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} \\] here \(l_{x}\) and \(l_{y}\) are the quantities of labor devoted to \(x\) and \(y\) production, respectively. Total labor available in region \(A\) is 100 units; that is, \\[ l_{x}+l_{y}=100 \\] 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 \(B:\) \\[ l_{x}+l_{y}=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 efficicntly 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: \(\mathrm{A}\) graphical analysis may be of some help here.

Short answer

The combined production possibilities frontier for Ruritania is maximized at goods combination totals of \( x = 15 \) and \( y = 15 \). If \( x \) total is 12, solve for split productions and calculate final \( y \).

Step-by-step solution

Set up Labour Constraints

For both regions, the labor constraint equation is given by \( l_x + l_y = 100 \). This equation means that the sum of labor allocated to goods \( x \) and \( y \) is 100 units for both regions \( A \) and \( B \).

Production Possibility Frontier for Region A

Substitute \( l_x = 100 - l_y \) into the production function for \( x_A \).- \( x_A = \sqrt{l_x} = \sqrt{100 - l_y} \) - \( y_A = \sqrt{l_y} \)The production possibilities curve for region \( A \) is expressed by relating \( x_A \) and \( y_A \):\[ x_A = \sqrt{100 - y_A^2}\]

Production Possibility Frontier for Region B

Substitute \( l_x = 100 - l_y \) into the production function for \( x_B \):- \( x_B = \frac{1}{2} \sqrt{l_x} = \frac{1}{2} \sqrt{100 - l_y} \) - \( y_B = \frac{1}{2} \sqrt{l_y} \)The production possibilities curve for region \( B \) is:\[x_B = \frac{1}{2} \sqrt{100 - 4y_B^2}\]

Efficiency Condition Between Regions A and B

For efficient allocation, the marginal rate of transformation (MRT) must be equalized between the two regions. Since the MRT is the slope of the production possibility frontier, it implies the rates at which \( x \) can be converted to \( y \) must be equal between regions. Given the different production functions, equilibrium will depend on specific numerical values.

Combined Production Possibility Curve for Ruritania

To find the combined PPF, sum the individual equations of \( x \) outputs given the \( y \) constrained to a fixed value, and vice versa. Consider maximum productions:- Maximum \( x_A = 10 \) and \( x_B = 5 \) (when \( l_y = 0 \)).- Maximum \( y_A = 10 \) and \( y_B = 5 \) (when \( l_x = 0 \)).Total maximum productions:\[ x = x_A + x_B = 10 + 5 = 15 \ y = y_A + y_B = 10 + 5 = 15\]If total \( x = 12 \):- \( x_A \), \( x_B \) split; solve for \( y_A \), \( y_B \) accordingly such that \( x_A + x_B = 12 \). Calculate respective \( y \) values and sum to find total \( y \).

Key concepts

Labor Allocation

In the context of economics, labor allocation refers to the distribution of a fixed amount of labor resources among different production activities. In Ruritania, both regions, A and B, have 100 units of labor each. These units can be devoted to the production of two goods, x and y. However, the way in which labor is allocated affects the output levels of those goods.

Understanding labor allocation involves solving labor equations such as \( l_x + l_y = 100 \) for both regions. This equation indicates that the sum of the labor units distributed toward x and y must equal the total labor available. How labor is allocated between the production of x and y determines the quantities that can be produced.

• In Region A: More labor can produce a higher output of either x or y, following its production function \( x_A = \sqrt{l_x} \) and \( y_A = \sqrt{l_y} \).
• In Region B: The same logic applies, but with different coefficients in their production functions \( x_B = \frac{1}{2} \sqrt{l_x} \) and \( y_B = \frac{1}{2} \sqrt{l_y} \).

Decisions about how labor is allocated is critical as it directly impacts the production output and efficiency of a region.

Marginal Rate of Transformation

The Marginal Rate of Transformation (MRT) is an important concept representing the rate at which one good must be sacrificed to produce an additional unit of another good. It is essentially the slope of the Production Possibility Frontier (PPF).

For regions A and B in Ruritania, achieving efficient production involves equalizing the MRTs between the two regions. This means that the opportunity cost of switching labor between the production of goods x and y should be the same for both regions. When the MRTs are equal, it indicates that resources are being used optimally across the regions.

• In Region A: The MRT between x and y is derived from their respective PPF equation \( x_A = \sqrt{100-y_A^2} \).
• In Region B: The MRT takes into account the halved coefficients in their production formulas \( x_B = \frac{1}{2} \sqrt{100-4y_B^2} \).

By understanding MRT, regions can make informed decisions about reallocating labor to maintain production balance and economic efficiency.

Production Functions

Production functions describe the relationship between input resources and output goods. In Ruritania, both regions have distinct production functions for goods x and y, which define how labor inputs are transformed into outputs.

The production functions for both regions are given as follows:

• For Region A:
  - x is produced with the function \( x_A = \sqrt{l_x} \), meaning output depends on the square root of the labor allocated to x.
  - y is produced with the function \( y_A = \sqrt{l_y} \), indicating a similar relationship for y.


• For Region B:
  - x is produced as \( x_B = \frac{1}{2} \sqrt{l_x} \). The output is half of what Region A produces for the same labor input.
  - y is produced as \( y_B = \frac{1}{2} \sqrt{l_y} \), demonstrating a similar pattern to x but for y.

By knowing the production functions, we can model and predict how changes in labor allocation affect the output, helping in planning and resource management.

Economic Efficiency

Economic efficiency is achieved when resources are used in a way that maximizes the production of goods and services. In Ruritania, ensuring economic efficiency involves making sure that labor is allocated in such a way that total outputs of both goods, x, and y, are optimized.

In the context of immobile labor between regions, Ruritania must ensure efficiency within each region individually. Here, maximizing the output from the given production functions while also maintaining equal MRT across regions signals efficient use of resources.

• Economically efficient allocation means that it is not possible to increase the production of one good without decreasing the output of another.
• This principle ensures that Ruritania produces on its combined PPF, which represents the maximum possible output combinations of goods x and y given its constraints.

Striving for economic efficiency ensures that Ruritania can maintain high productivity and maximize utility from its limited labor resources.