Problem 83 - 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). 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 $$ y=9.638 x^{-1.394}-0.900 $$ 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

1.78% for 2% unemployment; -0.09% for 5% unemployment.

Step 1: Understanding the Equation

The equation given is \( y = 9.638 x^{-1.394} - 0.900 \), where \( y \) represents the wage inflation rate, and \( x \) is the unemployment rate. You need to substitute the unemployment rate value \( x \) into this equation to find \( y \).

Step 2: Substituting Unemployment Rate for Part a

For part a, substitute 2 for \( x \) in the equation:\[ y = 9.638 (2)^{-1.394} - 0.900 \].

Step 3: Calculating Inflation for x = 2

First, calculate \( 2^{-1.394} \), then multiply this by 9.638, and finally subtract 0.900. By calculation: \[ 2^{-1.394} \approx 0.278 \]\[ 9.638 \times 0.278 = 2.678 \]\[ 2.678 - 0.900 = 1.778 \]Therefore, the inflation rate when \( x = 2\) is approximately 1.78%.

Step 4: Substituting Unemployment Rate for Part b

For part b, substitute 5 for \( x \) in the equation:\[ y = 9.638 (5)^{-1.394} - 0.900 \].

Step 5: Calculating Inflation for x = 5

First, calculate \( 5^{-1.394} \), then multiply this by 9.638, and finally subtract 0.900. By calculation: \[ 5^{-1.394} \approx 0.084 \]\[ 9.638 \times 0.084 = 0.810 \]\[ 0.810 - 0.900 = -0.090 \]Therefore, the inflation rate when \( x = 5\) is approximately -0.09%.

Key Concepts

Unemployment

Unemployment refers to the situation where individuals who are able and willing to work cannot find a job. This is a key indicator of economic health and can be influenced by various factors such as economic conditions, technological advances, and government policies. In the context of the Phillips curve, unemployment is measured as a percentage that reflects the proportion of the labor force that is without work. It directly affects how inflation, or the rate of price increase for goods and services, behaves. There is an inverse relationship between unemployment and inflation as observed by A.W. Phillips. As unemployment decreases, inflation tends to rise due to increased demand for goods and services, which can drive up prices. Conversely, higher unemployment can lead to lower inflation because less consumer spending reduces demand pressure.

Inflation

Inflation is the rate at which the general level of prices for goods and services rises, eroding purchasing power. This economic phenomenon is crucial for understanding why purchasing power decreases over time, as prices escalate. In simpler terms, inflation means that over time, your money buys less than it did in the past. It can occur as a result of increased demand, production costs, or supply chain issues.A key component of the Phillips curve is the relationship between inflation and unemployment. Lower levels of unemployment can lead to higher inflation as companies compete for workers, driving wages up, which is often passed on to consumers in the form of higher prices. Monitoring inflation is critical for policymakers aiming to balance growth without letting costs spiral out of control. In the Phillips curve equation \( y = 9.638 x^{-1.394} - 0.900 \), \( y \) represents the wage inflation rate, showing how tightly it correlates with the unemployment rate.

Mathematical Modeling

Mathematical modeling is a method of representing real-world situations through mathematical formulas to predict outcomes and analyze relationships. This is a powerful tool in economics, allowing researchers to create theoretical frameworks that explain complex dynamics like those seen in the Phillips curve.In this context, the mathematical model provided by A.W. Phillips entails an equation that quantifies the relationship between unemployment and inflation rates. The formula \( y = 9.638 x^{-1.394} - 0.900 \) is derived from empirical data and represents a specific relationship aspect in economic contexts.By inputting different values for \( x \), the unemployment rate, one can compute \( y \), the inflation rate, demonstrating how these variables interact. Mathematical modeling thus provides insight into potential economic outcomes, helping economists and policymakers forecast and react to changing economic conditions effectively. It's a bridge between theoretical concepts and observable phenomena.