A Tradeoff between the Output and Net Foreign Asset Effects of Pension Reform

Abstract

We use an overlapping generations model of a small open economy with perfect capital mobility to evaluate the long-term effects of pension reforms on output and net foreign assets. We compare reforms that achieve similar fiscal targets and show the existence of a tradeoff. Reforms that increase the retirement age have an expansionary effect on output, but a small (and oftentimes) negative effect on the net foreign asset position. In contrast, reforms that cut pension benefits improve the net foreign asset position but reduce output. Only mixed reforms that increase the retirement age and cut pension benefits sufficiently can boost both output and net foreign assets.

Appendices

Appendix 1 Proofs

1.1 Proof of Lemma 1

• a) In the function A ^h(R, τ), the first factor $w-\tau \cdot \left(\frac{L}{L-R}\right)>0$ for all $R\in \left(0,\overline{R}\right)$; this follows directly from assumption 1 (which is satisfied if and only if $R<L\cdot \left(1-\frac{\tau }{w}\right)$). The second factor $L\cdot \left(\frac{1-{e}^{-rR}}{1-{e}^{-rL}}\right)-R>0$ for all $R<\overline{R}<L$. Hence A ^h(R, τ) > 0 for all $R\in \left(0,\overline{R}\right)$.

• b) The first derivative of A ^h(R, τ) with respect to R is given by: $\frac{\partial {A}^h}{\partial R}=-\frac{\tau \cdot L}{r\cdot {\left(L-R\right)}^2}\cdot \left[\frac{L\cdot \left(1-{e}^{-rR}\right)}{\left(1-{e}^{-rL}\right)}-R\right]+\left(\frac{1}{r}\right)\cdot \left(w-\frac{\tau \cdot L}{L-R}\right)\cdot \left[\frac{r\cdot L\cdot {e}^{-rR}}{\left(1-{e}^{-rL}\right)}-1\right].$ Since $\underset{R\to 0}{ \lim}\left(\frac{\partial {A}^h}{\partial R}\right)=\frac{\left(w-\tau \right)}{r}\cdot \left[\frac{r\cdot L}{1-{e}^{-rL}}-1\right]>0$ (note that $\frac{r\cdot x}{1-{e}^{-rx}}-1>0$ for x > 0), the function A ^h is increasing in R when R approaches 0. In addition, $\underset{R\to \overline{R}}{ \lim}\left(\frac{\partial {A}^h}{\partial R}\right)=-\frac{\tau \cdot L}{r\cdot {\left(L-\overline{R}\right)}^2}\cdot \left[\frac{L}{1-{e}^{-rL}}-\frac{\overline{R}}{1-{e}^{-r\overline{R}}}\right]\cdot \left(1-{e}^{-r\overline{R}}\right)<0$; thus, the function A ^h is decreasing in R when R approaches $\overline{R}$. Note that in the expression for $\frac{\partial {A}^h}{\partial R}$, the first term in the right hand side is always negative, and the second term can be either positive or negative depending on the value of R (the function $\frac{r\cdot L\cdot {e}^{-rR}}{1-{e}^{-rL}}-1$ is decreasing in R and becomes negative for values sufficiently close to L).

• c) The second derivative of A ^h(R, τ) with respect to R is given by:

  $$ \frac{\partial^2{A}^h}{{\left(\partial R\right)}^2}=-\frac{2\cdot \tau \cdot L}{r\cdot {\left(L-R\right)}^3}\cdot \left[\frac{L\cdot \left(1-{e}^{-rR}\right)}{\left(1-{e}^{-rL}\right)}-R\right]-\frac{2\cdot \tau \cdot L}{r\cdot {\left(L-R\right)}^2}\cdot \left[\frac{r\cdot L\cdot {e}^{-rR}}{\left(1-{e}^{-rL}\right)}-1\right]-\left(w-\frac{\tau \cdot L}{L-R}\right)\cdot \frac{r\cdot L\cdot {e}^{-rR}}{\left(1-{e}^{-rL}\right)}. $$

The first and third terms in the right hand side are non-positive; the second term is negative for all R < R _m , but it becomes positive as the value of R approaches L. Cases in which the second term is positive and more than offsets the sum of the first and third terms (in absolute value) cannot be ruled out. For example, the function A ^h becomes convex when R approaches $\overline{R}$, and $\overline{R}$ approaches L. In this case, $\underset{R\to \overline{R}}{ \lim}\left(w-\frac{\tau \cdot L}{L-R}\right)=0$; the first and third terms in the expression for $\frac{\partial^2{A}^h}{{\left(\partial R\right)}^2}$ equal 0, and the second term is positive. Thus, $\frac{\partial^2{A}^h}{{\left(\partial R\right)}^2}>0$.

