# Growth theory

## Introduction

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 important, for example, if you want to understand why 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 are 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

### Definition

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

*are flows while*

**Y***is a stock. During a short period of time, we can assume that the amount of capital is constant.*

**K*** The aggregate production function, *or simply the production function is a function that relates L,

*and Y. Specifically, we assume that*

**K***is a function of*

**Y***and K:*

**L****Y = AL, K)**

In most cases, we will not specify exactly what the function * f *looks like. However, we always assume that

*is increasing in*

**f***and K, that is, when we use more labour and/or more capital, we will produce more goods.*

**L**### The marginal product of labor and capital

We define * the marginal product of labor, MPL *as the derivative of

*with respect to the*

**f***- that is, as (approximately) how much*

**L***will increase when*

**Y***increases by one unit. We also define*

**L***as the derivative of*

**the marginal product of capital, MPK***with respect to K. Note that*

**f***and*

**MPL***will depend on both*

**MPK***and*

**L***and*

**K (MPL***are functions, not variables).*

**MPK**• Since * f *is increasing in L,

*must be positive for all values of*

**MPL***and K.*

**L**• * MPL *assumed to be

*in*

**decreasing***- the more work that is used, the lower the marginal product of labor.*

**L**• * MPL *assumed to be

*in*

**increasing***- the more capital, the higher the marginal product of labor.*

**K**• In the same way, * MPK *must be positive for all values of

*and*

**L**

**K.**• * MPK *is assumed to be decreasing in

*and increasing in*

**K**

**L.**When we view * Y *as a function of

*holding*

**L***fix,*

**K***will be increasing in*

**Y***but at a decreasing pace (due to the fact that*

**L***is positive but decreasing in*

**MPL**

**L).****Fig. 9.1: Production function.**

We define * labour productivity *as

*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.*

**Y/L,**### Production function and Growth

From the simple production function * Y *=

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

**f(L,**• 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 /need not be the same function over time. It is possible that * Y *increases even though

*and*

**L***are fixed. When*

**K***changes so that the same amount of the factors of production will produce more output we say that we have*

**f***or*

**technological progress***With technological progress,*

**productivity growth.***and*

**MPL***will typically increase for given values of L and K, that is, the productivity of the factors increase.*

**MPK**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

*is the amount of human capital and*

**H***amount of physical capital.*

**K**• 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

*With this view, increasing the amount of human capital will lead to productivity growth.*

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

*growth in*

**L,***and growth in productivity.*

**K**