Problem 84 - Brief Applied Calculus

Question

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

a. Inflation at 3% unemployment is 5.57%. b. Inflation at 8% unemployment is 2.17%.

Step 1: Understand the Phillips Curve Equation

The Phillips curve equation given is \( y = 45.4 x^{-1.54} - 1 \), where \( y \) is the inflation rate and \( x \) is the unemployment rate. Substitute the unemployment rate into this equation to find the inflation rate.

Step 2: Calculate Inflation for 3% Unemployment Rate

Substitute \( x = 3 \) into the equation: \[ y = 45.4 \times (3)^{-1.54} - 1 \]. Calculate \( (3)^{-1.54} \), multiply by 45.4, and subtract 1 to find \( y \).

Step 3: Perform Calculation for 3% Unemployment

First, calculate \( 3^{-1.54} \) which is approximately \( 0.1447 \). Multiply by 45.4 to get approximately \( 6.5702 \). Finally, subtract 1 to get \( y \approx 5.5702 \). Thus, the estimated inflation rate is about 5.57%.

Step 4: Calculate Inflation for 8% Unemployment Rate

Substitute \( x = 8 \) into the equation: \[ y = 45.4 \times (8)^{-1.54} - 1 \]. Calculate \( (8)^{-1.54} \), multiply by 45.4, and subtract 1 to find \( y \).

Step 5: Perform Calculation for 8% Unemployment

First, calculate \( 8^{-1.54} \) which is approximately \( 0.0699 \). Multiply by 45.4 to get approximately \( 3.1735 \). Finally, subtract 1 to get \( y \approx 2.1735 \). Thus, the estimated inflation rate is about 2.17%.

Key Concepts

Unemployment Rate

The unemployment rate measures the percentage of the labor force that is jobless and actively seeking employment. It is a key indicator of economic health and is used to assess how well an economy is performing.

A higher unemployment rate generally signals that there are more people who are unemployed and looking for work, which can imply economic distress. Conversely, a lower unemployment rate suggests fewer unemployed individuals and is often associated with a strong and growing economy.

Understanding the unemployment rate helps us to connect with the Phillips Curve concept. The Phillips Curve suggests an inverse relationship between unemployment and inflation, meaning as unemployment decreases, inflation tends to rise, and vice versa. This happens because low unemployment can increase demand for goods and services, allowing businesses to charge higher prices, which in turn increases inflation.

Inflation Rate

The inflation rate represents how much the overall price level of goods and services in an economy is increasing over a given period, usually annually. It is typically expressed as a percentage and provides a measure of purchasing power declines as prices rise.

Inflation can be caused by various factors, such as increased demand, higher production costs, or expansionary monetary policies. It is important because it affects consumers' ability to purchase goods and services, influencing the cost of living and economic planning.

According to the Phillips Curve, inflation is influenced by the unemployment rate. When unemployment is low, inflation is likely to be higher, as more people working typically means more money to spend, driving prices up. Conversely, high unemployment often results in lower inflation because there is less spending power among consumers.

Consumer Price Index

The Consumer Price Index (CPI) is a measure that examines the weighted average prices of a basket of consumer goods and services, such as transportation, food, and medical care. This index is calculated by taking price changes for each item in the predetermined basket and averaging them.

CPI is vital because it is one of the primary indicators used to measure inflation. If the CPI is rising, it means the cost of living is increasing, indicating inflation. This is crucial for households because it affects the cost of living expenses.

In the context of the Phillips Curve, the CPI can be used to communicate inflation rates. Economists use the CPI to understand how inflation behaves in relation to changes in the unemployment rate. For students studying this, interpreting the Phillips Curve's equation using the CPI helps visualize how inflation rates may respond to different levels of unemployment.