Using the same production function as in the previous question, you know that if K and L are both multiplied by the same positive scalar
𝜆
, then
𝑦
=
𝑌
𝐿
(output per person) ______.

Question

Using the same production function as in the previous question, you know that if K and L are both multiplied by the same positive scalar 
𝜆
, then 
𝑦
=
𝑌
𝐿
 (output per person) ______.

BlackTom AI · Accepted Answer

Constant returns to scale: Y(λK, λL) = λ Y(K, L). This property is the key concept.

1) Relevant knowledge points and concepts
- Production function homogeneity and returns to scale. If a production function exhibits constant returns to scale (CRS), scaling all inputs by λ scales output by the same λ: Y(λK, λL) = λY(K,L).
- Output per person (per worker) is y = Y/L. We must see how y changes when both K and L are scaled by λ.

2) Step-by-step reasoning
Apply CRS to the scaled inputs and compute the new output per person:
- New aggregate output after scaling inputs: Y'(K,L) = Y(λK, λL).
- By CRS, Y(λK, λL) = λ Y(K, L).
- New population (labor) is λL, so new output per person y' = Y(λK, λL) / (λL).

3) Calculations with intermediate steps
y' = Y(λK, λL) / (λL)
   = [λ Y(K, L)] / (λL)        (by CRS)
   = λ/λ · [Y(K, L)/L]
   = 1 · y
   = y.

4) Why the correct answer is correct
The algebra shows the λ factors cancel: scaling both capital and labor by the same proportion scales aggregate output by λ but also increases the number of workers by λ, leaving output per worker unchanged. Thus y remains constant.

5) Why the other options are incorrect
- "is multiplied by the same factor λ": would mean y' = λy, but the λ in output is divided by the λ in labor, so they cancel.
- "is multiplied by factor 1/λ": would imply y' = y/λ, which contradicts the CRS result.
- "changes, but it's impossible to tell by how much": incorrect because CRS gives a definite outcome — no change.

6) Conclusion
Therefore output per person remains constant, matching the provided correct answer.

Using the same production function as in the previous question, you know that if K and L are both multiplied by the same positive scalar 𝜆 , then 𝑦 = 𝑌 𝐿 (output per person) ______.Single choice

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