Pensions

Abstract

In this chapter, we first review empirical facts of public pension systems in OECD countries. Subsequently, we introduce a public pension system in the standard two-period overlapping generations (OLG) model of Chap. 2. We consider two different social security systems, pay-as-you-go (PAYG) versus fully funded. While a fully funded pension system does not have any effect on aggregate savings if capital markets are perfect, aggregate savings fall significantly in a PAYG system. Since public pensions are likely to distort household labor supply decisions, we endogenize labor supply below. In addition, we extend the two-period model to a more realistic 70-period model in which the retirement period is smaller than the working period. Next, we derive the optimal amount of pensions in a PAYG system and study how the demographic transition and aging of the population affect the sustainability of social security. We also discuss the findings of the literature on quantitative pension studies in detail. Finally, we introduce the concept of fiscal space and point out its sensitivity with respect to the aging that takes place in many industrialized countries at present.

Appendices

Appendix 6.1: Accidental Bequests

To understand why accidental bequests in (6.56e) are given by the sum of next-period assets $\tilde k'$ and the interest $r_{t+1}\tilde k'$, we will consider a simplified two-period model. In particular, let N _t(1) and N _t(2) denote the sizes of the young and old generations in period t. The survival probability of the young is denoted by ϕ ₁, and population grows at rate n, implying:

$$\displaystyle \begin{aligned} N_{t+1}(2)&=\phi_1 N_t(1), \end{aligned} $$

(6.61a)

$$\displaystyle \begin{aligned} N_{t+1}(1)&=(1+n) N_t(1). \end{aligned} $$

(6.61b)

Consequently, total population is given by

$$\displaystyle \begin{aligned} N_t = N_t(1)+N_t(2).\end{aligned}$$

Total accidental bequests are confiscated by the government and redistributed lump-sum to the total population in the amount N _t+1 tr _t+1 in period t + 1:

$$\displaystyle \begin{aligned} Beq_{t+1}= N_{t+1} tr_{t+1}.\end{aligned}$$

For the rest of the model, we stipulate that the standard equilibrium conditions of the two-period OLG model in Chap. 3 hold. Therefore, the aggregate capital stock K _t+1 at the beginning of period t + 1 is equal to total savings of the young generation at the end of period t, N _t(1)s _t; consumption of the young $c^1_{t+1}$ is equal to their wage income plus transfers minus savings; and consumption of the old $c^2_{t+1}$ is equal to savings plus interest and transfers. Finally, the factor prices are equal to their marginal products. These conditions are summarized by the following equations:

$$\displaystyle \begin{aligned} N_t(1) s_t &= K_{t+1}, \end{aligned} $$

(6.62a)

$$\displaystyle \begin{aligned} N_{t+1}(1) s_{t+1} &=K_{t+2}, \end{aligned} $$

(6.62b)

$$\displaystyle \begin{aligned} c^1_{t+1} &= w_{t+1} l_{t+1} + tr_{t+1} -s_{t+1}, \end{aligned} $$

(6.62c)

$$\displaystyle \begin{aligned} c^2_{t+1} & = (1+r_{t+1})s_t + tr_{t+1}, \end{aligned} $$

(6.62d)

$$\displaystyle \begin{aligned} r_{t+1} &= F_K\left(K_{t+1},l_{t+1}N_{t+1}(1)\right) -\delta, \end{aligned} $$

(6.62e)

$$\displaystyle \begin{aligned} w_{t+1} & =F_L\left(K_{t+1},l_{t+1}N_{t+1}(1)\right). \end{aligned} $$

(6.62f)

As before, we assume that production is characterized by constant returns to scale, implying:

$$\displaystyle \begin{aligned} Y_t = F(K_t,l_{t}N_{t}(1) ) = F_K\left(K_{t},l_{t}N_{t}(1)\right) K_t + F_L\left(K_{t},l_{t}N_{t}(1)\right) l_t N_t(1).\end{aligned}$$

In the goods market equilibrium:

$$\displaystyle \begin{aligned} K_{t+1}(1-\delta) + Y_{t+1} = N_{t+1}(1) c^1_{t+1} + N_{t+1}(2) c^2_{t+1} + K_{t+2}.\end{aligned} $$

(6.63)

