The Marginal Product of Capital

Whether the world’s capital stock is efficiently allocated across countries – equivalently, whether aggregate marginal products of capital are roughly equal – and the answer is not settled by the huge dispersion in capital-labour ratios because poor countries also have less of the factors capital works with. “Given the enormous cross-country differences in observed capital-labor ratios (they vary by a factor of 100 in the data used in this paper) it may seem obvious that the MPK must vary dramatically as well,” which would force the conclusion that international capital markets are badly frictioned (§1, p. 1). But, as the paper credits Lucas (1990) with pointing out, “poor countries also have lower endowments of factors complementary with physical capital, such as human capital, and lower total factor productivity (TFP). Hence, large differences in capital-labor ratios may coexist with MPK equalization.” The stakes are set out immediately: if returns differ, “the world foregoes an opportunity to increase global GDP by reallocating capital from low to high MPK countries. The policy implications are far reaching.”

Interest-rate comparisons, output-on-capital regressions, and calibration each fail in a different way, and “the effort to generate reliable comparisons of cross-country MPK differences has not yet paid off.” Interest rates are problematic because “in financially repressed/distorted economies interest rates on financial assets may be very poor proxies for the cost of capital actually borne by firms,” with default a further wedge: emerging-market promised yields commonly exceed US yields by a factor of two or three, “but given the much higher risk these bonds carry it is possible that the expected cost of capital from the perspective of the borrower is considerably less” (§1, pp. 1-2, and fn. 3). Regressing output growth on capital growth “typically relies on unrealistic identification assumptions.” Calibration requires choosing a functional form and accurately measuring the complementary factors, which “is quite ambitious, [so] one may not want to rely on this method exclusively.” The paper positions itself against all three: its estimates “are extremely direct, impose extremely little structure on the data, and are extremely simple to calculate.”

That with constant returns and perfectly competitive domestic capital markets, the marginal product of capital equals the capital share times output divided by the capital stock. The chain is short: competitive domestic capital markets make the rental rate equal the marginal product, so aggregate capital income is the marginal product times the capital stock; dividing by output, the capital share equals the marginal product times capital over output (§2.1, p. 5). “Relative to alternative estimates in the literature, this method of calculating MPK requires no functional form assumptions (other than linear homogeneity), much less that we come up with estimates of human capital, TFP, or other factors that affect a country’s MPK,” and the assumptions it does make “are typically shared by the other approaches… so the set of restrictions we impose is a strict subset of those imposed elsewhere” (§2.1, pp. 6-7). The paper does not claim the restrictions are innocuous: “For example, we rule out adjustment costs to the stock of capital – which in certain models could drive a wedge between the rental rate on capital and the MPK” (§2.1, fn. 12).

Because it is one minus the labour share, so it includes payments to land and natural resources, while the capital stock in the denominator is built from investment flows and therefore contains reproducible capital only. The mismatch biases the level of the estimated return upward, and the bias is not uniform: “since the agricultural and natural-resource sectors represent a much larger share of GDP in poor countries, the overestimate of the MPK when using the total capital share is much more severe in such countries, and cross-country differences will once again be inflated (both in absolute and in relative terms)” (§2.1, pp. 5-6). The paper also flags that even a uniform natural-capital share would distort absolute comparisons – with 20 percent natural capital everywhere, a true US return of 8 percent and Indian return of 16 percent, a spread of 8 points, would be reported as 10 and 20 percent, a spread of 10 points (§2.1, fn. 11) – which matters because “interest-rate spreads, for example, are absolute differences.” It extends the point beyond its own exercise: the observation “extend[s] to a criticism of much work that has automatically plugged in standard capital-share estimates in empirical applications of models where all capital is reproducible” (§1, fn. 6).

