# Documentation and Methods

### Methodology for Measuring International Agricultural Total Factor Productivity (TFP) Growth

The documentation and methods are organized in the following sections:

See Summary Findings and details for the Update and Revision History and References also available.

#### Overview

Improving agricultural productivity has been the world's primary means of assuring that the needs of a growing population don't outstrip the ability to supply food. Over the past 50 years, productivity growth in agriculture has allowed food to become more abundant and cheaper (see New Evidence Points to Robust But Uneven Productivity Growth in Global Agriculture, Amber Waves, September 2012). The most informative measure of agricultural productivity is total factor productivity (TFP). TFP takes into account all of the land, labor, capital, and material resources employed in farm production and compares them with the total amount of crop and livestock output. If total output is growing faster than total inputs, we call this an improvement in total factor productivity ("factor" = input). TFP differs from measures like crop yield per acre or agricultural value-added per worker because it takes into account a broader set of inputs used in production. TFP encompasses the average productivity of all of these inputs employed in the production of all crop and livestock commodities.

"Growth accounting" provides a practicable way of measuring changes in agricultural TFP over time given available data on agricultural outputs, inputs, and their prices. The approach described here gives internationally consistent and comparable agricultural TFP growth rates, but not TFP levels. Most of the data on production and input quantities used in this analysis come from FAOSTAT database of the United Nations Food and Agriculture Organization (FAO). In some cases FAO input and output data are supplemented with data from national statistical sources. The methodology and data are also fully described in Fuglie (2012, 2015).

#### How These Estimates Differ From Other ERS Productivity Accounts for the United States

To facilitate international comparisons in ERS' International Agricultural Productivity (IAP) data product, certain simplifying assumptions must be made. As such, the estimates of TFP growth reported here may not be exactly the same as TFP growth estimates reported in other studies using different assumptions or methods. In particular, the TFP estimates reported here for the United States differ somewhat from those reported in ERS' Agricultural Productivity in the U.S. data product. The principal differences are (i) the Agricultural Productivity in the U.S. data use prices received by U.S. farmers to measure output growth, whereas a common set of global average agricultural prices are used in ERS' IAP data product; (ii) in Agricultural Productivity in the U.S., agricultural labor is quality-adjusted by skill level, whereas there is insufficient data for such quality adjustments in ERS’ IAP data product; and (iii) the Agricultural Productivity in the U.S. accounts use a perpetual inventory method to measure farm capital stock (i.e., current capital stock is a function of past capital expenditures, appropriately discounted for depreciation), whereas the current inventory method (based on the number of major pieces of machinery in-use on farms) is used in ERS’ IAP data product. Generally, the TFP index reported in the Agricultural Productivity in the U.S. data product should provide a more accurate measure of the rate of technical change in U.S. agriculture. However, the International Agricultural Productivity data series reported here are better suited for making comparisons of agricultural TFP growth between the United States and other countries.

#### Model

Total factor productivity (TFP) is defined as the ratio of total output to total inputs. Let total output be given by Y and total inputs by X. Then TFP is simply:

$TFP=Y/X$

(1)

It is often difficult to provide meaningful definitions of real output or real input due to the heterogeneity of outputs produced and inputs used. However, it is possible to provide meaningful definitions of output growth and input growth between any two periods of time using index number theory (Caves, Christensen and Diewert, 1982). Changes in TFP over time are found by comparing the rate of change in total output with the rate of change in total input. Expressed as logarithms, changes in equation (1) over time can be written as

$\frac{d\ln(TFP)}{dt}=\frac{d\ln(Y)}{dt}-\frac{d\ln(X)}{dt}$

(2)

which simply states that the rate of change in TFP is the difference between the rate of change in aggregate output and input.