1.2 Proof of Lemma 2

The function A ^h(b, R) and its first and second derivatives with respect to R are given by:

$$ \begin{array}{l}{A}^h\left(b,R\right)=\frac{1}{r}\cdot \left[w-b\cdot \frac{L}{R}\right]\cdot \left[L\cdot \left(\frac{1-{e}^{-rR}}{1-{e}^{-rL}}\right)-R\right];\\ {}\frac{\partial {A}^h\left(b,R\right)}{\partial R}=\frac{b\cdot {L}^2}{r\cdot {R}^2}\cdot \frac{\left[1-{e}^{-rR}\cdot \left(1+r\cdot R\right)\right]}{\left(1-{e}^{-rL}\right)}+\frac{w}{r}\cdot \left[\frac{r\cdot L\cdot {e}^{-rR}}{1-{e}^{-rL}}-1\right];\\ {}\frac{\partial^2{A}^h\left(b,R\right)}{{\left(\partial R\right)}^2}=-\left(w-\frac{b\cdot L}{R}\right)\cdot \left(\frac{L\cdot r\cdot {e}^{-rR}}{1-{e}^{-rL}}\right)-\frac{2\cdot b\cdot {L}^2}{r\cdot {R}^3}\cdot \frac{\left[1-{e}^{-rR}\cdot \left(1+r\cdot R\right)\right]}{\left(1-{e}^{-rL}\right)}<0.\end{array} $$

The first derivative could be positive or negative: the first term is always positive, but the second term becomes negative for R values that are sufficiently high. As the second derivative is always negative, the function is strictly concave.

The first derivative can also be expressed as follows:

$$ \frac{\partial {A}^h\left(b,R\right)}{\partial R}=\left(\frac{1}{r}\right)\cdot \left(w-\frac{b\cdot L}{R}\right)\cdot \left[\frac{r\cdot L\cdot {e}^{-rR}}{1-{e}^{-rL}}-1\right]+\frac{b\cdot L}{r\cdot R}\cdot \left[\frac{L\cdot \left(1-{e}^{-rR}\right)}{R\cdot \left(1-{e}^{-rL}\right)}-1\right]. $$

In the limit, when R approaches L, $\underset{R\to L}{ \lim}\left(w-\frac{b\cdot L}{R}\right)\cdot \left[\frac{r\cdot L\cdot {e}^{-rR}}{1-{e}^{-rL}}-1\right]<0$ and $\underset{R\to L}{ \lim}\left[\frac{L\cdot \left(1-{e}^{-rR}\right)}{R\cdot \left(1-{e}^{-rL}\right)}-1\right]=0$. Hence, $\underset{R\to L}{ \lim}\frac{\partial {A}^h\left(b,R\right)}{\partial R}<0$.

The following “envelope” condition can be used to establish a relation between the derivatives of the functions A ^h(R, b) and A ^h(R, τ) = A ^h[R, τ(b, R)]:

$$ \begin{array}{l}\frac{\partial {A}^h\left(R,b\right)}{\partial R}=\frac{\partial {A}^h\left(R,\tau \right)}{\partial \tau}\cdot \frac{\partial \tau \left(R,b\right)}{\partial R}+\frac{\partial {A}^h\left(R,\tau \right)}{\partial R};\\ {}\frac{\partial {A}^h\left(R,b\right)}{\partial R}-\frac{\partial {A}^h\left(R,\tau \right)}{\partial R}=\frac{b\cdot {L}^2}{r\cdot \left(L-R\right)}\cdot \left[\frac{L\cdot \left(1-{e}^{-rR}\right)}{R\cdot \left(1-{e}^{-rL}\right)}-1\right]>0.\end{array} $$

At the point $R={R}_m^{*}$, $\frac{\partial {A}^h\left({R}_m^{*},b\right)}{\partial R}>\frac{\partial {A}^h\left({R}_m^{*},\tau \right)}{\partial R}=\alpha \cdot {\left(\frac{\alpha }{r}\right)}^{\frac{1}{1-\alpha }}$ (slope of K line in Fig. 1). The convexity of the function A ^h(R, b) implies that if R ^* is the value of R such that $\frac{\partial {A}^h\left({R}^{*},b\right)}{\partial R}=\alpha \cdot {\left(\frac{\alpha }{r}\right)}^{\frac{1}{1-\alpha }}$, then ${R}^{*}>{R}_m^{*}$.

