Solow Model (continued)

# Solow Model (continued)
## Macroeconomics
### Kenji Sato
### 2018-4-19

---

## Quiz

Consider a Solow economy. 
How fast does the wage grow along the balanced growth path?

Recall

`$$w = A\left[f(k^*) - k^*f'(k^*)\right]$$`

---

---

## Sudden increase of s

---

## Sudden change of `$C/AL$` and `$S/AL$`

---

## Transition to new steady state

---

## New steady state

---

## Welfare gain in the long run

Whether or not the consumption becomes larger than the original steady-state 
value in the long run depends on the initial saving rate.

The steady-state value for consumption per unit of effective labor

`$$c^* = \frac{C}{AL}$$`

is maximized when

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

---

## Welfare gain in the long run

Let `$s_G$` be the saving rate that the corresponding steady state `$k^*_G$`
satisfies

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

* `$s_G$` and `$k^*_G$` and the Golden-Rule
* If an economy in the steady state with `$s_1 < s_G$` experiences a hike 
  in the saving rate to `$s_2 < s_G$`, then the consumption per unit of effective 
  labor initially drops but eventually gets larger than the original steady 
  state value
  
---

## Dynamic inefficiency for `$s > s_G$`

Let's do a similar exercise with `$s > s_G$`.
  
Suppose that the economy's saving rate is `$s_1 > s_G$` and it's on the 
balanced growth path. The saving rate suddenly drops to `$s_2 > s_G$`.

The economy increases the amount of consumption immediately and continue 
to enjoy larger consumption.

This means that if the consumers are allowed to chose their saving rate,
they wouldn't choose `$s > s_G$` because by reducing `$s$` they can consume more
at any point in time.

---

## Exercise

What happens in response to the following changes of parameter?

* Increase/decrease of `$\delta$`
* Increase/decrease of `$g$`
* Increase/decrease of `$n$`

Describe

* long-run growth and 
* transition path to the new steady state

---

---

## Cobb-Douglas production function

Let's suppose
`$$\begin{aligned}
  F(K, AL) = K^\alpha (AL)^{1-\alpha}, \quad 0 < \alpha <1
\end{aligned}$$`

Corresponding intensive form:

`$$\begin{aligned}
  f(k) = k^\alpha
\end{aligned}$$`

The steady state for the Solow model

`$$\begin{aligned}
  k^* = \left(
    \frac{s}{\delta + g + n}
  \right)^{\frac{1}{1 - \alpha}}
\end{aligned}$$`

---

## What's `$\alpha$`?

`$\alpha$` is the capital share!

Exercise: Prove that

`$$\begin{aligned}
  \alpha = \frac{(r + \delta) K}{Y}
\end{aligned}$$`
---

---

## Growth rate in transition

Consider a transition path.

The growth rate of output

`$$\begin{aligned}
\frac{\frac{d}{dt}\left( Y/L \right)}{Y/L} &=
\frac{\frac{d}{dt}\left( Ak^\alpha \right)}{Ak^\alpha} =
g + \alpha \frac{\dot k}{k}
\end{aligned}$$`

Since the second term is large and positive when `$k$` is small, 
relatively poor countries tend to grow rapidly.

---

## Testable prediction

If the Solow model is correct, the growth rate declines as an economy matures. This trend should be widely observed and we call this situation "convergence."

You can estimate the following model to test if the average growth rate is negatively correlated with the initial wealth.

`$$\begin{aligned}
   \ln \left(Y_t/L_t \right) - \ln \left( Y_0/L_0 \right)
   =
   \beta_0 + \beta_1 \ln(Y_0 / L_0)
\end{aligned}$$`

---

---

## Per-capita income

Per-capita income in the steady state is

`$$\begin{aligned}
  \frac{Y}{L} 
  &= 
  A \left(
    \frac{s}{\delta + g + n}
  \right)^{\frac{\alpha}{1 - \alpha}}\\
  &=
  A(0)e^{gt}
  \left(
    \frac{s}{\delta + g + n}
  \right)^{\frac{\alpha}{1 - \alpha}}
\end{aligned}$$`

Taking log,

`$$\begin{aligned}
  \ln \frac{Y}{L} = \text{const.} + gt 
    +\frac{\alpha}{1 - \alpha} \ln s
    -\frac{\alpha}{1 - \alpha} \ln (\delta + g + n)
\end{aligned}$$`

---

## Testable prediction

`$$\begin{aligned}
  \ln \frac{Y}{L} = \text{const.} + gt 
    +\frac{\alpha}{1 - \alpha} \ln s
    -\frac{\alpha}{1 - \alpha} \ln (\delta + g + n)
\end{aligned}$$`

The Solow model predicts that, other things being equal, one percent increase in `$s$` (e.g. from 30% to 30.3%) results in an increase of `$Y/L$` by `$\frac{\alpha}{1 - \alpha}\times 100$` percent.

Common wisdom suggests `$\alpha \simeq 1/3$` and the elasticity of output with respect to the saving rate should thus be

`$$\frac{\alpha}{1-\alpha} \simeq \frac{1}{2}$$`