Linear Production Function

The linear production function describes a linear relationship between the inputs and the output.

For example:

Q = aL

Q = aL + bK

Chart:

The linear production function has constant returns to scale.

Proof:
Y = aK + bL
Y’ = a (cK) + b (cL)
= c (aK + bL)
= c Y

The elasticity of substitution is ∞.

Proof:

The elasticity of substitution is the change in K/L divided the change in the MTRS, when K/L changes.

The MTRS is constant. The change in the MTRS is always 0.

By definition, the numerator is not 0.

The elasticity of substitution is ∞.

Proof of The Law of Cosines