Capital tax competition and social security

Abstract

The classic capital tax policy externality is studied in the presence of a social security program where both the benefits and taxes depend on wages in an overlapping generations economy with many countries and mobile capital. We study the response and welfare implications of a coordinated capital tax rate increase across countries competing for the mobile tax base on the initial generations, the transition, and the steady state. The tax increase is initially completely capitalized, but some of the burden is shifted to labor on the transition path and in the steady state. Several new welfare effects are uncovered including an effect involving the parameters of the social security program. Sufficient conditions are provided so that all current and future generations are better off from the reform. However, social security may reduce the gain to capital tax reform.

Appendices

Appendix A: Proposition 1: optimal second-best policy rules

Writing out the social welfare function of (6),

First, consider the optimal policy rule of the GTA. It chooses an infinite sequence {τ _t ,g _t } _t to maximize the social welfare function subject to a sequence of budget constraints given by (7), taking as given the policy sequence of the other agency, the policy sequences of all other countries, and the sequence of interest rates. The first-order conditions for the optimal policy sequence at time t are given by

where λ _t is the Lagrange multiplier for its budget constraint at time t. Combine conditions to obtain

(A.1)

Use u ₁=b(1+r)u ₂, $u^{*}_{1} = b(1+r^{*})u^{*}_{2}$, and the definition of Φ to rewrite (A.1) as (9a).

The SSA chooses the infinite sequence {γ _t ,β _t } _t to maximize the same social welfare function subject to its budget constraint, (8), taking as given the policy sequence of the GTA, the policy sequences of all other countries, and the sequence of interest rates. The first-order conditions for an interior solution are

where μ is the Lagrange multiplier for the social security constraint. Combine these conditions to obtain (9b), $\phi u_{1} +(1-\phi)u^{*}_{1} = (\phi u_{2} + (1-\phi) u^{*}_{2})$.

Appendix B: Proposition 2: response to the capital tax policy reform

2.1 B.1 Preliminary results

Let . This can also be written as , by using S ₂=−βR(1−S ₁). If Δ>0, then the steady state equilibrium is stable when it exists. Let D=ϕS _r −(1+n)K _r >0. Thus, . If

$$ S_{1} > \beta R(1 + \beta R - \gamma)^{-1},$$

(B.1)

then ds/dw>0. Furthermore, when α>0, it follows that γ>β/(1+n), from the social security budget constraint. Thus, 1+βR−γ<1+βR−β/(1+n), or

$$ [1 + \beta R - \gamma ]^{-1} > \bigl[1 + \beta R - \beta /(1+n)\bigr]^{-1}.$$

(B.2)

If (B.1) is satisfied, then S ₁>βR[1+βR−β/(1+n)]⁻¹, from (B.2). It then follows that $\phi [\beta R - [1 +\beta R - \tilde{\beta}(1+n)]S_{1}] < 0$. Hence, D>Δ.

2.2 B.2 Initial response

Equations (3) and (8) for t=1 determine (r ₁,γ ₁). Equation (3) implies that dr ₁/dτ=−1 and w ₁ is unaffected by the reform. It follows from this and (8) that dγ ₁/dτ=0. To see this, consider the social security budget in the first period, (1+n)γ ₁ W(r ₁+τ ₁)=α ₁+β ₁ w ₀. The right-hand side is fixed and the d(r ₁+τ ₁)/dτ ₁=0. It follows immediately that dγ ₁/dτ=0. The young at t=1 face (4) for t=1 and (8) for t=2. Differentiate this system and solve to obtain the following:

(B.3)

(B.4)

where we have used dr ₁/dτ=−1. Note that 0<ϕS _r /D<1 since K _r <0. The wage rate in the second period, W(r ₂+δ+τ ₂), responds according to

$$dw_{2}/d\tau = - k(1+ dr_{2}/d\tau) = - \phi kS_{r}/D < 0,$$

where we have used W _r =−k.

