Home > Harrod-Domar Model, Macro > The per-capita Harrod-Domar model

## The per-capita Harrod-Domar model

We can extend the basic Harrod-Domar model by taking into account the population size. This is important because it is per capita income that gives us a relative idea of living standards in each country. We might hear that China has become the world’s second largest economy, and its overall GDP (national income) is higher than most developed countries, but when you divide the total national income by the population you find that China has a much lower per capita income.

To take population into consideration (and the fact that population will grow over time) we can go back to the equation $\theta Y_{t+1}=(1-\delta)\theta Y_t +sY_t$.

Define population as $P_t$ and divide that equation throughout by it, $\frac {\theta Y_{t+1}}{P_t}=\frac{(1-\delta)\theta Y_t}{P_t} +\frac{sY_t}{P_t}$.

Now we have population in there can start thinking in terms of per capita income rather than absolute income. The way we will do this is to use small letters, ie define $y_t = \frac{Y_t}{P_t}$ so the equation above becomes $\frac {\theta Y_{t+1}}{P_t}=(1-\delta)\theta y_t +sy_t$.

This is tidier but the left hand side (LHS) is a bit of a nuisance because we have Y in terms of year t+1 and P in terms of t, so we can’t just use the small letter to express it in per capita terms. So there is a trick to deal with this LHS, multiply top and bottom by the same thing, ie multiply it by $\frac{P_{t+1}}{P_{t+1}}$.

This means you get $\frac {\theta Y_{t+1}}{P_t+1} \frac {P_{t+1}}{P_t}=(1-\delta)\theta y_t +sy_t$.

Now we are in per capita terms for year t+1, so we have $\theta y_{t+1} \frac {P_{t+1}}{P_t}=(1-\delta)\theta y_t +sy_t$.

Now divide both sides by $\theta y_t$ to get $\frac{y_{t+1}}{y_t} \frac {P_{t+1}}{P_t}=(1-\delta) +\frac{s}{\theta}$.

Look carefully at those terms on the LHS. $\frac{y_{t+1}}{y_t}$ is basically 1 + the rate of per capita income growth. $\frac {P_{t+1}}{P_t}$ is the rate of population growth.

We can define per capita income growth as $\frac{y_{t+1}}{y_t} = (1+g*)$ and population growth as $\frac {P_{t+1}}{P_t} = (1+n)$.

This makes our equation above $(1+g*)(1+n)=(1-\delta) +\frac{s}{\theta}$ so we can say
$(1+g*)=\frac{(1-\delta)}{(1+n)} +\frac{s}{\theta (1+n)}$.

This is the Harrod-Domar equation with population growth.

Growth depends on the ability to save and invest, the ability to convert capital into output (which depends inversely on $\theta$), the depreciation rate and the rate of population growth.

Lets try an example of this.
Suppose an economy has a capital-output ratio of 2.2, a depreciation rate of 8%, a saving rate of 25% and a population growth rate of 2%, what would you predict the per capita growth rate of income to be?

Substitute the values into $(1+g*)=\frac{(1-\delta)}{(1+n)} +\frac{s}{\theta (1+n)}$ to get $(1+g*)=\frac{(1-0.08)}{(1.02)} +\frac{0.25}{(2.2)(1.02)} = 1.0134$.

So if (1+g*)=1.0134 then g*=0.0134 which is equivalent to a 1.34% per capita growth rate.