Firm X produces output $x$ and Firm Y produces outputyusing capital, $K$, and labor, $L$, as inputs. Firm X has marginal rate of technical substitution $M R T S ^ { x } { } _ { K } L = K _ { x } / L _ { x _ { 0 } }$, while firm $Y$ has marginal rate of technical substitution $M R T S _ { K , L } = 9 K _ { y } / L _ { y }$. Which of the following allocations satisfies input efficiency?
A) Firm X has 5 units of $K$ and 3 units of $L$, and Firm Y has 3 units of $K$ and 1 unit of $L$.
B) Firm X has 5 units of $K$ and 5 units of $L$, and Firm Y has 1 unit of $K$ and 9 units of $L$.
C) Firm X has 9 units of $K$ and 1 units of $L$, and Firm Y has 5 units of $K$ and 5 units of $L$.
D) Firm X has 5 units of $K$ and 5 units of $L$, and Firm Y has 1 unit of $K$ and 3 units of $L$.

Question

Quizplus · Accepted Answer

Input efficiency requires that the marginal rate of technical substitution for each input be equal across firms. In this case, we have:
- For Firm X: MRTSxK,L=Kx/Lx0
- For Firm Y: MRTSK,L=9Ky/Ly
We need to identify the allocation where the MRTS for capital and labor are equal for both firms. Let's check each option:
A) Firm X: MRTSxK,L=5/3 Firm Y: MRTSK,L=27/1 - not equal
B) Firm X: MRTSxK,L=1 Firm Y: MRTSK,L=9/5 - equal
C) Firm X: MRTSxK,L=9 Firm Y: MRTSK,L=25/5 - not equal
D) Firm X: MRTSxK,L=1 Firm Y: MRTSK,L=3/1 - not equal

Therefore, only option B satisfies input efficiency. Firm X uses capital and labor in equal proportion, while Firm Y uses more labor than capital.

Firm X Produces Output xxx And Firm Y Produces Outputyusing Capital KKK

Firm X Produces Output $x$ And Firm Y Produces Outputyusing Capital $K$