Growth rates: CAGR, logs and the rule of 70

A 20% rise means the new value is 1.20 times the old one. Call 1.20 the growth factor: it is 1 plus the growth rate written as a decimal. A fall of 20% has factor 0.80. When growth happens in steps, the factors multiply:

• Wages rise 20% and then fall 15%. The factor is 1.20 × 0.85 = 1.02, a net rise of 2%, not 5% (Hyderabad 2013).
• A 20% pay cut is followed by a rise of what percentage to get back to where you started? 1 ÷ 0.80 = 1.25, so 25%, not 20% (JNU 2018).
• A shopkeeper sells two pipes at ₹12 each, one at a 20% profit and one at a 20% loss. The profit is 20% of the cost of the first (₹10, so a gain of ₹2) and the loss is 20% of the cost of the second (₹15, so a loss of ₹3). Overall he is ₹1 down (JNU 2018).

In each case the two percentages are of different bases, and that is why they do not cancel. The same logic gives the two famous doubling puzzles. A lily patch doubles every day and covers the lake on day 48, so it covered half the lake on day 47, not day 24 (JNU 2012). A fortune doubles every two years and is ₹10 lakh after 50 years, so it was ₹5 lakh after 48 (JNU 2017).

If growth multiplies, then “average growth” should mean the one steady rate that gives the same final result. That is the geometric mean of the growth factors (Lesson 3), the n-th root of their product:

The GM of 2, 4 and 8 is (2 × 4 × 8)^1/3 = 64^1/3 = 4 (CUET 2025). It needs positive values: if any value is zero the whole product, and so the GM, is zero (JNU 2015). That is no problem for growth factors, which are positive as long as nothing falls by more than 100%. Hyderabad 2015 asks which mean to use on seven yearly growth rates to find the compound annual rate: the geometric, and Hyderabad 2014 gives the formula above.

The arithmetic average of growth rates is never below the compound rate, and the gap widens as the rates get more erratic. Try it. The steady path and the bumpy one below have the same arithmetic average of 6%, but they end in different places.

JNU 2012 asks: a population rises 20% in ten years; is its annual exponential growth rate 2%, more than 2%, or less than 2%? The tempting 20 ÷ 10 = 2% ignores that each year’s growth is on a bigger base. If growth compounds continuously at rate r, then e^10r = 1.20, so r = ln(1.20) ÷ 10 = 1.82%. If it compounds once a year, 1.20^1/10 − 1 = 1.84%. Either way it is less than 2%.

For small rates the two nearly agree with the rate itself, because ln(1 + g) ≈ g. That is why growth rates of logarithms are used as growth rates throughout economics. It also gives the quick way to find how long a quantity takes to double. Since ln 2 = 0.693, doubling takes about 70 ÷ g years when g is the growth rate in percent; many people use 72 because it divides more evenly.

The Taylor series guide shows where the approximation comes from. For now remember the doubling problems from the start of this lesson: at 6% an economy doubles in about twelve years, and growth that looks modest each year is large over a generation.

When an economy has sectors, its overall growth rate is a weighted mean of the sectors’ growth rates, with each sector’s share of GDP as the weight. A sector with 25% of GDP growing at 6% contributes 0.25 × 6 = 1.5 percentage points (Hyderabad 2011).

The weights themselves change, and that is what JNU 2018 tests. An economy has agriculture, industry and services at 50, 20 and 30 percent of GDP, growing steadily at 2%, 5% and 6% a year.