The purpose of this chapter is to try to explain ** growth in GDP ** . The models in this chapter are very different from the rest of the models in this book as they use only the production function and factors of production to explain growth.

** Growth models are importan ** t, for example, if you want to understandwhy some countries grow faster and have a higher living standard than other countries. By growth, we mean the percentage change in ** real GDP ** . We use real GDP to eliminate the effect of inflation. In this chapter, it is perfectly OK to think of inflation as being zero in which case real and nominal GDP bare the same.

In this chapter we begin by describing ** the aggregate production function ** . The rest of this chapter will look at some different growth theories.

## The aggregate production function

Imagine the national economy during a short period of time (say one week). We denote:

**L**: the total amount of work used during this period (by all individuals in the economy).**K**: the total amount of capital used.**Y**: the total amount of finished goods produced during this period (real).

It is still the case that ** L ** and ** Y ** are flows while ** K ** is a stock. During a short period of time, we can assume that the amount of capital is constant.

** The aggregate production function ** , or simply the production function is a function that relates ** L ** , ** K ** and ** Y ** . Specifically, we assume that ** Y ** is a function of ** L ** and ** K ** :

** Y ** = ** f ** ( ** L ** , ** K ** )

In most cases, we will not specify exactly what the function ** f ** looks like. However, we always assume that ** f ** is increasing in ** L ** and ** K ** , that is, when we use more labour and/or more capital, we will produce more goods.

### The marginal product of labor and capital

We define ** the marginal product of labor ** , ** MPL ** as the derivative of ** f ** with respect to the ** L ** – that is, as (approximately) how much ** Y ** will increase when ** L ** increases by one unit. We also define ** the marginal product of capital ** , ** MPK ** as the derivative of ** f ** with respect to ** K ** . Note that ** MPL ** and ** MPK ** will depend on both ** L ** and ** K ** ( ** MPL ** and ** MPK ** are functions, not variables).

- Since
**f**is increasing in**L**,**MPL**must be positive for all values of**L**and**K**. **MPL**assumed to be**decreasing**in**L**– the more work that is used, the lower the**marginal product of labor**.**MPL**assumed to be**increasing**in**K**– the more capital, the higher the**marginal product of labor**- In the same way,
**MPK**must be positive for all values of**L**and**K**. **MPK**is assumed to be decreasing in**K**and increasing in**L**.

When we view ** Y ** as a function of ** L ** holding ** K ** fix, ** Y ** will be increasing in ** L ** but at a decreasing pace (due to the fact that ** MPL ** is positive but decreasing in ** L ** ).

Fig. 9.1: Production function.

We define ** labour productivity ** as

**Y**

**/**

**L**, that is, as

**GDP**per hour worked. Labour productivity tells us how much we can produce using one hour of labour and it depends on the amount of capital as well as the technology.

### Production function and Growth

From the simple production function ** Y **

**=**

**f****(**

**L****,**

**K****),**we can identify three sources of growth:

- An increase in
**L**. - An increase in
**K**. - A change in the function
**f**

The first two represent growth of the factors of production. ** L ** may increase if the population grows, if we have more individuals in the workforce, or if unemployment falls. K increases if investment are large as they are if total savings is large.

The function ** f ** need not be the same function over time. It is possible that ** Y ** increases even though ** L ** and ** K ** are fixed. When ** f ** changes so that the same amount of the factors of production will produce more output we say that we have ** technological progress ** or ** productivity growth ** . With technological progress, ** MPL ** and ** MPK ** will typically increase for given values of L and K, that is, the productivity of the factors increase.

Education and growth in human capital are important aspects of growth in ** GDP ** . Human capital is treated in different ways in the literature:

- You can think of human capital as being included in
**K**– with this view education is a type of investment. - You can add another variable in the production function:
**Y**=**f**(**L**,**K**,**H**) where**H**is the amount of human capital and**K**amount of physical capital. - The amount of human capital may affect the function
**f**. The more human capital, the more can be produced from the same amount of**L**and**K**. With this view, increasing the amount of human capital will lead to productivity growth.

** Growth Accounting ** is the activity in which we try to figure out how much of the growth in

**GDP**is due to growth in

**L**, growth in

**K**and growth in productivity.

## Growth Theories

### The classical growth theory

The production function will not provide us with a theory or explanation of growth. It is only a convenient tool which helps us breaking down growth into its components. However, there are many growth theories that try to go a step further. The oldest of these theories is the so-called ** classical growth theory ** which is primarily associated with Thomas Robert Malthus.

** The classical growth theory ** should not be confused with the classical model that we will look at in the next chapter. Also, ** the classical growth theory ** , which was developed in the late 1700s, has little or no relevance today. We present it so that you can better understand more modern growth theories.