Inserting the above equations into the equation for the goods market equilibrium, we derive

$$\displaystyle \begin{aligned} \begin{aligned} &K_{t+1} +w_{t+1} l_{t+1} N_{t+1}(1) + r_{t+1} K_{t+1}\\ &= w_{t+1} l_{t+1} N_{t+1}(1) + N_{t+1} (1) tr_{t+1} - N_{t+1}(1) s_{t+1} \\ &+N_{t+1}(2) \left[ 1+r_{t+1} \right] s_t + N_{t+1}(2) tr_{t+1}+K_{t+2}, \end{aligned} \end{aligned}$$

and therefore,

$$\displaystyle \begin{aligned} \begin{aligned} K_{t+1} + r_{t+1} K_{t+1}&= N_{t+1} tr_{t+1}+ N_{t+1}(2) \left[ 1+r_{t+1} \right] s_t\\ &= N_{t+1} tr_{t+1}+ \phi_1 N_{t}(1) \left[ 1+r_{t+1} \right] s_t\\ &= N_{t+1} tr_{t+1}+ \phi_1 \left[ 1+r_{t+1} \right] K_{t+1}.\end{aligned} \end{aligned}$$

Consequently,

$$\displaystyle \begin{aligned} Beq_{t+1} = N_{t+1} tr_{t+1} = (1-\phi_1) \left[ 1+r_{t+1} \right] K_{t+1}.\end{aligned} $$

(6.64)

Appendix 6.2: Computation of the Large-Scale OLG Model in Sect. 6.4

Solving for the stationary equilibrium is a computational challenge. Recall how we solved the two-period OLG model with a PAYG pension system and defined contribution benefits in Sect. 6.3.2. We were able to directly solve the first-order conditions of the households together with the equilibrium equations. In particular, the optimality condition with respect to the labor supply was given by (6.33c). In the large-scale OLG model in Sect. 6.4, the computational problem is much more complicated. We cannot solve for the optimal labor supply for all workers alive because of their high numbers. For this reason, consider the first period of life, s = 1. In this cohort, we have four different types of workers with the productivity types η _i 𝜖 _j with i = 1, 2 and j = 1, 2. They all accumulate different amounts of savings. At age 2, they may change their productivity type such that the number of heterogeneous workers increases to 4 × 2 = 8. Since the workers accumulate different amounts of savings depending on their employment history, the workers in a cohort are different (with respect to their savings) even for the same productivity type 𝜖 _i η _j. Prior to retirement at age s = 45, we observe 4 × 2⁴⁴ = 7.0 ⋅ 10¹³ different workers. We cannot solve this problem with the help of direct computation for such a large number of agents. Instead, we use value function iteration.

2.1 Value Function Iteration

To describe the optimization problem, we use a recursive representation of the consumer’s problem, following Stokey, Lucas, and Prescott (1989). This specification is very amenable to the solution methods described below. Let $V_t(k_t^s,s,\epsilon ,\eta )$ be the value of the objective function of the s-year-old agent with wealth $k_t^s$, age s, permanent efficiency type 𝜖, and individual productivity η in period t. The value function V _t(.) is equal to the optimized discounted expected lifetime utility. Thus, for the individual during the last period of his life, s = 70, the stationary value function is simply given by:

$$\displaystyle \begin{aligned} V_t(k_t^{70},70,\epsilon,\eta) = \max_{c_t^{70},k_{t+1}^{71}} u(c_t^{70},1) \end{aligned}$$

subject to $k_{t+1}^{71}\ge 0$ and the budget constraint (6.55) noticing that $c^s_t=A_t \tilde c_t^s$ and $k^s_t=A_t \tilde k^s_t$. Obviously, the optimal policy is given by $\tilde k_{t+1}^{71}=0$ and

$$\displaystyle \begin{aligned} \tilde c_t^{70}= \tilde pen_t + \left[1+(1-\tau^K) r_t\right] \tilde k_t^{70}+\tilde tr_t.\end{aligned}$$

The household completely consumes its income (from pensions and interest) and wealth during the last period of life.

In the second-to-last period of life, s = 69, the value function is given by the following equation:

$$\displaystyle \begin{aligned} V_t(k_t^{69},69,\epsilon,\eta) = \max_{c_t^{69},k_{t+1}^{70},c_{t+1}^{70},k_{t+2}^{71}} \left\{ u(c_t^{69},1) + \beta \mathbb{E}_t u(c_{t+1}^{70},1) \right\} \end{aligned}$$

subject to the budget constraint (6.55) in periods t and t + 1. It will be convenient to transform the above equation into one with stationary values. For this reason, divide the equation by $A_t^{\iota (1-\sigma )}$, which results in

$$\displaystyle \begin{aligned} \begin{aligned} & \tilde V_t(\tilde k_t^{69},69,\epsilon,\eta) \\ &= \max_{\tilde c_t^{69},\tilde k_{t+1}^{70},\tilde c_{t+1}^{70},\tilde k_{t+2}^{71}} \left\{ u(\tilde c_t(69),1) + (1+\gamma)^{\iota(1-\sigma)} \beta \mathbb{E}_t u(\tilde c_{t+1}^{70},1) \right\}, \end{aligned} \end{aligned}$$

with $\tilde V_t\equiv V_t/A_t^{\iota (1-\sigma )}$. In addition, we have used the fact that

$$\displaystyle \begin{aligned} \frac{u(c_{t+1}^{70},1)}{A_t^{\iota (1-\sigma)}}=\frac{(c_{t+1}^{70})^{\iota (1-\sigma)}}{1-\sigma} \frac{1}{A_t^{\iota (1-\sigma)}}= \frac{(\tilde c_{t+1}^{70})^{\iota (1-\sigma)}} {1-\sigma} (1+\gamma)^{\iota(1-\sigma)}.\end{aligned}$$

