Macroeconomic Modelling

# Macroeconomic Modelling
## Macroeconomics
### Kenji Sato
### Day 3

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## Stocks and flows

https://www.youtube.com/watch?v=nRlYGDBGcRA

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## Stocks and flows in macroeconomics

**Stock variables** measure the amount of something at some point in time.
**Flow variables** measure the volume of economic activities that happen during a period.  
.pull-left[
Stocks
- Capital
- Savings
- Labor force
- Knowledge
]

.pull-right[
Flows 
- Investment (Capital formation)
- Saving
- Consumption
- Production
- Depreciation
]

---

## Rule of Thumb

If you see `$\dot{X}$` of some variable `$X$` in the set-up of the model, 
`$X$` is a stock variable.

We must specify evolution of all the stock variables to complete model setting.
(NB: You might have the time-derivative of flow variables as equilibrium conditions.)

Example (capital stock):

`$$\dot K(t) = I(t) - \delta K(t)$$`

`$K$` is a stock variable, while `$I$` is a flow variable. `$\delta K$` is flow.

---

---

## Production

Input factors:

- `$K(t)$`: Physical capital
    - Machines, buildings, etc.
- `$L(t)$`: Labor
    - Working age population, total hours worked
- `$A(t)$`: Knowledge
    - Scientific knowledge, efficiency of labor

---

## Output

Output:

- `$Y(t)$`: Output

The economy has access to a production technology that 
can transform `$(K, L)$` together with `$A$` into the final output `$Y$`.

Let's symbolize the production technology as `$\tilde F$`

`$$Y(t) = \tilde{F} (K(t), L(t), A(t))$$`

---

## Expenditure approach to GDP

Output `$Y$` (GDP) is devided between

- `$C$`: Consumption
- `$I$`: Investment
- `$G$`: Government expenditure
- `$NX$`: Net export (Export minus import)

Fundamental macroeconomic identity:

`$$Y = C + I + G + NX$$`

---

## Consumer Behavior

Consumers divide their disposable income between consumption, `$C$`, and saving, `$S$`.

`$$Y - T = C + S,$$`

where `$T$` is tax.

---

## Saving and investment

Assume balanced budget:
`$$G = T$$`

From

`$$\begin{aligned}
  &Y = C + I + G + NX\\
  &Y - T = C + S
\end{aligned}$$`

we obtain

`$$S = I + NX$$`

Saving is divided between domestic investment and 
(net) foreign investment.

---

## Assumptions of closed, free economy

We assume that the there is no international trade; i.e.,

`$$NX = 0.$$`

We therefore have

`$$S = I$$`
For the time being we ignore government.

`$$G = T = 0.$$`

---

## Consumers' decision

Ecnomic agents divide their income between consumption and saving.

`$$Y = C + S$$`

All the saving is used as gross capital formation (gross investment).

`$$S = I$$`

This investment (= saving) forms the future capital.

---

## Capital formation

The amount of capital is determined by the capital accumulation equation.

`$$\dot K(t) = I(t) - \delta K(t)$$`

`$$\text{(Net Investment)} = \text{(Gross Investment)} - \text{(Depreciation)}$$`

---

## Income approach to GDP

National income is (roughly) divided between capital income and labor income.

`$$Y = (r + \delta) K + w L,$$`

- `$r$` is real rental rate (received by capitalist) 
- `$\delta$` is depreciation rate (received by _capital_)
- `$w$` is real wage rate

The following two figures add to one.

`$$\frac{(r + \delta) K}{Y} = \text{Capital share},
\quad \frac{wL}{Y} = \text{Labor share}$$`

---

---

## Kaldor's facts

[Kaldor (1961)](https://scholar.google.co.jp/scholar?q=Capital+accumulation+and+economic+growth+kaldor&btnG=&hl=en&as_sdt=0%2C5) observed the following facts, which are referred 
to as the stylized facts. [Jones and Romer (2010)](https://www.aeaweb.org/articles?id=10.1257/mac.2.1.224)

1. Labor productivity has grown at a sustained rate.
2. Capital per worker has also grown at a sustained rate.
3. The real interest rate or return on capital has been stable.
4. The ratio of capital to output has been stable.
5. Capital and labor have captured stable shares of national income.
6. Among the fast growing countries of the world, there is an appreciable variation in the rate of growth.

---

## Kaldor's facts (1 - 5)

Borrowing the mathematical notations defined earlier, 
1 - 5 of the stylized facts can be translated as

1. `$Y / L$` grows at a sustained rate.
2. `$K / L$` grows at a sustained rate.
3. `$r$` is constant.
4. `$K / Y$` is constant.
5. `$(r + \delta)K / Y$` and `$wL / Y$` are constant.

---

## Penn World Table

```r
pwt90 <- haven::read_dta("~/Data/pwt90.dta")
usa <- pwt90 %>% 
  filter(country == "United States") %>% 
  mutate(capsh = (1 - labsh), 
         caprtrn = capsh * rgdpo / rkna - delta)
p <- ggplot(usa)
```

We estimate `$r$` with

`$$r = \left(
  1 - \frac{wL}{Y}
\right) \frac{Y}{K} - \delta$$`

---

## 1. Per-Labor Output `$Y / L$`

```r
p + geom_line(aes(year, cgdpo / emp), size = 2)
```

---

## 2. Caiptal-Labor ratio `$K / L$`

```r
p + geom_line(aes(year, rkna / emp), size = 2)
```

---

## 3. Rental rate

```r
p + geom_line(aes(year, caprtrn), size = 2) + ylim(0, 0.15)
```

---

## 4. Capital-Output ratio `$K / Y$`

```r
p + geom_line(aes(year, rkna / rgdpe), size = 2) + ylim(0, 5)
```

---

## 5. Labor Share `$wL / Y$`

```r
p + geom_line(aes(year, labsh), size = 2) + ylim(0, 1.2)
```

---

---

## Assumptions on the production function

In the Solow model,

- The technical change is purely labor augmenting, i.e.
  `$$\tilde{F}(K, L, A) = F(K, AL)$$`
- The production function, `$F$`, is homogeneous of degree one, i.e., for 
  all `$c \ge 0$`,
  `$$F(cK, cAL) = cF(K, AL)$$`

`$AL$` is called **effective labor**.

---

## Assumption on the consumer behavior

Under the closed economy assumption, `$Y = C + S$`.

- A constant fraction, `$s$`, of output is saved, and the rest, `$1 - s$`, is consumed; i.e.,
  `$$C = (1 - s) Y \quad \text{and}\quad S = sY$$`

The saving rate, `$s$`, is constant.

---

## Assumption on growth of `$A$` and `$L$`

Knowledge `$A$` and labor (or population) `$L$` grow exogenously.

`$$\begin{aligned}
  \dot A &= g A\\
  \dot L &= n L
\end{aligned}$$`

- `$g$` is the growth rate of technology and 
- `$n$` the population growth rate.

`$g$` and `$n$` are constant.

---

## Summary: Solow Model

The following parameters characterize the Solow model:

- `$F$`: production function
- `$s$`: saving rate
- `$\delta$`: depreciation rate
- `$g$`: technology growth rate
- `$n$`: population growth rate

---