Agriculture is a multi-output, multi-input production process, so Y and X are vectors. When the underlying technology is represented by a constant-returns-to-scale Cobb-Douglas production function, producers maximize profits so that the output elasticity with respect to an input equals the cost share of that input, and markets are in long-run competitive equilibrium so that total revenue equal total cost, then equation (2) can be written as

$\ln\left(\frac{TFP_t}{TFP_{t-1}}\right)=\sum_{i}R_i\ln\left(\frac{Y_{i,t}}{Y_{i,t-1}}\right)-\sum_{j}S_j\ln\left(\frac{X_{j,t}}{X_{j,t-1}}\right)$

(3)

where Ri is the revenue share of the ith output and Sj is the cost-share of the jth input. Total output growth is estimated by summing over the growth rates for each output commodity weighted by its revenue share. Similarly, total input growth is found by summing the growth rate of each input, weighted by its cost share. TFP growth in Eq. (3) is thus the value-share-weighted difference between total output growth and total input growth.

One difference among growth accounting methods is whether the revenue and cost share weights are fixed or vary over time. Paasche and Laspeyres indexes use fixed weights whereas the Tornqvist-Thiel and other chained indexes use variable weights. Allowing the weights to vary reduces potential "index number bias." Index number bias arises when producers substitute among outputs and inputs depending on their relative profitability or cost. In other words, the growth rates in Yi and Xj are not independent of changes Ri and Sj. For example, if labor wages rise relative to the cost of capital, producers are likely to substitute more capital for labor, thereby reducing the growth rate in labor and increasing it for capital. In agriculture, cost shares of agricultural capital and material inputs tend to rise in the process of economic development while the cost share of labor tends to fall.

To reduce potential index number bias in TFP growth estimates, cost shares are varied by decade whenever such information is available. For outputs, base year prices (or equivalently, base year revenue shares) are fixed, since these depend on FAO’s measure of constant, gross agricultural output (described in more detail below in the Output subsection under Data). The base period for output prices is 2004-2006.

Direct estimates of cost shares were assembled for 22 countries from 16 studies. These 22 countries account for about two-thirds of world agricultural output. For another set of countries where input prices are not available or market-determined, (Sub-Saharan Africa and transition economies of the former Soviet Union and Eastern Europe), three studies provide econometric estimates of production elasticities, which were used in place of cost shares. These regions account for another 8 percent of world agricultural output. For remaining countries, representing about 25 percent of world agricultural output, cost shares are approximated by applying cost shares from a "like" country. The section below on Input Cost Shares provides details on the data sources and assumptions.

The framework outlined above provides a simple means of decomposing the relative contribution of TFP and inputs to the growth in output. Using the function g(.) to signify the annual rate of growth in a variable, the growth in output is simply the growth in TFP plus the growth rates of the inputs times their respective cost shares:

$g(Y)=&space;g(TFP)&space;+\sum_{j=1}^{J}S_jg(X_j)$

(4)

Equation (4) is a cost decomposition of output growth since each Sjg(Xj) term gives the growth in cost from using more of the jth input to increase output. It is also possible to focus on a particular input, say land (which we designate as X1), and decompose growth into the component due to expansion in this resource and the yield of this resource:

$g(Y)=&space;g(X_1)&space;+&space;g\left&space;(&space;\frac{Y}{X_1}&space;\right&space;)$

(5)

This decomposition corresponds to what is commonly referred to as extensification (land expansion) and intensification (land yield growth). We can further decompose yield growth into the share due to TFP and the share due to using other inputs more intensively per unit of land:

$g(Y)=g(X_1)+&space;g(TFP)&space;+&space;\sum_{j=2}^{J}S_jg\left(\frac{X_j}{X_1}\right)$

(6)

Equation (6) is a resource decomposition of growth since it focuses on the quantity change of a physical resource (land) rather than its contribution to changes in cost of production.

Figure 1 gives a graphical depiction of the growth decomposition described in equation (6). The height of the bars indicate the growth rate of real output. Growth in real output is first decomposed into growth attributable to agricultural land expansion (extensification) and growth attributable to raising yield per hectare (intensification). Finally, yield growth itself is decomposed into input intensification (i.e., more capital, labor and fertilizer per hectare of land), and TFP growth, where TFP reflects the efficiency with which all inputs are transformed into outputs. Improvements in TFP are driven by technological change, improved technical and allocative efficiency in resource use, and scale economies. The decomposition of output growth into these components is both intuitively appealing and has direct policy relevance: land expansion and input intensification are strongly influenced by changes in resource endowments and relative prices, whereas TFP growth is strongly influenced by long-term investments in agricultural research and extension services, education, and infrastructure, and improved resource quality and institutions.

