Question

In the country of Ruritania there are two regions, \(A\) and \(B\). Two goods $(X \text { and } Y)$ are produced in both regions. Production functions for region \(A\) are given by $$\begin{array}{l}X_{A}=\sqrt{L_{X}} \\ Y_{A}=\sqrt{L_{Y}}\end{array}.$$ \(L_{X}\) and \(L_{Y}\) are the quantity of labor devoted to \(X\) and \(Y\) production, respectively. Total labor available in region \(A\) is 100 units. That is, $$\boldsymbol{L}_{X}+\boldsymbol{L}_{Y}=\mathbf{1 0 0}$$ Using a similar notation for region \(B\), production functions are given by $$\begin{array}{l} X_{B}=\frac{1}{2} \sqrt{L_{X}} \\\Y_{B}=\frac{1}{2} \sqrt{L_{Y}}\end{array}$$ There are also 100 units of labor available in region \(B:\) $$\boldsymbol{L}_{x}+\boldsymbol{L}_{Y}=100$$ a. \(\quad\) 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 labor cannot move from one region to the other)? c. Calculate the production possibility curve for Ruritania (again assuming labor is immobile between regions). How much total \(Y\) can Ruritania produce if total \(X\) output is \(12 ?\) Hint: A graphical analysis may be of some help here.

Short answer

Answer: The total output of good Y in Ruritania will be 9 units when the total output of good X is 12.

Step-by-step solution

Production possibility curve for Region A

To find the production possibility curve for region A, we need to eliminate either LX or LY from the production functions using the labor constraint. The labor constraint for region A is: LX + LY = 100 From this equation, we can write LY as: LY = 100 - LX Now, substitute LY in the production function of YA: YA = sqrt(100 - LX) This is the production possibility curve for region A, which shows the relationship between production of good X and good Y in region A based on their labor allocation.

Production possibility curve for Region B

Similarly, for region B, the labor constraint is: LX + LY = 100 LY can be written as: LY = 100 - LX Now, substitute LY in the production function of YB: YB = 0.5 * sqrt(100 - LX) This is the production possibility curve for region B, which shows the relationship between production of good X and good Y in region B based on their labor allocation.

Efficient allocation condition

For efficient allocation of production between regions A and B, the marginal rate of transformation (MRT) between goods X and Y must be equal in both regions. MRT is defined as the ratio of the marginal product of good X to the marginal product of good Y. Let's find the MRT for region A: MRT_A = (d(XA)/d(LX)) / (d(YA)/d(LY)) MRT_A = (0.5/sqrt(LX)) / (-0.5/sqrt(100-LX)) For region B, the MRT is: MRT_B = (d(XB)/d(LX)) / (d(YB)/d(LY)) MRT_B = (0.25/sqrt(LX)) / (-0.25/sqrt(100-LX)) The efficient allocation condition is: MRT_A = MRT_B

Production possibility curve for Ruritania

To find the production possibility curve for Ruritania, we need to add the production functions of goods X and Y for both regions A and B, considering the efficient allocation condition from the previous step. The production functions for Ruritania are: XR = XA + XB YR = YA + YB Substituting the efficient allocation condition and the production possibility curves for regions A and B: XR = sqrt(LX_A) + 0.5*sqrt(LX_B) YR = sqrt(100 - LX_A) + 0.5*sqrt(100 - LX_B)

Total output of good Y for a given output of good X

Now, let's find the total output of good Y when the total output of good X is 12. Given: XR = 12 From the production possibility curve for Ruritania, substituting XR: 12 = sqrt(LX_A) + 0.5*sqrt(LX_B) Using the labor constraint equations, we can express LX_B in terms of LX_A: LX_B = 100 - (100 - LX_A) LX_B = LX_A Now, substituting LX_B in the equation and solving for LX_A: 12 = sqrt(LX_A) + 0.5*sqrt(LX_A) 12 = 1.5*sqrt(LX_A) sqrt(LX_A) = 8 LX_A = 64 Substituting LX_A in the equation for LY_A: LY_A = 100 - 64 = 36 Substituting LX_A and LY_A in the production functions for region A: YA = sqrt(36) = 6 Now, substituting LX_A in the equation for LX_B: LX_B = LX_A = 64 Substituting LX_B in the equation for LY_B: LY_B = 100 - 64 = 36 Substituting LX_B and LY_B in the production functions for region B: YB = 0.5 * sqrt(36) = 3 Finally, calculating the total output of good Y in Ruritania: YR = YA + YB = 6 + 3 = 9 Therefore, Ruritania can produce 9 units of good Y when the total output of good X is 12.