Appendix 2. Model for Numerical Simulations

Households

In the model, which is presented in stationary form,¹⁵ the utility of a household born at time t is determined by consumption (c) and leisure (l), and is given by

$$ {U}_t={\displaystyle \sum_{s=1}^{T_t+{T}_t^R}{\beta}^{s-1}\cdot \left\{ \log \left({c}_{t+s-1}^s\right)+\gamma \cdot \log \left({l}_{t+s-1}^s\right)\right\},} $$

(1)

where the working life lasts T _t  years and retirement lasts ${T}_t^R$ years. The household is endowed with a fixed number of hours per year, normalized to one, and distributed between work (n) and leisure (l): ${l}_{t+s-1}^s=1-{n}_{t+s-1}^s,\mathrm{where}\ {n}_{t+s-1}^s=0$ during retirement.

A household accumulates assets (A) according to the following budget constraint:

$$ \left(1+\xi \right)\cdot {A}_{t+s}^{s+1}=\left[1+{r}_{t+s-1}\cdot \left(1-{\tau}_{t+s-1}^I\right)\right]\cdot {A}_{t+s-1}^s+{H}_{t+s-1}^s-\left(1+{\tau}_{t+s-1}^c\right)\cdot {c}_{t+s-1}^s, $$

(2)

where ${H}_{t+s-1}^s=\left(1-{\tau}_{t+s-1}-{\tau}_{t+s-1}^I\right)\cdot {W}_{t+s-1}\cdot {e}^s\cdot {n}_{t+s-1}^s$ represents net (of taxes) wage income during the working phase, and ${H}_{t+s-1}^s={b}_{t+s-1}^s$ is the pension benefit during retirement. ξ denotes the annual rate of labor-augmenting technological progress.

The household takes as given the payroll (τ), income (τ ^I ), and consumption (τ ^c) tax rates, and the interest (r) and wage (W) rates.¹⁶ During the household’s working life, next year’s assets are determined by adding savings to this year’s assets; savings, in turn, are computed as the sum of the net return on assets plus net wage income, minus gross consumption. The household’s labor productivity per hour varies with age, according to an exogenous skill premium (e ^s) defined as the relative productivity of an s-year old household to that of a one-year old (unskilled) household.

At retirement, the first pension benefit is calculated based on average past wage earnings, indexed by the economy-wide wage growth; in stationary form, it is given by: ${b}_{t+{T}_t}^{T_t+1}=\psi \cdot \frac{1}{T_t}\cdot {\displaystyle \sum_{j=1}^{T_t}{W}_{t+j-1}\cdot {e}^j\cdot {n}_{t+j-1}^j}$, where (ψ) stands for the replacement parameter. Subsequent pension benefits are indexed annually with wage growth: ${b}_{t+s-1}^s={b}_{t+{T}_t}^{T_t+1},s={T}_t+2,\ldots {T}_t+{T}_t^R.$ ¹⁷ The household’s optimization problem is to choose the sequence of consumption, leisure, and asset holdings ${\left\{{c}_{t+s-1}^s,{l}_{t+s-1}^s,{A}_{t+s-1}^s\right\}}_{s=1}^{T_t+{T}_t^R}$ to maximize its utility (1) subject to the budget constraint (2), the pension benefit formula, and the no intergenerational bequest or inheritance condition ${A}_t^1={A}_t^{T_t+{T}_t^R+1}=0$ (for details, see below). Given a total population (P _t ) and a population of age s (${P}_t^s$) in year t, aggregate household effective labor supply (${N}_t^h$), assets (${A}_t^h$), and consumption (${C}_t^h$), are respectively given by:

$$ {N}_t^h={\displaystyle \sum_{s=1}^{T_t}{e}^s\cdot {n}_t^s\cdot \frac{P_t^s}{P_t}};\kern0.3em {A}_t^h={\displaystyle \sum_{s=1}^{T_t+{T}_t^R}{A}_t^s\cdot \frac{P_t^s}{P_t}};\ \mathrm{and}\ {C}_t^h={\displaystyle \sum_{s=1}^{T_t+{T}_t^R}{c}_t^s\cdot \frac{P_t^s}{P_t}}. $$

