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## Documentation and Methods

Note: Updates of the State-level statistics are suspended. (Quality of national statistics is preserved.)

This data set provides estimates of productivity growth for the aggregate farm sector for the period 1948-2009, and estimates of the growth and relative levels of productivity for the individual States for the period 1960-2004.

Major findings of the data include:

• Agricultural output growth slowed during the 2000-2007 subperiod (the subperiods are measured from cyclical peak to peak in aggregate economic activity) to a 0.77 percent average annual rate. Measured productivity growth slowed as well, averaging 0.63% per annum. Still, productivity growth accounted for 80% of the growth of output. Output growth bounced back to a 1.88-percent average annual rate over the 2007-2009 subperiod. Aggregate input use actually declined over this period. Productivity growth was responsible for positive output growth of the sector (see table 1 ).
• 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 (Ball et al., 2001, 2007).
• All but four States exhibited positive rates of growth in agricultural output over the 1960-2004 period. Output expanded most rapidly in Arkansas, growing an average 2.8 percent per year (see table 3 ).
• Every State exhibited a positive average annual rate of productivity growth over the entire 45-year period. Average annual rates of growth ranged from 2.6 percent for Oregon to 0.5 percent for Oklahoma. California and Florida had the highest relative levels of productivity in 2004 (see table 22 ).

 chart data

### Changes Introduced With This Release

In this update we change the base year from 1996 to 2005. The productivity estimates have been revised with this release. The revisions were necessitated by the adoption of new sources for data on employment, hours worked, and compensation per hour. The data on self-employed and unpaid family workers are taken from the decennial Census of Population and the annual Current Population Survey. We introduced the new data series when our original data source-the Farm Labor Survey administered by USDA's National Agricultural Statistics Service-was discontinued after 2002. 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) beyond 2002, but has also required that we revise these series for the years prior to 2002.

Contract labor services are becoming increasingly important in US farm production. Since farmers contract with labor brokers to assemble crews, there is little 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 US agricultural production we must construct an appropriate deflator. 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 (NAWS).

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. Theex ante rate is calculated as the nominal yield on investment grade corporate bonds adjusted for expected, rather than actual, price inflation. Inflation is modeled as an autoregressive process.

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.

### The Role of Productivity Growth in 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.

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. The estimates of depreciable capital are derived by representing capital stock at each point of time as a weighted sum of past investments where the weights correspond to the relative efficiencies of assets of different ages. The same pattern of decline in efficiency is used for both capital stock and the rental price of capital services, so that the requirement for internal consistency of a measure of capital input is met. The index of land input is based on the physical characteristics (i.e., quality) of a parcel of land and its location relative to population centers. The contribution 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 USDA time series of total factor productivity indexes now spanning the period 1948-2009 (see tables 1-2). Measures of economic performance of the individual States are also compiled for the period 1960-2004. The State series provides estimates of the growth and relative levels of total factor productivity (see tables 3-22).

### Farm Sector Productivity Growth in the U.S.

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 1948-2009 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 inlabor inputin the sector, a pattern that persists through every subperiod. Over the entire 1948-2009 period, labor input declined at an average annual rate of 2.5 percent. When weighted by its 20-percent share in total costs, the contraction in labor input contributes an annual average -0.52 percentage point per year to output growth.

Capital input (excluding land)in the sector exhibits a different history. Its contribution to output growth alternates between positive and negative over the 1948-2009 period. On average, however, capital input increases over the period. Its positive growth contributes an annual average 0.02 percentage point to output growth.

Land input declined throughout the study period. Its contribution to output growth averaged a -0.08 percentage point per year over the full 62-year period.

Material input's contribution was negative in three of the five most recent subperiods, but averaged a substantial positive rate equal to 0.69 percent per year over the postwar period. Though large, this positive contribution just offsets the negative contributions through labor and land. The net contribution of all four inputs was a mere 0.11 percentage point per year, leaving responsibility for positive growth in farm sector output substantially to productivity growth.