That frictionless international credit markets equalise the value of the marginal product of capital divided by the price of capital goods, so physical marginal products can legitimately differ across countries with no friction whatever, provided capital goods cost more in poor countries. In a J-good economy with constant returns in each good and domestic competition, an investor comparing a piece of equipment to a world interest rate equates the domestic output price times the physical marginal product, divided by the domestic price of capital, to the world return net of depreciation. Hence “frictionless international credit markets imply that the value of the marginal product of capital in any particular final good, divided by the price of capital, is constant across countries” (§2.2, pp. 7-8). The intuition the paper gives is the memorable one: “poor-country investors in physical capital need to be compensated by a higher physical MPK for the fact that capital is more expensive there (relative to output). Or, yet in other words, the physical MPK measures output per unit of physical capital invested, while for the purposes of cross-country credit flows one wants to look at output per unit of output invested.” Notably the one-sector measures retain a use in this world: the paper shows the capital share times real income over the capital stock is a harmonic-type average of sectoral physical marginal products, so it “offer[s] some quantitative assessment of cross-country differences in the average physical MPK” (§2.2, p. 9).

Penn World Table 6.1 for output, capital and prices; Bernanke and Gürkaynak (2001) for the total capital share; World Bank (2006) wealth data for the reproducible-capital share. Output is 1996 PPP GDP; the capital stock is built by perpetual inventory from real investment series with a depreciation rate of 0.06; the price ratio is a weighted average of final-good domestic prices over a weighted average of domestic equipment prices, with a common goods list across countries (§2.3, pp. 9-10). The total capital share is one minus the labour share, with the labour share being corporate-sector employee compensation plus adjustments for the self-employed and non-corporate employees, following and extending Gollin (2002); the paper reports that “when we use Gollin’s estimates we get very much the same results” (§2.3, fn. 17). Direct measures of the reproducible-capital share “do not appear to be available,” so the paper constructs one by multiplying the total capital share by reproducible capital’s share of national wealth – valid because, if all units of wealth pay the same return, reproducible capital’s share of capital income is proportional to its share of wealth (§2.3, p. 10). Bernanke and Gürkaynak’s coverage is the binding constraint, giving 53 countries, one fewer (Hong Kong) for the reproducible-capital-corrected calculations. On depreciation, the paper notes a bias if depreciation rates differ across countries but argues it partly cancels: countries with high depreciation have overstated capital stocks and hence understated returns, yet the arbitrage condition also says they should have higher returns, so “variation in [depreciation] biases both sides of (2) in the same direction” (§2.3, fn. 15).

Roughly half of total wealth unweighted and nearly 70 percent reproducible when weighted by capital stocks, with the reproducible share correlating 0.70 with log GDP per worker; the paper validates it against three independent sources. The World Bank method capitalises estimated rents from each form of capital at a fixed discount rate, with rents for subsoil resources also requiring projections of rent growth and a depletion horizon, timber rents valued at local market prices net of production costs with sustainability adjustments, cropland and pasture valued as output less a crop-specific percentage cost, and protected areas valued at the opportunity cost of crop and pasture land (§2.3, pp. 10-11). The weakest component is stated plainly: “Due to data limitations, no good estimates of the value of urban land are available. A very crude estimate values urban land at 24% of the value of reproducible capital.” Cropland is the most strongly negatively correlated component with income (-0.73). The three checks: the US Office of Management and Budget puts land at 20 to 26 percent of total capital between 1960 and 2003 with no clear trend, against the World Bank’s 26 percent on a comparable basis; the paper’s approach implies a US land share in GDP of 8 percent, close to what Caselli and Coleman’s (2001) sectoral shares imply; and Goldsmith (1985) reports 1978 land shares in total capital averaging about 20 percent across mostly rich countries, ranging 12 to 27 percent with Japan an outlier at 51 percent – which the paper attributes to the crude urban-land imputation understating Japanese land values given its population density (§2.3, fn. 18).

Returns average 27 percent in the 29 lower-income countries with a standard deviation of 9 percent, against 11 percent and 3 percent in the 24 higher-income countries, with the split falling neatly at Portugal. The relationship with income is negative overall, but “the non-linearity in the data cannot be ignored: there is a remarkably neat split whereby the MPK-naive is highly variable and high on average in developing countries (up to Malaysia), and fairly constant and low on average among developed countries (up from Portugal)” (§2.4, p. 13). Within neither subsample is there a statistically significant relationship between the naive return and output. The paper states the reading this invites without endorsing it: “If we were to stop here, it would be tempting to conclude that capital flows fairly freely among the rich countries, but not towards and among the poor countries. This looks like a big win for the credit friction answer to the Lucas question.”