Firms

Firms maximize profits net of capital depreciation, ${\Pi}_t^f$, using a constant-returns-to-scale production function with labor-augmenting technological progress ${\Pi}_t^f=\mathrm{Z}\cdot {\left({K}_t^f\right)}^{\alpha}\cdot {\left({N}_t^f\right)}^{1-\alpha }-\left({r}_t+\delta \right)\cdot {K}_t^f-{W}_t\cdot {N}_t^f$. In this expression, Z denotes the level of total factor productivity, α is the share of capital in domestic output, ${K}_t^f$ denotes the capital stock, and δ is the rate of capital depreciation. Both output and factor markets are perfectly competitive and firms face given wages (W _t ) and rental rates (r _t ). The first order conditions require that W _t and r _t  + δ equal, respectively, the marginal product of labor and of capital:

$$ {W}_t=\mathrm{Z}\cdot \left(1-\alpha \right)\cdot {\left(\frac{K_t^f}{N_t^t}\right)}^{\alpha };\kern0.3em {r}_t+\delta =\mathrm{Z}\cdot \alpha \cdot {\left(\frac{K_t^f}{N_t^t}\right)}^{-\left(1-\alpha \right)}. $$

Government

The government collects payroll, income, and consumption taxes from households. Tax revenues are used to finance (exogenous) public consumption (G) and pension benefits, and to redeem government debt (D), implying the following budget constraint:

$$ {D}_{t+1}\cdot \left(1+\xi \right)\cdot \frac{P_{t+1}}{P_t}=\left(1+{r}_t\right)\cdot {D}_t-\left[{\tau}_t^I\cdot \left({r}_t\cdot {A}_t^h+{W}_t\cdot {N}_t^h\right)+{\tau}_t^c\cdot {C}_t^h-{G}_t\right]-{\tau}_t\cdot {W}_t\cdot {N}_t^h+{\displaystyle \sum_{s={T}_t+1}^{T_t+{T}_t^R}{b}_t^s\cdot \frac{P_t^s}{P_t}}, $$

where the non-pension primary balance (in square brackets), and the pension system’s balance (last two terms) are shown separately.

Equilibrium

Given initial conditions and an exogenous path of international interest rates, an equilibrium is defined as a set of allocations for households and firms, prices, and government variables, that simultaneously place all households and firms on their maximizing paths, ensure the intertemporal solvency of the government, and clear all markets. The market-clearing conditions and the economy’s aggregate flow constraint are respectively given by: ${N}_t={N}_t^f={\displaystyle \sum_{s=1}^{T_t}\left({e}^s\cdot {n}^s\cdot \frac{P_t^s}{P_t}\right)},$ ${K}_t^f+{D}_t+{A}_t^{*}={A}_t^h={\displaystyle \sum_{s=1}^{T_t+{T}_t^R}\left({A}_t^s\cdot \frac{P_t^s}{P_t}\right)},$ and ${A}_{t+1}^{*}\cdot \left(1+\xi \right)\cdot \frac{P_{t+1}}{P_t}-{A}_t^{*}={r}_t\cdot {A}_t^{*}+{Y}_t-{C}_t-{G}_t-\left[{K}_{t+1}\cdot \left(1+\xi \right)\cdot \frac{P_{t+1}}{P_t}-\left(1-\delta \right)\cdot {K}_t\right]$.¹⁸ In the last expression, the left-hand side is the current account balance, where A ^* denotes the economy’s net foreign assets. The right-hand side is the economy’s balance of saving (${r}_t\cdot {A}_t^{*}+{Y}_t-{C}_t-{G}_t$) minus gross investment (${K}_{t+1}\cdot \left(1+\xi \right)\cdot \frac{P_{t+1}}{P_t}-\left(1-\delta \right)\cdot {K}_t$), where ${Y}_t={Y}_t^f$ and ${C}_t={C}_t^h$ are the equilibrium aggregate output and consumption, respectively. Note that due to the absence of current transfers, non-debt foreign assets, and factor income other than interest in the model, the primary current account balance (CAP) is always equal to net exports (NX): ${CAP}_t={NX}_t={Y}_t-{C}_t-{G}_t-\left[{K}_{t+1}\cdot \left(1+\xi \right)\cdot \frac{P_{t+1}}{P_t}-\left(1-\delta \right)\cdot {K}_t\right]$.

