Problem 92 - Applied Calculus

Question

BUSINESS: Phillips Curves Unemployment and inflation are inversely related, with one rising as the other falls, and an equation giving the relation is called a Phillips curve after the economist A. W. Phillips (1914-1975). Between 2000 and 2010 , the Phillips curve for the U.S. unemployment rate \(x\) and Consumer Price Index (CPI) inflation rate \(y\) was $$ y=45.4 x^{-1.54}-1 $$ where \(x\) and \(y\) are both percents. Use this relation to estimate the inflation rate when the unemployment rate is a. 3 percent b. 8 percent

Step-by-Step Solution

Answer

Inflation is approximately 8.20% at 3% unemployment, and 2.32% at 8% unemployment.

Step 1: Understanding the Phillips Curve Equation

The given Phillips curve equation is \( y = 45.4 x^{-1.54} - 1 \), where \( x \) represents the unemployment rate in percent and \( y \) represents the inflation rate in percent. We will substitute different unemployment rates into this equation to find the corresponding inflation rates.

Step 2: Calculate Inflation Rate for 3% Unemployment

Substitute \( x = 3 \) into the equation to find \( y \): \[y = 45.4 \times (3)^{-1.54} - 1\] First, calculate \( 3^{-1.54} \).Then multiply it by 45.4.Finally, subtract 1 from the result.

Step 3: Perform Calculation for 3% Unemployment

Start by calculating \( 3^{-1.54} \).Using a calculator, you get approximately \( 0.202595 \).Then, calculate \( 45.4 \times 0.202595 \).This equals approximately \( 9.196603 \).Finally, subtract 1: \[y = 9.196603 - 1 = 8.196603\] The inflation rate when the unemployment rate is 3% is approximately 8.20%.

Step 4: Calculate Inflation Rate for 8% Unemployment

Substitute \( x = 8 \) into the equation to find \( y \):\[y = 45.4 \times (8)^{-1.54} - 1\] First, calculate \( 8^{-1.54} \).Then multiply it by 45.4.Finally, subtract 1 from the result.

Step 5: Perform Calculation for 8% Unemployment

Start by calculating \( 8^{-1.54} \).Using a calculator, you get approximately \( 0.07318 \).Then, calculate \( 45.4 \times 0.07318 \).This equals approximately \( 3.320572 \).Finally, subtract 1: \[y = 3.320572 - 1 = 2.320572\] The inflation rate when the unemployment rate is 8% is approximately 2.32%.

Key Concepts

Unemployment Rate

The unemployment rate is a key indicator that reflects the percentage of people in the labor force who are not working but are actively seeking employment and are available to start working. This metric provides valuable insight into the health of an economy. A low unemployment rate generally suggests a thriving economy with plenty of available jobs, while a high unemployment rate may indicate economic troubles and scarce job opportunities.
Understanding the concept of the unemployment rate is crucial as it directly impacts consumer spending, governmental economic policies, and individual livelihoods. Factors influencing unemployment can include:

• Economic cycles – during a recession, unemployment tends to rise.
• Technological advancements – can lead to job displacement in certain sectors.
• Government policies – such as taxes or labor laws, can affect hiring practices.

In the context of the Phillips Curve, as the unemployment rate decreases, the inflation rate often increases, suggesting an inverse relationship between these two economic variables.

Inflation Rate

Inflation rate measures the rate at which the general level of prices for goods and services is rising, and subsequently, the purchasing power of currency is falling. Central banks attempt to control inflation by adjusting interest rates and other monetary policies. Stable inflation is essential for a healthy economy.
The inflation rate is calculated as the percentage change in the price level, often reported on an annual basis. A moderate inflation rate is usually considered beneficial, encouraging spending and investment. However, too high or too low inflation can be problematic:

• High inflation erodes purchasing power and can distort spending, savers, lenders, and borrowers’ financial decisions.
• Deflation, or negative inflation, can lead to decreased consumer spending and may slow economic growth.

The Phillips Curve illustrates the typical inverse relationship between inflation and unemployment. When unemployment is low, inflation tends to rise as demand for goods increases. Conversely, high unemployment usually coincides with lower inflation.

Consumer Price Index

The Consumer Price Index (CPI) is a measure that examines the weighted average of prices of a basket of consumer goods and services, such as transportation, food, and medical care. The CPI is calculated by taking price changes for each item in the predetermined basket of goods and averaging them. Changes in the CPI are used to assess price changes associated with the cost of living. The CPI is a crucial economic indicator.

• It is often used to calculate inflation or deflation, showing how much prices have increased or decreased over time.
• It can influence government economic policy decisions, wage adjustments, and social security payments.
• Investors and business leaders use CPI data to make informed financial decisions, as changes in the cost of living may impact consumer behavior.

In the context of the Phillips Curve, CPI data is essential because it directly measures the inflation rate, one half of the inflation-unemployment relationship. Understanding how CPI relates to the inflation rate helps in interpreting the economy's broader economic conditions and trends.