Phillips Curve: Inflation-Unemployment Trade-off Calculation

The Original Phillips Curve

In 1958, economist A.W. Phillips published a study of the UK economy that uncovered a remarkably stable, inverse empirical relationship between the rate of change of money wages and the unemployment rate. This relationship was soon extended to price inflation and dubbed the "Phillips Curve."

The initial finding suggested a simple policy menu for governments: they could choose a combination of unemployment and inflation. If they were willing to tolerate higher inflation, they could achieve lower unemployment, and vice-versa.

The economic intuition behind this is straightforward:

• Low unemployment implies a "tight" labor market. Firms must compete for a limited pool of workers, which gives workers more bargaining power to demand higher wages.
• To maintain their profit margins, firms then pass these higher wage costs on to consumers in the form of higher prices, leading to inflation.
• Conversely, high unemployment implies a "slack" labor market, weakening workers' bargaining power and easing wage pressures, which in turn leads to lower inflation.

This simple relationship can be expressed with an equation like:

where ${\pi }_t$ is the inflation rate, $u_t$ is the unemployment rate, $c$ is a constant representing other factors affecting inflation, and $\alpha$ is a positive parameter that captures the steepness of the trade-off.

The Role of Expectations and the Breakdown of the Curve

This simple, stable trade-off appeared to hold for a time, but it famously broke down during the 1970s. Many economies, including the US, experienced "stagflation"—a painful combination of high unemployment and high inflation, something the original Phillips curve suggested was impossible.

Economists like Milton Friedman and Edmund Phelps had already predicted this breakdown. Their crucial insight was that the original curve had omitted a key variable: inflation expectations. Workers and firms care about real wages, not just nominal wages. If they expect inflation to be high, they will build those expectations into their wage and price-setting decisions. A worker will demand a higher nominal wage increase if they expect prices to rise quickly, just to maintain their purchasing power.

This leads to the modern, expectations-augmented Phillips curve. The following video from MIT OpenCourseWare provides a comprehensive and rigorous explanation of its derivation and evolution.

As the video explains, the modern Phillips curve is best expressed as a relationship between the unemployment gap and unanticipated inflation:

Let's unpack this equation:

• ${\pi }_t$ is the actual inflation rate.
• ${\pi }_t^e$ is the expected inflation rate. The term on the left, ${\pi }_t-{\pi }_t^e$, is the "inflation surprise."
• $u_t$ is the actual unemployment rate.
• $u_n$ is the natural rate of unemployment. The term in the parentheses, $u_t-u_n$, is the "unemployment gap"—the deviation of unemployment from its natural rate.
• $\alpha$ is a parameter that measures how responsive inflation is to the unemployment gap.

This equation tells us that when unemployment is at its natural rate ($u_t=u_n$), inflation will be equal to expected inflation (${\pi }_t={\pi }_t^e$). To push unemployment below its natural rate, policymakers must accept inflation that is higher than what people expected.

Adaptive Expectations and the Long-Run Phillips Curve

A common and simple way to model expectations is to assume they are adaptive, meaning people expect this year's inflation to be the same as last year's: ${\pi }_t^e={\pi }_{t-1}$. Substituting this into our equation gives:

The term ${\pi }_t-{\pi }_{t-1}$ is the change in inflation. This form of the equation has a profound implication: there is no longer a trade-off between the level of inflation and unemployment. Instead, the trade-off is between the change in inflation and unemployment.

To keep unemployment below its natural rate (i.e., keep $u_t<u_n$), inflation must continuously accelerate (${\pi }_t>{\pi }_{t-1}$). This is why this is sometimes called the accelerationist Phillips curve.

This insight leads to the distinction between the short-run and long-run Phillips curve.

• Short-Run Phillips Curve (SRPC): For a given level of expected inflation, there is a downward-sloping trade-off.
• Long-Run Phillips Curve (LRPC): In the long run, unemployment must return to its natural rate, $u_n$. Expectations will eventually adjust to any sustained level of inflation. Therefore, the LRPC is a vertical line at $u_n$. There is no long-run trade-off.

