By the Old Keynesian's and the New

How does monetary policy affect the real economy in the New Keynesian models? To answer this question, we need to understand how monetary policy pins down inflation. I didn't recognize these two questions can be pursued separately until I read Cochrane's "Determinacy and Identification with Taylor Rules". The key is to recognize that we can have money as unit of count in a frictionless economy. New Keynesian models are examples of cashless economy, and money is the unit of count, and nominal variables are well-defined in absence of price stickiness.

Therefore, to understand how monetary policy pins down inflation, the New Keynesian Phillips Curve (NKPC) is not essential. Gali (2008) claims that:"The NKPC determines inflation given a path for the output gap, whereas the DIS equation determines the output gap given a path for the (exogenous) natural rate and the actual real rate. In order to close the model, supplement those two equations with one or more equations determining how the nominal interest rate i evolves over time, i.e., with a description of how monetary policy is conducted." However, the key point is that three differential equations do not uniquely pin down three processes. "Determinacy in the new-Keynesian model does not fundamentally reply on frictions". The fundamental source of indeterminacy is the absence of initial condition.

The key insight of Cochrane's critique is that New Keynsian model selects a certain initial condition (or boundary condition, or expectation) such that the inflation does not go to infinity as time goes by. Economically, it sounds odd. It means that the agents will choose a current inflation such that it will not explode in the future. In reality, inflation as a nominal variable does explode. In a model with price stickiness, exploding inflation will lead to exploding real variables, which seems to violate transversality condition. However, this is not a result. It is merely an assumption that price will always be sticky even if inflation is exploding. This assumption is not realistic since we do see price becomes more and flexible when inflation explodes, and we do see a currency is abandoned when inflation goes out of control, which basically means perfect price flexibility. In the real world, exploding nominal variables do happen, and it never leads to exploding real variables. So ruling out exploding paths is not sensible.

Exploding paths are not only sensible, they are also interesting to study. Exploding path does not necessarily leads to exploding inflation. The central bank can revert to a non-exploding path half way. The Fed in the 70s definitely allows inflation to grow out of bound, and then Volker reverted it. I feel this is a more accurate description than Fed under Volker chooses a Taylor rule such that the current inflation has to be certain value, otherwise it will explode. It was exploding before Volker, and Volker stopped the explosion.

So what pins down the initial condition? Cochrane suggests fiscal theory of price level. From the experience of Volker disinflation, it seems to be the right answer. How does Volker disinflation happen? Volker refused to monetize government bond (printing money to buy bond to keep price high and interest low), so nominal interest rate sky-rocketed. High interest rate generated political pressure to cut deficit. Inflation decreased as a result. Suppose the congress did not cut deficit, will disinflation still happen? Following Cochrane, let us think in the perfect foresight friction-less framework, without ruling the explosive equilibria. In this framework, real rate is exogenously pinned down by endowment processes, and current inflation is pinned down by nominal rate from the Fisher equation.  Face current high inflation, Fed unexpectedly hikes nominal rate, which leads to higher inflation, which in turn leads to even higher nominal rate. Inflation explodes rather than recedes. Current New Keynesian's approach will tell you a very different story: "Agents realize the inflation has to go down to certain level, otherwise the inflation will explode." The inflation was exploding before Volker. He cannot threaten the economy by make it explode.

Fiscal theory assumes an exogenous path of real prime surplus and current nominal bond level. This pins down the current price level. The government can choose a future nominal bond level, which pins down future price level. The change of two dates pins down inflation and nominal interest rate. In essence, choosing nominal bond level and choosing nominal interest rate is equivalent, given exogenous real surplus. So we can still think the central bank is following an interest rate rule. The only difference is that it uniquely pins down inflation. Comparing fiscal theory and Taylor rule-New Keynesian equilibrium determination, the only difference is that fiscal theory specifies the path of real prime surplus(Specifying nominal bond level is equivalent to specifying nominal interest rate). Therefore, it is all too transparent to understand why Taylor rule does not uniquely pin down the price and inflation: given a path of nominal rate(or how nominal rate react to inflation), there are many ways that the government can vary its a path of real prime surplus. Those path where inflation goes out of bound could simply because that the government issues too much nominal debt without backing by increase of real surplus. A key point for empirical research: Government does not need to run deficit to set off  an inflation. Government only needs to sell too many nominal bonds. In fiscal theory, nominal government debt is unit of count. If you have too many, the unit will change. Nominal interest rate is the spread in which the total amount of debt grows: A higher nominal rate seems to help to fuel an inflation, since high nominal rate increases refinancing cost, so the next period nominal bond level goes even higher. Paradoxically, a low nominal rate seems to stop inflation. Less controversially, a low real deficit also stop inflation.

Mathematically, fiscal theory is an interest rate rule with the initial conditions. So the dichotomy of monetary dominance and fiscal dominance is not hard to distinguish mathematically. If one believes Taylor rule alone does not pin down equilibrium, then one always need fiscal theory. Whether i. the congress chooses real surplus and nominal debt level, and let the market to sort out nominal rate, or ii. the Fed chooses nominal debt level and nominal rate, and force the congress to choose real surplus to avoid default, or iii. the congress chooses real surplus and Fed chooses nominal rate, let the market to sort out nominal debt level is hard to distinguish mathematically. Empirically, we can assume that congress does not want cut spending, so a cut means it was dominated by the Fed.

The new Keynesian's intuition is quite different from old in selecting equilibrium. In old Keynesian models, today's inflation is pinned down by yesterday's inflation and output. Old Keynesian needs nominal rate to over-react to inflation, so that today's output falls to cancel the shock to the inflation process (which is unit root). Otherwise, the equilibrium path will be exploding. In New Keynesian models, today's inflation depends on expectation of tomorrow's inflation. An over-reacting monetary policy will send the inflation to infinity, except the current inflation is exactly at the saddle point. By the old Keynesian and the new, I have to say that these two intuitions cannot be more different.

(the above graph is a simulation of King(2000) with the initial inflation and output slightly deviate from the saddle point (0,0). As shown, it leads to explosion.)