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 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: \(\mathrm{A}\) graphical analysis may be of some help here.

Short answer

The total amount of good y that Ruritania can produce if it's producing 12 units of good x is 9 units.

Step-by-step solution

Production Possibility Curve for Region A

To find the production possibility curve for region A, we'll use the given production functions and the constraint on labor availability. \[ x_A = \sqrt{l_x} \] \[ y_A = \sqrt{l_y} \] Given that the total labor available in region A is 100 units, we have: \[ l_x + l_y = 100 \] Solve the labor constraint for one of the variables, for example, \(l_y = 100 - l_x\). Then substitute this into the \(y_A\) equation: \[y_A = \sqrt{100 - l_x}\] The production possibility curve for region A is given by: \[ y_A = \sqrt{100 - l_x} \]

Production Possibility Curve for Region B

Similarly, to find the production possibility curve for region B, we'll use the given production functions and the constraint on labor availability. \[ x_B = \frac{1}{2}\sqrt{l_x} \] \[ y_B = \frac{1}{2}\sqrt{l_y} \] Given that the total labor available in region B is 100 units, we have: \[ l_x + l_y = 100 \] Solve the labor constraint for one of the variables, for example, \(l_y = 100 - l_x\). Then substitute this into the \(y_B\) equation: \[ y_B = \frac{1}{2}\sqrt{100 - l_x} \] The production possibility curve for region B is given by: \[ y_B = \frac{1}{2}\sqrt{100 - l_x} \]

The Condition for Efficient Production Allocation

For efficient production allocation between region A and B, given that labor is immobile, the marginal rate of transformation must be equal in both regions. The Marginal Rate of Transformation (MRT) for a good can be defined as the rate at which a good can be transformed into another good with the available resources. It can be expressed as the negative relationship between the production functions of the two goods. For region A, MRT_A = \(- \frac{dy_A}{dx_A} = - \frac{0.5}{\sqrt{100 - l_x}}\) For region B, MRT_B = \(- \frac{dy_B}{dx_B} = - \frac{1}{\sqrt{100 - l_x}}\) For efficient production allocation, MRT_A = MRT_B. \[ - \frac{0.5}{\sqrt{100 - l_x}} = - \frac{1}{\sqrt{100 - l_x}} \]

Production Possibility Curve for Ruritania

To find the production possibility curve for Ruritania, we need to sum the production functions of region A and region B for each good. Let \(x_T = x_A + x_B\) and \(y_T = y_A + y_B\). Note that when \(x_T = 12\), we need to determine \(y_T\). Though graphical analysis can be a quick method, we can also find the result by solving for \(l_{x_A}\) and \(l_{x_B}\). \(x_A = \sqrt{l_{x_A}} = 12 - \frac{1}{2}\sqrt{l_{x_B}}\) (since \(x_T = x_A + x_B\)) Squaring both sides: \(l_{x_A} = 144 - 12\sqrt{l_{x_B}} + \frac{1}{4}l_{x_B}\) Now we have two labor constraints: \(l_{x_A} + l_{y_A} = 100\) \(l_{x_B} + l_{y_B} = 100\) Substituting \(l_{x_A}\) from the equation above and the labor constraints: \(144 - 12\sqrt{l_{x_B}} + \frac{1}{4}l_{x_B} + l_{y_A} = 100\) We now have two equations and two unknowns (\(l_{x_B}\) and \(l_{y_A}\)): \(l_{y_A} = 100 - l_{x_A}\) \(l_{y_A} = 56 + 12\sqrt{l_{x_B}} - \frac{1}{4}l_{x_B}\) Solving these two equations, we get: \(l_{x_B} = 64\) \(l_{y_A} = 36\) Now, we can find the total \(y\) production for Ruritania, \(y_T\): \(y_T = y_A + y_B = \sqrt{100 - l_{x_A}} + \frac{1}{2}\sqrt{100 - l_{x_B}}\) Substitute the values of \(l_{x_A}\) and \(l_{x_B}\): \(y_T = \sqrt{36} + \frac{1}{2}\sqrt{36} = 6 + 3 = 9\) So Ruritania can produce a total of 9 units of \(y\) if total \(x\) output is 12, assuming labor is immobile between regions.

Key concepts

Marginal Rate of Transformation (MRT)

The Marginal Rate of Transformation (MRT) is a fundamental concept in microeconomics, representing the rate at which one good can be substituted for another in production while keeping total output constant. It reflects the trade-offs and costs of opportunity intrinsic to resource allocation. Understanding MRT is crucial for analyzing how economies adjust outputs of various goods based on available resources and technology.

In the context of the exercise from Ruritania, determining the MRT for both regions A and B allows us to understand how labor resources are allocated between producing good x and good y. Intuitively, it’s about how much of good y you must give up to produce an additional unit of good x, holding total labor constant. It's a measure of the slope of the production possibility curve at any given point, which in a sense, maps the 'price' of transforming one good into another within a specific region.

Efficient Production Allocation

Efficient production allocation is reached when resources are distributed in a way that it’s not possible to produce more of one good without producing less of another, also known as the point of productive efficiency. At this point, the economy operates on its production possibility curve, and the marginal rates of transformation across regions or segments equalize.

For Ruritania's two regions, achieving efficient production allocation means equating MRTs in both regions. This ensures that the cost of opportunity for producing goods x and y is the same, preventing resources from being misallocated. When resources (like labor in our problem) are limited, efficient allocation is key for maximizing the overall production and thus, welfare within the economy.

Labor Constraint

In the context of microeconomic theory, a labor constraint refers to the finite amount of labor resources available for producing goods and services. This constraint limits the maximum possible outputs, shaping the production possibility curve for an economy or region.

In the Ruritania example, both regions A and B have a labor constraint of 100 units each, which confines the production capacities for goods x and y. As labor cannot be transferred between regions, it's essential for Ruritania to allocate this scarce resource efficiently to maximize output. By understanding and applying the labor constraint equation, producers and planners can identify the feasible combinations of Good x and y each region can produce, significantly impacting their overall economic planning and strategy.

Microeconomic Theory

Microeconomic theory is the branch of economics that focuses on the behaviors of individuals and firms in making decisions regarding the allocation of limited resources. It encompasses various principles, like demand and supply, elasticity, marginal utility, and the production possibility frontier, which are essential in determining how economies function at the fundamental level.

The exercise about Ruritania incorporates principles of microeconomic theory by evaluating how regions with specific constraints allocate labor to produce different goods efficiently. The production possibility curves, MRT, and labor constraints are all rooted in microeconomic theory, helping us to understand the basic choices that underpin the broader economic system. By grasping these principles, students can better understand how optimal decisions are made in the face of scarcity, which is the essential problem that economics seeks to address.