The production function
𝑌
=
𝐾
1
−
𝛿
𝐿
𝛿
+
𝐴
exhibits ________ returns to scale in K and L. The parameter A is a positive scalar.

Question

The production function 
𝑌
=
𝐾
1
−
𝛿
𝐿
𝛿
+
𝐴
 exhibits ________ returns to scale in K and L. The parameter A is a positive scalar.

BlackTom AI · Accepted Answer

Returns to scale are determined by how output changes when all inputs are scaled by the same factor t. A function is homogeneous of degree r if Y(tK,tL) = t^r Y(K,L). If r = 1 it is constant returns, r < 1 decreasing returns, r > 1 increasing returns.

1) Relevant knowledge points and concepts
- Cobb–Douglas term K^{1−δ} L^{δ} has exponents summing to (1 − δ) + δ = 1, so by itself it exhibits constant returns to scale.
- Adding a fixed positive scalar A breaks homogeneity: the +A is independent of K and L and does not scale with t.
- Use the definition: compare Y(tK,tL) to t Y(K,L) for t > 1 to determine returns.

2) Complete reasoning step by step
Start with Y(K,L) = K^{1−δ} L^{δ} + A. Scale inputs by t > 0:
Y(tK,tL) = (tK)^{1−δ} (tL)^{δ} + A.

3) Calculations with intermediate steps
(tK)^{1−δ} (tL)^{δ} = t^{(1−δ)+δ} K^{1−δ} L^{δ} = t K^{1−δ} L^{δ}.
So
Y(tK,tL) = t K^{1−δ} L^{δ} + A.
Compute t Y(K,L) = t(K^{1−δ} L^{δ} + A) = t K^{1−δ} L^{δ} + tA.
Compare:
t Y(K,L) − Y(tK,tL) = (t K^{1−δ} L^{δ} + tA) − (t K^{1−δ} L^{δ} + A) = tA − A = A(t − 1).

4) Why the correct answer is correct
For any t > 1 and A > 0, A(t − 1) > 0, so t Y(K,L) > Y(tK,tL). That means scaling inputs by t yields less than proportional scaling of output: decreasing returns to scale.

5) Why other options are incorrect
- "constant": would hold only if A = 0 (pure Cobb–Douglas). Here A > 0 so homogeneity is lost.
- "increasing": would require Y(tK,tL) > t Y(K,L) for t > 1, which contradicts the inequality above.
- "None of these answers is correct.": not true because the function clearly shows decreasing returns.

6) Conclusion
Therefore the production function exhibits decreasing returns to scale in K and L. This matches the provided answer: decreasing.

The production function 𝑌 = 𝐾 1 − 𝛿 𝐿 𝛿 + 𝐴 exhibits ________ returns to scale in K and L. The parameter A is a positive scalar.Single choice

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