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