How to Calculate Catch-Up Time Under Convergence
Catch-up time equals the log of the income ratio divided by the log of the ratio of growth factors, giving the years a faster-growing poor country needs to reach a richer one.
The Catch-Up Time formula
Calculator
Enter two income levels and two growth rates to get the years until the poorer country catches the richer one.
Real output per person, in the same currency and prices as the other country.
The level the poorer country is trying to reach.
Long-run average growth in real GDP per person, not one strong year.
The richer country keeps growing while the poorer one chases it.
Holding both growth rates steady, the two income levels meet in 54.06 years.
- Income ratio today
- 8 to 1
- Growth advantage
- 4%
- Income per person when they meet
- $116,674
- Does the gap close?
- The poorer country catches up
The richer country produces 8 times as much per person right now, and this ratio is what convergence closes.
The poorer country grows 4% a year faster, and only this difference drives catch-up, not either rate on its own.
Both countries pass through about $116,674 per person on the date they meet, which is far above where either sits today.
Equal growth rates freeze the income ratio forever, so the answer turns on the growth difference and nothing else.
How to calculate Catch-Up Time, step by step
- 1Put both incomes in the same units. Use real GDP per person for each country in one currency, adjusted for purchasing power, so the ratio measures output and not an exchange rate quirk.
- 2Divide the richer income by the poorer one. That ratio is the size of the gap. A ratio of 8 means the rich country produces eight times as much per person as the poor one.
- 3Take the natural log of the ratio. The log turns a multiplicative gap into a distance that steady growth rates close at a constant pace, which is what makes a closed-form answer possible.
- 4Build the ratio of growth factors. Add 1 to each growth rate as a decimal, divide the poorer country's factor by the richer country's, then take the natural log of that result.
- 5Divide the two logs. The log of the income ratio divided by the log of the growth-factor ratio gives the number of years until the two income levels are equal.
- 6Check the sign before trusting the answer. If the poorer country does not grow faster, the denominator is zero or negative and no length of time closes the gap. Convergence is a prediction to test, not an outcome to assume.
Worked example: Catch-Up Time
Country A produces $5,000 per person and grows 6% a year. Country B produces $40,000 per person and grows 2% a year. The gap is 40,000 ÷ 5,000 = 8 to 1, and ln(8) = 2.079442. The growth factors are 1.06 and 1.02, so their ratio is 1.06 ÷ 1.02 = 1.039216 and ln(1.039216) = 0.038467. Catch-up time = 2.079442 ÷ 0.038467 = 54.06 years, and at that point both countries produce about $116,674 per person. Watch what happens on the way there. After 20 years A has $16,036 and B has $59,438, so the ratio has fallen from 8 to 1 down to 3.71 to 1 while the dollar gap has widened from $35,000 to $43,402. Relative convergence and a growing absolute gap happen at the same time, which is why the two sides of this argument often talk past each other.
Catch-Up Time questions
Why does the dollar gap widen while the ratio falls?
Because a growth rate applies to a base. Two percent of $40,000 is $800 while six percent of $5,000 is only $300, so the richer country adds more dollars per year at first even though it grows more slowly. The ratio starts closing immediately; the dollar gap keeps widening until the poorer country's income has grown large enough.
Does the convergence hypothesis mean every poor country catches up?
No. The data reject that unconditional version, since many low-income countries have grown more slowly than rich ones for decades. What holds up is conditional convergence: each country moves toward its own steady state set by saving, population growth, schooling and institutions, so countries with different fundamentals converge to different levels and never meet.
Which growth rates belong in the formula?
Long-run average growth in real GDP per person, not a single year. One boom year produces a catch-up time no country would ever hit, and per person growth already nets out population change, so subtracting population growth again double counts it.
What if the two countries grow at the same rate?
The growth-factor ratio equals 1, its log is zero, and the formula divides by zero, so no catch-up date exists. Read that as the honest answer rather than a broken calculation: equal growth rates hold the income ratio fixed forever, and the gap never closes.
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