Questions on Time Inconsistency

Here are some questions:

Firstly, what's the word "time inconsistency" mean in this case?

Suppose you ask the policy maker what inflation and output pair will he prefer ex ante, between (pi-star,y-bar) and (pi-high,y-bar), where pi-high is the Nash equilibrium inflation derived in lectures and is greater than pi-star the inflation target. The policy maker will readily tell you that he prefers (pi-star,y-bar), because in his loss function it gives lower inflation with no output loss. But once you let the public fix their inflation expectations at pi-star, the policy maker will no longer wish to deliver pi-star, as he wants to push output beyond y-bar by increasing inflation. When the time comes and agents decide fix their expectations at pi-star, the policy maker's optimal action is inconsistent with the plan he promises before the expectations are fixed. So ex post, what the policy maker wants to do is inconsistent with what he prefers ex ante. The public foreseeing this will not fix inflation expectations at pi-star in the first place, and we get inflation bias results.

Also in topic 7, we get output-inflation trade off but not in the topic 5. Is it because in topic 5 we consider only 1 time game and we also assume no shock but in topic 7, shock and time dimension are taking into consideration? Could you explain it intuitively?

Topic 5 uses essentially the Lucas or new classical supply function, so it is impossible to get any extra output beyond potential, unless you are willing to throw away the assumption of rational expectations. But we note that in real life, monetary policy does have real effects (on output gap, say) and it is probably due to nominal rigidities that allow the output-inflation tradeoff to exist at least in the short-run. So to be more realistic, replacing the Lucas supply curve with the New Keynesian supply function seems the most natural thing to do. The introduction of short-run output-inflation tradeoff also makes the question of optimal monetary policy much more interesting...how much short-run output benefit can you get (by active monetary policy), without generating excessive inflation expectations which will be bad for you in the long-run...and so on. So it is the fact that the inflation expectations enter with the particular lag structure according to New Keynesian story which allows us to draw implications as done in the lectures.

Even if topic 5 is extended to multi-period setup, as long as you keep the Lucas supply function, there would not be any interesting role for output stabilisation (i.e. you would not want to respond to aggregate supply shock, but let all the pain falls on output and accept no change in inflation).

In topic 7, we expect different result between rule and discretionary policy while topic 5 seems to offer indifferent result. Is that because in case of topic 5, one time game, when central banker who has similar preference to the society announces that he would commit to some rule, his main strategy is to cheat anyways? As it's the finite game, customer realized this. So output wouldn't be able stimulated, only inflation would be raised. How's this different from the topic 7 ka, why rule and discretion in topic 7 yield the different result?

Under topic 5 model, the only source of inefficiency is the policy maker's wish to target output beyond the potential. His wish to do it generates inefficiently high inflation expectations and hence actual inflation. The solution is to get rid of this vicious cycle somehow. One way is to impose a rule, so that the policy maker cannot cheat, and the inflation expectations can then be credibly controlled. Second solution is to pick someone who doesn't want to cheat (i.e. who targets only the potential output). Another related solution is to pick someone who cares infinitely more about inflation. In short, once you fix the time inconsistency problem, you've restored the efficient outcome.

Under topic 7, it's different. We showed that there is also a time inconsistency problem there, so that makes rule-type commitment valuable for essentially the same reason as in topic 5. But rules add another sort of values as well, which is more subtle. Not only can rules be used to anchor inflation expectations and solve time inconsistency problems, they can also be used to manipulate the future inflation expectations as well. And we showed that, when this is done optimally, the outcome is better than that attained by a discretionary policy maker who may target only zero output gap. The discretionary guy cannot influence future inflation expectations, so he cannot manipulate the output-inflation tradeoff facing him, so he is at a disadvantage for that reason. So the New Keynesian setup of topic 7 offers quite a few interesting implications, despite its simple dynamic form.

I have 2 Questions.For Time Inconsistency,
1. Is it necessary that at Nash equilibrium state, y has to be always y_bar when we plug in the loss function?

Under topic 5 yes...as long as you assume rational expectations. This follows from the Phillips curve alone, if Pi-e=Pi (rational expectations assumption), then we must have y=y-bar.

2. Why don't we have to minnimize our loss function with respect to y? Or Do we concentrate on only inflation?

The Phillips curve says that you face a constraint, so the moment you choose y, pi will already be automatically determined (given inflation expectations). Likewise, once you pick pi, y is already chosen implicitly. This is the same as in any consumption problem where you have a utility function and a budget set. We could minimise our loss with respect to either y or pi, as long as we get the right terms of tradeoff (from Phillips curve) into our first-order conditions.

Could you clearify about the shocks?What is the positive supply shock and negtive supply shock i.e. Ut should be positive or negative (also rise or fall) corresponding with the shocks and output in which case should be rise or fall? Finally, in AD-AS diagram, what direction will AS shift in order to repesent the shock (Ut) i.e. AS will shift to the right or left if there is a positive supply shock?

I think if there is ever a convention, by positive supply shock we normally mean a shock that lowers inflation and/or raise output. In other words, a positive supply shock means AS shifts to the right. So here, a positive supply shock in this language corresponds to u_t being negative (because u_t is the shock that adds to inflation, according to how we write our Phillips curve). But I think of this as a language issue...as long as we are clear whether u_t is moving up or down, there's no confusion.

As u_t goes up, we have higher inflation for any given output (or equivalently, lower output for any given inflation), so AS has to be shifting to the left. And I think it's conventional to call this type of shock a negative supply shock.