#### Data

FAO’s 1961-2013 annual time series of crop and livestock commodity outputs and land, labor, livestock, farm machinery, inorganic fertilizers and animal feed inputs are the primary data used to construct the national, regional and global quantity measures. In some cases these are modified or supplemented with data from other sources (such as national statistical agencies) when they are considered to be more accurate or up-to-date, as described below.

##### Output

For agricultural output, FAO publishes estimates of annual production of 198 crop and livestock commodities by country since 1961. FAO also aggregates production into a measure of gross agricultural output using a common set of global average commodity prices from 2004-2006 and expresses this in constant 2005 international dollars. FAO excludes production of animal forages but includes crop production that is used for animal feed and seed in estimating gross agricultural output. The FAO also provides a measure of output net of domestic production used for feed and seed. However, the net production measure does not exclude imported grain that may be used as feed or seed, or grain that is exported and used in another country for these purposes.

Because current (or near current) prices are fixed to aggregate quantities and measure changes in real output over time, the FAO gross agricultural output is equivalent to a Paasche quantity index. The set of common commodity prices is derived using the Geary-Khamis method. This method determines an international price pi for each commodity which is defined as an international weighted average of prices of the i-th commodity in different countries, after national prices have been converted into a common currency using a purchasing power parity (PPPj) conversion rate for each j-th country. The weights are the quantities produced by the country. The computational scheme involves solving a system of simultaneous linear equations that derives both the pi prices and PPPj conversion factors for each commodity and country. The FAO updates these prices every five years and recalculates its index of gross production value back to 1961 using its most recent set of international prices. See Rao (1993) for a thorough description and assessment of these procedures.

The FAO value of gross agricultural output in constant 2005 international dollars is the basis for a consistent measure of output for each country and the world. However, due to the influence of weather and other factors, agricultural production is volatile from year to year, and it can be difficult to disentangle short-run fluctuations from long-term trends. To relieve the data of some of these fluctuations, the output series are smoothed for each country using the Hodrick-Prescott filter (setting λ=6.25 as recommended for annual data by Ravn and Uhlig, 2002). Even with smoothing there is still considerable curvature in the output series, although much of the year-to-year fluctuation in output has been removed from the data. The smoothed series provides a better indicator of longrun productivity trends.

##### Inputs

Inputs are divided into six categories: farm labor, agricultural land, two forms of capital inputs (farm machinery and livestock), and two types of intermediate inputs (inorganic fertilizers and animal feed). The primary source of information is FAO, which published annual estimates beginning in 1961 (and for farm labor beginning in 1980) for each country, except for former Soviet Socialist Republics (SSRs) for which data begin in 1992. The time series for each SSR are extended back to 1980 using data from Shend (1993) and further to 1965 using data from Lerman et al. (2003).

Farm labor is the total number of adults (males and females) who are economically active in agriculture. FAO currently publishes farm labor estimates and projections for each country of the world from 1980 to 2020, although previously FAO also published estimates for 1961-1979. FAO estimates are used for each country except China, Nigeria, and transition economies (former Soviet Union and Eastern Europe). Labor data for 1961-1980 are estimated using the agricultural labor force growth rates derived from the 2006 version of FAO labor force statistics which include estimates from 1961 onward. For China, agricultural labor estimates are from the Statistical Yearbooks of the National Bureau of Statistics of China. For transition economies, national agricultural statistical sources are used, as reported in EUROSTAT for the Baltic countries and Eastern Europe, CISSTAT for Russia, Belorussia and Moldova, the International Labor Organization’s LABORSTA for Ukraine, and the Asian Development Bank for Asiatic former Soviet republics. Pre-1992 labor estimates for these countries are from Shend (1980) and Lerman et al. (2003).

