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This paper uses panel data for the 1980-2004 period to estimate the contributions of public research to US agricultural productivity growth. Local and social internal rates of return are estimated accounting for the effects of R & D spill-in, extension activities and road density. R & D spill-in proxies were constructed based on both geographic proximity and production profile to examine the sensitivity of the rates of return to these alternatives. We find that extension activities, road density, and R & D spill-ins, play an important role in enhancing the benefit of public R & D investments. We also find that the local internal rates of return, although high, have declined through time along with investments in extension, while the social rates have not. Yet, the social rates of return are not robust to the choice of spill-in proxy.

Since the pioneering work by Griliches [

Previous studies on the contribution of R & D to productivity growth can be grouped into four main categories. First, there are international studies [

Methodological differences aside, many of these studies point to significant technology spillovers across geographic boundaries. While the contribution of R & D spill-ins^{1} to US agricultural productivity growth from nearby states is widely recognized, it is less clear why productivity growth in some states with similar characteristics and with similar potential R & D spill-ins is faster than in other states. Nor is it clear through which channels technical knowledge is disseminated. Some studies ( [

The objectives of this paper are, first, to study the interaction between local (i.e., own) R & D and R & D spill-ins, extension activities, and road density^{2}. Second, we estimate the own as well as social internal rates of return to investment in research in each state. Third, we develop alternative estimates of potential R & D spill-in variables based on geographic proximity and production profile similarity, to investigate sensitivity of the rates of return to these alternative proxies. Finally, we evaluate how changes in extension activities and in road density affect the estimated internal rates of return.

We model technology in agriculture by a dual cost function using a panel of US states. Knowledge stocks, measured as the accumulation of past research expenditures, are treated as a public (i.e., exogenous) capital input. We treat R & D spill-ins, extension, and infrastructure differently from own R & D because we think that while own R & D is fully usable by state these “efficiency” variables are only partially usable and enter the cost function through interaction terms with local R & D stock.

We find that, although sensitive to the alternative proxies for knowledge spill-ins, the internal rates of return to R & D investments in US agriculture have been persistently high. Moreover, the rates of return are enhanced through the interaction of own R & D with extension activities, knowledge spill-ins, and road density.

A number of model specifications have been used to assess the contribution of public R & D to US agricultural productivity. Some have first constructed an index of productivity growth and, in a two-step procedure, related this index to R & D investments [

While local investment in public agricultural research is viewed as a major driver of technological advancement, investment in research in other states, especially those with similar production characteristics, also contributes to local productivity growth. This effect is generally referred to as a research “spill-in” from other states. We assume that research spill-ins, along with extension activities and the road network, interact with local public research to enhance diffusion and absorption of technical information. An intensive road network can provide farmers with an easier and less costly way to acquire new technologies by attending workshops or other extension activities. It can also save on the time it takes the extension staff to contact producers around the state. Given the development of internet technology and broadband investment, extension staff now have more ways to directly strengthen and speed the dissemination and absorption of technical information. Similarly, research spill-ins from nearby states as well as from states with similar production profiles could provide a “cluster” effect and generate a multipliable impact with local R & D on productivity growth. In this way, these factors may act as catalysts in stimulating diffusion and utilization of technical information.

We proceed by estimating a translog cost function using state-by-year panel data. We then derive estimates of productivity growth that capture the impact of local R & D investments as well as the magnifying effects of R & D spill-ins, extension activities, and infrastructure. Given its importance, we pay particular attention to construction of R & D spill-in variables. Finally, we estimate state- level internal rates of return to public agricultural research.

