If inflation is a continuous process (that is, prices rising daily or even hourly, as in a hyperinflation), calculating inflation rates at discrete intervals (such as months, quarters, and years) may be misleading. We desire a continuous analogue to the equation in problem 1. In Pt = P0ept let e represent the base of natural logarithms, p the instantaneous rate of inflation, and the other variables remain as defined earlier.

(a) Prove that p is the instantaneous rate of inflation in the preceding equation. This requires the use of calculus. You’re trying to prove the following:

p = (1/Pt)(dPt/dt)

(b) The logarithmic price change is given by p = (ln Pt - ln P0)/t Derive this equation from the immediately preceding one.

(c) If P0 = 1.00, P12 = 129.75, and the time interval between these periods is twelve months, find p, using the log price change formula.

(d) For (c), you should have gotten p = 40.5 percent per month. Now calculate the instantaneous rate of inflation per year equivalent to the instantaneous rate of inflation of 40.5 percent per month. Use the equation for log price change, but this time let P1 = 129.75 and t = 1.