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## Findings, Documentation, and Methods

This data product provides estimates of productivity growth for the U.S. farm sector for 1948-2013, and estimates of the growth and relative levels of productivity for U.S. States for 1960-2004.

Note that the national data series has been revised (see details below), and updates of the State-level statistics are suspended in light of reduced ERS resources and the discontinuance of key source data series. The quality of the national statistics is preserved.

### Summary of Recent Findings

It is widely agreed that increased productivity is the main contributor to economic growth in U.S. agriculture. The level of U.S. farm output was about 2.7 times its 1948 level in 2013, growing at an average annual rate of 1.52 percent. Aggregate input use increased at a modest 0.05 percent annually in the same period, so the positive growth in farm sector output was very substantially due to productivity growth, which increased at an average 1.47 percent per year. But what exactly is productivity?

Single-factor measures of productivity, such as corn production per acre (yield or land productivity) or per hour of labor (labor productivity), have been used for many years because the underlying data are often easily available. While useful, such measures can also mislead. For example, yields could increase simply because farmers are adding more of other inputs, such as chemicals, labor, or machinery, to their land base. USDA produces measures of total factor productivity (TFP), taking account of the use of all inputs to the production process.

Specifically, annual productivity growth is the difference between growth of agricultural output and the growth of all inputs taken together. Productivity, therefore, measures changes in the efficiency with which inputs are transformed into outputs. USDA also produces State-level productivity measures (annual growth rates as well as cross-State differences in levels) of productivity, or differences in output per unit of combined inputs. Input measures are adjusted for changes in their quality, such as improvements in the efficacy of chemicals and seeds, changes in the demographics of the farm workforce, or innovations in machinery design. As a result, agricultural productivity is driven by innovations in onfarm tasks, changes in the organization and structure of the farm sector, and research aimed at improvements in farm production. In the short-term, measured agricultural productivity can also be affected by random events like weather.

• The level of U.S. farm output more than doubled between 1948 and 2013, growing at an average annual rate of 1.52 percent. Aggregate input use increased at a modest 0.05 percent annually in the same period, so the positive growth in farm sector output was very substantially due to productivity (sometimes referred to as total factor productivity, or TFP) growth, which increased at an annual 1.47 percent over the 1948-2013 period.
• During the 2007-13 subperiod, growth in output maintained an average annual rate of about 0.9 percent, comparable to that in the 2000-07 subperiod (the subperiods are measured from cyclical peak-to-peak in aggregate economic activity). During 2007-13, however, input use declined 0.54 percent per year (compared to increasing 0.32 percent per year over 2000-07), and measured average annual TFP growth rate rebounded from 0.6 percent during 2000-07 to 1.45 percent during 2007-13.

The full detail of data are available in table 1 , and a summary of agricultural output, input, and TFP change by subperiods is available in table 2 .

• U.S. agricultural productivity growth compares favorably to agricultural productivity growth in other industrialized countries, and to productivity growth in the overall U.S. economy, see Ball et al. (2010), for example.
• Analysis of the individual States reveals a positive and generally substantial average annual rate of productivity growth (see table 21 ). There was considerable variance, however. The median rate of growth over the 1960-2004 period was 1.67 percent per year. However, 9 of the 48 States had productivity growth rates averaging more than 2 percent per year. Only Oklahoma and Wyoming had average annual growth rates less than 1 percent per year. The reported average annual rates of growth ranged from 0.58 percent for Oklahoma to 2.58 percent for Oregon.
• Cumulated over the entire 45-year period covered by the State accounts, productivity growth in Oklahoma was responsible for only a 30 percent increase in that State's production. Over the same period, TFP growth in Oregon caused output in that State to increase by 319 percent between 1960 and 2004.
• The wide disparity in productivity growth rates had a significant impact on the rank ordering of States (see table 22 ). A key economic question is whether States with lower levels of productivity tend to grow faster than the technology leaders: are there forces (e.g., the diffusion of technical knowledge from the leading States to the more backward ones) that lead to convergence over time in the levels of productivity? See the discussion in the States section for details.

### Changes Introduced With This Release

To reflect the changes made by our data sources, such as NASS, ERS, BEA, and BLS, some output and input data have been revised back as far as 2000. In addition, methodologies for capital measurement and purchases of livestock have changed with this release, and some input estimates are revised back to 1948 when data are available, for consistency.

Changes in the measurement of labor input were necessitated by the adoption of new sources for data on employment, hours worked, and compensation per hour. Our original data source (the Farm Labor Survey administered by USDA's National Agricultural Statistics Service) was discontinued. The data on self-employed and unpaid family workers are now taken from the decennial Census of Population and the annual Current Population Survey. The National Income and Product Accounts (NIPA) are the source for data on employment, hours worked, and compensation of hired farm workers. The adoption of the new data sources has allowed us to extend our estimates of labor input (and hence productivity) through 2013, but has also required that we revise these series for prior years. In addition, the American Community Survey (ACS), was fully implemented in 2005 by the Census Bureau, is now part of the Decennial Census Program and has replaced the long-form sample questionnaire of the Census of Population. Therefore, we now rely on ACS micro data rather than the decennial Census of Population when developing estimates of labor input.

