The Golden Rule Level of Capital

The steady state level of capital per worker that maximizes steady state consumption $c^*$ is called the Golden Rule level of capital and is denoted $k^{*}_{gold}$ . We have seen previously that consumption is maximised for a capital stock of 1.43 in our economy, which is hence the Golden Rule level.

When comparing steady states, we must keep in mind that more capital has two opposing effects on steady state consumption. Whilst more capital means more output, it also means more output must be used to replace capital which depreciates.

If the capital stock is below the Golden Rule level, the production function is steeper than the $δ k^*$ line, so consumption (which is the gap between the two curves) grows with $k^*$. In other words, an increase in the capital stock raises output more than depreciation, so consumption rises.

In contrast, if the capital stock is above the Golden Rule level, the production function is flatter than the $δ k^*$ line, so consumption shrinks as $k^*$ rises. An increase in the capital stock will reduce consumption as the increase in output will be smaller than the increase in depreciation.

At the Golden Rule level of capital $k^{*}_{gold}$, the slope of the production function (or $MPK$) is equal to the slope of the depreciation function (which is $δ$). Hence, the Golden Rule is described by:

$MPK = δ$

We can verify this graphically by varying the steady state level of capital: