The effects of population aging on optimal redistributive taxes in an overlapping generations model

Abstract

The impact of population aging on the steady-state solution to an Ordover and Phelps (J. Public Econ. 12:1–26, 1979) overlapping generations optimal nonlinear income tax problem with two types of worker and quasilinear-in-leisure preferences is investigated. A decrease in the rate of population growth, which leads to an aging population, increases the relative price of consumption per person in retirement, which tends to decrease optimal consumption for retirees of both skill types. Nevertheless, it is also shown that the optimal marginal income tax rates are independent of the rate of population growth. In addition, the steady-state interest rate unambiguously declines when the rate of population growth declines. Resulting adjustments in production plans have an ambiguous effect on the aggregate wage rate. This article identifies factors contributing to an increase in the aggregate wage when the population ages, namely normality of consumption in retirement, complementarity between capital and labor in production, and a large capital deepening effect relative to the increase in dependency owing to demographic change. Depending on the sign of this wage effect, ambiguities may arise in the direction of change in the optimal steady-state consumption and production plans. However, when the dependency effect is sufficiently strong, it is possible to sign the direction of change in all production and consumption plans. Moreover, regardless of the direction of change in optimal consumption plans, the absolute value of the changes in consumption plans are smaller for low-skilled workers than for high-skilled when utility is time-separable and preferences exhibit decreasing absolute risk aversion. Adopting, instead, a quasilinear-in-consumption specification of preferences sharpens the comparative statics of consumption allocations, but introduces ambiguity into the effect of the rate of population growth on the optimal marginal income tax rate.

Appendix

In order to carry out any comparative static exercise, it is first necessary to show that the problem at hand has a unique solution. Lemma A.1 establishes that this is so for the Steady-State Optimal Nonlinear Income Tax Problem.

Lemma A.1

The Steady-State Optimal Nonlinear Income Tax Problem has a unique solution.

Proof of Lemma A.1

Solving (14) for a ₂ v(c ₂,x ₂) and substituting into (13) yields

$$\mathcal{W} = \lambda_1 a_1 v(c_1,x_1) - \lambda_1y_1 + \lambda_2\bigl[a_2 v(c_1,x_1)-y_1+y_2\bigr] -\lambda_2y_2. $$

(A.1)

Employing the normalization λ ₁+λ ₂=2 along with (A.1) yields

$$\mathcal{W} = \lambda_1 a_1 v(c_1,x_1)+ (1-\lambda_1)a_2 v(c_1,x_1) + a_2 v(c_1,x_1)- 2y_1. $$

(A.2)

Solving (14) for a ₂ v(c ₁,x ₁) and substituting into the penultimate term in (A.2) yields,

$$\mathcal{W} = \bigl[\lambda_1 a_1 + (1-\lambda_1) a_2\bigr] v(c_1,x_1) + a_2 v(c_2,x_2) - y_1 -y_2. $$

(A.3)

Thus, the Steady-State Optimal Nonlinear Income Tax Problem is equivalent to maximizing the objective (A.3) subject to the constraint (11). The curvature and boundary conditions on v and f guarantee a unique solution for the vector (c ₁,c ₂,x ₁,x ₂,k,y ₁+y ₂). Using the binding self-selection constraint (14) and y=y ₁+y ₂, it is straightforward to compute unique solution values of y ₁ and y ₂ from the uniquely determined (c ₁,c ₂,x ₁,x ₂,k,y). □

Proof of Lemma 1

Rearranging (A.3) yields

$$\mathcal{W} = \bigl[ a_1 + (1-\lambda_1) (a_2-a_1)\bigr] v(c_1,x_1) + a_2 v(c_2,x_2) - y. $$

(A.4)

Substituting (17) and (18) into (A.4) yields (16). In so doing, one constraint appearing in the Steady-State Optimal Nonlinear Income Tax Problem has been substituted into its objective, and the variables y ₁ and y ₂ have been eliminated. However, the variable y is inserted and the constraint (11) remains. The lemma follows. □

Proof of Proposition 1

Part (i) follows directly from (3) and (22). Part (ii) follows from dividing (19) by (20) for individuals of each type. By (21),

$$\mu=\frac{1}{f_y}. $$

(A.5)

Part (iii) follows from substituting (A.5) and (18) into (19) for individuals of type 2 and rearranging.