Nigeria is the largest economy and the largest agricultural producer in Sub-Saharan Africa, although socio-economic data for this country, particular of its population and labor force, remain subject to considerable uncertainty. Fuglie and Rada (2013) determined that FAO farm labor force estimates for this country were likely to be grossly undercounted. However, recent national labor force surveys indicate a more accurate count may lie in between these estimates. Sackey et al. (2012) cite surveys from the mid-2000s that find about 40 percent of Nigeria’s labor were primarily employed in agriculture, compared with less than 25 percent estimated by FAO. Although Sackey et al. (2012) caution that population and labor force surveys from Nigeria are subject to a wide degree of uncertainty and possible error, the pattern that emerges is one of a sharp reduction in the agricultural labor force share in the 1970s, followed by a slower rate of decline in the share since the 1980s. This is consistent with Collier (1988), who suggested that Nigeria’s 1970s oil boom led to a deterioration in the agricultural terms of trade and encouraged farm labor flight to urban areas, while the post-boom economic contraction and structural adjustment imposed in the 1980s significantly slowed this process.

To derive a more plausible estimate of agricultural labor in Nigeria, we make use of historical population census and labor force survey data from Nigeria to derive the share of total labor employed in agriculture, and how this share has likely evolved over time. Multiplying this share by the International Labor Organization’s estimate of Nigeria’s total labor force gives the size of the agricultural labor force. Figure 1 shows our estimate of the agricultural employment share for Nigeria and the census and survey estimates from which it is derived. Note that two of the labor force survey estimates (from 1983 and 2010) show substantially lower agricultural employment shares than the other surveys. The 1983 and 2010 estimates are assumed to be unrepresentative outliers and are not used to derive the new estimate of the agricultural labor force for Nigeria. Note that in absolute size, the new estimate implies that the agricultural labor force was growing at around 2% per year since the 1960s, except from around 1970 to the mid-1980s when the agricultural labor force remained roughly constant. Given the importance of Nigeria to Africa’s economy, improving the reliability of economic data from this country would be of great benefit to understanding patterns of productivity growth in the region.

##### Figure 2. Estimating the share of agricultural employment in Nigeria's labor force

Agricultural land is the area in permanent crops (perennials), annual crops, and permanent pasture. Cropland (permanent and annual crops) is further divided into rainfed area and area equipped for irrigation. The areas of rainfed cropland, irrigated area, and permanent pasture are then aggregated into a quality-adjusted measure that gives greater weight to irrigated cropland and less weight to permanent pasture to account for relative land productivity (see the next section on Land Quality). However, for agricultural cropland in Sub-Saharan Africa total area harvested for all crops is used rather than the FAO cropland series (Fuglie and Rada, 2013). For China we use sown crop area (National Bureau of Statistics of China) for cropland, given unreasonable discontinuities in the cropland series of both the FAO and Chinese government sources (Fan and Zhang, 2002). For New Zealand, FAO cropland series prior to 2002 fails to reflect changes in a consistent definition over time. We therefore use the area in grain, seed, fodder, and horticultural crops from Statistics New Zealand (2003) for 1961-2001, and FAO data from 2002 onward. For similar reasons, cropland in Indonesia prior to 1990 is based on national agricultural statistics as reported in Fuglie (2010).

Farm machinery is the total metric horse-power (CV) of major farm equipment in use. It is the aggregation of the number of 4-wheel riding tractors, 2-wheel pedestrian tractors, power harvester-threshers, and milking machines, expressed in "40-CV tractor-equivalents." The average CV per machine is assumed to be 40 CV per 4-wheel tractor, 12 CV per 2-wheel tractor, 20 CV per power harvester-thresher, and 1 CV per milking machine. However, due to insufficient information no adjustment is made for differences across countries or over time in farm machinery sizes within these categories, except for China, which reports farm machinery inventories in power units (National Statistical Bureau of China). Also, for Indonesia, the FAO figure for the number of power thresher-harvesters in use includes both pedal and power threshing machines. We include only power thresher-harvesters from Indonesian national statistics, as reported in Fuglie (2010).

