Solow Model

# Solow Model
## Macroeconomics
### Kenji Sato
### Day 4

---

---

## Review for the problem set

- `$Y/L$` grows at a sustained rate (Item 1)
- `$K/Y$` is constant (Item 4)

jointly imply

- `$K/L$` grows at a sustained rate. (Item 2)

---

## Proof

Notice that

`$$\begin{aligned}
  \frac{K}{L} = \underbrace{\frac{K}{Y}}_{\text{is constant}} 
                \times
                \underbrace{\frac{Y}{L}}_{\text{has a constant growth rate}}
\end{aligned}$$`

We have `$K/L$` has a constant growth rate.

`$K/L$` and `$Y/L$` grow at the same rate.

---

## Growth rate

Let `$X$` be a time-series variable and `$g_X$` the growth rate of `$X$`, i.e.,

`$$g_X = \frac{\dot X}{X}$$`

Interpret this with the "discrete-time approximation."

`$$\begin{aligned}
  \frac{\dot X(t)}{X(t)} 
  \simeq
  \frac{[X(t+\Delta t) - X(t)] / \Delta t}{ X(t)}
\end{aligned}$$`

---

## Growth rate

It might be tempting to define the growth rate with

`$$\begin{aligned}
  & \frac{X(t+\Delta t) - X(t)}{X(t)}  & (\text{bad formula!})
\end{aligned}$$`

but this is wrong. You always get zero when `$\Delta t \to 0$`.

Convert the change in `$X$` into an equivalent change for a year (or month/quarter etc.) and then compute the growth rate.

This is why we have `$\Delta t$` in

`$$\begin{aligned}
  \frac{\dot X(t)}{X(t)} 
  \simeq
  \frac{[X(t+\Delta t) - X(t)] / \Delta t}{ X(t)}
  =
  \frac{\text{Change in one year}}{\text{Original level}}
\end{aligned}$$`

---

## Growth rate formulas

Let `$X, Y$` be a time-series variable and `$\alpha$` a constant number.

We have the following formulas:

1. ** `$g_{XY} = g_X + g_Y$`**. Equivalently, 
`$$\frac{\frac{d}{dt}(XY)}{XY} = \frac{\dot X}{X} + \frac{\dot Y}{Y}$$`
1. ** `$g_{X/Y} = g_X - g_Y$`**
1. ** `$g_{X^\alpha} = \alpha g_X$`**

---

[Your turn] Prove these facts using:

`$$\begin{aligned}
  \frac{d}{dt}\left(
    \ln X
  \right)
  =
  \frac{\dot X}{X}
\end{aligned}$$`

---

---

## Solow Model

The following parameters characterize the Solow model:

- `$F(K, AL)$`: production function
- `$s$`: saving rate
- `$\delta$`: depreciation rate
- `$g$`: technology growth rate
- `$n$`: population growth rate

---

---

## 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

A constant fraction, `$s$`, of output is saved, and the rest, `$1 - s$`, is consumed; i.e.,

`$$C = (1 - s) Y \quad \text{and}\quad S = sY$$`

Under the closed economy assumption, `$S = I$`.

The saving rate, `$s$`, is constant. The capital accumulation equation can be written as:

`$$\begin{aligned}
  \dot K
  &= I - \delta K\\
  &= S - \delta K\\
  &= sY - \delta K\\
  &= sF(K, AL) - \delta K
\end{aligned}$$`

---

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

Knowledge `$A$` and labor (or population) `$L$` grow at exogenously
given constant rates.

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

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

We assume that

`$$\delta + g + n > 0$$`

---

## Capital per unit of effective labor

Variables of interest are

* `$Y$`: Output
* `$K$`: Capital
* `$C$`: Consumption
* `$Y/L$`: Output per capita
* `$K/L$`: Capital per capita
* `$C/L$`: Consumption per capita

Instead of analyzing the behavior of them directly, we study that of
**capital per effective labor**

`$$k = \frac{K}{AL}$$`

---

## Output per unit of effective labor

`$$\begin{aligned}
  \frac{Y}{AL} 
  = \frac{F(K, AL)}{AL}
  = F\left(
    \frac{K}{AL}, \frac{AL}{AL}
  \right)
  = F(k, 1)
\end{aligned}$$`

Define
`$$y = \frac{Y}{AL} \quad
\text{and} \quad
f(k) = F(k, 1).$$`

We have the **intensive-form production function**
`$$y = f(k)$$`

---

## Assumption on `$f$`

The following assumptions are made for `$f$`:

* `$f(0) = 0$`
* `$f'(k) > 0$`
* `$f''(k) < 0$` 
* `$f'(0) > (\delta + g + n) / s$`
* `$f'(\infty) < (\delta + g + n) / s$`

Usually we assume stronger conditions called Inada conditions:

* `$f'(0) = \infty$`
* `$f'(\infty) = 0$`

---

## Shape of `$f$`

`$f$` is increasing, has diminishing marginal productivity, and satisfies Inada conditions.

