Question

For each of the following document if the function has increasing/decreasing or constant returns to scale, explain why:

a) y=x1+x2
b) y=2×1+3×2
c) y=x₁²/³x₂²/³​
d) y=min{2x₁​;x₂​}
e) y=min{x₁​;x₂​}²
f) y=(xᵖ₁+xᵖ₂​) ¹/β

Answer 1

Various functions demonstrate different types of returns to scale: y=x1+x2 and y=2x1+3x2 have constant returns to scale. The function y=x₁²/³x₂²/³ has increasing returns to scale. Functions y=min{2x₁​;x₂​} and y=min{x₁​;x₂​}² have constant returns to scale. The function y=(xⁿ₁+xⁿ₂​) ¹/β's returns scale can vary.

Step-by-step explanation:

We are evaluating the type of returns to scale for various functions. The function y=x1+x2 demonstrates constant returns to scale since doubling the inputs x1 and x2 will simply double the output y. The function y=2x1+3x2 also shows constant returns to scale for the same reason: scaling the inputs will linearly scale the output by the same factor.

The function y=x₁²/³x₂²/³​ has increasing returns to scale. At higher levels of inputs, the output increases at an increasing rate due to the nature of the exponents being less than 1, which is characteristic of production functions with increasing returns to scale.

The function y=min{2x₁​;x₂​} shows constant returns to scale as long as the minimum value remains within the same term after scaling the inputs. The function y=min{x₁​;x₂​}² also has constant returns to scale for the same reason.

Lastly, the function y=(xⁿ₁+xⁿ₂​) ¹/β is more complex to analyze without specific values for p and β, but generally, if β=p, then it represents constant returns to scale. If β>p, it indicates increasing returns to scale and if β