The FAO reports continuous time series data for 4-wheel tractors, harvest-threshers and milking machines, but not 2-wheel walking tractors. For many developing countries, particularly in Asia, 2-wheel tractors have been a major component of farm mechanization. For 2-wheel tractors, FAO reports numbers in use for 1970s but then discontinued this series until recommencing it in 2002. For interim years, national farm machinery statistics were collected on 2-wheel tractors in use from the agricultural censuses of China, Japan, South Korea, Taiwan, Thailand, Philippines, Indonesia, Indian, Bangladesh, Pakistan, and Sri Lanka, and interpolated between census years. These countries constitute most of the global use of 2-wheel tractors in use on farms.

Presently, FAO farm machinery statistics only extend to 2009 (and for many countries they may not extend past 2005). To extend estimates of farm machinery to 2012, national statistics on the number of tractors and combine-harvested from more recent years were collected for a number of countries: Bangladesh (Hassan, 2013), China (National Statistical Bureau of China, 2014), Europe (Eurostat), India (Singh et al., 2015), Japan (Ministry of Agriculture, Forestry and Fisheries, 2012), Russia (Russian Federation Federal State Statistics Service, 2015), and the United States (National Agricultural Statistical Service, 2014).

To extend estimates of farm machinery stocks held on farms beyond the last available census or survey estimate, two approaches were used. The first approach uses annual data on new machinery sales, taking into account obsolescence of older machinery, assuming a 15-year useful lifespan for new farm machinery. Data on annual sales of new farm tractors and combine-harvesters during 1991-2013 were collected from farm machinery manufacturers (sources: VDMA, Verband Deutscher Maschinen- und Anlagenbau, or the Mechanical Engineering Industry Association, and John Deere corporate reports) for the United States, Canada, Brazil, Argentina, Mexico, South Africa, and European countries. Estimates of farm-held machinery stocks were extended from the latest available census year by adding the number of new machinery sales since the census year and subtracting the number of tractors purchased 15 years earlier. In other words, if Mc is the stock of machines held on farms in census year c, then the number estimated to be held on farms in year c+1 is:

$M_{c+1}=&space;M_{c}+S_{c+1}-S_{c+1-15}$

(7)

where Sc+1 is the number of new machinery sales in year c+1 and Sc+1-15is the number of sales 15 years prior to year c+1. Farm machinery stock in year c+2 is estimated as Mc+2 = Mc+1 + Sc+2 – Sc+2-15, and so on for subsequent years. Individual types of farm machinery were then aggregated into the total stock of metric horse power held on farms using the weights described in the Data section.

The second approach, used for countries for which we do not have information on annual machinery sales, uses an econometric model to estimate the change in farm-held machinery stocks over time since the latest available data on farm machinery stocks. The econometric model is based on the Kislev-Peterson model of farm machinery adoption and farm size. Kislev and Peterson (1982) hypothesized that as non-farm wages rose, farm labor would be induced to migrate to the non-farm sector. This would stimulate farm consolidation and mechanization to replace the labor leaving farms. Thus, farm mechanization would be correlated with non-farm wages and farm size. While the Kislev-Peterson model was developed in specific reference to the United States, in a comparative historical assessment of agricultural mechanization, Binswanger (1986) found a "remarkable similarity in the early mechanization experiences of developed and developing countries." He showed that farm machinery was typically first used for power-intensive operations such as tillage and transport, while mechanization of control-intensive operations like weeding and fruit picking came later, especially in response to rising wages.

Using panel data on countries from 1990 to 2003, we estimated the following fixed effects model:

$ln\left&space;(&space;\frac{CV}{Worker}&space;\right&space;)=&space;\alpha&space;_{c}&space;+&space;\beta_{\gamma&space;}ln\left&space;(&space;\frac{Cropland}{Worker}\right&space;)+\gamma&space;_{\gamma&space;}ln\left&space;(&space;\frac{GDP}{Population}&space;\right&space;)$

(8)

where CV/Worker = metric horse power of farm machinery per agricultural worker, Cropland/worker = hectares of cropland per agricultural worker (representing average farm size), GDP/Population = GDP per capita in constant 2005 PPP\$ (a proxy for non-farm wages), and α, β, and γ, are parameters to be estimated. The parameter values for β and γ vary by region r (the five regions include developing countries in Asia, Latin America, Sub-Saharan Africa, West Asia-North Africa, Transition countries of the former Soviet Union and eastern Europe, and all other developed countries). Since this is a fixed effect model, the intercept term varies for each country c to account for unobserved country-specific factors.

