In the following production functions output q is a function

In the following production functions, output q is a function of capital K and labour L. Do the following functions exhibit increasing, constant, or decreasing returns to scale?

Explain your answers.

1) q = L^1/2K^1/2

2) q = 4L^1/2 + 4K

3) q = log(L), where L is the only input

Solution

1) q = L^1/2K^1/2

Q=(LK)

z(LK) ~ ((zL)(zK))
z(LK) ~ z(LK)
z ~ z
1 = 1 = Constant returns to scale.

2) q = 4L^1/2 + 4K

z(4L + 4K) ~ (4zL + 4zK)
4zL + 4zK ~ 4zL + 4zK
4zL ~ 4zL
z ~ z
z ~ 1
z > 1 decreasing returns to scale

3) q = log(L)

As the labor (L) increases, the value of output (Q) increases.

As log function increases as the value vaires. Log2 = 0.3010, log3 = 0.4771 etc.

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