Question

Consider an economy with just one technique available for the production of each good. \\[ \begin{array}{lcc} \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 equals 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 frontier? When is land fully employed? Labor? Both? 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

Answer: Introducing capital as a third factor of production changes the production possibility frontier by introducing additional constraints on the production of Food and Cloth. The PPF will still be concave, but the endpoints and shape will be determined by the availability of capital, along with labor and land. The relative price of Food as a function of its output will continue to have an increasing relationship, but the curve may change due to the capital requirements for producing each good.

Step-by-step solution

Understanding the given table

The table presents the labor and land requirements for producing one unit of each good. For example, producing one unit of Food requires one unit of labor and two units of land.

a) Production possibility frontier with unlimited land and limited labor

Since labor is limited to 100 units and land is unlimited, we can start by determining how much of each good can be produced using all available labor. If all labor is allocated to Food production, the economy can produce 100 units of Food (100 units of labor divided by 1 unit of labor per Food). Similarly, if all labor is allocated to Cloth production, the economy can produce 100 units of Cloth. This defines the endpoints of the line of the production possibility frontier. Sketch this line on a graph with Food on the horizontal axis and Cloth on the vertical axis.

b) Production possibility frontier with unlimited labor and limited land

Since land is limited to 150 units and labor is unlimited, we can start by determining how much of each good can be produced using all available land. If all land is allocated to Food production, the economy can produce 75 units of Food (150 units of land divided by 2 units of land per Food). Similarly, if all land is allocated to Cloth production, the economy can produce 150 units of Cloth. This defines the endpoints of the line of the production possibility frontier. Sketch this line on a graph with Food on the horizontal axis and Cloth on the vertical axis.

c) Production possibility frontier with limited labor and land

Given that both labor and land are limited (100 and 150 units, respectively), we need to consider both resources at the same time. The intercepts are the points where only one of the goods is produced. If all labor is allocated to Food production, the economy can produce 100 units of Food. If all land is allocated to Cloth production, the economy can produce 150 units of Cloth. Alternatively, if all land is allocated to Food production, the economy can produce 75 units of Food. And if all labor is allocated to Cloth production, the economy can produce 100 units of Cloth. The production possibility frontier in this case is the concave curve connecting these points. Sketch this curve on a graph with Food on the.horizontal axis and Cloth on the vertical axis.

d) Explaining the concavity of the production possibility frontier

The concavity of the production possibility frontier of part (c) is due to the increasing opportunity cost of producing one good as more of that good is produced. This is because the resources (land and labor) are not evenly divisible between the two goods. As more of one good is produced, both labor and land are allocated more unequally between the two goods, resulting in a lower marginal output of the good with less resource allocation.

e) Sketching the relative price of Food as a function of its output in part (c)

When sketching the relative price of Food as a function of its output, consider that the slope of the production possibility frontier represents the opportunity cost of producing Food in terms of Cloth. Since the PPF has a concave shape in part (c), the slope becomes steeper as more Food is produced, indicating that the relative price of Food increases as more Food is produced. Sketch a curve showing the relationship between the output of Food and its relative price, with Food output on the horizontal axis and the relative price on the vertical axis.

f) Relative price of Food when trading 4 units of Food for 5 units of Cloth

If consumers insist on trading 4 units of Food for 5 units of Cloth, they value one unit of Food at 1.25 units of Cloth. The relative price of Food is \(\frac{1.25}{1} = 1.25\). Consumers are expressing their preference for the trade ratio in the form of exchange rates, which directly reflects the relative price of the goods.

g) Production at price ratios of \(p_{F} / p_{c} = 1.1\) and \(p_{F} / p_{c} = 1.9\)

The production levels in both cases would be exactly the same because the concave shape of the PPF in part (c) causes the opportunity cost and relative prices to rise as production increases. For both ratios, producers would choose to produce on the PPF where the opportunity cost equals the relative price. Therefore, at both relative prices, producers would be indifferent between producing the same combination of Food and Cloth, leading to the same level of production for both goods.

h) Production possibility frontier with capital requirements and limited capital

Now, we have 100 units of capital available and capital requirements per unit of Food and Cloth are 0.8 and 0.9, respectively. We must determine the endpoints of the production possibility frontier by allocating all available capital to one of the two goods while still meeting labor and land constraints. Calculate the maximum amount of each good that can be produced given the capital, labor, and land constraints. This will give you the endpoints of the production possibility frontier, which will also be concave due to the unequal requirements of the two goods. Sketch this curve on a graph with Food on the horizontal axis and Cloth on the vertical axis. Finally, analyze the relative price of Food as a function of its output as calculated in step 6 and sketch the new curve representing this relationship.

Key concepts

Opportunity Cost

Opportunity cost is a fundamental concept in economics that describes the value of the next best alternative that is foregone when a choice is made. It's the cost associated with not choosing the next best option. For instance, in the context of the production possibility frontier (PPF), when an economy decides to produce more food, it must use resources that could have been used to produce cloth. As a result, the opportunity cost of producing more food is the amount of cloth that could have been produced with those same resources.

Understanding opportunity cost helps in making informed decisions about resource allocation. It is crucial for businesses and individuals to consider opportunity costs to optimize their outcomes. For students grappling with the idea, a simple way to think about it is by considering everyday decisions, like choosing to study an extra hour for an exam instead of watching a movie. The opportunity cost here is the enjoyment and relaxation that the movie would have provided.

Resource Allocation

Resource allocation involves the distribution of resources among competing uses to maximize efficiency and output. In our exercise, we're looking at how an economy allocates its limited resources—labor, land, and capital—to produce two goods, food and cloth. The PPF demonstrates this allocation visually, with each point on the frontier representing an efficient allocation of resources.

In practice, determining the best resource allocation can be complex, as it involves trade-offs. For example, if an economy fully employs its labor to produce food, it cannot produce any cloth, and vice versa. This is illustrated in the exercise when we discuss limited resources of labor and land and calculate the maximum outputs of food and cloth. Equally important for students to consider is how changes in the availability of resources, like the introduction of capital in our exercise, affect this allocation and subsequently the PPF.

Economic Trade-Offs

Economic trade-offs are crucial to understanding how choices affect an economy's resource utilization and output. The trade-offs are vividly represented by the PPF, which shows the combinations of goods that can be produced with a fixed set of resources. As we move along the PPF, to increase the production of one good, we must decrease the production of another due to limited resources, which is the trade-off.

In the provided exercise, sketching different PPFs based on various combinations of labor and land constraints exemplifies how shifting resources from the production of one good to another results in different trade-offs. A key takeaway for students should be that these trade-offs are guided by opportunity costs. For instance, if the economy is at a point along the PPF where the trade-off between food and cloth is favorable, moving away from that point will increase the opportunity cost. As a learning tip, students can reinforce their understanding of economic trade-offs by applying these concepts to real-life scenarios, such as budgeting time or money.