The golden rule and the data

We consider the Solow model with technical progress, in the notation of the note on simulating the Solow model. Output is \(Y = F(K, AL)\), where \(F\) is a neoclassical production function with constant returns to scale, the population \(L\) grows at rate \(n\) and labour efficiency \(A\) at rate \(x\). We denote by \(\hat k = K/(AL)\) capital per effective worker and by \(f(\hat k) = F(\hat k, 1)\) output per effective worker. Households save a constant fraction \(s\) of their income, capital depreciates at rate \(\delta\), and capital per effective worker obeys the law of motion

At the steady state \(\hat k^{\star}\), investment per effective worker exactly offsets the depreciation and the dilution of capital: \(s f(\hat k^{\star}) = (n + x + \delta)\hat k^{\star}\). Capital, output and consumption then all grow at the same rate, which we denote

and the capital-output ratio is constant, equal to \(\hat k^{\star}/f(\hat k^{\star}) = s/(g + \delta)\).

Factor markets are perfectly competitive. Capital is rented to firms at the price \(R = f'(\hat k)\), its marginal product, and labour is paid the wage \(w = A\bigl(f(\hat k) - \hat k f'(\hat k)\bigr)\). The owner of capital bears its wear and tear: the net return, the real interest rate, is

Consumption per effective worker at the steady state is what remains of output once investment has been made:

Since \(\hat k^{\star}\) is an increasing function of \(s\), choosing the saving rate amounts to choosing the steady-state capital stock. Consumption per head, \(c = A\hat c\), follows by multiplying by labour efficiency, which is exogenous: maximising \(\hat c^{\star}\) therefore amounts to maximising consumption per head at every date along the balanced growth path. The golden rule capital stock, \(\hat k_{\mathrm{gold}}\), sets the derivative of \(\hat c^{\star}\) with respect to \(\hat k^{\star}\) to zero:

and this is indeed a maximum since \(f\) is concave. At the golden rule, the net return on capital equals the growth rate of the economy. Below \(\hat k_{\mathrm{gold}}\), the net return on capital exceeds the growth rate, and accumulating more would raise long-run consumption: the economy is in under-accumulation. Above it, the opposite holds, and we speak of over-accumulation. The golden rule saving rate is the one that leads to \(\hat k_{\mathrm{gold}}\):

The last ratio is the share of capital income in total income, evaluated at the golden rule capital stock: whatever the production function, the golden rule saving rate equals the capital share at the golden rule (exercise 1). With a Cobb-Douglas production function, \(f(\hat k) = \hat k^{\alpha}\), this share is \(\alpha\) whatever the capital stock, and we recover the result of the course, \(s_{\mathrm{gold}} = \alpha\).

These four statements are equivalent in the model, but they do not use the same data. Statements 2 and 4 involve the capital stock, its depreciation rate and the growth rate. They actually boil down to the same empirical test, since the net return is measured by \(r = \alpha Y/K - \delta\), and \(r > g\) is then equivalent to \(K/Y < \alpha/(g + \delta)\). We shall call it the stock test. Statement 3 involves only two shares of income, that of capital and that of investment; we shall call it the flow test. The two tests coincide only if the observed capital-output ratio is the steady-state one, \(K/Y = s/(g + \delta)\). The rest of the note shows that this is not the case in the data, and that the two tests then return very different verdicts.

The flow test has a simple interpretation. Output is entirely distributed to the factors, since returns to scale are constant and factors are paid their marginal products: \(Y = wL + RK\). It is also entirely consumed or invested: \(Y = C + I\). Equating the two expressions, we obtain

Consumption equals wages, plus the net flow that capital provides to the economy: the income it yields, minus the investment it absorbs. With a Cobb-Douglas production function, capital income is a constant fraction of income, \(RK = \alpha Y\), and investment is \(I = sY\), so that

At the golden rule, \(s = \alpha\): the economy reinvests exactly what capital yields, and consumption equals wages. In under-accumulation, capital yields more than it absorbs, and consumption exceeds wages. In over-accumulation, the opposite holds: maintaining and expanding the capital stock costs more than it yields, and consumption falls short of wages. The economy would then gain by saving less, and bringing the saving rate back to \(\alpha\) raises consumption immediately and at every future date (exercise 2).