In examining the results in table 2 , some researchers (e.g., Alston et al., 2010) find evidence of a slowdown in the rate of productivity growth. If productivity growth falters, growing global demand will lead to sharp increases in food prices and to increased environmental stress as more marginal lands are brought into production. Meanwhile, funding for agricultural research, a major driver of productivity growth, has lagged, both domestically and internationally.

Long run trends in productivity growth can be obscured by short-term factors such as extremes in weather or changes in agricultural policies. For example, output and TFP growth both fell sharply away from trend in 1983, 1988, and 1993, years of drought and/or flooding. Productivity also fell in 2007, as the expansion of corn-based ethanol production prompted a departure from the normal corn-soybean crop rotation. Corn acreage increased in 2007. But yields per acre were essentially flat despite increases in chemical (i.e., fertilizer and pesticides) consumption. Still, statistical analysis of the data does not provide evidence of a longrun productivity slowdown.

One way of estimating the productivity growth rate is to fit a trend line to the logarithm of TFP. First, however, we test whether the growth rate has changed over time. Stated differently, we test the null hypothesis of a stable linear model against the alternative of "breaks" in the parameters in the time series regression. We reject the hypothesis of a fixed slope coefficient over time, indicating the presence of a structural break or breaks.

Next, we test whether the TFP time series is stationary. This is an important step given that it is not admissible to use classical techniques for inference in models with non-stationary components. A number of alternative tests have been proposed. These are discussed in Dickey and Fuller (1979), Phillips and Perron (1988), and Kwiatkowski, Phillips, Schmidt, and Shin (1992). The test results indicate that the TFP series is stationary with trend. Zivot and Andrews (1992) extend these results to the case where the effects of a structural break(s) are present. Using the Zivot and Andrews (1992) procedure, we find the series to be stationary in levels, with a structural break occurring in 1985.

As a final step, we compare the rates of productivity growth for the subperiods 1948-1985 and 1985-2009. The trend rate of growth for the latter period is somewhat higher. However, after incorporating an intercept shift, this difference is not statistically significant. We conclude that there is no statistical significant evidence indicating a longrun productivity slowdown.

### Measuring State Productivity

A properly constructed measure of productivity growth for the aggregate farm sector is certainly important. It 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 State Rank in 2004 Level in 2004 Rank in 1960 Level in 1960 Average annual change, 1960-2004 Rank Change (%) 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.

The wide disparity in productivity growth rates over the 1960-2004 period resulted in substantial changes in the rank order of States. Florida and California remain at the top of the pack, although California rose from second in 1960 to first in 2004. The largest relative gains in TFP were made by Indiana, Rhode Island, and Oregon. Indiana jumped from 27th to 7th among the 48 States, Rhode Island rose from 35th to 8th, and Oregon advanced from 46th to 15th. This finding is consistent with Gerschenkron's (1952) notion of the advantages of relative backwardness. Those States/regions that lagged particularly far behind the technology leaders had the most to gain from the diffusion of technical knowledge, and, hence, exhibited the most rapid rates of productivity growth.

### 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:

### 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, seed, and livestock 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 , where 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 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 time dummy coefficients.

Finally, 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:

where is the Box-Cox transformation of the dependent price variable, ; 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.

The corresponding pesticide quantity index is formed implicitly by taking the ratio of aggregate expenditures for pesticides to the hedonic price index.

Other purchased inputs collectively account for about 15 percent of the input service flow. We compute price and implicit quantity indexes 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 constructed by aggregating across each category of intermediate input described above.

### Capital Input

Measures of capital input and capital service prices for each State are estimated from the capital stock and rental price for each asset type for each State. The perpetual inventory method is used to develop stocks of depreciable capital from data on investment. 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.

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. We assume that the relative efficiency of capital goods declines with age, giving rise to the need for replacement of productive capacity. The proportion of investment to be replaced is equal to the decline in efficiency during each period. These proportions represent mortality rates for capital goods of different ages. Replacement requirements in each period are the weighted sum of past investments, where the weights are the mortality rates. The change in capital stock in any period is equal to the acquisition of new investment goods less replacement requirements.

