# Neoclassical economics

## Neoclassical exogeneous growth models

### The Harrod-Domar model

#### Introduction to growth models

We have output as a function of capital.

$$Y=f(K)$$

We also have capital dynamics.

$$\dot K=I-\delta K$$

$$I=S=sY$$

This gives us:

$$\dot K = sY-\delta K$$

#### Introduction

The production function is:

$$Y=cK$$

This gives us:

$$\dot K=(sc-\delta )K$$

#### Growth

$$\dot Y=c\dot K$$

$$\dfrac{\dot Y}{Y}=c\dfrac{\dot K}{Y}$$

$$\dfrac{\dot Y}{Y}=c\dfrac{(sc-\delta )K}{cK}$$

$$\dfrac{\dot Y}{Y}=sc-\delta$$

#### Per-capita growth

Per capita income is:

$$y=\dfrac{Y}{L}$$

$$k=\dfrac{K}{L}$$

### The Solow-Swan model

#### Recap of growth models

As with the Harrod-Domar model we have output as a function of capital:

$$Y=f(K)$$

Capital dynamics:

$$\dot K=I-\delta K$$

$$I=S=sY$$

This gives us:

$$\dot K = sY-\delta K$$

#### Recap of the Harrod-Domar model

The production function of the Harrod-Domar model is:

$$Y=cK$$

And long-term growth of:

$$\dfrac{\dot Y}{Y}=sc-\delta$$

#### The Solow-Swan production function

We use a new production function:

$$Y=K^\alpha (AL)^{1-\alpha }$$

We add dynamics for technology and labour.

$$A_t=A_0e^{gt}$$

$$L_t=L_0e^{nt}$$

$$Y_t=K_t^\alpha (A_0e^{gt}L_0e^{nt})^{1-\alpha }$$

#### Effective captial

$$k_t=\dfrac{K_t}{A_tL_t}$$

$$Y_t=\dfrac{Y_t}{A_tL_t}$$

The dynamics of effective capital is:

$$\dot k_t=sk_t^{\alpha }-(n+\delta +g)k_t$$

In equilibrium effective capital is stable.

$$\dot k_t^*=sk_t^{\alpha }-(n+\delta +g)k_t^8=0$$

$$sk_t^{*\alpha }=(n+\delta +g)k_t^8$$

$$k_t^*=\biggr (\dfrac{s}{n+g+\delta }\biggr )^{\dfrac{1}{1-\alpha }}$$

### The Mankiw-Romer-Weil model

We add human capital to the Solow-Swan model.

### The Golden Rule savings rate

The Golden Rule savings rate is the rate which maximises long term consumption per capita.

If the savings rate is $$0$$ there is no capital and no income. If the savings rate is $$1$$ then then there is no consumption.

### The Ramsey-Cass-Koopmans model

This is based on the Solow-Swan model, with an endogeneous savings rate.

## Neoclassical endogeneous growth models

### The AK model

#### Recap of growth models

As with the Harrod-Domar model we have output as a function of capital:

$$Y=f(K)$$

Capital dynamics:

$$\dot K=I-\delta K$$

$$I=S=sY$$

This gives us:

$$\dot K = sY-\delta K$$

#### Recap of the Harrod-Domar and Solow-Swan models

In the Solow-Swan model the production function was:

$$Y=K^\alpha (AL)^{1-\alpha }$$

In the Harrod-Domar model the production function was:

$$Y=cK$$

In the Solow-Swan model we also added population and technology growth

#### The AK model

In the AK model the production function is:

$$Y=AK$$

We keep population growth from the Solow-Swan model.

#### Per-capita income

$$\dot K = sAK-\delta K$$

$$\dot K = (sA-\delta )K$$

$$k=\dfrac{K}{L}$$

$$\dot k =\dfrac{\dot K}{L}-\dot L\dfrac{K}{L^2}$$

$$\dot k =\dfrac{(sA-\delta )K}{L}-\dot L\dfrac{K}{L^2}$$

$$\dot k =(sA-\delta )k-k\dfrac{\dot L}{L}$$

$$\dot k =(sA-\delta -n)k$$

$$\dfrac{\dot k}{k} =sA-\delta -n$$