Kenji Sato

January 25, 2017

We next study the overlapping generations model introduced by Diamond (1965). Unlike the Ramsey-type models, OLG models assume finitely lived agents.

New individuals are continually being born and old individuals are continually dying. There is turnover in the population.

Applications of OLG models include

- pension system,
- public finance, and
- environmental protection, etc,

in which confilict of interests between generations may arise.

Older generations never want to abolish pension system but younger generations might do. In Japan, where the fertility rate is slightly more than 1.4, young generations doubt that the current pension system is sustainable and many expect not to receive what you pay. There is conflict of interests between generations.

In this case, OLG model is a natural choice.

When issueing public bond for government investment, the government usually limits the bond duration up to the duration of depreciation of the invested capital. This is an incarnation of the benefit principle that the beneficiaries share the cost.

If bond duration is longer, younger generation will have to repay the debt that they receive no benefit. Carelessly issueing public bond may risk the future generations while immediate beneficiaries will receive large benefit and die without paying its full cost. The analysis of such problems go well with OLG models.

The OLG model presented here is one of the most simplest. OLG models in the wild have much more complex specifications. For instance, a model of Conesa, Kitao and Krueger (2009, AER) has 81 generations, compared to 2 generations in our OLG model.

Time is discrete.

Individuals live for two periods. \(L_{t}\) individuals are born in period \(t\). The population growth rate is \(n\): \(L_{t}=(1+n)L_{t-1}\).

\(t=0\) | \(t=1\) | \(t=2\) | \(t=3\) | \(\cdots\) | |
---|---|---|---|---|---|

Generation -1 | Old | ||||

Generation 0 | Young | Old | |||

Generation 1 | Young | Old | |||

Generation 2 | Young | Old | |||

Generation 3 | Young | Old |

Young individuals, who are born without financial wealth, supply unit labor and earn wage. They divide their income between consumption and saving. When they become old, they simply consume what you can buy with the saving and the interest they will have earned. Note the difference in notations from the textbook.

- Let \(c_{t}^{Y}\) be the consumption of the representative consumer in generation \(t\) when they are young, (per capita consumption)
- \(c_{t+1}^{O}\) be the consumption of the representative consumer in generation \(t\) when they are old.

The utility function of generation \(t\) is given by

\[ u_{t}=\frac{\left(c_{t}^{Y}\right)^{1-\theta}}{1-\theta}+\frac{1}{1+\rho} \frac{\left(c_{t+1}^{O}\right)^{1-\theta}}{1-\theta},\quad\theta>0,\ \rho>-1 \]

Young generations’ budget constraint

\[ c_{t}^{Y}+s_{t} = w_{t} \]

Old generations’ budget constraint

\[ c_{t+1}^{O}=(1+r_{t+1})s_{t} \]

\(r_{t+1}\) is the interest rate between period \(t\) and \(t+1\). The saving contract is made in period \(t\) and the interest is paid in period \(t+1\).

\[ \begin{aligned} &\max_{c_{t}^{Y},c_{t+1}^{O},s_{t}} \frac{\left(c_{t}^{Y}\right)^{1-\theta}}{1-\theta} + \frac{1}{1+\rho}\frac{\left(c_{t+1}^{O}\right)^{1-\theta}}{1-\theta}\\ &\text{subject to }\\ &\qquad c_{t}^{Y}+s_{t}=w_{t},\\ &\qquad c_{t+1}^{O}=(1+r_{t+1})s_{t}. \end{aligned} \]

The Lagrangian:

\[ \mathcal{L}=\frac{\left(c_{t}^{Y}\right)^{1-\theta}}{1-\theta} + \frac{1}{1+\rho}\frac{\left(c_{t+1}^{O}\right)^{1-\theta}}{1-\theta} + \lambda\left(w_{t}-c_{t}^{Y}-s_{t}\right) + \mu\left((1+r_{t+1})s_{t}-c_{t+1}^{O}\right) \]

**Excercise**: Derive the first-order condition.

\[ \frac{c_{t+1}^{O}}{c_{t}^{Y}}=\left(\frac{1+r_{t+1}}{1+\rho}\right)^{1/\theta} \]

This is analogous to the Euler equation in the Ramsey model.

