A company has developed the production function of Q=80L.5 K0.6 where L represents labor and K represents Capital.
a. If L is increased by 2% with K unchanged, what is the resulting percentage change in output?
b. Describe the nature of returns to scale for this production function.
To calculate the resulting percentage change in output when labor (L) is increased by 2% with capital (K) unchanged, we can start by understanding the production function equation:
Q = 80L^0.5 * K^0.6
a. To find the resulting percentage change in output, we need to evaluate the impact of the 2% increase in labor on the production function and calculate the new output.
First, let's calculate the new labor value after a 2% increase:
New L = L + (2% * L)
= L + (0.02 * L)
= L * (1 + 0.02)
= L * 1.02
Now, substitute the new labor value in the production function equation:
Q_new = 80(L_new)^0.5 * K^0.6
= 80(L * 1.02)^0.5 * K^0.6
= 80(1.02^0.5) * L^0.5 * K^0.6
Now, let's calculate the percentage change in output:
Change in Q = (Q_new - Q) / Q * 100
Substituting the values:
Change in Q = (80(1.02^0.5) * L^0.5 * K^0.6 - 80L^0.5 * K^0.6) / (80L^0.5 * K^0.6) * 100
After simplifying the equation, we get:
Change in Q = (1.02^0.5 - 1) * 100
Now, calculate the value:
Change in Q = (1.010153 - 1) * 100
= 0.010153 * 100
= 1.0153%
Therefore, the resulting percentage change in output when labor (L) is increased by 2% with capital (K) unchanged is approximately 1.0153%.
b. To describe the nature of returns to scale for this production function, we need to analyze how the percentage change in output compares to the percentage change in inputs.
If the percentage change in output is proportional to the percentage change in inputs, meaning they both increase (or decrease) by the same factor, it is called constant returns to scale (CRS). In other words, doubling the inputs will result in double the output and vice versa.
If the percentage change in output is greater than the percentage change in inputs, it is known as increasing returns to scale (IRS). Increasing inputs will lead to a more than proportionate increase in output.
If the percentage change in output is less than the percentage change in inputs, it is called decreasing returns to scale (DRS). Increasing inputs will result in a less than proportionate increase in output.
In this case, we need to examine the exponents of the labor (L) and capital (K) terms in the production function:
Q = 80L^0.5 * K^0.6
The exponents are less than 1, which implies diminishing marginal returns to both labor and capital. Consequently, the production function exhibits decreasing returns to scale. As inputs increase, the output will increase, but at a decreasing rate.
a. To find the percentage change in output resulting from a 2% increase in labor (L) with capital (K) unchanged, we need to use the formula for the percentage change in output:
Percentage Change in Output = ((New Output - Original Output) / Original Output) x 100%
Given that the production function is Q=80L^0.5K^0.6, let's denote the original output as Q1 and the new output as Q2. We can calculate the percentage change in output as follows:
Q1 = 80L^0.5K^0.6
When L is increased by 2%, the new labor value (L2) would be:
L2 = L + (2% of L) = L + 0.02L = 1.02L
Substituting L2 and K into the production function, we get the new output (Q2):
Q2 = 80(1.02L)^0.5K^0.6
Now, we can calculate the percentage change in output:
Percentage Change in Output = ((Q2 - Q1) / Q1) x 100%
= ((80(1.02L)^0.5K^0.6 - 80L^0.5K^0.6) / 80L^0.5K^0.6) x 100%
b. To describe the nature of returns to scale for this production function, we need to analyze the exponents of labor (L) and capital (K) in the production function. In this case, the exponents are 0.5 and 0.6 respectively.
When the exponents sum up to 1 (0.5 + 0.6 = 1), it indicates constant returns to scale. Constant returns to scale imply that if all inputs are increased by the same proportion, output increases proportionally.
If the exponents sum to less than 1, such as in this case (0.5 + 0.6 < 1), it indicates decreasing returns to scale. Decreasing returns to scale imply that if all inputs are increased by the same proportion, the increase in output is less than proportionate.
Therefore, for the given production function, the nature of returns to scale is decreasing.