The share of purchased contract labor services in total production cost has increased over time in US farm production. Since farmers typically contract with labor brokers to assemble crews, there is a scarcity of data on hours worked. Only data on nominal expenditures for contract labor are collected. In order to account for the contribution of contract labor services to output growth, we must construct an appropriate deflator for these expenditures. Since the compensation of contract workers will likely vary with differences in demographic characteristics such as age, experience, gender, and education, we construct a deflator for contract labor using hedonic methods based on data from the National Agricultural Workers Survey.

In this release, purchases of livestock are not included in intermediate input as they were in previous releases. Rather, these animals represent "goods in progress." Acquisitions are, therefore, recorded as additions to stocks, while the cost of livestock purchases is deducted from livestock receipts.

We also introduced changes in the way we measure capital input. There has been a long-standing debate over whether an ex post or ex ante measure of the user cost of capital should be used in growth accounting. In the ex post approach (see, for example, Jorgenson and Griliches, 1967; Christensen and Jorgenson, 1969; Jorgenson, Gollop, and Fraumeni, 1987), it is assumed that the rate of return is equalized across assets. Then this unknown rate can be found by using the condition that the sum of returns across assets (where the return on an asset is the product of its user cost and the flow of services it yields) equals observed gross profits. The alternative ex ante approach (see Coen, 1975; Penson, Hughes, and Nelson, 1977; Diewert, 1980; Romain, Penson, and Lambert, 1987; Ball et al., 2008) employs a rate of return derived from financial market data, together with estimates of expected rather than actual asset price inflation. We adopt the latter approach. The ex ante rate is calculated as the nominal yield on investment grade corporate bonds adjusted for expected, rather than actual, price inflation. The ex ante rate is calculated as the nominal yield on investment grade corporate bonds adjusted for expected, rather than actual, price inflation. We introduce the use of asset-specific rates of price inflation as recommended by Shumway et al. (2014). Earlier (see Ball et al., 1997; 1999), the USDA used a broad measure of inflation, the implicit deflator for gross domestic product, to calculate the real rate of return, based on the theory that expected real rates of return should be equal across all assets.

Our estimates of the stock of land are based on county-level data on land area and value obtained from the Census of Agriculture. Data for the inter-census years are obtained through interpolation using spline functions. The Census reports the value of farm real estate (i.e., land and structures), as opposed to the value of land. Historically, the value of farm real estate was partitioned into components using information from the Agricultural Economics and Land Ownership Survey (AELOS). However, the AELOS was last published in 1999. More recently, we have relied on data from the annual Agricultural Resource Management Survey (ARMS) to partition real estate values into its components.

Pesticides and fertilizer are important intermediate inputs, but their data require adjustment since these inputs have undergone significant changes in input quality over the study period. Since input price and quantity series used in a study of productivity must be denominated in constant-efficiency units, we construct price indexes for fertilizers and pesticides from hedonic regression results. The corresponding quantity indexes are formed implicitly as the ratio of the value of each aggregate to its price index.

Finally, we report price indexes and implicit quantities (i.e., values of expenditures at constant 2005 prices) of the economic aggregates (e.g., output; capital; labor; intermediate goods), see the tab for table 1a . At the State level, these data are reported in panel format (see table 23 ). They can be used for both time series and cross section analysis.

### The Role of Productivity Growth in U.S. Agriculture

The rise in agricultural productivity has long been chronicled as the single most important source of economic growth in the U.S. farm sector. Though their methods differ in important ways, the major sectoral productivity studies by Kendrick and Grossman (1980) and Jorgenson, Gollop, and Fraumeni (1987) share this common conclusion. In a recent study, Jorgenson, Ho, and Stiroh (2005) find that productivity growth in agriculture averaged 1.9 percent over the 1977-2000 period. Output grew at a 3.4 percent average annual rate over this period. Thus productivity growth accounted for almost 80 percent of the growth of output in the farm sector. Moreover, only three of the forty-four sectors covered by the Jorgenson et al. (2005) study achieved higher rates of productivity growth than did agriculture.

The U.S. Department of Agriculture (USDA) has been monitoring agriculture's productivity performance for decades. In fact, in 1960, USDA was the first agency to introduce multifactor productivity measurement into the Federal statistical program. Today, the Department's Economic Research Service (ERS) routinely publishes total factor productivity (TFP) measures based on a sophisticated system of farm production accounts. Its TFP model is based on the transcendental logarithmic (translog) transformation frontier. It relates the growth rates of multiple outputs to the cost-share weighted growth rates of labor, capital, and intermediate inputs. See Ball et al. (1997, 1999) for a complete description of the USDA model.

