Consider the aggregate income of a given economy and call it Y.
The aggregate real output is the number of goods (and services) produced per year and is represented by Q.
The aggregate price is Y/Q = p and is called the price level so that pQ = Y.
Y can also be expressed as the product of the money stock m and money velocity V so that Y = mV
The equation of exchange is the statement that (1) pQ = mV
Income growth is measured in percentages. From the equation of exchange, it follows, then, that
(pQ)’/(pQ) = (mV)’/(mV), where f’ = df/dt
This leads to (2) p’/p +Q’/Q = m’/m + V’/V
A corollary to (2) is the equation (3) p’/p – V’/V = m’/m – Q’/Q
where the demand terms are grouped on the LHS of (3) and the supply terms are the RHS.
p , the aggregate price, is the demand for Q , the aggregate goods flow. V , the money velocity, is the demand for m , the money stock. Y is always the product of a demand term and an appropriate supply term.
If m’/m – Q’/Q = c , then it must be true that p’/p – V’/V = c as well. In other words, if supply terms grow at different rates then demand terms will change at the same differing rate.
A frequent error made in discussions of the Quantity Theory of Money is the statement that an increase in money supply leads to inflation, leaving out consideration of the other 2 terms, ie., Q’/Q and V’/V. This causes many to conclude, erroneously, that the Quantity Theory of Money is “not true”.
Real Product Growth
Real product Q is the product Q = aL where a is productivity which is defined as product per worker. L is the work force. L , in turn. is the product of population n , and employment level E , so that Q = anE , and Q’/Q = a’/a+n’/n+E’/E . Both a and n vary slowly over time so that it is possible to write
a’/a = α and n’/n = β with α,β constant so that Q’/Q = α+β+ E’/E. From this, it can be seen that Q’/Q depends directly on employment growth, E’/E.