In short, the ** classical growth theory ** may be described as follows:

- Due to technological development, the amount of capital increases and the
**marginal product of labor rises**. **GDP**per capita rises. With higher living standards, the population will increase.- As population increases, the
**labor**productivity will fall (more individuals but the sameamount of capital). **GDP**per capita will fall again. When**GDP**per capita has fallen to a level just high enough to keep the population from starving, the increase in population will cease.

Destruction of capital, for example, through a war, works in the opposite way. The ** marginal product of labor falls ** , ** GDP ** per capita falls and the population decreases. This will again lead to an increase in the ** marginal product of labor ** and ** GDP ** per capita return to the "survival rate".

The main point of the model is that population growth will always eliminate the positive effects of technological development and ** GDP ** per capita will always return to the survival level. This very "dismal" growth theory was prominent in the early 1800s, and economics to this day is sometimes called the "dismal science".

Today we know that the predictions of the model where incorrect. During the rest of the 1800s Europe experienced a growth in ** GDP ** per capita. Although the population growth was high, it was not nearly sufficient to eliminate the positive effects of technological development.

### The neo-classical growth model

The main purpose of another important growth model, the neo-classical growth model, is to explain how it is possible to have a permanent growth in ** GDP ** per capita. The model was developed by Robert Solow in the 1960s and it is sometimes called the Solow growth model or the exogenous growth model.

The neo-classical growth model should not be confused with the neoclassical synthesis, which we will study in chapter 10. "Neo" means "new" – ** the neo-classical growth theory ** is a “new version” of the classical growth model.

The crucial difference between the classical and neo-classical growth model is that population is endogenous in the former and exogenous in the latter. In the classical model, population will increase or decrease depending on whether ** GDP ** per capita is higher than or lower than the survival level. In the neo-classical model population growth is not affected by ** GDP ** per capita (however, the population growth will affect the growth in ** GDP ** per capita).

In the neo-classical model, it is the ** technological progress ** only that affects the ** GDP ** per capita in the long run. We will have a permanent increase in ** GDP ** per capita when there is a technological development that increases productivity of labour. Permanent ** growth ** in ** GDP ** then requires continuous technological progress.

It is not possible for the government, except temporarily, to affect the growth rate in the neo-classical growth model. The government might be able to affect ** GDP ** per capita (and thus is the growth rate) but the growth rate always returns to the level determined by the technological progress. The same is true for savings. An increase in savings may have a temporary effect on ** GDP ** but it will have no effect in the long run.

## Endogenous growth theory

** Endogenous growth theory ** or new growth theory was developed in the 1980s by Paul Romer and others. In the neo-classical model, technological progress is an exogenous variable. The neo-classical growth model makes no attempt to explain how, when and why technological progress takes place.

The main objective of the ** endogenous growth theory ** is to make the technological progress an endogenous variable to be explained within the model, hence the name endogenous growth theory.

There are many different explanations for technological progress. Most of them, however, have a lot of common characteristics:

- They are based on
**constant return to scale for capital**. Thus,**MPK**is not a decreasing function of**K**in these models. - They consider technological development as a public good.
- They focus more on human capital.
- It is possible for the government to affect the growth rate. Higher savings also leads to highergrowth, not just higher
**GDP**per capita.

- They predict convergence of
**GDP**per capita between countries in the long run. This is aconsequence of the public good property of the technological developments.

## Separation of growth and fluctuation

It is often useful to separate the evolution of a variable that grows over time into a trend and fluctuations around the trend. The graphs below show such a separation for real ** GDP ** .

** Fig. 9.2: ** ** Growth and the fluctuation around the trend. **

The left diagram shows a stylized graph of real ** GDP ** over time. It demonstrates the two important characteristics in real ** GDP ** . ** GDP ** fluctuates over time and ** GDP ** grows over time – at least over a longer period of time. The left graph is the sum of the middle graph and the right graph. The middle graph shows the ** trend ** in ** GDP ** . The trend represents the second characteristic of ** GDP ** – the fact that ** GDP ** grows over time. The right graph shows the fluctuations around the trend (cycles) of ** GDP ** . These fluctuations around the trend represent the first property of ** GDP ** .

In macroeconomics it is common to study trends and cycles separately. The purpose of ** growth theory ** is to investigate the trend while most of macroeconomics apart from growth theory is about the cycles. The trend is about the very long run perspective of the economy while cycles are about the short and medium run. The rest of this is all about cycles and not at all about trends. Therefore, when you think of ** GDP ** in the remaining chapters, you should think of ** GDP ** as in the right-hand graph: ** GDP ** has cycles but no trend. Basically, we will study ** GDP ** where the trend has been removed.