The applied USDA model is quite detailed. The changing demographic character of the agricultural workforce is used to build a quality-adjusted index of labor input. Similarly, much asset-specific detail underlies the measure of capital input. The index of land input, for example, is constructed by aggregating county-level data. The contributions of feed, seed, energy, and agricultural chemicals are captured in the index of intermediate inputs. An important innovation is the use of hedonic price indexes in constructing measures of fertilizer and pesticide consumption. The result is a time series of total factor productivity indexes for the aggregate farm sector spanning the period 1948 to 2011 (see table 1 ). State-specific measures are available for the 1960-2004 period. The State series provides estimates of both the growth and relative levels of total factor productivity (see tables 3-23).

#### Aggregate U.S. Farm Sector Productivity

Input growth typically has been the dominant source of economic growth for the aggregate economy and for each of its producing sectors. Jorgenson, Gollop, and Fraumeni (1987) find this to be the case for the aggregate economy for every subperiod over 1948-79. Denison (1979) draws a similar conclusion for all but one subperiod, covering the longer period 1929-76. In their sectoral analysis, Jorgenson, Gollop, and Fraumeni find that output growth relies most heavily on input growth in 42 of 47 private business sectors in the 1948-79 period, and in a more aggregated study (Jorgenson and Gollop, 1992) that extends through 1985, in 8 of 9 sectors.

Agriculture turns out to be one of the few exceptions: productivity growth dominates input growth. This is confirmed in table 2  that reports the sources of output growth in the farm sector for the entire 1948-2013 period and 12 peak-to-peak subperiods. (The subperiods are not chosen arbitrarily, but are measured from cyclical peak to peak in aggregate economic activity. Since the data reported for each subperiod are average annual growth rates, the unequal lengths of the subperiods do not affect the comparisons across subperiods. This convention and these subperiods have been adopted by the major productivity studies). Applying the USDA model, output growth equals the sum of contributions of labor, capital, and materials inputs and TFP growth. The contribution of each input equals the product of the input's growth rate and its respective share in total cost.

The singularly important role of productivity growth in agriculture is made all the more remarkable by the dramatic contraction in labor input in the sector, a pattern that persists through every subperiod. Over the full 1948-2013 period, labor input declined at an average annual rate of 2.22 percent. When weighted by its 20-percent share in total costs, the contraction in labor input contributes an annual average -0.49 percentage point per year to output growth.

Capital input in the sector exhibits a different history. Its contribution to output growth alternates between positive and negative over the 1948-2013 period. On average, however, capital, like labor, contracts over the full period. Its negative growth contributes an annual -0.06 percentage point to output growth.

The negative contributions of both labor and capital are all the more notable given the positive contributions offered through improvements in both labor and capital quality, the recomposition of labor hours and capital stocks to higher marginal productivity sub-types. As revealed in table 2 , farms have shifted to higher-quality labor. This is primarily due to a more highly educated labor force. (As discussed earlier, labor hours are cross-classified by sex, employment class, age, and education. Analysis of the changing composition of hours over the 1948-2013 period reveals that, among these four sources, education was the only dimension making a positive contribution to labor quality. As overall labor hours declined, demographic shifts in the sex, employment class, and age composition of workers left higher proportions of hours worked in cells representing lower marginal productivity sex, class, age cohorts. In contrast, the decline in labor hours was coincident with an increase in the proportion of more highly educated workers. This was sufficient to offset the negative effects of the changing sex, class, and age composition of hours and results in the persistent pattern of improving labor quality throughout the full 1948-2013 period.) Increased labor quality made a positive contribution to output growth in 11 of 12 subperiods, averaging 0.12 percentage points per year. Quality improvements in capital added another 0.02 percentage points, yet neither improved labor nor capital quality was sufficient to offset the contraction in the corresponding input. Labor quality offsets less than 20 percent of the decline in raw labor hours; improvements in capital quality offset only about 26 percent of the decline in capital stocks.

Material input's contribution was positive in 9 of the 12 subperiods, but averaged a substantial positive rate equal to 0.60 percent per year. Though large, this positive contribution just offsets the negative contributions through labor and capital. The net contribution of all three inputs was 0.05 percentage points per year, leaving responsibility for positive growth in farm sector output to productivity growth in all but the 1973-79 subperiods.

Examining table 1 makes clear that the 1973-79 period is an outlier. Output, labor, and capital growth rates did not deviate much from trend. Materials input, however, exhibited significant positive growth at a rate far in excess of the incremental growth in output, accounting single-handedly for the measured decline in TFP growth. This anomaly appears to be due to rapid growth in export demand during this subperiod which resulted in both the increased consumption of intermediate inputs as well as a significant withdrawal of goods from inventory. Both led to a reduction in productivity growth.