#### Estimating Replacement

To estimate replacement, we must introduce an explicit description of the decline in efficiency. This function, d, may be expressed in terms of two parameters, the service life of the asset, say L, and a curvature or decay parameter, say ß. Initially, we will hold the value of L constant and evaluate the efficiency function for various values of ß. One possible form for the efficiency function is given by:

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

This function is a form of a rectangular hyperbola that provides a general model incorporating several types of depreciation as special cases.

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.

Once the frequency of occurrence of a particular service life has been determined, the efficiency function for that service life is calculated using the assumed value of ß. This process is repeated for all possible service lives. An aggregate efficiency function is then constructed as a weighted sum of the individual efficiency functions, using the frequency of occurrence as weights. This function not only reflects changes in efficiency, but also the discard distribution around the mean service life of the asset.

Finally, beginning inventories of crops and livestock are also included in capital input. We estimate the stock of inventories using the perpetual inventory method, assuming zero replacement.

#### Capital Rental Prices

The behavioral assumption underlying the derivation of the rental price of capital is that firms buy and sell assets so as to maximize the present value of the firm. This implies that 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:

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

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:

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

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

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

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 Moody's AAA-rated 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.

Indexes of capital input in each State are constructed by aggregating over the different capital assets using as weights the asset-specific rental prices. Service prices for capital input are formed implicitly as the ratio of the total current dollar value of capital service flows to the quantity index. The resulting measure of capital input for each State is adjusted for changes in input 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.

### Labor Input

The USDA labor accounts for the aggregate farm sector incorporate a demographic cross-classification of the agricultural labor force. 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. ERS developed a set of similarly formatted but otherwise demographically distinct matrices of labor input and labor compensation by State by combining the aggregate farm sector matrices with State-specific demographic information available from the decennial census of population. The result is State-by-year matrices of hours worked and hourly compensation with cells cross-classified by sex, age, education, and employment class and with each matrix consistent with the USDA hours-worked and compensation totals.

Labor compensation (opportunity cost) data for self-employed and unpaid family workers are not observed. As a result, for each State and year, self-employed and unpaid family workers in each State are imputed using the mean wage earned by hired workers with the same demographic characteristics.

Indexes of labor input are constructed for each State over the 1960-2004 period 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 hours.

### Ongoing and Planned Research

#### Productivity Growth in U.S. Agriculture and the Role of Public R&D

We use the production accounts for the States to estimate both the Luenberger productivity indicator and its dual, the Bennet-Bowley productivity indicator. This work takes a broader view of the production process to account for the relationship between productivity change and changes in prices and profits. This allows us to decompose changes in profitability in agriculture into a normalized price change indicator and a Bennet-Bowley productivity indicator. We then investigate the relationship between productivity growth and public investment in research and development. The relationship between price change and R&D is negative, and there is a weak negative relationship between R&D and profits, which is consistent with our decomposition of profit change into price and productivity components. Contact Eldon Ball.

#### Impact of Local Public Goods on U.S. Agricultural Productivity Growth

This study looks at the impact of public R&D expenditure on U.S. agricultural productivity growth. We estimate a dual cost function using state-by-year panel data. Capital stocks include "own" R&D and "spill-in" R&D from other states. The latter is measured based on both geographic location and product mix. We also examine the roles of extension, infrastructure, and human capital in the dissemination of technology. Our results indicate that higher levels of "public goods" result in lower cost of production, although the social rates of return differ markedly across states. Contact: Sun Ling Wang.

#### Productivity Growth Decomposition in U.S. Agriculture

This study uses State-level panel data to estimate total factor productivity (TFP) change components. Our focus is on the measurement and decomposition of TFP change into technical change and scale components as well as distortions in output and input prices. A multi-output, multi-input transformation function is used to derive the components of TFP change. These components are then computed from the multi-output translog cost function estimated using a system approach. We also consider an estimation approach in which the TFP change equation is added to the cost system to avoid the discrepancy between parametric and non-parametric measures of TFP change. Our results indicate that at the mean TFP change is driven by technological change. That is, distortions and scale components do not contribute much to the mean TFP change. However, these components vary substantially as evidenced from their percentile values, as well as their values by state. Output (input) price distortions are found to have positive (negative) impact on TFP change for most of the states. We argue that output price distortions are the result of government intervention in output markets. More specifically, the distortions are caused by the divergence between observed and shadow output prices (i.e., effective prices that include the effects of government subsidies). Input price distortions more likely reflect quasi-fixity of inputs. Contact: Eldon Ball.