Using (3) in conjunction with the definition of IMTR ₁ given in (25) yields

$$\mathit{IMTR}_1= 1-\frac{1}{a_1 f_y v_c(\tilde{c}_1,\tilde{x}_1)}. $$

(A.6)

Substituting (A.5) and (19) into (A.6) yields

$$\mathit{IMTR}_1=1-\frac{1}{\frac{a_1}{\beta_1}} = \frac{a_1 -\beta_1}{a_1}. $$

(A.7)

Recalling the definition of β ₁ from (17) and rearranging yields (25). □

Proposition A.2

The optimality conditions (11) and (19)–(22) define a continuously differentiable solution function $F \colon \mathbb{R}_{+} \to \mathbb{R}^{7}_{++}$ of the problem (16) with n ↦ $(\tilde{c}_{1},\tilde{x}_{1},\tilde{c}_{2},\tilde{x}_{2},\tilde{y},\tilde{k},\tilde{\mu})$. For all n∈ℝ₊, the derivative DF of F at n is given by

$$DF(n)=\bigl(A^{-1}b\bigr)(n), $$

(A.8)

where

(A.9)

and

$$b(n) = \begin{bmatrix}0\\-(1+n)^{-2}\tilde{\mu}\\0\\-(1+n)^{-2}\tilde{\mu}\\0\\1\\\tilde{k}-(1+n)^{-2}\tilde{x}\end{bmatrix}, $$

(A.10)

and where all expressions on the right-hand sides of (A.9) and (A.10) are evaluated at the solution to (16).

Proof of Proposition A.2

By Lemma A.1, the first-order necessary conditions define a solution function. Differentiating the first-order conditions and the resource constraint yields

$$A[dc_1 \quad dx_1 \quad dc_2 \quad dx_2 \quad dy \quad dk \quad d\mu ]^\top = b \> dn, $$

(A.11)

where dependence on the parameter n is now expunged from the notation. The first zero in the final line of (A.9) follows from (22). In order to establish the Proposition, it suffices to show that the matrix A is invertible. To that end, introduce the partition

(A.12)

where H is the upper 6 × 6 block of A, p is a column of length 6 containing all but the last element of the seventh column of A, and the zero in (A.12) is a scalar.

The matrix H is block-diagonal. I now show that each of its blocks is invertible, so that H ⁻¹ exists.¹³ Specifically,

(A.13)

where each block in the partition of H is 2 × 2 and

(A.14)

and

(A.15)

Strict concavity of v and f imply that Δ _i >0, i=1,2,f. Indeed, the curvature properties imply that H is negative-definite.

It is straightforward to check that¹⁴

(A.16)

where

$$\theta= \frac{1}{p^\top H^{-1} p}. $$

(A.17)

Incidentally, because H is negative-definite, so is H ⁻¹; therefore, θ<0. □

Proof of Result 1

Using the bottom line of (A.16), (A.8) and (A.10) yields

(A.18)

Substituting (A.13)–(A.15) into (A.18) and performing the matrix multiplication gives (26).

Normality of x implies that the terms inside the summation sign on the right-hand side of (26) are negative. Linear homogeneity of f implies that f _y is homogeneous of degree zero. Hence, by Euler’s Theorem

$$yf_{yy}+kf_{ky}=0.$$

(A.19)

But f _yy <0, so f _ky >0. Hence, the entire expression inside the square bracket is negative when x is normal. Because θ<0, the first term is positive. Clearly, (1+n)² k>x is sufficient for the final term to be positive as well. □

Proof of Results 2–4

It is possible to use equations (A.13)–(A.16) to directly compute the results presented in Results 2–4. However, it is instructive to use a more heuristic solution method. The top six lines of (A.11) can be written

$$H\begin{bmatrix}dc_1 \\ dx_1 \\dc_2 \\dx_2 \\dy \\dk\end{bmatrix}= \begin{bmatrix}d\mu\\(1+n)^{-1} \,d\mu-(1+n)^{-2}\mu \,dn\\d\mu\\(1+n)^{-1} \,d\mu-(1+n)^{-2}\mu \,dn \\-f_y \,d\mu \\dn\end{bmatrix}. $$

