Does the production function 𝑓(𝑥1, 𝑥2) = 𝑚𝑖𝑛{𝑥1, 3𝑥2} exhibit IRS, CRS, or DRS?
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).
To determine the type of returns to scale exhibited by a production function, we need to examine how the output changes when we proportionally increase the inputs.
Let's consider the production function 𝑓(𝑥1, 𝑥2) = 𝑚𝑖𝑛{𝑥1, 3𝑥2}.
To test for returns to scale, we will compare the output when the inputs are doubled, tripled, or in general, increased by a constant factor.
1. Doubling the inputs:
𝑓(2𝑥1, 2𝑥2) = 𝑚𝑖𝑛{2𝑥1, 3(2𝑥2)} = 𝑚𝑖𝑛{2𝑥1, 6𝑥2}
2. Tripling the inputs:
𝑓(3𝑥1, 3𝑥2) = 𝑚𝑖𝑛{3𝑥1, 3(3𝑥2)} = 𝑚𝑖𝑛{3𝑥1, 9𝑥2}
From the above calculations, we can see that if we double or triple the inputs, the output is not proportional to the increase in inputs. Therefore, the production function 𝑓(𝑥1, 𝑥2) = 𝑚𝑖𝑛{𝑥1, 3𝑥2} exhibits diminishing returns to scale (DRS).
Note: In general, a production function exhibits:
- Increasing returns to scale (IRS) if doubling inputs results in more than doubling the output.
- Constant returns to scale (CRS) if doubling inputs results in exactly doubling the output.
- Diminishing returns to scale (DRS) if doubling inputs results in less than doubling the output.
To determine whether the production function 𝑓(𝑥1, 𝑥2) = 𝑚𝑖𝑛{𝑥1, 3𝑥2} exhibits increasing returns to scale (IRS), constant returns to scale (CRS), or decreasing returns to scale (DRS), we need to analyze how the output changes with respect to changes in inputs.
First, let's define the terms:
- IRS (Increasing Returns to Scale): If doubling all inputs results in more than a doubling of output, the production function exhibits IRS.
- CRS (Constant Returns to Scale): If doubling all inputs results in exactly a doubling of output, the production function exhibits CRS.
- DRS (Decreasing Returns to Scale): If doubling all inputs results in less than a doubling of output, the production function exhibits DRS.
In this case, the production function is 𝑓(𝑥1, 𝑥2) = 𝑚𝑖𝑛{𝑥1, 3𝑥2}. To analyze the returns to scale, we need to consider how the output changes when the inputs are multiplied by a constant factor.
Let's assume that we double the inputs and examine the resulting output:
𝑓(2𝑥1, 2𝑥2) = 𝑚𝑖𝑛{2𝑥1, 3(2𝑥2)} = 𝑚𝑖𝑛{2𝑥1, 6𝑥2}
Comparing this with the original production function, we can see that doubling the inputs result in more than a doubling of the second input (3𝑥2 to 6𝑥2). Therefore, the output would be determined by the minimum of the first input (𝑥1) and the doubled second input (6𝑥2).
In other words, doubling the inputs does not result in more than a doubling of output. Instead, the output is only determined by the minimum of 𝑥1 and 6𝑥2.
Therefore, the production function 𝑓(𝑥1, 𝑥2) = 𝑚𝑖𝑛{𝑥1, 3𝑥2} exhibits constant returns to scale (CRS).