Calculating average inflation rate per year

[ainster31]

Homework Statement

An economist has predicted that for the next 5 years, annual inflation will be 8%, and then there will be 5 years at a 6% inflation rate. This is equivalent to what average price change per year for the entire 10-year period?

Homework Equations

The Attempt at a Solution

This is the solution given:

To buy $1 worth of goods today will require:
F = P (F/P, f%, n) n years hence.
F = $1 (1 + 0.08)⁵ = $1.47 5 years hence.

For the subsequent 5 years the amount required will increase to:
$1.47 (F/P, f%, n) = $1.47 (1 + 0.06)⁵ = $1.97

Thus for the ten year period $1 must be increased to $1.97. The average price change per year is:
($1.97 - $1.00)/10 yrs = 9.7% per year

Isn't this wrong? You can't just divide the average price change per year because it compounds, right?

I've tried verifying the answer and it appears wrong to me:

$$F=$1(1+0.097)^{ 10 }\\ F=$2.52$$

which is not equal to the $1.97 we were expecting.

 

[SteamKing]

The OP states that the inflation rate is 8% for the first 5 years, and then the rate drops to 6% for the next 5 years. These were not the numbers used in your calculations.

 

[ainster31]

Yeah, it seems the textbook messed that up too. I've corrected the numbers.

 

[Ray Vickson]

Homework Statement

An economist has predicted that for the next 5 years, annual inflation will be 8%, and then there will be 5 years at a 6% inflation rate. This is equivalent to what average price change per year for the entire 10-year period?

Homework Equations

The Attempt at a Solution

This is the solution given:

To buy $1 worth of goods today will require:
F = P (F/P, f%, n) n years hence.
F = $1 (1 + 0.08)⁵ = $1.47 5 years hence.

For the subsequent 5 years the amount required will increase to:
$1.47 (F/P, f%, n) = $1.47 (1 + 0.06)⁵ = $1.97

Thus for the ten year period $1 must be increased to $1.97. The average price change per year is:
($1.97 - $1.00)/10 yrs = 9.7% per year

Isn't this wrong? You can't just divide the average price change per year because it compounds, right?

I've tried verifying the answer and it appears wrong to me:

$$F=$1(1+0.097)^{ 10 }\\ F=$2.52$$

which is not equal to the $1.97 we were expecting.

Why are you using an 'arithmetic' average (0.97/10) in a problem having 'geometric' growth?

 

[ainster31]

Why are you using an 'arithmetic' average (0.97/10) in a problem having 'geometric' growth?

Yeah, this textbook is terrible. It has way too many errors. I'm not sure how high these authors were when they wrote this textbook:

Donald G. Newnan is Professor Emeritus of Industrial and Systems Engineering at San Jose State University.

Ted G. Eschenbach is a consultant and Professor Emeritus of Engineering Management at the University of Alaska Anchorage.

Jerome P. Lavelle is Associate Dean in the College of Engineering at North Carolina State University.

 

[haruspex]

Yes, it's quite obvious that the answer must be between 6% and 8%. In fact, because the two periods are the same there's a very easy shortcut.

 

[Chestermiller]

Try $(1+i)^{10}=1.97$

 

[haruspex]

Try $(1+i)^{10}=1.97$

More simply, (1+i)² = 1.06*1.08.

 

[Chestermiller]

More simply, (1+i)² = 1.06*1.08.

Ha! Very cute.

Chet