Crop yield responses to prices: a Bayesian approach to blend experimental and market data

Abstract

1. Introduction

Agricultural production plays a critical role in the set of human practices that impact the environment and the use of natural resources. The recent expansion of agricultural production prompts a higher pressure on both of them. For example, it has been argued that increased demand for agricultural products originating from biofuel policies and higher per-capita income in developing countries have induced land-use changes at a global scale. These land-use changes have in turn generated additional direct and indirect greenhouse gas (GHG) emissions (Righelato and Spracklen, 2007; Fargione et al., 2008; Searchinger et al., 2008; Dumortier et al., 2011).

A direct consequence of higher demand for agricultural products is an increase in commodity prices in international markets. The quantity of new land required to satisfy this extra demand depends critically on the ability of crop yields to react to these higher prices (Keeney and Hertel, 2009). Furthermore, small changes in crop yields have a large impact on the payback period of GHG emissions induced by agriculture, and on the quantity of new land brought into agriculture to satisfy the increasing demand (Dumortier et al., 2011). Finally, the allocation of land to competing enterprises is often quite sensitive to price shocks. Therefore, there is a pressing need for accurate and updated estimates of how crop yields respond to prices.

The two main methods to estimate yield elasticities are the primal and dual approaches, both based on the Neoclassical theory of the firm. In the primal approach, elasticities are calculated after a direct estimation of the production technology. In the dual approach, elasticities are obtained from the estimation of input demand and output supply function equations, and the underlying production parameters are recovered indirectly.

Applications of the primal approach can be traced back to the pioneering work by Houck and Gallagher (1976) who found clear evidence of a positive own-price yield elasticity for US corn over the period 1951–1971. However, Menz and Pardey (1983) pursued a similar analysis but for a longer period and found that the yield-price elasticity was not significant for the following 10-year period. Reed and Riggins (1982) also followed the Houck and Gallagher’s approach for 10 extension regions in Kentucky for the period 1960–1979, estimating a negative and non-significant effect of corn prices on yield. Low-corn yield elasticities were also found by Ash and Lin (1987) in the Corn Belt and Lake States regions. Choi and Helmberger (1993) found a positive relationship for US corn, soybean and wheat over the period 1964–1988. Kaufmann and Snell (1997) modelled US corn yields as a function of a large group of climatic and economic variables and obtained results consistent with those of Houck and Gallagher, but with elasticities close to zero.

Keeney and Hertel (2008) have provided possible explanations for the lack of response found in some studies. In the case of studies heavily relying on a primal specification (e.g. Menz and Pardey, 1983; Ash and Lin, 1987), they attribute the lack of response to the plateau-like relationship between yields and fertiliser. For studies estimating single-equation models (e.g. Reed and Riggins, 1982), they suggest that the lack of response arises from the failure to acknowledge land substitution effects. The wide range of parameter estimates may also be explained in part by geography. Some crop producers can increase acres planted by converting pasture to cropland. Others are located in areas where almost all land that is suitable for cropping is already in crops. This latter group will be more likely to focus on input use as a way to increase output. The choice between extensive and intensive margin will depend on local geography and any attempt to aggregate up from these micro-level responses will reflect these geographic differences.

More recently, Miao, Khanna and Huang (2015), using US county-level data, fitted a reduced-form model of crop yields as a function of prices, climate variables and county-level fixed effects, and found a positive response to price increases. Similarly, Goodwin et al. (2012) analysed the response of corn yields to own prices in Iowa, Illinois and Indiana and estimated positive and significant (long-run) elasticities using data by crop reporting district. Furthermore, they looked at the intra-seasonal effect of prices changes early in the growing season on crop yields and also found a positive (though small) significant effect. They conducted focus group analysis and reinforced these latter findings. With a slightly different methodological approach, but also setting crop yield models as functions of weather variables and instrumented prices, Berry and Schlenker (2011) find corn and soybeans yield-price elasticities statistically non-different from zero. Also, in an unpublished work, Scott (2013) finds similar results for corn, soybeans and wheat, but pursuing an indirect estimation relying on fertiliser input use elasticities with respect to crop prices.