Equivalently, the above dynamic equation can be restated as a recursive equation as follows:

$$\displaystyle \begin{aligned} \begin{aligned} & \tilde V_t(\tilde k_t^{69},69,\epsilon,\eta)= \max_{\tilde c_t^{69},\tilde k_{t+1}^{70}} \left\{ u(\tilde c_t^{69},1) \right. \\ & \left. + (1+\gamma)^{\iota(1-\sigma)} \beta \mathbb{E}_t \tilde V_{t+1}(\tilde k_{t+1}^{70},70,\epsilon,\eta) \right\}. \end{aligned} \end{aligned}$$

For the household aged s, we can use this recursive formulation more generally as follows:

(6.65)

subject to (6.55) and (6.46). Equation (6.65) is also known as the Bellman equation. We will exploit its recursive nature to compute the optimal policy functions of the households (for given factor prices {w _t, r _t}, government transfers $\tilde tr_t$, and pension policies $\tilde pen_t$).

We solve for the optimization problem of the household starting in the last period of its life, working our way back to the first period of the household’s life. As we do not know the exact value of $\tilde k_t^{70}$, we compute the value function $\tilde V(\tilde k,70,\epsilon ,\eta )$ over a range of $\tilde k\in [\tilde k_{min},\tilde k_{max}]$. Since we assume a liquidity constraint $\tilde k\ge 0$, we choose $\tilde k_{min}=0$. Finding a good value for the upper limit of the interval $\tilde k_{max}$ is more difficult. Since we do not yet know the equilibrium values of w, r, $\tilde tr$, or $\tilde pen$, we do not know the maximum income of households that is required to obtain a possible guess for the upper limit of savings. We, instead, advocate for a rather pragmatic procedure. We will calibrate our model such that the real interest rate r is approximately equal to 4% as observed in the US economy. In addition, we know the mass of working agents in our model, which is approximately 2/3. Assuming that average productivity is equal to one and agents work approximately 30% of their available time (we will calibrate the model accordingly below), we obtain a rough estimate of $\tilde L \approx 0.2$. From the first-order condition of the firm, we know that $r=\alpha \tilde k^{\alpha -1} \tilde L^{1-\alpha }-\delta$. Consequently, we can compute an approximate value of $\tilde k=6.19$. Since savings initially increase over the working life, wealth $\tilde k_t^s$ will also be hump-shaped over the life-cycle and may well exceed average wealth. We, therefore, choose an initial value of $\tilde k_{max}=20$ and find it to be non-binding in our computations.

Since we cannot compute the value function of the 70-year-old at each point of the interval $[\tilde k_{min},\tilde k_{max}]$ (the number of all points is infinite), we only perform this calculation at certain grid points. We choose equispaced grids with n _k = 100 points. In the computation of the value function of the 69-year-old, we will need the value of the value function at age s = 70 between grid points. To obtain this, we will interpolate linearly between grid points if necessary.

Computing the value function $\tilde V(\tilde k, s, \epsilon ,\eta )$ at a grid point $(\tilde k^i,70,\epsilon ,\eta )$, i = 1, …, n _k, is straightforward and follows from the budget constraint and the definition of the utility function:

$$\displaystyle \begin{aligned} \tilde V_t(\tilde k^i, 70,\epsilon,\eta) =\frac{\left( \tilde pen_t + \left[1+(1-\tau^K) r_t\right] \tilde k^i +\tilde tr_t \right)^{\iota (1-\sigma)}}{1-\sigma}.\end{aligned}$$

Notice that the value of the value function is the same for all productivity types (𝜖, η) since they no longer affect income as long as households have accumulated the same wealth $\tilde k^i$.

