§7 The Phillips Curve and Inflation

The Theory

• Recall the wage-setting and price-setting equations:

  $$W= P^e F(u, z)$$

  $$P = (1 + m)W$$

• Assume a line functional form for $F$

  $$F(u, z) = 1 - \alpha u + z$$

• Combine all of them to solve out for $P$

  $$P = P^e (1 + m) (1 - \alpha u + z)$$

The Phillips Curve

• Divide by $P_{-1}$ both sides

  $$\frac{P}{P_{-1}} = \frac{P^e}{P_{-1}}(1 + m)(1 - \alpha u + z)$$

• Note that

  $$\frac{P}{P_{-1}} = 1 + \pi; \quad \frac{P^e}{P_{-1}} = 1 + \pi^e$$

• So that

  $$1 + \pi = (1 + \pi^e)(1 + m)(1 - \alpha u + z)$$

• Take logs on both sides and use the approximation $\ln(1 + x) \approx x$ for small $x$, to obtain the (formal) Phillips Curve:

  $$\pi = \pi^e + (m + z) - \alpha u$$

The Phillips Curve till the late 60s

• Assume

  $$\pi^e = \overline{\pi}$$

• Then we get the curve that Phillips estimated

  $$\pi = \underbrace{\overline{\pi} + (m + z)}_{\text{constant}} - \alpha u$$

The Role of Expected Inflation

• Suppose

  $$\pi_{t}^{e}=(1 - \theta)\overline{\pi}+\theta\pi_{t - 1}$$

• Then

  $$\begin{aligned} \pi_{t}&=\pi^{e}+(m + z)-\alpha u\\ &=(1 - \theta)\overline{\pi}+\theta\pi_{t - 1}+(m + z)-\alpha u \end{aligned}$$

• We used to have $\theta = 0$.

• During 70s and 80s (and early 90s) we got very close to $\theta=1$…

  $$\pi_{t}=\pi_{t - 1}+(m + z)-\alpha u$$

  or

  $$\pi_{t}-\pi_{t - 1}=(m + z)-\alpha u$$

The Phillips curve and the natural rate of unemployment

• Suppose that $\pi^e=\pi$. Replacing into the Phillips curve, we have

  $$\pi=\pi+(m + z)-\alpha u^n$$

  to imply

  $$u^n=\frac{m + z}{\alpha}$$

• Replacing back into the Phillips curve we have:

  $$\pi=\pi^e-\alpha(u - u^n)$$

— Apr 15, 2025

§7 The Phillips Curve and Inflation by Lu Meng is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Permissions beyond the scope of this license may be available at About.