The early 2000s saw the emergence of bio-fuels as a major source of demand for grains and oilseeds. Corn used in ethanol production in 2007 accounted for roughly one-quarter of total demand. The land area planted to corn increased some 15 million acres between 2006 and 2007, resulting in the cultivation of more marginal lands. Consumption of agricultural chemicals (i.e., fertilizer and pesticides) increased more than 10 percent. Yet yields per acre were largely unchanged.

In spite of these anomalous subperiods, TFP growth was truly extraordinary over the 1948-2013 period. As indicated in table 2, it averaged 1.47 percent per year. Cumulated over the full 65 year period, this average annual rate (compounded annually) implies that farm sector productivity in 2013 was 161 percent above its 1948 level. Given the cumulative 0.05 percent annual increase in total input growth between 1948 and 2013, productivity growth caused agricultural output to grow significantly in every subperiod so that by 2013, farm output was 169 percent above its level in 1948.

#### Measuring State Productivity

A properly constructed national measure of productivity growth for the aggregate farm sector provides a useful summary statistic indicating how economic welfare is being advanced through productivity gains in agriculture, but it may mask important State-specific or regional trends. For this reason, USDA has constructed estimates of the growth and relative levels of productivity for the 48 contiguous States for the 1960-2004 period (estimates are not made for Alaska and Hawaii). These indexes, expressed relative to the level of TFP in Alabama in 1996, are presented in table 19 along with their percentage rates of growth. In the table below, we rank the States by their level of TFP in 2004. We also include in the table each State's rank in 1960 and the average annual percentage growth from 1960 to 2004.

States ranked by level and growth of total factor productivity
StateRank in 2004Level in 2004Rank in 1960Level in 1960Average annual change, 1960-2004
RankChange (%)
California 1 1.7979 2 0.8643 25 1.66
Florida 2 1.6304 1 0.8649 38 1.44
Iowa 3 1.5297 4 0.6733 17 1.87
Illinois 4 1.5297 7 0.6456 11 1.96
Delaware 5 1.4345 6 0.6498 22 1.80
Idaho 6 1.4285 15 0.5891 8 2.01
Indiana 7 1.4220 27 0.5211 5 2.28
Rhode Island 8 1.4192 35 0.4766 2 2.48
Georgia 9 1.3891 12 0.5986 14 1.91
Massachusetts 10 1.3877 28 0.5069 4 2.29
Arizona 11 1.3847 3 0.7057 33 1.53
Arkansas 12 1.3705 16 0.5864 12 1.93
North Carolina 13 1.3554 11 0.6023 19 1.84
Connecticut 14 1.3209 29 0.5028 6 2.20
Oregon 15 1.3154 46 0.4231 1 2.58
New Jersey 16 1.2831 10 0.6161 24 1.67
Maryland 17 1.2457 19 0.5578 20 1.83
Minnesota 18 1.2359 23 0.5462 18 1.86
Ohio 19 1.2075 38 0.4673 7 2.16
Alabama 20 1.1791 5 0.6599 40 1.32
Nebraska 21 1.1619 17 0.5746 30 1.60
Maine 22 1.1458 31 0.4966 15 1.90
Washington 23 1.1457 25 0.5362 23 1.73
New York 24 1.1327 14 0.5898 36 1.48
Mississippi 25 1.1306 36 0.4738 10 1.98
South Carolina 26 1.1247 20 0.5531 28 1.61
Wisconsin 27 1.1130 22 0.5523 31 1.59
Michigan 28 1.1058 47 0.3832 3 2.41
Vermont 29 1.0762 26 0.5274 26 1.62
South Dakota 30 1.0760 21 0.5530 35 1.51
Pennsylvania 31 1.0601 34 0.4781 21 1.81
Colorado 32 1.0325 9 0.6359 45 1.10
North Dakota 33 1.0278 41 0.4461 16 1.90
Missouri 34 1.0212 30 0.5012 27 1.62
New Hampshire 35 1.0204 45 0.4231 9 2.00
Kansas 36 1.0124 8 0.6377 46 1.05
Louisiana 37 0.9904 44 0.4241 13 1.93
Virginia 38 0.9660 32 0.4933 34 1.53
Nevada 39 0.9640 18 0.5594 42 1.24
Utah 40 0.9638 33 0.4874 32 1.55
Kentucky 41 0.9403 40 0.4634 29 1.61
New Mexico 42 0.8925 37 0.4728 37 1.44
Texas 43 0.8873 24 0.5376 43 1.14
Montana 44 0.8145 42 0.4447 39 1.38
Oklahoma 45 0.7693 13 0.5962 48 0.58
Tennessee 46 0.7648 39 0.4661 44 1.13
West Virginia 47 0.5777 48 0.3278 41 1.29
Wyoming 48 0.5712 43 0.4281 47 0.66
These data are available in Excel as Table 22 .

