I think I have it.
Suppose that real income does not impact the demand to hold money. That will lead to Kling's result--nominal income adjusts with real income. The quantity of money and the demand to hold money only impact the price level.
To see this, start with the standard approach. The demand for money depends on the price level, real income, the interest rate on other assets, the interest rate on money, and other unspecified things--
+ + - +
Md = Md(P, y, Rb, Rd, X)
(I haven't figured out superscripts on my blog editor yet.)
If the demand for money is a demand for real money balances--a command over goods and services--then this can be written--
Md = P md(y, Rb, Rd, X)
The nominal demand for money is proportional to the price level and the real demand for money.
If the demand for money is also proportional to real income, then:
Md = Pyk(Rb, Rd, X)
Where "k" is the "Cambridge k," introduced by Alfred Marshall. Marshall actually defines it in terms of nominal income, Y, where Y = Py.
Md = kY
This approach is only sensible if money is a normal good. The demand for money (or really the demand for the flow of services from the real quantity of money) is positively related to income. And further, that money is exactly on the cusp between necessity and luxury, so that the demand for money (services) is unit income elastic.
If that it not true, then you are left with Md = Py k(y, Rb, Rd, X), which is not a simplification. At some point, I saw empirical evidence that the income elasticity of money demand was .9. Of course, that all depends on exactly how money is measured, but that was enough for me. Why should it be exactly one? And so, sometimes I use "k" and "V" to communicate to other economists, but I understand that they must be used with care.
But, let us stick with tradition.
Md = Pyk(Rb, Rd, X).
Marshall mentions that k probably depends on interest rates, as shown here, but says that the effect is small and he will ignore it. I have often thought that it is no coincidence that in my integral calculus class we used "k" to refer to the constant found when finding the anti-derivative. Anyway, I think interest rates should appear as an opportunity cost of holding money, (Rb - Rd), which is negatively related to real money demand.
In equilibrium, Ms = Md. Leaving off the subscript for the quantity of money--
M = Pyk.
M(1/k) = Py
And so, we see that if V is defined as 1/k, and k is derived from money demand, then we have the equation of exchange.
MV = Py
It is an equilibrium condition following from Ms = Md.
Since V depends on the interest rate on assets other than money, the interest rate on money, and any number of other unspecified things, it is hardly a constant. To my way of thinking, monetary disequilibrium generally has financial effects that impact the opportunity cost of holding money, and so "correct" monetary disequilibrium almost immediately, while interfering with the ability of interest rates to coordinate saving and investment--the Wicksell approach.
But back to Kling.
Suppose that y doesn't impact money demand.
Md = P md(Rb-Rd, X)
In equilibrium, Ms = Md
So, P = M/md
Nominal income still equals the price level multiplied by real income, Y = Py.
So, Y = (M/md) y
Changes in real income lead to proportional changes in nominal income. Changes in the quantity of money lead to changes in the price level.
With V = Y/M, as an identity rather than an equilibrium condition, then,
V = y/md.
Velocity is proportional to real income, given the demand for real money balances.
And, of course, V = Py/M. So, if P is given because of price stickiness, and real income is given by productive capacity, them V must be inversely proportional to the quantity of money? Right? Wrong!
The fundamental problem with Kling's view is that in reality the demand for money does depend on real income. (It isn't just on the cusp between being a normal and an inferior good with an income elasticity of zero.)
More importantly, in disequilibrium, shortages or surpluses of money will still disrupt production if the price level is sticky. I have no simple algebra to describe that process. The process is that if there is a shortage of money at the current price level, each individual can obtain more money by spending less. Nominal expenditure falls. Firms sell less. The usual way to respond to lower sales is to reduce production and prices. Prices are assumed to be sticky (and let us say, stuck for now.) And so, output falls.
With the standard, more realistic approach, the reduced output and income results in lower money demand, and so the demand for money drops, resolving the disequilibrium. Ceteris paribus, productive capacity has not changed, and so there are at least notional surpluses of goods and resources, and continued downward pressure on prices. As prices make the needed adjustment, real output and real income recover.
With Kling's version, however, reduced output and income has no impact on money demand. Until prices drop enough to bring the real supply of money to the demand for money, the contraction continues. Of course, this shows why the assumption in so absurd. Nothing prevents real income from falling to zero. So people will continue to demand the same amount of services from holding money, while entirely giving up the services provided by food, clothing, and shelter. Not likely. Of course, while absurd globally, that doesn't mean that the demand for money couldn't possibly be independent of income, or even that money cannot be inferior, over some ranges.
So let me grant that Kling is right. If the demand to hold money is independent of real income, and real income has fallen due to a drop in productive capacity, and nominal income has fallen in proportion to real income at the existing price level, increasing the money supply, causing monetary disequilibrium, and so causing the price level and nominal income to rise to its previous level would be undesirable.
However, I don't believe that this situation applies in the real world. Further, my view of current conditions is that the real demand for money rose, nominal expenditure fell. Because prices are sticky, real expenditure fell. Real output fell with real expenditure and it is currently well below the admittedly depressed level of productive capacity. Increasing the quantity of money and nominal expenditure will return real output to productive capacity. And further, as a policy rule, stable growth of nominal expenditure, and particularly, expected stable growth of nominal expenditure, is not a bad macroeconomic environment for adjustment to shifts in productive capacity.