Balanced Growth Equilibrium

It is defined assuming constant population growth rate (p), labor augmenting technological progress (ξ), working life (T _t  = T), retirement period (${T}_t^R={T}^R$), and a fiscal policy characterized by constant tax rates and unchanged public expenditure and debt-to-output ratios. This equilibrium is expressed as a steady state solution in terms of detrended variables in the stationary-transformed model.

It is important to stress that along a balanced growth equilibrium path, the (detrended) current account balance is constant but not necessarily zero. The current account balance CA is fully consistent with a stable path of net foreign assets A ^*, and satisfies the following condition: CA = A ^* ⋅ [(1 + ξ) ⋅ (1 + p) − 1]. The optimal net foreign assets A ^* are determined endogenously by household’s decisions, government policies, and the conditions in the international capital markets.

Note that the primary current account balance is consistent with the intertemporal sustainability of the net external debt (−A ^*): CAP = NX =  − A ^* ⋅ [(1 + r) − (1 + ξ) ⋅ (1 + p)]. Thus, if the interest rate is greater than the rate of output growth (1 + r) > (1 + ξ) ⋅ (1 + p) and the economy is externally indebted, it must run a trade surplus in the steady state.

Parameter Values

The Table below shows our choice of parameter values. Parameter values corresponding to the utility and production functions (α and β), and the rate of depreciation (δ) are standard in the literature. The leisure parameter (γ) is set so that the fraction of time worked by a representative household is 0.293.¹⁹ The values for the rate of labor-augmenting technological progress (ξ) and population growth (p) match those observed historically in Spain.²⁰ The international interest rate (r) and the discount factor (β) jointly determine the economy’s current account and net foreign investment position; we set them to obtain a current account deficit of 3% of GDP in the baseline (Table 1). The choice of parameter values related to fiscal and pension systems is guided by historical observations from Spain.²¹ Also based on Spanish data, the working life (T _t  = T) and retirement period (${T}_t^R={T}^R$) are set so that households enter the labor force when they are 23 years old, retire at age 63, and die at 81 years old. The labor skill profile (e ^s) is hump-shaped, as reported by Hansen (1993).²² Unless otherwise indicated, the qualitative results presented below are robust to plausible changes in parameter values.

Household’s Optimization Problem

It can be expressed as a sequence of two dynamic optimization problems, as follows:

$$ \underset{{\left\{{c}_{t+s-1}^s,{l}_{t+s-1}^s,{A}_{t+s}^{s+1}\right\}}_{s=1}^{T_t}}{Max}{\displaystyle \sum_{s=1}^{T_t}{\beta}^{s-1}\cdot \left\{ \log \left({c}_{t+s-1}^s\right)+\gamma \cdot \log \left({l}_{t+s-1}^s\right)\right\}}+{\beta}^{T_t}\cdot V\left({A}_{t+{T}_t}^{T_t+1},{b}_{t+{T}_t}^{T_t+1}\right) $$

subject to (2); ${b}_{t+{T}_t}^{T_t+1}=\psi \cdot \frac{1}{T_t}\cdot {\displaystyle \sum_{j=1}^{T_t}{W}_{t+j-1}\cdot {e}^j\cdot {n}_{t+j-1}^j}$; ${b}_{t+s-1}^s={b}_{t+{T}_t}^{T_t+1},s={T}_t+2,\dots {T}_t+{T}_t^R$; and ${A}_t^1={A}_t^{T_t+{T}_t^R+1}=0.$ $V\left({A}_{t+{T}_t}^{T_t+1},{b}_{t+{T}_t}^{T_t+1}\right)$ is the household’s value function or discounted indirect utility when it retires at time t + T _t having reached the age of T _t  + 1 years. Upon retirement, the household’s optimization problem can be expressed recursively, and a closed-form solution for the value function (V) follows from the log utility assumption.²³ We first show the derivation of the value function; then we show the first order conditions of the household’s optimization problem.