One remarkable similarity exists across all States for the full 1960-2004 period. Every State exhibited a positive and generally substantial average annual rate of TFP growth. There is considerable variance, however. The median TFP growth rate over the 1960-2004 period was 1.67 percent per year. However, 9 of the 48 States had productivity growth rates averaging more than 2 percent per year. Only Oklahoma and Wyoming had average annual rates of growth less than 1 percent per year. The reported average annual rates of growth ranged from 0.58 percent for Oklahoma to 2.58 percent for Oregon (see map above). Cumulated over the entire 45-year period, productivity growth in Oklahoma was responsible for only a 30-percent increase in that State's output. Over the same period, TFP growth in Oregon resulted in a 319-percent increase in the State's agricultural output.

A key question is whether States with lower levels of productivity tend to grow faster than the technology leaders: are there forces (e.g., the diffusion of technical knowledge from the leading States to the more backward ones) that lead to convergence over time in the levels of productivity?

A number of studies find evidence of convergence. McCunn and Huffman (2000) found evidence of "catching-up" in levels of TFP (i.e., β-convergence), although they rejected the hypothesis of declining cross-sectional dispersion (i.e., σ-convergence). Ball, Hallahan, and Nehring (2004) also found evidence of convergence in levels of productivity after controlling for differences in relative factor intensities (i.e., embodiment). (Their tests for convergence are conditional on these variables. In the literature on the empirics of growth (see Barro and Sala-i-Martin, 1992), this is referred to as conditional convergence.)

More recently, Ball, San Juan, and Ulloa (2014) examined the relation between the business cycle and convergence in levels of agricultural productivity. They found evidence of convergence in TFP levels across the different phases of the business cycle, but the speed of convergence was greater during periods of contraction in economic activity than during periods of expansion.

Finally, the expected pattern of convergence across the business cycle finds some empirical support. This pattern is the result of the pro-cyclical nature of innovation and the time lags in the diffusion of technical information. In contrast with, say the manufacturing sector, however, the magnitude of the effects of the business cycle through the rate of convergence appears to be smaller in the agricultural sector. The authors attribute this to public funding of R&D in the agricultural sector. Since innovations resulting from public R&D can be considered public goods that firms can adopt relatively quickly, the diffusion of technical information will be more rapid in agriculture and this points to a smaller impact of the business cycle on TFP convergence.

### Methods

Ball et al. (2004) estimated each State's growth and relative level of productivity for the period 1960-99 using an index number approach, and this method is used to extend the series through 2004. A productivity index is generally defined as an output index divided by an index of inputs. The individual State productivity indices are formed from Fisher quantity indices of outputs and inputs. In comparing relative levels of productivity, we first construct bilateral Fisher indices of output and input among States. Unfortunately, there is no guarantee of transitivity in such comparisons, i.e. direct comparisons between two States may give different results than making indirect comparisons through other States. Eltetö and Köves (1964) and Szulc (1964) proposed independently a method ("EKS" index) which achieves transitivity while minimizing the deviations from the bilateral comparisons.

The EKS index is based on the idea that the most appropriate index to use when comparing two States is the binary Fisher index. However, when the number I of States in a comparison is greater than two, the application of the Fisher index number procedure to the I(I-1)/2 possible pairs of States gives results that do not satisfy Fisher's circularity test. The problem, therefore, is to obtain results that satisfy transitivity, and that deviate the least from the bilateral Fisher indexes.

Let ${Q}^{jk}_{F}$ denote the bilateral Fisher quantity index for State j relative to State k. If ${Q}^{jk}_{EKS}$ denotes the multilateral quantity index, then the EKS method suggests that ${Q}^{jk}_{EKS}$ should deviate the least from the bilateral quantity index ${Q}^{jk}_{F}$. Thus, ${Q}^{jk}_{EKS}$ should minimize the distance criterion:

$\sum_{j=1}^{I}\sum_{k=1}^{I}{\left(\ln{Q}^{jk}_{EKS}-{\ln{Q}^{jk}_{F}\right)}^{2}$

Using the first-order conditions for a minimum, it can be shown that the multilateral quantity index with the minimum distance is given by:

${Q}^{jk}_{EKS}={\left(\prod_{i=1}^{I}{Q}^{ji}_{F}\cdot{Q}^{ik}_{F}&space;\right)}^{1/I},&space;j,k,=1,...I$

The EKS quantity index may, therefore, be expressed as the geometric mean of the I indirect comparisons of j and k through other States.

We have constructed EKS indices of relative levels of output and input among all 48 States for a single base year. We have also constructed these quantity indices for each State for the period 1960-2004. We obtain indexes of output and input quantities in each State relative to those in the base State for each year by linking these time-series quantity indexes with estimates of relative output and input levels for the base period. Tables 3-22 present indexes of relative output, input, and productivity levels among the States for the period 1960-2004, with a base equal to unity in Alabama in 1996.