For the retired households with age s = 69, 68, …, 46, we have to solve an optimization problem in one variable. To do so, let us consider the Bellman equation at age s = 69 at a grid point $(\tilde k^i,69,\epsilon ,\eta )$ ⁷¹:

$$\displaystyle \begin{aligned} \tilde V_t(\tilde k^i,69,\epsilon,\eta) = \max_{\tilde c_t^{69},\tilde k_{t+1}^{70}} \left\{ u(\tilde c_t^{69},1) + (1+\gamma)^{\iota(1-\sigma)}\beta \mathbb{E}_t \tilde V_t(\tilde k_{t+1}^{70},70,\epsilon,\eta) \right\}. \end{aligned}$$

If we substitute for $\tilde c_t^{69}$ from the budget constraint (6.55), the maximand within the brackets is a function of $\tilde k_{t+1}^{70}$. There are various numerical procedures to solve such a maximization problem. In the Gauss program Ch6_optimal_pension.g, we use the so-called golden section search.⁷² The basic idea of this method is to bracket the maximum $\tilde k^{-}\le \tilde k_{t+1}^{70} \le \tilde k^{+}$ and let this interval shrink against zero. The limit points are easy to choose, e.g., $\tilde k^{-}=0$ for the lower limit and $\tilde k^{+}$ as the value for which $\tilde c_t=0$. In the next step, two points within the interval are selected, and the one with the lower value for the right-hand side of the above equation becomes the new limit point of the interval. The golden section search method optimizes the choice of these new points. This procedure is iterated until the interval length is sufficiently small, e.g., 10⁻⁶, and we then stop.

For the working agent, the maximization problem is more complicated because he also chooses his optimal labor supply $l_t^s$. There are various numerical methods that are able to compute this two-dimensional optimization problem. We have chosen to transform the problem into two nested one-dimensional optimization problems and apply golden section search in the outer loop over the next-period capital stock $\tilde k_{t+1}$ and direct computation from the first-order condition with respect to labor l _t with the help of the Gauss-Newton algorithm in the inner loop.

Once we have solved the individual optimization problem, we can aggregate individual savings and labor supply to derive aggregate quantities and update our initial guesses of $\tilde K$, $\tilde L$, and the factor prices w and r. The budgets of the government and the pension system imply the values for $\tilde tr$ and τ ^p. We update the old values by taking a weighted average of the two and iterating until convergence. The complete algorithm is described in Algorithm 6.1.

Algorithm 6.1 Computation of the Stationary Equilibrium of the OLG Model in Sect. 6.4.1

We compute the transition dynamics for the US economy as described in Algorithm 9.2.1 in Heer and Maußner (2009). We first choose a number of transition periods under the assumption that the transition is complete by 2250.⁷³ Next, we compute the initial and final steady states and project a trajectory for the endogenous values of $\{\tilde K_t,\tilde L_t,\bar l_t,\tau ^w_t,\tau ^p_t,\tilde tr_t\}_{t=2015}^{2250}$. As our initial guess, we postulate a linear adjustment path for all endogenous variables. We assume that the economy is in steady state in and prior to 2015. For given path of $\{\tilde K_t,\tilde L_t,\bar l_t,\tau ^w_t,\tau ^p_t,\tilde tr_t\}_{t=2015}^{2250}$, we compute the individual policy functions in each year and aggregate individual labor supply and consumption. With the help of the consistency conditions and the fiscal budget constraints, we are able to provide a new guess for the path of $\{\tilde K_t,\tilde L_t,\bar l_t,\tau ^w_t,\tau ^p_t,\tilde tr_t\}_{t=2015}^{2250}$. Again, we use a simple dampening iterative scheme, as described in Section 3.9 of Judd (1998), to update the sequence $\{\tilde K_t,\tilde L_t,\bar l_t,\tau ^w_t,\tau ^p_t,\tilde tr_t\}_{t=2015}^{2250}$ until the sequence converges (with an accuracy equal to 10⁻⁶).

The run time of the computer program Ch6_optimal_pension.g depends sensitively on the amount of grid points n _k and is considerable. Using Windows 7 on a computer with a 64-BIT system, 32 MB RAM, and an Intel(R) Xeon(R) 2.90 GHz processor, the stationary equilibrium is computed within 4 min, while the computation of the transition takes approximately 14 h.

Appendix 6.3: Data Sources

The data on population are taken from the UN, while the pensions-related data are retrieved from the OECD.

• Old-age dependency ratio The data presented in Fig. 6.1 is published by the UN in its World Population Prospects: The 2017 Revision, ‘File POP/13-D: Old-age dependency ratio 65+/(25–64) by region, subregion and country, 1950–2100 (ratio of population 65+ per 100 population 25–64)’. The UN provides projections of the OADR for a ‘low’, ‘medium’, and ‘high’ fertility variant. If not mentioned otherwise, we use the ‘medium’ fertility variant.

• Pension spending The data displayed in Fig. 6.3 are taken from the OECD (2015), Pensions Statistics: Pensions at a Glance (Accessed on February 15, 2018).

  http://data.oecd.org/socialexp/pension-spending.htm.

• Pension replacement rates The data in Fig. 6.4 presents the gross pension replacement rates of men as a percentage of pre-retirement earnings and can be retrieved from OECD (2017), Gross pension replacement rates (indicator). doi: 10.1787/3d1afeb1-en (Accessed on February 15, 2018).

  https://data.oecd.org/pension/gross-pension-replacement-rates.htm.