There are numerous applications of the dual approach in supply response models,¹ but only a few of them estimate yield elasticities. Arnade and Kelch (2007) found positive yield responses to own-price changes for corn, soybeans and other grains in Iowa using a dual approach that included shadow land price equations. Guyomard, Baudry and Carpentier (1996) jointly estimated supply elasticities for several crops to study the effects of the Common Agricultural Policy (CAP) in Europe and also obtained positive responses of yields to own prices. However, Arnade, Kelch and Leetmaa (2002) analysed the case of French corn and found a negative own-price yield elasticity. Sckockai and Moro (2006) estimate a system of input demand and output supply derived from the dual expected utility problem, and allowing for price uncertainty, using a panel data of Italian specialised farms. They found that the elasticity of total supply of corn to own prices is positive and higher than the corn area elasticity to own price, implying a positive elasticity of corn yields with respect to prices. Lansink (1999) estimates a system of input demands and output supply, including land allocation response equations, using farm-level data from Dutch farms from 1975 to 1992 using a cost function approach that accounts for output price uncertainty. The estimated cereal and oilseed total supply elasticity with respect to own price is 1.17 and the area own-price response is also positive but smaller (0.26), indicating that the yields positively respond to own prices. This analysis does not report elasticities for the individual crops. Ball et al. (1997) use duality with profit maximisation applied to nine EU countries and find a positive elasticity of total supply of cereals to own prices while holding constant land allocation to the different crops. This result implies a positive yield-price elasticity for aggregate output.

While it is generally accepted that the dual approach is preferable over the primal approach to estimate supply response models (Colman, 1983; Just, 1993), only a handful of papers have employed the dual approach to estimate yield response to prices. One possible explanation might be the heavy influence of uncertain events during the production process (e.g. weather and other unexpected factors, such as pests), translating into high levels of spatial and temporal production variability commonly observed in real-world data. This issue is especially problematic in the case of spatial price variation that tends to be significantly smaller than spatial production variability. Also, the temporal price variation may not be sufficient to identify the changes in yields caused by production shocks. As a result, when the focus is the estimation of crop yield responses to price changes, the dual approach (which relies on market-based data) may not be able to recover all features of the production technology.

One aspect of the technology; plant yield response to an increase in input use, must be uncovered to measure yield response to price. The accuracy with which this response is estimated will obviously influence the accuracy of the measure of yield-price elasticity. Historic, market-based yield data are heavily influenced by the weather uncertainty described above, which makes it extremely difficult to extract this term from the econometric analysis of historic data. But the yield response to input use can be well characterised using experimental crop-trial data. All that is needed is an objective method to blend the information from the experimental data into the econometric approach.

The contribution of this paper is twofold. First, it provides updated estimates of crop yield elasticities with respect to output and input prices. Second, it shows how one can exploit the information contained in both market-based and crop-trial data for estimation purposes. Estimation is performed by means of Bayesian methods, which are nicely suited for the present purpose because they allow us to combine the dual approach (based on market data) with the primal approach (which relies on experimental data) in a relatively straightforward manner.

We start by setting up the dual model conditioned on market-based data, with the objective of estimating crop yield responses to prices. The latter are functions of technology parameters that can be recovered by resorting to the duality theorem and estimated by directly fitting functions of yield responses to input use. Therefore, we also set up and fit models of yield responses to inputs as permitted by the data available, so as to estimate production parameters (yield marginal effects) that are theoretically consistent with their counterparts recovered from the dual model. This is guaranteed by Lau’s (1976) ‘Hessian identities’ and the appropriate choice of functional forms. Bayesian estimation is theoretically appealing, by weighting the contribution that each data source makes to recover each parameter. In addition, explicitly and simultaneously fitting yield responses using experimental data as part of the overall estimation procedure help to overcome, at least in part, the issues raised in the two preceding paragraphs. Importantly, the procedures proposed here may prove useful for other applications, e.g. to estimate demand elasticities simultaneously incorporating aggregate consumption data and data from consumer experiments or retail trials.

2. Data

The market-based data are time series of agricultural input and output quantities per hectare and prices.² The experimental data contain panel observations of corn yield responses to nitrogen, phosphate and potash fertiliser application rates, as well as corn yield responses to seed population.

2.1. Market-based data

The market-based data consist of state-level time series from 1960 to 2004 for Iowa. The series include four variable outputs (corn, soybeans, other crops and livestock products), three variable inputs (hired labour, intermediate inputs and fertiliser) and five quasi-fixed netputs (farmland, agricultural capital, family labour, conservation reserve programme (CRP) land and farm-related output).

