GROWTH and COMPOUNDING

LINEAR GROWTH

GDP_t = GDP₀ + A*t

where t is time and A is the addition per time period (quarter, year, etc) and GDP₀ is the intial value at t=0. E.g., in 1991 nominal GDP was 5986.2 - in ten years, 2001, it was 10082.2. For linear growth, A is calculated as

A = (10082.2-5986.2)/11 = 372

meaning that the average per year addition to nominal GDP was 372 billion in the period 1991 to 2001. Rarely do we model growth with linear growth models.

Compounded growth

GDP_t = GDP₀*(1+g)^t where GDP₀ is the intial value at time t = 0, and g the growth rate per period.

For example, nominal GDP in 2000 was 9824.6 and 10082.2 in 2001.

10082.2 = (1 + 2.62%)^1 * 9824.6

In 1991 nominal GDP was 5986.2 - in ten years, 2001, it was 10082.2 -

10082.2 = (1+5.35%)^10 * 5986.2

To calculate the 5.35%, we note

10082.2/5986.2 = (1+g)^10 or

(10082.2/5986.2)^(1/10) = (1 + g) or

g = (10082.2/5986.2)^(1/10) – 1

5.35% is the annual compounded growth rate.

Exponential growth

GDP = GDP₀*e^gt = GDP₀*exp(g*t) where g is the continuously compounded growth rate or exponential growth rate. When one speaks of a growth rate (or an interest rate) one must state just which growth rate (annual, exponential, quarterly, etc). The common assumption is that the meaning is ANNUAL compounded rate..

For the 10 year example above, we can calculate g as

g = ln(GDP_t/GDP₀)/10 or ln(10082.2/5986.2)/10 = 5.21%

Note that 5.21% exponential (or continuously compounded) rate is exactly equal to 5.35% ANNUAL rate just as the binary number 11 is exactly equal to the decimal number 3.

Exponential compounding results from compounding an annual rate n periods per year and letting n grow infinite. For example, you might have a savings account with 4% interest per year but compouned quarterly. The total interest per year is slightly more than 4% due to compounding quarterly.

(1 + .04/4)^4 = 1.04060401

If we compounded daily,

(1 + 0.04/365)^365=1.04080849

If we compounded every minute,

(1 + 0.04/525600)^525600=1.040810773

For continous compounding, we take the limit as n goes to infinity of

(1 + g/n)^n

and our math friends tell us it is exp(g). exp(.04)=1.040810774

Continuous (exponential) growth is often used because it is easy to manipulate. If we have a 'real' growth rate and an inflation rate, and we measure them in exponential terms, then we can add them. For example the 5.23% nominal rate of GDP growth above could be 3% real growth with 2.23% inflation.