Production accounts used in constructing these indices are derived from State and aggregate accounts for the farm sector constructed by USDA. The accounts are consistent with a gross output model of production. Output is defined as gross production leaving the farm, as opposed to real value added. The existence of the value-added function requires that intermediate inputs be separable from primary inputs (capital and labor). This places severe restrictions on marginal rates of substitution that are not likely to be realistic. Moreover, even if the value-added function exists, the exclusion of intermediate inputs assigns all measured technical progress to capital and labor inputs, ruling out increased efficiency in the use of purchased inputs. Accordingly, inputs are not limited to capital and labor, but include intermediate inputs as well. Both State and aggregate accounts view all of agriculture within their respective boundaries as if it were a single farm. Output includes all off-farm deliveries but excludes intermediate goods produced and consumed on the farm. The difference is that output in the aggregate accounts is defined as deliveries to final demand and intermediate demands in the non-farm sector. State output accounts include these deliveries plus interstate shipments to intermediate farm demands.

The next section is organized by component measures. Except where indicated, the methods described apply equally to States and aggregate farm sector:

### Output

The output measure begins with disaggregated data for physical quantities and market prices of crops and livestock compiled for each State. The output quantity for each crop and livestock category consists of quantities of commodities sold off the farm, additions to inventory, and quantities consumed as part of final demand in farm households during the calendar year. Off-farm sales in the aggregate accounts are defined only in terms of output leaving the sector, while off-farm sales in the State accounts include sales to the farm sector in other States as well.

One unconventional aspect of our measure of total output is the inclusion of goods and services from certain non-agricultural or secondary activities. These activities are defined as activities closely linked to agricultural production for which information on output and input use cannot be separately observed. Two types of secondary activities are distinguished. The first represents a continuation of the agricultural activity, such as the processing and packaging of agricultural products on the farm, while services relating to agricultural production, such as machine services for hire, are typical of the second.

The total output of the industry represents the sum of output of agricultural goods and the output of goods and services from secondary activities. We evaluate industry output from the point of view of the producer; that is, subsidies are added and indirect taxes are subtracted from market values.

### Intermediate Input

Intermediate input consists of goods used in production during the calendar year, whether withdrawn from beginning inventories or purchased from outside the farm sector or, in the case of the State production accounts, from farms in other States. Open-market purchases of feed and seed inputs enter both State and aggregate farm sector intermediate goods accounts. Withdrawals from producers' inventories are also measured in output, intermediate input, and capital input. Beginning inventories of crops and livestock represent capital inputs and are treated in the discussion of capital below. Additions to these inventories represent deliveries to final demand and are treated as part of output. Goods withdrawn from inventory are symmetrically defined as intermediate goods and recorded in the farm input accounts.

Data on current dollar consumption of petroleum fuels, natural gas, and electricity in agriculture are compiled for each State for period 1960-2004. Prices of individual fuels are taken from the Energy Information Administration's Monthly Energy Review. The index of energy consumption is formed implicitly as the ratio of total expenditures (less State and Federal excise tax refunds) to the corresponding price index.

Pesticides and fertilizers have undergone significant changes in input quality over the study period. Since input price and quantity series used in a study of productivity must be denominated in constant-efficiency units, we construct price indexes for fertilizers and pesticides using hedonic methods. Under this approach, a good or service is viewed as a bundle of characteristics which contribute to the productivity (utility) derived from its use. Its price represents the valuation of the characteristics "that are bundled in it", and each characteristic is valued by its "implicit" price (Rosen, 1974). However, these prices are not observed directly and must be estimated from the hedonic price function.

A hedonic price function expresses the price of a good or service as a function of the quantities of the characteristics it embodies. Thus, the hedonic price function for, say pesticides, may be expressed as Wp = W(X,D), where Wp represents the price of pesticides, X is a vector of characteristics or quality variables, and D is a vector of other variables.

Kellogg et al. (2002) have compiled data on characteristics that capture differences in pesticide quality. These characteristics include toxicity, persistence in the environment, and leaching potential, among others.

Other variables (denoted by D) are also included in the hedonic equation, and their selection depends not only on the underlying theory but also on the objectives of the study. If the main objective of the study is to obtain price indexes adjusted for quality, as in our case, the only variables that should be included in D are time or State dummy variables, which will capture all price effects other than quality. After allowing for differences in the levels of the characteristics, the part of the price difference not accounted for by the included characteristics will be reflected in the coefficients on the dummy variables.

Economic theory places few if any restrictions on the functional form of the hedonic price function. We adopt a generalized linear form, where the dependent variable and each of the continuous independent variables is represented by the Box-Cox transformation. This is a mathematical expression that assumes a different functional form depending on the transformation parameter, and which can assume both linear and logarithmic forms, as well as intermediate non-linear functional forms.