#### Productivity and International Competitiveness of Agriculture in the European Union and the United States

This study looks at international competitiveness of agriculture in the United States and the European Union. At the outset it is necessary to define a measure of competitiveness. We define international competitiveness as the price of output in the Member States of the European Union relative to that in the United States. We then decompose relative price movements into changes in relative input prices and changes in relative productivity levels. Our price comparisons indicate that the United States was more competitive than its European counterparts throughout the period 1973-2002, except for the years 1973-74 and 1983-85. Our results also suggest that the relative productivity level was the most important factor in determining international competitiveness. Over time, however, changes in competitiveness were strongly influenced by variations in exchange rates through their impact on relative input prices. During the periods 1979-1984 and 1996-2001, the strengthening dollar helped the European countries improve their competitive position, even as their relative productivity performance lagged. Contact Eldon Ball.

#### Quality-Adjusted Price and Quantity Indices for Pesticides Revisited

The use of quality-adjusted pesticide price and quantity indices is critical in calculating agricultural productivity and in estimating aggregate supply models. Indices need to be adjusted for quality differences across pesticides and years because there are important inherent differences in pesticide characteristics that prevent the direct comparison of observed prices of pesticides over time and across regions. We develop quality-adjusted measures by estimating hedonic pesticide price functions; hedonic functions express the price of a good or service as a function of the quantities of the characteristics it embodies-in this case, pesticide potency, hazardous characteristics, and persistence. When we control for such pesticide characteristics in a hedonic price function, we can then derive quality-adjusted pesticide price indices for States and major crops 1960-2005, updating a previous database that ended in 1999. Contact Richard Nehring.

#### Quality-Adjusted Price and Quantity Indices for Contract Labor Services

In the hedonic framework a good or service is viewed as a bundle of characteristics which contribute to output or utility derived from its use. We estimated a hedonic wage function in terms of farm work experience, gender, age, education, and legal status, along with geographic and time dummy variables for contract labor. The parameter estimates can be interpreted as shadow values of the characteristics. The results indicate slower growth for the quality adjusted contract labor price compared to the unadjusted series-that is, some of the growth in the unadjusted series was due to changes in the demographics of farm workers.  As a result of this analysis, the rate of growth of contract labor services increased in our productivity accounts, and measured productivity growth was slightly reduced. That is, some of what we previously attributed to TFP growth was instead attributed to improving quality of contract labor services. Contact Sun Ling Wang.

### References

Alston, J., M. Anderson, J. James, and P. Pardey. Persistence Pays: U.S. Agricultural Productivity growth and the Benefits from R&D Spending. New York: Springer, 2010.

Ball, V. E., W. Lindamood, R. Nehring and C. San Juan Mesonada. "Capital as a Factor of Production in OECD Agriculture: Measurement and Data," Applied Economics 40 (May-June 2008): 1253-1278.

Ball, V. E., J-P Butault, C. San Juan Mesonada, and R. Mora. "Productivity and International Competitiveness of European Union and United States Agriculture, 1973-2002," Agricultural Economics (September 2010): 611-627.

Ball, V. E., C. Hallahan, and R. Nehring. "Convergence of Productivity: An Analysis of the Catch-up Hypothesis within a Panel of States," American Journal of Agricultural Economics 86 (December 2004): 1315-1321.

Ball, V. E., J-C Bureau, J-P Butault, and R. Nehring. "Levels of Farm Sector Productivity: An International Comparison," Journal of Productivity Analysis 15 (2001): 5-29.

Ball, V. E., F. Gollop, A. Kelly-Hawke, and G. Swinand. "Patterns of Productivity Growth in the U.S. Farm Sector: Linking State and Aggregate Models," American Journal of Agricultural Economics 81 (February 1999): 164-179.

Ball, V. E., J-C Bureau, R. Nehring, and A. Somwaru. "Agricultural Productivity Revisited," American Journal of Agricultural Economics 79 (August 1997): 1045-1063.