Input quantities and prices and quasi-fixed netput quantities (except for CRP land) were provided by Eldon Ball at USDA-ERS.³ All netput (variable and quasi-fixed) quantities are expressed in per-hectare units. The quantity of hired labour is a productivity-weighted index of hours worked and hourly compensation. Intermediate inputs is an aggregate variable including pesticides, energy (petroleum fuels, natural gas and electricity), and other purchased intermediate inputs (seeds, contract labour services, custom machine services, machine and building maintenance and repairs and irrigation). Pesticide and fertiliser prices come from hedonic price functions, and the corresponding quantity index is calculated as the ratio between total pesticide and fertiliser expenditures and its price index. Energy quantities are the ratios between total expenditures and the price indexes of the individual fuels; other purchased inputs are calculated in a similar fashion. In the case of fertiliser, Ball’s quantities are allocated to different crops based on data available on per hectare fertiliser use by crops by state (USDA-ERS, 2015b).

Farmland quantity consists of a quality-weighted index, calculated as the county total farmland value divided by a price index. Agricultural capital input quantity is the sum over different assets weighted by their own rental rates and in turn adjusted for quality. Self-employed and family labour opportunity cost is calculated by applying the mean wage earned by hired workers of similar demographic characteristics to the reported worked hours.

Output quantities for corn, soybean and other crops (wheat, oat, hay, silage corn, rye and barley) are from USDA-NASS (USDA-NASS, 2015) and transformed into per-hectare units. Quantities of other crops are calculated as a revenue-weighted average of each production quantity (Arnade and Kelch, 2007). Prices of corn and soybeans are from the Chicago Mercantile Exchange (CME) futures markets. Corn (soybean) prices equal the average of the nearest December (November) maturity futures price on 15 March and 30 March (Choi and Helmberger, 1993). Livestock prices are from Ball’s livestock products price index. While livestock future prices exist, they are not available for all categories, implying that an index would include a mixture of future and current prices, still providing measurement error with respect to the expected price. Future prices for other crops are not available; therefore, a price index is computed as the ratio of the total revenue from these crops to the weighted average of production.

Two sources of information are used to control for farm programmes in supply response. First, the area enrolled in the CRP is included as a quasi-fixed input, because land enrolments may change as farmers observe changes in expected output prices, contracts expire or enrolment requirements change (USDA-FSA, 2015). Second, output prices are taken to be the maximum between the loan rate and the CME future prices, because price floors imposed by federal farm programmes are expected to affect farmers’ decisions.⁴

2.2. Experimental data

The experimental dataset comprises per-hectare yield responses to applications of nitrogen, phosphate and potash fertiliser and seed density. These inputs are among the ones firstly targeted when farmers seek to increase expected yields, and account for about 70 per cent of all the pre-harvest input cost (Plastina, 2015). Ideally, the yield responses should be measured at the farm level to precisely reflect the decision scenarios faced by farmers. In practice, however, obtaining farm-level experimental data is unfeasible. Due to such limitations, in the present study we used the sources of experimental information described below. While they may not provide the ideal data, they are assumed to be representative enough to help identify the parameters of interest.

Yield responses to nitrogen and to seed density come from simulated data using EPIC.⁵ The nitrogen (seed) EPIC data consist of 133 (144) observations of yield and input quantities for each year (30) and for each of the 22 most representative soil types of Iowa, implying a balanced panel of 87,780 (95,040) observations.⁶ Input quantities ranged from 100 to 300 kg/ha in the case of nitrogen, and from 50,000 to 180,000 kernels per ha for seed density. These datasets are also accompanied by the area of each soil type in Iowa, which allows us to weight soil types by how representative they are in the state.

The corn yield response to phosphate and potash fertiliser data come from peer-reviewed published work based on field experiments conducted throughout the state of Iowa from 1975 to 2007 and for the most representative types of soils (Mallarino, Webb and Blackmer, 1991; Bordoli, 1996; Barker, 1998; Borges and Mallarino, 2001; Dodd and Mallarino, 2005; Clover, 2008; Mallarino et al., 2009). Each observation in the datasets also contains data on soil type and soil level of phosphate and potash.⁷

In summary, we collectively refer to the seed density and nitrogen fertiliser simulated data from EPIC, and the phosphate and potash fertiliser field-trial data, as experimental data.

3. Empirical model

The next two subsections discuss the dual and the primal approaches, in connection to the estimation of yield-price elasticities.

