Mathematical Representation of a Concave Production Function

Strictly Concave Function

Concave Decreasing Function

A function is described as concave when its slope decreases as the input variable increases. This means the function's graph becomes less steep or more steeply negative. Consider the four functions depicted in the graphs below, each shown for positive values of an input variable. Which graph represents a function that is concave across the entire displayed domain?

Production Function Analysis

A function is plotted on a graph. For any positive value of the input, the function's slope is always positive, but the value of the slope steadily decreases as the input gets larger. Based on this description, the function must be concave.

Analyzing a Function from its Slope

A firm's production schedule shows how its total output changes as it adds more units of a single input, holding all other inputs constant. The table below shows the additional output (marginal product) generated by each successive unit of input.

Based on this data, what can be concluded about the shape of the firm's total production function over this range of input?

Match each function type with its corresponding mathematical condition related to its second derivative, denoted as $f''(x)$.

Economic Interpretation of a Concave Function

A production function that exhibits a continuously declining marginal product for each additional unit of input is an example of a(n) ________ function.

A function is defined by several points given in the table below. By analyzing the change in the function's value as the input increases, determine the shape of the function over the given domain.

You are given a table of input (x) and output (y) values for a function. Arrange the following steps in the correct logical order to determine if the function represented by these data points is concave.