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In macroeconomics, the 'classical dichotomy' refers to the idea that real and nominal variables can be analyzed separately. To be precise, an economy exhibits the classical dichotomy if real variables such as output, unemployment, and real interest rates can be completely analyzed without considering what is happening to nominal variables. In particular, this means that GDP and other real variables can be determined without knowing the level of the nominal money supply or the rate of inflation. Therefore, in an economy that exhibits the classical dichotomy, the money supply only affects nominal variables like the price level. An economy exhibits the classical dichotomy if money is superneutral; it is also likely to be quantitatively close to the classical dichotomy if money is neutral.
The classical dichotomy was central to the thinking of early economists (money as a veil). It is a feature of many classical and new classical theories of macroeconomics. Keynesians and monetarists reject the classical dichotomy, because they argue that prices are sticky. That is, they think prices fail to adjust in the short run, so that an increase in the money supply raises aggregate demand and thus alters real macroeconomic variables.

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Controversy

Mathematical representation

Controversy

Don Patinkin (1954) challenged the classical dichotomy as being inconsistent, with the introduction of the 'Real balance effect' of changes in the nominal money supply. The early classical writers postulated that money is inherently equivalent in value to that quantity of real goods which it can purchase. Therefore, in Walrasian terms, a monetary expansion would raise prices by an equivalent amount, with no real effects (employment, growth). Patinkin postulated that this inflation could not come about without a corresponding disturbance in the goods market, through the 'real balance effect'. As the money supply is increased, the real stock of money balances exceeds the 'ideal' level, and thus expenditure on goods is increased to re-establish the optimum balance. This raises the price level in the goods market, until the excess demand is satisfied, at the new equilibrium. He thus argued that the classical dichotomy was inconsistent, in that it did not explicitly allow for this adjustment in the goods market. Later writers (Archibald & Lipsey, 1958) argued that the dichotomy was perfectly consistent, as it did not attempt to deal with the 'dynamic' adjustment process, it merely stated the 'static' initial and final equilibria.

Mathematical representation

If an economy exhibits the classical dichotomy, then comparative statics analysis can be performed using a jacobian matrix in block triangular form. That is, suppose we write
:$mathbf\{J\}dy\; =\; dx$
where $dx$ represents some exogenous shocks (changes in productivity, aggregate demand, money supply, etc., ordered so that all real shocks come first), and $dy$ represents the change in the endogenous variables (output, employment, prices, etc., again listing real variables first). Then the matrix 'J' can be partitioned into submatrices as follows:
:$mathbf\{J\}=$
egin{bmatrix}
A & 0 \
B & C \
end{bmatrix}

In other words, you should be able to calculate how all the real variables change by inverting the submatrix $A$ only, thus excluding all nominal variables like money supply and prices from the analysis.