Together, these definitions yield the so-called equation of exchange Now, define m, v, x, p, and q as the natural logarithms of M, V, X, P, and Q.
Then the equation becomes These quantities are functions of time, t, which we will take to be an integer which counts the quarters of years.
is defined by If we apply the forward difference operator, we get and so The velocity of money changes due to changes in technology and regulation.
Thus one approximates The velocity term is not intended to reflect current conditions in the business cycle.
for an extended period, then the growth rate of real GDP will be near to its long-run average,
And thus that the growth rate of nominal GDP will be close to their sum However, it is not obvious what that desired value of inflation should be.
McCallum takes the long-run average rate of growth of real GDP to be 3 percent per year which amounts to on a quarterly basis.