The graph below illustrates this dynamic. An attempt to hold unemployment at a level below the natural rate (moving from A to B) leads to higher inflation. This raises inflation expectations, which shifts the entire short-run curve upwards (e.g., from ${\pi }^e=0$ to ${\pi }^{e^{\prime }}$). The economy moves from B to C, and then potentially to D and E, with ever-accelerating inflation.

Calculating the Inflation-Unemployment Trade-Off

Now we can put this model to work. The primary use of this equation is to calculate the cost of disinflation—the policy of reducing the rate of inflation. The equation shows that to reduce inflation (${\pi }_t<{\pi }_{t-1}$), unemployment must rise above its natural rate ($u_t>u_n$). The amount of excess unemployment required to reduce inflation by one percentage point is known as the sacrifice ratio.

Let's work through a problem. Suppose an economy is described by the following Phillips curve:

Question: How much cyclical unemployment is necessary to reduce inflation by 5 percentage points in one year?

The following video provides a clear, step-by-step solution to this exact type of problem.

Let's formalize the calculation shown in the video.

1. Identify the natural rate of unemployment ($u_n$): The natural rate is the unemployment rate at which inflation is stable, meaning ${\pi }_t-{\pi }_{t-1}=0$.

   $$0=-0.5(u_t-0.06)$$

   Solving for $u_t$ gives $u_t=0.06$, or $u_n=6\%$.

2. Calculate the required unemployment for disinflation: We want to reduce inflation by 5 percentage points, so the target change in inflation is ${\pi }_t-{\pi }_{t-1}=-0.05$.

   $$-0.05=-0.5(u_t-0.06)$$

3. Solve for $u_t$:

   $$\frac{-0.05}{-0.5}=u_t-0.06$$ $$0.1=u_t-0.06$$ $$u_t=0.1+0.06=0.16$$

   The unemployment rate must rise to 16% for one year to achieve this 5-point reduction in inflation.

4. Calculate the cyclical unemployment: Cyclical unemployment is the difference between the actual unemployment rate and the natural rate.

   $$\text{Cyclical Unemployment}=u_t-u_n=16\%-6\%=10\%$$

   This means the economy needs to endure 10 percentage points of unemployment above the natural rate for one year to bring inflation down by 5 points.

Practice Problem

Now it's your turn. Use the methods we've just covered to solve a similar problem. This problem comes from a set of practice questions and uses slightly different notation, but the logic is identical.

Conclusion

In this lesson, we have formalized the crucial, dynamic relationship between inflation and unemployment. We saw how the simple trade-off first observed by A.W. Phillips gave way to a more sophisticated model that incorporates the central role of inflation expectations. In doing so, we connected back to the concept of the natural rate of unemployment from our previous lesson. Most importantly, we have used the modern Phillips curve equation to perform a concrete policy analysis: calculating the unemployment cost required to achieve a desired reduction in inflation.

Key Takeaways:

• The original Phillips curve suggested a stable, long-run trade-off between inflation and unemployment.
• This relationship broke down when inflation expectations became de-anchored.
• The expectations-augmented Phillips curve, ${\pi }_t-{\pi }_t^e=-\alpha (u_t-u_n)$, shows that the trade-off is between the unemployment gap and unanticipated inflation.
• In the long run, the Phillips curve is vertical at the natural rate of unemployment ($u_n$), meaning there is no long-run trade-off.
• We can use the Phillips curve equation to calculate the amount of cyclical unemployment needed to achieve a specific disinflation target.

In our next lesson, we will broaden our macroeconomic perspective by defining potential output, calculating the output gap, and identifying phases of the business cycle. This will allow us to see how short-run fluctuations in unemployment and GDP are connected, setting the stage for building more comprehensive models of the economy.