3.1. The per hectare dual demand–supply system component

Empirical applications of duality theory usually approximate the multi-output profit (value) function by a flexible functional form. Alternatively, the approximation can be done to the input demands and output supplies arising from a standard expected profit maximisation problem, which is referred to as the virtually ideal production system (Chambers and Pope, 1994; O’Donnell, Shumway and Ball, 1999).⁸ We follow the latter approach because it allows us to incorporate available information about input allocations.⁹

For the market data discussed in the previous section, we choose a linear approximation for the deterministic part of the per-hectare demand and supply curves.¹⁰ This choice implies a normalised quadratic flexible functional form for the profit function,¹¹ which yields a system of $n$ = 8 estimation equations.¹²

System (2.1)–(2.8) are expressed in standard netput notation, i.e. inputs are represented as negative per-hectare quantities. For hired labour and intermediate inputs, there are only aggregate demand equations, because the data contain no allocation information for them; however, there is one equation for the fertiliser allocated to each crop because allocation data are available. Also, to impose homogeneity of degree zero on input demands and output supplies as required by theory, input and output prices in systems (2.1)–(2.8) are normalised by the prices of the numeraire good, consisting of livestock products. The livestock supply equation is omitted from the estimation system to avoid singularity of the error’s variance–covariance matrix; however, its parameters can be recovered by means of the parameter restrictions and the maximisation problem’s objective function.

As a consequence of the constant returns to scale assumption embedded in the construction of Eldon Ball’s dataset, all input and output quantity variables (⁠$x$⁠, $y$⁠, $K$⁠) are expressed in per-hectare terms. This assumption proves particularly useful to estimate yield elasticities, because the estimated crop supply responses can be directly linked to yield responses.

Each equation in systems (2.1)–(2.8) has a disturbance term reflecting factors unknown to the econometrician, but not necessarily unobserved by the firm. It can be shown that the implied error structure is consistent with McElroy’s (1987) additive general error model (AGEM) applied to the case of the profit function (see Appendix C in supplementary data at ERAE online for more details).

3.2. The production function component

In principle, the marginal price effects could be computed by fitting only the dual system (2.1)–(2.8) to estimate the profit parameters $Θπ$⁠. However, the variances of the resulting estimates are often too large for real-world applications (Rosas and Lence, 2017, 2018). We address this issue by using experimental data to increase the precision of the estimates of the production parameters $ΘH$ shared by the dual and primal approaches. To be consistent with the Neoclassical theory of the firm, the linkage between the primal and dual estimates is established through Lau’s Hessian identities (equation (2.10)).

A quadratic technology is adopted for theoretical consistency, because it is the one implied by the dual model. The availability to this study of experimental datasets on corn yield responses to nitrogen (⁠$N$⁠), phosphate (⁠$P$⁠), potash (⁠$K$⁠) and seed density $(S)$⁠, which in turn are independent from each other, implies that a model of the response of yields to each of these inputs can be estimated. Lack of experimental data on yield responses to other inputs prevented us from incorporating them into the estimation.

The marginal effects of other inputs estimated from the experimental data are computed from expressions similar to equation (2.18).¹⁵

4. Estimation methods

Estimation is performed by means of Bayesian methods, which prove to be very convenient for this particular application. The Bayesian approach is especially suited to impose the constraints resulting from the use of the market-based and the experimental databases. Furthermore, it introduces such constraints in a way that takes into account the degree of information that each dataset provides to the recovery of common parameters, rather than deterministically. Whereas, this feature is most important for parameters common to both the primal and the dual models, parameters not shared across models are also affected by the use of the information from the additional datasets.

The shared parameters involve the corn yield responses to intermediate inputs (⁠$ϕ21$⁠) and fertilisers (⁠$ϕ31)$⁠, because they can be computed from equations (2.15) and (2.16) by applying the Hessian identities (2.10) to the profit parameters estimated using market data, and also obtained from equation (2.18) based on experimental data. Lack of experimental data on corn yields as a function of hired labour prevents us from employing a similar approach to estimate the corn yield response to hired labour (⁠$ϕ11$⁠). Hence, the latter is recovered by means of market-based data only, by applying the bisection method to equations (2.10) and (2.15), conditional on the dual parameters drawn in the same iteration.

4.1. Dual system estimation

The $k$ parameters in vector $β$ must satisfy a set of $k2+k3+k4<k$ equality constraints. Such constraints stem from two sources, namely, (i) the cross-equation restrictions given by the symmetry conditions and (ii) the restrictions on the marginal input effects on corn output (⁠$ϕi3$⁠) associated with the experimental data. In consequence, the set of dual parameters to be estimated can be classified in the following subsets:

• Subset $Θπ1$⁠: free, denoted by the $k1$-vector $β⁎$

• Subset $Θπ2$⁠: constrained by symmetry, denoted by the $k2$-vector $β2$

• Subset $Θπ3$⁠: restricted by equations (2.13) and (2.14), denoted by the $k3$-vector $β3$

• Subset $Θπ4$⁠: constrained by knowledge of $ϕ21$ and $ϕ31$ through equations (2.15) and (2.16), denoted by the $k4$-vector $β4$

Succinctly, the Bayesian approach consists of estimating each subset of parameters conditioning on the other subsets.