Problems

6.1

Recompute the solution to Numerical Examples in Sect. 6.3 with the following changes:

• 1. Check the robustness of the results with respect to the intertemporal elasticity of substitution 1∕σ, with σ ∈{2, 4}.

• 2. Assume that capital depreciates completely so that the real interest rate is represented by

  $$\displaystyle \begin{aligned} r_t= \alpha k_t^{\alpha-1} l_t^{1-\alpha}-1.0.\end{aligned}$$

6.2

Contribution-Based Pensions in the Three-Period OLG Model Assume that an agent lives three periods. Each period length is equal to 20 years. In the first two periods, the agent is working, and in the third period, he receives a pension. Each generation has mass 1∕3. We will only consider the steady state.

Lifetime utility is given by

$$\displaystyle \begin{aligned} U= \sum_{s=1}^3 \beta^{s-1} u(c^s,1-l^s).\end{aligned} $$

(6.66)

Instantaneous utility is presented by

$$\displaystyle \begin{aligned} u(c,1-l)=u(c,1-l)=\frac{\left( c (1-l)^\iota \right)^{1-\sigma}}{1-\sigma},\end{aligned} $$

with ι = 2.0 and σ = 2.0. Assume that β = 0.90. Time is allocated to either work or leisure.

During the first two periods, households work; in the third period, they retire (l ³ ≡ 0). Agents are born without assets, k ¹ = 0. In addition, the workers pay contributions to the pension system equal to τ = 10% of their gross labor income. Therefore, the budget constraint at age s = 1, 2 is given by

$$\displaystyle \begin{aligned} (1-\tau) w l^s +(1+r) k^s = c^s + k^{s+1}. \end{aligned} $$

During retirement, agents receive pensions that depend on past earnings

$$\displaystyle \begin{aligned} d=\sum_{s=1}^2 \tau w l^s.\end{aligned} $$

In particular, the pension system does not pay any interest on accumulated contributions. The pension depends on accumulated contributions as follows:

$$\displaystyle \begin{aligned} pen(d)=pen_{min}+\rho_{pen} d.\end{aligned} $$

Therefore, the budget constraint of the retired worker is given by:

$$\displaystyle \begin{aligned} pen(d) +(1+r) k^3 = c^3. \end{aligned} $$

In addition, we assume that the government runs a balanced budget:

$$\displaystyle \begin{aligned} \tau w \frac{l^1+l^2}{3} = \frac{1}{3} \left( pen_{min}+\rho_{pen} d\right).\end{aligned} $$

Assume that ρ _pen = 0.5, implying

$$\displaystyle \begin{aligned} pen_{min}=0.5 \tau w (l^1+l^2).\end{aligned} $$

Production is modeled as in Sect. 6.3:

$$\displaystyle \begin{aligned} Y=K^\alpha L^{1-\alpha},\end{aligned} $$

with

$$\displaystyle \begin{aligned} L= \frac{l^1+l^2}{3}, \;\;\; K= \frac{k^2+k^3}{3},\end{aligned}$$

and α = 0.36.

Factors are rewarded by their marginal products:

$$\displaystyle \begin{aligned} \begin{aligned} w_t & = (1-\alpha) \left(\frac{K_t}{L_t}\right)^{\alpha},\\ r_t& =\alpha \left(\frac{K_t}{L_t}\right)^{\alpha-1}-\delta. \end{aligned} \end{aligned}$$

The depreciation rate is set equal to δ = 0.5.

• 1. Solve the problem with the help of direct computation (solving a system of non-linear equations). Show that the first-order conditions are given by

  $$\displaystyle \begin{aligned} \begin{aligned} \lambda^1 &=\left(c^1\right)^{-\sigma} \left( 1-l^1\right)^{\iota (1-\sigma)} ,\\ \lambda^2 &=\left(c^2\right)^{-\sigma} \left( 1-l^2\right)^{\iota (1-\sigma)} ,\\ \lambda^3 &=\left(c^3\right)^{-\sigma}, \\ \iota \left(c^1\right)^{1-\sigma} \left( 1-l^1\right)^{\iota (1-\sigma)-1} & = \lambda^1(1-\tau) w +\beta^2 \lambda^3 \rho_{pen} \tau w,\\ \iota \left(c^2\right)^{1-\sigma} \left( 1-l^2\right)^{\iota (1-\sigma)-1} & = \lambda^2 (1-\tau) w +\beta \lambda^3 \rho_{pen} \tau w,\\ \lambda^1 &= \beta \lambda^2 (1+r),\\ \lambda^2 &= \beta \lambda^3 (1+r). \end{aligned} \end{aligned}$$

  For given aggregate variables w and r, the 7 first-order conditions together with the 3 budget constraints are a system of non-linear equations in 10 unknowns c ^s and λ ^s, s = 1, 2, 3, l ¹, l ², k ², k ³. Use numerical methods to solve the system. Start with educated guesses for K and L, compute the individual policy functions, and update K and L accordingly. Compute the implied gross pension replacement rate with respect to the earnings in the second period. How does the abolition of social security affect output, labor, and welfare?

