Question

The production of barstools \((q)\) is characterized by a production function of the form \\[ q=K^{1 / 2} \cdot U^{2}=V K-L \\] a. What is the average productivity of labor and capital for barstool production \((A P,\) will depend on \(K\), and \(A P_{K}\) will depend on \(L\) ? b. Graph the \(A P_{L}\) curve for \(K=100\) c. For this particular function, show that \(M P_{L}=^{\wedge} A P_{L}\) and \(M P_{K}=\sim A P_{K} .\) Using that infor mation, add a graph of the \(M P\), function to the graph calculated in part (b) (again for \(K=100) .\) What is unusual about this curve? d. Sketch the \(q=10\) isoquant for this production function. e. Using the results from part (c), what is the \(R T S\) on the \(^{\wedge}=10\) isoquant at the points: \(K=L=10 ; L=25, K=4 ;\) and \(K=4, L=25 ?\) Does this function exhibit a diminishing \(R T S ?\)

Short answer

In conclusion, we found the average productivity of labor and capital for the given production function and graphed the APL and MPL curves for K=100. We also sketched the q=10 isoquant and calculated the RTS at given points. In this specific production function, there is no diminishing return to scale or diminishing marginal productivity of labor, which is unusual for real-world production functions. This indicates that the given production function exhibits constant or increasing RTS and constant MPL.

Step-by-step solution

a. Average Productivity of Labor and Capital

Given the production function: \\[ q=K^{1 / 2} \cdot U^{2}=V K-L \\] To find the average productivity of labor (APL), we need to divide the output (q) by the labor input (L). Thus, APL can be found as follows: \\[ AP_L = \frac{q}{L} = \frac{K^{1/2} \cdot L^{2}}{L} = K^{1/2} \cdot L \\] Similarly, to find the average productivity of capital (APK), we need to divide the output (q) by the capital input (K). So APK can be found as follows: \\[ AP_K = \frac{q}{K} = \frac{K^{1/2} \cdot L^{2}}{K} = L^{2} \cdot K^{-1/2} \\]

b. Graph APL Curve for K=100

To graph the APL curve for K=100, substitute K=100 into the APL equation: \\[ AP_L = 100^{1/2} \cdot L = 10 \cdot L \\] The APL curve is a straight line with a slope of 10, and it starts from the origin (0,0).

c. Marginal Productivity of Labor and Capital and Graph MPL for K=100

To find the marginal productivity of labor (MPL), we need to differentiate the production function q with respect to L: \\[ MPL = \frac{dq}{dL} = 2 \cdot K^{1/2} \cdot L \\] Similarly, to find the marginal productivity of capital (MPK), we need to differentiate the production function q with respect to K: \\[ MPK = \frac{dq}{dK} = \frac{1}{2} \cdot K^{-1/2} \cdot L^{2} \\] Now we can show that MPL=^ APL and MPK= ~ APK: For MPL and APL: \\[ \begin{aligned} MPL &= 2 \cdot K^{1/2} \cdot L \\ AP_L &= K^{1/2} \cdot L \end{aligned} \\] MPL can be equal to APL if the multiplier is 2. For MPK and APK: \\[ \begin{aligned} MPK &= \frac{1}{2} \cdot K^{-1/2} \cdot L^{2} \\ AP_K &= L^{2} \cdot K^{-1/2} \end{aligned} \\] MPK can be equal to APK if the multiplier is 1/2. Now to graph the MPL function for K=100, substitute K=100 into the MPL equation: \\[ MPL = 2 \cdot 100^{1/2} \cdot L = 20 \cdot L \\] The MPL curve is a straight line with a slope of 20, and it also starts from the origin (0,0). What's unusual about this curve is that it has a constant slope, indicating that there is no diminishing marginal productivity of labor, which is a rare situation in real-world production functions.