We assume that each state produces three outputs, livestock (V), crops (C) and other farm related goods and services (O), using four variable inputs including land (A), labor (L), materials (M), and capital (K), and one fixed input, own agricultural R & D stock (RD). We include interactions between own R & D and extension activities (ET), road density (RO), and R & D spill-ins (SR), which we term “efficiency variables” (E). These variables have the potential of increasing the marginal productivity of local R & D capital. The translog variable cost function is:

ln T V C = α 0 + ∑ n = 1 10 ∑ i = 1 4 α n i D n ln w i + ∑ l = 1 3 β l ln y l + γ R D ln R D + 1 2 γ R D R D ln R D ln R D + 1 2 ∑ i = 1 4 ∑ j = 1 4 α i j ln w i ln w j + 1 2 ∑ l = 1 3 ∑ k = 1 3 β l k ln y l ln y k + ∑ i = 1 4 ∑ l = 1 3 δ i l ln w i ln y l + ∑ i = 1 4 θ i R D ln w i ln R D + ∑ l = 1 3 ϕ l R D ln y l ln R D + ∑ h = 1 3 ξ h R D ln E h ln R D + ∑ h = 1 3 ∑ i = 1 4 ρ i h ln E h ln w i + ∑ i = 1 4 ρ i W ln P ln w i (1)

where the w’s are input prices, the y’s are output quantities, RD is the own-state R & D stock, the E’s are efficiency variables, the D’s are regional dummy variables, and P is a measure of rainfall. We introduce regional dummies in the first-order terms to allow for differences in cost shares across the production regions. The regions are the USDA’s farm production regions defined in

Symmetry and linear homogeneity in prices are imposed during estimation. Using Shephard’s lemma, the cost share for input i is:

S i = ∑ n = 1 10 α n i D n + ∑ j = 1 4 α i j ln w j + ∑ l = 1 3 δ i l ln y l + θ i R D ln R D + ∑ h = 1 3 ρ i h ln E h + ρ i W ln P (2)

Data source: USDA.

The estimated system of equations includes the total variable cost Equation (1) and the input cost share Equation (2). Additive disturbances are appended to each share equation and the variable cost function. These disturbances are presumed temporally independent, multivariate normal with zero mean and nonzero contemporaneous covariances. The contemporaneous covariance matrix of the disturbance terms of the cost and share equations is singular since the cost shares must sum to unity at every sample point. Hence, a single share equation is dropped in estimation. The system of equations is estimated using the Iterative Seemingly Unrelated Regression (ITSUR) algorithm in SAS. The estimation results are independent of the equation dropped under the maintained assumptions on the error structure.

Price responsiveness can be measured by the input price elasticities of derived demand (η):

η i i = α i i + s i 2 − s i s i , i = 1 , ⋯ , N , (3)

η i j = α i j + s i s j s i , i , j = 1 , ⋯ , N , i ≠ j , (4)

where S_{i} and S_{j} are the fitted cost shares for inputs i and j. The marginal cost elasticity (e) is also estimated:

e l = ∂ ln T V C ∂ ln Y l = β l + ∑ k = 1 3 β l k ln y k + ∑ i = 1 4 δ i l ln w i + ϕ l R D ln R D , l = 1 , 2 , 3 (5)

as are the cost elasticities (ε) with respect to local R & D stocks and the efficiency variables (E_{h}) ― spill-in stocks (SR), extension activities (ET), and road density (RO):

ε R D = ∂ ln T V C ∂ ln R D = γ R D + γ R D R D ln R D + ∑ i = 1 4 θ i R D ln w i + ∑ l = 1 3 ϕ l R D ln y l + ∑ h = 1 3 ξ h R D ln E h (6)

ε E h = ∂ ln T V C ∂ ln E h = ∑ h = 1 3 ξ h R D ln R D + ∑ i = 1 4 ρ i E h ln w i (7)

As noted above, one of the effects that we would like to highlight in this study is the interaction between local R & D stocks and the efficiency variables. This cross effect is:

M E E h R D = ∂ ε R D ∂ ln E h = ∑ h = 1 3 ξ h R D (8)

If ε_{RD} or ε_{E} is negative, then an increase in local R & D stock or any of the efficiency variables E_{h} reduces total variable cost, given input prices and output levels. If M E E h R D is negative then the efficiency variables have a further cost reducing effect; they magnify the cost-reducing impact of own R & D, as hypothesized.