• 2. Assume that the government switches from a defined contribution to a defined benefit system and that it applies the same gross pension replacement rate with respect to the earnings in the second period as above (for the case with τ = 10%). What are the effects on labor supply, savings, output, and the social security contribution rate τ?

6.3

Quasi-Hyperbolic Discounting Recompute the three-period OLG model from Problem 6.2. However, instead assume (1) that pensions are not contribution-based but provided lump-sum (ρ _pen = 0) and (2) that the household behaves inconsistently and in a naive way. Therefore, let the household at age 1 assume that its lifetime utility is represented by

$$\displaystyle \begin{aligned} U= u(c^1,1-l^1)+ \mu \beta u(c^2,1-l^2) +\mu \beta^2 u(c^3,1-l^3),\;\; \mu<1, \end{aligned} $$

(6.67)

rather than by Eq. (6.66), where μ denotes the hyperbolic discounting parameter. The household maximizes its utility for μ = 0.85 in period 1, where the first-order conditions are given by:

$$\displaystyle \begin{aligned} \begin{aligned} \lambda^1 &=\left(c^1\right)^{-\sigma} \left( 1-l^1\right)^{\iota (1-\sigma)} ,\\ \lambda^2 &=\left(c^2\right)^{-\sigma} \left( 1-l^2\right)^{\iota (1-\sigma)} ,\\ \lambda^3 &=\left(c^3\right)^{-\sigma}, \\ \iota \left(c^1\right)^{1-\sigma} \left( 1-l^1\right)^{\iota (1-\sigma)-1} & = \lambda^1(1-\tau) w,\\ \iota \left(c^2\right)^{1-\sigma} \left( 1-l^2\right)^{\iota (1-\sigma)-1} & = \lambda^2 (1-\tau) w,\\ \lambda^1 &=\mu \beta \lambda^2 (1+r),\\ \lambda^2 &= \beta \lambda^3 (1+r). \end{aligned} \end{aligned}$$

Solve this system of equations for 10 unknowns c ^s and λ ^s, s = 1, 2, 3, l ¹, l ², k ², k ³ and denote the solutions by $\tilde c^s$, $\tilde \lambda ^s$, $\tilde k^s$, and $\tilde l^s$.

In period 2, however, the household behaves in an inconsistent way and, for a given k ², re-maximizes⁷⁴

$$\displaystyle \begin{aligned} U= u(c^2,1-l^2)+ \mu \beta u(c^3,1-l^3).\end{aligned}$$

The first-order conditions with respect to c ², c ³, l ², and k ³ are given by:

$$\displaystyle \begin{aligned} \begin{aligned} \lambda^2 &=\left(c^2\right)^{-\sigma} \left( 1-l^2\right)^{\iota (1-\sigma)} ,\\ \lambda^3 &=\left(c^2\right)^{-\sigma}, \\ \iota \left(c^2\right)^{1-\sigma} \left( 1-l^2\right)^{\iota (1-\sigma)-1} & = \lambda^2 (1-\tau) w,\\ \lambda^2 &= \mu \beta \lambda^3 (1+r). \end{aligned} \end{aligned}$$

Denote the solutions to these equilibrium conditions as $\hat c^s$ and $\hat \lambda ^s$, s = 2, 3, $\hat l^2$, and $\hat k^3$. The household suffers from inconsistent behavior and chooses smaller savings for μ < 1 in period 2 than it would have chosen in period 1.⁷⁵

In general equilibrium, aggregate capital and labor are given by

$$\displaystyle \begin{aligned} \begin{aligned} K &=\frac{ \tilde k^2 + \hat k^3}{3},\\ L &= \frac{ \tilde l^1 + \hat l^2}{3}. \end{aligned} \end{aligned}$$

In a large-scale OLG model with individual income uncertainty, İmrohoroğlu, İmrohoroğlu, and Joines (2003) show that quasi-hyperbolic discounting at the rate of 15% lowers the capital stock by approximately 20% at any social security contribution rate.⁷⁶ Can you verify this result in the above example for pension replacement rates θ _pen ∈{0, 50%}?

6.4

Hicksian Compensation Consider Fig. 6.5. Recompute the dynamics of the model under the assumption that the two cohorts alive at the time of the policy changes in period 1 are compensated such that the consumption equivalent change in comparison to the cohorts in the economy without the policy change is zero. Further assume that future generations receive transfers or pay lump-sum taxes such that (1) they all have equal lifetime utility and (2) the net present value of these transfers is equal to the payments of transfers to the cohorts in period 1. How large is the Hicksian efficiency gain?

