Homogeneous & Linear Homogenous Production Function

A homogeneous production function (HPF) of degree k implies that all factors of production are increased in a given proportion t, outputs are also increased in the proportion t^k
Q (tx₁,tx₂) = t^k Q (x₁,x₂) .
Value of k                     Return to Scale
k = 1                              Constant
k > 1                              Increasing
k < 1                              Decreasing

Properties of HPF:
1. Partial derivatives of homogeneous function of degree k are homogeneous of degree k-1.
2. Isoquants of HPF possess the similar properties of indifference curves of utility function.
3. Expansion path of HPF is also the straight line through the origin, as RTS = f₁/f₂ depends on the ratio in which inputs are used.
Linear Homogeneous Production Function: 
Linear Homogeneous Production Function (LHPF) of 1st degree implies that if all factor of production are increased in a given proportion, outputs are also increased in same proportion i.e.
Q (tx₁,tx₂) = t. Q (x₁, x₂)
A function which is homogenous of degree one is said to be linearly homogeneous. This does not imply that the production function is linear.