Caves, D.W., L.R. Christensen, and W.E. Diewert. "Multilateral Comparisons of Output, Input, and Productivity Using Superlative Index Numbers," The Economic Journal 92 (1982): 73-86.

Christensen, L.R. and D.W. Jorgenson. "The Measurement of U.S. Real Capital Input, 1929-1967," Review of Income and Wealth 15 (1969): 293-320.

Coen, R. "Investment Behavior, the Measurement of Depreciation, and Tax Policy," American Economic Review 65 (March 1975): 59-74.

Denison, Edward. "Explanations of declining productivity growth," Survey of Current Business 59 (August 1979): 1-24.

Dicky, D. and W.A. Fuller, "Distribution of the Estimates for Autoregressive Time Series With a Unit Root, Journal of the American Statistical Association 74 (1979): 427-431.

Diewert, W.E. "Aggregation problems in the Measurement of Capital," in D. Usher, ed. The Measurement of Capital, Studies in Income and Wealth No. 45, The University of Chicago Press, Chicago, 1980.

Diewert, W. E. "Exact and Superlative Index Numbers," Journal of Econometrics 4 (1976): 115-146.

Elliot, G., U. K. Muller, "Efficient Tests for General Persistent Time Variation in Regression Coefficients."Review of Economic Studies, 73 (2006): 907-40.

Eltetö, O. and P. Köves, "On an Index Computation Problem in International Comparisons," Statisztikai Szelme 42 (1964): 507-18.

Eswaren, H., F. Beinroth, and P. Reich. "A Global Assessment of Land Quality," in Keith Wiebe, ed. Land Quality, Agricultural Productivity, and Food Security. Cheltenham, UK: Edward Elgar, 2003.

Gerschenkron, A. "Economic Backwardness in Historical Perspective," in B.F. Hoselitz (ed.) The Progress of Underdeveloped Areas. Chicago: University of Chicago Press, 1952.

Jorgenson, D., M. Ho, and K. Stiroh. Productivity: Information Technology and the American Growth Resurgence. Cambridge: The MIT Press, 2005.

Jorgenson, D., and F. Gollop. "Productivity Growth in U.S. Agriculture: A Postwar Perspective." American Journal of Agricultural Economics 74 (August 1992): 745-50.

Jorgenson, D., F. Gollop, and B. Fraumeni. Productivity and U.S. Economic Growth. Cambridge MA: Harvard University Press, 1987.

Jorgenson, D. and Z. Griliches. "The Explanation of Productivity Change," Review of Economic Studies 34 (1967): 249-283.

Kellogg, R., R. Nehring, A. Grube, D. Goss, and S. Plotkin. "Environmental Indicators of Pesticide Leaching and Runoff from Farm Fields," in V.E. Ball and G. Norton, eds. Agricultural Productivity: Measurement and Sources of Growth, Boston: Kluwer Academic Publishers, 2002.

Kendrick, J.W., and E.S. Grossman. Productivity in the United States, Trends and Cycles. Baltimore MD: The Johns Hopkins University Press, 1980.

Kwiatkowski, D., Peter C.B. Phillips, P. Schmidt, and Y. Shin, "Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root," Journal of Econometrics 54 (1992): 159-78.

Penson, J.B., D.W. Hughes and G.L. Nelson. "Measurement of Capacity Depreciation Based on Engineering Data," American Journal of Agricultural Economics 35 (1977): 321-329.

Phillips, P.C.B. and P. Perron. "Testing for a Unit Root in Time Series Regression." Biometrika 75(1988): 335-346.

Romain, R., J.B. Penson, and R. Lambert. "Capacity Depreciation, Implicit Rental Prices, and Investment Demand for Farm Tractors in Canada." Canadian Journal of Agricultural Economics 35 (1987): 373-78.

Szulc, B. "Indices for Multiregional Comparisons," Przeglad Statystyczny 3 (1964): 239-54.

Zivot, E., and D. Andrews, "Further Evidence on the Great Crash, the Oil-Price Shock, and the Unit-Root Hypothesis." Journal of Business & Economic Statistics, 10 (1992): 251-70.

Last updated: Monday, May 20, 2013

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