To evaluate the benefits of public research, we proceed to calculate the internal rate of return (IRR)^{3}, which is the rate of discount that makes the net present value of all cash flows (including both inflows and outflows) from a particular investment equal to zero. In other words, the IRR of an investment is the rate of discount at which the present value of the stream of benefits equals the initial investment. In this framework of analysis, benefits are measured as cost savings (−DTVC) . Furthermore, an investment in public R & D (R) at time t is assumed to increase the stock of local R & D (RD) in t + τ ( τ = 0 , ⋯ , s ) at a rate of:

Δ R D t + τ Δ R t = ω τ (9)

Therefore, the local internal rate of return is the rate r_{1} that solves the following formula:

1 = ∑ τ = 0 s − Δ T V C t + τ Δ R t ⋅ 1 ( 1 + r 1 ) τ = ∑ τ = 0 s − Δ T V C t + τ Δ R D t + τ ⋅ Δ R D t + τ Δ R t ⋅ 1 ( 1 + r 1 ) τ = ∑ τ = 0 s − Δ T V C t + τ Δ R D t + τ ⋅ ω τ ( 1 + r 1 ) τ (10)

The impact of a one-dollar increase in a state’s local public agricultural R & D stock (RD) on that state’s total variable cost (TVC) can be approximated as (for simplicity the time subscript t is dropped):

Δ T V C Δ R D = ∂ ln T V C ∂ ln R D T V C R D (11)

To obtain the local internal rate of return we substitute (6) into (10), and solve for r_{1}:

1 = − ∑ τ = 0 s [ γ R D + γ R D R D ln R D t + τ + ∑ i = 1 4 θ i R D ln w t + τ , i + ∑ l = 1 3 ϕ l R D ln y t + τ , l + ∑ h = 1 3 ξ h R D ln E t + τ , h ] ⋅ T V C t + τ R D t + τ ⋅ ω τ ( 1 + r 1 ) τ (12)

Given that R & D investments in agriculture have the characteristics of an impure public good^{4}, the relevant concept in evaluation should include not only the local benefits (cost-savings) but also the benefits reaped by other states through R & D spillovers (i.e., the social rate of return). Taking into account both effects, the social internal rate of return, r_{2}, is derived by solving for r_{2} in the following equation:^{ }

1 = ∑ τ = 0 s − Δ T V C f , t + τ Δ R f , t ⋅ 1 ( 1 + r 2 ) τ − ∑ g ≠ f q − 1 ∑ τ = 0 s Δ T V C g , t + τ Δ R f , t ⋅ 1 ( 1 + r 2 ) τ (13)

where f indicates the state that makes the investment in public R & D, g indexes the states hypothesized to benefit from the spillovers from the research investment in state f, and q indicates the total number of states that benefit from the research investment in state f (including f). The first term in (13) is similar to Equation (12), and represents the local benefits. The second term in (13) captures the social benefits in other states generated by state f’s local research investment. The second term in Equation (13) can be alternatively expressed as:

− ∑ g ≠ f q − 1 ∑ τ = 0 s Δ T V C g , t + τ Δ R f , t ⋅ 1 ( 1 + r 2 ) τ = − ∑ g ≠ f q − 1 ∑ τ = 0 s Δ T V C g , t + τ Δ S R g , t + τ ⋅ Δ S R g , t + τ Δ R f , t ⋅ 1 ( 1 + r 2 ) τ (14)

The R & D spill-in stock for state f ( S R f ) is constructed as a weighted sum of contemporaneous local R & D stocks in other states:

S R f , t = ∑ g ≠ f q − 1 Ω f g R D g , t (15)

where Ω_{fg} are the weights used to capture the g^{th} state’s R & D stock contribution to state’s f spill-in stock. Therefore, the change in spill-ins stocks in state g at time t + τ from an investment in R & D in state f at time t is:

Δ S R g , t + τ Δ R f , t = Δ S R g , t + τ Δ R D f , t + τ Δ R D f , t + τ Δ R f , t = Ω g f ω τ (16)

The impact of research spill-ins from other states on state g’s total variable cost can be approximated as follows (excluding time indexes for simplicity):

Δ T V C Δ S R = ∂ ln T V C ∂ ln S R T V C S R (17)

We obtain the social internal rate of return by substituting (6) and (7) into (11) and (17), correspondingly, and substituting those results and (16) into (14), and solving for r_{2}:

