Question

Suppose that a country can produce two goods: food and clothing. To produce one unit of food, it requires one worker. To produce one unit of clothing, it requires two workers. The total amount of workers available in the economy is fixed and equal to 100 . Denoting with \(L\) the total amount of workers, with \(F\) the units of food produced and with \(C\) the units of clothing produced, the resource constraint for this economy can be written as: \(L=a_{F} F+a_{C} \mathrm{C}\), where \(a_{F}\) is the amount of workers needed to produce one unit of food and \(a_{C}\) the amount of workers needed to produce one unit of clothing. Show how to construct the production possibility frontier from that resource constraint. In a graph with \(C\) on the vertical axis and \(F\) on the horizontal axis, plot the PPF of this economy. What is the slope of the PPF?

Short answer

The slope of the PPF is -0.5.

Step-by-step solution

Set Resource Constraint

The total number of workers is given by the equation \(L = a_F F + a_C C\). For this problem, we know that \(a_F = 1\) and \(a_C = 2\). Therefore, the constraint becomes \(100 = 1 \cdot F + 2 \cdot C\).

Solve for Clothing Units (C)

To express clothing in terms of food, we rearrange the constraint equation: \(C = \frac{100 - F}{2}\). This equation shows how much clothing can be produced given the level of food production.

Determine Intercepts of PPF

Find the intercepts of the PPF by setting \(F = 0\) and \(C = 0\):- For \(F = 0\), \(C = \frac{100}{2} = 50\).- For \(C = 0\), \(F = 100\).

Draw the Production Possibility Frontier (PPF)

Plot the points on a graph, with \(C\) on the vertical axis and \(F\) on the horizontal axis. The two intercepts are \((0, 50)\) and \((100, 0)\). Connect these points with a straight line to form the PPF.

Calculate the Slope of the PPF

The slope of the PPF can be calculated using the formula \(-\frac{\Delta C}{\Delta F}\), which is the change in clothing per unit change in food. Since the intercept change is from \((0, 50)\) to \((100, 0)\), the slope is \(-\frac{50 - 0}{0 - 100} = -0.5\).

Key concepts

Resource Constraint

In economics, a resource constraint refers to the limited availability of resources which restricts the quantity of goods that can be produced. This concept is a cornerstone in understanding production possibilities. Imagine a country with a fixed number of workers, say 100, and the ability to produce two products: food and clothing. Each unit of food requires one worker to produce, while each unit of clothing requires two workers.

Given this setup, the resource constraint equation is:

• For food production: 1 worker per unit = \(a_F F\).
• For clothing production: 2 workers per unit = \(a_C C\).

Thus, the total resource constraint becomes:\[L = a_{F} F + a_{C} C\] Plug in the values:\[100 = 1 imes F + 2 imes C\]
This equation represents all possible combinations of food and clothing that can be produced utilizing the entire workforce of 100 workers.

Opportunity Cost

Opportunity cost is the loss of potential gain from other alternatives when one option is chosen. In the scenario of our country producing food and clothing, if more resources (workers) are dedicated to producing food, then fewer resources are available for clothing production and vice versa.

When you allocate resources to produce one more unit of food, you incur an opportunity cost in terms of clothing forgone. The opportunity cost for producing food can be found by the slope of the production possibility frontier (PPF). In this case, the opportunity cost of producing one additional unit of food is represented by how much clothing production must be reduced.

The calculated slope of the PPF, \(-0.5\), indicates that for every additional unit of food produced, half a unit of clothing is foregone. This trade-off quantifies the opportunity cost for decision-making in resource allocation.

Graphical Representation

Graphical representation is a technique used to better visualize relationships and constraints in economics. In terms of the production possibility frontier (PPF), this is particularly valuable. The PPF graphically displays all possible combinations of two goods that an economy can produce within its given resource limits.

Consider our example graph where clothing units (\(C\)) are on the vertical axis and food units (\(F\)) on the horizontal axis. With the calculated intercepts at \((0, 50)\) and \((100, 0)\), these points demonstrate the maximum output of one good when no units of the other are produced. Drawing a straight line between these intercepts gives the PPF.

• Intercept \((0, 50)\) signifies producing all clothing, using all 100 workers.
• Intercept \((100, 0)\) represents producing all food, utilizing all 100 workers.

The PPF illustrates feasible production levels and helps in identifying the efficient use of resources.

Economic Model

An economic model is a simplified version of reality that economists use to understand and predict economic behavior. The production possibility frontier (PPF) is a classic example of such a model. It helps visualize and understand complex concepts like resource constraints and opportunity costs.

Our PPF model, with food and clothing, showcases critical economic principles. It simplifies the real-world conditional factors into a two-product framework to study more dynamically. With this model, economists and students can better comprehend how changes in resources or technology distinctly affect production abilities and economic outcomes.

• Illustrates resource constraints
• Demonstrates opportunity costs
• Visualizes trade-offs in production

This economic model is an essential tool for enabling clearer insights into decision-making and policy formulation regarding resource allocation within an economy.