Problem 45 - 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). 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 in percents. Find the derivative of this function at each \(x\) -value and interpret your results. a. 2 percent b. 5 percent

Step-by-Step Solution

Answer

At 2% unemployment, \( \frac{dy}{dx} \approx -3.210 \); at 5%, \( \frac{dy}{dx} \approx -0.362 \).

Step 1: Understand the Function

The function given is the Phillips curve equation: \( y = 9.638x^{-1.394} - 0.900 \). This equation describes the relationship between the unemployment rate \( x \) and wage inflation rate \( y \). We need to find the derivative \( \frac{dy}{dx} \) at specific \( x \)-values to understand how \( y \) changes with respect to \( x \).

Step 2: Differentiate the Function

To find the derivative \( \frac{dy}{dx} \), apply the power rule to the term \( 9.638x^{-1.394} \). The derivative of a power function \( ax^n \) is \( nax^{n-1} \). Here, the derivative is \( -1.394 \times 9.638 \times x^{-1.394 - 1} \). The constant \(-0.900\) has a derivative of 0.

Step 3: Simplify the Derivative Expression

The derivative expression is \( \frac{dy}{dx} = -1.394 \times 9.638 \times x^{-2.394} \). Simplify the expression to find: \( \frac{dy}{dx} = -13.436752 \times x^{-2.394} \).

Step 4: Evaluate the Derivative at \( x = 2 \)

Substitute \( x = 2 \) into \( \frac{dy}{dx} = -13.436752 \times x^{-2.394} \). Calculating this gives: \( \frac{dy}{dx} = -13.436752 \times (2)^{-2.394} \approx -3.210 \).

Step 5: Interpret the Result for \( x = 2 \)

At an unemployment rate of 2%, the derivative \( -3.210 \) indicates the wage inflation rate \( y \) is decreasing at a rate of about 3.21% for each 1% increase in unemployment. This shows a steep decrease in wage inflation as unemployment is quite low.

Step 6: Evaluate the Derivative at \( x = 5 \)

Substitute \( x = 5 \) into \( \frac{dy}{dx} = -13.436752 \times x^{-2.394} \). Calculate this to get: \( \frac{dy}{dx} = -13.436752 \times (5)^{-2.394} \approx -0.362 \).

Step 7: Interpret the Result for \( x = 5 \)

At an unemployment rate of 5%, the derivative \( -0.362 \) indicates that the wage inflation rate \( y \) is decreasing at a rate of about 0.36% for each 1% increase in unemployment. This shows a smaller decrease in wage inflation as unemployment is higher.

Key Concepts

Unemployment Rate

The unemployment rate is a key indicator of the economic health of a country. It represents the percentage of the labor force that is currently without work but is actively seeking employment. Understanding the unemployment rate is crucial because:

• It reflects the availability of jobs.
• Impacts the economy's overall productivity.
• Affects consumer spending, as more people without income lead to less spending.

In the context of the Phillips Curve, the unemployment rate is crucial because it helps to analyze the trade-off between unemployment and inflation. When the unemployment rate is low, indicating more people are employed, businesses may struggle to find talent. This scarcity of labor can lead to higher wages as companies compete for employees. Conversely, when the unemployment rate is high, there is less pressure on businesses to increase wages because there are more job seekers available. Therefore, the unemployment rate is inversely related to wage inflation in the Phillips Curve, as seen in the given equation.

Wage Inflation Rate

The wage inflation rate is the rate at which wages increase over time. It can be influenced by various factors, such as demand for labor, cost of living adjustments, and economic policies. Wage inflation can affect both economic conditions and individual well-being. In the Phillips Curve scenario, wage inflation is inversely related to the unemployment rate. Here’s why:

• When unemployment is low, wages tend to rise because workers have the power to demand higher pay.
• Conversely, when unemployment is high, there are more individuals vying for the same positions, keeping wage growth subdued.

The specific relationship in our formula, given by Phillips, shows how wage inflation is a function of the unemployment rate. This model was important for policymakers to understand how they could manage labor markets and control inflation by influencing employment levels.

Differentiation

Differentiation in mathematics involves finding the derivative of a function, which represents the rate of change of a quantity. In the context of the Phillips Curve, differentiation helps us understand how changes in the unemployment rate (\(x\)) affect the wage inflation rate (\(y\)). By taking the derivative of the equation \( y = 9.638x^{-1.394} - 0.900 \), we can determine how sensitive wage inflation is to changes in the unemployment rate:

• The power rule was used for differentiation, stating that if a function is in the form \( ax^n \), its derivative is \( nax^{n-1} \).
• Applying this rule to our function, we computed the derivative: \( \frac{dy}{dx} = -13.436752 \times x^{-2.394} \).
• This derivative tells us the rate at which the wage inflation rate is changing with respect to the unemployment rate.

Understanding differentiation in this context allows economists to interpret the dynamic relationship between unemployment and wage inflation, which is pivotal for making informed economic policy decisions.