Value Function at Retirement

The value function is the solution of the following problem: $V\left({A}_{t+{T}_t}^{T_t+1},{b}_{t+{T}_t}^{T_t+1}\right)=\underset{{\left\{{c}_{t+s-1}^s,{A}_{t+s}^{s+1}\right\}}_{s={T}_t+1}^{T_t+{T}_t^R}}{Max}{\displaystyle \sum_{s={T}_t+1}^{T_t+{T}_t^R}{\beta}^{s-1}\cdot \log \left({c}_{t+s-1}^s\right)}$ subject to (2); ${b}_{t+{T}_t}^{T_t+1}=\psi \cdot \frac{1}{T_t}\cdot {\displaystyle \sum_{j=1}^{T_t}{W}_{t+j-1}\cdot {e}^j\cdot {n}_{t+j-1}^j}$; ${b}_{t+s-1}^s={b}_{t+{T}_t}^{T_t+1},s={T}_t+2,\dots {T}_t+{T}_t^R$; ${A}_t^{T_t+{T}_t^R+1}=0$; and given ${A}_{t+{T}_t}^{T_t+1}$ and ${b}_{t+{T}_t}^{T_t+1}$. The household’s asset holdings at retirement (${A}_{t+{T}_t}^{T_t+1}$), and the annual pension benefit (${b}_{t+{T}_t}^{T_t+1}$) are given, as they are determined by household’s past decisions. We obtain the value function by backward induction, starting with the household’s problem in its last year of life.

• 1. The household’s problem at date $t+{T}_t+{T}_t^R-1$ (household’s age is $s={T}_t+{T}_t^R$) is given by

  $$ V\left({A}_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R},{b}_{t+{T}_t}^{T_t+1}\right)=\underset{A_{t+{T}_t+{T}_t^R}^{T_t+{T}_t^R+1}}{Max} \log \left\{\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-1}\right)\cdot {A}_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R}+{b}_{t+{T}_t}^{T_t+1}-\left(1+\xi \right)\cdot {A}_{t+{T}_t+{T}_t^R}^{T_t+{T}_t^R+1}\right\}. $$

The household consumes all its remaining assets in its last period of life, as it leaves no bequests and the no-Ponzi condition (${A}_{t+{T}_t+{T}_t^R}^{T_t+{T}_t^R+1}=0$) is satisfied. Thus, the solution is given by

$$ V\left({A}_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R},{b}_{t+{T}_t}^{T_t+1}\right)= \log \left\{\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-1}\right)\cdot {A}_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R}+{b}_{t+{T}_t}^{T_t+1}\right\}. $$

• 2. The household’s problem at date $t+{T}_t+{T}_t^R-2$ (household’s age is ${T}_t+{T}_t^R-1$) is given by

  $$ \begin{array}{l}V\left({A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1},{b}_{t+{T}_t}^{T_t+1}\right)=\underset{A_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R}}{Max} \log \left\{\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-2}\right)\cdot {A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1}+{b}_{t+{T}_t}^{T_t+1}-\left(1+\xi \right)\cdot {A}_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R}\right\}\\ {}\begin{array}{lll}\hfill \begin{array}{ccc}\hfill \begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill \end{array}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}+\beta \cdot V\left({A}_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R},{b}_{t+{T}_t}^{T_t+1}\right).\end{array} $$

Plug the solution of $V\left({A}_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R},{b}_{t+{T}_t}^{T_t+1}\right)$ found in 1 to obtain the following expression:

$$ \begin{array}{l}V\left({A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1},{b}_{t+{T}_t}^{T_t+1}\right)=\underset{A_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R}}{Max} \log \left\{\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-2}\right)\cdot {A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1}+{b}_{t+{T}_t}^{T_t+1}-\left(1+\xi \right)\cdot {A}_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R}\right\}\\ {}\begin{array}{ccc}\hfill \begin{array}{ccc}\hfill \begin{array}{ccc}\hfill \begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill \end{array}+\beta \cdot \log \left\{\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-1}\right)\cdot {A}_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R}+{b}_{t+{T}_t}^{T_t+1}\right\}.\end{array} $$

