Gross Domestic Product (GDP) is the market value of all...

Gross Domestic Product (GDP) is the market value of all final goods and services produced within a country during a given time period. The following fifthdegree polynomial approximates the per capita GDP (the average GDP per person) for the United States for the years 1933 to 1950 $g(x)=0.294 x^{5}-12.2 x^{4}+169 x^{3}-912 x^{2}+2025 x+4508$ where $g(x)$ is in 1996 dollars and $x$ is the number of ycars since $1933 .$ Note that when dollar amounts are measured over time, they are converted to the dollar value for a specific base year. In this case, the base ycar is $1996 .$ (Source: Economic History Scrvices) (a) Use this model to calculate the per capita GDP (in 1996 dollars) for the years $1934,1942,$ and 1949 What do you observe? (b) Explain why this model may not be suitable for predicting the per capita GDP for the year 20024 (c) Use your graphing utility to find the year(s), during the period $1933-1950$, when the GDP reached a local maximum.

Gross Domestic Product (GDP) is the market value of all final goods and services produced within a country during a given time period. The following fifthdegree polynomial approximates the per capita GDP (the average GDP per person) for the United States for the years 1933 to 1950 $g(x)=0.294 x^{5}-12.2 x^{4}+169 x^{3}-912 x^{2}+2025 x+4508$
where $g(x)$ is in 1996 dollars and $x$ is the number of ycars since $1933 .$ Note that when dollar amounts are measured over time, they are converted to the dollar value for a specific base year. In this case, the base ycar is $1996 .$ (Source:
Economic History Scrvices)
(a) Use this model to calculate the per capita GDP (in 1996 dollars) for the years $1934,1942,$ and 1949 What do you observe?
(b) Explain why this model may not be suitable for predicting the per capita GDP for the year 20024
(c) Use your graphing utility to find the year(s), during the period $1933-1950$, when the GDP reached a local maximum.

Key Concepts

Extrapolation and Model Limitations

Extrapolation is the process of estimating values outside the range of the data used to build a model. Economic models like polynomial approximations are generally valid only within the time period for which they were developed, as trends and underlying conditions can change, making predictions far outside the original data range potentially unreliable.

Local Maximum

A local maximum in a function is a point where the function value is higher than those immediately around it. In economic models, identifying a local maximum can help pinpoint a year or period during which a particular economic indicator, such as GDP, peaked relative to its nearby observations.

Polynomial Modeling

Using polynomial functions to model economic data allows for the approximation of complex trends over a limited time period. Such models are constructed to fit historical data, and parameters are chosen based on observed values, although they may not capture underlying economic mechanisms or long-term trends.

Per Capita GDP

Per capita GDP is the average economic output per person, calculated by dividing the total GDP by the population. It is useful for comparing the economic well-being of different populations or tracking changes in an economy’s standard of living over time.

Gross Domestic Product (GDP)

GDP represents the total monetary value of all final goods and services produced within a country over a specific time period. It is a primary indicator used to gauge the economic health of a nation, reflecting both the size and the performance of an economy.

Base Year Adjustment

When comparing dollar values over time, it is important to adjust for inflation by converting amounts into the dollars of a specific base year. This standardization allows for meaningful comparisons by accounting for changes in the purchasing power of money.

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