The Bayesian estimation of model (3.5) seeks to obtain the marginal posterior density functions of the parameters in $β⁎$ and $Σ$⁠, conditional on the data. To this end, we assume that $ε~MVN(0,Ω)$ and use a Gibbs sampler to generate random draws from these marginal posteriors (Casella and George, 1992). Upon estimation of the $β⁎$ parameters, the elements of subset $Θπ2$ are recovered from the equality $β2=Rβ2−1(r−Rβ⁎β⁎)$⁠. See steps 10–12 in Appendix D (in supplementary data at ERAE online).

The parameters in subset $Θπ3$ are computed directly from equations (2.13) and (2.14), which describe the marginal effects of own and input prices on corn yield. The constrained parameters are $[d1]1$⁠, $[b1]2$ and $[b1]3$ which can be computed directly from these equalities (without the need to draw from their conditional posterior densities), conditional on the most recent draw of $ϕi1$ for $i$ ∈ {1, 2, 3} and the corresponding dual parameters of subsets $Θπ1$ and $Θπ2$⁠.²⁰ See step 13 of Appendix D (in supplementary data at ERAE online).

Finally, parameters in subset $Θπ4$ are restricted due to equations (2.15) and (2.16), which make explicit how underlying production parameters can be recovered by the duality theorem. As they also involve the experimental data, its explanation is postponed until Section 3.3, after discussing the estimation of the primal model.

4.2. Direct estimation of yield response to input quantities

The corn per-hectare production regressions exemplified by the nitrogen model (2.17) are estimated with the experimental data on input quantities per hectare and the corresponding corn yields.²¹ The yield elasticity with respect to input quantity could be estimated on its own with the yield response model (2.17); however, when we ‘combine’ the equation corresponding to the primal model (2.18) with our per-hectare supply and demand system, we allow the experimental data to directly provide information about these parameters. Given the experimental design, estimation is performed as a pooled regression because the input quantities and the soil and time dummies (⁠$D1$ and $D2$⁠) are independent (Greene, 2003: 285). Importantly, error term $τi$ is independent from $τj≠i$ and from $ε$ in the dual system (3.2). As will become clear in the next subsection, this facilitates the estimation of the yield response parameters $ϕ21$ and $ϕ31$ that are common to the primal and dual models.

The information about corn yield responses to applications of nitrogen fertiliser, $ϕN1$⁠, contained in both the experimental datasets as well as other sources, implies constraints on the parameters of regression (equation (2.17)). Thus, the set of parameters to be estimated from primal model (equation (2.17)) can be classified into two subsets:

• Subset $ΘHN1$⁠: free, denoted as $ηN⁎$

• Subset $ΘHN2$⁠: constrained by the market data through the linear equations (2.18) and (3.1).

According to the linear equation (2.18), knowledge of the marginal effect $ϕN1$ implies one constrained parameter (subset $ΘHN2$⁠), arbitrarily set to be $η1N$⁠. The remaining parameters are free (subset $ΘHN1$⁠), denoted as $ηN⁎$⁠.

To estimate the marginal probability functions of the free parameters $ηN⁎$ and $στN2$⁠, we use a Gibbs sampler to draw random numbers from the marginal conditional posterior distribution of $ηN⁎$ (⁠$στN2$⁠), conditional on the value of $ϕN1$ and the previous draw of $στN2$ (⁠$ηN⁎$⁠). These posteriors assume a joint non-informative prior distribution. See steps 2 and 3 in Appendix D (in supplementary data at ERAE online) for a detailed explanation.

The estimation of the yield response model for phosphate (⁠$P$⁠), potash (⁠$K$⁠) and seed density (⁠$S$⁠) is analogous, but variable names change respectively $N$ for $P$⁠, $K$ and $S$⁠. See steps 4–9 in Appendix D in supplementary data at ERAE online.

