Let the production function be
𝑌
=
𝐾
𝛿
𝐿
1
−
𝛿
and
𝛿
=
0.4
. Moreover,
𝐾
=
1200
,
𝐿
=
245
. Then the ratio
MPL
APL
=
______.

(Recall,
M
P
L
=
∂
𝑌
∂
𝐿
and
A
P
L
=
𝑌
𝐿
.)

Round your answer to the nearest hundredth.

Question

Let the production function be 
𝑌
=
𝐾
𝛿
𝐿
1
−
𝛿
 and 
𝛿
=
0.4
. Moreover, 
𝐾
=
1200
,
𝐿
=
245
. Then the ratio 
MPL
APL
=
______.

(Recall, 
M
P
L
=
∂
𝑌
∂
𝐿
 and 
A
P
L
=
𝑌
𝐿
.)

Round your answer to the nearest hundredth.

BlackTom AI · Accepted Answer

Cobb–Douglas form Y = K^δ L^{1−δ} implies a simple relation between marginal and average products of labor.

1. Relevant concepts
- Marginal product of labor (MPL) is the partial derivative ∂Y/∂L.
- Average product of labor (APL) is Y/L.
- For a Cobb–Douglas specification Y = K^δ L^{1−δ}, exponents determine how MPL and APL relate.

2. Step-by-step reasoning
- Compute APL: APL = Y/L = (K^δ L^{1−δ})/L = K^δ L^{−δ}.
- Compute MPL by differentiating Y with respect to L:
  MPL = ∂/∂L (K^δ L^{1−δ}) = K^δ * ∂/∂L (L^{1−δ}) = K^δ * (1−δ) L^{−δ} = (1−δ) K^δ L^{−δ}.
- Form the ratio:
  MPL/APL = [(1−δ) K^δ L^{−δ}] / [K^δ L^{−δ}] = 1 − δ.

3. Intermediate algebraic steps shown
- APL = K^δ L^{−δ}.
- MPL = (1−δ) K^δ L^{−δ}.
- Division cancels K^δ L^{−δ}, leaving MPL/APL = 1 − δ.

4. Why the correct answer is correct
- The cancellation shows the ratio depends only on the exponent δ, not on the levels K or L. With δ = 0.4, MPL/APL = 1 − 0.4 = 0.6.

5. Why other numeric answers would be incorrect
- Any value other than 0.6 would neglect the algebraic cancellation or miscompute the derivative. Using the provided K and L explicitly (e.g., computing K^{0.4} and L^{−0.4}) and then dividing yields the same 0.6 because those factors cancel; hence level-based alternatives are wrong.

6. Conclusion
- The ratio MPL/APL = 1 − δ = 0.6. Rounded to the nearest hundredth: 0.60, which matches the provided answer.

Let the production function be 𝑌 = 𝐾 𝛿 𝐿 1 − 𝛿 and 𝛿 = 0.4 . Moreover, 𝐾 = 1200 , 𝐿 = 245 . Then the ratio MPL APL = ______. (Recall, M P L = ∂ 𝑌 ∂ 𝐿 and A P L = 𝑌 𝐿 .) Round your answer to the nearest hundredth.数值题

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