-loaded (given early, so the child earns returns on the transferred wealth) if the child's marginal utility of wealth is high and the parent faces no liquidity constraints. Estate taxes amplify this: because gifts within the annual exclusion are tax-free, a long series of annual gifts dominates a single bequest.

Transfer-wealth gradient: The model generates the empirical pattern that richer parents give more (both inter-vivos and at death), but the ratio of transfers to wealth is non-monotone. For estates well above the lifetime exemption $\bar{E}$, the gift tax is tax-exclusive (the donor pays tax on top of the gift) while the estate tax is tax-inclusive (tax is paid from the estate), making inter-vivos gifts slightly more efficient at the same statutory rate. The annual gift tax exclusion provides an additional advantage: a long series of annual gifts within $\bar{g}$ transfers wealth completely tax-free. The model quantifies this tradeoff between the immediate tax benefit of gifts and the parent's desire to retain precautionary wealth.

Crowding out of child's saving: Inter-vivos transfers may reduce the child's own saving (crowding out). The degree of crowding out depends on whether transfers are anticipated or unexpected, and on the child's own borrowing constraints.

A natural application is the child's home purchase (Issue #1730), where parental gifts relax the down-payment constraint, and the child's entrepreneurial entry (Issue #1736), where gifts relax the collateral constraint.

Key References

• De Nardi, M. (2004). "Wealth Inequality and Intergenerational Links." Review of Economic Studies, 71(3), pp. 743-768.
• Barczyk, D. and Kredler, M. (2014). "Altruistically Motivated Transfers under Uncertainty." Quantitative Economics, 5(3), pp. 705-749.
• Nishiyama, S. (2002). "Bequests, Inter Vivos Transfers, and Wealth Distribution." Review of Economic Dynamics, 5(4), pp. 892-931.
• McGarry, K. (1999). "Inter Vivos Transfers and Intended Bequests." Journal of Public Economics, 73(3), pp. 321-351.
• Kopczuk, W. (2013). "Taxation of Intergenerational Transfers and Wealth." Handbook of Public Economics, Vol. 5, pp. 329-390.
• Poterba, J. M. (2001). "Estate and Gift Taxes and Incentives for Inter Vivos Giving in the US." Journal of Public Economics, 79(1), pp. 237-264.

Implementation Notes

• Start with the paternalistic (joy-of-giving) specification, where the parent's problem is self-contained (no need to solve the child's value function inside the parent's optimization). State space reduces to $(a^p, z^p, t)$ plus possibly $G^{\text{cum}}$ for tax tracking.
• From the child's perspective, transfers arrive exogenously, so the child solves a standard IndShockConsumerType problem with an extra income source.
• Dynastic linkage: solve the child's problem backward, then the parent's problem backward taking the child's policy as given. For the altruistic version, an outer iteration on the transfer policy is needed.
• Gift/estate tax introduces a kink at $g = \bar{g}$ (annual exclusion) and a cumulative state variable $G^{\text{cum}}$ for the lifetime exemption. A simplification: assume all gifts are within the annual exclusion (valid for most non-wealthy households), dropping $G^{\text{cum}}$ from the state.
• BequestWarmGlowConsumerType already has warm-glow bequest utility $\Phi(a)$; adding $v(g)$ for inter-vivos gifts is a natural extension.
• For inequality analysis, the model can be embedded in HARK's Market framework with a stationary distribution over dynasties, yielding wealth Gini coefficients and intergenerational mobility matrices.