4.3. Estimation of corn yield response to input use

Focusing first on yield response to fertiliser applications, the value of the marginal effect of fertiliser on corn yields (⁠$ϕ31$⁠) that conditions models in Sections 3.1 and 3.2 can be estimated by means of the Metropolis–Hastings algorithm (Chib and Greenberg, 1996). In particular, we employ the general purpose sampling algorithm called t-walk (Christen and Fox, 2010; Lieberman and Fox, 2015). This algorithm requires an objective function to evaluate whether to accept or reject values of the proposed parameters. Because information about parameter $ϕ31$ is provided by both the market and experimental datasets, which in turn are independent from each other, the t-walk objective function is the sum of the log-likelihoods from each model (equations D.1 and D.4 in Appendix D in supplementary data at ERAE online), conditional on the values of all of the remaining parameters. Such likelihood gives the joint probability that the proposed value of $ϕ31$ is generated from these datasets.^22,23 See steps 14–17 of Appendix D (in supplementary data at ERAE online) for a precise description of the steps employed. Therefore, the procedure will more (less) often accept candidates that come from the dataset with higher (lower) likelihood, i.e. the more (less) likely parameter values. Equation (3.1) is used to transform the yield effect of each nutrient into the yield effect of the aggregate fertiliser applied.

5. Results

The MCMC procedure was conducted using three chains of 50,000 samples each. Distribution convergence was checked for all of the reported parameters, satisfactory accomplishing the potential scale reduction factor diagnostic (Brooks and Gelman, 1998). First, we present the estimates of corn yield elasticities with respect to prices, and then we compare results of our combined approach with the ones using the traditional dual approach.

Figure 1 depicts the corn yield elasticity estimates with respect to selected prices, which according to equations (2.13) and (2.14) are the parameters expected to be most influenced by the data included to aid the duality approach estimation. The histograms represent the 95 per cent highest posterior density interval (HPDI), also known as most credible interval, of the marginal posterior density function of each elasticity. Table 1 reports the corresponding summary estimates.²⁴

Fig. 1.

Corn yield elasticities with respect to selected prices. Note: Histograms show the highest probability density intervals at the 95 per cent.

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The median²⁵ of the posterior distribution of the corn yield own-price elasticity is 0.21. Not only is the median positive but also the entire credible interval lies above zero. This value also falls in the range of previous estimates from the literature, in particular, among the most recent figures. Appendix F in supplementary data at ERAE online contains a full set of prior estimates. Our elasticity estimate of 0.21 lies in the centre of the prior range and is very close to the estimates obtained by Choi and Helmberger (1993) (0.27), Lyons and Thompson (1981) (0.22), Arnade and Kelch (2007) (0.19), Goodwin et al. (2012) (0.19–0.27) and Miao Khanna and Huang (2015) (0.23–0.26). A positive own-price elasticity can be caused by farmers expecting higher prices for their corn and, as a consequence, reacting by changing farm management practices, for example by applying more fertiliser, planting more and better seeds per hectare and hiring more labour, among others.

Point estimates of the corn yield elasticities with respect to fertiliser and intermediate input prices have the expected negative sign, consistent with observing farmers cutting their input usage as their prices increase. According to the posterior distribution medians shown in Table 1, corn yields are slightly more responsive to changes in intermediate inputs than to fertilisers prices (with median elasticities of −0.15 and −0.10, respectively). However, the hypothesis that yields are non-responsive to intermediate input prices cannot be rejected because zero is within the elasticity 95 per cent HPDI. In contrast, higher fertiliser prices are found to exert a statistically significant reduction in yields because zero is not included in the 95 per cent HPDI. The median elasticity with respect to hired labour is positive, which is not to be expected, but it is negligible and not statistically significant. This result may be caused by the fact that labour is more difficult to cut (inelastic) than reducing other inputs such as fertiliser or seeds.

Another set of parameters of interest are the corn yield elasticities with respect to the quantities of hired labour, intermediate inputs and fertiliser. Such elasticities are based on the marginal productivities of hired labour (⁠$ϕ11$⁠), intermediate inputs (⁠$ϕ21$⁠) and fertiliser (⁠$ϕ31$⁠), respectively. Parameter $ϕ11$ is recovered from the dual approach only, whereas parameters $ϕ21$ and $ϕ31$ are estimated by relying on both the dual and primal approaches. Importantly, parameters $ϕ21$ and $ϕ31$ serve as the ‘bridge’ through which the two approaches complement each other. Figure 2 shows histograms of the marginal posterior densities for the elasticities of corn yield with respect to input quantities, whereas the median, 95 per cent HPDI, mean and standard deviation are reported in the bottom three rows of Table 1. As expected, they are non-negative, implying that, at the optimum, corn yields are non-decreasing in the use of these inputs. The response of corn yield with respect to intermediate inputs is 0.13 and with respect to fertiliser use is 0.26. As it can be inferred from the very narrow 95 per cent HPDIs, the estimates are highly significant, which is a consequence of an average higher log-likelihood for the experimental data.

