(Solution) - In the country of Ruritania there are two regions A -(2025 Original AI-Free Solution)

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In the country of Ruritania there are two regions, A and B. Two goods (X and Y) are produced in both regions. Production functions for region A are given by
XA = ?LX
YA = ?LY
LX and LY are the quantity of labor devoted to X and Y production, respectively. Total labor available in region A is 100 units. That is,
LX + LY = 100
Using a similar notation for region B, production functions are given by
XB = 1/2 ?LX
YB = 1/2 ?LY
There are also 100 units of labor available in region B:
LX + LY = 100
a. Calculate the production possibility curves for regions A and B.
b. What condition must hold if production in Ruritania is to be allocated efficiently between regions A and B (assuming that labor cannot move from one region to the other)?
c. Calculate the production possibility curve for Ruritania (again assuming that labor is immobile between regions). How much total Y can Ruritania produce if total X output is 12?