Question

Power Goat Lawn Company uses two sizes of mowers to cut lawns. The smaller mowers have a 22 -inch deck. The larger ones combine two of the 22 -inch decks in a single mower. For each size of mower, Power Goat has a different production function, given by the rows of the following table. $$\begin{array}{lccc} \hline & \begin{array}{c} \text { Output per Hour } \\ \text { (square feet) } \end{array} & \begin{array}{c} \text { Capital Input } \\ \text { (# of 22" mowers) } \end{array} & \text { Labor Input } \\ \hline \text { Small mowers } & 5000 & 1 & 1 \\ \text { Large mowers } & 8000 & 2 & 1 \\ \hline \end{array}$$ a. Graph the \(q=40,000\) square feet isoquant for the first production function. How much \(k\) and \(l\) would be used if these factors were combined without waste? b. Answer part (a) for the second function. c. How much \(k\) and \(l\) would be used without waste if half of the 40,000 -square-foot lawn were cut by the method of the first production function and half by the method of the second? How much \(k\) and \(l\) would be used if one fourth of the lawn were cut by the first method and three fourths by the second? What does it mean to speak of fractions of \(k\) and \(l\) ? d. Based on your observations in part (c), draw a \(q=40,000\) isoquant for the combined production functions.

Short answer

In summary, we have analyzed the Power Goat Lawn Company's two production functions: small mowers and large mowers. We have graphed the isoquant for the q=40,000 square feet output for both production functions and combined them to determine the optimal isoquant for the company. The key points in our analysis are: - For small mowers, the 40,000-square-foot isoquant is represented by the equation k * l = 8, with possible combinations of (1,8), (2,4), (4,2), and (8,1). - For large mowers, the 40,000-square-foot isoquant is represented by the equation k * l = 10, with possible combinations of (1,10), (2,5), (5,2), and (10,1). - Combining the production functions, we found optimal points: (4,4.5) and (5,3.5) without waste for different fractions of cutting the lawn. - The combined isoquant for q=40,000 includes points from both production functions, as well as the optimal points found in part (c).

Step-by-step solution

Part (a): Small Mowers Isoquant

To graph the q=40,000 square feet isoquant for the first production function, we need to find the relationship between capital input (k) and labor input (l). We know that output per hour for small mowers is 5,000 square feet with k=1 and l=1. Without any waste, if the output per hour is 40,000 square feet, we can set up the following equation: $$40,000 = 5,000 \cdot k \cdot l$$ $$k \cdot l = 8$$ This equation represents the isoquant for the 40,000-square-foot output. To graph the isoquant, we need to find different combinations of k and l. Some possible points would be (1,8), (2,4), (4,2), and (8,1).

Part (b): Large Mowers Isoquant

For the second production function (large mowers), the output per hour is 8,000 square feet with k=2 and l=1. Following the same procedure as in part (a), we can set up an equation for the 40,000-square-foot output: $$40,000 = 8,000 \cdot \frac{k}{2} \cdot l$$ $$k \cdot l = 10$$ The possible points for this isoquant would be (1,10), (2,5), (5,2), and (10,1).

Part (c): Combining Production Functions

Here, we have two situations: 1. Half of the lawn (20,000 square feet) is cut using the first method, and the other half is cut using the second method. We can set up the equations for k and l for each method individually and then combine the results. $$20,000 = 5,000 \cdot k_1 \cdot l_1$$ $$k_1 \cdot l_1 = 4$$ $$20,000 = 8,000 \cdot \frac{k_2}{2} \cdot l_2$$ $$k_2 \cdot l_2 = 5$$ We can't speak of fractions of k and l, so we can approximate the inputs. For example, k_1=2, l_1=2, k_2=2, and l_2=2.5. Thus, k=4 and l=4.5 without waste. 2. One-fourth of the lawn (10,000 square feet) is cut using the first method, and three-fourths (30,000) are cut using the second method. $$10,000 = 5,000 \cdot k_1 \cdot l_1$$ $$k_1 \cdot l_1 = 2$$ $$30,000 = 8,000 \cdot \frac{k_2}{2} \cdot l_2$$ $$k_2 \cdot l_2 = 7.5$$ Approximating the inputs, we can use k_1=2, l_1=1, k_2=3, and l_2=2.5. Thus, k=5 and l=3.5 without waste.

Part (d): Combined Isoquant

Knowing the possible points for parts (a) and (b) and considering the combinations from part (c), we can draw q=40,000 isoquant for the combined production functions. Some of the possible points are (4,4.5), (5,3.5), (1,8), (2,4), (4,2), (8,1), (1,10), (2,5), (5,2), and (10,1).

Key concepts

Production Function

Understanding the production function is essential for comprehending how businesses convert inputs into desired outputs. In economics, a production function represents the relationship between the quantities of productive factors used (inputs) and the amount of product obtained (output). It could be thought of as a recipe that details how much labor and capital are required to produce an item.

The most common variables in a production function are capital and labor. Capital refers to physical assets such as machinery and equipment, while labor is human effort, measured in hours or number of workers. A production function can be represented mathematically as Q = f(L, K), where Q is the total output, L is labor input, and K is capital input.

In the Power Goat Lawn Company's scenario, two different production functions are used for small and large mowers respectively, each with varying output capacities per hour. Learning to graph the relationship between labor, capital, and output allows us to create an isoquant, which represents all the combinations of inputs that yield the same level of output.

Capital Input

The term capital input in the context of production refers to the assets used in the process of creating goods or services. Capital can include physical tools, machinery, buildings, and technology. In the example of Power Goat Lawn Company, the capital input is quantified by the number of 22-inch mowers utilized. In each production scenario, more mowers equate to higher output capability, assuming labor input remains constant. Capital is considered essential for leveraging the productivity of labor; more advanced or larger quantities of capital can significantly enhance the output per hour of operation.

Capital input is not immediately consumed in the production process; instead, it provides ongoing service over many production cycles. Power Goat's decision to use one or two mowers can heavily influence the total output, making capital a critical factor in maximizing efficiency and productivity for each acre of lawn cut.

Labor Input

Alternatively, labor input refers to the human contribution to the production process, commonly measured in man-hours, number of workers, or even skill level. In the exercise, labor input is represented by the total number of workers operating the mowers. Each size of mower requires one worker to operate, indicating that labor is not used in greater quantity but could potentially be used more effectively depending on the method employed.

An important aspect of labor input is that, unlike capital, it is expended at the moment of production – workers must be present and actively contributing to generate output. As we determine the efficiency of Power Goat's operations, it’s clear that labor input is a variable that, when combined with capital, dictates the total output per hour. When exploring optimization under various production functions, it becomes evident that finding the right balance of labor input in conjunction with capital input is key to eliminating waste and increasing productivity.

Output per Hour

When assessing the productivity of a company, a key indicator is the output per hour. This measures the quantity of goods or services produced in an hour of labor and is essential in evaluating the efficiency of the production process. For Power Goat, the output is quantified in square feet of lawn cut per 22-inch mower.

This metric provides direct insight into the effectiveness of their production methods. Higher output per hour indicates a more efficient use of inputs or a more productive production function. This can result from either technological improvements, increased skill levels of workers, or more efficient use of capital inputs. In the practice problems, we can compare the output of 5,000 square feet per hour for small mowers and 8,000 square feet per hour for large mowers. These differences highlight the impact of varying the combination of labor and capital inputs, specifically by using more capital (large mowers).

Exercises such as this illustrate the balance and choices faced by companies: whether to increase the quantity of input or improve their quality to achieve a more favorable output per hour.