Thus the general functional form of our model is given by:

$W_p(\lambda_0)=\sum_{n}^{&space;}\alpha_nX_n(\lambda&space;_n)+\sum_{d}^{&space;}\gamma&space;_dD_d+\varepsilon,$

where $W_p(\lambda_0)$  is the Box-Cox transformation of the dependent price variable, Wp > 0; that is,

Similarly, is the Box-Cox transformation of the continuous quality variable where if and if . Variables represented by D are time dummy variables, not subject to transformation;  and  are unknown parameter vectors, and  is a stochastic disturbance.

Finally, price and implicit quantity indexes are calculated for the remaining intermediate inputs, a variety of purchased services such as contract labor services, custom machine services, machine and building maintenance and repairs, and irrigation from public sellers of water. Indexes of total intermediate input are calculated for each State and the aggregate farm sector by aggregating across each category of intermediate input described above.

### Labor Input

The USDA labor accounts for the aggregate farm sector incorporate the demographic cross-classification of the agricultural labor force developed by Jorgenson, Gollop, and Fraumeni (1987). Matrices of hours worked and compensation per hour have been developed for laborers cross-classified by sex, age, education, and employment class (employee versus self-employed and unpaid family workers).

In addition, the ERS has developed a set of similarly formatted but otherwise demographically distinct matrices of labor input and labor compensation by State. This is accomplished using the RAS procedure popularized by Jorgenson, Gollop, and Fraumeni (1987) by combining the aggregate farm sector matrices with State-specific demographic information available from the decennial Census of Population (U.S. Department of Commerce). The result is a complete State-by-year panel data set of annual hours worked and hourly compensation matrices with cells cross-classified by sex, age, education, and employment class and with each matrix controlled to the USDA hours-worked and compensation totals, respectively.

Indexes of labor input were constructed for each State and the aggregate farm sector using the demographically cross-classified hours and compensation data. Labor hours having higher marginal productivity (wages) are given higher weights in forming the index of labor input than are hours having lower marginal productivities. Doing so explicitly adjusts indexes of labor input for “quality” change in labor hours as originally defined by Jorgenson and Griliches (1967).

### Capital Input

This study requires measures of capital input and capital service prices for each State and the aggregate farm sector. Construction of these series begins with estimating the capital stock and rental price for each asset type. For depreciable assets, the perpetual inventory method is used to develop stocks from data on investment. For land and inventories, capital stocks are measured as implicit quantities derived from balance sheet data. Implicit rental prices for each asset are based on the correspondence between the purchase price of the asset and the discounted value of future service flows derived from that asset.

#### Depreciable Assets

Under the perpetual inventory method, capital stock at the end of each period, , is measured as the sum of all past investments, each weighted by its relative efficiency, ${d}_{\tau$:

$K_t=\sum_{\tau&space;=0}^{\infty&space;}d_\tau&space;I_t_-_\tau$

where ${d}_{\tau$ is approximated by a hyperbolic efficiency function

${d}_{\tau}=(L-\tau)/(L-\beta\tau),&space;&space;0\leq&space;\tau\leq&space;L$
${d}_{\tau}=0,&space;\tau\geq&space;L$

and where L is the service life of the asset, τ represents the asset’s age, and β is a curvature or decay parameter. The value of ß is restricted only to values less than or equal to one. For values of ß greater than zero, the efficiency of the asset approaches zero at an increasing rate. For values less than zero, efficiency approaches zero at a decreasing rate.

Little empirical evidence is available to suggest a precise value for ß. However, two studies (Penson, Hughes and Nelson, 1977; Romain, Penson and Lambert, 1987) provide evidence that efficiency decay occurs more rapidly in the later years of service, corresponding to a value of ß in the 0 to 1 interval. For purposes of this study, it is assumed that the efficiency of a structure declines slowly over most of its service life until a point is reached where the cost of repairs exceeds the increased service flows derived from the repairs, at which point the structure is allowed to depreciate rapidly (ß=0.75). The decay parameter for durable equipment (ß=0.5) assumes that the decline in efficiency is more uniformly distributed over the asset's service life.

Consider now the efficiency function that holds ß constant and allows L to vary. This concept of variable service lives is related to the concept of investment, where investment is a bundle of different types of capital goods. Each of the different types of capital goods is a homogeneous group of assets in which the actual service life, L, is a random variable reflecting quality differences, maintenance schedules, etc. For each asset type, there exists some mean service life, , around which there exists some distribution of actual service lives. In order to determine the amount of capital available for production, the actual service lives and their frequency of occurrence must be determined. It is assumed that the underlying distribution is the normal distribution truncated at points two standard deviations above and below the mean service life.