With parameter estimates of β and γ, (shown as β hat and γ hat in the formula below) then the percent change in CV/Worker can be estimated for countries and years for which data on CV are missing (the symbol ∆ln(quantity) below refers to the percent rate of change in the quantity in parentheses):

$\Delta&space;ln\left&space;(&space;\frac{CV}{Worker}&space;\right&space;)=&space;\hat{\beta&space;_{\gamma&space;}}\Delta&space;ln\left&space;(&space;\frac{Cropland}{Worker}&space;\right&space;)+\hat{\gamma&space;_{\gamma&space;}}\Delta&space;ln\left&space;(&space;\frac{GDP}{Population}&space;\right&space;)$

(9)

The annual growth rate in the total stock of farm machinery is simply the growth rate in CV per agricultural worker given by the formula above plus the growth rate in total agricultural workers. Using this growth rate estimated for each year, the stock of farm machinery is then extended from the last census observation or FAO estimate available.

The following table gives the econometric estimates of β and γ

 Region β γ ASIA 0.2753 1.5334 SSA 0.1046 0.0000 LAC 0.1064 0.1168 WANA 1.4186 0.6628 Transition 1.0201 0.1808 Developed 0.5275 0.3410

All coefficients are statistically significant at the 1 percent level except the estimate of γ for Sub-Saharan Africa (which is therefore fixed at zero in the regression). The results suggest that rising non-farm wages has been relatively more important in Asia in stimulating farm mechanization compared with other regions.

Livestock Capital is the aggregate value of animals used for breeding, milking, egg laying, wool production, and to provide animal traction. To approximate livestock capital, total inventories of animals on farms, measured in "cattle equivalents" are used. Inventories include dairy cows, other cattle, water buffalo, camels, horses, other equine species (asses, mules, and hinnies), small ruminants (sheep and goats), pigs, and poultry species (chickens, ducks, and turkeys), with each species weighted by its relative size. The weights for aggregation are based on Hayami and Ruttan (1985, p. 450): 1.38 for camels, 1.25 for water buffalo, dairy cows and horses, 1.00 for other cattle and other equine species, 0.25 for pigs, 0.13 for small ruminants, and 12.50 per 1,000 head of poultry.

Fertilizer is the amount of major inorganic nutrients applied to agricultural land annually, measured as metric tons of N, P2O5, and K2O nutrients. The source of the data is the International Fertilizer Association, except for small countries, which is from FAO.

Animal Feed is the total amount of crop (except fodder), animal, and fish products used for feed, measured in metric tons of dry-matter (DM) equivalents. Data on commodities used for animal feed are from the FAO Commodity Balance Sheets. Parameters for the DM for each type of feed are from the National Research Council (1982).

Other Inputs. While these six inputs account for the major part of total agricultural input usage, there are a few types of inputs for which complete country-level data are lacking, namely, use of chemical pesticides, seed, veterinary pharmaceuticals, energy, and services from farm structures. However, more detailed input data are available from several of the national studies from which input cost shares are derived (see section below on Input Cost Shares). To account for these inputs, we assume that their growth rate is correlated with one of the six input variables just described and include their cost with the related input. For instance, services from capital in farm structures as well as irrigation fees are included with the agricultural land cost share; the cost of chemical pesticide and seed is included with the fertilizer cost share; costs of veterinary medicines are included in the animal feed cost share, and energy costs are included in the farm machinery cost share. So long as the growth rates of the observed input and its unobserved counterparts are similar, then the model captures the growth of the unobserved inputs in the aggregate input index.

##### Land Quality

The FAO agricultural database provides time-series estimates of agricultural land by country and categorizes this as either permanent pasture or cropland (which is further divided in arable and permanent crop land). It also provides an estimate of area equipped for irrigation. The productive capacity of land among these categories and across countries can be very different, however. For example, some countries count vast expanses of semi-arid lands as permanent pastures even though these areas produce very limited agricultural output. Using such data for international comparisons of agricultural productivity can lead to serious distortions, such as significantly biasing downward the econometric estimates of the production elasticity of agricultural land (Peterson, 1987).

