Pension reform with migration and mobile capital: is a Pareto improvement possible?

Abstract

This paper shows that in a two-country two-overlapping-generations model with migration, capital mobility and an immobile production factor (land), a locally welfare-improving pension reform at the cost of the neighboring country is possible if land plays a minor role in production. Furthermore, differences in the size of the PAYG pension schemes between the countries distort the international allocation of labour and capital. As a result, a Pareto-improving pension reform is possible if countries employ PAYG pension schemes of different size, provided that a federal government exists that redistributes benefits and losses of the reform both intergenerationally and internationally.

Appendix

Derivation of Eq. 10 and 11

Equations 3 and 7 imply

$$ \frac{\tilde{K}_{t}}{K_{t}}= \left( \frac{\tilde{L}_{t}}{L_{t}} \right)^{\frac{\beta }{1-\alpha}} $$

(29)

Lifetime income, i.e., the present value of an agent’s total income, is $w_{t}(1-\tau )+(1+n_t)w_{t+1}\tau /(1+\bar {r})$. Using the fact that there is no migration in equilibrium ( n _t = 0) and the wages are constant ( w _t = w _{t + 1} ≡ w) lifetime income may be written as $w(1+r-\tau \bar {r})/(1+\bar {r})$. As a higher lifetime income is equivalent to a higher utility, (this follows from the assumption, that utility function is strictly increasing in both arguments) perfect labour mobility implies:

$$ w\frac{1+\bar{r}-\tau \bar{r}}{1+\bar{r}}=\tilde{w}\frac{1+\bar{r}-\tilde{\tau}\bar{r}}{1+\bar{r}} $$

(30)

Using (29), Eq. 30 can be rewritten as

$$ w(1+\bar{r}-\tau \bar{r}) =w \left( \frac{\tilde{L}}{L} \right)^{-\frac{1-\alpha-\beta}{1-\alpha}} (1+\bar{r}-\tilde{\tau}\bar{r}) $$

(31)

Equation 10 then immediately follows. Equation 11 can be derived by inserting Eq. 10 into (29).

1.1 Proof of lemma 1

Proof

The equality of utilities is equivalent to the equality of present values of lifetime income, so:

$$ w_{t} (1-\tau)+\tau w_{t+1} \frac{L_{t+1}}{L_{t} (1+\bar r)}=\tilde w_{t} (1-\tilde \tau)+\tilde \tau \tilde w_{t+1} \frac{\tilde L_{t+1}}{\tilde L_{t} (1+\bar r)} $$

(32)

Substituting w _t and $\tilde w_{t}$ from Eq. 2 and using Eqs. 3 and 7 to eliminate capital we can rewrite this as:

$$ \frac{1}{L_{t}}\left( (1-\tau)L_{t}^{\frac{\beta}{1-\alpha}}+\tau \frac{L_{t+1}^{\frac{\beta}{1-\alpha}}}{1+\bar r} \right)= \frac{1}{\tilde L_{t}}\left( (1-\tilde\tau) \tilde L_{t}^{\frac{\beta}{1-\alpha}}+\tilde\tau \frac{\tilde L_{t+1}^{\frac{\beta}{1-\alpha}}}{1+\bar r} \right) $$

(33)

If L _{t + 1} = L _t , then the system is in its steady state. Suppose that this is not the case. Write L _t = L _s + ΔL _t , where L _s is a steady state and ΔL _t is a deviation from the steady state. Then $\tilde L_{t}=\tilde L_{s}-\Delta L_{t}$, where $\tilde L_s=\Lambda -L_{s}$. Rewrite Eq. 33

$$\begin{array}{@{}rcl@{}} \underbrace{(1-\tau)(L_{s}+\Delta L_{t})^{\frac{\beta+\alpha-1}{1-\alpha}}}_{\Psi_{1}}+\underbrace{\frac{\tau}{1+\bar r} \left(\frac{L_{s}+\Delta L_{t+1}}{L_{s}+\Delta L_{t}}\right)^{\frac{\beta}{1-\alpha}}}_{\Psi_{2}} \underbrace{\left(\frac{1}{L_{s}+\Delta L_{t}}\right)^{\frac{1-\alpha-\beta}{1-\alpha}}}_{\Psi_{3}}=\\ \underbrace{(1-\tilde\tau) (\tilde L_{s}-\Delta L_{t})^{\frac{\beta+\alpha-1}{1-\alpha}}}_{\tilde\Psi_{1}} +\underbrace{\frac{\tilde\tau}{1+\bar r} \left(\frac{\tilde L_{s}-\Delta L_{t+1}}{\tilde L_{s}-\Delta L_{t}}\right)^{\frac{\beta}{1-\alpha}}}_{\tilde\Psi_{2}} \underbrace{\left(\frac{1}{\tilde L_{s}-\Delta L_{t}}\right)^{\frac{1-\alpha-\beta}{1-\alpha}}}_{\tilde \Psi_{3}} \end{array} $$

(34)

All values except ΔL _t and ΔL _{t + 1} in Eq. 34 are constant, and the equation holds for ΔL _t = ΔL _{t + 1} = 0. Suppose ΔL _t > 0. Then Ψ ₁ and Ψ ₃ decrease and $\tilde \Psi _{1}$, $\tilde \Psi _{3}$ grow comparing to the equilibrium values. In order to keep the equality Ψ ₂ shall grow and/or $\tilde \Psi _{2}$ shall decrease. This is possible only if ΔL _{t + 1} is larger than ΔL _t . Hence, at time t + 1 the deviation from the equilibrium values is even larger. Similarly ΔL _t < 0 leads to a smaller ΔL _{t + 1}, implying that the equilibrium in the model is a saddle point equilibrium. □

1.2 Continuity of consumption

Since, we are interested in parameter values α + β ≤ 1 only the continuity of consumption from below will be shown.

Lemma 2

Suppose conditions of lemma 1 are satisfied. Then the steady-state consumptions c ^o, c ^yare continuous from below in α and β at the points α + β = 1.

Proof

Consider Eqs. 9 and 10. In lemma 1 we suppose that $\tau <\tilde \tau$, then L(α, β) = Λ if α + β = 1. Choose increasing sequences α _k and β _k , α _k → α, β _k → β when k → ∞. It is obvious that l i m _{k → ∞} L(α _k , β _k ) = Λ = L(α, β). Hence, labour in the home country is continuous on α and β from below at the point α + β = 1. Equation 13 then implies that wages are a continuous function of α and β. □