Determine whether returns to scale is decreasing, increasing or constant:
Q = 2K + 3L + KL
Take a shot. Hint: increase both K and L by, say 10%. Does Q rise by more than 10% (increasing), less than 10% (decreasing), or exactly 10% (constant)
Increasing returns to scale
increasing
To determine whether returns to scale is decreasing, increasing, or constant in the given equation Q = 2K + 3L + KL, we can follow the hint provided and observe the impact of increasing both K and L by a certain percentage.
Let's assume we increase both K and L by 10%. This means we increase K and L by 0.1K and 0.1L, respectively.
After increasing K and L, the new equation becomes:
Q' = 2(K + 0.1K) + 3(L + 0.1L) + (K + 0.1K)(L + 0.1L)
Simplifying the equation:
Q' = 2.1K + 3.1L + 1.1KL + 0.1K^2 + 0.1KL + 0.1L^2
Now, let's compare the new output (Q') with the original output (Q) to determine the impact on Q:
(Q' - Q) / Q * 100
Substituting the values:
((2.1K + 3.1L + 1.1KL + 0.1K^2 + 0.1KL + 0.1L^2) - (2K + 3L + KL)) / (2K + 3L + KL) * 100
Simplifying further:
((0.1K + 0.1L + 0.1KL + 0.1K^2 + 0.1KL + 0.1L^2)) / (2K + 3L + KL) * 100
We can see that there are a few terms with the factor of 0.1 (representing the percentage increase) and some cross-product terms (KL), but let's focus on the main terms for simplicity:
(0.1K + 0.1L) / (2K + 3L + KL) * 100
From the simplified equation, we can determine whether the percentage change in Q is more than 10% (increasing), less than 10% (decreasing), or exactly 10% (constant).
If the numerator (0.1K + 0.1L) is greater than 10% of the denominator (2K + 3L + KL), then returns to scale are increasing. If the numerator is less than 10% of the denominator, returns to scale are decreasing. If the numerator is exactly 10% of the denominator, returns to scale are constant.
By calculating the values, you can conclude whether returns to scale are decreasing, increasing, or constant in this specific equation.