7 The Overlapping Generations Model
The Ramsey-Cass-Koopmans model considers a representative household that lives infinite horizons. In many circumstances, however, the assumption of a representative household is not appropriate. One important set of circumstances that may require departure from this assumption is in the analysis of an economy in which new households are born over time. The arrival of new households in the economy is not only a realistic feature, but it also introduces a range of new economic interactions. In particular, decisions made by older generations will affect the prices faced by younger generations. These economic interactions have no counterpart in the neoclassical growth model. They are most succinctly captured in the overlapping generations (OLG) models introduced and studied by Paul Samuelson and later by Peter Diamond. The OLG model considers infinite agents who only live finite periods. In particular, new individuals are continually being born, and old individuals are continually dying.
The OLG model is useful for a number of reasons. First, it captures the potential interaction of different generations of individuals in the marketplace. Second, it provides a tractable alternative to the infinite-horizon representative agent models. Third, some of the key implications are different from those of the neoclassical growth model (e.g. dynamic inefficiency). Finally, the OLG model provides a flexible framework to study the effects of macroeconomic policies such as national debt and social security.
7.1 The Model
In this economy, time is discrete and runs to infinity. Each individual lives two periods. For the generation born in period \(t\), they live for period \(t\) and \(t+1\). In period \(t\), they are young generation, and become old generation in period \(t+1\). As individuals live only two periods, the economy always have two generations in any period. \(L_{t}\) individuals are born in period \(t\). As in Ramsey model, population grows at rate \(n\), i.e.,
\[ L_{t}=(1+n) L_{t-1} \]
Thus, there are \(L_{t}\) young generation and \(L_{t-1}\left(=L_{t} /(1+n)\right)\) old generation.
7.1.1 Consumers
Each consumer supplies 1 unit of labor at wage rate \(W_{t}\) when he/she is young and divides the labor income between first-period consumption and saving with interest rate \(R_{t}=(1+r_t)\). In the second period, the individual simply consumes the saving and any interest he/she earns. Let \(c_{1 t}\) and \(c_{2 t}\) denote the consumption in period \(t\) of young and old individuals. A representative individual born in period \(t\) solves
\[ \max _{\left\{c_{1 t}, c_{2 t+1}\right\}} \frac{c_{1 t}^{1-\theta}}{1-\theta}+\beta \frac{c_{2 t+1}^{1-\theta}}{1-\theta} \]
subject to budget constraint
\[ \begin{aligned} c_{1 t}+s_{t} & \leq W_{t} \\ c_{2 t+1} & \leq R_{t+1} s_{t} \end{aligned} \]
The above problem can be written more compactly by substituting the budget constraints as
\[ \max _{s_{t}} \frac{\left(W_{t}-s_{t}\right)^{1-\theta}}{1-\theta}+\beta \frac{\left(R_{t+1} s_{t}\right)^{1-\theta}}{1-\theta} \]
and consumptions are give by
\[ \begin{aligned} c_{1 t} & =W_{t}-s_{t} \\ c_{2 t+1} & =R_{t+1} s_{t} \end{aligned} \]
First order condition for the optimal saving is
\[ \left(W_{t}-s_{t}\right)^{-\theta}=\beta R_{t+1}\left(R_{t+1} s_{t}\right)^{-\theta} \]
Thus the optimal saving \(s_{t}\) is given by
\[ s_{t}=s\left(R_{t+1}\right) W_{t} \]
where \(s\left(R_{t}\right)=\frac{1}{1+\beta^{-\frac{1}{\theta}} R_{t+1}^{1-\frac{1}{\theta}}}\) indicates the saving rate. Note that, for \(\theta=1\) (the utility is logarithm), the saving rate is just a constant \(\frac{\beta}{1+\beta}\). Later we will show that in this case the OLG model is equivalent to the Solow model with saving rate \(\beta /(1+\beta)\). Moreover, optimal consumptions are given by
\[ \begin{aligned} c_{1 t}=\left[1-s\left(R_{t+1}\right)\right] W_{t} \\ c_{2 t+1}=R_{t+1} s\left(R_{t+1}\right) W_{t} \end{aligned} \]
7.1.2 Firms
A representative firm hires labor \(L_{t}\) and rents capital \(K_{t}\) to produce final goods according to the production function \(Y_{t}=F\left(K_{t}, A_{t} L_{t}\right)\), where the technology \(A_{t}\) is assumed to follow
\[ A_{t}=(1+g)^{t} A_{t-1} \]
We assume that the capital is fully depreciated. The firm aims to maximize the profit by choosing \(L_{t}\) and \(K_{t}\). The optimization problem is
\[ \max _{\left\{L_{t}, K_{t}\right\}} F\left(K_{t}, A_{t} L_{t}\right)-W_{t} L_{t}-R_{t} K_{t} \]
The first order conditions w.r.t. \(\left\{L_{t}, K_{t}\right\}\) are given by
\[ \begin{aligned} R_{t}=F_{K}\left(K_{t}, A_{t} L_{t}\right) \\ W_{t}=F_{A L}\left(K_{t}, A_{t} L_{t}\right) A_{t} \end{aligned} \]
We assume the production function is constant return to scale. Let \(f(\tilde{k})=F\left(\frac{K}{A L}, 1\right)\), where \(\tilde{k}=\frac{K}{A L}\). The input demands can be expressed as
\[ \begin{aligned} R_{t}=f^{\prime}\left(\tilde{k}_{t}\right) \\ W_{t}=\left[f\left(\tilde{k}_{t}\right)-f^{\prime}\left(\tilde{k}_{t}\right) \tilde{k}_{t}\right] A_{t} \end{aligned} \tag{7.1}\]
7.1.3 Competitive Equilibrium
In the competitive equilibrium, consumers and firms achieve the individual optimum. Each market clears. In particular, capital market clearing condition implies
\[ K_{t+1}=L_{t} s\left(R_{t+1}\right) W_{t} \]
According to (7.1), the last equation can be rewritten as
\[ \tilde{k}_{t+1}=\frac{1}{(1+g)(1+n)} s\left(f^{\prime}\left(\tilde{k}_{t+1}\right)\right)\left[\frac{f\left(\tilde{k}_{t}\right)-f^{\prime}\left(\tilde{k}_{t}\right) \tilde{k}_{t}}{f\left(\tilde{k}_{t}\right)}\right] f\left(\tilde{k}_{t}\right) \]
The above equation fully describes the dynamics of capital stock.
