# Chapter 8 Labour supply, employment and productivity

In chapter 7 we introduced the relationship between GDP, i.e. aggregate output, and employment. There we saw that the short-term labour demand of firms depends in particular on their production plans and thus on their sales expectations. In this chapter we will take a closer look at how the current level of unemployment is determined by the interaction of demand, labour supply and labour productivity. To do this, we assume for the purposes of this chapter that demand, and thus the production planned by firms, is exogenous. Firms match the volume of output to demand by maintaining a corresponding level of employment.

The “labour supply” of our economy is given by the number of persons in the **labour force**. The labour force includes all persons who are self-employed, employed or looking for employment. Thus:

\[\begin{equation} \text{Labour force} = \text{Employed} + \text{Unemployed} \tag{8.1} \end{equation}\]

We define the unemployment rate here as the ratio of unemployed persons to the labour force:

\[\begin{equation} \text{Unemployment rate} = \frac{\text{Unemployed}}{\text{Labour force}} \cdot 100 = \\ \frac{\text{Labour force} - \text{Employed}}{\text{Labour force}} \cdot 100 \tag{8.2} \end{equation}\]

**Unemployment - A matter of definition?**

The calculation of the unemployment rate just introduced is only one of many different definitions of unemployment. In fact, official statistics also show differently defined key figures of unemployment. These can even be different depending on the purpose for which the statistics are used. For example, if the purpose is to calculate state social benefits, the definition of unemployed would be defined relatively narrowly as the group of people entitled to benefits. If, on the other hand, it is a matter of general well-being, people without benefit entitlement who suffer from unemployment would also be included in the unemployment rate. It is therefore important to pay attention to the exact definition.

We assume here that the labour force is constant and exogenous. Since the number of employed persons and job seekers is determined by the demand for labour of firms, the unemployment rate thus depends on aggregate output. How many employees firms need to produce GDP is determined by labour productivity. Labour productivity measures the amount of output that can be produced with one unit of labour, measured in hours or full-time employed workers. Labour productivity itself depends on various factors, especially technological knowledge and the capital stock used in the economy.^{31} If we define the labour unit as one person employed full-time in the production period, we can represent the emergence of GDP as follows:

\[\begin{equation} \text{GDP} = \text{Labour productivity (per person)} \cdot \text{Employment} \tag{8.3} \end{equation}\]

We have already used this simple relationship in chapter 6 to illustrate the effect of a change in government spending on employment, assuming there that labour productivity is exactly 1. In our model, the relationship between output, \(Y\), labour productivity, \(y\), and employment, \(L\), is determined by the following equation:

\[\begin{equation} Y = yL \tag{8.4} \end{equation}\]

The relationship captured in equation (8.4) could be called a short-run production function. Strictly speaking, this is a limited production function. Since our model is short-run, we assume that the capital stock in our economy (machinery and equipment) is constant but not fully utilised. As employment increases, so does the utilisation rate of the capital stock. As employment falls, the degree of utilisation falls. A medium- and long-term model would then of course have to take into account the effects of investment and depreciation on the capital stock and technological progress. However, this is not done here for the short run.

If we rearrange equation (8.4) for employment, \(L\), we get employment as a function of aggregate output, which in our model has so far been determined on the demand side:

\[\begin{equation} L = \frac{Y}{y} \tag{8.5} \end{equation}\]

To calculate the number of unemployed and the unemployment rate, we now also need the number of labour force, \(N\). The number of unemployed, \(U\), is thus calculated as \(U = N - L\). The unemployment rate, \(u\), is thus given by \(u = U/N\).

The level of employment in the goods market equilibrium, \(Y^*\), is therefore:

\[\begin{equation} L^* = \frac{Y^*}{y} \tag{8.6} \end{equation}\]

The unemployment rate in the goods market equilibrium is calculated as:

\[\begin{equation} u^* = \frac{N - L^*}{N} \tag{8.7} \end{equation}\]

or

\[\begin{equation} u^* = \frac{U^*}{N} \tag{8.8} \end{equation}\]

The interactive app accessible through the link below shows this relationship, where we have initially set labour productivity to a value of 1. As in the example from chapter 6, production is plotted on the vertical axis and employment on the horizontal axis in the figure. If the output of firms, i.e. GDP, \(Y\), changes, the number of workers required to produce it also changes. With higher labour productivity, the same amount of output could be produced with a smaller number of workers. In the following interactive app, labour productivity can be changed.

We can now link the production side of our model economy with the demand side from the previous chapters. In the following app, all parameters of the simple income-expenditure model can be changed with an interest elastic IS curve and the production function.

Organisational efficiency or worker well-being, for example, could also be discussed as other determinants of labour productivity.↩︎