Inheritance taxation and gifts Endogenous gift shifting, revenue-equivalent reform, and hidden-transfer bounds
Problem ID: H24-183.
Status: Unresolved; restricted results and explicit gaps.
A cohort model that includes surviving donors
The target compares tax regimes through a fixed cohort's terminal recipient resources, revenue and cardinal family welfare. Advani and Sturrock [ifs], Section 7.5 and Box 7.5, distinguish mechanical costings from gift, saving and asset-form responses. The optimization and regression criterion in the catalogue are editorial. We derive a tractable gift-timing response, a revenue-equivalent comparison, and sharp bounds for unreported transfers. These results do not identify an optimal real-world tax system.
Fix baseline parental transferable resources $P>0$ with ${\mathbb{E}} P^2<\infty$ and ${\operatorname{Var}}(P)>0$. Each donor chooses a lifetime gift $g\in[0,P]$; the residual $P-g$ remains in the estate. The donor dies before a fixed horizon $H$ with indicator $D$, independent of baseline resources and other recipient income $Y_0$, and probability $d\in(0,1]$. The mortality law is unchanged by the regimes considered. Assume ${\mathbb{E}} Y_0^2<\infty$ and ${\operatorname{Cov}}(Y_0,P)=0$. All families are included, with estate receipts equal to zero for survivors.
A regime sets gift and estate rates $t_g,t_e\in[0,\bar t]$, $\bar t<1$. There are no exemptions or reporting gaps in this first benchmark, and no uncertain policy announcement. A donor values expected recipient receipts minus real gift-management cost $\kappa g^2/(2P)$, $\kappa>0$, choosing \[
\max_{0\leq g\leq P}
(1-t_g)g+d(1-t_e)(P-g)-\frac{\kappa g^2}{2P}. \tag{1}
\] The cost is paid from separate donor consumption resources sufficient to cover it and is included once in family welfare. Thus the principal $P$ in (1) is not spent a second time on this cost. Prices and saving outside this transfer allocation are fixed; this is a partial-equilibrium timing model with a stated resource account.
Recipient terminal resources and mean tax receipts are \begin{align}
Y^H&=Y_0+(1-t_g)g+D(1-t_e)(P-g),\tag{2}\\
R&={\mathbb{E}}[t_gg+Dt_e(P-g)].\tag{3}
\end{align} Revenue funds a public service worth one unit per collected unit in the mean cardinal family-welfare account. For $d=1$, this account is ${\mathbb{E}} V={\mathbb{E}} Y^H+R-{\mathbb{E}}[\kappa g^2/(2P)]$; the constant baseline income and donor consumption resources can be added without changing comparisons. This equal weighting is normative, and not inferred from tax data.
Gift behavior and the descriptive association
Theorem (Exact gift response and population slope): Define \[
\alpha={\operatorname{clip}}_{[0,1]}
\frac{(1-t_g)-d(1-t_e)}\kappa,
\quad
T=(1-t_g)\alpha+d(1-t_e)(1-\alpha),
\quad r=t_g\alpha+dt_e(1-\alpha).
\tag{4}
\] The unique donor optimum is $g=\alpha P$. The regression slope and revenue are \[
\beta=\frac{{\operatorname{Cov}}(Y^H,P)}{{\operatorname{Var}}(P)}=T,
\qquad R=r{\mathbb{E}} P,\qquad T+r=\alpha+d(1-\alpha). \tag{5}
\] All expectations in these formulas exist.
Proof: For each $P>0$, the objective in (1) has second derivative $-\kappa/P<0$. Its unconstrained maximizer divided by $P$ is the fraction in (4), so projection gives the unique constrained optimum. Substituting into (2) and conditioning on $P,Y_0$ gives ${\mathbb{E}}[Y^H\mid P,Y_0]=Y_0+TP$ by independence of $D$. Because the conditional residual has zero conditional mean, its covariance with $P$ is zero. The assumed zero covariance of $Y_0$ then gives ${\operatorname{Cov}}(Y^H,P)=T{\operatorname{Var}}(P)$. Equation (3) gives $R=r{\mathbb{E}} P$. Adding the two coefficients in (4) proves the last identity. All quantities are bounded linear functions of $P$ and $Y_0$, apart from the gift cost, which equals $\kappa\alpha^2P/2$ and is integrable. ∎
The final identity matters when some donors survive. A shift from estates to gifts changes the amount transferred
by the horizon, so $\beta$ is not then simply one minus revenue per unit parental wealth. Conditioning instead on $D=1$ would answer a different cohort question.
Proposition (A behavioral reversal of static estate-tax effects): Suppose $d=1$ and $0<t_e-t_g<\kappa$. Then \[
\alpha=\frac{t_e-t_g}{\kappa},\qquad
r=t_e-\frac{(t_e-t_g)^2}{\kappa},\qquad \beta=1-r.
\tag{6}
\] Holding $t_g$ fixed, an increase in $t_e$ reduces revenue and increases the slope whenever $\kappa/2<t_e-t_g<\kappa$, provided the rates remain interior to their legal range.
