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Concave Decreasing Function
A function that is both decreasing and concave exhibits a slope that becomes increasingly negative as the input variable x increases. This means that while the slope value is decreasing, the curve itself becomes steeper because the absolute value of the slope is growing. Mathematically, this property is defined by a negative second derivative, . This shape is contrasted with the convex shape typically used for indifference curves.
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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.
Unit of Input Additional Output (Marginal Product) 1st 20 2nd 18 3rd 15 4th 11 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 .
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.
Input (x) Output (y) 0 0 1 10 2 18 3 24 4 28 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.
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A firm is phasing out an old product line. The production level is reduced by 100 units in the first week, by an additional 150 units in the second week, and by an additional 200 units in the third week. If this trend of accelerating reduction continues, which term best describes the mathematical shape of the function representing the product's remaining production level over time?
Analyzing Production Shutdown Data
If a function is both decreasing and concave, its curve becomes less steep as the input variable increases.
For a function that is both decreasing and concave, match each characteristic of the function to its correct mathematical description.
Interpreting Functional Properties
Modeling with Decreasing Concave Functions
A company's quarterly profit is declining. The statements below describe the amount of profit lost in three consecutive quarters. Arrange these events in a sequence that would define the company's total profit over time as a concave and decreasing function.
A function that is both decreasing and concave models a rate of change that is negative and becomes even more negative as the input value increases. This implies that the curve representing the function becomes progressively _______ as the input value grows.
A company is tracking the declining market share of one of its products. The data collected over five consecutive months is shown below:
- Month 1: 20%
- Month 2: 18%
- Month 3: 15%
- Month 4: 11%
- Month 5: 6%
Based on this data, which of the following best describes the function representing the market share over time?
Evaluating Pollution Reduction Models