---

## Capital accumulation per `$AL$`

Divide by `$AL$`

`$$\dot K = sF(K, AL) - \delta K$$`

to get

`$$\frac{\dot K}{AL} = sf(k) - \delta k$$`
and then

`$$\frac{\dot K}{K} k = sf(k) - \delta k$$`

How can we express `$\dot K/K$` with `$k$`?

---

## `$\dot K / K =\ ?$`

Recall that

`$$k = \frac{K}{AL}$$`

The growth rate of `$k$` is

`$$\begin{aligned}
\frac{\dot k}{k}
&= \frac{\dot K}{K} - \frac{\dot A}{A} - \frac{\dot L}{L}\\
\end{aligned}$$`

and thus

`$$\begin{aligned}
  \frac{\dot K}{K}
  =
  \frac{\dot k}{k} + g + n
\end{aligned}$$`

---

## Capital accumulation equation

We now have 
`$$\begin{aligned}
  \frac{\dot K}{K} k &= sf(k) - \delta k\\
  \frac{\dot K}{K}   &= \frac{\dot k}{k} + g + n
\end{aligned}$$`

Therefore,

`$$\dot k = sf(k) - (\delta + g + n) k$$`

This is _the_ fundamental differential equation that governs the dynamics 
of the Solow economy.

---

## Dynamics

`$$\dot k = sf(k) - (\delta + g + n) k$$`

* `$sf(k) > (\delta + g + n) k \Longrightarrow k$` increases.
* `$sf(k) = (\delta + g + n) k \Longrightarrow k$` stays unchanged.
* `$sf(k) < (\delta + g + n) k \Longrightarrow k$` decreases.

Due to the assumptions on `$f$`, it is shown that

* `$k$` grows when `$k$` is sufficiently small
* `$k$` decreases when `$k$` is sufficiently large

There is a threashold value `$k^*$`, below which `$k$` grows and above which `$k$` decreases.

---

## Graphical exposition

---

## Dynamics for `$k(0) < k^*$`

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## Dynamics for `$k(0) > k^*$`

---

## Dynamics for `$k(0) = k^*$`

---

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## Long-run equilibrium

Since `$k(t)$` alwyas converges to steady state `$k^*$`, `$k^*$` can be considered as 
the long-run equilibrium.

The economy with

`$$k(t) = k^*$$`

is a nice benchmark case for our study.

In the steady state, there is no growth in `$k$`.

But what about `$Y, K, C$` and `$Y/L, K/L, C/L$`?

---

## Aggregate variables

In the steady state,

`$$K = AL k^*,\quad
Y = ALf(k^*), \quad \text{and} \quad
C = AL (1 - s)f(k^*)$$`

They all grow at the rate of

`$$g + n$$`

---

## Per-capita variables

In the steady state,

`$$\frac{K}{L} = Ak^*, \quad
\frac{Y}{L} = Af(k^*), \quad \text{and}, \quad
\frac{C}{L} = A(1-s)f(k^*)$$`

They all grow at the rate of

`$$g$$`

---

## Balanced Growth

In the steady state, all the important variables grow at constant rates.

This situation is called **balanced growth**.

We use phrases "in the steady state" and "along the balanced growth path"
interchangeably.

---

## Interest rate

The firm's profit maximization in the competitive factor market
dictates that marginal product of capital equals rate of return
on capital.

`$$\frac{\partial F}{\partial K} = r + \delta$$`

Notice that

`$$\frac{\partial}{\partial K}F(K, AL)
= AL \frac{\partial}{\partial K}F(k(K, AL), 1)
= AL \frac{d F(k, 1)}{dk} \frac{\partial k}{\partial K}$$`

`$$\Longrightarrow
\frac{\partial F}{\partial K} = f'(k)$$`

since `$dk/dK = 1/AL$`.

---

## Interest rate (cont'd)

In the steady state,

`$$f'(k^*) = r + \delta,$$`
which is constant over time.

---

## Wage

The firm's profit maximization in the competitive factor market
dictates that marginal product of labor equals wage rate.

`$$\frac{\partial F}{\partial L} = w$$`

By Euler's theorem on homogeneous functions,

`$$F(K, AL) = K \frac{\partial F}{\partial K} + L \frac{\partial F}{\partial L}$$`

---

## Wage (cont'd)

Divide by `$AL$` to get

`$$f(k) = k f'(k) + \frac{w}{A}$$`

and

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

It is sometimes useful to introduce thewage per unit of effective labor

`$$\hat{w} := \frac{w}{A} = f(k) - kf'(k)$$`

---

## Exercise: Stylized facts

Verify that the steady state of the Solow model can explain the stylized facts.

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.

---

## Quiz

How fast does the wage grow along the balanced growth path?

Recall

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