To determine whether the production function exhibits increasing returns to scale (IRS), constant returns to scale (CRS), or decreasing returns to scale (DRS), we need to examine how the function behaves when we change the scale of inputs.
The production function 𝑓(𝑥1, 𝑥2) = min{𝑥1, 3𝑥2} implies that the output level is determined by the minimum value between 𝑥1 and 3𝑥2. This means that the production function is limited by the quantity of the input that is available in the smallest amount.
If we uniformly increase both 𝑥1 and 𝑥2 by a factor of k, then the new inputs will be 𝑘𝑥1 and 𝑘𝑥2. Let's consider the two cases individually:
1. If 𝑘𝑥1 ≤ 3𝑘𝑥2, then the output will be 𝑘𝑥1.
2. If 𝑘𝑥1 > 3𝑘𝑥2, then the output will be 3𝑘𝑥2.
In both cases, the output increases by a factor of k when the inputs increase by a factor of k. Therefore, the production function exhibits constant returns to scale (CRS).
Does the production function 𝑓(𝑥1, 𝑥2) = 𝑚𝑖𝑛{𝑥1, 3𝑥2} exhibit IRS, CRS, or DRS?
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