The Golden Rule Capital Stock

Why the Golden Rule Is a Different Steady State

The Solow model can have different steady-state capital stocks, depending on saving and other parameters. But a steady state is not automatically the one that provides the highest sustainable consumption. The Golden Rule capital stock answers a distinct question: which steady-state capital stock maximizes consumption available per effective worker?

This distinction matters because capital must be maintained and expanded to keep pace with depreciation, population growth, and labor-augmenting technological progress. A larger capital stock raises output, but it also requires more investment just to keep the stock steady. The Golden Rule balances those two effects.

Definition: In the Solow model, the Golden Rule capital stock, denoted \(k_{\mathrm{GR}}\), is the steady-state capital stock per effective worker that maximizes steady-state consumption per effective worker, given the production function, population growth, technological progress, and depreciation.

The Model and Steady-State Consumption

Use the continuous-time Solow model with labor-augmenting technological progress. Let \(K\) be aggregate capital, \(L\) labor, and \(A\) labor efficiency. Capital per effective worker is \(k=K/(AL)\), and output per effective worker is \(y=f(k)\). Assume \(f\) is differentiable, increasing, and strictly concave over positive capital: \(f'(k)>0\) and \(f''(k)<0\). Let the population growth rate be \(n\), the growth rate of labor efficiency be \(g\), the depreciation rate be \(\delta\), and the saving rate be \(s\), with all rates measured continuously.

In this model, saving finances investment. Capital per effective worker rises with investment and falls as depreciation, population growth, and technological progress dilute capital relative to effective labor. Its law of motion is

\dot{k}=s f(k)-(n+g+\delta)k.

Write \(b=n+g+\delta\) for the break-even investment rate per unit of capital. Break-even investment is the amount of investment needed to keep \(k\) constant. Thus a steady state \(k^*\) satisfies \(s f(k^*)=b k^*\). Output per effective worker is \(f(k^*)\), while investment per effective worker is \(b k^*\). The remainder is consumption.

c^*=f(k^*)-b k^* =f(k^*)-(n+g+\delta)k^*.

Here \(c^*\) is steady-state consumption per effective worker. When \(g>0\), consumption per ordinary worker grows along the balanced growth path; it is consumption per effective worker that is constant and is maximized by the Golden Rule. The expression for \(c^*\) does not say that all capital stocks occur under one fixed saving rate. It compares the consumption sustainable at alternative steady-state capital stocks.

Worked Example: Finding a Golden Rule Stock With Square-Root Production

Suppose output per effective worker is \(f(k)=12\sqrt{k}\), population grows at \(n=0.02\), labor efficiency grows at \(g=0.01\), and capital depreciates at \(\delta=0.03\). Find the Golden Rule capital stock and its steady-state consumption per effective worker.

Method. First calculate break-even investment per unit of capital, then maximize the steady-state consumption function. This method avoids treating output itself as consumption.

The break-even rate is

b=n+g+\delta=0.02+0.01+0.03=0.06.

At a steady-state capital stock \(k\), consumption per effective worker is therefore \(c(k)=12\sqrt{k}-0.06k\). Its derivative is

c'(k)=\frac{6}{\sqrt{k}}-0.06.

Set the derivative equal to zero to find the interior maximum:

\frac{6}{\sqrt{k_{\mathrm{GR}}}}=0.06, \qquad \sqrt{k_{\mathrm{GR}}}=\frac{6}{0.06}=100, \qquad k_{\mathrm{GR}}=100^2=10{,}000.

Output at this stock is \(12\sqrt{10{,}000}=12(100)=1{,}200\). Break-even investment is \(0.06(10{,}000)=600\). Hence Golden Rule consumption is \(1{,}200-600=600\) per effective worker.

Check. The marginal product is \(f'(k)=6/\sqrt{k}\), so \(f'(10{,}000)=6/100=0.06=b\). Also \(c''(k)=-3k^{-3/2}<0\) for \(k>0\), confirming that this stationary point is a maximum. The calculation avoids the common mistake of reporting output, \(1{,}200\), as consumption.

