Question

If adding an additional input does not produce additional output, what is the slope of the production function at this point? [LO 12.5\(]\)

Short answer

The slope of the production function is zero.

Step-by-step solution

Identify the concept of a production function

A production function shows the relationship between input quantities and the maximum output that can be produced with those inputs. It is typically represented mathematically and can have different forms depending on the type of production process.

Understand marginal product

The slope of the production function is related to the concept of the marginal product. The marginal product refers to the additional output produced by an additional unit of input. It is derived from the derivative of the production function with respect to the input.

Analyze the given scenario

The problem states that adding an additional input does not produce additional output. This implies that the marginal product of the input is zero.

Determine the slope of the production function

Since the marginal product is zero, the derivative of the production function with respect to the input is zero. Therefore, the slope of the production function at this point is zero.

Key concepts

Marginal Product

In economics, understanding how each additional unit of input affects the output is crucial. This is where the concept of Marginal Product (MP) comes into play. Imagine you are running a factory that produces toys, and you decide to hire one more worker. The Marginal Product measures what that extra worker contributes to the total production. In simple terms, the Marginal Product is the extra output that results from adding one more unit of an input. It is calculated by taking the change in total output divided by the change in the number of units of input. When the additional worker you hired results in 10 more toys being made in a day, those 10 toys are the marginal product of labor. The concept can apply to various inputs such as labor, materials, and more. Understanding the Marginal Product helps managers make informed decisions on resource allocation. If the marginal product is significant, it might be beneficial to increase input. However, if the output does not increase much, it may indicate diminishing returns, highlighting the need to optimize resources better.

Input-Output Relationship

The Input-Output Relationship is a foundational concept in production economics. It refers to the interconnection between the quantities of inputs used and the resulting total output. Think of it as a recipe in cooking, where the ingredients (inputs) directly affect the dish you make (output). In mathematical terms, this relationship is captured by a production function, which shows how different inputs contribute to the making of goods or services. Adjusting one or more inputs gives us different levels of output, helping businesses understand efficiency and productivity. An important thing to note is how changes in input can lead to changes in output—not always in a linear manner. Sometimes, adding more of a particular input might initially increase output significantly, but after a certain point, its effect may decrease, illustrating diminishing returns. This insight helps businesses understand how to manage resources effectively, ensuring optimal output levels are achieved without unnecessary overuse of resources.

Derivative in Economics

The concept of derivatives is fundamental in economics for understanding how things change and at what rate. In the context of a production function, the derivative with respect to an input measures the Marginal Product. This tells us how much additional output can be expected from a small increase in input. Mathematically, taking the derivative of the production function helps us find the slope of the curve at any given point. If you imagine a graph of production output over units of input, the slope at any particular point reflects the efficiency or productivity of that input unit. When the derivative, or slope, is zero, it means that adding more input does not result in any extra output—a critical situation reflecting zero marginal product. This can signal optimal use or potential overuse of resources. Hence, derivatives provide essential insights into maximizing production efficiency and guiding strategic planning in resource management.