OLG Model

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# OLG Model
## Macroeconomics (2017Q2)
### Kenji Sato
### 2017-07-24

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## Announcement: TA sessions

TA sessions are held as per the schedule below:

1. July 25 (Tue) 10:40-12:10 @I223
2. July 27 (Thu) 13:20-14:50 @I209

You can participate in either of the two or both.

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## Anticipated increase in `$n$`

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## Why this path is optimal

Mathematical proof is based on Kemp and Long (1977, Economic Record): 
The costate variable `$\lambda(t) = e^{- \rho t} u'(c(t))$` cannot jump 
at the time of policy change.

If there is anticipated discontinuous drop of `$c(t)$` at `$t = t_0$`, 
the agents anticipate a hike in `$\lambda(t) = e^{- \rho t} u'(c(t))$`.

If the price of asset, `$\lambda$`, is known to jump in the future, 
economic agents all try to buy the asset and the _present price_ would 
rise.

What you observe thus is an immediate drop of `$c$`.

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## Summary

Keep in mind the following points:

* There is no discontinuity in `$k$`
* There is no anticipated discontinuity in `$u'(c)$` or `$c$`
* The path must converge to the new steady state.

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## Suggested exercises for the final

The Ramsey model has the following parameters.

* `$\delta$`, `$g$`, `$n$`, `$\rho$`, `$\theta$`
* `$K(0)$`: Initial capital **← added**
* `$L(0)$`: Labor **← added**
* `$A(0)$`: Technology **← added**

What are the short-run/long-run effects of anticipated/unanticipated changes 
in these parameters that last permanently/temporarily?

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## Suggested exercises for the final

What is the effects of taxation in the Ramsey model?

* Capital taxation similar to Romer's Problem 2.10, 95f.

`$$r(t) = (1 - \tau) f'(\hat{k}(t))$$`

* Income taxation

`$$\hat{G}(t) = \tau \hat{w}(t)$$`

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 # Diamond OLG Model

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## Overlapping Generations

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.

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## Overlapping Generations (cont'd)

Applications of OLG models include

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

in which **confilict of interests between generations** may arise.

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## Example 1: Pension system

Older generations never want to abolish pension system but younger generations might do.

Young generations doubt that the current pension system is sustainable and many expect not to receive what you pay. (In Japan, the fertility rate is slightly above 1.4)

There is conflict of interests between generations. 
In this case, OLG model is a natural choice.

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## Example 2: Public finance

When issuing 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 issuing public bond may risk the future generations while immediate beneficiaries will receive large benefit because they will die before paying its full cost.** The analysis of such problems go well with OLG models.

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## Note

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

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## Demography

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       |

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## Young generations

Young individuals are endowed with

* no financial wealth
* unit labor force

They supply their full labor force irrespective of the wage rage (perfectly inelatic labor supply). They divide their labor income between consumption and saving.

Let `$c_{t}^{Y}$` be the consumption of the representative consumer in generation 
`$t$` when they are young (per capita consumption).

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## Old generations

Young consumers get old in the next period.

Old consumers simply consume what you can buy with the savings and the interest they earned.

Let `$c_{t+1}^{O}$`  be the consumption of the representative consumer in generation 
`$t$` when they are old.

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## Households' utility

The utility function of generation `$t$` is given by

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

They are subject to the following budget constraints.

`$$\begin{aligned}
c_{t}^{Y}+s_{t}	&= w_{t} &(\textrm{Young})\\
c_{t+1}^{O} &= (1+r_{t+1})s_{t} &(\textrm{Old})
\end{aligned}$$`

`$r_{t+1}$` 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 `$t+1$`.

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## Households' optimization

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

**Excercise: Set up the Lagrangian**

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## Lagrangian

The Lagrangian:

`$$\begin{multline}
\mathcal{L}=\frac{\left(c_{t}^{Y}\right)^{1-\theta}}{1-\theta} +
\frac{1}{1+\rho}\frac{\left(c_{t+1}^{O}\right)^{1-\theta}}{1-\theta} \\+
\lambda\left(w_{t}-c_{t}^{Y}-s_{t}\right) +
\mu\left((1+r_{t+1})s_{t}-c_{t+1}^{O}\right)
\end{multline}$$`

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

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## 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.

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## Saving rate function

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

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

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## Saving rate function (cont'd)

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

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

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## Income and substitution effects

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.
  
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## Income and substitution effects (cont'd)

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

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 `$r$`.

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## Firm

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 `$\hat{k}=K/AL$`.

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## Factor markets

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

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

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## Capital market

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, 
`$$K_{t+1}=s_{t}L_{t} \quad
\textrm{or}
\quad K_{t+1}=s(r_{t+1})\hat{w}_{t}A_{t}L_{t}$$`

We get

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

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## Product market

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.

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## Dynamics

The dynamics of the economy is governed by the following system.

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

Notice that both right-hand and left-hand sides contain `$k_{t+1}$`.

This system is said to be an implicit system. `$k_{t+1}$` may not be a function of `$k_{t}$` in which case, there is multiple possibility of time path.

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## Dynamics (cont'd)

To obtain a clear result, we assume that

`$$f(k)=k^{\alpha},\quad0<\alpha<1$$`

and the logarithmic utility function: i.e., `$\theta=1$`.

Prove that the dynamics is characterized by

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