Find the first order condition of this optimization problem and solve for ${A}_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R}$,

$$ {A}_{t+{T}_t+{T}_t^R-1}^{T_t+{T}_t^R}=\frac{\beta \cdot {\displaystyle \prod_{i=1}^2\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)\cdot {A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1}-\left[1+\xi -\beta \cdot \left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-1}\right)\right]\cdot {b}_{t+{T}_t}^{T_t+1}}}{\left(1+\xi \right)\cdot \left(1+\beta \right)\cdot \left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-1}\right)}; $$

plug this expression into the value function $V\left({A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1},{b}_{t+{T}_t}^{T_t+1}\right)$ and solve, as follows:

$$ V\left({A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1},{b}_{t+{T}_t}^{T_t+1}\right)=\left(1+\beta \right)\cdot \log \left\{{\displaystyle \prod_{i=1}^2\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)\cdot }{A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1}+\left(2+{\tilde{r}}_{t+{T}_t+{T}_t^R-1}+\xi \right)\cdot {b}_{t+{T}_t}^{T_t+1}\right\}-{\Omega}_1, $$

where Ω₁ is a constant: ${\Omega}_1= \log \left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-1}\right)+\left(1+\beta \right)\cdot \log \left(1+\beta \right)+\beta \cdot \log \left(1+\xi \right)-\beta \cdot \log \left(\beta \right).$

• 3. The household’s problem at date $t+{T}_t+{T}_t^R-3$ (household’s age is $s={T}_t+{T}_t^R-2$):

  $$ \begin{array}{l}V\left({A}_{t+{T}_t+{T}_t^R-3}^{T_t+{T}_t^R-2},{b}_{t+{T}_t}^{T_t+1}\right)=\underset{A_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1}}{Max} \log \left\{\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-3}\right)\cdot {A}_{t+{T}_t+{T}_t^R-3}^{T_t+{T}_t^R-2}+{b}_{t+{T}_t}^{T_t+1}-\left(1+\xi \right){A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1}\right\}\\ {}\begin{array}{ccc}\hfill \begin{array}{ccc}\hfill \begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill \end{array}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}+\beta \cdot V\left({A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1},{b}_{t+{T}_t}^{T_t+1}\right).\end{array} $$

Replacing $V\left({A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1},{b}_{t+{T}_t}^{T_t+1}\right)$ from 2, we can write the previous expression as follows:

$$ \begin{array}{l}V\left({A}_{t+{T}_t+{T}_t^R-3}^{T_t+{T}_t^R-2},{b}_{t+{T}_t}^{T_t+1}\right)=\underset{A_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1}}{Max} \log \left\{\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-3}\right)\cdot {A}_{t+{T}_t+{T}_t^R-3}^{T_t+{T}_t^R-2}+{b}_{t+{T}_t}^{T_t+1}-\left(1+\xi \right)\cdot {A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1}\right\}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}+\beta \cdot \left(1+\beta \right)\cdot \log \left\{{\displaystyle \prod_{i=1}^2\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)}\cdot {A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1}+\left(2+{\tilde{r}}_{t+{T}_t+{T}_t^R-1}+\xi \right)\cdot {b}_{t+{T}_t}^{T_t+1}\right\}-\beta \cdot {\Omega}_1.\end{array} $$

Find the first order condition and solve for ${A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1}$,

$$ {A}_{t+{T}_t+{T}_t^R-2}^{T_t+{T}_t^R-1}=\frac{\beta \cdot \left(1+\beta \right)\cdot {\displaystyle \prod_{i=1}^3\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)\cdot {A}_{t+{T}_t+{T}_t^R-3}^{T_t+{T}_t^R-2}+\left[\beta \cdot \left(1+\beta \right)\cdot {\displaystyle \prod_{i=1}^2\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)}-\left(1+\xi \right)\cdot \left(2+{\tilde{r}}_{t+{T}_t+{T}_t^R-1}+\xi \right)\right]\cdot {b}_{t+{T}_t}^{T_t+1}}}{\left(1+\xi \right)\cdot \left(1+\beta +{\beta}^2\right)\cdot {\displaystyle \prod_{i=1}^2\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)}} $$

