where [tex]$Q$[/tex] is total output, [tex]$K$[/tex] is capital stock, [tex]$L$[/tex] is labour stock and [tex]$A$[/tex] and are positive constants, exhibit constant returns to scale. (ii) What returns to scale does [tex]$Q=5 K^{-0.25} L^{0.6}$[/tex] exhibit?
Answer (1)
The production function Q = A K a L b exhibits constant returns to scale when a + b = 1 .
The production function Q = 5 K − 0.25 L 0.6 exhibits decreasing returns to scale because the sum of the exponents is less than 1.
The sum of exponents for the second production function is 0.6 − 0.25 = 0.35 .
Since 0.35 < 1 , the second production function exhibits decreasing returns to scale, therefore the answer is decreasing returns to scale. decreasing returns to scale
Explanation
Problem Analysis We are given two production functions and asked to determine their returns to scale. The first production function is a general form, and we need to find the condition for it to exhibit constant returns to scale. The second production function is specific, and we need to determine whether it exhibits increasing, decreasing, or constant returns to scale.
Constant Returns to Scale Condition (i) Let's analyze the first production function: Q = A K a L b , where Q is total output, K is capital stock, L is labor stock, and A , a , and b are positive constants. For constant returns to scale, if we increase both capital and labor by a factor of λ , the output should also increase by the same factor λ . Mathematically, this means Q ( λ K , λ L ) = λ Q ( K , L ) .
Applying the Condition Substitute λ K for K and λ L for L in the production function: Q ( λ K , λ L ) = A ( λ K ) a ( λ L ) b = A λ a K a λ b L b = A λ a + b K a L b = λ a + b ( A K a L b ) = λ a + b Q ( K , L ) For constant returns to scale, we require Q ( λ K , λ L ) = λ Q ( K , L ) . Therefore, λ a + b Q ( K , L ) = λ Q ( K , L ) . This implies that a + b = 1 .
Analyzing the Second Production Function (ii) Now, let's analyze the second production function: Q = 5 K − 0.25 L 0.6 . To determine the returns to scale, we increase K and L by a factor of λ :
Q ( λ K , λ L ) = 5 ( λ K ) − 0.25 ( λ L ) 0.6 = 5 λ − 0.25 K − 0.25 λ 0.6 L 0.6 = 5 λ − 0.25 + 0.6 K − 0.25 L 0.6 = λ 0.35 ( 5 K − 0.25 L 0.6 ) = λ 0.35 Q ( K , L )
Determining Returns to Scale Since the exponent of λ is 0.35 , and 0.35 < 1 , the production function exhibits decreasing returns to scale. If the exponent were greater than 1, it would exhibit increasing returns to scale, and if it were equal to 1, it would exhibit constant returns to scale.
Final Answer In summary, the first production function Q = A K a L b exhibits constant returns to scale when a + b = 1 . The second production function Q = 5 K − 0.25 L 0.6 exhibits decreasing returns to scale.
Examples
Understanding returns to scale is crucial for businesses when making decisions about expanding production. For example, if a company knows its production exhibits increasing returns to scale, it might be incentivized to increase its capital and labor inputs to achieve a more than proportional increase in output, leading to higher efficiency and profitability. Conversely, if a company faces decreasing returns to scale, expanding production might not be as beneficial, and the company might need to focus on improving efficiency or adopting new technologies to overcome the diminishing returns. This concept is also applicable in agriculture, where farmers consider the optimal amount of fertilizer and labor to maximize crop yield.
