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Chapter 11: Problem 1

Digging clams by hand in Sunset Bay requires only labor input. The total number of clams obtained per hour \((q)\) is given by \\[ q=100 \vee Z \\] where \(L\) is labor input per hour. a. Graph the relationship between \(q\) and \(L\) b. What is the average productivity of labor in Sunset Bay? Graph this relationship and show that \(A P_{L}\) diminishes for increases in labor input. c. Show that the marginal productivity of labor in Sunset Bay is given by \(\boldsymbol{M P}_{L}=50 / V \boldsymbol{Z} .\) Graph this relationship and show that \(M P,

Short Answer

Expert verified
Answer: The marginal productivity of labor is less than the average productivity of labor for all values of labor input in Sunset Bay due to the law of diminishing marginal returns. This law states that as more of a variable input (in this case, labor) is added to a fixed input, the additional output produced by the last unit of variable input will eventually decline. In other words, as more labor is added to the fixed resource (digging clams), the productivity of each additional unit of labor will decrease, resulting in a lower marginal productivity of labor compared to the average productivity of labor.

Step by step solution

01

To graph the relationship between the total number of clams obtained per hour (\(q\)) and labor input per hour (\(L\)), simply plot the given equation \(q = 100\sqrt{L}\) with \(L\) on the x-axis and \(q\) on the y-axis. This will show the direct correlation between labor input and the number of clams obtained. #b1. Finding the average productivity of labor#

The average productivity of labor (\(AP_L\)) is found by dividing the total number of clams obtained per hour (\(q\)) by the labor input per hour (\(L\)). So, \(AP_L = \frac{q}{L} = \frac{100\sqrt{L}}{L} = \frac{100}{\sqrt{L}}\). #b2. Graphing the average productivity of labor and proving it diminishes for increases in labor input#
02

To graph the relationship between the average productivity of labor (\(AP_L\)) and labor input per hour (\(L\)), simply plot the equation \(AP_L = \frac{100}{\sqrt{L}}\) with \(L\) on the x-axis and \(AP_L\) on the y-axis. To show that \(AP_L\) diminishes for increases in labor input, observe the downward slope of the graph, which indicates that as \(L\) increases, \(AP_L\) decreases. #c1. Finding the marginal productivity of labor#

The marginal productivity of labor (\(MP_L\)) is the derivative of the total number of clams obtained per hour (\(q\)) with respect to the labor input per hour (\(L\)). So, given the equation \(q = 100\sqrt{L}\), we can find the derivative with respect to \(L\) to get \(MP_L = \frac{dq}{dL} = \frac{50}{\sqrt{L}}\). #c2. Graphing the marginal productivity of labor and proving that \(MP_L < AP_L\) for all values of \(L\)#
03

To graph the relationship between the marginal productivity of labor (\(MP_L\)) and labor input per hour (\(L\)), simply plot the equation \(MP_L = \frac{50}{\sqrt{L}}\) with \(L\) on the x-axis and \(MP_L\) on the y-axis. To show that \(MP_L < AP_L\) for all values of \(L\), compare the graphs of \(MP_L\) and \(AP_L\). You will notice that the graph of \(MP_L\) lies below the graph of \(AP_L\) for all values of labor input. #c3. Explaining why \(MP_L < AP_L\) for all values of \(L\)#

The reason why \(MP_L < AP_L\) for all values of \(L\) is because of the law of diminishing marginal returns, which states that as more of a variable input (in this case, labor) is added to a fixed input, the additional output produced by the last unit of the variable input will eventually decline. In other words, as more labor is added to the fixed resource (digging clams), the productivity of each additional unit of labor will decrease, resulting in a lower marginal productivity of labor compared to the average productivity of labor.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Productivity of Labor
In microeconomics, when we discuss the productivity of labor, we're essentially measuring the output that is produced per unit of labor input. Average Productivity of Labor (APL) is a critical concept that helps us understand how effectively labor is used within the production process.

To calculate APL, divide the total output produced by the number of labor hours used. Using the Sunset Bay example, where the number of clams harvested per hour is represented as \(q = 100\sqrt{L}\), the average productivity of labor comes out to be \(AP_{L} = \frac{q}{L} = \frac{100\sqrt{L}}{L} = \frac{100}{\sqrt{L}}\).

It's important to notice that as more labor is added, APL typically decreases. This relationship can be visually represented on a graph, showing a downward trend as labor increases, indicating that each additional labor hour contributes less to average output than the previous ones.
Marginal Productivity of Labor
Next to the average, there is a nuanced concept known as Marginal Productivity of Labor (MPL). Marginal productivity measures the change in output that results from adding one more unit of labor. Essentially, it tells us the additional number of clams a worker can gather when one more hour of work is invested.