Along the balanced growth path, investment offsets depreciation and makes capital grow at rate \(g\), so that \(I = (g + \delta)K\), while capital income is \(RK = (r + \delta)K\). The net flow then reads

and the flow reading coincides with the comparison of the net return and the growth rate. Out of the steady state, however, the two expressions part ways. The net flow, \((\alpha - s)Y\), keeps the sign of \(\alpha - s\) at every date. The gap \(r(t) - g = \alpha\hat k(t)^{\alpha - 1} - \delta - g\), on the other hand, depends on the capital accumulated at date \(t\), and has the sign of \(\alpha - s\) only once the economy is close enough to its steady state. To see this, let us simulate two economies that save 20% of their income while the capital share is \(1/3\): they are in under-accumulation. The first starts from half its steady-state capital stock, the second from twice the golden rule capital stock. We use the exact solution of the Solow equation, presented in the note on its simulation.

alpha, s = 1/3, 0.2
n, x, delta = 0.01, 0.02, 0.04
g = n + x
k_star = (s/(g + delta))**(1/(1 - alpha))
k_gold = (alpha/(g + delta))**(1/(1 - alpha))

def path(k0, t):
    """Exact solution of the Solow equation with a Cobb-Douglas technology."""
    z = (s/(g + delta)
         + (k0**(1 - alpha) - s/(g + delta))*np.exp(-(1 - alpha)*(g + delta)*t))
    return z**(1/(1 - alpha))

t = np.linspace(0, 80, 801)
k_low = path(k_star/2, t)
k_high = path(2*k_gold, t)
r_star = alpha*k_star**(alpha - 1) - delta
print(k_star, k_gold, r_star - g)

Both measures converge to the same value, \(r^{\star} - g \approx 0.047\), but not in the same way. The net flow, divided by capital to be comparable with \(r - g\), remains positive throughout both transitions. In the economy that starts from a high capital stock, however, the net return is initially lower than the growth rate, and remains so as long as capital exceeds the golden rule level. With the exact solution, we compute the date \(T\) at which \(\hat k(T) = \hat k_{\mathrm{gold}}\) (exercise 4):

T = -np.log((alpha - s)/((2*k_gold)**(1 - alpha)*(g + delta) - s))/((1 - alpha)*(g + delta))
print(T)

For a little over 19 years, the stock test diagnoses over-accumulation in an economy that accumulates too little. This economy has inherited an abundant capital stock that it does not renew: its net return is low because its capital is high, not because it saves too much. The flow test is never wrong. This is one of the reasons why economists prefer to compare capital income with investment rather than an interest rate with the growth rate¹.

To confront the golden rule with the data, we need, for each country, the capital share \(\alpha\), the investment rate \(s\), the depreciation rate \(\delta\), the growth rate \(g\) and the capital-output ratio \(K/Y\). All these quantities are available in version 11.0 of the Penn World Tables (PWT), which covers 185 countries between 1950 and 2023. The following code reads the data sheet of the PWT Excel workbook.

pwt = pd.read_excel('pwt110.xlsx', 'Data')

We keep the period 1960-2023 and the countries for which the relevant variables are available every year:

FIRST, LAST = 1960, 2023
VARIABLES = ['labsh', 'csh_i', 'pl_i', 'pl_gdpo', 'delta', 'pop', 'rgdpna', 'rnna']
data = (pwt[(pwt['year'] >= FIRST) & (pwt['year'] <= LAST)]
        .set_index(['countrycode', 'year'])[VARIABLES]
        .sort_index())
complete = data.notna().all(axis=1).groupby('countrycode').sum() == LAST - FIRST + 1
data = data.loc[complete[complete].index]
print(complete.sum())

The sample contains 79 countries. For each of them, we compute the quantities of the model as averages over the period:

• the capital share, \(\alpha\), is one minus the share of labour compensation in GDP, labsh, measured at current national prices;
• the investment rate, \(s\), is the share of gross capital formation in GDP. The PWT provide it at current purchasing power parities (csh_i), that is valuing investment and output at the same international prices. To compare it with the capital share, measured at national prices, we multiply it by the ratio of the price level of investment to that of output (pl_i and pl_gdpo), which gives the investment share at national prices;
• the depreciation rate, \(\delta\), is the average depreciation rate of the capital stock, delta;
• the growth rate, \(g = n + x\), is the sum of the average annual growth rate of population, pop, and that of real GDP per head, rgdpna/pop, between 1960 and 2023;
• the capital-output ratio, \(K/Y\), divides the capital stock, rnna, by GDP, rgdpna, both at constant 2021 national prices.