2.3 B.3 Steady state response

Substitute the steady state social security budget for the payroll tax rate into the equilibrium condition to obtain,

To derive the steady state responses, totally differentiate this last equation to obtain

$$dr/d \tau = -1 + \phi S_{r}/\varDelta < 0. $$

(B.5)

Since Δ<D, it follows that dr ₂/dτ<dr/dτ<0. The response of the payroll tax rate is

$$d \gamma / d \tau = ( k/w)\bigl[ \gamma - \beta /(1+ n)\bigr]\bigl[1 + (dr/d \tau)\bigr] > 0. $$

(B.6)

Under (B.1), dγ/dτ<dγ ₂/dτ. To see this, compare (B.4) with (B.6) after using (B.3) and (B.5). The local wage W(r+δ+τ) responds to the reform according to

$$dw/d \tau = - k\bigl(1 + dr/d \tau\bigr) = - \phi kS_{r}/ \varDelta < 0 .$$

Since D>Δ, dw/dτ<dw ₂/dτ.

2.4 B.4 Transition response

Substitute the social security constraint into the equilibrium condition for the payroll tax to get

$$\begin{aligned}&\phi S\bigl(W(r_{t} + \tau) - \bigl(\alpha + \beta W(r_{t-1} +\tau)\bigr)/(1+ n), \alpha + \beta W(r_{t}, \tau) , r _{t+1}\bigr)\\&\quad =(1+ n)K(r _{t+1} + \tau) .\end{aligned}$$

This is a second-order difference equation. Differentiate

$$D(dr _{t+1}/ d \tau) - k \phi [ S _{1} +\beta S _{2}]( dr_{t}/d \tau) +k \phi\beta\bigl( S _{1}/(1+ n)\bigr)(dr_{t-1}/d \tau) = \phi S_{r} -\varDelta .$$

Divide by D and rewrite using the lag operator, L, where Lx _t =x _t−1,

$$\bigl(1 - A _{1} L + A _{2} L ^{2}\bigr)( dr _{t+1}/ d \tau) = A _{3} ,$$

where A ₁=kϕ[S ₁+βS ₂]/D=ϕk((1+βR)S ₁−βR)/D, A ₂=kϕβS ₁/(1+n)D, and A ₃=[ϕS _r −Δ]/D. Following Bernheim (1981), the A _i are evaluated at the initial steady state equilibrium, and hence constant. If A ₁<1 in the absence of social security, then the equilibrium is locally stable. If A ₁<β/γ(1+n) (< or =) 1 in the presence of the social security program, then the equilibrium will be stable. The roots to this second-order equation are given by

$$\lambda = (1/2)\bigl[ A _{1} (+ \ \mbox{or}\ -) \bigl(A _{1}^{2} - 4 A _{2}\bigr)^{1/2}\bigr] .$$

If $A_{1}^{2} - 4A_{2} > 0$, the roots are real, which is the case we will pursue. It is straightforward to show that the stable root is given by $\lambda_{1} = (1/2)[A_{1} + (A_{1}^{2} - 4 A_{2})^{1/2}]$. If Δ>0, then λ ₁<1. This is easy to confirm by expanding λ ₁. Since 1+βR−β/(1+n)<1+βR, it follows that S ₁>βR/(1+βR) by (B.1) so that A ₁>0. Therefore, 0<λ ₁. The stable solution to the difference equation is given by

$$dr_{t}/d \tau = dr/d \tau - (1 + dr/d \tau)(1 - \varDelta / D)( \lambda_{1})^{t-2} $$

(B.7)

or,

$$1+ dr_{t}/d \tau = (1+ dr/d \tau)\bigl(1 - (1 - \varDelta / D) \lambda_{1}^{t-2}\bigr) ,$$

for t=2,…, where we have used (B.3) and (B.5). Clearly, the response converges to the steady state response, dr/dτ. Thus,

$$- 1 = dr _{1}/ d \tau < dr _{2}/ d \tau <\cdots< dr_{t}/d \tau < \cdots < dr/d \tau < 0 .$$

Since dw _t /dτ=−k(1+dr _t /dτ), we also have the result that

$$dw _{1}/ d \tau = 0 > dw _{2}/ d \tau > \cdots > dw_{t}/d \tau > \cdots > dw/d \tau .$$

The response of the payroll tax rate on the transition path is

$$d \gamma _{t}/d \tau = ( k/w)\bigl[ \gamma(1 + dr_{t}/d \tau) -\bigl(\beta /(1+ n)\bigr)(1 + dr_{t-1}/d \tau)\bigr] ,$$

from the social security budget constraint. Since γ>β/(1+n) for α>0, and 1+dr _t /dτ>1+dr _t−1/dτ, by (B.7), this response is positive, dγ _t /dτ>0. Furthermore, if A ₁<β/(1+n)γ<1, then dγ ₂/dτ>dγ/dτ. The response of the payroll tax rate converges to the steady state response and we have the result, dγ ₂/dτ>⋯>dγ _t /dτ>⋯>dγ/dτ>0. When α=0, we obtain, dγ ₂/dτ>⋯>dγ _t /dτ>⋯>dγ/dτ=0.

