# Homogeneous & Linear Homogenous Production Function

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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 tk
Q (tx1,tx2) = tk 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.

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