To account for the contributions to growth from different land types, irrigated cropland, rain-fed cropland, and permanent pastures are converted into "rainfed cropland equivalents" based on their relative productivity. Productivity weights vary regionally. In order not to confound the land quality weights with productivity change itself, the weights are estimated using country-level data from the beginning of the period of study (i.e., using average annual data from 1961-65). Let Regioni be a set of indicator variables representing five global regions (i=1,2,…5). For each country, Regioni takes a value of either 1 if the country is in the region and zero otherwise. Regions as (1) developed and former Soviet bloc countries, (2) Asia-Pacific, (3) Latin America and the Caribbean, (4) West Asia and North Africa, and (5) Sub-Saharan Africa. Define agricultural yield as total output Y divided by the sum of cropland and pasture area. We then regress agricultural yield against the proportions of agricultural land in rain-fed cropland (Rainfed), irrigated cropland (Irrig), and permanent pasture (Pasture). Multiplying the land-use proportions by the regional indicator variables allows the coefficients to vary among regions:

$ln\left(\frac{Y}{CroplandPasture}\right)=\sum_{i=1}^{5}\alpha_i(Rainfed*Region_i)+\\\Sigma_i\beta_i(Pasture*Region_i)+\Sigma_i\gamma_i(Irrig*Region_i)$

(10)

The coefficient vectors α, β and γ provide the quality weights for aggregating the three land types into an aggregate land input index. Countries with a higher proportion of irrigated land are likely to have higher average land productivity, as will countries with more cropland relative to pasture. The estimates of the parameters in equation (10) reflect these differences and provide a ready means of weighting the relative qualities of these land classes.

The regression estimates show that, on average, one hectare of irrigated land was between 1.1 to 3.0 times as productive as rainfed cropland, which in turn was 10-20 times as productive as permanent pasture. The results give plausible weights for aggregating agricultural land across broad quality classes. The approach to account for land quality differences among countries is similar to one developed by Peterson (1987), who derived land quality weights by regressing average cropland values in U.S. states against the share of irrigated and unirrigated cropland and long-run average rainfall. He then applied these regression coefficients to data from other countries to derive an international land quality index. The advantage of the present model is that it is based on international rather than U.S. land yield data and provides results for a larger set of countries.

This adjustment for changes in different classes of land allows us to further refine the resource decomposition of output growth in equation (6) to isolate the contribution of irrigation apart from expansion in agricultural area to output growth. Letting X1 be the quality-adjusted quantity of land (and for simplicity, dropping the Region subscripts on the land quality parameters), then a change in X1 is given by

$\Delta&space;X_1=\alpha\Delta(Cropland)+\beta&space;\Delta&space;(Pasture)+(\gamma&space;-&space;\alpha)\Delta(Irrig)$

(11)

The first two right-hand-side terms indicate the expansion in land area (with growth in pasture area adjusted for quality to put it in comparable terms with cropland expansion). The third term isolates the contribution of irrigation expansion:(γ-α)*100% gives the percent augmentation to yield, holding other factors fixed, from equipping a hectare of cropland with supplemental irrigation. Dividing equation (11) by X1 converts the expression into percentage changes so that it shows the respective contributions of changes in rainfed cropland, pasture area and irrigation to output growth. Combined with equation (6), the resource decomposition expression shows the contributions to agricultural growth from expansion of agricultural land, extension of irrigation, intensification of other inputs per hectare, and improvements in TFP:

$g(Y)=\left[\theta_c\alpha&space;g(X_{1c})+\theta_p\beta&space;g(X_{1p})+\theta_w(\gamma-\alpha)g(X_{1w})\right]+\sum_{j=2}^{J}S_jg\left(\frac{X_j}{X_1}\right)+g(TFP)$

(12)

where θc,θp,and θw are the shares of quality-adjusted agricultural land in crops (X1c), pasture (X1p), and irrigated area (X1w), respectively (note:X1=X1c+X1p+ X1w). The first two terms [θcαg(X1c)+θpβg(X1p)] give the share of output growth attributable to land expansion (holding yield fixed), while the third term [θw(γ-α)g(X1w)] indicates the share of output growth due to the extension of irrigation (holding other inputs fixed). The fourth term of equation (12) gives the contribution to growth of input intensification and the last term the contribution of growth in total factor productivity.