Proof: Substitute the interior gift fraction into (4) and simplify. Differentiating gives $\partial r/\partial t_e=1-2(t_e-t_g)/\kappa$ and $\partial\beta/\partial t_e=-\partial r/\partial t_e$. These have the stated strict signs on the specified interval. ∎
This is an exact condition for a reversal in this model. It is not evidence that observed tax changes lie on that part of the response curve. The mechanism is endogenous timing, with no change in initial total wealth.
Revenue-equivalent treatment of gifts and estates
Theorem (Removing a pure timing distortion when all estates mature): Assume $d=1$. For any rate pair in $[0,\bar t]^2$, let $r$ be its effective revenue rate in (4). The harmonized regime $(t_g,t_e)=(r,r)$ is legal, raises the same expected revenue, and has the same regression slope. It sets gifts to zero and weakly increases mean family welfare by exactly \[
\frac{\kappa\alpha^2}{2}{\mathbb{E}} P. \tag{7}
\] The gain is strict if the initial regime induces positive gifts. Consequently any feasible original regime under a lower revenue bound and a mean-welfare-loss tolerance has a harmonized replacement feasible under the same constraints and with the same slope.
Proof: For $d=1$, $r=t_g\alpha+t_e(1-\alpha)$ is a convex combination of legal rates, so $0\leq r\leq\bar t$. At harmonized rates the numerator of (4)'s unconstrained gift fraction is zero, hence the unique gift is zero. The harmonized effective revenue is $r$ and its slope is $1-r$, the same as the original by (5). In either regime mean recipient resources plus public service equal ${\mathbb{E}} Y_0+{\mathbb{E}} P$: all principal is received by the horizon and taxes are counted as transfers. The only changing resource cost is $\kappa\alpha^2{\mathbb{E}} P/2$, proving (7). Equal revenue and slope, with weakly greater welfare, preserve the stated feasibility constraints. ∎
Under these restrictive equal-weight, fixed-saving assumptions, the entire slope-minimization problem over the bounded rate square is elementary: $t_g=t_e=\bar t$ has $\beta=1-\bar t$, maximal revenue $\bar t{\mathbb{E}} P$, and no gift distortion. Every rate pair has $r\leq\bar t$, so no lower slope is attainable. If the baseline is also an admissible rate pair, this harmonized regime weakly improves its mean welfare; therefore for any tolerance $\delta\geq0$ the target is feasible exactly when $\bar R\leq\bar t{\mathbb{E}} P$. This conclusion follows from the preceding bounds and is deliberately confined to this model. Differential social weights, useful early gifts, exemptions, endogenous saving, and concealed wealth invalidate its simple dominance argument.
In particular, even at equal tax rates $t$ when $d<1$, \[
\alpha={\operatorname{clip}}_{[0,1]}\frac{(1-d)(1-t)}\kappa,
\] which is positive. Early gifts then insure receipt by $H$ and are not merely tax avoidance. The theorem's $d=1$ restriction cannot be silently removed.
Sharp bounds for unreported terminal transfers
Now leave the behavioral benchmark and consider a single policy's measurement problem. Suppose the observable law of $(P,Y^{\rm obs})$ is known, both variables have finite second moments, and the true outcome is $Y=Y^{\rm obs}+U$. Let $0\leq U\leq h(P)$ almost surely for a known nonnegative function with ${\mathbb{E}} h(P)^2<\infty$. No further restriction links $U$ to observables. Write $\mu={\mathbb{E}} P$, $\sigma^2={\operatorname{Var}}(P)>0$, and $\beta_{\rm obs}={\operatorname{Cov}}(Y^{\rm obs},P)/\sigma^2$.
Theorem (Exact hidden-transfer identified interval): The sharp identified set for $\beta$ is \[
\left[
\beta_{\rm obs}+\frac{{\mathbb{E}}[(P-\mu)h(P){\mathbf 1}\{P<\mu\}]}{\sigma^2},\quad
\beta_{\rm obs}+\frac{{\mathbb{E}}[(P-\mu)h(P){\mathbf 1}\{P>\mu\}]}{\sigma^2}
\right]. \tag{8}
\]
Proof: The unobserved slope component is ${\mathbb{E}}[(P-\mu)U]/\sigma^2$. Conditional on the observed $P$, multiplication by $P-\mu$ preserves the order $0\leq U\leq h(P)$ above the mean and reverses it below. Integrating these pointwise lower and upper bounds proves containment in (8). The lower endpoint is attained by $U_-=h(P){\mathbf 1}\{P<\mu\}$, the upper by $U_+=h(P){\mathbf 1}\{P>\mu\}$. For every $t\in[0,1]$, the hidden transfer $(1-t)U_-+tU_+$ remains in $[0,h(P)]$ and yields the corresponding convex combination of endpoints. It changes no observed variable. Thus every point is attainable. Cauchy--Schwarz and the moment assumptions justify all integrals and covariances. ∎
The interval is sharp for the declared measurement class, not for the structural timing model. Applying separate intervals policy by policy would generally lose the coupling needed for joint revenue and welfare constraints. A full robust design must retain common hidden resources, behavioral parameters and reporting responses across reforms. Those restrictions, endogenous mortality or migration, and comparison with recipient accession taxes remain unresolved. No empirical estimates or numerical experiments are claimed.
References
Completion Estimate
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