Discrete-time Models

# Discrete-time Models
## Macroeconomics (2017Q4)
### Kenji Sato
### 2018-01-22

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## Continuous-time v. Discrete-time

We have studied continuous-time macroeconomic models. Here is a summary of our modelling strategy: We

1. model economic activities in a very short period of time `$[t, t + \Delta t)$`
1. derive a system of differential equation(s) by taking `$\Delta t \to 0$`
1. describe the behavior of the economy diagramatically
1. observe what the economy looks like in transition and in the limit

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## Continuous-time v. Discrete-time (cont'd)

Continuous-time models are convenient because solutions are often simpler than discrete-time models.

For instance, see growth rate formulas:

`$$\begin{aligned}
  e^{a} e^{b} &= e^{a + b} \\
  (1 + a)(1 + b) &= 1 + a + b + ab
\end{aligned}$$`

Also, phase-diagramatic analysis may not work in the discrete-time models.

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## Continuous-time v. Discrete-time (cont'd)

In spite of the downsides of discrete-time models, many papers use them probably because

* its interpretation is straightforward;
* time series analysis works better (note that all available data are in discrete time).

Since analysts can choose whichever is convenient, you also need to understand both schemes.

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## Notation and goal

We will use the third choice.

* Time is discrete and extends from `$0$` to `$\infty$`.
* Period between time points `$t - 1$` and `$t$` is called "Period `$t$`".
* For stock variables, notation like `$K_t$` denotes the end-of-period balance.

We need to find a rule that governs the development of

`$$K_0, K_1, \dots, \quad Y_1, Y_2, \dots$$`

etc. Let's first study the discrete-time Solow model.

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## Exogenous growth

Let's suppose that knowledge `$A$` and labor `$L$` grow exogenously. In discrete-time models, 
this assumption is expressed as

`$$\begin{aligned}
  A_t &= (1 + g) A_{t-1}\\
  L_t &= (1 + n) L_{t-1}, \qquad t = 1, 2, \dots
\end{aligned}$$`

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## Production and investment

The firms produce output, employing what is available at the beginning of the period, or
equivalently, what was available at the end of the previous period.

`$$Y_t = F(K_{t-1}, A_{t-1} L_{t-1})$$`

`$F$` is a CRS production function and `$Y$` denotes aggregate production.

We assume a closed free economy with a constant saving rate

`$$S_t = I_t = sY_t$$`

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## Capital Accumulation

Net investment `$K_{t} - K_{t-1}$` is by definition

`$$\begin{aligned}
K_t - K_{t-1} 
  &= I_t - \delta K_{t-1}\\
  &= sY_t - \delta K_{t-1}\\
  &= sF(K_{t-1}, A_{t-1} L_{t-1}) - \delta K_{t-1}
\end{aligned}$$`

where `$\delta$` is the depreciation rate. By rearranging,

`$$K_t = sF(K_{t-1}, A_{t-1} L_{t-1}) + (1 - \delta) K_{t-1}$$`

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## Effective labor

Define

`$$\hat k_t = \frac{K_t}{A_t L_t}$$`

and

`$$\hat{y}_t = f\left( \hat k_t \right) = F\left(\hat{k}_t, 1    \right)$$`

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## Capital accumulation

Divide by `$A_{t-1} L_{t-1}$` both sides of
`$$K_t = sF(K_{t-1}, A_{t-1} L_{t-1}) + (1 - \delta) K_{t-1}$$`

The left-hand side

`$$\frac{K_t}{A_{t-1} L_{t-1}}
=
\frac{K_t}{A_{t} L_{t}} \frac{A_{t} L_{t}}{A_{t-1} L_{t-1}}
=
\hat{k}_t (1 + g)(1 + n)$$`

The right hand side

`$$\frac{sF(K_{t-1}, A_{t-1} L_{t-1}) + (1 - \delta) K_{t-1}}{A_{t-1} L_{t-1}}
= sf\left(\hat{k}_{t-1}\right) + (1 - \delta) \hat{k}_{t-1}$$`

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## Capital accumulation

We obtain

`$$\hat{k}_t
= \frac{sf\left(\hat{k}_{t-1}\right) + (1 - \delta) \hat{k}_{t-1}}{(1+g)(1+n)}$$`

`$$\hat{k}_t - \hat{k}_{t-1}
= \frac{sf\left(\hat{k}_{t-1}\right) - (\delta + g + n + gn) \hat{k}_{t-1}}{(1+g)(1+n)}$$`

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## Break-even investment

`$$\hat{k}_t > \hat{k}_{t-1} \Leftrightarrow sf\left(\hat{k}_{t-1}\right) > (\delta + g + n + gn) \hat{k}_{t-1}$$`

`$$\hat{k}_t < \hat{k}_{t-1} \Leftrightarrow sf\left(\hat{k}_{t-1}\right) < (\delta + g + n + gn) \hat{k}_{t-1}$$`

Break-even investment in discrete-time model is `$(\delta + g + n + gn)\hat{k}$`, which is slightly 
larger than the continuous-time counterpart.

This difference comes from the difference between nominal and effective rates. When `$g$` and `$n$` are very small,
then we can ignore the difference:

`$$\delta + g + n + gn \simeq \delta + g + n$$`

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## Dynamics

Analysis analogous to continuous-time Solow model may not work since 
we haven't ruled out a scenario that the sign conditions alternate

`$$k_{t-1} - k_t < 0, \quad k_t - k_{t+1} >0 , \quad k_{t+1} - k_{t+2} < 0, \dots$$`

This doesn't happen in continuous-time models in which stock variables develop continuously.

In discrete-time models, the following "staircase" diagram might work better.

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## Policy function

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## Policy function

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## 45 degree line

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## `$t = 1 \to 2$`

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## 45 degree line

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## Continue this argument to get convergence

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## Steady state

We get convergence to the steady state value

`$$\hat{k}_t \to \hat{k}^*$$`

In the steady state, balanced growth is achieved:

`$$K_t = A_t L_t \hat{k}^*$$`

`$$Y_t = A_t L_t f\left( \hat{k}^* \right)$$`

Exercise: Verify the same convergence from `$\hat{k}_0 > \hat k^*$`.