CLASS #17: POLICY GAMES: MONETARY POLICY AND THE UNEMPLOYMENT/INFLATION TRADE OFF

There is always a temporary tradeoff between inflation and unemployment; there is no permanent tradeoff. The temporary tradeoff comes not from inflation per se, but from unanticipated inflation, which generally means, from a rising rate of inflation.
Milton Friedman

• The empirical trade off between inflation and unemployment: the Phillips curve.
• The Phillips curve across different monetary policy regimes: the expectation augmented Phillips curve.
• Can monetary policy exploit this trade off?

The Phillips Curve: an empirical regularity

• Phillips 1958 article, using UK data: negative relationship between unemployment and changes in nominal wages.
• US during the 1960's: negative relationship between unemployment and inflation.
• US after 1970: the relationship disappears. Why?

• Monetary policy in the Classical Model: the Misperception theory (Lucas' islands model)
  • Say the government doubles the money supply (and people are not aware of it).
  • There is more money in the economy. At the ongoing nominal prices, demand rises.
  • If people knew about the doubling of the money supply, they would simply double the price at which they sell their products.
  • However, since they do not know, each producer may think that the observed increase in the demand for their own product is due to a change in tastes in favor of her own product, and decide to work more and increase supply:
    
    $$\ln{Y_t}=\ln{\bar{Y}}+b(\ln{P_t}-\ln{P_t^e})$$
    
    

  • The government (the central bank) cannot fool people all the time. If it tries to, people will interpret all increases in demand as due increases in the money supply, and will not change their behavior.
  • We know we obtain something very similar from surprise changes in the growth rate of money supply with sticky wages.
  • Is the Phillips Curve exploitable for policy purposes?

The Expectations Augmented Phillips Curve

From the misperception theory:

$$\ln{Y_t}=\ln{\bar{Y}}+a(\ln{P_t}-\ln{P_t^e})$$

Let us add and subtract $\ln{P_{t-1}}$ from the right hand side:

$$\ln{Y_t}=\ln{\bar{Y}}+a(\ln{P_t}-\ln{P_{t-1}}-(\ln{P_t^e}-\ln{P_{t-1}}))=$$

$$=\ln{\bar{Y}}+a(\pi_t-\pi_t^e)$$

and let us remember Okun's Law:

$$\ln{Y_t}-\ln{\bar{Y}}=-b(u_t-\bar{u})$$

We obtain a ``theoretical version" of the Phillips curve:

$$u_t-\bar{u}=-\theta(\pi_t-\pi_t^e)$$

where $\theta=\frac{a}{b}.$

• The negative relationship is there, for given inflationary expectations.
• Suppose the Central Bank tries to exploit this relationship by increasing the growth rate of money supply, and thereby increasing inflation:
  the public is at first fooled, but then changes inflationary expectations.
  The Phillips curve shifts upward.
  The surprise is over.
• The US after 1970: inflationary expectations change and the Phillips curve shifts; in the late 80's it shifts back.
• Like in the Classical misperception theory: in the long run the trade off is no longer there.

Policy Games

• Can the Central Bank exploit the Phillips curve to achieve lower unemployment?
• What is the fight for the head of the European Central Bank all about?
• Why do the media talk of ``conservative" Central Bankers, of reputation of central bankers, etc.?
• Independent central bakers.

Let us call $y=\ln(Y)$ and introduce a disturbance e in the aggregate supply equation:

$$y=\bar{y}+a(\pi-\pi^e)+e$$

Let us postulate the following link between inflation and the growth rate of money supply:

$$\pi=\mu+v$$

where v is a -velocity- disturbance in the equation. Lets us say that the Central Bank wants to maximize output (say, for political economy reasons) and minimize inflation, that is, its objective function is:

$$max_\mu \lambda(y-\bar{y})-\frac12 \pi^2$$

Of course, the Central Bank takes into account that there is something called Phillips curve out there, so that it cannot minimize both. After substituting the constraint into the objective function we obtain:

$$max_\mu \lambda(a(\mu+v-\pi^e)+e)-\frac12 (\mu+v)^2$$

The timing of events is as follows:

• The private sector forms inflationary expectations (and wages are set according to those expectations).
• The supply shock e is realized.
• The Central Bank sets $\mu$ (can respond to e).
• Monetary shock v is realized.

The private sector is contracting their wages, debt contracts... they want to predict inflation as good as they can, otherwise it is going to be bad news for somebody (for workers if inflation is above expectations, for entrepreneurs is inflation is below expectations). Say that the Central Bank could commit to an inflation rate $\pi^c$, announce it today, and have the public believe it.
Since the public is forming expectations on the basis of the Central Bank's announcement, it is clear that $\pi_t^e=\mu^c$. So nobody is fooled, and the Central Bank cannot ``run the Phillips curve".
The objective function becomes:

$$max_\mu \lambda(av+e)-\frac12 (\mu^c+v)^2$$

Of course, the best thing to do for the Central Bank is to set $\mu^c=0$, so at least it has zero inflation, even if it has no gain on the unemployment side.
Expected utility is:

$$E(U)=-\frac12 \sigma^2_v$$

Let us consider the more realistic case where the Central Bank cannot commit itself (discretionary policy). Once the private sector has formed the inflationary expectations $\pi_t^e$, the Central Bank takes them as given as chooses $\mu$ so to maximize its objective function.
The first order condition is:

$$a\lambda-\mu=0$$

which delivers:

$$\mu=a\lambda$$

Actual inflation will then be:

$$\pi=a\lambda+v$$

Agents understand the model and set expectations equal to:

$$\pi^e=a\lambda$$

What is the result?

• From
  
  $$\lambda(a(\mu+v-\pi^e)+e)-\frac12 (\mu+v)^2$$
  
  
  since $\pi_t^e=\mu=a\lambda$, the Central bank has made no progress in increasing output.
• However, inflation is higher than zero:
  
  $$\pi=a\lambda+v$$
  
  

• Expected utility is:
  
  $$E(U)=-\frac12 (a^2\lambda^2+\sigma^2_v)$$
  
  

• Why do we obtain such a perverse result?
• Time inconsistency
  If the Central Bank could commit itself in advance, it would chose zero inflation all the time. But it cannot commit, and once the private sector has chosen expected inflation the incentive to cheat is too strong. The private sector knows it, and it takes preventive action by setting inflationary expectations so high that it is not convenient for the government to cheat.
• Note that the lower $\lambda$, the lower inflation: if the Central Banker dislikes inflation a lot, the private sector does not in turn have to set expectations so high.
• Also, the lower a, the lower inflation: if the Phillips curve is vertical the Central Bank cannot cheat anyway. But the flatter it is, the greater the incentive to cheat, the greater the ``punishment" in terms of high inflation.
• Conservative Central Bankers are low $\lambda$ Central Bankers: people who dislike inflation a lot (for political or intellectual reasons)
• Repeated games: Reputation.
  The public perceives that a Central Banker who has never cheated will not cheat.
• Independent central bakers.