If your firm's production function has constant returns to scale, then if you double all your inputs, your firm's output will

Question

BlackTom AI · Accepted Answer

Constant returns to scale (CRS) is defined by the property F(tK, tL) = t·F(K, L) for any scalar t > 0. Therefore, doubling all inputs (t = 2) gives F(2K, 2L) = 2·F(K, L), so output doubles.

1) Relevant knowledge points and concepts
- Production function F(K, L): maps capital K and labor L into output.
- Returns to scale: CRS means proportional scaling of inputs yields the same proportional scaling of output.
- Productivity: typically measured as output per unit of input (e.g., output per worker F(K,L)/L or output per unit of capital).

2) Step-by-step reasoning
- Start from the CRS property: F(tK, tL) = tF(K, L).
- Set t = 2 to represent doubling each input: F(2K, 2L) = 2F(K, L).
- Thus, output after doubling inputs is exactly twice the original output.

3) Intermediate calculations (explicit)
- Original output = Y = F(K, L).
- New output after doubling inputs = Y' = F(2K, 2L) = 2·F(K, L) = 2Y.
- Productivity per worker originally = y = F(K, L)/L.
- New productivity per worker = y' = F(2K, 2L)/(2L) = (2F(K, L))/(2L) = F(K, L)/L = y.
- Similarly, output per unit of capital or average output per combined unit remains unchanged.

4) Why the correct answer is correct
Because CRS ensures proportional scaling of inputs yields an identical proportional scaling of output. Doubling inputs doubles output, and since both numerator (output) and denominator (inputs) double in any per-input productivity ratio, the ratio (productivity) remains the same.

5) Why other options are incorrect
- "Less than double" would describe decreasing returns to scale (F(2K,2L) < 2F(K,L)), not CRS.
- "More than double" would describe increasing returns to scale (F(2K,2L) > 2F(K,L)), not CRS.
- Any statement claiming productivity changes contradicts the algebra above showing per-input ratios are invariant under CRS.

6) Conclusion
Therefore, under constant returns to scale, doubling all inputs doubles output but leaves productivity unchanged. This confirms the correct answer: "double but productivity will not change."

If your firm's production function has constant returns to scale, then if you double all your inputs, your firm's output will单项选择题

类似问题

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