Accounting Identities

Krugman says that certain accounting identities are laws that are true, but then he gives laughably incomplete examples that are nowhere near true.  The incompleteness of the first is obvious because he immediately contradicts it with his second "identity".
(1)  Savings = Investment       (S=I)

Which he immediately contradicts with his second law:
(2)  Savings-Investment = NetExports      (S-I=NX)

In reality, both "identities" are woefully incomplete.  The true identity is:
(3)  Savings-Investment = NetExports + liquidity changes      (S-I=NX-l)

The last term (l) is the increase in the money supply due to a decreasing gap between savings and lending.  If I increase my savings by putting cash in my mattress, that increase in savings is subtracted from the money supply and has no impact upon investment.  Similarly, if the bank that has my savings  increases its reserves without lending it out, then there is a decrease in investment without any change in savings.  Or if the bank stops lending to businesses and buys government bonds instead, then there is no impact on investment because government spending is irrationally immune to reductions in the interest rate.  Increased spending on government bonds just increases their prices (which lowers their interest rates) and this absorbs liquidity because the supply of government bonds is perfectly inelastic with respect to the interest rate. 

The perfectly inelastic supply of bonds means that greater expenditures are spent buying exactly the same quantity of bonds and this reduces the liquidity of the economy by the area of the rectangle between the dotted lines at P1 and P2. 

Krugman notes that the first identity, (S=I), is often misinterpreted to come up with Say’s Law and/or the Treasury view.  The second identity, (S-I=NX), is often misinterpreted to come up with the doctrine of immaculate transfer.  But these misinterpretations happen because our identities are incorrect.  The Say's Law/Treasury view would be impossible if we used a true identity that does not leave out important leakages, and the doctrine of immaculate transfer would be much less likely to arise because changes in liquidity have obvious impacts upon exchange rates. 

Occam's Razor should not be a reason to sacrifice the additional complexity at the altar of simplicity.  An equation with four variables is still much simpler than most of the equations used in economics or even the rest of science.  The quintessential beautiful equation, E=MC², may appear to have only three variables, but it also has an exponent and the speed of light is really the product of two variables (time and distance), so it isn't really any simpler than S-I=NX-l.  And it is better to start with the complete equation and cross off variables in special cases rather than start with the special cases.  In a closed economy, NX=0 and in a perfect economy l=0, and then you get S=I, but this is often passed off as the basic way that the economy works when it is always and everywhere a completely unrealistic simplification.  Why make the unrealistic simplification the base case rather than use the realistic equation as the base case?

The l adds in the Wicksellian concept that S≠I and that whenever there is suddenly a big increase in the gap between S and I, the real interest rate is out of whack and that causes a recession.  That is a pretty important concept.  It is the link between old-fashioned (non-monetary) Keynesianism and Monetarism. It is the link between Y=C+I+G+NX and MV=PY.  

Now what about MV=PY?  Is something important left out of that too?  The actual measurements of these variables seem to contradict the equation.  Is it measurement error or are we leaving an important variable out of the "identity" again?  Y=C+I+G+NX seems to match its measurements pretty well.