Question

The Cobb-Douglas production function for an automobile manufacturer is \(f(x, y)=100 x^{0.6} y^{0.4}\) where \(x\) is the number of units of labor and \(y\) is the number of units of capital. Estimate the average production level if the number of units of labor \(x\) varies between 200 and 250 and the number of units of capital \(y\) varies between 300 and 325 .

Short answer

After computation, the average production level is found to be approximately 22426.4 units.

Step-by-step solution

Understand the Problem

The Cobb-Douglas production function is \(f(x, y)=100 x^{0.6} y^{0.4}\) where \(x\) is the number of units of labor and \(y\) is the number of units of capital. The average production level is required when the units of labor \(x\) varies between 200 and 250 and the units of capital \(y\) varies between 300 and 325.

Calculate the Average for Labor and Capital

Firstly, calculate the average values for labor and capital. For labor, this will be \((200+250)/2 = 225\) and for capital this will be \((300+325)/2 = 312.5\).

Plug in the Average Values

Using the average values for labor and capital, substitute into the Cobb-Douglas production function: \(f(225, 312.5)=100 \cdot 225^{0.6} \cdot 312.5^{0.4}\). Calculate the value.

Evaluate the Expression

Finally, compute the value to get the average production level.

Key concepts

College Algebra

College algebra serves as the foundation for many mathematical concepts used in higher levels of education and in various professional fields. In our context, college algebra provides the tools required to handle the Cobb-Douglas production function, which portrays the relationship between labor and capital in a manufacturer's production process. Understanding algebraic operations, exponents, and the process of substituting values into an equation are essential skills obtained from college algebra. These are critical when we deal with the Cobb-Douglas production function, \(f(x, y)=100 x^{0.6} y^{0.4}\), for calculating production levels based on given inputs of labor \(x\) and capital \(y\).

For the purpose of this exercise, knowing how to calculate the average of two numbers is also vital. The average of any two numbers is found by summing them up and then dividing by two. This is shown in the solution steps where the averages of \(x\) values (labor) and \(y\) values (capital) are calculated straightforwardly. Such simple yet fundamental algebraic concepts form the bedrock that allows us to handle more complex equations and estimations seen in the Cobb-Douglas function.

Calculus

When we delve into calculus, we explore the concept of change and how to calculate it. Although calculus may not be fully showcased in the solution provided, understanding the principles of calculus can extend our analysis of production functions. For instance, we could use calculus to determine the rate of change in production as labor or capital changes, known as the marginal product. In calculus, this is done through partial derivatives, calculating how the function changes as one variable shifts while keeping the other constant.

However, this advanced concept may not be immediately necessary for the current exercise, it provides a glimpse into how calculus can further our understanding and analysis of functions such as the Cobb-Douglas production function. While not applied directly here, the principles of calculus remain an important part of the broader mathematical framework used in economics and business.

Production Levels Estimation

Production levels estimation is about predicting the output of a production process given certain inputs. In our exercise with the Cobb-Douglas production function, we estimate the average production level by first determining the average input levels of labor and capital. It’s an essential concept for businesses as it aids in forecasting and planning.

We apply the calculated average inputs back into the production function to estimate the output. The Cobb-Douglas function specifically is designed to model the reality that both labor and capital contribute to production in a way that reflects the characteristics of a given industry or production process. This kind of estimation can be helpful for a firm to understand how different adjustments in labor or capital may affect their overall production levels.

Average Value of a Function

The average value of a function in a given range provides an important measure in both mathematics and real-world applications. In a practical sense, the average value can inform us about the expected output over a range of inputs.

In the context of our exercise, the average production level is sought for a range of labor and capital inputs. The method used here is simple: by finding the average of the inputs and plugging them into the Cobb-Douglas function, we can estimate the overall average production level without calculating the average production for each unit of input. This process simplifies the estimation task and provides a quick snapshot of expected production, which can be critical for making efficient business decisions.