Problem 91 - Applied Calculus

Question

91-92. 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). Phillips used data from 1861 to 1957 to establish that in the United Kingdom the unemployment rate \(x\) and the wage inflation rate \(y\) were related by $$ y=9.638 x^{-1394}-0.900 $$ where \(x\) and \(y\) are both percents. Use this relation to estimate the inflation rate when the unemployment rate was a. 2 percent b. 5 percent

Step-by-Step Solution

Answer

Inflation is about 1.741% at 2% unemployment and 0.276% at 5% unemployment.

Step 1: Understanding the problem

We are given a Phillips curve which relates the unemployment rate \( x \) with the inflation rate \( y \) using the equation \( y = 9.638 x^{-1.394} - 0.900 \). We need to use this equation to estimate the inflation rate for different unemployment rates.

Step 1: Calculate inflation rate for 2% unemployment

Substitute \( x = 2 \) into the given equation:\[ y = 9.638 \times (2)^{-1.394} - 0.900 \]Calculate \( (2)^{-1.394} \) and then multiply by 9.638. Finally, subtract 0.900 from the result to find \( y \).

Step 2: Perform the calculations for 2% unemployment

Calculate \( (2)^{-1.394} \approx 0.274 \). Now multiply by 9.638:\[ y = 9.638 \times 0.274 - 0.900 \]\[ y \approx 2.641 - 0.900 \]\[ y \approx 1.741 \]Thus, the inflation rate when the unemployment rate is 2% is approximately 1.741%.

Step 3: Calculate inflation rate for 5% unemployment

Substitute \( x = 5 \) into the given equation:\[ y = 9.638 \times (5)^{-1.394} - 0.900 \]Calculate \( (5)^{-1.394} \) and then multiply by 9.638. Finally, subtract 0.900 from the result to find \( y \).

Step 4: Perform the calculations for 5% unemployment

Calculate \( (5)^{-1.394} \approx 0.122 \). Now multiply by 9.638:\[ y = 9.638 \times 0.122 - 0.900 \]\[ y \approx 1.176 - 0.900 \]\[ y \approx 0.276 \]Thus, the inflation rate when the unemployment rate is 5% is approximately 0.276%.

Key Concepts

Unemployment Rate

The unemployment rate is an important economic indicator that measures the percentage of the labor force that is jobless and actively seeking employment. It provides insight into the health of an economy. Understanding unemployment involves:

• Definition: It indicates the level of joblessness among people able and willing to work.
• Calculation: The unemployment rate is calculated by dividing the number of unemployed individuals by the total labor force, then multiplying by 100 to get a percentage.
• Types of Unemployment: Three main types are frictional, structural, and cyclical unemployment, each arising from different causes.

An increase in unemployment usually suggests economic challenges, while a decrease points to economic growth. Within the Phillips Curve model, a lower unemployment rate often correlates with higher inflation, revealing an inverse relationship between these economic factors.
In our exercise, a decrease in the unemployment rate from 5% to 2% led to an increase in the inflation rate, illustrating this concept.

Inflation Rate

The inflation rate is the rate at which the general level of prices for goods and services rises, eroding purchasing power. It is a key indicator of economic stability. Key components of inflation include:

• Definition: Inflation is the percentage increase in the price level from one period to another.
• Measurement: Often measured by the Consumer Price Index (CPI) or the Producer Price Index (PPI).
• Types of Inflation: Demand-pull, cost-push, and built-in inflation are the most common types, each with unique triggers and implications.
• Effects: Inflation can decrease the real value of money, influence interest rates, and affect purchasing power and cost of living.

Within the Phillips Curve context, inflation is interconnected with unemployment, showing how demand in the economy can drive price levels. For example, in the given equation, as the unemployment rate decreases, the inflation rate predicted increases, which is demonstrated by calculations in the solution when moving from a 5% to a 2% unemployment rate.

Economic Models

Economic models are simplified frameworks designed to illustrate complex economic processes. They help economists understand, predict, and communicate economic phenomena. Here's a closer look:

• Purpose: Models simplify reality to help analyze how economies function.
• Types of Models: These range from basic supply and demand models to complex mathematical representations like the Phillips Curve.
• Applications: Used in policy-making, teaching, and research to test hypotheses and forecast future economic trends.

The Phillips Curve is a classic example, representing the relationship between inflation and unemployment, suggesting that as unemployment falls, inflation could rise. This model highlights trade-offs policymakers face, like deciding between stabilizing inflation or reducing unemployment. In practical applications, such models aid in understanding and navigating complex economic environments, as illustrated by the estimation tasks in the exercise, using the Phillips Curve equation to predict inflation rates based on changing unemployment.