##### Input Cost Shares

The FAO (and supplementary) quantity data allow us to calculate the growth rates for six categories of production inputs (land, labor, machinery capital, livestock capital, and material inputs represented by fertilizer and feed), but to combine these into an aggregate input measure requires information on their cost shares or production elasticities. For this we draw upon 19 studies that have estimated nationally or regionally representative cost shares or production elasticities for agricultural inputs. These costs shares are assumed to be representative of not only those nations but also for other countries in the same region. For instance, the cost shares from India were applied to other countries in South Asia, the cost shares for Indonesia were applied to other countries in Southeast Asia and the Pacific, the cost shares for Mexico were assigned to other countries in Central America and the Caribbean, and the cost shares for Brazil were applied to other countries in South America as well as the North Africa-West Asia region. These assignments were based on judgments about the resemblance among the agricultural sectors of these countries. Countries assigned to the cost shares from Brazil tended to be middle-income countries having relatively large livestock sectors, for example. For agricultural capital, some of these studies only reported an aggregate cost share for all capital services. To partition capital services into machinery and livestock capital services, the average proportions of capital stock in machinery, livestock and tree capital for low, middle and high income countries reported in Butzer, Mundlak and Larson (2012) are used. The cost share of capital services from trees is assigned to land.

While the lack of direct observations on input cost shares for most countries introduces uncertainty in the TFP estimation, the countries for which cost shares are observed represent about 65 percent of the global agricultural economy. This proportion rises to three-fourths when Sub-Saharan Africa and the former Soviet Union are included – regions where econometrically-estimated production elasticities are used in place of cost shares. Thus, countries to which input cost shares were imputed represent only one-quarter of world agricultural output. Another argument in support of this approach is that there is a significant degree of congruence among the cost shares reported for these country studies. For the developing countries for which cost shares data are available (India, Indonesia, China, Brazil and Mexico), farm-supplied inputs (land, labor, and livestock capital) account for between 60 and 90 percent of total costs, while inputs supplied by industry (machinery, or fixed capital, and purchased materials such as fertilizers and processed animal feed), accounted for a far smaller share of resources. The cost share of inputs supplied by industry rises with the income of a country, and accounts for a third or more of total costs in the more highly industrialized countries. The use of modern inputs in transition countries, on the other hand, fell sharply after reforms were initiated in the early 1990s. These patterns of input use is reflected in cost shares estimated or imputed for these countries.

##### Country and Regional Productivity

Using the methodology and data described above agricultural TFP indexes are estimated for nearly every country of the world on an annual basis beginning in 1961 (and since 1965 for the independent states of the former Soviet Union). However, some countries have dissolved or are too small to have complete data. For the purpose of estimating long-run productivity trends, some national data are aggregated to create consistent political units over time. For example, data from the nations that formerly constituted Yugoslavia are added together to make comparisons with productivity before Yugoslavia’s dissolution. Similarly, data were aggregated for Czechoslovakia, Ethiopia and the former Soviet Union (TFP series for individual SSR’s begin in 1965). Because some small island nations have incomplete or zero values for some agricultural data, three composite territories were constructed by adding up available data for island states in the Lesser Antilles, Micronesia, and Polynesia. Altogether, the countries included in the analysis account for more than 99.7 percent of FAO’s global gross agricultural output. The only areas not included in the analysis that have significant agricultural production are the West Bank and Gaza.

In addition to individual countries, data are aggregated and TFP indexes estimated at the regional level. Input and output quantity aggregation is straight forward since they are all measured in the same units (although not adjusted for quality differences in the inputs). Regional cost shares are the weighted averages of the national cost shares for the countries in a region.

See References for a list of citations mentioned above.