Which of the following production functions exhibits constant returns to scale with respect to K and L?

Select all correct answers.

Question

Which of the following production functions exhibits constant returns to scale with respect to K and L?

Select all correct answers.

BlackTom AI · Accepted Answer

Constant returns to scale (CRS) means that for any t > 0, Y(tK, tL) = t · Y(K, L). Equivalently, the production function is homogeneous of degree 1 in (K, L). I will test each candidate by replacing K → tK and L → tL and checking the factor that pulls out.

1) Y = K^α L^{−α}
- Scale: Y(tK,tL) = (tK)^α (tL)^{−α} = t^{α} t^{−α} K^α L^{−α} = t^0 Y(K,L) = Y(K,L).
- Degree = 0, not degree 1 → not CRS.

2) Y = A + K^{3/4} L^{1/4}
- Scale: Y(tK,tL) = A + (tK)^{3/4}(tL)^{1/4} = A + t^{3/4+1/4} K^{3/4} L^{1/4} = A + t · K^{3/4} L^{1/4}.
- Because of the additive constant A, Y(tK,tL) ≠ t·Y(K,L) in general (unless A = 0). So not CRS.

3) Y = K^{−1.1} L^{2.1}
- Scale: Y(tK,tL) = (tK)^{−1.1} (tL)^{2.1} = t^{−1.1+2.1} K^{−1.1} L^{2.1} = t^{1.0} Y(K,L).
- Mathematically this is homogeneous of degree 1 (scales by t). However, in standard economic interpretation production exponents are assumed nonnegative so marginal products are nonnegative; the negative exponent on K implies output falls with more K (economically implausible). If the question restricts to economically sensible (nonnegative-exponent) CRS functions, this would be excluded. Under that interpretation it is not an admissible CRS production function.

4) Y = K^{−0.1} + L^{−0.9}
- Scale: Y(tK,tL) = t^{−0.1}K^{−0.1} + t^{−0.9}L^{−0.9} which is not equal to t·Y(K,L) (no common factor t). Not CRS.

5) Y = K + 7L
- Scale: Y(tK,tL) = tK + 7 tL = t(K + 7L) = t·Y(K,L).
- This is homogeneous of degree 1 and satisfies CRS (and is economically well behaved: linear, nonnegative marginal products).

6) Y = K^{3/4} L^{−1/4}
- Scale: Y(tK,tL) = t^{3/4−1/4} K^{3/4} L^{−1/4} = t^{1/2} Y(K,L), degree 1/2 → not CRS.

Why the correct answer is correct: K + 7L explicitly multiplies by t when both inputs are scaled by t, satisfying the CRS definition exactly. Other options fail because their homogeneity degree is not 1, include an additive constant, or (in the case with negative exponents) violate standard economic assumptions about inputs. Therefore the only admissible CRS among the choices is Y = K + 7L, which matches the provided answer.

Which of the following production functions exhibits constant returns to scale with respect to K and L? Select all correct answers.多项选择题

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