Appendix C: Proposition 3: welfare effects of the coordinated capital tax reform

3.1 C.1 Initial generations

The per capita sum of the utility of the agents alive in the first period of the reform is

(C.1)

where we have appended the first two social security budget constraints since they affect utility in the first period of the economy. The Lagrange multipliers are μ _t . Substitute the GTA’s budget constraint for g ₁ and g ₂ into (C.1). Differentiate (C.1) and use the definitions of θ and ε, dr ₁/dτ=−1, ϕs ₀=(1+n)k ₁, and the policy rules of Proposition 1 to obtain the effect of the reform on social welfare for the initial generations,

evaluated at the initial steady state. The initial old spender is better off because of the spending effect that occurs in the first period of the reform, $\theta\varepsilon U ^{*\,0}_{4} > 0$. The initial old saver is better off if the positive spending effect outweighs the negative interest rate effect, which occurs if $\theta\varepsilon m ^{0}_{2} > s/k = (1+n)/\phi$. The initial young spender is better off if

$$1 + \theta\varepsilon m ^{*\,1}_{1} -\theta\varepsilon R^*m^{*\,1}_{2}( dr _{2}/d \tau) > \gamma_{1} -\beta_{2}/(1+n),$$

where $R^{*}m^{*\,1}_{2} =(U^{*\,1}_{4}/U^{*\,1}_{2})(U^{*\,1}_{2}/U^{*\,1}_{1})$. A sufficient condition is that the current spending effect outweigh the social security effect on wages, $\theta\varepsilon m^{*\,1}_{1} > \gamma_{1} -\beta_{2}/{(1+n)}$.

The initial young saver is better off under the reform if

$$k _{1}\bigl[1 + \theta\varepsilon m ^{1}_{1} - \bigl(\gamma_{1} -\beta_{2}/(1+ n)\bigr)\bigr] + \bigl(s _{1} -k _{2}\theta\varepsilon m ^{1}_{2}\bigr) R(dr _{2}/ d \tau) > 0 .$$

Sufficient conditions for this to be satisfied are $\theta\varepsilon m ^{1}_{1} > \gamma_{1} - \beta_{2}/(1+ n)$ and $\theta\varepsilon m ^{1}_{2} > s/k = (1+n)/ \phi$.

3.2 C.2 Welfare effects on the transition path

Define Ω _t in the same manner as Ω ₁ in (C.1). It is then straightforward to derive the following result, by using the same steps,

The old spender on the transition path is unambiguously better off since they only experience the current spending effect, $-\theta\varepsilon U^{*\,t-1}_{4}(dr_{t}/d \tau) > 0$. The old saver is better off if the spending effect outweighs the interest rate effect, $\theta\varepsilon U^{t-1} _{4} >{(1+ n)U^{t-1} _{2}/\phi}$, or $\theta\varepsilon m^{t-1} _{2} >(1+ n)/ \phi$. The welfare of the young saver improves if $(1+n)U^{t}_{2} < \phi U^{t}_{4}\theta\varepsilon$, or

$$ \theta\varepsilon m^{t} _{2} > (1+ n)/ \phi,$$

(C.2)

and, $- (U^{t}_{3}\theta\varepsilon +U^{t}_{1})(dr_{t}/d\tau) > U^{t}_{1}(\gamma - \beta /(1+r))$, or

$$ - \bigl(1 + m^{t}_{1}\theta\varepsilon\bigr)( dr_{t}/d \tau) > \gamma -\beta /(1+ r).$$

(C.3)

The young spender’s welfare improves if $-(U^{*\,t} _{3}\theta\varepsilon +U^{*\,t} _{1})( dr_{t}/d \tau) > {U^{*\,t} _{1}(\gamma - \beta /(1+ r))}$, or

$$ - \bigl(1 + m^{*\,t} _{1}\theta\varepsilon\bigr)( dr_{t}/d \tau) > \gamma -\beta /(1+ r).$$

(C.4)

3.3 C.3 Steady state welfare effects

The steady state welfare results can be easily derived by letting t go to infinity in conditions (C.2)–(C.4).