The Production Function

Ouput is based on the production function, which relates output $Y$ to the size of the capital stock and of the labor force $L$:

$Y = F(K, L)$

where $K$ is the total value of all physical capital in the economy, such as machines, tractors and office buildings, $L$ is the total number of workers in the economy and $Y$ is real output.

The model assumes constant returns to scale, meaning that the prodction function satisfies the following property:

$zY = F(zK, zL)$

If we scale both the amount of capital and labor by some constant $z$, output will increase by that same constant. This assumption is considered realistic and will simplify our analysis.

A famous function which has this property is the Cobb-Douglas production function, which is formulated as follows:

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

where $\alpha$ determines the relative importance of capital and labour. To understand $\alpha$ intuitively, change it's value using the slider below.

As you can see, decreasing $\alpha$ increases the relative importance of labour $L$, and increasing $\alpha$ increases the relative importance of capital $K$.