Problem 33 - Applied Calculus

Question

Show that the Cobb-Douglas function $$Q=b K^{\alpha} L^{1-\alpha} \text { where } \quad 0<\alpha<1$$ satisfies the equation $$K \frac{\partial Q}{\partial K}+L \frac{\partial Q}{\partial L}=Q.$$

Step-by-Step Solution

Answer

The equation is satisfied: \(K \frac{\partial Q}{\partial K} + L \frac{\partial Q}{\partial L} = Q\).

Step 1: Calculate the partial derivative with respect to K

The Cobb-Douglas production function is given as \(Q = b K^{\alpha} L^{1-\alpha} \). To find the partial derivative with respect to \(K\), apply the power rule: \[ \frac{\partial Q}{\partial K} = b \cdot \alpha \cdot K^{\alpha - 1} \cdot L^{1-\alpha}. \] This expression represents the marginal product of capital.

Step 2: Find K * (∂Q/∂K)

Multiply the partial derivative from Step 1 by \(K\): \[ K \cdot \frac{\partial Q}{\partial K} = K \cdot \left(b \cdot \alpha \cdot K^{\alpha - 1} \cdot L^{1-\alpha}\right). \] This simplifies to:\[ K \cdot \frac{\partial Q}{\partial K} = b \alpha K^{\alpha} L^{1-\alpha}. \]

Step 3: Calculate the partial derivative with respect to L

For the partial derivative with respect to \(L\), use the function \(Q = b K^{\alpha} L^{1-\alpha} \) and apply the power rule:\[ \frac{\partial Q}{\partial L} = b \cdot (1-\alpha) \cdot K^{\alpha} \cdot L^{-\alpha}. \] This expression represents the marginal product of labor.

Step 4: Find L * (∂Q/∂L)

Multiply the partial derivative from Step 3 by \(L\):\[ L \cdot \frac{\partial Q}{\partial L} = L \cdot \left(b \cdot (1-\alpha) \cdot K^{\alpha} \cdot L^{-\alpha}\right). \]This simplifies to:\[ L \cdot \frac{\partial Q}{\partial L} = b (1-\alpha) K^{\alpha} L^{1-\alpha}. \]

Step 5: Add the results from Steps 2 and 4

Combine the results from Step 2 and Step 4:\[ K \cdot \frac{\partial Q}{\partial K} + L \cdot \frac{\partial Q}{\partial L} = b \alpha K^{\alpha} L^{1-\alpha} + b (1-\alpha) K^{\alpha} L^{1-\alpha} = b K^{\alpha} L^{1-\alpha} (\alpha + 1 - \alpha). \]

Step 6: Simplify the equation

Simplify the expression obtained in Step 5:\[ K \cdot \frac{\partial Q}{\partial K} + L \cdot \frac{\partial Q}{\partial L} = b K^{\alpha} L^{1-\alpha} = Q. \]Thus, the equation is shown to be true.

Key Concepts

Marginal Product of Capital

The concept of the marginal product of capital is a key consideration in production economics, particularly when using a Cobb-Douglas function. It represents the additional output generated by adding one more unit of capital, while holding all other inputs constant. To calculate this, we take the partial derivative of the output function with respect to capital (K). Given the Cobb-Douglas function \[ Q = b K^{\alpha} L^{1-\alpha}, \]we find the partial derivative with respect to K as follows:\[ \frac{\partial Q}{\partial K} = b \cdot \alpha \cdot K^{\alpha - 1} \cdot L^{1-\alpha}. \]This expression tells us that the marginal product of capital depends not just on the amount of capital, but also on labor and the given parameters of the function. Recognizing this relationship is vital as it helps firms understand how changes in capital affect output. To further analyze its impact, we consider the product of this derivative and the amount of capital, resulting in:\[ K \cdot \frac{\partial Q}{\partial K} = b \alpha K^{\alpha} L^{1-\alpha}. \]This step is crucial for evaluating the contribution of capital to the total output in the context of a Cobb-Douglas function.

Marginal Product of Labor

Much like the marginal product of capital, the marginal product of labor is a critical measure in assessing how output varies with changes in labor input, while keeping capital constant. This is especially relevant in a Cobb-Douglas production framework. To calculate the marginal product of labor, we take the partial derivative of the output function with respect to labor (L). For the Cobb-Douglas function, this derivative is:\[ \frac{\partial Q}{\partial L} = b \cdot (1-\alpha) \cdot K^{\alpha} \cdot L^{-\alpha}. \]This equation indicates how the output changes with variations in labor, providing essential information for businesses in decision-making processes regarding workforce adjustments. Similar to capital, we evaluate the effect of labor on output by multiplying this derivative by L:\[ L \cdot \frac{\partial Q}{\partial L} = b (1-\alpha) K^{\alpha} L^{1-\alpha}. \]Understanding this expression helps illustrate how labor contributes incrementally to production, offering insights into labor efficiency and cost management.

Partial Derivatives

Partial derivatives are fundamental tools in calculus used to understand how a function's output changes with respect to one variable, while other variables are kept constant. In the context of the Cobb-Douglas production function, partial derivatives allow us to dissect the individual contributions of capital and labor to production.For a function like \[ Q = b K^{\alpha} L^{1-\alpha}, \]we calculate the partial derivative with respect to both K and L to derive the marginal products. The calculation\[ \frac{\partial Q}{\partial K} = b \cdot \alpha \cdot K^{\alpha - 1} \cdot L^{1-\alpha} \]and\[ \frac{\partial Q}{\partial L} = b \cdot (1-\alpha) \cdot K^{\alpha} \cdot L^{-\alpha} \]provides insights into the rate of output change when each input is modified independently. These derivatives are pivotal in economics because they help in optimizing resource allocation and improving efficiency. Businesses use them to forecast production levels and make strategic adjustments to their input levels to maximize output.