Which of the following production functions exhibit constant returns to scale? In each case
ys output and K and L are inputs. (1) y = K1/2L2/3 (2) y = 3K1/2L1/2 (3) y = K1/2 + L1/2
(4) y = 2K + 3L.
To determine which of the production functions exhibit constant returns to scale, we need to evaluate their properties.
Constant returns to scale occurs when increasing both inputs, K and L, by the same proportion results in a proportional increase in output, ys. Mathematically, a production function exhibits constant returns to scale if it satisfies the following condition:
f(tK, tL) = t * f(K, L)
where f(K, L) represents the production function with inputs K and L, t is a scaling factor.
Let's evaluate each of the given production functions:
(1) y = K^(1/2) * L^(2/3)
To determine if this production function exhibits constant returns to scale, we need to check if the equation holds:
f(tK, tL) = t * f(K, L)
Substituting the function, we have:
(tK)^(1/2) * (tL)^(2/3) = t * K^(1/2) * L^(2/3)
Simplifying, we have:
t^(1/2) * t^(2/3) * K^(1/2) * L^(2/3) = t * K^(1/2) * L^(2/3)
t^(7/6) * K^(1/2) * L^(2/3) = t * K^(1/2) * L^(2/3)
Since the equation holds, we can say that production function (1) exhibits constant returns to scale.
(2) y = 3K^(1/2) * L^(1/2)
Following the same process as before:
(tK)^(1/2) * (tL)^(1/2) = t * 3K^(1/2) * L^(1/2)
t^(1/2) * t^(1/2) * K^(1/2) * L^(1/2) = t * 3K^(1/2) * L^(1/2)
t * K^(1/2) * L^(1/2) = t * 3K^(1/2) * L^(1/2)
The equation holds, so production function (2) also exhibits constant returns to scale.
(3) y = K^(1/2) + L^(1/2)
Following the same process:
(tK)^(1/2) + (tL)^(1/2) ≠ t * K^(1/2) + L^(1/2)
The equation does not hold, so production function (3) does not exhibit constant returns to scale.
(4) y = 2K + 3L
Following the same process:
(t * 2K) + (t * 3L) ≠ t * 2K + 3L
The equation does not hold, so production function (4) does not exhibit constant returns to scale.
In conclusion, production functions (1) and (2) exhibit constant returns to scale, while production functions (3) and (4) do not.