A firm's output (y) is determined by a single input (x), where x represents a positive quantity of resources used. Based on the fundamental principle that more input should result in more output, which of the following mathematical relationships is an implausible representation of this firm's production process for any positive amount of input?

Evaluating Production Function Models

For a production process described by the function y = f(x), where y is the total output and x is the amount of a single input, the fundamental requirement that more input leads to more output implies that the rate of change of the output must itself be continually increasing as more input is added.

The Rationale for an Increasing Production Function

The Limits of the Increasing Production Function Principle

For each of the following mathematical relationships between a positive input (x > 0) and an output (y), match it with the correct statement describing its suitability as a production model based only on the principle that output must increase with input.

Consider the production function y = 40x - x^2, where y is the total output and x is the amount of a single, positive input. For this relationship to represent a valid production scenario where more input leads to more output, the quantity of input x must be less than ____.

Evaluating a Production Model's Validity

A firm's production process is modeled by the function y = f(x), where y is the total output and x is the quantity of a single input (x > 0). According to the principle that more input should always result in more output, which of the following functions could represent this production process for any and all positive amounts of input?

A production process relates a single input (x) to a total output (y). A fundamental property of this process is that any increase in the amount of input used must result in an increase in the total output produced. Which of the following graphs, showing output on the vertical axis and input on the horizontal axis, correctly illustrates a relationship that adheres to this property for all positive levels of input shown?