Home > Macro, Solow Model > The Golden Rule level of capital

## The Golden Rule level of capital

An increase in the saving rate allows you to reach a higher steady state level of capital per worker and output per worker.

However, if you just keep on increasing the saving rate, you start to defeat the point of growth, you want to have more output available for consumption now, as that is what determines living standards. If you have high output but everybody is saving all their income for the future then nobody has anything to spend on consumption now.

So there is a point where you can maximise the benefit for the population in terms of consumption. If you save too little, and consume too much now, you don’t increase your capital stock by enough to get much growth. If you save too much, you might increase your capital stock but you aren’t leaving enough for consumption now. Given that the economy is going to head to steady state in the long run, if you choose just the right saving rate, you can strike the right balance and optimise your living standards when you get there.

How do you do this?

First think about the basic concept that all national income is either saved or consumed.

$y=c+i$ means income per worker is equal to consumption per worker plus investment per worker (remember investment per worker depends on saving per worker). So $c = y - i$.

When we are in steady state, $c*=y*-i*$. In steady state, investment per worker is equal to depreciation per worker, so we can rewrite this as $c{*}= y{*}- \delta k{*}$.

This gives us an important idea: in steady state, consumption per worker is equal to the difference between output per worker and depreciation per worker, it is what is left of national income per worker once depreciation per worker has been taken care of. So in the Solow diagram, it is the difference between the output per worker curve, and the depreciation line.

The brown arrow shows the difference between output per worker and depreciation at different points. The difference will be at its widest when the slope of the production function (output per worker curve) is parallel to the depreciation line.

On this diagram the saving rate has been chosen so that it intersects the depreciation line at the point where we have the golden rule level of capital per worker. This is the saving rate that would get the economy into steady state at a point that would maximise consumption per worker in steady state.

The slope of the production function is the marginal productivity of capital, it tells you the amount of extra output you get from adding another unit of capital – here of course we are thinking in terms of capital and output ‘per worker’.

We can say then that the condition for the golden rule is that $MP_K = \delta$, marginal productivity of capital equals the depreciation rate, as $\delta$ is the slope of the depreciation line, $\delta k$.

An important note here is that the depreciation line giving ‘break even’ investment is actually a line of slope $\delta + g_n$ when you take into consideration population growth. As the population grows, then the amount of capital you need to ‘break even’ increases over and above depreciation, because you have to not only replace the capital that has depreciated, but you have to add some new capital for the new workers to use, if you are going to keep the same amount of capital per worker to break even. This is called capital widening, as your population grows you need to add more capital just to keep your capital per worker ratio constant. Only once you have added enough to equip the new workers with the same level of capital that the existing workers had, are you starting to get capital deepening which is increasing the overall level of capital per worker.

So in this case, the condition for the golden rule would be that $MP_K = \delta + g_n$

In the version of the Solow model that uses labour augmenting technological progress, the condition becomes $MP_K = \delta + g_n + g_{\pi}$.