Plug this previous expression into the value function $V\left({A}_{t+{T}_t+{T}_t^R-3}^{T_t+{T}_t^R-2},{b}_{t+{T}_t}^{T_t+1}\right)$ and solve,

$$ V\left({A}_{t+{T}_t+{T}_t^R-3}^{T_t+{T}_t^R-2},{b}_{t+{T}_t}^{T_t+1}\right)=\left(1+\beta +{\beta}^2\right)\cdot \log \left\{{\displaystyle \prod_{i=1}^3\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)}\cdot {A}_{t+{T}_t+{T}_t^R-3}^{T_t+{T}_t^R-2}+\left[{\displaystyle \prod_{i=1}^2\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)+\left(2+\xi +{\tilde{r}}_{t+{T}_t+{T}_t^R-1}\right)}\right]\cdot {b}_{t+{T}_t}^{T_t+1}\right\}-{\Omega}_2, $$

where Ω₂ is a constant: ${\Omega}_2= \log \left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-1}\right)+ \log \left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-2}\right)+\left(1+\beta +{\beta}^2\right)\cdot \log \left(1+\beta +{\beta}^2\right)$ +β ⋅ (1 + β) ⋅  log (1 + ξ) − β ⋅ (1 + β) ⋅  log [β ⋅ (1 + β)] + β ⋅ Ω₁.

• 4. Repeating the procedure backwards, the value function at date t + T _t  (household’s age is T _t  + 1) is given by

  $$ V\left({A}_{t+{T}_t}^{T_t+1},{b}_{t+{T}_t}^{T_t+1}\right)=\left({\displaystyle \sum_{j=1}^{T_t^R}{\beta}^{j-1}}\right)\cdot \log \left\{{A}_{t+{T}_t}^{T_t+1}\cdot {\displaystyle \prod_{i=1}^{T_t^R}\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)}+{b}_{t+{T}_t}^{T_t+1}\cdot \right.\left\{{\displaystyle \prod_{i=1}^{T_t^R-1}\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)}\left.+\left(1+\xi \right)\cdot \left[1+\xi +{\displaystyle \sum_{j=1}^{T_t^R-2}{\displaystyle \prod_{i=1}^j\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)}}\right]\right\}\right\}-\Omega, $$

  where Ω is a constant. The derivatives of the value function with respect to changes in asset holdings (V _A ) and pension benefits (V _b ) are given by

  $$ \begin{array}{l}{V}_A(.)=\frac{\left({\displaystyle \sum_{j=1}^{T_t^R}{\beta}^{j-1}}\right)\cdot {\displaystyle \prod_{i=1}^{T_t^R}\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)}}{A_{t+{T}_t}^{T_t+1}\cdot {\displaystyle \prod_{i=1}^{T_t^R}\left(1+{r}_{t+{T}_t+{T}_t^R-i}\right)}+{b}_{t+{T}_t}^{T_t+1}\cdot \left\{{\displaystyle \prod_{i=1}^{T_t^R-1}\left(1+{r}_{t+{T}_t+{T}_t^R-i}\right)}+\left(1+\xi \right)\cdot \left[1+\xi +{\displaystyle \sum_{j=1}^{T_t^R-2}{\displaystyle \prod_{i=1}^j\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)}}\right]\right\}},\\ {}{V}_b(.)=\frac{\left({\displaystyle \sum_{j=1}^{T_t^R}{\beta}^{j-1}}\right)\cdot \left\{{\displaystyle \prod_{i=1}^{T_t^R-1}\left(1+{r}_{t+{T}_t+{T}_t^R-i}\right)}+\left(1+\xi \right)\cdot \left[1+\xi +{\displaystyle \sum_{j=1}^{T_t^R-2}{\displaystyle \prod_{i=1}^j\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)}}\right]\right\}}{A_{t+{T}_t}^{T_t+1}\cdot {\displaystyle \prod_{i=1}^{T_t^R}\left(1+{r}_{t+{T}_t+{T}_t^R-i}\right)}+{b}_{t+{T}_t}^{T_t+1}\cdot \left\{{\displaystyle \prod_{i=1}^{T_t^R-1}\left(1+{r}_{t+{T}_t+{T}_t^R-i}\right)}+\left(1+\xi \right)\cdot \left[1+\xi +{\displaystyle \sum_{j=1}^{T_t^R-2}{\displaystyle \prod_{i=1}^j\left(1+{\tilde{r}}_{t+{T}_t+{T}_t^R-i}\right)}}\right]\right\}}.\end{array} $$

Table 2 Baseline parameter values