Problem 5 - Applied Calculus

Question

The gross domestic product, \(G\), of Switzerland was 310 billion dollars in 2007 . Give a formula for \(G\) (in billions of dollars) \(t\) years after 2007 if \(G\) increases by (a) \(3 \%\) per year (b) 8 billion dollars per year

Step-by-Step Solution

Answer

(a) \( G(t) = 310 \times 1.03^t \); (b) \( G(t) = 310 + 8t \).

Step 1: Understanding the Problem

We need to find a formula representing the GDP of Switzerland starting from 2007, with two different growth scenarios: a percentage growth and an absolute growth.

Step 2: Analyzing Scenario (a)

For scenario (a), the GDP increases by 3% per year. This means the GDP follows an exponential growth model. The formula for exponential growth is \( G(t) = G_0 (1 + r)^t \), where \( G_0 \) is the initial GDP (310 billion), \( r \) is the growth rate (0.03), and \( t \) is the number of years after 2007.

Step 3: Constructing the Formula for Scenario (a)

Using the formula from Step 2, we have: \( G(t) = 310 \cdot (1 + 0.03)^t = 310 \cdot 1.03^t \).

Step 4: Analyzing Scenario (b)

For scenario (b), the GDP increases linearly by 8 billion dollars per year. This implies a linear growth pattern. The formula for linear growth is \( G(t) = G_0 + kt \), where \( k \) is the yearly increase (8 billion).

Step 5: Constructing the Formula for Scenario (b)

Using the formula from Step 4, we have: \( G(t) = 310 + 8t \).

Key Concepts

Exponential Growth

When discussing GDP increases, exponential growth refers to a scenario where the GDP grows by a fixed percentage each year. This type of growth is characterized by a compounding effect, meaning each year's growth is built upon the previous year. The formula for exponential growth is given by:
\[ G(t) = G_0 (1 + r)^t \]

• \( G_0 \) represents the initial value, which in our case is the GDP of Switzerland in 2007, amounting to 310 billion dollars.
• \( r \) is the growth rate expressed as a decimal. For example, a 3% growth rate is represented as 0.03.
• \( t \) is the time in years since 2007.

When growth is exponential, even a modest increase rate like 3% can lead to significant increases over time. Each year the base increases, so does the increment derived from the percentage, leading to what we call compound growth. For instance, in scenario (a), the formula becomes \( G(t) = 310 \times 1.03^t \). This model is widely applicable in real-world economics, reflecting how investments can grow over time.

Linear Growth

Linear growth offers a much simpler model of increase where the GDP increases by a constant amount each year. Instead of compounding on the previous year's value, it simply adds the same quantity annually. The formula for linear growth is:
\[ G(t) = G_0 + kt \]

• \( G_0 \) is again the initial GDP, which is 310 billion dollars in 2007.
• \( k \) is the constant increase, given as 8 billion dollars per year in this scenario.
• \( t \) is the number of years elapsed since 2007.

In this model, every year contributes the same increase to the total GDP, making it easier to predict future amounts. For example, in scenario (b), the formula to calculate GDP becomes \( G(t) = 310 + 8t \). While exponential growth is more common in financial and economic contexts, linear growth can often provide a more straightforward approximation of future values, especially over shorter periods.

Economic Mathematics

Economic mathematics involves mathematical techniques and models to solve economic questions. It provides a framework for analyzing data, understanding economic theories, and predicting future trends. Core tools in this area include:

• Exponential and Linear Models: These are used to represent different types of growth, as seen in GDP growth models.
• Formulas and Equations: The ability to create and manipulate formulas such as \( G(t) = G_0 (1 + r)^t \) for exponential growth or \( G(t) = G_0 + kt \) for linear growth is fundamental.
• Statistical Analysis: This includes measuring economic variables and trends over time.

Understanding economic mathematics allows economists and students to interpret complex interactions within a country's economy. It provides insight into how small annual increases can cumulatively lead to significant economic changes. Mastering these tools is crucial for anyone looking to delve deeper into economics, providing the analytical skills to evaluate scenarios like GDP changes effectively.