Each correction shrinks both the level gap and the dispersion; together they reverse the ordering. The reported averages, with standard deviations in parentheses, are: naive, 11.4 (2.7) rich against 27.2 (9.0) poor; natural-capital-corrected, 7.5 (1.7) against 11.9 (6.9); price-corrected, 12.6 (2.5) against 15.7 (5.5); both corrections, 8.4 (1.9) against 6.9 (3.7) (§2.4, Table 2, p. 14). The statistical status of each comparison is reported: “The differences between the poor and rich countries are significant for the first three rows of the table. For the case using both corrections, the difference is only significant at the 10% level.” The paper’s characterisation is that “taking both adjustments together eliminates the variance almost completely and the rich countries actually have a higher marginal product on average than the poor countries.”

Yes, and it runs the opposite way to the friction story: a positive and significant relationship between the fully corrected return and output, with the very poorest countries having the lowest returns. The paper offers an interpretation consistent with its policy conclusion rather than with credit rationing: “This would be consistent with a model where capital flows out of the country were forbidden, and where some capital flows into the country were not responsive to the rate of return. This may describe the poorest countries in our sample, where aid flows represent a significant proportion of investment capital (and all of the inward flow of capital)” (§2.4, p. 15).

By imposing Cobb-Douglas technology, solving for the common world return at which the existing world capital stock is exactly exhausted, and comparing counterfactual world output to actual – and it is explicitly not a policy proposal. The counterfactual requires a functional form the measurement did not, and also requires assuming the reproducible-capital share is constant across sectors within a country (though it may vary across countries); because the reallocation scales capital in every sector of a country by the same factor, marginal products of labour and natural capital move proportionally too, so no labour or land reallocation is needed to sustain it (§3, pp. 15-17). Two disclaimers matter. “We stress that this is not a normative exercise: our capital reallocation is not a policy proposal. The observed distribution of output is an equilibrium outcome given certain distortions that prevent MPK equalization. The point of this exercise is to assess the welfare losses the world experiences relative to a frictionless first best, not that the first best is easily achievable by moving some capital around.” And the estimate is conservative in one direction: removing the frictions “would almost certainly also lead to an increase in the world aggregate capital stock. Our calculations clearly abstract from this additional benefit, and are therefore a lower bound on the welfare cost of such frictions” (§3, fn. 22).

Under the naive measure, a near-tripling of poor-country capital per worker; under both corrections, poor countries lose capital. Unweighted, the average developing country’s capital-labour ratio rises 274.5 percent under the naive measure while the average rich country’s falls 12.9 percent; population-weighted the figures are 205.8 and -19.3 percent (§3.1, Table 3, pp. 18-19). Either correction alone cuts the weighted poor-country gain to roughly 50-60 percent with rich countries losing about 5 percent; with both corrections the poor countries lose 10.6 percent unweighted and 14.5 percent weighted while rich countries gain slightly. Output moves correspondingly: the average developing country gains 76.7 percent under the naive measure against a 3.0 percent “loss” for the average developed country, falling to gains under 25 percent with either single correction, and “for the scenario with both adjustments, average output in the two groups is essentially unchanged” – 0.0 percent unweighted for poor countries, -2.4 percent weighted (§3.2, Table 4, pp. 20-21). One feature of the naive counterfactual is worth noting because it shows the exercise is not mechanical: “despite this substantial amount of reallocation, many developing countries would still have less physical-capital per worker, reflecting their lower average efficiency levels.”

2.9 percent of world output with no corrections, 1.4 percent with the price correction alone, 0.6 percent with the natural-capital correction alone, and 0.1 percent with both. The 2.9 percent figure is put in perspective by noting that “the 28 developing countries in our sample account for 12 percent of the aggregate GDP of the sample. This result implies that the deadweight loss from inefficient allocation of capital is in the order of one quarter of the aggregate (and hence also per capita) income of developing countries” (§3.3, pp. 20-22, Table 5). Then: “The natural-capital adjustment alone reduces the dead weight loss to less than a quarter of the base case. The price adjustment alone reduces the dead weight losses by over half. Taken together, the dead weight loss is negligible.” The paper explains why the natural-capital correction dominates here even though the price correction dominated for the returns themselves: the natural-capital adjustment works twice over, once by narrowing the rich-poor return gap and again by lowering the capital share used in the counterfactual production function, which “reduces the sensitivity of output to reallocations of capital and reduces the dead weight losses further.”

