Optimal Monetary Policy - NK model

There are two main distortions in the New Keynesian model:

1. markups due to imperfect competition (leads to low employment in the long run): we assume that it is controlled by the fiscal authority.

2. price dispersion due to nominal rigidities (short run): central bank's responsibility

Now we will think about what is the optimal value of interest rate. Till now IR was an exogenous variable but now we will convert it to endogenous. Objectives of CB are controlling inflation and smoothening business cycle i.e. keeping inflation gap and output gap small.

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The above-mentioned function is called the loss function. CB basically wants to minimize the loss function. "Omega" is CB's weight on the 2nd mandate. The formulae's have the square term so that when we derivate we can get the interior solution. A truly micro-founded model should have maximized household welfare. But we do not do that because (a) algebra gets way too messy (b) this objective is equivalent to the household utility maximization.

What are the two ways CB could make the choice of optimal IR:

(a) Choose the whole sequence from t=0 or CB solves entire problem at the beginning of time and commits to a policy. This is called commitment.

(b) CB solves the one period model each period. This is called discretion.

Recall the 3 equations i.e. AS (NKPC), AD (Euler), Taylors rule. The first two equations cannot be changed by the central bank since they have been obtained by the firm and household optimization. Further, from the AD function, we can observe that the only parameter which will be affected by the choice of IR will be the output gap and inflation gap.

We will discuss the discretion policy of CB now. For this, the loss function will be minimized subjected to the NKPC constraint. The IR will affect the output gap and inflation gap but at the same time, we cannot move away from the above-mentioned constraint. So, what we do is take the derivatives of constraint w.r.t. inflation gap and output gap putting the values which they wish to have in equilibrium and put it in dynamic IS curve to get the IR. 

On solving the lagrangian with the mentioned constraint wrt inflation gap and the output gap, we get the relation between the inflation gap and the output gap which depends on the omega and slope of new Keynesian Philips curve.

We can see that the loss function attains the global minimum to objective function when the inflation gap and the output gap are equal to zero. This also satisfies the FOCs, so since CB optimizes every period, they will always choose this solution. What this means is that they do not want any inflation or output gap at eqm. and it makes sense because these lead to inefficiency wedges. CB's will set therefore IR such that inflation gap and output gaps are zero because then at flexible price equilibrium there would be no deadweight losses. 

Putting these conditions in dynamic IS curve, we get IR equals natural R. But how can CB achieve IR = natural R. If CB will set this IR then the model runs into convergence issue. 

Looking at the Taylor rule, the Taylor principle states that the coefficient of inflation gap should be greater than 1. In other words, if people are expecting higher inflation then CB should increase IR more than natural R in order to stop it from exploding.

Therefore, to adhere to optimal policy we need to adhere to Taylor principle i.e. IR should be sufficiently sensitive to the increase in inflation. The way we will have our equilibrium to be the flexible price equilibrium is if our CB is extremely credible to omit any small changes in the inflation gap or output gap. In the real world, we do not know the natural R, and we are not at flexible price equilibrium. So, we do not know which direction to move in. If we just stick to Taylor rule then studies have shown that we may move towards optimal monetary policy without knowing it.