1 = − ∑ τ = 0 s [ γ R D + γ R D R D ln R D f , t + τ + ∑ i = 1 4 θ i R D ln w f , t + τ , , i + ∑ l = 1 3 ϕ l R D ln y f , t + τ , l + ∑ h = 1 3 ξ h R D ln E f , t + τ , h ] ⋅ T V C f , t + τ R D f , t + τ ⋅ ω τ ( 1 + r 2 ) t + τ + ∑ g ≠ f q − 1 ∑ τ = 0 s ( ξ S R R D ln R D g , t + τ + ∑ i = 1 4 ρ i , S R ln w g , t + τ , i ) ⋅ T V C g , t + τ S R g , t + τ ⋅ Ω g f ω τ ( 1 + r 2 ) t + τ (18)

Our data consist of a panel of state-level observations spanning the years 1980 to 2004. This section provides a brief overview of data sources and aggregation procedures. Details on the data construction are in the USDA productivity web page [

State-specific aggregates of output and labor, capital and intermediate inputs are Törnqvist indexes over detailed output and input accounts. Törnqvist output indexes are formed by aggregating over agricultural goods and services using revenue-share weights based on shadow prices. The changing demographic character of the agricultural labor force is used to build a quality adjusted index of labor input. The measure of capital input begins with data on the stock of capital for each component of capital input. For depreciable assets, the capital stocks are the accumulation of all past investments adjusted for discards of worn-out assets and loss of efficiency of assets over their service life. For land and inventories, capital stocks are measured as implicit quantities derived from balance sheet data. Indexes of capital input are formed by aggregating over the various capital assets using cost share-weights based on assets-specific rental prices. Törnqvist indexes of energy consumption are calculated for each state by weighting the growth rates of petroleum fuels, natural gas, and electricity consumption by their share in the overall value of energy input. Fertilizers and pesticides are important intermediate inputs. Price indexes for fertilizers and pesticides are constructed using hedonic methods. The corresponding quantity indexes of fertilizers and pesticides are formed implicitly by taking the ratio of the value of each aggregate to its hedonic price index. A Törnqvist index of intermediate inputs is constructed for each state by weighting the growth rates of each category of intermediate inputs by their value share in the overall value of intermediate inputs. Finally, following Caves, Christensen, and Diewert [

There are many different methods used to construct knowledge stocks. In studies of the impact of private research in manufacturing, research stocks are frequently constructed from data on research expenditures using the perpetual inventory method. However, as noted by Griliches [^{5}.

We use a knowledge stock variable developed by Huffman that uses the trapezoidal distribution proposed by Huffman and Evenson [

In this study, we use two public research stock variables, an own-state variable and an R & D spill-in variable. Most studies that include potential spill-ins assume that discoveries from public research in a given state are an impure public good and use one particular approach to calculate them. Most impose the simplifying assumption that research benefits are regionally confined and apply simple aggregation over USDA production regions (see [

Because this is a key variable in the calculation of social returns, and because other studies estimated rates of return using just one of these approaches, we construct four alternative measures of the R & D spill-in variable. Our objective is to provide information on the sensitivity of the estimated rates of return to alternative models for the R & D spill-in stocks. The first two approaches we use are based on geographic proximity^{6}, while the last two reflect “production profile” similarities across states^{7}. The differences in the R & D spill-in stocks reside in the weights used in (15), as described below:

Model 1: Ω_{fg} = 1 for state j in the same USDA production region (

Model 2: Ω_{fg} = 1/dist_{fg} for an R & D spill-in variable generated based on the geographic distance among states. This approach, inspired by gravity-type trade models [

Model 3: Ω_{fg} = 1 for states f and g within the same production profile cluster. We use cash receipts from twelve categories of outputs to generate a production profile for each state. The twelve outputs categories are: meat animals, dairy products, poultry/eggs, miscellaneous, food grains, feed crops, cotton, tobacco, oil crops, vegetables, fruits/nuts, and all other crops. We use cluster analysis to group the states with similar production profiles. While there are several clustering techniques, we use the complete linkage clustering method following Sorensen [^{8}. In complete linkage clustering, the distance between two clusters is the maximum distance between an observation in one cluster and an observation in the other cluster, considering multiple elements. It avoids the drawback of the single linkage method that may force states to be grouped together due to closeness in one single element while many other elements are very different. The procedure is implemented using the SAS econometric package and results are presented in