d. Sketch the q=10 Isoquant for the Production Function

To sketch the q=10 isoquant for the production function, set q=10: \\[ 10 = K^{1/2} \cdot L^{2} \\] Solve for L: \\[ L = \sqrt{\frac{10}{K^{1/2}}} \\] The q=10 isoquant is an upward-opening parabola with the vertex at K=0, L=0.

e. RTS Points and Diminishing RTS

To find the RTS, we need to calculate the ratio of MPL to MPK: \\[ RTS = \frac{MPL}{MPK} = \frac{2 \cdot K^{1/2} \cdot L}{\frac{1}{2} \cdot K^{-1/2} \cdot L^{2}} = \frac{4 \cdot K}{L} \\] Now, we can find the RTS at each given point: 1. K=L=10: \\[ RTS = \frac{4 \cdot 10}{10} = 4 \\] 2. L=25, K=4: \\[ RTS = \frac{4 \cdot 4}{25} = \frac{16}{25} \\] 3. K=4, L=25: \\[ RTS = \frac{4 \cdot 4}{25} = \frac{16}{25} \\] At all given points, the RTS is either constant or increasing as we move along the isoquant. This indicates that the production function does not exhibit diminishing RTS. This is consistent with the fact that the MPL curve is upward-sloping, implying no diminishing marginal productivity of labor.

Key concepts

Average Productivity

Average productivity is a crucial measure in a production process. It tells us how much output (or products) is being produced on average per unit of input. In our exercise, we evaluate the Average Productivity of Labor (APL) and Average Productivity of Capital (APK). These metrics allow us to judge how effectively labor and capital are being used in the production of barstools.

To calculate APL, we take the total output, represented by the production function, and divide it by the amount of labor used. Mathematically, it’s expressed as:

• \( AP_L = \frac{q}{L} = K^{1/2} \cdot L \)

This shows that APL depends on both the capital (\(K\)) and labor (\(L\)), and reveals the efficiency of labor used.

Similarly, APK is calculated by dividing the total output by the input of capital:

• \( AP_K = \frac{q}{K} = L^{2} \cdot K^{-1/2} \)

Here, we see how the productivity of capital varies with the different levels of labor input. These formulas help businesses and economists understand the effectiveness of labor and capital in the production process.

Marginal Productivity

Marginal productivity refers to the additional output produced by using an extra unit of input. It tells us how much more we can produce if we increase either labor or capital by one unit. Understanding marginal productivity is key to optimizing production because it helps identify how changes in input volume affect output.

In the context of our problem, Marginal Productivity of Labor (MPL) and Marginal Productivity of Capital (MPK) are calculated by taking the derivative of the production function with respect to labor and capital respectively. Let's see the formulas:

• MPL is given by the derivative \( MPL = \frac{dq}{dL} = 2 \cdot K^{1/2} \cdot L \)
• MPK is given by the derivative \( MPK = \frac{dq}{dK} = \frac{1}{2} \cdot K^{-1/2} \cdot L^{2} \)

These calculations show how output changes as we tweak the labor and capital inputs. Notably, in this exercise, it can be shown that MPL equals APL when adjusted for a constant multiplier, which provides insight into how the function behaves differently from typical production functions.

Returns to Scale

Returns to Scale (RTS) examine how the output responds when all inputs are increased proportionally. This concept is significant for understanding long-term production possibilities and how efficiently a firm can scale its operations.

To determine RTS, we examine the ratio of marginal products (MPL to MPK) at various production levels, which is shown by:

• \( RTS = \frac{MPL}{MPK} = \frac{4 \cdot K}{L} \)

In our exercise, calculating RTS at specified points reveals insights about the production function's behavior. For example, we see that at some points, RTS does not decrease, suggesting that we may not encounter diminishing returns commonly expected in such functions.

Analyzing returns to scale helps producers make informed decisions about expanding production, as it shows whether increasing scale leads to proportionate increases in output, which is crucial when planning for growth and development of production processes.