It sides with Lucas against credit frictions but adds a third proximate cause he did not have: the higher relative price of equipment in poor countries. “Since we find essentially no difference in (properly measured) MPKs between poor and rich countries, we end up siding with Lucas on the (un)importance of international credit frictions as a source of differences in capital-labor ratios. But our results also call for some qualifications to Lucas’ preferred explanation” (§4, p. 23). Decomposing the variance of log capital per worker into a price-and-share term and a complementary-factor/efficiency term gives variances of 0.82 and 0.61 with a covariance of 0.52; splitting the covariance equally attributes 54 percent to prices and 46 percent to the Lucas factors, with a population-weighted split of about 50-50 (§4, p. 25). Two caveats are attached to this decomposition and should travel with it. It rests on “a very special case, in which each country produces only one output good,” which the paper itself labels “admittedly very strong.” And the two terms are correlated at 0.73, which the paper reads as a warning against treating the split as causal: since Hsieh and Klenow (2003) attribute high relative capital-goods prices to low productivity in capital-producing sectors, “the ultimate cause of differences in capital per worker may therefore be productivity differences if productivity differences are the ultimate cause of differences in capital costs and the share of capital. However, failure to account for these factors will falsely suggest that financial frictions play a large role.”

Tentative evidence that the cost of credit frictions has declined, offered with two explicit reasons for caution. The caveats come first in the text: the estimates depend on capital stocks built from investment series, so “the capital stock numbers become increasingly unreliable as we proceed backward in time,” and capital-share estimates are from a single year, “so changes over time are not reflected. This is a particular problem in the case of the calculations corrected for natural capital. The share of natural resources was particularly volatile during this period” (§5, pp. 24-25). With those caveats: the naive deadweight loss shows “little – or perhaps a slightly increasing – long-run trend”; once prices are accounted for the losses “have fallen somewhat over time”; and adding the natural-capital correction makes “the trend to be clearly downward.” The paper’s own characterisation is “tentative evidence,” consistent with increasing world financial integration, and it flags that the 1980s acceleration “may reflect historically low MPKs in developing countries during that decade’s crisis.”

That increased aid to developing countries is unlikely to raise their capital stocks or output, because returns are already equalised, so extra inflows are offset by private outflows. The mechanism is stated as an equilibrium response, not a behavioural claim about aid: “our result that financial rates of return are fairly similar in rich and poor countries… implies that any additional flow of resources to developing countries is likely to be offset by private flows in the opposite direction seeking to restore rate-of-return equalization” (§1, p. 5). The conclusion sharpens this with a conditional that is easy to miss and important: aid is unlikely to matter “unless they are accompanied by a return to financial repression, and in particular to an effective ban on capital outflows in these countries. Even in that case, increased aid flows would be a move towards inefficiency, and not increased efficiency, in the international allocation of capital” (§6, p. 25). The paper also distinguishes its conclusion from Gourinchas and Jeanne (2003), who find small welfare gains from capital-account openness despite large calibrated return differentials: “Our point is that, even though differences in physical MPKs are large, differences in rates of return are small, so we should not even expect much of a reallocation of capital in the first place” (§1, fn. 10).

No – it argues on unobservables that the poor-country returns are probably still overstated, which would mean poor countries already have too much capital. Rewriting the arbitrage condition with an effective tax on physical-capital income and an effective tax on financial-capital income, the paper reasons that “anecdotal evidence from developing countries suggests that physical capital installed domestically is more easily targeted by the tax authorities than various forms of financial investment, especially in offshore accounts,” a gap that widens “if one takes a broad view of physical-capital income taxation that includes expropriation by rent seeking governments” (§6, pp. 26-27). Since the physical-capital tax is likely large in poor countries and the financial one likely smaller, and since the paper has found the price-corrected physical return varies little, it follows that “aid flows and financial repression in developing countries may already have created a situation in which there is ’too much’ capital there. This seems an area of potentially fruitful future research.” The paper is explicit that it has no direct data on either tax rate.