The Overlapping Generations Model

Abstract

This chapter investigate the standard overlapping generations (OLG) model with two periods. It serves as one of the main tools to study problems in modern public finance such as pensions, unemployment insurance, and debt. The standard OLG model will be shown to be possibly Pareto-inefficient. In addition, we discuss the problem of stability and note that it is less relevant for the large-scale OLG models that we consider in later chapters than for simple two-period models. With the help of an example, we show that the transition in the OLG model might take place over very long time horizons exceeding several decades. In addition, important extensions of the standard two-period OLG model such as bequests and growth are introduced.

Appendices

Appendix 3.1: Kuhn-Tucker First-Order Conditions in the Model with Altruistic Bequests

The model with altruistic bequests involves the constraint

$$\displaystyle \begin{aligned} beq_{t+1} \ge 0.\end{aligned}$$

Households cannot leave negative bequests.

To be more precise, the optimization problem needs to be solved using the Kuhn-Tucker method. The Lagrange function is given by

$$\displaystyle \begin{aligned} \begin{aligned}{\mathscr{L}} & = \sum_{t=0}^\infty \frac{1}{(1+R)^t} \Bigg\{ \left[ u(c^1_{t}) + \beta u(c^2_{t+1})\right]+\\& \lambda_t \left[ w_t+beq_t- c^1_t -\frac{c^2_{t+1}+(1+n)beq_{t+1}}{1+r_{t+1}}\right] +\mu_t beq_{t+1} \Bigg\} \end{aligned} \end{aligned}$$

with the first-order conditions (with respect to beq _t+1):

$$\displaystyle \begin{aligned} \frac{1}{1+R} \lambda_{t+1} &= \lambda_t \frac{1+n}{1+r_{t+1}}-\mu_t, \end{aligned} $$

(3.42a)

$$\displaystyle \begin{aligned} \mu_{t} \cdot beq_{t+1} &=0, \end{aligned} $$

(3.42b)

$$\displaystyle \begin{aligned} \mu_{t}& \ge 0. \end{aligned} $$

(3.42c)

Two cases can be distinguished:

• 1. beq _t+1 > 0: In this case, μ _t = 0, and

  $$\displaystyle \begin{aligned} \frac{1}{1+R} \lambda_{t+1} = \lambda_t \frac{1+n}{1+r_{t+1}}.\end{aligned}$$

• 2. beq _t+1 = 0: μ _t ≥ 0, and

  $$\displaystyle \begin{aligned} \frac{1}{1+R} \lambda_{t+1} \le \lambda_t \frac{1+n}{1+r_{t+1}}.\end{aligned}$$

The conditions are restated in (3.33).

Appendix 3.2: Utility Function and Economic Growth

In the OLG model with economic growth, we assumed the functional form of life-time utility to be represented by (3.36). In this appendix, we study the sensitivity of savings with respect to the choice of the arguments $c^1_t$ and $c^2_{t+1}$ in the utility function.

There are basically two definitions of the variables $c^1_t$ and $c^2_t$, we might choose from. (1) We can define $c^1_t$ and $c^2_{t+1}$ as the per capita consumption of the generation born in period t, $c^1_t=C^1_t/N_t$ and $c^2_{t+1}=C^2_{t+1}/N_t$, where $C^s_t$ denotes total consumption of the s-year old household in period t. This seems to be a natural formulation of preferences. (2) We can employ the arguments $\tilde c^1_t\equiv C^1_t/(A_t N_t)$ and $\tilde c^2_{t+1}\equiv C^2_{t+1}/( A_{t+1}N_t )$. We can interpret this alternative behavior in the sense that consumption habits adjust to technological change. The newest computer in 2000 provides the current household with the same utility as the newest computer in 2020 does. As an implication, households do not grow happier over time, which is in accordance with the so-called Easterlin paradox. According to this paradox, a higher level of a country’s per capita gross domestic product does not correlate with greater self-reported levels of happiness among its citizens.²²

Next, we turn to the question how these two formulations of preferences affect the savings behavior of the household. For this reason, let us generalize life-time utility (3.36) to the case with a constant intertemporal elasticity of substitution 1∕σ and (1) let us use individual consumption (or, equally, consumption per household member) as its arguments:

$$\displaystyle \begin{aligned} U_t = \frac{\left(C^1_t/N_t \right)^{1-\sigma}-1}{1-\sigma} +\beta \frac{\left(C^2_{t+1}/{N_{t}} \right)^{1-\sigma}-1}{1-\sigma}. \end{aligned} $$

(3.43)

