The share of households in the United States that are...

The share of households in the United States that are earning middle-class incomes has been in decline since the 1970 s. The percentage of households with annual incomes within $50 \%$ of the median income is given by $$ P(t)=50.3 t^{-0.09} \quad(1 \leq t \leq 6) $$ where $t$ is measured in decades, with $t=1$ corresponding to $1980 .$ a. What percentage of households had annual incomes within $50 \%$ of the median income in 2010? b. How fast was the percentage of households with annual incomes within $50 \%$ of the median income decreasing in $2010 ?$

The share of households in the United States that are earning middle-class incomes has been in decline since the 1970 s. The percentage of households with annual incomes within $50 \%$ of the median income is given by
$$
P(t)=50.3 t^{-0.09} \quad(1 \leq t \leq 6)
$$
where $t$ is measured in decades, with $t=1$ corresponding to $1980 .$
a. What percentage of households had annual incomes within $50 \%$ of the median income in 2010?
b. How fast was the percentage of households with annual incomes within $50 \%$ of the median income decreasing in $2010 ?$

Key Concepts

Power Function Modeling

A power function is a type of mathematical function where a constant base is raised to a variable exponent, or vice versa. In economic contexts, such functions can be used to describe non-linear relationships, such as the percentage of households earning middle-class incomes over time. They provide a way to capture trends like decay or growth in data, and their parameters determine the shape and rate of change of the modeled phenomenon.

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable into a mathematical function to calculate the corresponding value of the dependent variable. In this context, it means plugging a specific time (t) into the percentage function to determine the proportion of households with incomes within a defined range at that moment in time.

Differentiation

Differentiation is a fundamental concept in calculus used to compute the rate at which a function changes with respect to a variable. For power functions, the derivative can be determined using the power rule, which expresses how a small change in the independent variable affects the function’s output. This is crucial when assessing how rapidly economic measures, such as income distribution percentages, are changing over time.

Rate of Change Interpretation

The rate of change interpretation involves understanding the derivative as the instantaneous rate at which a variable is increasing or decreasing. In economic modeling, such as determining trends in household income distributions, the derivative provides insight into how quickly the proportion of households in a particular income bracket is changing at a given time, helping to forecast and analyze economic health.

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