This article is an excerpt from Abolish Money (From Economics)! A description of the contents of the book is found here.
The problem with this standard “all else equal” (ceteris paribus for people who want to sound sophisticated) phrasing is hand-waving. We need a more formal definition before we discuss the idea; as otherwise, we end up debating our own personal definitions of what that phrasing means.
I grabbed the text Monetary Economics: Theory and Policy by Bennett T. McCallum off my bookshelf, and looked up the definition of money neutrality.
Within Section 5.6, he discusses what he refers to as the “Classical Model.” Money neutrality is defined in two parts (on page 95).
- “The other experiment that we consider at present is that of an exogenous, policy-induced increase in the stock of money from M0 to M1.” [Note: should be superscripts; could not easily get that formatted properly here...]
- “In the new equilibrium, then, the values of y,r,n, W/P, and M/P are the same as before the increase in M. It follows that the values of P [the price level] and W [wages] must have risen in proportion to M, for otherwise either W/P or M/P would have changed. In short, the increase in the money stock changes all nominal variables in the same proportion and results in no change in any of the real variables. This property of the system is referred to as money neutrality. In the classic model, in other words, we have a case of the neutrality of money [emphasis in the original].”
Some might argue the above phrasing is just a long-winded way of writing out “increase the money supply, holding all else equal.” Not so; the key is that we need to specify how the money supply changes in the first place.
Furthermore, the model discussed did not have a well-defined concept of time. We can extend the definition of money neutrality to short-run and long-run neutrality.
- If money is neutral in the short run, the mathematical relationships described above will hold over all time periods (no matter how short).
- If money is neutral in the long run, the above mathematical relationships may not hold over short periods of time, but the time series tend to converge to the predicted values.
What makes this definition useless when talking about real-world economies is that it assumes that “money” is exogenous; it is an external variable set at an arbitrary level by policy makers. This definition is meaningless in a model where the money supply is endogenous. We need to find some other definition, which ends up having important practical differences.
I have seen a lot of post-Keynesians object to the notion of long-run money neutrality, presumably because that is what the Monetarists believed. However, entering into that argument based on this definition requires accepting that money is exogenous – which runs counter to post-Keynesian doctrine. Since this definition does not apply to real-world economies, it makes no sense to argue whether it is true or not.
Neutral Endogenous Money?I am unaware of any attempt to define money neutrality when the money supply is endogenous. However, it is clear that it is very difficult to do so.
Since the level of money is determined by the model and the exogenous variables, we are in a position that we cannot hold “all else equal”: we must change an exogenous variable in order to change the money stock within the model economy.
We could attempt to use the following definition.
(Failed Definition) Fix two scenarios that are defined by two differing sets of exogenous variables. Money is neutral if the real variables within the economy always converge to the same levels, while the level of money can be different in the two scenarios.
This definition fails because it is rather obvious that we can find scenarios in which real variables will obviously converge towards different levels. Examples:
- If one scenario were that the country involved launches a nuclear war that annihilates all life on Earth, one might expect that economic output would be somewhat lower than the case where war was avoided.
- If the two scenarios involved the government taxing 20% of GDP versus 90% of GDP (note that taxes are imposed in real terms), supply side economists would insist that total output would be lower in the high tax case.
In other words, we cannot allow for any possible changes to exogenous variables.
I can think of two reasonable replacements for money neutrality that appears to capture the intent of the idea.
- Interest rate neutrality. Shifting the expected path of the policy rate by a fixed amount does not have a long-term effect on real variables.
- Inflation target neutrality. Assuming that we are in an inflation-targeting regime, changing the inflation target level does not have a long-term effect on real variables.
In the first case, why do we care about monetary policy if it has no real effects? Do we really believe that it is impossible for a central bank to cause a recession by pursuing a policy of ultra-high interest rates? Moreover, from the perspective of a mainstream economist, it is hard see how we can permanently change the level of interest rates without changing something else. If the central bank’s reaction function resembles something like a Taylor Rule, such a change would end up being equivalent to changing the inflation target (the second case). 
The second possibility appears more plausible. For example, would we expect greater long-term Canadian prosperity if the target inflation rate is 3% instead of 2%? 
Even if a small change in the inflation target did matter, we have no ability to know what the exact outcome would be, and so we end up with presumed “inflation neutrality” because of our ignorance. Of course, there are presumably limits – a sustained inflation rate of 10% might lead to quite different economic outcomes than 3%. (The theory being that there is a psychological aversion to “high” rates of inflation, and people waste resources on strategies designed to protect themselves from it.)
Velocity and Money NeutralityOne possibility is to express long-run money neutrality in terms of velocity: that is, the velocity of money tends to revert towards some “equilibrium” or “steady state” value. This seems to be equivalent to what is wanted for money neutrality: if one scenario results in having double the money supply of the other, nominal GDP would also be double.
Such behaviour is predicted by some Stock-Flow Consistent (SFC) models. The argument is that economic actors tend to want to hold financial assets near target levels that are some multiple of their nominal incomes. (This is known as a stock-flow norm, as the desired level for the stock of financial assets is a multiple of the flow of nominal income.) As a result, the ratios of stocks of financial asset divided by nominal GDP typically tend towards steady state values. In the case of money, this ratio is the reciprocal of the velocity of money, and hence the velocity converges towards a steady state value.
Since this is just a restatement of a stock-flow norm, it is unclear why “money neutrality” should be privileged as a special theoretical concept.
Concluding RemarksOnce we accept that the level of money balances is determined by private sector behaviour, it is nearly impossible to come up with a satisfactory definition for money neutrality. The concept is purely an artifact of the mistaken classical view that the money supply is exogenous.
 Bennett T. McCallum, Monetary Economics: Theory and Policy, Macmillan Publishing Company, 1989.
 Technically, we could shift the interest rate generated by a Taylor Rule by changing the real interest rate term within the equation to a different value from the model’s natural real rate of interest. However, the net result would be a persistent miss of the inflation target, and so is mathematically equivalent to a change in the inflation target while using the correct natural rate of interest.
 Some theories suggest that there would be a difference. Mainstream economists are currently excited about the lower zero bound for interest rates, and raising the inflation target would allegedly lower the risk of the policy rate hitting 0%. Alternatively, many post-Keynesians invoke Verdoorn’s Law, which I would summarise as: greater short-term growth increases long-term growth rates, since businesses will invest more, raising productivity. Although reasonable, it is also possible that raising the inflation target could just raise inflation expectations (and administered prices) without affecting real variables.