# Neoclassical economics

## Introduction

### Walras equilibrium

### The Arrowâ€“Debreu model

## Classical dichotomy

## 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:

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

#### Steady state

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

\(\dot K = sAK-\delta K\)

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

\(k=\dfrac{K}{L}\)

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

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

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

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

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

## Overlapping generations model

### Introduction