# Is a production function bilinear?

I believe the following is the multiplicative property of bilinearity:

$$Y=F(K,AL)$$ $$c_1 F(K,AL) = F(c_1 K, AL)$$ $$c_2 F(K,AL) = F(K, c_2 AL)$$

But when we have multiplied through the production function with a constant we have done so through each term as below

$$c_3 F(K,AL) = F(c_3 K, c_3 AL)$$

e.g.

$$\frac{1}{AL} F(K,AL) = F\left( \frac{K}{AL} , \frac{AL}{AL} \right) = F(k,1) = f(k)$$

What is the name of this property?

Such a function is called homogeneous of degree 1.

Constant returns of scale

As observed above, in mathematical terms the function is homogeneous of degree 1. But in terms of economic theory, this is called a production function with constant returns of scale.

It means that if you change the inputs in a determinate proportion, the output varies in the same proportion.