Home > Macro, Solow Model > The algebra of the Solow model

## The algebra of the Solow model

When I looked at the Harrod-Domar model on this blog I basically presented two forms,
a basic version: $g =\frac{s}{\theta} -\delta$,

and a version in per capita terms: $(1+g*)=\frac{(1-\delta)}{(1+n)} +\frac{s}{\theta (1+n)}$.

Because the Solow model is based around the concept of two factors of production (capital and labour), the presence of labour is central to the model because this is why you get diminishing returns to capital as you add more capital, so it makes sense to think about the Solow as a sort of ‘per capita’ model, by thinking in terms of output and capital per worker.

To form a Solow equation you go back to the basic equation of capital accumulation: $K_{t+1}=K_t+I_t-\delta K_t$ which can be rewritten as $K_{t+1}= (1-\delta)K_t + I_t$.

Because investment is equal to the saving rate multiplied by income, you can substitute that value here to get $K_{t+1}= (1-\delta)K_t + sY_t$.

Now we are going to use the same small letter notation for ‘per worker’ that we did in the per-capita Harrod-Domar model. Again we are treating ‘population’ and ‘workers’ as the same thing here, which is not an exactly correct assumption, but that is something we can deal with in the later versions of the Solow model.

So we can define our small letters in terms of population, ie $k_t = \frac{K_t}{P_t}$ and $y_t = \frac{Y_t}{P_t}$.

Now if you divide the equation above throughout by population you get $\frac{K_{t+1}}{P_t}= \frac{(1-\delta)K_t}{P_t} + \frac{sY_t}{P_t}$.

Now we can start using small letters, $\frac{K_{t+1}}{P_t}= (1-\delta)k_t + sy_t$.

This is tidier but the left hand side (LHS) is a bit of a nuisance because we have K 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 gives us $\frac{K_{t+1}}{P_t+1} \frac{P_{t+1}}{P_t}= (1-\delta)k_t + sy_t$ ,
which means we can use a small k on the LHS, $k_{t+1} \frac{P_{t+1}}{P_t}= (1-\delta)k_t + sy_t$.

We can define population growth as $\frac {P_{t+1}}{P_t} = (1+n)$,

so $k_{t+1} (1+n) = (1-\delta)k_t + sy_t$.

This is a tidier equation. It is saying that next year’s capital stock (per worker) depends on the amount of this year’s capital stock that is not depreciated, plus the extra capital stock we have added through investment. However, the LHS includes population growth, which is a drag on the per capita stock – the faster population grows, the faster we need to increase capital to keep per capita stock constant. These models consider population growth to be constant.

Eventually we will reach a point where the amount of new capital accumulated in a year will be just enough to keep the per capita capital stock constant, when you have taken depreciation and population growth into account. This will be the steady state level of capital per capita, which we can call k*, which is associated with a steady state level of output per capita, which we can call y*.

At the steady state point, the amount of capital per worker next year will be the same as the capital per worker this year, so $k_{t+1} = k_t = k*$ and the amount of output per worker next year will be the same as the output per worker this year, $y_{t+1} = y_t = y*$.

So at steady state, $k*(1+n) = (1-\delta)k* + sy*$.

This allows us to find an expression which links k* to y*, ie $k*(1+n) - (1-\delta)k* = sy* \Rightarrow (n +\delta) k* = sy* \Rightarrow k* = \frac{s}{(n +\delta)}y*$.

Alternatively, $y* = \frac{(n +\delta)}{s}k*$.