Question

An economy can produce good 1 using labor and capital and good 2 using labor and land. The total supply of labor is 100 units. Given the supply of capital, the outputs of the two goods depend on labor input as follows: $$\begin{array}{cccc} \text { Labor Input } & \text { Output } & \text { Labor Input } & \text { Output } \\ \text { to Good 1 } & \text { of Good 1 } & \text { to Good 2 } & \text { of Good 2 } \\ \hline 0 & 0.0 & 0 & 0.0 \\ 10 & 25.1 & 10 & 39.8 \\ 20 & 38.1 & 20 & 52.5 \\ 30 & 48.6 & 30 & 61.8 \\ 40 & 57.7 & 40 & 69.3 \\ 50 & 66.0 & 50 & 75.8 \\ 60 & 73.6 & 60 & 81.5 \\ 70 & 80.7 & 70 & 86.7 \\ 80 & 87.4 & 80 & 91.4 \\ 90 & 93.9 & 90 & 95.9 \\ 100 & 100 & 100 & 100 \end{array}$$ a. Graph the production functions for good 1 and good 2 b. Graph the production possibility frontier. Why is it curved?

Short answer

Each good is plotted on separate curves with labor input on the x-axis and output on the y-axis. The PPF is plotted by allocating the total labor to each good in varying amounts and plotting these output combinations, resulting in a curved line due to increasing opportunity costs.

Step-by-step solution

Plot the Production Functions for Good 1 and Good 2

Use a graph with two separate curves, one for each good. Plot the labor input on the x-axis and output of the good on the y-axis. For Good 1, plot the points from the given data (e.g., Labor Input 0, Output 0; Labor Input 10, Output 25.1; etc.). Do the same for Good 2. Connect the points for each good to form two curves.

Identify the Production Possibility Frontier

The production possibility frontier (PPF) can be found by plotting the maximum possible output combinations of Good 1 and Good 2 when all labor units are efficiently allocated between both goods. This involves finding various combinations of Good 1 and Good 2 that can be produced using all 100 units of labor.

Graph the Production Possibility Frontier

To graph the PPF, determine points where labor is allocated differently between the two goods while ensuring the total labor used does not exceed 100 units. Use the outputs from the production functions for the different allocations of labor. Plot these combinations on a graph with Good 1 on the x-axis and Good 2 on the y-axis, and connect these points to form the frontier.

Explain the Shape of the Production Possibility Frontier

The PPF is curved because of the principle of increasing opportunity costs, which in this case is due to the different efficiencies of producing each good. Both goods have increasing marginal costs because resources (labor, capital, and land) are not perfectly adaptable to the production of both goods. As production of one good increases, resources that are less efficient at producing that good must be used, increasing the opportunity cost and resulting in a bowed-out shape for the frontier.

Key concepts

Production Functions

The concept of production functions is essential for understanding how an economy uses resources to produce goods and services. A production function represents the relationship between inputs used in production – such as labor, capital, and land – and the resulting output. In our exercise, two goods are produced using different combinations of labor and either capital or land.

Graphically representing these relationships shows how much output can be produced with varying amounts of labor. As evidenced by the numerically increasing outputs for increased labor input, we can see that each additional unit of labor contributes to a greater output, up to a point. This reflects the properties of the production function, where initially, labor contribution to output might increase, but over time, it can face diminishing returns, meaning that each additional unit of labor results in progressively smaller increases in output. Understanding production functions is crucial for businesses and economies to estimate potential production and allocate resources efficiently.

Opportunity Cost

Another pivotal concept in economics, encapsulated within the exercise, is opportunity cost. This term refers to the value of the next best alternative that is foregone when a decision is made to use resources in a certain way. When resources such as labor are scarce, choosing to produce more of Good 1 means that less labor is available to produce Good 2, and vice versa.

This trade-off is fundamental to decision-making and is graphically showcased by the downward slope of the production possibility frontier. The slope indicates how much of Good 2 must be given up to produce more of Good 1. The steeper the slope at any point, the higher the opportunity cost of Good 1 in terms of Good 2. This concept helps individuals and societies make informed decisions on resource allocation by understanding the true cost of their economic choices.

Efficient Resource Allocation

The efficient allocation of resources is another cornerstone of economic analysis and refers to using resources in a way that maximizes the production of goods and services. In our textbook exercise, efficient resource allocation requires distributing the available 100 units of labor between the production of Good 1 and Good 2 in such a way that no additional output of either good can be achieved without decreasing the output of the other.

The production possibility frontier (PPF) illustrates the concept of efficiency. Any point on the PPF represents a situation where resources are allocated efficiently, and the economy is producing at its maximum potential given the resources. Points inside the PPF indicate inefficiency because more of at least one good can be produced without sacrificing the production of the other. Points outside the PPF are unattainable with the current available resources. Efficient resource allocation adjusts with changes in the availability and productivity of resources, technology, and societal preferences.

Marginal Costs

Marginal costs play a critical role in the analysis of production and decision-making. It is the cost of producing one additional unit of a good. In the context of our exercise, marginal costs are associated with the allocation of labor between the two goods. As more labor is assigned to the production of one good, the marginal cost of producing that good increases, because each additional unit of labor contributes less to the output than the previous unit due to the concept of diminishing marginal returns.

Graphically, the increasing marginal costs are reflected in the bowed-out shape of the production possibility frontier. As more of one good is produced and resources become more specialized, switching these specialized resources to the production of the other good becomes increasingly costly. This relationship is crucial for understanding why the PPF is not a straight line but convex to the origin. Understanding marginal costs helps businesses in pricing decisions and in determining the level of production that maximizes profit.