Question

The production of barstools \((q)\) is characterized by a production function of the form \\[ q=K^{1 / 2} \cdot L^{1 / 2}=\sqrt{K \cdot L} \\] a. What is the average productivity of labor and capital for barstool production \(\left(A P_{L}\) will \right. 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 \(M P_{L}=\frac{1}{2} A P_{L}\) and \(M P_{K}=\frac{1}{2} A P_{K}\). Using that information, add a graph of the \(M P_{L}\) 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 \(q=10\) 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 this exercise, we first calculated the average productivity of labor and capital for the given production function. We then graphed the AP_L curve for K=100 and showed that MP_L = (1/2) AP_L and MP_K = (1/2) AP_K. We then sketched the isoquant at q=10 and calculated the RTS for three given points. We found that the given function exhibits diminishing RTS since the RTS decreases as more labor is added.

Step-by-step solution

Calculate the AP_L and AP_K formulas

For the average productivity of labor (AP_L), divide the production function q by the amount of labor L, and similarly for the average productivity of capital (AP_K), divide the production function q by the amount of capital K: \[AP_L = \frac{q}{L}\] \[AP_K = \frac{q}{K}\] Substitute the production function \(q = \sqrt{K \cdot L}\) into the above formulas: \[AP_L = \frac{\sqrt{K \cdot L}}{L}\] \[AP_K = \frac{\sqrt{K \cdot L}}{K}\] #b. Graph the AP_L curve for K=100#

Substitute K=100

We are asked to graph the AP_L curve for K=100, so substitute K=100 into the AP_L formula: \[AP_L = \frac{\sqrt{100 \cdot L}}{L}\]

Simplify the AP_L formula

Next, simplify the formula: \[AP_L = \frac{10\sqrt{L}}{L}\]

Graph the curve

To graph the AP_L curve, plot AP_L (\(y\)-axis) against the labor input L (\(x\)-axis). #c. Show that MP_L=(1/2)AP_L and MP_K=(1/2)AP_K and graph MP_L for K=100#

Calculate the Marginal Productivities: MP_L and MP_K

We'll find the marginal productivity of labor (MP_L) and capital (MP_K) by partially differentiating the production function with respect to L and K, respectively: \[MP_L = \frac{\partial q}{\partial L}\] \[MP_K = \frac{\partial q}{\partial K}\] \[MP_L = \frac{\partial \sqrt{K \cdot L}}{\partial L}\] \[MP_K = \frac{\partial \sqrt{K \cdot L}}{\partial K}\]

Solve for MP_L and MP_K

Calculating the derivates: \[MP_L = \frac{1}{2} \cdot \frac{K^{1/2}}{L^{1/2}}\] \[MP_K = \frac{1}{2} \cdot \frac{L^{1/2}}{K^{1/2}}\]

Show that MP_L = (1/2)AP_L and MP_K = (1/2)AP_K

Comparing the calculated formulas of MP_L and AP_L, and MP_K and AP_K, it's clear that: \[MP_L = \frac{1}{2}AP_L\] \[MP_K = \frac{1}{2}AP_K\]

Graph MP_L for K=100

To graph the MP_L curve for K=100, substitute K=100 into the MP_L formula: \[MP_L = \frac{1}{2} \cdot \frac{100^{1/2}}{L^{1/2}} = \frac{5}{\sqrt{L}}\] Plot MP_L (\(y\)-axis) against labor input L (\(x\)-axis) on the same graph as the AP_L curve from part (b). #d. Sketch the q=10 isoquant#

Calculate the isoquant

Set the production function \(\sqrt{K \cdot L}\) equal to 10, and solve for K in terms of L: \[10 = \sqrt{K \cdot L}\] \[K = \frac{100}{L}\] This equation represents the isoquant for q=10. Plot the isoquant as a curve, with capital K on the \(x\)-axis and labor L on the \(y\)-axis. #e. Calculate RTS for given points and determine if the function exhibits diminishing RTS#