Deriving the Golden Rule Condition

To derive the condition, regard steady-state consumption as a function of the steady-state capital stock. Break-even investment per effective worker is \(bk\), so the amount left for consumption is \(c(k)=f(k)-bk\). Differentiate with respect to \(k\). At an interior maximum, the marginal change in output must equal the marginal increase in break-even investment.

\frac{dc(k)}{dk}=f'(k)-b=0 \quad\Longrightarrow\quad f'(k_{\mathrm{GR}})=n+g+\delta.

Strict concavity makes \(c(k)\) strictly concave: \(c''(k)=f''(k)<0\). Therefore, if the first-order condition has a positive solution, it identifies the unique maximum. Intuitively, if \(f'(k)>b\), one more unit of capital adds more output than the extra break-even investment it requires, so consumption can rise by increasing capital. If \(f'(k)<b\), the additional investment required exceeds the additional output, so reducing capital raises consumption.

Theorem: For an interior Golden Rule steady state in the continuous-time Solow model with differentiable, strictly concave production, the gross marginal product of capital equals break-even investment per unit of capital: \(f'(k_{\mathrm{GR}})=n+g+\delta\). Equivalently, the net marginal product after depreciation is \(f'(k_{\mathrm{GR}})-\delta=n+g\).

The distinction between the gross and net conditions is important. The production function’s derivative \(f'(k)\) is the gross marginal product: it does not subtract the depreciation of the additional capital. Subtracting \(\delta\) gives the net return after depreciation. The equivalent net condition says that this return equals the growth rate of effective labor, \(n+g\). It does not say that the gross marginal product equals \(n+g\).

The condition is an interior optimum. If a model’s feasible range or parameter choices do not permit a positive stock satisfying it, the maximum must instead be considered at a boundary. In the usual Solow application with an increasing, strictly concave production function whose marginal product falls from very high values to values below \(b\), there is a unique interior Golden Rule stock.

Worked Example: Cobb-Douglas Production and the Marginal-Product Check

Suppose \(f(k)=24k^{1/3}\), \(n=0.02\), \(g=0.02\), and \(\delta=0.04\). Find the Golden Rule stock and consumption, then verify the marginal-product condition.

Method. Differentiate the production function, equate its marginal product to the full break-even rate, and then calculate output less break-even investment. Using the full rate avoids omitting either capital dilution or depreciation.

First, \(b=0.02+0.02+0.04=0.08\). The marginal product is \(f'(k)=8k^{-2/3}\). Thus

8k_{\mathrm{GR}}^{-2/3}=0.08, \qquad k_{\mathrm{GR}}^{2/3}=\frac{8}{0.08}=100, \qquad k_{\mathrm{GR}}=100^{3/2}=1{,}000.

At \(k_{\mathrm{GR}}=1{,}000\), \(k_{\mathrm{GR}}^{1/3}=10\), so output per effective worker is \(24(10)=240\). Break-even investment is \(0.08(1{,}000)=80\). Therefore, steady-state consumption per effective worker is \(240-80=160\).

Check. The marginal product is \(8(1{,}000)^{-2/3}=8/100=0.08\), exactly the break-even rate. Also, \(f''(k)=-(16/3)k^{-5/3}<0\), so the stationary point maximizes consumption. The calculation avoids substituting \(k=1{,}000\) into the marginal product without checking the exponent.

Common Trap: Leaving Depreciation Out

A frequent error is to set gross marginal product equal to \(n+g\). That condition omits the investment needed to replace depreciated capital. The correct gross condition includes depreciation, \(f'(k)=n+g+\delta\). Alternatively, subtract depreciation from the gross marginal product first, and then use the net condition \(f'(k)-\delta=n+g\). These are equivalent; mixing the two versions is not.

Worked Example: Correcting an Omitted-Depreciation Error

An analyst uses \(f(k)=20\sqrt{k}\), \(n=0.02\), \(g=0.02\), and \(\delta=0.06\), but equates the gross marginal product to \(n+g\). Find the stock implied by the analyst’s condition, then calculate the correct Golden Rule stock and consumption.

Method. Calculate the analyst’s result as stated, then recalculate using the full break-even rate. Comparing the two makes the depreciation omission explicit.