Total savings of the young household with N _t members is represented by

$$\displaystyle \begin{aligned} S_t = A_t N_t w_t -C^1_t,\end{aligned}$$

and total consumption of the household in period 2 is equal to savings plus interest earnings

$$\displaystyle \begin{aligned} C^2_{t+1} = (1+r_{t+1}) S_t,\end{aligned}$$

such that the intertemporal budget constraint can be formulated in terms of individual consumption $c^1_t\equiv C^1_t/N_t$ and $c^2_{t+1}\equiv C^2_{t+1}/N_t$:

$$\displaystyle \begin{aligned} c^1_t + \frac{c^2_{t+1}}{1+r_{t+1}} = A_t w_t.\end{aligned} $$

(3.44)

Accordingly, we can formulate the Lagrange function of the household as follows:

$$\displaystyle \begin{aligned}{\mathscr{L}} = \frac{\left(c^1_t \right)^{1-\sigma}-1}{1-\sigma} +\beta \frac{\left(c^2_{t+1} \right)^{1-\sigma}-1}{1-\sigma} + \lambda_t \left[A_t w_t -c^1_t- \frac{c^2_{t+1}}{1+r_{t+1}} \right]. \end{aligned}$$

The first-order conditions of the household’s maximization problem follow from the derivation of the above Lagrangean with respect to $c^1_t$ and $c^2_{t+1}$:

$$\displaystyle \begin{aligned} \lambda_{t} &= \left(c_t^1\right)^{-\sigma}, \end{aligned} $$

(3.45a)

$$\displaystyle \begin{aligned} \frac{\lambda_{t}}{1+r_{t+1}} &= \beta \left(c_{t+1}^2\right)^{-\sigma}, \end{aligned} $$

(3.45b)

and, therefore,

$$\displaystyle \begin{aligned} \left( \frac{c_{t+1}^2}{c_t^1} \right)^{\sigma} = \beta (1+r_{t+1}).\end{aligned} $$

(3.46)

Inserting this first-order condition in the intertemporal budget constraint (3.44), we derive optimal individual consumption

$$\displaystyle \begin{aligned} c^1_t = \frac{A_t w_t }{1+\beta^{\frac{1}{\sigma}} (1+r_{t+1})^{\frac{1}{\sigma}-1} } \end{aligned} $$

(3.47)

and, hence, savings

$$\displaystyle \begin{aligned} \begin{aligned} s_t & \equiv \frac{S_t}{A_t N_t}\\ & = w_t - \frac{c^1_t}{A_t}\\ & = w_t - \frac{w_t }{1+\beta^{\frac{1}{\sigma}} (1+r_{t+1})^{\frac{1}{\sigma}-1}}\\ & = \left[ 1- \frac{1 }{1+\beta^{\frac{1}{\sigma}} (1+r_{t+1})^{\frac{1}{\sigma}-1}} \right] w_t. \end{aligned} \end{aligned}$$

Notice that we would have derived the same amount of optimal savings if we had used the arguments $c^1_t\equiv C^1_t/(A_t N_t)$ and $c^2_{t+1} \equiv C^2_{t+1}/(A_t N_t)$. In Chaps. 6 and 7, we will use this notation in the life-time utility of the household in a growing economy.

(2) Let us consider the second specification with lifetime utility

$$\displaystyle \begin{aligned} \tilde U_t = \frac{\left(C^1_t/(A_t N_t) \right)^{1-\sigma}-1}{1-\sigma} +\beta \frac{\left(C^2_{t+1}/{(A_{t+1}N_{t})} \right)^{1-\sigma}-1}{1-\sigma}. \end{aligned} $$

(3.48)

In this case, consumption habits adjust to the level of the current technology A _t. Let us now define $\tilde c^1_t \equiv C^1_t/(A_t N_t)$ and $\tilde c^2_{t+1}\equiv C^2_{t+1}/(A_{t+1}N_{t})$. With this definition, we can formulate the budgets at age 1 and 2 as follows (using A _t+1∕A _t = 1 + γ):

$$\displaystyle \begin{aligned} \begin{aligned} s_t & = w_t - \tilde c^1_t\\ \tilde c^2_{t+1} & = \frac{1+r_{t+1}}{1+\gamma} s_t, \end{aligned}\end{aligned} $$

and the intertemporal budget constraint

$$\displaystyle \begin{aligned} w_t = \tilde c^1_t + \frac{1+\gamma}{1+r_{t+1}} \tilde c^2_{t+1}.\end{aligned} $$

(3.49)

The household maximizes its Lagrangean function

$$\displaystyle \begin{aligned}{\mathscr{L}} = \frac{\left(\tilde c^1_t \right)^{1-\sigma}-1}{1-\sigma} +\beta \frac{\left(\tilde c^2_{t+1} \right)^{1-\sigma}-1}{1-\sigma} + \lambda_t \left[w_t -\tilde c^1_t- \frac{1+\gamma}{1+r_{t+1}} \tilde c^2_{t+1} \right],\end{aligned} $$

with respect to $\tilde c^1_t$ and $\tilde c^2_{t+1}$ resulting in the first-order conditions:

$$\displaystyle \begin{aligned} \lambda_{t} &= \left(\tilde c_t^1\right)^{-\sigma}, \end{aligned} $$

