Macroeconomics Q4
========================================================
author: Kenji Sato
date: 2016-12-21
autosize: true

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Recap 
========================================================

The master equation:

$$
\dot k = s f(k) - (\delta + g + n) k
$$

Important variables in the steady state $k^*$:

$$
\frac{K}{L} = A k^*
$$

$$
\frac{Y}{L} = Af(k^*)
$$

both grow at the rate of $g$.


Balanced Growth
====================

Let $k = k^*$.

The common growth rate for 

$$K/L,\quad Y/L,\quad C/L$$ 
is 
$$g = \dot A / A.$$


The common growth rate for 
$$K,\quad Y,\quad C$$ 
is $$g + n = \dot A/A + \dot L/L.$$


The situation in which important variables share growth rate is 
called **balanced growth**.

When $k=k^*$, the economy is on the **balanced growth path**.


Comparative Statics/Dynamics
==============================

Comparative Statics/Dynamics is a common excercise of macroeconomics. 

It is important to understand what happens after a (small) parameter change.

**What happens after an increase of the saving rate?**

- $s$ is an important policy variable for the government.
  - government's consumption-investment decision, 
  - decision of tax/debt finance, or
  - changing tax treatments of saving and investment
  
  may have impact on $s$.
  
  
Preparation
==============
class: small-code

```{r}
library(ggplot2)
library(ggthemes)

s0 = 0.3
s1 = 0.4

alpha = 0.3
delta = 0.05
g = 0.02
n = 0.01

f = function(k) {
  return(k^alpha)
}

k = seq(0.0, 25.0, by=0.01)
df = data.frame(k=k, f=f(k), s0f=s0*f(k), s1f=s1*f(k))

head(df)
```

Comparative Statics/Dynamics
==============================
class: small-code

```{r}
fig = ggplot(df) + 
  geom_line(aes(x=k, y=f)) + # Production Function
  geom_line(aes(x=k, y=s0f), color='blue', size=1.5) + # For s0
  geom_line(aes(x=k, y=s1f), color='red', size=1.5) +  # For s1
  geom_line(aes(x=k, y=(delta+g+n)*k))  # Break-Even
```

```{r, fig.width=9, fig.asp=0.75, fig.align='center', echo=FALSE}
fig + 
  annotate("text", x=20, y=s0*f(20)*0.85, label="s0=0.3", size=13, color="blue") + 
  annotate("text", x=20, y=s1*f(20)*1.15, label="s1=0.4", size=13, color="red") +
  theme_gdocs()
```


Comparative Statics
=======================================

- $k^*$ is larger when $s$ gets larger.
  - **Policy that increases the saving rate increases GDP per capita**: 
    $$
     \frac{Y}{L} = A f(k^*)
    $$
  - There is a **level effect**
  
- What about growth rate?
  - In the long run, there is no change: fixed at $g$.There is **no growth effect**.
  - In the shorter run, there is some change.

Comparative Dynamics
================================

- Suppose that the economy is on the balanced growth path.
  - i.e., $k = k^*$ 
- At time $t_0$, the economy experiences a sudden increase in $s$ (from s0 to s1).
  - $k$ gradually moves toward the new $k^*$ **because the new saving level is greater than the break-even level of investment**.
  - Investment per capita immediately **jumps** up to a point on the new (red) saving curve. (**jump variable**)
  - Consumption per capita immediately falls because of the rise of investment. (**jump variable**)
  
  
Simulation for k
================================
class: small-code 

```{r}
k0 = (s0 / (g + n + delta))^(1 / (1 - alpha)) # steady state 
t0 = 10  # Change of policy

solow_update = function(t, k){
  if (t < t0){
    return(k + dt * (s0 * f(k) - (delta + g + n) * k))
  } else {
    return(k + dt * (s1 * f(k) - (delta + g + n) * k))
  }
}

dt = 0.01   # controls precision of approximation
t_max = 100   # simulation for t_max years
t = seq(from=0, to=t_max, by=dt)
simulation = as.data.frame(t)

simulation[1, "k"] = k0
for (i in 2:nrow(simulation)){
  simulation[i, "k"] = solow_update(simulation[i-1, "t"], 
                                    simulation[i-1, "k"])
}
```


Simulation for k
================================
class: small-code

```{r, fig.width=7, fig.asp=0.75, fig.align='center'}
ggplot(simulation, aes(x=t, y=k)) + geom_line() + theme_gdocs()
```

At $t = t_0$, $k$ starts to increase and

it stops increasing when it attains the new steady state value.


Simulation for K/L (log scale for y-axis)
===========================================
class: small-code

Let $A(0) = 1$. Plot using a logarithmic scale for y-Axis.

```{r, fig.width=8, fig.asp=0.75, fig.align='center'}
simulation$KL = simulation$k * exp(g*simulation$t)
ggplot(simulation, aes(x=t, y=KL)) + geom_line() + scale_y_log10() + theme_gdocs()
```

Simulation for C/L (log scale for y-axis)
============================================
class: small-code

Exercise: Reproduce the following graph.

```{r, fig.width=9, fig.asp=0.75, fig.align='center', echo=FALSE}
simulation$s = ifelse(simulation$t < t0, s0, s1)
simulation$CL = (1 - simulation$s) * f(simulation$k) * exp(g*simulation$t)
ggplot(simulation, aes(x=t, y=CL)) + geom_line() + 
  scale_y_log10() + theme_gdocs()
```

Transition Dynamics
=====================

Note that the growth rate, $g_{K/L}$ of $K/L$ satisfies

$$
g_{K/L}(t) = g + g_k(t), 
$$

where $g$ is the exogenous growth rate of $A$, $g_k$ is the growth rate of $k$.