The production function has marginal product \(f'(k)=10/\sqrt{k}\). The analyst uses \(n+g=0.04\), giving

\frac{10}{\sqrt{k}}=0.04, \qquad \sqrt{k}=\frac{10}{0.04}=250, \qquad k=250^2=62{,}500.

That is the stock implied by the incorrect condition, not the Golden Rule stock. The full break-even rate is \(0.02+0.02+0.06=0.10\). The correct condition gives

\frac{10}{\sqrt{k_{\mathrm{GR}}}}=0.10, \qquad \sqrt{k_{\mathrm{GR}}}=100, \qquad k_{\mathrm{GR}}=10{,}000.

At the correct stock, output is \(20\sqrt{10{,}000}=20(100)=2{,}000\), break-even investment is \(0.10(10{,}000)=1{,}000\), and consumption is \(2{,}000-1{,}000=1{,}000\).

Check and correction. At \(k=10{,}000\), \(f'(k)=10/100=0.10\), as required. At the analyst’s stock, \(f'(62{,}500)=10/250=0.04\), but the correct break-even rate there is still \(0.10\). Using the actual resource constraint, consumption at that stock would be \(20(250)-0.10(62{,}500)=5{,}000-6{,}250=-1{,}250\), not a positive sustainable consumption level. The mistake is omitting depreciation, not an arithmetic error in squaring 250.

Interpreting the Result

The Golden Rule is a benchmark for the capital stock that maximizes sustainable consumption, not a claim that every economy is currently at that stock. The production function determines how much additional output capital provides; population growth, technological progress, and depreciation determine how much investment is needed to maintain capital per effective worker. The first-order condition equates these marginal effects.

Use the full break-even rate \(n+g+\delta\) when matching it to the gross marginal product.
Subtract depreciation only once. If using the net return \(f'(k)-\delta\), set it equal to \(n+g\), not \(n+g+\delta\).
Calculate consumption, not just output. At a steady state, subtract break-even investment from output.
Check the maximum. With a strictly concave production function, the consumption function is strictly concave, so an interior solution to the first-order condition is a maximum.

These checks keep the Golden Rule calculation distinct from simply solving for a steady state under a specified saving rate. The Golden Rule identifies the stock with the highest sustainable consumption; the marginal-product condition explains why that stock is optimal.

Check Your Understanding

For each problem, calculate the break-even rate before applying the marginal-product condition. Answers are provided for checking.

Let \(f(k)=15\sqrt{k}\), \(n=0.01\), \(g=0.02\), and \(\delta=0.03\). Find the Golden Rule stock and consumption per effective worker. (Answer: \(k_{\mathrm{GR}}=15{,}625\); consumption \(=937.5\).)
Let \(f(k)=18k^{1/3}\), \(n=0.02\), \(g=0.01\), and \(\delta=0.03\). Find the Golden Rule stock and consumption per effective worker, and verify the gross marginal product. (Answer: \(k_{\mathrm{GR}}=1{,}000\); consumption \(=120\); \(f'(k_{\mathrm{GR}})=0.06\).)
For \(f(k)=10\sqrt{k}\), let \(n=0.04\), \(g=0.01\), and \(\delta=0.05\). Find the Golden Rule stock and consumption per effective worker. (Answer: \(k_{\mathrm{GR}}=2{,}500\); consumption \(=250\).)
For \(f(k)=20\sqrt{k}\), let \(n=0.02\), \(g=0.02\), and \(\delta=0.04\). An analyst sets the gross marginal product equal to \(n+g\). What stock does that incorrect condition imply, and what is the correct Golden Rule stock? (Answer: incorrect stock \(=62{,}500\); correct stock \(=15{,}625\).)
Suppose a differentiable, strictly concave production function has \(f'(k)=0.07\) at a proposed Golden Rule stock. The economy has \(n=0.02\), \(g=0.01\), and \(\delta=0.04\). Is the proposed stock consistent with the interior Golden Rule condition? State the equivalent net-return check. (Answer: Yes; \(n+g+\delta=0.07\), and \(f'(k)-\delta=0.03=n+g\).)

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