Model 4: Ω_{fg} = 1/Tecdist_{fg} for an R & D spill-in variable generated based on the technical distance among states within the same cluster from Model 3. Tecdist_{fg} is the technological distance measured by the inverse of the Spearman correlation coefficient on the production mix among states. The higher is the correlation relationship, the smaller is the technical distance among states within the same cluster.

Descriptive statistics for the four R & D spill-in variables, along with other efficiency variables described below, are presented in

Data source: Developed by authors.

Note: FTE indicates full time equivalent staff numbers.

note that Model 2 yields a higher level of potential R & D spill-in stocks than all other models mainly because it includes more states in its cluster.

Extension is measured by total extension full-time equivalent staff days per year (FTEs). Extension FTEs have declined between 1980 and 2010 at national and regional levels [^{9}. Ahearn et al. [^{10}. Data on FTEs by state were drawn from [

of this variable^{11}.

We construct a road density index to examine the impact of road infrastructure on dissemination of local R & D. The state road density index was constructed using total annual road miles, excluding local (i.e. city street) miles for each state, obtained from [

Weather is treated as a control variable in this model. While several alternative weather indexes have been applied to studies in the past, such as the Palmer index and the Stallings weather index, we use total precipitation in inches from March to November [

We estimate the variable cost function (1) and the cost share Equation (2) using

the four alternative measures of R & D spill-ins defined above. Prior to estimation, we investigate the time series properties of the data. We conduct panel unit root tests proposed by Levin, Lin, and Chu [

We then estimate a total of 100 parameters based on 1200 observation for each model subject to symmetry and linear homogeneity in input prices. The curvature and monotonicity properties of the cost function were inspected after estimation. Monotonicity was satisfied globally. Concavity in prices implies a negative semi-definite Hessian. We find that this condition holds locally.

In

The impacts of public R & D, extension activities, roads, and R & D spill-ins on agricultural productivity growth can be examined through the alternative cost elasticities and the marginal effects of the efficiency variables on R & D’s cost saving effect. The cost elasticities of own R & D, extension activities, road density, and R & D spill-ins are all negative (see

Note 1: The LLC panel unit root test is based on the method proposed by Levin, Lin, and Chu (2002). Our tests include a constant term for every variable except LnK. In the case of LnTVC, LnET, a time trend was included. Note 2: SR1, SR2, SR3 SR4 are alternative R & D spillins based on the estimates from Model 1 through Model 4. Note 3: V stands for livestock, C for crops, O for other farm related goods and services, A for land, L for labor, M for materials, K for capital, RD for own agricultural R & D stock, ET for extension, RO for road density, SR for R & D spillins.

Note 1: The spillin RD stocks are based on production region, geographical distance, un-weighted production profile, and correlation weighted production cluster for Model 1 to Model 4, respectively. Note 2: V stands for livestock, C for crops, O for other farm related goods and services, A for land, L for labor, M for materials, K for capital, RD for own agricultural R&D stock, ET for extension, RO for road density, SR for R & D spillins. Note 3: ***indicates significant at 1% level. **indicates significant at 5% level. *indicates significant at 10% level.

Note: The spillin RD stocks are based on production region, geographical distance, un-weighted production profile, and correlation weighted production cluster for Model 1 to Model 4, respectively.

Note: The spillin RD stocks are based on production region, geographical distance, un-weighted production profile, and correlation weighted production cluster for Model 1 to Model 4, respectively.

and R & D spill-ins significantly enhance the cost-reducing effect of local R & D expenditures. Among the efficiency variables, extension activities have the greatest impact while road density has the smallest impact. This effect, paired with the decreasing trend of the extension variable through time, is important in understanding the evolution of the own-state internal rate of return, as will be seen later.