Following our calculation from Sunset Bay, the MPL is the derivative of \(q = 100\sqrt{L}\) with respect to labor, which results in \(MP_{L} = \frac{dq}{dL} = \frac{50}{\sqrt{L}}\). This measurement is pivotal for employers to decide how much labor they should employ to maximize productivity and reduce waste.

In simplistic terms, MPL helps us understand the efficiency of the 'last worker hired'. This concept is key to making informed operational decisions in any labor-intensive industry.
Law of Diminishing Marginal Returns
One of the fundamental principles behind the behavior of average and marginal productivity is the Law of Diminishing Marginal Returns. This economic theory suggests that when additional units of a variable resource (like labor) are added to a fixed resource (like land or capital), beyond a certain point, the additional output produced will start to decrease.

In the context of Sunset Bay's clam digging, when employing more and more workers, each new worker adds less to the total output than the previous one. This is because they may start to get in each other's way or run out of optimal space to work efficiently. This law is beautifully intuitive but also mathematically demonstrable through the relationship between MPL and APL and their respective graphs, stressing the importance of strategic resource allocation.
Graphing Economic Functions
Visual representations in economics are more than just lines on a graph; they convey significant insights about the relationships between different economic variables. Graphing economic functions can offer a clear view of concepts such as productivity, cost, and revenue.

For instance, in our clam-digging scenario, graphing the function for APL or MPL against labor input exposes the underlying economic relationship. The resulting curves illustrate not just quantities but also the rate at which productivity is increasing or decreasing. Recognizing how to interpret and construct these graphs is an invaluable skill for understanding how individual changes in labor or other inputs influence the overall output and efficiency of any production activity.

Graphs serve as a bridge between abstract theories and practical, real-world applications, making them indispensable tools in the realm of economics.

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Most popular questions from this chapter

Suppose the production function for widgets is given by \\[ q=K L-.8 K^{2}-. I V \\] where \(q\) represents the annual quantity of widgets produced, \(K\) represents annual capital input, and \(L\) represents annual labor input. a. Suppose \(K-10\); graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point? b. Again assuming that \(K=10\), graph the \(M P_{L}\) curve. At what level of labor input does \\[ M P_{L}=0 ? \\] c. Suppose capital inputs were increased to \(K-20 .\) How would your answers to parts and (b) change? d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale?

Show that for the constant returns-to-scale CES production function \\[ q=\left[K f+L_{P} Y^{\wedge}\right. \\] a. \(M P_{K}=\left(^{\wedge} | \sim^{P} \text { and } M P_{L}=\left(j-k^{\prime \prime}\right.\right.\) b. \(\quad R T S=[-) \quad\) Use this to show that \(\mathrm{cr}=1 /(1-\mathrm{p})\) \\[ \left.\right|^{K} \boldsymbol{I} \\] c. Determine the output elasticities for Xand \(L\). Show that their sum equals 1 d. Prove that Hence, show Note: The latter equality is useful in empirical work, because in some cases we may approximate \(d q / d L\) by the competitively determined wage rate. Hence, \(a\) can be estimated from a regression of \(\ln (q / L)\) on in \(w\)

As in Problem \(11.8,\) again use Euler's theorem to prove that for a constant returns-to-scale production function with only two inputs \((K \text { and } L), /^{\wedge}\) must be positive. Interpret this result.

Show that Euler's theorem (see footnote 5 of Chapter 7 ) implies that for a constant returnsto-scale production function \([q=f(K, L)]\) Use this result to show that for such a production function, if \(M P_{L}>A P, M P_{K}\) must be negative. What does this imply about where production must take place? Can a firm ever produce at a point where \(A P_{L}\) is increasing?

Constant returns-to-scale production functions are sometimes called homogeneous of degree 1 More generally, as we showed in footnote 1 of Chapter \(5,\) a production function would be said to be homogeneous of degree \(k\) if \\[ f(t K, t L)=t y f(K, L) \\] a. Show that if a production function is homogeneous of degree \(k\), its marginal productiv ity functions are homogeneous of degree \(k-1\) b. Use the result from part (a) to show that marginal productivities for any constant returns-to-scale production function depend only on the ratio \(K / L\) c. Use the result from part (b) to show that the \(R T S\) for a constant returns-to-scale pro duction function depends only on the ratio \(K / L\) d. More generally, show that the \(R T S\) for any homogencous function is independent of the scale of operation - all isoquants are radial expansions of the unit isoquant. Hence, such a function is homothetic. e. Show that the results from part (d) apply to any monotonic transformation of a homo geneous function. That is, show that any such transformation of a homogeneous func tion is homothetic.
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