Abstract
We use recent international data on cost shares by industry to conduct the first robust test of Leontief’s hypothesis of factor-specific productivity differences. We strongly reject this hypothesis. Hence tests of the Heckscher–Ohlin–Vanek paradigm cannot be based upon simple modifications that define factors in efficiency units. We also discuss a theory of productivity differences that describes the factor content of trade well.
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Notes
These parameters depend upon Country c and Country 0, but that dependence is suppressed for notational convenience.
Some data were available that might have undercut one’s belief in Leontief’s conjecture. Exploiting cross-country data on a wide sample of industries, Arrow et al. (1961) estimated elasticities of substitution that were typically quite different from unity. Since factor prices are not equalized, it could be inferred that an industry’s factor cost shares differed across countries. Also, scholars such as Rosefielde (1974) had long studied input-output matrices from other countries.
See OECD (2015). The URL http://www.oecd-ilibrary.org/industry-and-services/data/stan-input-output/input-output-database_data-00650-en?isPartOf=/content/datacollection/stan-in-out-data-en was retrieved on 12 October 2015.
Our definition has a slight drawback. Factor shares always sum to unity, but there are a few subsidized industries where payments to social capital are negative. The most striking case is “Motor Vehicles, Trailers, and Semi-trailers” in Indonesia. Capital’s share is 1.6, labor’s is 0.8, and social capital’s is \(-\)1.4. Some might consider it an advantage to identify rare cases of highly subsidized industries. These cases give the data fat tails.
We drop “Steam and Hot Water Supply” since that industry is not active in the United States.
See Everitt (1998, pp. 182–83) for a discussion of this distribution.
The binomial distribution is discrete. Each marginal significance level is the two-sided probability of a more extreme value than that observed in the data.
A standard graduate textbook (Feenstra 2004, p. 61) assigns an exercise that illustrates Gabaix’s algebra. It is unfortunate that Gabaix (1997) was never published and is not readily available. In essence, Gabaix shows that Trefler’s calculations are not identified. When the measured factor content of trade in labor services is zero, then Trefler (1993) computed either labor productivity parameters or GDP per capita.
Trefler’s (1993) productivity parameters are the solution to an invertible system of linear equations. Its kernel has the property that each country’s factor-specific productivity is simply national income per unit of that factor. For example, the productivity parameters for labor are just output per worker, and those for capital are the inverses of the capital-output ratios. Since any invertible linear mapping is continuous, the imputed productivity parameters are quite near output per factor when the system’s image is in a neighborhood of zero.
Let \(Y_c\) be the output of Country c. In steady state, investment is \(I_c = \delta K_c\) where the variables have their usual meanings. The data report \(\kappa _c = I_c/Y_c\), the share of investment in GDP. Hence \(Y_c/K_c = \delta /\kappa _c\), and the depreciation rate is common across all countries. Thus the productivity parameters for capital should be correlated with the inverses of the investment shares of GDP.
Schott (2003) contends that the rubrics in these data are too broad. He argues that countries produce highly disaggregated goods in different diversification cones, depending upon their level of economic development. We must be agnostic about this claim, but we note that there are very few zeros in each country’s vector of imports and exports at this level of aggregation.
Let \(Ax=b\) be a system of n equations in f unknowns x. Then the set of all solutions is \(x=A^+b+(I-A^+A) z,\) where z is an arbitrary \(f \times 1\) vector. If the rank of A is at least f, then \(I-A^+A=0\). In fact, the Moore–Penrose inverse gives a solution to an inconsistent system of equations \(Ax \approx b\), a fact that will be useful below.
We are using the symmetry property of the Moore Penrose inverse: \((A^\prime )^+=(A^+)^\prime .\)
Brecher and Choudhri (1982) give a nice diagrammatic exposition of an economy with more goods than factors.
Again, for notational convenience, we will henceforth omit the dependence of \(\Theta _c(\cdot )\) on local factor prices.
When a sector is not active, every element in that row is zero. Then the pseudo-inverse has a corresponding column that also has every element equal to zero.
Thus the rates of tchnological progress lie in the column space of \(\Theta _0(w_0)\).
Because unit input requirements minimize costs, the envelope theorem implies that \(\Theta _0(w_0)\) is unaffected for small changes \(\hat{w}_0\). Given fixed factor supplies, higher factor returns are equivalent to more productive factors.
We constructed the dollar values of endowments from those in local currencies in the OECD data using purchasing power parity exchange rates from the World Bank’s International Comparison Program.
Note that \(\Theta _i^+\Theta _j \ne \Theta _j^+\Theta _i\), just as a regression of Y on X is different from a regression of X on Y. Also, \(\Theta _i^+\Theta _i = I_f\), so there are only 1056 such matrices in these data that are not trivial. Again, we will make all these data available to any interested researcher.
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Acknowledgments
The authors would like to thank two anonymous referees, an associate editor, Daniel Trefler, Xavier Gabaix, John Cochrane, and Matt Cole for helpful comments on earlier drafts. All the data and programs are available upon request.
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Fisher, E.O., Marshall, K.G. Leontief was not right after all. J Prod Anal 46, 15–24 (2016). https://doi.org/10.1007/s11123-016-0466-2
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DOI: https://doi.org/10.1007/s11123-016-0466-2
Keywords
- Factor-specific productivity differences
- Leontief conjecture
- Total factor productivity
- Empirical studies of trade