After an increase in saving rate, we get $\dot k > 0$ and thereby $g_k(t) > 0$.

**On the transition path, the growth of per capita capital is faster than on the BGP**.

It seems to be consistent with observations about NICs.

Golden rule
=====================

- There is a certain value for $s$ that maximizes steady state consumption. 
  - Such saving rate is called **Golden rule saving rate** and denoted by $s_G$.
  - For Cobb--Douglas production funtion $f(k) = k^\alpha$, $s_G = \alpha$. (Why?)

Golden rule (cont'd)
=====================

Note that 

$$\begin{aligned}
  c^* &= (1 - s) f(k^*) \\
      &= f(k^*) - (\delta + g + n) k^*
\end{aligned}$$

When $c^*$ is maximized, we should have (think of $c^*$ as a function of $k^*$)

$$
  f'(k^*) = \delta + g + n
$$

Let $k^*_G$ be the unique stock level that satisfies the above equation. **Golden rule capital stock**.


Simulation for C/AL
=======================
class: small-code

Exercise: Reproduce the following graph.

```{r, fig.width=6, fig.asp=0.75, fig.align='center', echo=FALSE}
simulation$CAL = (1 - simulation$s) * f(simulation$k)
ggplot(simulation, aes(x=t, y=CAL)) + geom_line() + theme_gdocs()
```


Observe that the new steady state value, $c^*$, for $c = C/(AL)$ is smaller after the parameter change considered above.


Steady state consumption
==========================================

- If $s_0 < s_1 \le s_G$, the parameter shift from $s_0$ to $s_1$ necessarily makes $(C/AL)^*$  larger after the shift. 
  - Confirm this fact with pen and paper, and with R.
- If $s_G \le s_0 < s_1$, $(C/AL)^*$ gets smaller. 
  - This is what we have observed.




Dynamic Inefficiency
===========================

- Saving rate greater than the golden-rule level is unrealistic.
  - If you lower the saving rate, you can increase consumption immediately 
    and forever.
  - There is **dynamic inefficiency**.
    

Growth Accounting
=====================

$Y = F(K, AL)$ implies that 

$$
\begin{aligned}
  \dot Y &= \frac{\partial Y}{\partial K} \dot K
           +
           \frac{\partial Y}{\partial (AL)} \frac{d}{dt}(AL)\\
         &= \frac{\partial Y}{\partial K} \dot K
           +
           \left[\frac{\partial Y}{\partial (AL)} A\right] \dot L
           +
           \left[\frac{\partial Y}{\partial (AL)} L\right] \dot A\\
         &=: \frac{\partial Y}{\partial K} \dot K
           +
           \frac{\partial Y}{\partial L} \dot L
           +
           \left[\frac{\partial Y}{\partial (AL)} L\right] \dot A.
\end{aligned}
$$

We therefore have

$$
\frac{\dot Y}{Y}
         = \frac{\partial Y}{\partial K} \frac{K}{Y} \frac{\dot K}{K} 
           +
           \frac{\partial Y}{\partial L} \frac{L}{Y} \frac{\dot L}{L}
           +
           \left[\frac{\partial Y}{\partial (AL)} \frac{AL}{Y} \right] g.
$$


Growth Accounting (cont'd)
============================

Define

$$\begin{aligned}
  \alpha_K (t) &:= \frac{K}{Y} \frac{\partial Y}{\partial K} & (\text{capital elasticity of output}) \\
  \alpha_L (t) &:= \frac{L}{Y} \frac{\partial Y}{\partial L} & (\text{labor elasticity of output})
\end{aligned}$$

**Veryfy for the Cobb--Douglas family that they are constant: $\alpha_K = \alpha$ and $\alpha_L = 1-\alpha$**.

By Euler's theorem on CRS functions,

$$
\alpha_K (t) + \alpha_L (t) = 1
$$

Growth Accounting (cont'd)
============================

1% increase in capital input results in $\alpha_K$% increase in output.

$$
  \alpha_K 
  =
  \frac{K}{Y} \frac{\partial Y}{\partial K}
  \simeq
  \dfrac{
    \dfrac{Y + \Delta Y}{Y}
  }{
    \dfrac{K + \Delta K}{K}
  }
$$



Growth Accounting (cont'd)
============================

By employing this notation, the decomposition of $\dot Y/Y$ becomes

$$
  \frac{\dot Y}{Y}
         = \alpha_K \frac{\dot K}{K} 
           +
           \alpha_L \frac{\dot L}{L}
           +
           R,
$$

where

$$
  R := \left[\frac{\partial Y}{\partial (AL)} \frac{AL}{Y} \right] g = \alpha_L g,
$$

called the **Solow residual**. 

**All terms other than $R$ can be obtained from data.**

Growth Accounting (cont'd)
============================

Equivalently,

$$
  g_{Y/L} = \alpha_K g_{K/L} + R = \alpha_K g_{K/L} + a_L g.
$$

In the steady state, $\alpha_K$ fraction of growth in output per worker is attributable to capital accumulation.
The rest is due to the technological progress.


After extended to incorporate **human capital accumulation**, the Solow model fits fairly well with data. See Mankiw, Romer and Weil (1992, QJE).