Let a function \(s(\cdot)\) be such that

\[ s_{t}=s(r_{t+1})w_{t}.s(\cdot) \]

represents the fraction of income allocated to saving.

\[ c_{t}^{Y} = \left[1-s(r_{t+1})\right]w_{t},\quad \text{and} \quad c_{t+1}^{O} = (1+r_{t+1})s(r_{t+1})w_{t} \]

imply

\[ \frac{(1+r_{t+1})s(r_{t+1})}{1-s(r_{t+1})} = \left(\frac{1+r_{t+1}}{1+\rho}\right)^{1/\theta} \]

We get

\[ s(r_{t+1}) = \frac{(1+r_{t+1})^{\frac{1-\theta}{\theta}}} {(1+\rho)^{\frac{1}{\theta}}+(1+r_{t+1})^{\frac{1-\theta}{\theta}}} \]

Observe that

\[ \begin{aligned} \frac{ds}{dr}>0&\Leftrightarrow\theta<1\\ \frac{ds}{dr}<0&\Leftrightarrow\theta>1. \end{aligned} \]

Recall a microeconomics result that response to a change in relative price is decomposed into income effect and substitution effect.

Change in \(r\) has both income effect (when r becomes larger, increase in financial income increase consumption both in their youth and old age) and substitution effect (change in relative price makes consumption in old more attractive).

When \(\theta<1\), they are willing to defer consumption. They take advantage of increased interest to get more consumption when old, that is, the substitution effect dominates. \(s(\cdot)\) is increasing function of \(r\).

Firms have access to technology Y=F(K,AL). They rent capital from households and employ labor force. Technology \(A\) is exogenously given. \(A_{t+1}=(1+g)A_{t}\).

As always, we consider the intensive form:

\[ \hat{y}=f(\hat{k}) \]

where \(\hat{y}=Y/AL\) and \(\hat{k}=K/AL\).

The factor markets are assumed to be competitive.

We get (under no depreciation \(\delta = 0\)) that

\[ r_{t+1} =f'(\hat{k}_{t+1}) \]

\[ w_{t} =A_{t}\left[f(\hat{k}_{t})-\hat{k}_{t}f'(\hat{k}_{t})\right], \] or

\[ \hat{w}_{t}=w_{t}/A_{t}=f(\hat{k}_{t})-\hat{k}_{t}f'(\hat{k}_{t}). \]

Period \(0\) capital \(K_{0}\) are owned by all old individuals \(L_{-1}\) of generation \(-1\).

Capital stock in period \(t+1\) is the amount saved by the young in generation \(t\). Thus, \[ K_{t+1}=s_{t}L_{t} \] or

\[ K_{t+1}=s(r_{t+1})\hat{w}_{t}A_{t}L_{t} \]

We get

\[ \hat{k}_{t+1} = \frac{s(r_{t+1})\hat{w}_{t}}{(1+g)(1+n)} = \frac{s\left(f'(\hat{k}_{t+1})\right)\left[f(\hat{k}_{t})-\hat{k}_{t}f'(\hat{k}_{t})\right]}{(1+g)(1+n)}. \]

Does the product market clear? To check this, prove

\[ Y_{t}+K_{t}=\left(c_{t}^{Y}L_{t}+s_{t}L_{t}\right)+c_{t}^{O}L_{t-1}. \]

We need \(K_{t}\) because the old guys dissave (and eat) all the capital they had saved when they were young. \(c_{t}^{O}\) contains this term.

Since the dynamical system,

\[ \hat{k}_{t+1} = \frac{s\left(f'(\hat{k}_{t+1})\right) \left[f(\hat{k}_{t})-\hat{k}_{t}f'(\hat{k}_{t})\right]} {(1+g)(1+n)}, \]

is an implicit system, it is difficult to solve. \(k_{t+1}\) may not be a function of \(k_{t}\) in which case, there is multiple possibility of time path

To obtain a clear result, we assume that

\[ f(k)=k^{\alpha},\quad0<\alpha<1 \]

and the log instantaneous utility function: i.e., \(\theta=1\).

Prove that the dynamics is characterized by

\[ \hat{k}_{t+1}=\frac{(1-\alpha)}{(1+g)(1+n)(2+\rho)}\hat{k}_{t}^{\alpha} \]

Looks like the Solow model.