Fig. 2.

Corn yield elasticities with respect to quantity of hired labour (HL), intermediate inputs (II) and quantity of fertilisers (F). Note: Histograms of marginal posterior density function showing highest probability density intervals at the 95 per cent.

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The elasticity of corn yield with respect to the quantity of hired labour, based on parameter $ϕ11$⁠, is estimated with a median of about zero (see Table 1). As experimental data on corn yield response to labour are not available, this parameter was recovered from the underlying production technology using the dual theorem in equation (2.15) and Hessian identities. The imprecision of this recovery, reflected on its much wider HPDI compared with the HPDIs for the elasticities with respect to the other input quantities, highlights the imprecision of the duality approach in recovering parameters of the underlying technology. Through equations (2.13) and (2.14), parameter $ϕ11$ impacts the precision of the proposed approach because it increases the variance of all posterior densities. The overall precision of the approach could be improved by assuming an informative prior for this parameter, for example, based on non-negative estimates from the literature.

To compare the results obtained with the proposed approach (which uses both market and experimental data, i.e. full data) with the standard dual approach (which uses no experimental data), the SUR model in equations (2.1)–(2.8) was estimated imposing only the symmetry restrictions. Therefore, a SUR model analogous to equation (3.2), but with only two groups of parameters (free and constrained by symmetry), was set up and estimated by means of a Gibbs sampler on the marginal posterior distributions.²⁶

Results in Figures 3 and 4 show that the two approaches provide different results. We illustrate the results with corn and soybeans yield elasticities; the former should be the ones impacted the most by the independent sources of information, whereas the latter should be much less so.

Fig. 3.

Corn yield elasticities with respect to selected prices. Comparison between ‘full data’ approach (light blue) and the ‘no experimental data’ approach (blue). Note 1: Histograms show the highest probability density intervals at the 95 per cent. Note 2: ‘Full data’ approach refers to the proposed approach combining market and experimental data, and ‘No experimental data’ approach uses only market data, which is how the duality approach is usually performed. Note 3: The ‘full data’ approach (‘no experimental data’ approach) median estimates of the corn yield-price elasticities are 0.21 (0.17) for corn price, 0.003 (−0.03) for hired labour price, −0.15 (−0.09) for intermediate input price and −0.10 (−0.02) fertiliser price. See Table G.6 in Appendix G in supplementary data at ERAE online for more details. Note 4: The ‘full data’ approach (‘no experimental data’ approach) mean estimates of the corn yield-price elasticities are 0.22 (0.17) for corn price, 0.004 (−0.03) for hired labour price, −0.16 (−0.09) for intermediate input price and −0.10 (−0.02) for fertiliser price. See Table G.6 in Appendix G in supplementary data at ERAE online for more details.

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Fig. 4.

Soybean yield elasticities with respect to selected prices. Comparison between ‘full data’ approach (light blue) and the ‘no experimental data’ approach (blue). Note 1: Histograms show the highest probability density intervals at the 95 per cent. Note 2: ‘Full data’ approach refers to the proposed approach combining market and experimental data, and ‘No experimental data’ approach uses only market data, which is how the duality approach is usually performed. Note 3: The ‘full data’ approach (‘no experimental data’ approach) median estimates of the soybean yield-price elasticities are 0.40 (0.45) for soybean price, −0.09 (−0.07) for hired labour price, −0.32 (−0.18) for intermediate input price and −0.01 (−0.01) fertiliser price. See Table G.6 in Appendix G in supplementary data at ERAE online for more details. Note 4: The ‘full data’ approach (‘no experimental data’ approach) mean estimates of the soybean yield-price elasticities are 0.40 (0.44) for soybean price, −0.09 (−0.07) for hired labour price, −0.32 (−0.18) for intermediate input price and −0.01 (−0.01) for fertiliser price. See Table G.6 in Appendix G in supplementary data at ERAE online for more details.