6.5

Compute the optimal pension in the model of Sect. 6.4.1 under the assumption that the government holds both government consumption $\tilde G$ and transfers $\tilde Tr$ constant at their benchmark equilibrium values. Adjust the wage income tax rate τ ^w such that the government budget (6.50) is balanced.

6.6

A Simple Auerbach-Kotlikoff Model ⁷⁷ Consider a 60-period OLG model in the tradition of Auerbach and Kotlikoff (1987). Three sectors can be depicted: households, production, and the government.

Households Every year, a generation of equal measure is born. The total measure of all generations is normalized to one. Their first period of life is period 1.

Households live J = 60 years. Consequently, the measure of each generation is 1/60. During their first T = 40 years, agents supply labor $l_t^s$ at age s in period t enjoying leisure $1-l_t^s$. After 40 years, retirement is mandatory ($l_t^s=0$ for s > 40) for the remaining T ^R = 20 years. Agents maximize lifetime utility at age 1 in period t:

$$\displaystyle \begin{aligned} \sum_{s=1}^{J} \beta^{s-1} u(c_{t+s-1}^s,1-l_{t+s-1}^s), \end{aligned}$$

where β denotes the discount factor. Instantaneous utility is a function of both consumption and leisure:

$$\displaystyle \begin{aligned} u(c,1-l)=\frac{\left( c (1-l)^\iota \right)^{1-\sigma}-1}{1-\sigma}. \end{aligned} $$

(6.68)

Agents are born without wealth, $k_t^1=0$, and do not leave bequests, $k_t^{61}=0$. Agents receive income from capital $k_t^s$ and labor $l_t^s$. The real budget constraint of the working agent is given by

$$\displaystyle \begin{aligned} k_{t+1}^{s+1}=(1+r_t) k_t^s +(1-\tau_t) w_t l_t^s-c_t^s, \;\;\; s=1,\ldots,T, \end{aligned}$$

where r _t and w _t denote the interest rate and the wage rate in period t, respectively. Wage income in period t is taxed at rate τ _t. We can also interpret $\tau _t w_t l_t^s$ as the worker’s social security contributions.

The first-order conditions of the working household are given by:

$$\displaystyle \begin{aligned} (1-\tau_t) w_t &=\frac{u_{1-l}(c_t^s,1-l_t^s)}{u_c(c_t^s,1-l_t^s)}=\iota \frac{c_t^s}{1-l_t^s},\\\frac{1}{\beta}&= \frac{ u_c(c_{t+1}^{s+1},1-l_{t+1}^{s+1}) } {u_c(c_t^s,1-l_t^s)} \left[1+r_{t+1}\right] \end{aligned} $$

(6.69)

$$\displaystyle \begin{aligned} & = \frac{ \left(c_{t+1}^{s+1}\right)^{-\sigma} \left(1-l_{t+1}^{s+1}\right)^{\iota(1-\sigma)} }{\left(c_{t}^s\right)^{-\sigma} \left(1-l_{t}^s\right)^{\iota(1-\sigma)}}\left[1+r_{t+1}\right]. \end{aligned} $$

(6.70)

During retirement, agents receive public pensions pen irrespective of their employment history, and the budget constraint of the retired worker is given by

$$\displaystyle \begin{aligned} k_{t+1}^{s+1}=(1+r_t) k_t^s + pen-c_t^s, \;\;\; s=T+1,\ldots,T+T^R. \end{aligned}$$

The first-order condition of the retired worker is given by (6.70) with $l_t^s=0$.

Production Firms are of measure one and produce output Y _t in period t with labor L _t and capital K _t. Labor L _t is paid wage w _t. Capital K _t is hired at rate r _t and depreciates at rate δ. Production Y _t is characterized by constant returns to scale and assumed to be Cobb-Douglas:

$$\displaystyle \begin{aligned} Y_t= K_t^\alpha L_t^{1-\alpha}. \end{aligned} $$

(6.71)

In factor market equilibrium, factors are rewarded with their marginal products:

$$\displaystyle \begin{aligned} w_t&=(1-\alpha) K_t^\alpha L_t^{-\alpha}, \end{aligned} $$

(6.72)

$$\displaystyle \begin{aligned} r_t&=\alpha K_t^{\alpha-1} L_t^{1-\alpha}-\delta. \end{aligned} $$

(6.73)

Government The government uses the revenues from taxing labor to finance its expenditures on social security:

$$\displaystyle \begin{aligned} \tau_t w_t L_t= \frac{T^R}{T+T^R} pen. \end{aligned} $$

(6.74)

Following a change in the provision of public pensions pen or in gross labor income w _t L _t, the labor income tax rate τ _t adjusts to keep the government budget balanced.

