The Phillips Curve: a deep dive

The Phillips Curve: Unemployment and Inflation

Background: A. W. Phillips and the Phillips Curve

• In 1958, A. W. Phillips, a New Zealander economist, analyzed data on unemployment and wage rate changes (inflation).
• The relationship he found was later termed the Phillips Curve.
• This lecture extends the theoretical model of unemployment and wages to account for the Phillips Curve, showcasing how data informs economic theory.

Labor Market and Wage Determination

• Wage Determination equation: $W = P^eF(u, z)$, where:
  • $W$ = Nominal wage
  • $P^e$ = Expected price level
  • $u$ = Unemployment rate
  • $z$ = Catch-all term for other wage determinants
• Specific form for $F(u, z)$:
  $F(u, z) = 1 - \alpha u + z$, where $\alpha$ is the sensitivity of wages to unemployment.
• Therefore, the wage equation becomes:
  $W = P^e(1 - \alpha u + z)$

Price Level and Inflation

• Price Determination equation: $P_t = (1 + m)W$, where:
  • $P_t$ = Price level at time t
  • $m$ = Markup over labor costs
• Substituting the wage equation into the price determination equation:
  $P<em>t = (1 + m)P^e</em>t(1 - \alpha u_t + z)$
• Dividing both sides by the previous period's price level ($P<em>{t-1}$): $\frac{P</em>t}{P<em>{t-1}} = \frac{P^e</em>t}{P<em>{t-1}}(1 + m)(1 - \alpha u</em>t + z)$
• Inflation is defined as:
  $\pi<em>t = \frac{P</em>t - P<em>{t-1}}{P</em>{t-1}} \Rightarrow \frac{P<em>t}{P</em>{t-1}} = 1 + \pi_t$
• Expected inflation is defined as:
  $\pi^e<em>t = \frac{P^e</em>t - P<em>{t-1}}{P</em>{t-1}} \Rightarrow \frac{P^e<em>t}{P</em>{t-1}} = 1 + \pi^e_t$
• Substituting inflation into the equation:
  $1 + \pi<em>t = (1 + \pi^e</em>t)(1 + m)(1 - \alpha u_t + z)$
• Solving for $\pi_t$ gives the Phillips Curve.

Phillips Curve: General Form

• The derived Phillips Curve equation:
  $\pi<em>t = \pi^e</em>t + (m + z) - \alpha u_t$
• Components and implications:
  • Expected inflation ($\pi^e$) directly affects actual inflation ($\pi$).
  • Increases in the markup ($m$) increase inflation.
  • Factors increasing wage determination ($z$) increase inflation.
  • Increases in the unemployment rate ($u$) decrease inflation.

Original Phillips Curve

• Assumptions:
  • Inflation varies around an average, $\bar{\pi}$ (pi bar).
  • Inflation is not persistent; last year's inflation is not a good predictor of this year's inflation.
• These assumptions imply that agents expect inflation to be the average value:
  $\pi^e_t = \bar{\pi}$
• The Original Phillips Curve:
  $\pi<em>t = \bar{\pi} + (m + z) - \alpha u</em>t$
• This suggests a trade-off for policymakers: high unemployment with low inflation versus high inflation with low unemployment.
• This relationship held throughout the 1960s.

The Breakdown of the Original Phillips Curve

• The negative relationship observed in the 1960s did not hold in the subsequent decades.

Inflation Expectations and the Phillips Curve

• In the 1960s, inflation expectations were steady and time-independent, described as "anchored":
  $\pi^e_t = \bar{\pi}$
• After the 1960s, inflation became more persistent; high inflation in one year was likely followed by high inflation the next year. Wage setters changed how they formed inflation expectations, and inflation expectations became de-anchored.
• De-anchored inflation expectations:
  $\pi^e<em>t = (1 - \theta)\bar{\pi} + \theta\pi</em>{t-1}$, where $\theta$ (theta) is the weight placed on last year’s inflation (a number between 0 and 1).
• When $\theta = 0$, inflation expectations are anchored.
• As $\theta$ increases, inflation depends more on last year's inflation:
  $\pi<em>t = [(1 - \theta)\bar{\pi} + \theta\pi</em>{t-1}] + (m + z) - \alpha u_t$
• When inflation expectations are completely unanchored ($\theta = 1$), wage setters put no weight on the average inflation.
• The Phillips Curve relationship becomes:
  $\pi<em>t = \pi</em>{t-1} + (m + z) - \alpha u<em>t$$\pi</em>t - \pi<em>{t-1} = (m + z) - \alpha u</em>t$
• Unemployment determines the change in inflation rather than the level of inflation itself.

Accelerationist Phillips Curve

• The relationship between the change in inflation and the unemployment rate is described by the Accelerationist Phillips curve:
  $\pi<em>t - \pi</em>{t-1} = (m + z) - \alpha u_t$