The results presented in

Next, we use the estimated coefficients and Equations (13) and (17) to calculate own and social rates of return to agricultural research by state and by year. The annual rates of return for all states by year are shown in _{1}) are robust across the different model specifications. Note also the sensitivity of the estimated social rates of return to alternative proxies for research spill-ins. Estimates from Models 3 and 4 that use the “production mix” approach to the construction of the R & D spill-in stocks are very close while those from Model 1 and 2, based on geographical proximity are very different. The rates in Model 1, estimated using the most common approach found in the literature (i.e., grouping states according to the USDA production regions) are the largest, while those from Model 2 are the smallest. This is consistent with the relative magnitude of the spill-in R & D stocks calculated across these models (see

Note 1: The spillin RD stocks are based on production region, geographical distance, un-weighted production profile, and correlation weighted production cluster for Model 1 to Model 4, respectively. Note 2: r_{1} indicates local internal rate of return, and r_{2} indicates social internal rate of return.

see that both the own-state internal rate of return r_{1} and the social rate of return r_{2} in all four models declined beginning in the mid-1980s. While the own-state internal rates of return (r_{1}) continued to decline over the sample period, the social internal rates of return (r_{2}) stabilized or exhibited a slight increase. The declining own-state internal rates of return estimated here are associated with declining extension staffing during these years. However, in the estimation of the social rates of return, this effect seems to be outweighed by research spill-ins. The average local rate of return across models ranged from 10.69% in Model 1 to 13.49% in Model 4. The average social rate of return ranged from 14.35% in Model 2 to 39.56% in Model 1. These IRR rates are lower than the ones in [^{12}.

Based on the estimated marginal effects (Equation (8)) of extension activities, road density, and R & D spill-ins, we calculate the impacts of these variables on the internal rates of return (Equations (12)-(17)). These results are presented in

This paper uses data for a panel of states developed by USDA to estimate the own and social internal rates of return to public R & D expenditures in agriculture. The social rates of return incorporate the interaction with R & D spill-ins from other states, extension activities, and road density. We construct four alternative measures of potential R & D spill-ins based on geographic proximity

Note 1: The spillin RD stocks are based on production region, geographical distance, un-weighted production profile, and correlation weighted production cluster for Model 1 to Model 4, respectively. Note 2: r_{1} indicates local internal rate of return, and r_{2} indicates social internal rate of return.

Note 1: The spillin RD stocks are based on production region, geographical distance, un-weighted production profile, and correlation weighted production cluster for Model 1 to Model 4, respectively. Note 2: r_{1} indicates local internal rate of return, and r_{2} indicates social internal rate of return.

and similarities in production to determine the sensitivity of the estimated rates of return to model specification. We estimate four models, using a different measure of potential R & D spill-in in each model. Our estimates indicate that extension activities, road density, and R & D spill-ins from other states play an important role in determining the efficacy of R & D expenditures. Among these variables, the impact of extension activities seems to be the strongest. These activities enhance productivity growth by facilitating dissemination of technical information.

We estimate own and social rates of return that, although high, are lower than the ones found in previous literature. Local internal rates are, on average across all years, states and models, 12 percent while social rates are 27 percent. The estimates of the own internal rates of return are robust across the alternative models, while the social internal rates of return deviate from each other depending on the particular measure of R & D spill-ins. The social rates of return based on USDA production regions are much higher than those estimated by the other models. This is important given the prevalence in the literature of the production mix approach for the calculation of knowledge spill-in stocks. We find that the decline in own rates is associated with declines in extension investments during this period. These findings can inform decisions about allocating public resources to alternative research and extension activities.

Wang, S.L., Plastina, A., Fulginiti, L.E. and Ball, E. (2017) Benefits of Public R & D in US Agriculture: Spill-Ins, Extension, and Roads. Theoretical Economics Letters, 7, 1873-1898. https://doi.org/10.4236/tel.2017.76128

Note 1: The spill-in RD stocks are based on production region, geographical distance, un-weighted production profile, and correlation weighted production cluster for Model 1 to Model 4, respectively. Note 2: V stands for livestock, C for crops, O for other farm related goods and services, A for land, L for labor, M for materials, K for capital.