The other critical variable in the efficiency function is asset lifetime L. For each asset type, there exists some mean service life, , around which there exists some distribution of actual service lives. It is assumed that the underlying distribution is the normal distribution truncated at points two standard deviations above and below the mean service life. Mean service lives correspond to 85% of the U.S. Department of the Treasury’s Bulletin F lives.

#### Capital Rental Prices

Firms will add to the capital stock so long as the present value of the net revenue generated by an additional unit of capital exceeds the purchase price of the asset. This can be stated algebraically as:

$\sum_{t=1}^{\infty}\left(p\frac{\partial{y}}{\partial{K}}-w_K\frac{\partial{R_t}}{\partial{K}}\right)({1}{+}{r})^{-t}>{w_K}$

where p is the price of output, wK is the price of investment goods, and r is the real discount rate. To maximize net present value, firms will continue to add to capital stock until this equation holds as an equality:

$p\frac{\partial{y}}{\partial{K}}=r{w}_{K}+{r}\sum_{t=1}^{\infty}{w}_{K}\frac{\partial{R}_{t}}{\partial{K}}\left(1+r\right)^{-t}=c$

where c is the implicit rental price of capital.

The rental price consists of two components. The first term, rwK, represents the opportunity cost associated with the initial investment.

The second term,${r}\sum_{t=1}^{\infty}{w}_{K}\frac{\partial{R}_{t}}{\partial{K}}\left(1+r\right)^{-t}$ is the present value of the cost of all future replacements required to maintain the productive capacity of the capital stock.

Let F denote the present value of the stream of capacity depreciation on one unit of capital according to the mortality distribution m:

$F=\sum_{t=1}^{\infty}{m}_{\tau}{\left(1+r&space;\right)}^{-\tau},$

where ${m}_{\tau&space;}=-\left(&space;{d}_{\tau}-{d}_{\tau-1\right),\:&space;&space;\tau=1,...,t.$

Since replacement at time t is equal to capacity depreciation at time t:

$\sum_{t=1}^{\infty}\frac{\partial{R}_{t}}{\partial{K}}\left(1+r\right)^{-t}=\sum_{t=1}^{\infty}&space;{F}^{t}=\frac{F}{(1-F)}$

so that

$c=\frac{r{w}_{K}}{(1-F)}.$

The real rate of return, r in the above expression, is calculated as the nominal yield on investment grade corporate bonds less the rate of inflation as measured by the implicit deflator for gross domestic product. An ex ante rate is then obtained by expressing observed real rates as an ARIMA process. We then calculate F holding the required real rate of return constant for that vintage of capital goods. In this way, implicit rental prices c are calculated for each asset type.

Finally, indexes of capital input for each State and the aggregate farm sector are constructed by aggregating over the different capital assets using as weights the asset-specific rental prices. As is the case for labor input, the resulting measure of capital input for each State and the aggregate farm sector is adjusted for changes in asset "quality."

#### Land Input

To obtain a constant-quality land stock, we first construct intertemporal price indexes of land in farms. The stock of land is then constructed implicitly as the ratio of the value of land in farms to the intertemporal price index. We assume that land in each county is homogeneous, hence aggregation is at the county level.

Differences in the quality of land across States and regions prevent the direct comparison of observed prices. To account for these quality differences, we calculate relative prices of land from hedonic regression results.

As noted above, the hedonic approach views land as a bundle of characteristics which contribute to output derived from its use. The World Soil Resources Office of the USDA's Natural Resource Conservation Service has compiled data on land characteristics (see Eswaren, Beinroth, and Reich (2003)). They develop a procedure for evaluating inherent land quality, and use this procedure to assess land resources on a global scale. Given the Eswaren, Beinroth, and Reich database, we use GIS to overlay State and county boundaries. The result of the overlay gives us the proportion of land area of each county that is in each of the soil stress categories. These characteristics include soil acidity, salinity, and moisture stress, among others. The "level" of each characteristic is measured as the percentage of the land area in a given region that is subject to stress. A detailed description of the characteristics included in the hedonic model is provided in Ball et al. (2007). The environmental attributes most highly correlated with land prices in major agricultural areas are moisture stress and soil acidity. In areas with moisture stress, agriculture is not possible without irrigation. Hence irrigation (i.e., the percentage of the cropland that is irrigated) is included as a separate variable. Because irrigation mitigates the negative impact of acidity on plant growth, the interaction between irrigation and soil acidity is included in the vector of characteristics.

In addition to environmental attributes, we also include a "population accessibility" score for each county in each State. These indices are constructed using a gravity model of urban development, which provides a measure of accessibility to population concentrations. A gravity index accounts for both population density and distance from that population. The index increases as population increases and/or distance from that population center decreases.

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Last updated: Monday, May 09, 2016

For more information contact: Eldon Ball, Sun Ling Wang, and Richard Nehring