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In the case of corn yields, Figure 3 shows that the dual approach tends to provide estimates of smaller magnitude for the yield elasticities with respect to selected prices. A plausible explanation for this result is attenuation bias (Greene, 2003: 85) if the market data are subject to sources of noise that prevent the dual theorem from holding exactly. Importantly, the null hypothesis that corn yield does not respond to prices can be rejected for corn and fertiliser prices when using the full data approach. In contrast, with the dual approach it is not possible to reject such hypothesis for any of the four prices, because zero is included in each HPDI. It is clear from Figure 3 that the greatest impact of the full data approach is on the estimation of the yield elasticity with respect to fertiliser prices. This finding conforms with intuition, as one would expect the information contained in the experimental fertiliser data to be reflected to the greatest extent on the marginal posterior probability of this elasticity.

Regarding soybean yields, for which no additional information about soybean production technology is used, Figure 4 reveals that both methods also provide different results and with the same pattern as for corn, i.e. lower values of the dual approach’s parameter estimates (with exception of soybean yields own-price elasticity). This shows how incorporating the information contained in experimental data has effects on all parameters of the dual model.

6. Conclusions

We estimate the crop yield elasticities with respect to output and input prices by blending market data with experimental data. Yield elasticities have become a focus of discussion because the literature (Keeney and Hertel, 2009; Dumortier et al., 2011; Gohin, 2014) show evidence that small deviations in the values assumed for these elasticities have great impacts on a country’s GHG emissions accounting and land-use change evaluations.

Each of the two standard methods used to calculate yield elasticities, the primal and dual approaches, has its own drawbacks (Colman, 1983; Just, 1993). The elasticity estimates provided here combine the two approaches by incorporating experimental data on production responses to complement the market data used by the dual method. The combined estimates are obtained by means of Bayesian methods, because they allow us to combine the information from the different datasets in a straightforward way. The relative weight assigned to each source of information is based on the log-likelihood function of each model.

The median own-price elasticity of corn yields is estimated to be equal to 0.21, which lies within the interval of the most recent empirical estimates found in the literature. This elasticity, as well as the elasticities with respect to input prices, are all of the expected signs, that is, yields increasing with own price and decreasing with input prices (with the exception of elasticity with respect to hired labour, which is estimated to be non-responsive and not statistically significant). Also, we estimate soybean yield elasticities with respect to own price with a median of 0.40 and with low dispersion of the posterior density, as opposed to the most recent estimates from the literature. Research on the difference between the maximum possible yield for corn and the actual yield achieved (i.e. the ‘yield gap’) has shown that EU corn yields in the main production areas are as close to maximum achievable yields as those in the USA. This finding suggests that farmers’ ability to increase yield in response to price signals should be similar in both regions.

We also estimated the corn yield elasticities with respect to the quantities of hired labour, intermediate inputs and fertiliser. As expected, they are estimated as non-negative, implying that, at the optimum, corn yields are non-decreasing in the use of these inputs. In particular, the response to intermediate inputs is 0.13, and to fertiliser use is 0.26, both statistically significant. However, the response to hired labour quantity is estimated not only with a low elasticity but also with a high imprecision as per the width of the 95 per cent HPDI. This imprecision propagates to other estimated parameters (mainly through equations (2.13) and (2.14)), a problem not shared by the estimation of the response to the other two inputs which are estimated with a smaller variance. In fact, one might argue that the higher precision in the estimation of the intermediate inputs and fertiliser use elasticities propagates to other parameters, as one can note from the width of their HPDIs.

Finally, examination of the posterior marginal density functions reveals that complementing the market data with experimental data for fertilisers and intermediate inputs has the largest impact on the elasticity of corn yields with respect to fertiliser prices, and that such impact is substantial.

The period covered by Ball’s dataset ends in 2004, which limits our ability to include more recent years in the analysis. Adoption of genetically engineered corn in Iowa was 30 per cent in 2001 and 54 per cent in 2004; but this was later to reach 93 per cent.²⁷ Introduction of stacked gene varieties was only 8 per cent in 2004, and this later reached 80 per cent. Ball’s data end before the dramatic increase in corn and soybean prices that began in 2006, as well as the run-up in corn seed costs that appears to have begun in 2005. The USDA-ERS is currently in the process of updating Ball’s dataset to the most recent years, but no updates are expected to be released before the summer of 2019.²⁸ The release of the more recent data will provide researchers with the opportunity to update the present study. In the meantime, we can only speculate about how the availability of traits with higher yield potential as well as significantly higher output prices might have influenced our results. It seems possible that higher output prices coupled with the relatively large price yield elasticity estimated for the earlier period prompted the widespread demand and development of more expensive seed traits. This would have resulted in a significant increase in yields and a mitigation of the need to add to world crop acres at the extensive margin.

Footnotes