(3.50a)

$$\displaystyle \begin{aligned} \lambda_{t} \frac{1+\gamma}{1+r_{t+1}}&= \beta \left(\tilde c_{t+1}^2\right)^{-\sigma},\end{aligned} $$

(3.50b)

and, therefore,

$$\displaystyle \begin{aligned} \left( \frac{\tilde c_{t+1}^2}{\tilde c_t^1} \right)^{\sigma} = \beta \frac{1+r_{t+1}}{1+\gamma}.\end{aligned} $$

(3.51)

Inserting this first-order condition in the intertemporal budget constraint (3.49), we derive optimal individual consumption

$$\displaystyle \begin{aligned} \tilde c^1_t = \frac{w_t }{1+\beta^{\frac{1}{\sigma}} \left(\frac{1+r_{t+1}}{1+\gamma}\right)^{\frac{1}{\sigma}-1} },\end{aligned} $$

(3.52)

and, hence, savings

$$\displaystyle \begin{aligned} \begin{aligned} s_t & \equiv \frac{S_t}{A_t N_t}\\ & = w_t - \tilde c^1_t\\ & = \left[ 1- \frac{1}{1+\beta^{\frac{1}{\sigma}} \left(\frac{1+r_{t+1}}{1+\gamma}\right)^{\frac{1}{\sigma}-1} } \right]w_t. \end{aligned} \end{aligned}$$

Evidently, the two specifications of the lifetime utility (3.43) and (3.48) imply different amounts of savings that are only equal to each other in the case of either no growth with γ = 0 or for σ = 1. Notice that the effect of the utility choice on savings depends on the intertemporal elasticity of substitution 1∕σ. Empirical evidence supports the hypothesis that this elasticity is below one, 1∕σ ≤ 1.0. In this case, specification (3.43) results in lower savings than the alternative (3.48) for positive growth γ > 0.

Problems

3.1

Show that (3.12a) and (3.12b) hold in the case of a production function with constant returns to scale.

3.2

Show that the optimal savings function in the Numerical Example in Sect. 3.2.6 with log-linear utility and Cobb-Douglas production is given by (3.20).

3.3

Consider the Numerical Example in Sect. 3.2.6 with log-linear utility and Cobb-Douglas production.²³ Analyze whether the allocation in the market economy is efficient.

Next, analyze the effects of a transfer from the young to the old that is administered by a government authority that maintains a balanced budget. Compute the optimal (possibly negative) transfer. How does the transfer depend on the population growth rate? Consider different values n ∈{0, 0.1, 0.2, …, 2.0}.

3.4

Derive (3.35).

3.5

Compute the solution for the transition in the 20-period OLG example model by backward shooting, i.e., starting by providing a guess for k _T and finding the solution for k _T−1 and so forth.

3.6

Compute the transition in the following 60-period finite-horizon Ramsey model:

Let U be given by a constant elasticity of substitution function

$$\displaystyle \begin{aligned}U(C_0, \dots, C_T) := \left \lbrace \sum_{t=0}^T C_t^\varrho \right \rbrace^{1/\varrho}, \qquad \varrho \in (-\infty, 1],\end{aligned} $$

and define $f(K_t):=K_t^\alpha , \; \alpha \in (0,1)$. Let the household maximize U(C ₀, …, C _t) subject to the budget constraint

$$\displaystyle \begin{aligned} K_{t+1} = K_t^\alpha - C_t, \qquad t=0, 1, \dots, T,\end{aligned} $$

(3.53)

implying the first-order conditions

$$\displaystyle \begin{aligned} \left[ \frac{C_t}{C_{t+1}} \right]^{1-\varrho} \alpha K_{t+1}^{\alpha-1} = 1, \qquad t=0, 1, \dots, T-1. \end{aligned} $$

(3.54)

If we eliminate consumption from the second set of equations using the first T + 1 equations, we arrive at a set of T non-linear equations in the T unknowns (K ₁, K ₂, …, K _T):

$$\displaystyle \begin{aligned} \begin{aligned} 0 &= \left( \frac{K_1^\alpha - K_2}{K_0^\alpha-K_1}\right)^{1-\varrho} - \alpha K_1^{\alpha-1},\\ 0 &= \left( \frac{K_2^\alpha - K_3}{K_1^\alpha-K_2}\right)^{1-\varrho} - \alpha K_2^{\alpha-1},\\ \vdots \\ 0 &= \left( \frac{K_T^\alpha}{K_{T-1}^\alpha-K_T}\right)^{1-\varrho} - \alpha K_T^{\alpha-1}. \end{aligned} \end{aligned} $$

(3.55)

Solve the problem for T = 59, α = 0.35, ϱ = 0.5, k ₀ = 0.1 and K ₆₀ = 0.