7.2 Dynamics
- Special Case: \(\theta=1\).
Assume that \(\theta=1\) (utility is logarithm) and the production function takes Cobb-Douglas form, i.e., \(F(K, A L)=K^{\alpha}(A L)^{1-\alpha}\). The saving rate in this case is \(s\left(f^{\prime}\left(\tilde{k}_{t+1}\right)\right)=\frac{\beta}{1+\beta}\). Equation (19) can be reduced into
\[ \tilde{k}_{t+1}=\frac{1}{(1+g)(1+n)} \frac{\beta}{1+\beta}(1-\alpha) \tilde{k}_{t}^{\alpha} \tag{7.2}\]
Note that this expression essentially has the same form as the one derived from the Solow model. Hence, when utility function takes logarithm form, the OLG model is degenerated to the Solow model.
7.3 A word on the general Case
Once we relax the assumptions of logarithmic utility and Cobb-Douglas production technology, a wide range of behaviors of the economy are possible, including multiple equilibria. See Michel and De La Croix (2002)
7.4 Dynamic Inefficiency
Even though in the OLG model, competitive equilibrium is achieved, it turns out that the competitive allocation is not necessarily dynamically efficient, just as in the Solow model. To see this, let us discuss the capital stock at the steady state. For simplicity, we still consider the special case where \(\theta=1\) and \(f(\tilde{k})=\tilde{k}^{\alpha}\).
From (7.2), we can obtain the steady-state capital stock \(k^{*}\) for the competitive equilibrium from
\[ f^{\prime}\left(k^{*}\right)=\alpha\left(k^{*}\right)^{\alpha-1}=(1+g)(1+n)\left(\frac{1+\beta}{\beta} \frac{\alpha}{1-\alpha}\right) . \tag{7.3}\]
Now consider a social planner’s problem:
\[ \begin{aligned} & \max _{\left\{c_{1 t}, c_{2 t}\right\}} \sum_{t=0} \beta^{t}\left(L_{t} \frac{c_{1 t}^{1-\theta}}{1-\theta}+L_{t-1} \phi \frac{c_{2 t}^{1-\theta}}{1-\theta}\right) \\ = & \max _{\left\{c_{1 t}, c_{2 t}\right\}} \sum_{t=0}[\beta(1+n)(1+g)]^{t}\left(\frac{\tilde{c}_{1 t}^{1-\theta}}{1-\theta}+\frac{1}{1+n} \phi \frac{\tilde{c}_{2 t}^{1-\theta}}{1-\theta}\right) \end{aligned} \]
where \(\phi>0\) is the weight that social planner puts on the old generation, \(\tilde{c}_{1 t}=c_{1 t} / A_{t}, \tilde{c}_{2 t}=c_{2 t} / A_{t}\). The resource constraint is
\[ L_{t} c_{1 t}+L_{t-1} c_{2 t}+K_{t+1}=Y_{t} \]
Detrending both side with \(A_{t} L_{t}\) gives us
\[ \tilde{c}_{1 t}+\frac{\tilde{c}_{2 t}}{1+n}+(1+n)(1+g) \tilde{k}_{t+1}=f\left(\tilde{k}_{t}\right) \]
FOCs w.r.t \(\left\{\tilde{c}_{1 t}, \tilde{c}_{2 r}, \tilde{k}_{t+1}\right\}\) are given by
\[ \begin{aligned} \tilde{c}_{1 t}^{-\theta}=\phi \tilde{c}_{2 t}^{-\theta}=\lambda_{t} \\ (1+n)(1+g) \lambda_{t}=\beta \lambda_{t+1} f^{\prime}\left(\tilde{k}_{t+1}\right) \end{aligned} \tag{7.4}\]
In the steady state, we have
\[ f^{\prime}(k^{gr})=\frac{(1+n)(1+g)}{\beta} \] which is the golden rule.
Comparing (7.3) with (7.4), the capital stock in competitive equilibrium is efficient only if
\[ \frac{(1+\beta) \alpha}{1-\alpha}=1 \]
Therefore, in general, the competitive equilibrium in the OLG model is not dynamically efficient in contrast to the Ramsey model. This is mainly due to the finite-horizon of households which prevents violating any transversality condition.