CLASS #18: REPUTATION AND TARGETING

• If a Central Bank cannot pre-commit, are there any solutions to the time inconsistency problem?
• What does it mean for the Banco de Mexico to acquire ``credibility''? What are the gains from credibility?
• Walsh, Chapter 8

• Let us assume that the aggregate supply equation is:
  
  $$y=\bar{y}+a(\pi-\pi^e)$$
  
  
  (for the time being there are no aggregate supply shocks), and let us postulate the following link between inflation and the growth rate of money supply:
  
  $$\pi=\mu$$
  
  
  (no velocity shocks, for simplicity).
• The objective function of the Central Bank is:
  
  $$max \sum_i=0^{\infty} \beta^i U_{t+i}$$
  
  
  where
  
  $$U_{t+i}=\lambda(y_{t+i}-\bar{y})-\frac12 \pi_{t+i}^2$$
  
  

• Note that, even if the Central Bank is forward looking, the problem is still a one-period problem: there is no link between today's action by the Central Bank and tomorrow's outcomes.
• Let us introduce such a link by assuming that the public remembers past Central Bank's actions. In particular, if in the past the Central Bank has ``cheated'', the public expects that it will cheat again this period. Formally, this implies that the public follows the rule:
  
  $$\pi_t^e=\bar{\pi}<a\lambda \mbox{ if } \pi_{t-1}=\pi_{t-1}^e$$
  
  
  
  
  $$\pi_t^e=a\lambda \mbox{ otherwise }$$
  
  
  where (remember!) $a\lambda$ is the equilibrium inflation under discretion.
• Note that this ``trigger strategy'' involves punishment only for one period. This implies that the Central Bank, when deciding if it is convenient to try to cheat the public or not, is weighting the benefit of cheating tomorrow with the costs of falling back on the discretionary equilibrium today.
• In particular, if the public expects inflation $\bar{\pi}$, the Central Bank can either choose to stick to it, or can deviate and choose inflation $\epsilon$.
• If it chooses to deviate, it is intuitive that it will choose the ``discretionary inflation'' level. This can be seen considering that the utility of the Central Bank today is:
  
  $$\lambda a(\epsilon -\bar{\pi})-\frac12 \epsilon^2$$
  
  
  and that this expression is maximized for
  
  $$\epsilon=a\lambda$$
  
  

• The utility from not deviating is $-\frac12 \bar{\pi}^2$. The overall advantage today from deviating (the ``temptation''), which is a function of $\bar{\pi}$, is:
  
  $$G(\bar{\pi})=(a\lambda)^2-a\lambda \bar{\pi}-\frac12 (a\lambda)^2+\frac12 \bar{\pi}^2=\frac12(a\lambda-\bar{\pi})^2$$
  
  

• Next period the Bank pays the cost of having deviated, and goes back to the equilibrium with discretion, for which the utility is:
  
  $$-\frac12 (a\lambda)^2$$
  
  
  So the overall cost from deviating is:
  
  $$C(\bar{\pi})=\beta(\frac12(a\lambda)^2-\frac12 \bar{\pi}^2)$$
  
  
  where the cost is discounted because it occurs next period
• The Central bank will choose not to cheat if the temptation is less than the loss
  
  $$G(\bar{\pi})<C(\bar{\pi})$$
  
  

• when is $G(\bar{\pi})=C(\bar{\pi})$? For
  
  $$\frac12(a\lambda-\bar{\pi})^2=\beta(\frac12(a\lambda)^2-\frac12 \bar{\pi}^2)$$
  
  
  which can be rewritten as:
  
  $$\frac12(a\lambda-\bar{\pi})^2=\frac{\beta}{2}(a\lambda)-\bar{\pi})(a\lambda)+\bar{\pi})$$
  
  
  or
  
  $$(a\lambda-\bar{\pi})( a\lambda-\bar{\pi}-\beta a\lambda+\beta\bar{\pi})=0$$
  
  
  This has two solutions. One is $\bar{\pi})=a\lambda$, and the other is
  
  $$\bar{\pi}=\frac{a\lambda (1-\beta)}{1+\beta}\equiv \pi^{min}$$
  
  
  If $\bar{\pi})\geq a\lambda$ there is no advantage from sticking to the rule: the public is ``too pessimistic'' anyway. If $\bar{\pi} \leq \pi^{}$ the public is ``too optimistic'': the temptation from the Bank to deviate is too great. These are clearly not equilibrium outcomes, in the sense that expectations in this range would not be rational. But for $\pi^{}\leq \bar{\pi} \leq a\lambda$ the expectations are rational: the Bank has no incentive to deviate. These are equilibria.
• What have we learned? Reputation equilibria are better than discretionary equilibria: equilibrium inflation is less than $a\lambda$
• Reputation equilibria are still worse than commitment equilibria: equilibrium inflation is greater than 0. Targeting
• Back to the one period game
• Suppose the Central Bank is forced to follow a target inflation. That is, it is penalized if it misses the target.
• The objective function becomes:
  
  $$\lambda(y-\bar{y})-\frac12 \pi^2-\frac12 h(\pi-\pi^T)^2$$
  
  
  so the Central Bank maximizes with respect to $\mu$ the expression:
  
  $$\lambda a(\mu-\pi^e)-\frac12 \mu^2-\frac12 h(\mu-\pi^T)^2$$
  
  

• the first order condition is:
  
  $$\lambda a -\mu-h(\mu-\pi^T)=0$$
  
  
  and the solution is:
  
  $$\mu=\frac{a\lambda+h\pi^T}{1+h}$$
  
  

• The lower the target, and the higher the weight placed on the target, the closer is inflation to the commitment equilibrium.