Question

a)Inflation erodes the value of money. Assuming an annual inflation rate of 4%, what would a 2010 dollar be worth in 2020? that is find an amount A in 2010 that would be worth $1.00 in 2010
b)Assuming inflamation rate of 4% when would a 2010 dollar be worth 50 cents?

Answer 1

A)
Due to inflation the costs of goods increase. But the value of paper currency remains same. So inflation erodes the value of money - currency.

Inflation rate = 4%        Amount in $1.00 in 2010
Every year the price goes up by 4%.  It is like interest rate compounded annually.

Let the Cost of an item: $ 1  in 2010
Its cost in 2020 - after 10 years : $ 1 (1+4/100)¹⁰ = $ 1 * 1.04¹⁰
                   = $ 1.48

Value of $ 1 in 2010 will reduce to $ 1 / 1.48  in 2020  = $ 0.67
[ this is because cost of an item increased to $1.48 from $1, the buying power of the dollar in 2020 is only equal to 67 cents as compared to 2010.]

b)

Let n be the number of years when a $1 note is worth only 50 cents.

[tex]\$ 0.50 = \frac{\$1}{(1 + 4/100)^n} \\ \\ (1 + \frac{4}{100})^n = \frac{\$ 1}{\$ 0.50} = 2 \\ \\ 1.04^{n} = 2 \\ \\ Taking\ Logarithms\ to\ base\ 10, \\ \\ n\ Log 1.04 = Log 2 \\ \\ n = \frac{Log 2}{Log 1.04} = 17.67 years \\ [/tex]