def calibrate(country):
    span = LAST - FIRST
    population = country['pop'].to_numpy()
    per_capita = (country['rgdpna']/country['pop']).to_numpy()
    n = (population[-1]/population[0])**(1/span) - 1
    x = (per_capita[-1]/per_capita[0])**(1/span) - 1
    return pd.Series({
        'alpha': 1 - country['labsh'].mean(),
        's': (country['csh_i']*country['pl_i']/country['pl_gdpo']).mean(),
        's_ppp': country['csh_i'].mean(),
        'delta': country['delta'].mean(),
        'n': n,
        'x': x,
        'g': n + x,
        'KY': (country['rnna']/country['rgdpna']).mean()})

countries = data.groupby('countrycode').apply(calibrate)
print(countries.median().round(3))

The median values are a capital share of 0.47, an investment rate of 0.24, a depreciation rate of 0.041, a growth rate of 3.7% per year and a capital-output ratio of 3.7. The capital share is much higher than the value of \(1/3\) usually assumed; we return to this below.

The stock test compares the observed capital-output ratio with the golden rule one, \(\alpha/(g + \delta)\), which amounts to comparing the net return on capital, \(r = \alpha Y/K - \delta\), with the growth rate. The flow test compares the investment rate with the capital share.

countries['KY_gold'] = countries['alpha']/(countries['g'] + countries['delta'])
countries['r'] = countries['alpha']/countries['KY'] - countries['delta']
stock = countries['KY'] < countries['KY_gold']
flow = countries['s'] < countries['alpha']
print(stock.sum(), flow.sum(), len(countries))
print((stock == (countries['r'] > countries['g'])).all())
print(list(countries.index[~stock]))
SELECTION = ['FRA', 'DEU', 'GBR', 'USA', 'JPN', 'KOR', 'CHN', 'IND']
print(countries.loc[SELECTION, ['alpha', 's', 'g', 'delta', 'KY', 'KY_gold', 'r']].round(3))
print(countries.loc['CHN', 'alpha'] - countries.loc['CHN', 's'])
print((countries['s_ppp'] < countries['alpha']).sum(),
      list(countries.index[countries['s_ppp'] >= countries['alpha']]))

The stock test finds under-accumulation in 71 countries out of 79, and over-accumulation in the other 8: Barbados, China, Cyprus, Ecuador, Iceland, Nigeria, Portugal and Taiwan. The flow test finds under-accumulation in all 79 countries. The verdict is almost the same if we use, as in the course, the investment share at purchasing power parities, csh_i: only Switzerland then moves to over-accumulation. The following table details a few countries.

The two scatter plots are very different. In the flow test, all countries lie well below the 45-degree line: the median investment rate is half the median capital share, and even China, which invests 37% of its income, remains below. In the stock test, countries are closer to the 45-degree line, and a few of them cross it. France is almost at the golden rule, with a capital-output ratio of 5.6 against a benchmark of 5.9, whereas the United States is far from it, with 3.5 against 5.7. China is the most striking case: the stock test classifies it in over-accumulation, the flow test in under-accumulation, with capital income exceeding investment by 4.6 percentage points of GDP.

The two tests can diverge only if the observed capital-output ratio departs from its steady-state value, \(s/(g + \delta)\), computed with the observed investment rate, growth rate and depreciation rate. This is what the following figure shows.

countries['KY_ss'] = countries['s']/(countries['g'] + countries['delta'])
ratio = countries['KY']/countries['KY_ss']
print(ratio.median(), ratio.quantile(0.1), ratio.quantile(0.9))
print((ratio > 1).sum())

The observed ratio exceeds the steady-state ratio in 64 countries out of 79, by 25% at the median; for one country in ten, the gap exceeds 88%. This gap is what brings countries closer to the 45-degree line in the stock test. Two reasons can explain it.

The first is that economies are not at their steady state. The relation \(K/Y = s/(g + \delta)\) holds only along the balanced growth path, where the capital-output ratio is constant. Yet it is not constant over the period:

KY = (data['rnna']/data['rgdpna']).unstack('countrycode')
print(KY.loc[[FIRST, LAST], ['FRA', 'DEU', 'USA', 'CHN']].round(2))

China's ratio doubled between 1960 and 2023, from 2.7 to 5.5, and Germany's rose from 2.7 to 4.1. These economies accumulated capital faster than they produced: they were in transition. As in the simulation of the previous section, the stock test then measures inherited capital as much as saving behaviour.