(A.20)

Given the block-diagonal structure of H, (A.20) can be decomposed into the following three matrix equations:

(A.21)

Using (A.14) and (A.15) to compute the solutions to (A.21) gives

(A.22)

and

$$\begin{bmatrix} dy \\ dk \end{bmatrix}=\frac{1}{\varDelta _f}\begin{bmatrix}-f_{kk}f_y \,d\mu - \mu f_{ky} \,dn\\[6pt]f_{ky} f_y \,d\mu + \mu f_{yy} \,dn\end{bmatrix}. $$

(A.23)

Equations (27)–(30) follow from “dividing” the appropriate entries in (A.22) and (A.23) through by dn.

The final sentence of Result 2 is immediate. □

Proof of Corollary 1

When utility is of the form (31), (27) can be written

$$\frac{d\tilde{c}_i}{d n}=\frac{h^{\prime\prime}(\tilde{x}_i)}{\varDelta _i} \frac{ d\tilde{\mu}}{dn},\quad i=1,2. $$

(A.24)

Also, (31) and (A.14) imply

$$\varDelta _i= \beta_i g^{\prime\prime}(\tilde{c}_i) h^{\prime\prime}(\tilde{x}_i), \quad i=1,2. $$

(A.25)

Moreover, when (31) holds, the first-order condition (19) becomes

$$\beta_i g^\prime(\tilde{c}_i) =\tilde{\mu}, \quad i=1,2. $$

(A.26)

Substituting (A.25) and (A.26) into (A.24) and simplifying yields

$$\frac{d\tilde{c}_i}{d n} = \frac{d \tilde{\mu}}{dn}\frac{1}{\tilde{\mu}} \frac{g^\prime(\tilde{c}_i)}{g^{\prime\prime}(\tilde{c}_i)}, \quad i=1,2. $$

(A.27)

Thus,

$$\biggl\vert \frac{d \tilde{c}_1}{dn}\biggr\vert \ge \biggl\vert \frac{d \tilde{c}_2}{dn}\biggr\vert \quad \longleftrightarrow \quad \biggl\vert \frac{g^\prime(\tilde{c}_1)}{g^{\prime\prime}(\tilde{c}_1)} \biggr\vert \ge \biggl\vert \frac{g^\prime(\tilde{c}_2)}{g^{\prime\prime}(\tilde{c}_2)} \biggr\vert \quad \longleftrightarrow \quad -\frac{g^{\prime\prime}(\tilde{c}_1)}{g^{\prime}(\tilde{c}_1)} \le - \frac{g^{\prime\prime}(\tilde{c}_2)}{g^{\prime}(\tilde{c}_2)}. $$

(A.28)

Because β ₂>β ₁, it follows from (A.26) and strict concavity of g that $\tilde{c}_{1} < \tilde{c_{2}}$. Thus, part (i) of the Corollary follows from (A.28).

The proof of part (ii) of the Corollary is analogous. □

Proof of Lemma 2

The steady-state resource constraint (11) and the self-selection constraint (14) imply

(A.29)

Solving the system (A.29) for x ₁ and x ₂ yields

(A.30)

Social welfare is given by

$$\mathcal{W} =\lambda_1\Biggl[g\biggl(c_1, \frac{y_1}{a_1}\biggr)+x_1\Biggr] + \lambda_2 \Biggl[g\biggl(c_2, \frac{y_2}{a_2}\biggr)+x_2\Biggr]. $$

(A.31)

Substituting (A.30) into (A.31) and simplifying yields

(A.32)

Applying the normalization λ ₁+λ ₂=2 to (A.32) yields the objective function in (33). □

Proof of Proposition 2

When preferences have the form (34), the objective function in (33) becomes

(A.33)

The first-order condition for the choice of c _i is, therefore,

$$u^\prime(c_i) = 1+n. $$

(A.34)

The Proposition follows immediately from applying the Implicit Function Theorem to (A.34). □