Calculate RTS at each point

The RTS is defined as: \(RTS = \frac{MP_K}{MP_L}\) Using the MP_L and MP_K formulas, calculate RTS at each of the three given points: \[RTS = \frac{\frac{1}{2} \cdot \frac{L^{1/2}}{K^{1/2}}}{\frac{1}{2} \cdot \frac{K^{1/2}}{L^{1/2}}} = \frac{K}{L}\] 1. \(K=L=10\) : \(RTS = \frac{10}{10}=1\) 2. \(L=25, K=4\): \(RTS = \frac{4}{25}=\frac{1}{6.25}\) 3. \(K=4, L=25\): \(RTS = \frac{25}{4}=6.25\)

Determine if the function exhibits diminishing RTS

A function exhibits diminishing RTS if the RTS decreases as more labor is added. In the given function, the RTS at point 1 is equal to 1, and it decreases to \(\frac{1}{6.25}\) at point 2, and increases to 6.25 at point 3. Thus, this function does exhibit diminishing RTS.

Key concepts

Average Productivity of Labor

Average productivity of labor (APL) is a fundamental metric in economics that quantifies the output produced per unit of labor input. In the context of our example, the production function for barstools given by \( q = \sqrt{K \cdot L} \) allows us to express APL as \( AP_L = \frac{q}{L} \), which simplifies to \( AP_L = \frac{\sqrt{K \cdot L}}{L} \). Simplified further, if we consider a fixed amount of capital, say \( K=100 \), the APL becomes \( AP_L = \frac{10\sqrt{L}}{L} \).
A deeper analysis of APL can help us understand how efficiently labor is being used in the production process. When this concept is graphed, we often observe the principle of diminishing returns, which indicates a point beyond which the addition of labor leads to a less than proportionate increase in output. By graphing \( APL \) for a fixed capital, we can visually interpret the efficiency of labor input and optimize production levels accordingly.

Marginal Productivity of Capital

The concept of marginal productivity of capital (MPK) is pivotal in the study of production functions. It measures the additional output obtained from one more unit of capital while holding the amount of labor constant. For our production function, \( q = \sqrt{K \cdot L} \), the MPK is found by differentiating this function with respect to capital, yielding \( MP_K = \frac{1}{2} \cdot \frac{L^{1/2}}{K^{1/2}} \).
Interestingly, for the given production function, the \( MP_K = \frac{1}{2}AP_K \). This relationship suggests that each additional unit of capital adds half the average productivity of capital to the total output. We can represent this relationship graphically by plotting MPK against different levels of capital on the graph and comparing it with the average productivity curve. This concept is crucial for firms to decide the amount of capital to employ for optimal production.

Isoquant

An isoquant is a graphical representation, in input space, of all the combinations of two inputs that result in the same level of output. It is the production function’s counterpart to the indifference curve from the consumer theory in economics. In our barstool production example, the equation \( 10 = \sqrt{K \cdot L} \), which is solved to \( K = \frac{100}{L} \), represents the isoquant for an output level of ten barstools. This curve will be downward sloping, indicating that as more labor is used, less capital is necessary to maintain the same level of output.
Understanding isoquants is critical because they illustrate the concept of substitution between inputs – the trade-offs between capital and labor that can produce the same output. Mapping isoquants helps businesses in making decisions about the combination of inputs to use for efficient production while considering the trade-offs and cost constraints.

Returns to Scale

The concept of returns to scale (RTS) describes how the change in inputs affects the level of output in the long run. 'Returns to scale' can be constant, increasing, or decreasing. For our production function, we evaluate RTS by the ratio \( RTS = \frac{MP_K}{MP_L} = \frac{K}{L} \). By analyzing the given data points:

• When \( K=L=10 \), \( RTS = \frac{10}{10}=1 \), suggesting constant returns to scale.
• At \( L=25, K=4 \), \( RTS = \frac{4}{25}=0.16 \), indicating decreasing returns to scale.
• Conversely, at \( K=4, L=25 \), \( RTS = \frac{25}{4}=6.25 \), we have increasing returns to scale.

Therefore, in this example, the production function exhibits varying returns to scale at different points. Knowledge of the RTS helps firms understand how efficiently they can scale their production with respect to the inputs. If the RTS is diminishing, as in our example, adding more labor and capital may not proportionately increase output, suggesting it may not always be beneficial to scale up production.