Question

Consider an economy with just one technique available for the production of each good. $$\begin{array}{lcc} \hline \text { Good } & \text { Food } & \text { Cloth } \\ \hline \text { Labor per unit output } & 1 & 1 \\ \text { Land per unit output } & 2 & 1 \\ \hline \end{array}$$ a. Suppose land is unlimited but labor cquals 100 . Write and sketch the production possibility fronticr. b. Suppose labor is unlimited but land equals \(150 .\) Write and sketch the production possibility frontier. c. Suppose labor equals 100 and land equals \(150 .\) Write and sketch the production possibility frontier. Hint: What are the intercepts of the production possibility fronticr? When is land fully employed? Labor? Both? d. Explain why the production possibility frontier of part (c) is concave. e. Sketch the relative price of food as a function of its output in part (c). f. If consumers insist on trading 4 units of food for 5 units of cloth, what is the relative price of food? Why? g. Explain why production is exactly the same at a price ratio of \(p_{F} / p_{C}=1.1\) as at \(p_{F} / p_{C}=1.9\) h. Suppose that capital is also required for producing food and clothing and that capital requirements per unit of food and per unit of clothing are 0.8 and \(0.9,\) respectively. There are 100 units of capital available. What is the production possibility curve in this case? Answer part (e) for this case.

Short answer

In each scenario, the PPF illustrates the trade-off between food and cloth production based on resource constraints, with varying slopes indicating opportunity costs and resource allocation needs.

Step-by-step solution

Understanding the Scenario

In this problem, we are analyzing an economy with two goods: Food and Cloth. Both require labor and land as resources for production. The challenge is determining how these resources limit the production of each good.

Step 2a: Construct PPF with Unlimited Land, 100 Labor

Given 100 units of labor, with the labor required per unit output for both goods being 1, the maximum output is 100 units collectively. As land is unlimited, the PPF is a straight line from 100 units of food (if only food is produced) to 100 units of cloth (if only cloth is produced), with the equation: \( Q_F + Q_C = 100 \).

Step 2b: Construct PPF with Unlimited Labor, 150 Land

With 150 units of land and the land required per unit being 2 for food and 1 for cloth, the maximum production capacity is bounded by land. The frontier is a line from 75 units of food to 150 units of cloth, with the equation: \( 2Q_F + Q_C = 150 \).

Step 2c: Construct PPF with 100 Labor, 150 Land

This combines the constraints of both labor and land. We calculate max units by \([100,0]\) if only labor is used for cloth and \([0,75]\) if only land is used for food, with the PPF equation: \( rac{Q_F}{50} + rac{Q_C}{100} = 1 \).

Analyze PPF Concavity

The PPF is concave because as more resources shift to producing one good, the opportunity cost of producing additional units of that good rises. This phenomenon is due to diminishing returns, where reallocating resources yields progressively smaller increases in output.

Sketch Relative Price of Food vs Output (Part C)

For part (c), sketch a graph with output of food on the x-axis and relative price of food on the y-axis. The relative price is determined by the slope of the PPF and will adjust as differing amounts of goods are produced, reflecting changed opportunity costs.

Determining Relative Price of Food with Trade (Part F)

If consumers trade 4 units of food for 5 units of cloth, the price of food relative to cloth is given by the equation \( p_F / p_C = 5/4 = 1.25 \). This reflects consumer preferences, where 1 unit of food is equivalent in value to 1.25 units of cloth.

Explaining Same Production at Different Price Ratios

The production remains constant at price ratios \( p_F / p_C=1.1 \) and \( p_F / p_C=1.9 \) because the focus is on efficient resource allocation rather than prices. Production limits depend on labor and land constraints, not these particular price points.

PPF with Additional Capital Requirement (Part H)

Now, add capital requirements per unit of food (0.8) and cloth (0.9). With 100 capital, the equation would be \( 0.8Q_F + 0.9Q_C = 100 \).Combine with previous labor and land constraints to find new intercepts, the intersections determine feasibility on the PPF layout.

Key concepts

Resource Allocation

When we discuss resource allocation in the context of a Production Possibility Frontier (PPF), we are referring to how limited resources such as labor and land are distributed between different goods to maximize production output. In the economy from our exercise, resources include labor, land, and capital, each with specific constraints. We have 100 units of labor, 150 units of land, and 100 units of capital, and different goods require different amounts of these resources. For example:

• Food requires more land per unit than cloth, which influences how resources are allocated depending on resource availability.
• The allocation is optimized to reflect the highest possible outputs of food and cloth without surpassing resource limits.

The PPF graphically represents these choices and shows the efficient combinations of goods that can be produced with these resource limits, highlighting trade-offs in production.

Opportunity Cost

Opportunity cost is a critical concept in economics and directly linked to the PPF. It represents the cost of foregone alternatives when a particular decision is made. In the case of our production scenario:

• If producers decide to increase the production of food, they have to utilize more labor and land for food, thereby reducing the available resources for cloth. This trade-off is the opportunity cost of producing more food.
• The slope of the PPF reflects the opportunity cost between the two goods, indicating how much cloth needs to be sacrificed to produce one more unit of food.

As we move along the curve, the opportunity cost often increases, signifying that shifting resources becomes progressively more costly in terms of the forgone production of the other good.

Diminishing Returns

Diminishing returns is a phenomenon that occurs when adding more of one input, while keeping others constant, leads to a decrease in the additional output created. In our case:

• As more labor and land units are allocated to produce more of one good, the output added by each additional unit decreases over time.
• This is because the most efficient combinations of inputs have already been utilized, and any subsequent resource allocation becomes less effective.

The concave shape of the PPF in part (c) of the exercise illustrates diminishing returns. As we increase the production of one good, the output increase becomes smaller and the opportunity cost rises.

Economic Models

Economic models, like the Production Possibility Frontier (PPF), are simplifications of reality used to understand complex economic processes. These models help in:

• Visualizing the relationship between different factors of production and outputs within an economy.
• Exploring efficiency, opportunity costs, and the effects of different resource allocations.

By assuming certain constraints and conditions, such as fixed labor, land, and capital in our exercise, the PPF provides insights on how economies can efficiently allocate resources. It shows potential maximum outputs under various scenarios, enabling predictions on how changes in resource availability could impact production and economic outcomes.