structural:expansion
The equations of Keynesian and classical economics

Keynes's initial statement of his economic model (in Chapter 14) is based on his Chapter 13 theory of liquidity preference. His restatement in Chapter 18 doesn't take full account of his Chapter 15 revision, treating it as a source of 'repercussions' rather than as an integral component. It was left to John Hicks to give a satisfactory presentation. Equilibrium between supply and demand of money depends on two variables – interest rate and income – and these are the same two variables as are related by the equation between the propensity to save and the schedule of the marginal efficiency of capital. It follows that neither equation can be solved in isolation and that they need to be considered simultaneously.

The 'first postulate' of classical economics was also accepted as valid by Keynes, though not used in the first four books of the General Theory. The Keynesian system can thus be represented by three equations in three variables as shown below, roughly following Hicks. Three analogous equations can be given for classical economics. As presented below they are in forms given by Keynes himself (the practice of writing r as an argument to V derives from his Treatise on money).

Here y is written as a function of N, the number of workers employed; p is the price (in money terms) of a unit of real output; V(r) is the velocity of money; and W is the wage rate in money terms. N, p and r are the 3 variables we need to recover. In the Keynesian system income is measured in wage units and is therefore not a function of the level of employment alone since it will also vary with prices. The first postulate assumes that prices can be represented by a single variable. Strictly it should be modified to take account of the distinction between marginal wage cost and marginal prime cost.

The classics took the second equation as determining the rate of interest, the third as determining the price level, and the first as determining employment. Keynes believed that the last two equations could be solved together for Y and r, which is not possible in the classical system. He accordingly concentrated on these two equations, treating income as 'almost the same thing' as employment on p. 247. Here we see the benefit he has gained by simplifying the form of the consumption function. If he had written it (slightly more accurately) as C(Y,p/W), then he would have needed to bring in the first equation to get a solution.

If we wish to examine the classical system our task is made easier if we assume that the effect of the interest rate on the velocity of circulation is small enough to be ignored. This allows us to treat V as constant and solve the first and third equations (the 'first postulate' and the quantity theory) together, leaving the second equation to determine the interest rate from the result. We then find that the level of employment is given by the formula

y ′ (N) y (N) = W M ^ V {\displaystyle {\frac {y'(N)}{y(N)}}={\frac {W}{{\hat {M}}V}}}.

The graph shows the numerator and denominator of the left-hand side as blue and green curves; their ratio – the pink curve – will be a decreasing function of N even if we don't assume diminishing marginal returns. The level of employment N ̂ is given by the horizontal position at which the pink curve has a value of W / (M ^ V) {\displaystyle W/({\hat {M}}V)}, and this is evidently a decreasing function of W.