12.4 FDH in Practice: Manufacturing Industry

In order to illustrate how FDH works, the Manufacturing Industry Productivity Database from the National Bureau of Economic Research (NBER), USA is considered. This database is downloadable from the website of NBER [http://www.nber.org]. It contains annual industry-level data on output, employment, payroll, and other input costs, investment, capital stocks, and various industry-specific price indices from 1958 on hundreds of manufacturing industries (indexed by 4 digits numbers) in the United States. We selected data from the year 1996 (458 industries) with the following 4 input variables, $ p=4$, and 1 output variable, $ q=1$ (summary statistics are given in Table 12.5):

$ X_1$
: Total employment,
$ X_2$
: Total cost of material,
$ X_3$
: Cost of electricity and fuel,
$ X_4$
: Total real capital stock,
$ Y$
: Total value added.

Table 12.6 summarizes the result of the analysis of US manufacturing industries in 1996. The industry indexed by 2015 was efficient in both input and output orientation. This means that it is one of the vertices of the free disposal hull generated by the 458 observations. On the other hand, the industry 2298 performed fairly well in terms of input efficiency (0.96) but somewhat badly (0.47) in terms of output efficiency. We can obtain the efficient level of inputs (or outputs) by multiplying (or dividing) the efficiency score to each corresponding observation. For example, consider the industry 2013, which used inputs $ X_1 = 88.1$, $ X_2 =
14925$, $ X_3 = 250$, and $ X_4 = 4365.1$ to yield the output $ Y =
5954.2$. Since its FDH input efficiency score was 0.64, this industry should have used the inputs $ X_1 = 56.667$, $ X_2 =
9600$, $ X_3 = 160.8$, and $ X_4 = 2807.7$ to produce the observed output $ Y =
5954.2$. On the other hand, taking into account that the FDH output efficiency score was 0.70, this industry should have increased its output upto $ Y = 4183.1$ with the observed level of inputs.


Table 12.5: Summary statistics for Manufacturing Industry Productivity Database (NBER, USA)
  Minimum Maximum Mean Median Std.Error
X1 0.8 500.5 37.833 21 54.929
X2 18.5 145130 4313 1957.2 10771
X3 0.5 3807.8 139.96 49.7 362
X4 15.8 64590 2962.8 1234.7 6271.1
Y 34.1 56311 3820.2 1858.5 6392


Table 12.6: FDH efficiency scores of 458 US industries in 1996
    Efficiency scores
  Industry input output
1 2011 0.88724 0.94203
2 2013 0.79505 0.80701
3 2015 0.66933 0.62707
4 2021 1 1
. . . .
. . . .
. . . .
75 2298 0.80078 0.7439
. . . .
. . . .
. . . .
458 3999 0.50809 0.47585
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