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Philips curve says that inflation and unemployment were inversely correlated. In the 1970s, stagflation happened. If Philips curve hold true, stagflation is impossible.
What were the factors that caused the Philips curve to stop working during the 1970s when the economy was hit by stagflation?
The general way people figured the Phillips curve worked was that a shock in aggregate demand or fiscal stimulus would cause labor demand to increase as government spending generated growth, which makes labor more scarce and causes firms to compete for workers by raising nominal wages. Wage costs then rise, and the firm passes on some of the cost to the consumer by raising the prices of their products.
So in this way, unemployment and inflation become tradeoffs. The problem is then this can be easily exploited, but also easily noticed. Lucas's critique states that people's expectations of inflation--and future inflations--will affect price levels as well. So if a policy maker tried to create inflation all the time to lower unemployment, what would happen is that the Phillips curve would actually shift around, so the tradeoff would hold, but you would be getting worse tradeoffs as people's expectations shifted.
By raising inflation constantly by the central bank and by politicians, you'd fall into a liquidity trap (when raising inflation fails to decrease interest rates and stimulate and economy). The reason why is because if everyone expected the rise in inflation, there'd be no reason to hire more workers, since real demand is staying the same. So instead of a short run rise in output, firms just went straight to raising prices instead of hiring more people.
That said, today, Lucas's critique has its limitations. Even transparent policies that are fully known to increase inflation by firms can still boost output in the short run. This is probably because after the 1970s in America, the new head of the Federal Reserve, Paul Volcker, decided that the central bank should commit more heavily to a set inflation rate in the long run, so that their policies would be more credible.
We can set up a game-theoretic model for optimal monetary policy, as such:
$$\min_{u, \pi} V(u, \pi) = -(u^2 + \pi^2)$$
The central bank wants to minimize a combination of unemployment and inflation. Their constraint is a simple Phillips curve:
$$u = u^* - k(\pi - \pi^e)$$
Where $k > 0$ is some weight for the degree of the tradeoff ($-k$ is the slope of the Phillips curve). $u^*$ is the natural rate of unemployment and fixed, while $\pi^e$ is the expected inflation.
Comparing the Ramsey and Nash equilibria of this game, where consumers and firms set expectations of inflation and the central bank tries to game that, we'll find that if firms trust the central bank to be honest about what inflation they are targeting, the central bank will always have the incentive to lie, if only acting for one period.
Substitute the Phillips curve into the central bank's maximization problem:
$$\begin{align} & V = [u^* - k(\pi_t - \pi^e)]^2 + \pi^2 \\ & \frac{\partial V}{\partial\pi} = -2[u^* - k(\pi_t - \pi^e)]k + 2\pi = 0\\ & \implies \pi_{\text{opt}} = \frac{k}{1+k^2}(u^* + k\pi^e)\\ & u_{\text{opt}} = u^* - k(\pi - \pi^e) \\ & \implies u_{\text{opt}} = \frac{1}{1+k^2}u^* + \frac{k}{1+k^2}\pi^e \\ \end{align}$$
Now assume the private sector knows the Fed's optimal monetary policy problem, and set $\pi^e = \pi_{\text{opt}}$. Say the Fed is credible.
$$\pi_{\text{opt}} = \pi^e = \frac{k}{1+k^2}(u^* + k\pi^e)$$ $$\boxed{\pi^e = ku^*}$$
$$u_{\text{opt}} = \frac{1}{1+k^2}u^* + \frac{k}{1+k^2}\pi^e$$ $$= \frac{1}{1+k^2}u^* + \frac{k}{1+k^2}ku^*$$ $$\boxed{u = u^*}$$
These are the best responses for the Fed...if they are worried about telling the truth. Investigating Ramsey's time inconsistency problem, the Fed can technically do better. Suppose it announces $\pi = 0$ as a target.
$$\pi_{\text{opt}} = \frac{k}{1+k^2}u^*$$ $$u_{\text{opt}} = \frac{1}{1+k^2}u^*$$
You can verify that the Fed has an incentive to deviate in a one-period game.
