Problem 46 - Brief Applied Calculus

Question

45-46. 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 in percents. Find the derivative of this function at each \(x\) -value and interpret your results. a. 3 percent b. 8 percent

Step-by-Step Solution

Answer

The derivatives are approximately -2.427 at 3% and -0.325 at 8% unemployment.

Step 1: Understand the Given Function

We are given the Phillips curve equation \( y = 45.4x^{-1.54} - 1 \). This equation models the relationship between the unemployment rate \( x \) and the CPI inflation rate \( y \). We need to find the derivative of \( y \) with respect to \( x \) at specific \( x \)-values: 3 and 8 percent.

Step 2: Find the Derivative of the Function

To find the derivative \( \frac{dy}{dx} \), we apply the power rule. The function can be rewritten as \( y = 45.4x^{-1.54} - 1 \), which means we differentiate each term.- The derivative of \( 45.4x^{-1.54} \) is \( -1.54 \times 45.4 \times x^{-1.54-1} \). Calculate the coefficient: \(-1.54 \times 45.4 = -69.916\).- Therefore, \( \frac{dy}{dx} = -69.916x^{-2.54} \).The derivative of \(-1\) is 0, which does not affect the calculation.

Step 3: Substitute x-values and Solve

We now substitute the given \( x \)-values into the derivative to evaluate \( \frac{dy}{dx} \).**a. When \( x = 3 \) percent:**\[ \frac{dy}{dx} = -69.916 \times 3^{-2.54} \]Let's compute \( 3^{-2.54} \) and then multiply it by \(-69.916\):- Calculate \( 3^{-2.54} \) to find it as approximately 0.0347.- Multiply: \( \frac{dy}{dx} = -69.916 \times 0.0347 \approx -2.427 \).**b. When \( x = 8 \) percent:**\[ \frac{dy}{dx} = -69.916 \times 8^{-2.54} \]Let's compute \( 8^{-2.54} \) and then multiply it by \(-69.916\):- Calculate \( 8^{-2.54} \) to find it as approximately 0.00464.- Multiply: \( \frac{dy}{dx} = -69.916 \times 0.00464 \approx -0.325 \).

Step 4: Interpret the Derivative Results

The derivative \( \frac{dy}{dx} \) measures the rate at which the inflation rate \( y \) changes with respect to changes in the unemployment rate \( x \).- At \( x = 3 \) percent, \( \frac{dy}{dx} \approx -2.427 \): A 1% increase in unemployment corresponds to an approximate 2.427% decrease in the inflation rate.- At \( x = 8 \) percent, \( \frac{dy}{dx} \approx -0.325 \): A 1% increase in unemployment corresponds to an approximate 0.325% decrease in the inflation rate.

Key Concepts

Phillips curve

The Phillips Curve is an economic theory that illustrates the inverse relationship between inflation rates and unemployment rates within an economy. Originally proposed by economist A. W. Phillips in 1958, this concept has become a fundamental tool in understanding macroeconomic dynamics.

According to the Phillips Curve, when unemployment is low, inflation tends to be high, and vice versa. This happens because low unemployment means more people have jobs and income, leading to increased demand for goods and services. This increase in demand can cause prices to rise, hence higher inflation.

• This relationship suggests a trade-off between inflation and unemployment in the short run.
• Policymakers often use it to predict the impact of economic policies.
• Helps in understanding periods of economic stability and instability.

However, in the longer term, the relationship can become inconsistent due to changes in inflation expectations. This means that over time, factors like people's expectations of future inflation can alter this inverse relationship.

Unemployment rate

The unemployment rate is a crucial indicator of economic health in any society. It measures the percentage of the labor force that is jobless and actively seeking employment. Understanding the unemployment rate helps in analyzing the Phillips Curve's implications.

When the unemployment rate is high, it typically means fewer people are working and possibly earning lower incomes, leading to decreased overall demand in the economy. In the context of the Phillips Curve, a higher unemployment rate is associated with lower inflation as demand drops.

• It serves as a key metric for evaluating the effectiveness of government policies.
• Helps understand the economic well-being of a population.
• Vital for identifying labor market trends and economic cycles.

It’s important to note that not all unemployment is bad, as some level of unemployment is considered natural and necessary for healthy economic dynamism. This concept includes frictional unemployment, which occurs when people transition between jobs or enter the workforce.

Consumer Price Index (CPI)

The Consumer Price Index (CPI) measures the average change over time in the prices paid by consumers for goods and services. It's a vital economic indicator that helps understand inflation, which is crucial when examining the Phillips Curve.

The CPI is used to assess price changes associated with the cost of living. By examining the Phillips Curve, we see how CPI inflation is inversely related to unemployment rates. As unemployment decreases, more money is available for spending, often leading to increases in demand and, consequently, higher CPI.

• It’s used to adjust wages and pensions against inflation.
• Helps economists gauge the purchasing power of a currency.
• Aids in making informed economic policy decisions.

When CPI shows a drastic rise, it can indicate inflation is increasing significantly, affecting the Phillips Curve's depiction of unemployment and inflation interplay.

Inverse relationship

An inverse relationship is one in which two variables move in opposite directions. In the context of the Phillips Curve, the inverse relationship refers to the way inflation and unemployment typically move in opposing directions.

In simpler terms, when unemployment falls, inflation tends to rise, and when unemployment rises, inflation tends to fall. This inverse relationship is pivotal in policy-making, helping economic policymakers navigate the balance between monetary policies that can impact unemployment and inflation.

• This concept is crucial for setting interest rates.
• Helps forecast potential economic changes and challenges.
• Aids businesses in strategic planning based on economic trends.

However, it must be noted that the inverse relationship shown by the Phillips Curve can be influenced by other factors, such as external economic shocks or supply-side changes, leading to periods where the inverse trend may not hold true.