Question

Economists use production functions to describe how the output of a system varies with respect to another variable such as labor or capital. For example, the production function \(P(L)=200 L+10 L^{2}-L^{3}\) gives the output of a system as a function of the number of laborers \(L\). The average product \(A(L)\) is the average output per laborer when \(L\) laborers are working; that is \(A(L)=P(L) / L\). The marginal product \(M(L)\) is the approximate change in output when one additional laborer is added to \(L\) laborers; that is, \(M(L)=\frac{d P}{d L}\). a. For the given production function, compute and graph \(P, A,\) and \(M\). b. Suppose the peak of the average product curve occurs at \(L=L_{0},\) so that \(A^{\prime}\left(L_{0}\right)=0 .\) Show that for a general production function, \(M\left(L_{0}\right)=A\left(L_{0}\right)\).

Short answer

In this exercise, we analyzed a production function \(P(L)=200L+10L^2-L^3\) with \(L\) being the number of laborers. Firstly, we derived its average product function by dividing the production function by the number of laborers: \(A(L)=200+10L-L^2\). Then, we calculated the marginal product function by differentiating the production function with respect to the number of laborers: \(M(L)=200+20L-3L^2\). We graphed the production function, average product function, and marginal product function to visualize their behaviors. Finally, we showed that for a general production function, the marginal product is equal to the average product at the peak of the average product curve. For the given production function, we proved that when \(A'(L_0)=0\), we have \(M(L_0)=A(L_0)\), which confirms the relationship between average product and marginal product at the peak of the average product curve.

Step-by-step solution

Analyzing the given function

Firstly, we have to analyze the given production function \(P(L)=200L+10L^2-L^3\). As we can observe, this function is a polynomial of degree 3, which means that its behavior will depend on the values of \(L\). Furthermore, we notice that the coefficient of the \(L^3\) term is negative, indicating that the graph of the function will approach negative infinity as \(L\) goes to positive infinity.

Calculating the average product function

Now, we need to find the average product function \(A(L)\). To do that, we need to divide the production function \(P(L)\) by the number of laborers \(L\):$$A(L)=\frac{P(L)}{L}=\frac{200L+10L^2-L^3}{L}.$$We can factor out \(L\) and simplify:$$A(L)=\frac{L(200+10L-L^2)}{L}=200+10L-L^2.$$

Calculating the marginal product function

Next, we need to find the marginal product function \(M(L)\), which represents the approximate change in output when an additional laborer is added. To do this, we need to differentiate the production function \(P(L)\) with respect to \(L\):$$M(L)=\frac{dP}{dL}=\frac{d(200L+10L^2-L^3)}{dL}.$$Using the power rule of differentiation, we get:$$M(L)=200+20L-3L^2.$$

Graphing the functions

At this point, we have the expressions for the production function \(P(L)\), the average product function \(A(L)\), and the marginal product function \(M(L)\). To graph these functions, you could use a graphing tool or software to plot the expressions: 1. \(P(L)=200L+10L^2-L^3\) 2. \(A(L)=200+10L-L^2\) 3. \(M(L)=200+20L-3L^2\) This will help you visualize how the output, average product, and marginal product change as the number of laborers \(L\) varies.

Proving the relationship between average product and marginal product at the peak

Finally, we need to show that for a general production function, the marginal product is equal to the average product at the peak of the average product curve, i.e., when \(A'(L_0)=0\). Given a general production function \(P(L)\), the average product function \(A(L)\) is defined as: $$A(L)=\frac{P(L)}{L}$$and the marginal product function \(M(L)\) is defined as:$$M(L)=\frac{dP}{dL}$$. At the peak of the average product curve, we have \(A'(L_0)=0\). To find the derivative of \(A(L)\), we can use the quotient rule. The quotient rule states that:$$\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$$Applying the quotient rule to \(A(L)\), we get:$$A'(L)=\frac{P'(L)L - P(L)}{L^2}$$When \(A'(L_0)=0\), we have:$$0 = \frac{P'(L_0)L_0 - P(L_0)}{L_0^2}$$Multiplying both sides by \(L_0^2\):$$0 = P'(L_0)L_0 - P(L_0)$$From the definition of the marginal product, we know that \(P'(L_0)=M(L_0)\). Hence:$$0 = M(L_0)L_0 - P(L_0)$$Now, isolating \(M(L_0)\) gives:$$M(L_0) = \frac{P(L_0)}{L_0}$$But from the definition of the average product, we have \(A(L_0)=\frac{P(L_0)}{L_0}\). Thus, we have shown that:$$M(L_0) = A(L_0)$$at the peak of the average product curve.

Key concepts

Average Product of Labor

When we talk about the average product of labor in the context of a production function, we are referring to the output produced per worker or unit of labor. This is a key concept in understanding how efficiently labor is being used in the production process. In the case of the production function P(L) = 200L + 10L^2 - L^3, the average product of labor A(L) is mathematically expressed by dividing the total output P(L) by the number of laborers L, resulting in A(L) = 200 + 10L - L^2.

To give an intuitive example, if a factory employs 100 workers and produces 5000 widgets, the average product of labor would be 5000/100, or 50 widgets per worker. This measure helps businesses and economists gauge the effectiveness of labor in the production process and identify optimal labor levels for maximizing output.

Marginal Product of Labor

The marginal product of labor is another critical concept in economic production theory. It represents the additional output that one more worker would produce. Understanding this aspect is key for businesses when deciding if hiring an extra hand is beneficial. In calculus terms, it's derived from the production function by taking the derivative of P(L) with respect to labor L, symbolized as M(L). For the function P(L) = 200L + 10L^2 - L^3, the marginal product of labor is obtained by differentiating, which gives us M(L) = 200 + 20L - 3L^2.

The concept of marginality is foundational in economics; it helps predict changes in production with incremental adjustments in inputs. For instance, if a factory is contemplating hiring an additional worker, the marginal product of labor can provide important insight into the likely increase in production that the new worker will contribute.

Differentiation

Differentiation, a core operation in calculus, involves finding the derivative of a function. In the context of our production function, it is the mathematical tool we use to calculate the marginal product of labor. Applied to P(L), differentiation gives us a formula that describes how the production output changes with each additional unit of labor. By differentiating, we are able to move from a general understanding of total output to more nuanced insights about the effects of changing labor inputs.

Differentiation involves rules and operations that allow us to break down complex expressions into simpler components, such as using the power rule for polynomials. It is the cornerstone for analysis in many fields, including economics, physics, and engineering, where understanding how one variable changes in relation to another is essential.

Quotient Rule

In calculus, the quotient rule is a technique for differentiating functions that are fractions of two other functions. It's especially relevant when dealing with the average product of labor, which is a division of total production by the labor force size. Formally, the quotient rule states that the derivative of a quotient is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all over the denominator squared. The quotient rule formula is \[ \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \].

Applying this to our average product function A(L) = P(L) / L, we can find the point at which the average product is at its peak, which is an important consideration for businesses looking to optimize their labor allocations. The quotient rule helps us navigate through these complex fractions, ensuring we can effectively analyze and maximize production functions.