Equilibrium An equilibrium for a given government policy pen and initial distribution of capital $\left \{k_0^s\right \}_{s=1}^{J}$ is a collection of individual policy rules $c(s,k_t^s,K_t,L_t)$, $l(s,k_t^s,K_t,L_t)$, and $k'(s,k_t^s,K_t,L_t)$, and relative prices of labor and capital {w _t, r _t}, such that:

• (i) Individual and aggregate behavior are consistent:

  $$\displaystyle \begin{aligned} L_t&=\sum_{s=1}^{40} \frac{l_t^s}{60},\\ K_t&=\sum_{s=1}^{60} \frac{k_t^s}{60}. \end{aligned} $$

  The aggregate labor supply L _t is equal to the sum of the labor supplies of each cohort, weighted by their mass 1∕J = 1∕60. Similarly, the aggregate capital supply is equal to the sum of the capital supplies of all cohorts.

• (ii) Relative prices {w _t, r _t} solve the firm’s optimization problem.

• (iii) Given relative prices {w _t, r _t} and the government’s policy pen, the individual policy rules c(.), l(.), and k′(.) solve the consumer’s optimization problem.

• (iv) The goods market clears:

  $$\displaystyle \begin{aligned} K_t^\alpha L_t^{1-\alpha} = \sum_{s=1}^{60} \frac{c_t^s}{60} + K_{t+1}-(1-\delta) K_{t}. \end{aligned} $$

• (v) The government budget is balanced.

Calibration The benchmark case is characterized by the following calibration: σ = 2, β = 0.99, α = 0.3, δ = 0.1, replacement rate $\theta ^p=\frac {pen}{(1-\tau ) w \bar l}=0.3$ (where $\bar l$ denotes the average labor supply in the economy). ι is chosen to imply a steady-state labor supply of the working agents approximately equal to $\bar l=35\%$ of available time and amounts to ι = 2.0.

• 1. Compute the steady state using numerical methods. To do so, you have to consider a system of 99 non-linear equations in the variables k ^s, s = 2, …, 60 and l ^s, s = 1, …, 40. Show that the shape of the consumption-age profile resembles that presented in Fig. 6.11 and displays a downward jump at the beginning of retirement. Show that the steady-state level of output is equal to Y = 0.3842.

• 2. Introduce a consumption tax of 10% that is retransferred lump-sum to the household. How does the consumption tax affect steady-state values?

• 3. Introduce a government sector that consumes 20% of steady-state output Y , G = 0.0768. Compare the following three tax instruments to finance G: (1) a tax on consumption τ ^c, (2) a tax on capital income τ ^K (assuming that depreciation is tax-deductible), and (3) a wage income tax rate τ ^w. Assume further that government consumption provides utility to the household that is additively separated from the utility in consumption and leisure such that the first-order conditions of the household are not affected by government consumption G. What are the steady-state effects of these three tax instruments on capital, labor, and output? What is the tax instrument that results in the lowest welfare losses? To answer these questions, compute the consumption equivalent change of each policy measure with respect to the case without government consumption.

• 4. Introduce consumption habits

  $$\displaystyle \begin{aligned} u(c^s,l^s;c^{s-1}) = \frac{\left[ (c^s-\kappa c^{s-1}) \left(1-l^s\right)^\iota \right]^{1-\sigma}}{1-\sigma},\;\;\; \kappa \in [0,1)\end{aligned}$$

  with κ = 0.7. In addition, assume that the utility of the newborn generation is given by $u(c^1,l^1)=\frac {\left [ c^1 \left (1-l^1\right )^\iota \right ]^{1-\sigma }}{1-\sigma }$. Notice that you cannot solve the problem recursively but have to solve the system of 99 variables simultaneously. Use your solution from the case without habits as an initial value for {l ¹, l ², …, l ⁴⁰, k ², k ³, …, k ⁶⁰}. Do consumption habits help to make the consumption-age profile smoother? In particular, does the downward jump in consumption at the beginning of retirement disappear?

• 5. In the literature, two types of PAYG pension systems are often distinguished, the Bismarck versus the Beveridge system. In the former, only employed workers contribute to the pension system, and the contribution is levied on the wage income. In the latter, all households contribute prior to their retirement age, and contributions are based on total income, i.e., including capital income. Moreover, while in the Bismarck system, pensions are closely linked to contributions, the Beveridge pension system provides a guaranteed minimum income during retirement and redistributes strongly.

  In the above model, introduce a pension system that provides a lump-sum payment to the retired worker that is financed by a tax on both labor and capital income. Compare this with the average lifetime utility under a PAYG pension system that collects contributions that are levied only on labor income.