the price P(t)=5(2)^t/2 of a good is rising du to inflation, where t is time in years.

Question

a) what is the doubling time?

b.) what is the annual inflation rate?

Answer 1

Try picking some easy integer numbers that simplify the expression for Price

At time t=0 P=5 P(0)=5 t=0 is the initial time and also makes the exponent zero, so 2⁰=1

t=2 P=5(2)=10 P(2)=10 10 is double 5. Pick t=2 because it makes the exponent simplest 2/2=1

time t=2 doubles the price. We got lucky finding the t that quickly.

Doubling means a 100% inflation rate for a 2 year period.

Annual inflation rate is less. You might guess or estimate the average of 50% for one year, but with annual compound interest,or continuously compounded interest, or somewhere in between, a less than 50% rate would double the price in two years. If you just had simple interest with no compounding, then 50% would be the annual inflation rate. Simple interest means no interest on interest, no compounding.

P=Ae^rt is the formula for continuously compounded interest If P=2A and t=2, then plug those values into the formula and solve for r=interest rate. 2A=Ae^2r A's cancel. 2=e^2t Take natural logs of both sides:

ln(2)=2r

r=ln2/2= approximately .693/2 = .about 35% annual interest. if compounded continuously

But inflation isn't exactly like interest. Yet, the answer is somewhere between 50% and 35%, with those bounds.

Take the original equation P=5(2)^t/2 You want the annual inflation rate. Let t=1 for one year, solve for P

P=5 times the square root of 2 or 5(1.414) approximately P=about 7.07

7.07 is 2.07 more than the original 5. A price level of 5 grew to 7.07 in one year. Calculate the percentage increase for that one year. 7.07-5=2.07 2.07/5 = about 41%, which is a little closer to 35 than 50%