Problem 51 - Applied Calculus

Question

During 2008 the US economy stopped growing and began to shrink. Table \(1.25^{48}\) gives quarterly data on the US Gross Domestic Product (GDP), which measures the size of the economy.(a) Fstimate the relative growth rate (percent per year) at the first four times in the table. (b) Economists often say an economy is in recession if the GDP decreases for two quarters in a row. Was the US in recession in \(2008 ?\) $$\begin{array}{c|c|c|c|c|c} \hline t \text { (years since 2008) } & 0 & 0.25 & 0.5 & 0.75 & 1.0 \\ \hline \text { GDP (trillion dollars) } & 14.15 & 14.29 & 14.41 & 14.2 & 14.09 \\\ \hline \end{array}$$

Step-by-Step Solution

Answer

Annual growth rates were 4%, 3.4%, and -5.7%, indicating recession in last two quarters of 2008.

Step 1: Calculate GDP Changes

First, determine how the GDP changes from quarter to quarter for the first four times in the given table. We have GDPs for 0 to 0.75 years: - From 0 to 0.25 years: 14.29 - 14.15 = 0.14 trillion dollars increase. - From 0.25 to 0.5 years: 14.41 - 14.29 = 0.12 trillion dollars increase. - From 0.5 to 0.75 years: 14.20 - 14.41 = -0.21 trillion dollars decrease.

Step 2: Compute Quarterly Growth Rates

Calculate the relative growth rate for each of these changes:- For 0 to 0.25 years: \(\frac{0.14}{14.15} \times 100 = 0.99\%\).- For 0.25 to 0.5 years: \(\frac{0.12}{14.29} \times 100 = 0.84\%\).- For 0.5 to 0.75 years: \(\frac{-0.21}{14.41} \times 100 = -1.46\%\).

Step 3: Convert Quarterly Rates to Annual Rates

Assume that these quarterly rates compound for four quarters in a year to convert them into annual growth rates. Approximate the annual rate using the formula \((1 + r)^{4} - 1\) where \(r\) is the quarterly rate in decimal:- Annual rate from 0 to 0.25 years: \((1 + 0.0099)^{4} - 1 \approx 0.040\) or \(4.0\%\).- Annual rate from 0.25 to 0.5 years: \((1 + 0.0084)^{4} - 1 \approx 0.034\) or \(3.4\%\).- Annual rate from 0.5 to 0.75 years: \((1 - 0.0146)^{4} - 1 \approx -0.057\) or \(-5.7\%\).

Step 4: Check for Recession

Check if the GDP decreases for two consecutive quarters, indicating a recession. The GDP at 0.5 years is 14.41 trillion, and at 0.75 years is 14.20 trillion, and at 1.0 years it is 14.09 trillion. The GDP indeed falls for two consecutive quarters (0.5 to 0.75 and 0.75 to 1.0 years), confirming a recession in 2008.

Key Concepts

Economic Recession

An economic recession is a period when a country's economy experiences a significant decline. It's like when someone is not feeling well and slows down. In terms of the economy, this means that activities such as employment, company profits, and production of goods start going down.
A common way to detect a recession is by observing the Gross Domestic Product or GDP. If GDP falls for two back-to-back quarters (a total period of six months), economists usually agree that the country is in a recession. In 2008, the US faced such a scenario.
During 2008, GDP numbers showed a decrease over successive quarters, indicating that the economy was indeed shrinking. This situation is similar to how a person might catch a cold and feel less energetic over a few days. The only difference here is that GDP is the thermometer that checks the economy's health.

Quarterly Growth Rate

The concept of a quarterly growth rate refers to how much a particular economic indicator, like the GDP, changes over a three-month period. Each quarter is a part of the year, and observing changes in these smaller periods helps economists spot trends and make comparisons.
To calculate this rate, you compare the GDP values of two quarters. For example, if the GDP increases from 14.15 trillion to 14.29 trillion, you'd calculate the change to find the growth rate. You do this by dividing the change by the initial value and multiplying by 100 to get a percentage. Formally, this formula is:

• Growth rate = \( \frac{{\text{{Change in GDP}}}}{{\text{{Starting GDP}}}} \times 100 \)

In our example, from 0.00 to 0.25 years, the GDP changed by 0.14 trillion dollars, leading to a quarterly growth rate of 0.99%.
Such calculations help in understanding the short-term health of an economy and is like checking your progress in a task every few days.

Annual Growth Rate

The annual growth rate takes the quarterly growth rate and projects it over a full year, assuming the conditions remain constant throughout the year. This helps to understand how an economy might grow or shrink if it follows the same trends for an entire year.
To calculate this, we use the formula, \( (1 + r)^{4} - 1 \), where \( r \) is the quarterly growth rate expressed as a decimal. For instance, if you have a quarterly growth rate of 0.99%, you would convert that to decimal form (0.0099) and then plug it into the formula.
This calculation is somewhat like looking ahead in a game to predict your final score if you continue scoring at the same rate. It helps economists prepare and make policies that can adjust for upcoming challenges or make the most of expected opportunities.
This annual perspective can be particularly telling, especially when quarterly rates fluctuate, as it offers a big-picture view of economic health.

Gross Domestic Product

Gross Domestic Product, or GDP, is like the report card for a country’s economy. It sums up the value of all the goods and services produced over a specific time.
Think about everything produced in an economy \( e.g., \) cars, healthcare, technology. The total monetary value of these products and services gives the GDP.
Monitoring the GDP is crucial because it gives insight into whether an economy is growing or shrinking. A growing GDP indicates a healthy economy where businesses flourish, more jobs are available, and people usually earn more money. Conversely, a shrinking GDP suggests economic issues such as falling production and potential unemployment.
The GDP data is usually reported on a quarterly basis, allowing for frequent check-ins regarding the economy's state. In 2008, the US GDP data showed a decrease over time, hinting at the impending recession.
So, GDP works as a comprehensive measure, helping everyone, from economists to policymakers, understand the productive health of a country. It strategizes ways to sustain growth or combat recession.