Homogeneous Production Functions and Returns to Scale Explained

Which of the production functions are homogeneous, and if so, of what degree?

In each case say whether there are constant, increasing, or decreasing returns to scale.

Answer:

The production functions (b), (d), (e), and (f) are homogeneous. The production function (b) is of degree 1/2, while the production functions (d), (e), and (f) are of degree 1. The production functions (a) and (c) are not homogeneous.

A production function is said to be homogeneous if it exhibits constant returns to scale. Let's analyze each production function:

(a) Y = 3LK + L^2: This production function is not homogeneous because the exponents of L and K are different.

(b) Y = (3LK + L^2)^(1/2): This production function is homogeneous of degree 1/2. When all inputs are scaled by a constant factor, the output will be scaled by the same factor raised to the power of 1/2.

(c) Y = 3K^2 + 10LK^0.5 + 5L^2: This production function is not homogeneous because the exponents of L and K are different.

(d) Y = 100K^0.3 L^0.7: This production function is homogeneous of degree 1. When all inputs are scaled by a constant factor, the output will be scaled by the same factor raised to the power of 1.

(e) Y = 50K^0.6 L^0.5: This production function is homogeneous of degree 1. When all inputs are scaled by a constant factor, the output will be scaled by the same factor raised to the power of 1.

(f) Y = 25K^0.3 L^0.5: This production function is homogeneous of degree 1. When all inputs are scaled by a constant factor, the output will be scaled by the same factor raised to the power of 1.