Derivation of the golden-rule savings rate[edit]

The following arguments are presented more completely in Chapter 1 of Barro and Sala-i-Martin^[3] and in texts such as Abel et al..^[4]


Let k be the capital/labour ratio (i.e., capital per capita), y be the resulting per capita output (${\displaystyle y=f(k)}$), and s be the savings rate. The steady state is defined as a situation in which per capita output is unchanging, which implies that k be constant. This requires that the amount of saved output be exactly what is needed to (1) equip any additional workers and (2) replace any worn out capital.


In a steady state, therefore: ${\displaystyle sf(k)=(n+d)k}$, where n is the constant exogenous population growth rate, and d is the constant exogenous rate of depreciation of capital. Since n and d are constant and ${\displaystyle f(k)}$ satisfies the Inada conditions, this expression may be read as an equation connecting s and k in steady state: any choice of s implies a unique value for k (thus also for y) in steady state. Since consumption is proportional to output (${\displaystyle c=(1-s)f(k)}$), then a choice of value for s implies a unique level of steady state per capita consumption. Out of all possible choices for s, one will produce the highest possible steady state value for c and is called the golden rule savings rate.


An important question for policy-makers is whether the economy is saving too much or too little. Given the interconnection of s and k in steady state, noted above, the question can be phrased: "How much capital per worker (k) is needed to achieve the maximum level of consumption per worker in the steady state?"


To discover the optimal capital/labour ratio, and thus the golden rule savings rate, first note that consumption can be seen as the residual output that remains after providing for the investment that maintains steady state: ${\displaystyle c=f(k)-(n+d)k}$


Differential calculus methods can identify which steady state value for the capital/labour ratio maximises per capita consumption. The golden rule savings rate is then implied by the connection between s and k in steady state (see above).


In detail, if ${\displaystyle k^{G}}$ is the golden rule steady state level of k, then ${\displaystyle k=k^{G}}$ requires ${\displaystyle dc/dk=0}$, i.e.${\displaystyle df/dk-(n+d)=0}$


${\displaystyle {\mbox{Golden rule for capital/labour ratio: }}{\frac {df}{dk}}=(n+d)}$


The Inada conditions ensure that this rule is satisfied by a unique ${\displaystyle k=k^{G}}$ and thus produces a unique ${\displaystyle y^{G}=f(k^{G})}$. Since steady state requires a particular level of investment, i.e., saved output: ${\displaystyle i^{G}=(n+d)k^{G}}$, then the golden rule savings rate must be whatever is required to generate this;


${\displaystyle {\mbox{Golden rule savings rate: }}s^{G}={\frac {(n+d)k^{G}}{f(k^{G})}}}$


Given the rule for optimal k, this may also be expressed as:


${\displaystyle {\mbox{Golden rule savings rate: }}s^{G}={\frac {mpk^{G}}{apk^{G}}}}$


in which ${\displaystyle mpk^{G}}$ is the marginal product of capital (${\displaystyle df(k)/dk}$) at the optimal value of k and ${\displaystyle apk^{G}}$ is the corresponding average product of capital (${\displaystyle f(k)/k}$).


The actual values of ${\displaystyle k^{G}}$, ${\displaystyle y^{G}}$, ${\displaystyle apk^{G}}$, and ${\displaystyle s^{G}}$ depend upon the precise specification of the production function ${\displaystyle f(k)}$. For example, a Cobb–Douglas specification with constant returns to scale has ${\displaystyle y=f(k)=k^{a}}$, hence ${\displaystyle apk=k^{(a-1)}}$ and ${\displaystyle mpk=ak^{(a-1)}}$. This gives ${\displaystyle s^{G}=a}$ and hence ${\displaystyle k^{G}=(a/(n+d))^{1/(1-a)}}$, ${\displaystyle y^{G}=(a/(n+d))^{a/(1-a)}}$.