Problem 42 - Calculus Early Transcendentals

Question

Cobb-Douglas production function The output \(Q\) of an economic system subject to two inputs, such as labor \(L\) and capital \(K\), is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b},\) where \(a, b,\) and \(c\) are positive real numbers. When \(a+b=1,\) the case is called constant returns to scale. Suppose \(a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=40.\) a. Graph the output function using the window \([0,20] \times[0,20] \times[0,500].\) b. If \(L\) is held constant at \(L=10,\) write the function that gives the dependence of \(Q\) on \(K.\) c. If \(K\) is held constant at \(K=15,\) write the function that gives the dependence of \(Q\) on \(L.\)

Step-by-Step Solution

Answer

Answer: When L is held constant at L=10, the function for the dependence of Q on K is \(Q(10, K) = 86.176K^{\frac{2}{3}}\). When K is held constant at K=15, the function for the dependence of Q on L is \(Q(L, 15) = 243.312L^{\frac{1}{3}}\).

Step 1: a. Graph the Cobb-Douglas production function

To graph the function \(Q(L, K) = 40L^{\frac{1}{3}}K^{\frac{2}{3}}\) with the range \([0,20] \times [0,20] \times [0,500]\), use a graphing tool like Desmos or WolframAlpha. Type the function with the specified range, and observe the 3D graph.

Step 2: b. Find the dependence of Q on K when L=10

To find the dependence of Q on K when L=10, substitute the value of L into the Cobb-Douglas production function: \(Q(10, K) = 40(10)^{\frac{1}{3}}K^{\frac{2}{3}}\). Now, we can simplify the expression: \(Q(10, K) = 40(2.1544)K^{\frac{2}{3}}\), and \(Q(10, K) = 86.176K^{\frac{2}{3}}\). So the function for the dependence of Q on K when L=10 is \(Q(10, K) = 86.176K^{\frac{2}{3}}\).

Step 3: c. Find the dependence of Q on L when K=15

To find the dependence of Q on L when K=15, substitute the value of K into the Cobb-Douglas production function: \(Q(L, 15) = 40L^{\frac{1}{3}}(15)^{\frac{2}{3}}\). Now, we can simplify the expression: \(Q(L, 15) = 40L^{\frac{1}{3}}(6.0828)\), and \(Q(L, 15) = 243.312L^{\frac{1}{3}}\). So the function for the dependence of Q on L when K=15 is \(Q(L, 15) = 243.312L^{\frac{1}{3}}\).

Key Concepts

Constant Returns to Scale in Cobb-Douglas

When talking about the Cobb-Douglas production function, a key concept is "constant returns to scale." This happens when the sum of the exponents of labor (L) and capital (K), which are the variables in the function, equals one.
This is mathematically represented as \(a + b = 1\).

• This means if you scale both inputs by the same factor, the output \(Q\) will increase by that same factor.
• The function will retain its proportionality, indicating an efficient use of resources.

In our given exercise, \(a = \frac{1}{3}\) and \(b = \frac{2}{3}\), leading to \(a + b = 1\).
This confirms the constant returns to scale for the function \(Q(L, K) = 40L^{\frac{1}{3}}K^{\frac{2}{3}}\). Understanding this helps us know that the production process is on par with proportional input and output changes, making it predictable and stable.

3D Graphing of Cobb-Douglas Function

Visualizing the Cobb-Douglas production function involves 3D graphing. By graphing, we can understand how changes in labor and capital inputs affect output.

• When you graph \(Q(L, K) = 40L^{\frac{1}{3}}K^{\frac{2}{3}}\), it is set within specific ranges: \([0, 20]\) for both inputs and \([0, 500]\) for the output.
• A 3D graph illustrates a surface where you can observe the curvature and how both inputs jointly influence \(Q\).
• Tools like Desmos or WolframAlpha are essential for plotting this graph, as they allow you to rotate and zoom to understand the graph better.

The visualization offers insight into the incremental output gains provided by adding units of labor or capital, showing the diminishing returns as you increase an input while holding the other constant.

Partial Dependence on Variables

In the context of the Cobb-Douglas production function, understanding partial dependence on a variable is crucial. This means isolating how changes in one input, while holding the other constant, affects the output.

Dependence on \(K\) (Capital) when \(L = 10\)

Substitute \(L = 10\) in the function: \(Q(10, K) = 40(10)^{\frac{1}{3}}K^{\frac{2}{3}} = 86.176K^{\frac{2}{3}}\).
Here, it shows how changing \(K\) affects \(Q\) when \(L\) is fixed.

Dependence on \(L\) (Labor) when \(K = 15\)

Substitute \(K = 15\): \(Q(L, 15) = 40L^{\frac{1}{3}}(15)^{\frac{2}{3}} = 243.312L^{\frac{1}{3}}\).
This indicates how variations in \(L\) impact \(Q\) with a constant \(K\).
These partial dependencies help in predicting outcomes and optimizing input usage by understanding isolated effects in a real-world scenario.