The second reason is that the capital stock is not observed. Statisticians build it with the perpetual inventory method, by cumulating past investments, valued at the prices of a reference year, and subtracting a depreciation computed from assumptions on the service lives of equipment. The result depends on these assumptions, on the initial stock and on the price indices of investment, whereas the capital share and the investment rate are shares of current income, read directly from the national accounts at the same prices. The stock test therefore accumulates measurement errors that the flow test avoids, since the latter involves neither the capital stock, nor depreciation, nor the growth rate.

Both tests rely on the capital share, measured here as one minus the labour share. This measure is delicate, for two reasons. First, the income of the self-employed rewards both their labour and their capital. Counted entirely as capital income, it leads to very low labour shares in poor countries, where self-employment is widespread. Gollin (2002) shows that once this income is split, the labour share lies between 0.65 and 0.80 in most countries. The PWT impute part of this income to labour, but the imputation remains fragile. Second, one minus the labour share contains everything that does not reward labour, including land rents and natural resource rents, which do not reward accumulated capital. The median capital share of our sample, 0.47, is indeed much higher than the value of \(1/3\) usually assumed. Let us see how the two verdicts change when the same capital share is imposed on all countries.

for a in (0.30, 1/3, 0.40):
    print(round(a, 3),
          (countries['KY'] < a/(countries['g'] + countries['delta'])).sum(),
          (countries['s'] < a).sum())
print(list(countries.index[countries['s'] >= 0.30]))

The table gives the number of countries in under-accumulation, out of 79. The verdict of the stock test is very sensitive to the capital share: with \(0.30\), it classifies only half of the countries in under-accumulation. The flow test is much more robust, because the investment rate is usually well below all these values: only the countries that invest more than 30% of their income, such as China, Japan, Korea or Singapore, change sides when the capital share is lowered to \(0.30\).

These are precisely the countries for which the verdict is disputed. Applying the criterion of Abel et al. (1989) with national accounts that better separate the income of the self-employed, and removing from capital income land rents, which are very high in Japan and Korea, Geerolf (2018) concludes that these two countries have over-accumulated, and that the criterion is not met by any advanced economy. The flow test is more robust than the stock test, but it is no more robust than the measurement of the capital share on which it rests.

The flow test depends neither on the measurement of capital, nor on the steady-state assumption, nor on the choice of an interest rate. With the PWT data, it concludes that the economies of our sample did not save too much between 1960 and 2023, by a wide margin for most of them. For the economies that invest the most, such as Japan or Korea, the verdict depends, however, on how the reward of accumulated capital is isolated in income.

Should we conclude that they ought to save more? Not necessarily. Raising the saving rate towards the golden rule increases long-run consumption, but reduces it first: present generations would pay for future generations. For the United States, raising the saving rate to the capital share would increase long-run consumption by 12%, but would reduce it immediately by 21%, and it would take 23 years to get back to the initial level (exercise 3). The golden rule says nothing about this trade-off between generations. In the Ramsey model, where saving results from the choices of households that prefer the present, the optimal steady state lies below the golden rule: this is the modified golden rule, where the net return on capital exceeds the growth rate. The under-accumulation revealed by the data is therefore not an anomaly. Over-accumulation, on the other hand, would be one, which is why it matters to be able to rule it out.

The golden rule, and the fact that it equates the saving rate and the capital share for any production function, are due to Phelps, E. (1961), "The Golden Rule of Accumulation: A Fable for Growthmen", American Economic Review, 51(4), 638-643. The criterion comparing capital income with investment, and its justification under uncertainty, are due to Abel, A., Mankiw, N. G., Summers, L. and Zeckhauser, R. (1989), "Assessing Dynamic Efficiency: Theory and Evidence", Review of Economic Studies, 56(1), 1-19. They conclude that the United States and six other large OECD countries are dynamically efficient. Geerolf, F. (2018), "Reassessing Dynamic Efficiency", working paper, UCLA, revisits this verdict with more recent data. On the measurement of the labour share, see Gollin, D. (2002), "Getting Income Shares Right", Journal of Political Economy, 110(2), 458-474. The returns on US equities and riskless securities are taken from Mehra, R. and Prescott, E. (1985), "The Equity Premium: A Puzzle", Journal of Monetary Economics, 15(2), 145-161.

The data come from version 11.0 of the Penn World Tables (Groningen Growth and Development Centre, 2025, doi:10.34894/FABVLR), described in Feenstra, R., Inklaar, R. and Timmer, M. (2015), "The Next Generation of the Penn World Table", American Economic Review, 105(10), 3150-3182. On the Solow model itself, see the notes on the Solow model and on its simulation.